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 |
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/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 1,236 | 1,293 | |
/-
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.Data.Fintype.Parity
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.FieldTheory.Finite.Basic
/-!
# Quadratic characters of finite fie... |
/-- For `a : F`, `quadraticChar F a = -1 ↔ ¬ IsSquare a`. -/
theorem quadraticChar_neg_one_iff_not_isSquare {a : F} : quadraticChar F a = -1 ↔ ¬IsSquare a := by
| Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 156 | 158 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | -- this has the same content as the subsingleton
| Mathlib/Algebra/Polynomial/Basic.lean | 744 | 744 |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic... | w (algebraMap (𝓞 K) K (seq K w₁ hB m)) < w (algebraMap (𝓞 K) K (seq K w₁ hB n)) := by
induction m with
| zero =>
exfalso
exact Nat.not_succ_le_zero n h
| succ m m_ih =>
cases eq_or_lt_of_le (Nat.le_of_lt_succ h) with
| inl hr =>
rw [hr]
exact (seq_next K w₁ hB (se... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 249 | 262 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | ext fun _ => le_succ_iff_eq_or_le
theorem Icc_succ_right (h : a ≤ succ b) : Icc a (succ b) = insert (succ b) (Icc a b) := by
simp_rw [← Ici_inter_Iic, Iic_succ, inter_insert_of_mem (mem_Ici.2 h)]
theorem Ioc_succ_right (h : a < succ b) : Ioc a (succ b) = insert (succ b) (Ioc a b) := by
simp_rw [← Ioi_inter_Iic,... | Mathlib/Order/SuccPred/Basic.lean | 437 | 445 |
/-
Copyright (c) 2022 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.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | intro ε εpos
rw [mem_Ioi] at εpos
exact lintegral_abs_det_fderiv_le_addHaar_image_aux1 μ hs hf' hf εpos
theorem lintegral_abs_det_fderiv_le_addHaar_image (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
(∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) := b... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,029 | 1,045 |
/-
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 Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | /-- Variant of `mul_eq_one_iff_inv_eq` with swapped equality. -/
@[to_additive]
| Mathlib/Algebra/Group/Basic.lean | 678 | 679 |
/-
Copyright (c) 2018 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.LinearAlgebra.BilinearForm.Properties
/-!
# Dual submodule with respect to a bilinear form.
## Main definitions and results
- `BilinForm.dualSubmodule`: T... | lemma dualSubmodule_dualSubmodule_of_basis
{ι} [Finite ι] (hB : B.Nondegenerate) (hB' : B.IsSymm) (b : Basis ι S M) :
B.dualSubmodule (B.dualSubmodule (Submodule.span R (Set.range b))) =
Submodule.span R (Set.range b) := by
classical
letI := FiniteDimensional.of_fintype_basis b
rw [dualSubmodule_spa... | Mathlib/LinearAlgebra/BilinearForm/DualLattice.lean | 130 | 137 |
/-
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.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsympt... | /-- If `l : Filter ℂ` is an "exponential comparison filter", then for any complex `a₁`, `a₂` and any
real `b₁ < b₂`, we have `(fun z ↦ z ^ a₁ * exp (b₁ * z)) =o[l] (fun z ↦ z ^ a₂ * exp (b₂ * z))`. -/
theorem isLittleO_cpow_mul_exp {b₁ b₂ : ℝ} (hl : IsExpCmpFilter l) (hb : b₁ < b₂) (a₁ a₂ : ℂ) :
(fun z => z ^ a₁ * ... | Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 168 | 178 |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Monad operations on `MvPolynomial`
... | apply map_C
| Mathlib/Algebra/MvPolynomial/Monad.lean | 230 | 231 |
/-
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.... |
/-- Together `Matrix.detInvertibleOfInvertible` and `Matrix.invertibleOfDetInvertible` form an
equivalence, although both sides of the equiv are subsingleton anyway. -/
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 103 | 105 |
/-
Copyright (c) 2022 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Tactic.Field... | theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : #(treesOfNumNodesEq n) = catalan n := by
induction n using Nat.case_strong_induction_on with
| hz => simp
| hi n ih =>
rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
· apply sum_congr rfl
rintro ⟨i, j⟩ H
| Mathlib/Combinatorics/Enumerative/Catalan.lean | 181 | 187 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | simp only [ΨSq, map_preΨ, map_Ψ₂Sq, ← coe_mapRingHom]
map_simp
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 572 | 573 |
/-
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... | (OrderEmbedding.ofStrictMono fun i => ⟨c.sizeUpTo i, Nat.lt_succ_of_le (c.sizeUpTo_le i)⟩) <|
| Mathlib/Combinatorics/Enumerative/Composition.lean | 252 | 252 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... | Mathlib/Logic/Equiv/Basic.lean | 1,605 | 1,607 | |
/-
Copyright (c) 2020 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.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
/-!
# Universal enveloping algebra
Given a commu... | g₁ = g₂ :=
have h' : (lift R).symm g₁ = (lift R).symm g₂ := by ext; simp [h]
(lift R).symm.injective h'
| Mathlib/Algebra/Lie/UniversalEnveloping.lean | 153 | 155 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Filippo A. E. Nuccio, Sam van Gool
-/
import Mathlib.Data.Fintype.Order
import Mathlib.Order.Interval.Finset.Basic
import Mathlib.Order.Irreducible
import Mathlib.Order.Upp... | unfold OrderEmbedding.birkhoffSet; split <;> simp [eq_iff_true_of_subsingleton]
| Mathlib/Order/Birkhoff.lean | 230 | 231 |
/-
Copyright (c) 2022 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.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | -- Porting note: had to add arguments to `nmul_one` in the next two lines
-- for the termination checker.
· simpa [nmul_one c] using H c hc
· simpa [nmul_one c] using hc.trans_le ha
termination_by a
@[simp]
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 527 | 533 |
/-
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.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsympt... | calc
(fun z => z ^ a) =Θ[l] fun z => Real.exp (re a * Real.log ‖z‖) :=
hl.isTheta_cpow_exp_re_mul_log a
_ =o[l] fun z => exp (b * z) :=
IsLittleO.of_norm_right <| by
simp only [norm_exp, re_ofReal_mul, Real.isLittleO_exp_comp_exp_comp]
refine (IsEquivalent.refl.sub_isLittleO ?_).sy... | Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 158 | 164 |
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
/-!
# Diagonal matrices
This file co... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 184 | 188 | |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
/-!
# Lie algebras of skew-adjoint endomorphisms o... | ← Function.Injective.eq_iff e.injective, map_mul, AlgEquiv.apply_symm_apply, map_neg]
@[simp]
theorem skewAdjointMatricesLieSubalgebraEquivTranspose_apply {m : Type w} [DecidableEq m]
| Mathlib/Algebra/Lie/SkewAdjoint.lean | 142 | 145 |
/-
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.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | Mathlib/Computability/Primrec.lean | 1,537 | 1,538 | |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib... | (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) <| μ.restrict (Ι a b))
(hF_int : IntervalIntegrable (F x₀) μ a b)
(hF'_meas : AEStronglyMeasurable F' <| μ.restrict (Ι a b))
(h_lip : ∀ᵐ t ∂μ.restrict (Ι a b),
LipschitzOnWith (Real.nnabs <| bound t) (F · t) (ball x₀ ε))
(bound_integrable : ... | Mathlib/Analysis/Calculus/ParametricIntegral.lean | 182 | 201 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Algebra.Order.Ring.Canonical
import Mathlib.Algebra.BigOperators.Group.List.Defs
/-!
# Big operators on a list in ordered rings
This file contains ... | Mathlib/Algebra/Order/BigOperators/Ring/List.lean | 29 | 35 | |
/-
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.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | theorem log_one : log 1 = 0 := by simp [log]
| Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 93 | 94 |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousMap.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and lower... | [Topology.IsLowerSet α] [IsLower β] {f : α → β} (hf : Monotone f) : Continuous f :=
IsUpperSet.monotone_to_upperTopology_continuous (α := αᵒᵈ) (β := βᵒᵈ) hf.dual
lemma lowerSet_le_lower {t₁ t₂ : TopologicalSpace α} [@Topology.IsLowerSet α t₁ _]
| Mathlib/Topology/Order/UpperLowerSetTopology.lean | 344 | 347 |
/-
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.Algebra.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... |
end Abs
end CauSeq
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 862 | 867 |
/-
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.Complex
/-!
# The `arctan` function.
Inequalit... | rw [arctan_inv_of_pos hx, arctan_inv_of_pos hy, show _ + _ = π - (arctan x + arctan y) by ring,
sub_eq_iff_eq_add, ← sub_eq_iff_eq_add', sub_eq_add_neg, ← arctan_neg, add_comm] at k
convert k.symm using 3
field_simp
rw [show -x + -y = -(x + y) by ring, show x * y - 1 = -(1 - x * y) by ring, neg_div_neg_eq]
... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 239 | 245 |
/-
Copyright (c) 2023 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Combinatorics.SimpleGraph.Subgraph
/-!
# Local graph operations
... | `G`. -/
lemma adj_replaceVertex_iff_of_ne_left {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj s w ↔ G.Adj s w := by simp [replaceVertex, hw]
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 65 | 68 |
/-
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.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,784 | 1,785 | |
/-
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 | 943 | 947 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | smul_mem' := by simp+contextual [Submodule.smul_mem]
lemma mem_coeffsIn : p ∈ coeffsIn σ M ↔ ∀ i, p.coeff i ∈ M := .rfl
| Mathlib/Algebra/MvPolynomial/Basic.lean | 904 | 907 |
/-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Analytic.CPolynomialDef
/-! # Properties of continuously polynomial functions
We expand the API a... | Mathlib/Analysis/Analytic/CPolynomial.lean | 416 | 426 | |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
/-!
# The layer cake formula / Cavalieri's principle / tail probability formula
In this file we prove the f... |
end LayercakeLT
section LayercakeIntegral
variable {α : Type*} [MeasurableSpace α] {μ : Measure α} {f : α → ℝ}
/-- The standard case of the layer cake formula / Cavalieri's principle / tail probability formula:
| Mathlib/MeasureTheory/Integral/Layercake.lean | 501 | 508 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... |
theorem einfsep_of_fintype [DecidableEq α] [Fintype s] :
s.einfsep = s.offDiag.toFinset.inf (uncurry edist) := by
| Mathlib/Topology/MetricSpace/Infsep.lean | 140 | 142 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | intro _ hx hy
rw [mem_interedges_iff] at hx hy
exact ht hx.2.1 hy.2.1
section DecidableEq
| Mathlib/Combinatorics/SimpleGraph/Density.lean | 85 | 90 |
/-
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.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
/-!
# Asco... | and `F : ι → (X → α)` a family which is equicontinuous on each `K ∈ 𝔖`. Then, the uniform
structures of uniform convergence on `𝔖` and pointwise convergence induce the same
uniform structure on `ι`.
This is a specialization of `EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi'` to
the case where `𝔖` covers `X... | Mathlib/Topology/UniformSpace/Ascoli.lean | 261 | 280 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | @sdiff_symmDiff_left αᵒᵈ _ _ _
| Mathlib/Order/SymmDiff.lean | 499 | 499 |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# The OnePoint Compactification
We ... | tendsto_nhds_infty'.trans <| and_iff_right (tendsto_pure_nhds _ _)
theorem continuousAt_infty {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} :
| Mathlib/Topology/Compactification/OnePoint.lean | 365 | 367 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Determinant
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Ri... | exact le_of_succ_succ_get?_contsAux_b nth_partDenom_eq
/-- Shows that the sequence of denominators is monotone, that is `Bₙ ≤ Bₙ₊₁`. -/
theorem of_den_mono : (of v).dens n ≤ (of v).dens (n + 1) := by
let g := of v
rcases Decidable.em <| g.partDens.TerminatedAt n with terminated | not_terminated
· have : g.part... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 294 | 309 |
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Complex.Cardinality
import Mathlib.Data.Complex.Modu... |
theorem finrank_real_of_complex (E : Type*) [AddCommGroup E] [Module ℂ E] :
Module.finrank ℝ E = 2 * Module.finrank ℂ E := by
rw [← Module.finrank_mul_finrank ℝ ℂ E, Complex.finrank_real_complex]
| Mathlib/Data/Complex/FiniteDimensional.lean | 52 | 56 |
/-
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
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.ConjTranspose
/-!... |
@[simp]
theorem blockDiagonal_apply_eq (M : o → Matrix m n α) (i j k) :
| Mathlib/Data/Matrix/Block.lean | 330 | 332 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Tactic.Contrapose
import Mathl... |
theorem pow_pred_clog_lt_self {b : ℕ} (hb : 1 < b) {x : ℕ} (hx : 1 < x) :
b ^ (clog b x).pred < x := by
| Mathlib/Data/Nat/Log.lean | 284 | 286 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Kim Morrison
-/
import Mathlib.Data.List.Chain
/-!
# Ranges of naturals as lists
This file shows basic results about `List.iota`, `List.range`, `List.range... | Mathlib/Data/List/Range.lean | 208 | 210 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 886 | 891 | |
/-
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.Group.Submonoid.BigOperators
import Mathlib.Algebra.Ring.Action.Subobjects
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.Prod... | exact ⟨L ++ M, List.forall_mem_append.2 ⟨HL1, HM1⟩, by
rw [List.map_append, List.sum_append, HL2, HM2]⟩
· rintro ⟨L, HL1, rfl⟩
| Mathlib/Algebra/Ring/Subsemiring/Basic.lean | 525 | 527 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | Mathlib/Data/Int/Bitwise.lean | 506 | 527 | |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | hI.dep_of_ssubset (ssubset_insert he.2) (insert_subset he.1 hI.subset)
theorem IsBasis.mem_of_insert_indep (hI : M.IsBasis I X) (he : e ∈ X) (hIe : M.Indep (insert e I)) :
e ∈ I :=
by_contra (fun heI ↦ (hI.insert_dep ⟨he, heI⟩).not_indep hIe)
| Mathlib/Data/Matroid/Basic.lean | 925 | 930 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Topological study of spaces `Π (n : ℕ), E n`
When `E n` are topological spaces, the space `Π (n : ... |
theorem mem_cylinder_comm (x y : ∀ n, E n) (n : ℕ) : y ∈ cylinder x n ↔ x ∈ cylinder y n := by
simp [mem_cylinder_iff_eq, eq_comm]
theorem mem_cylinder_iff_le_firstDiff {x y : ∀ n, E n} (hne : x ≠ y) (i : ℕ) :
x ∈ cylinder y i ↔ i ≤ firstDiff x y := by
constructor
· intro h
by_contra!
exact apply_fi... | Mathlib/Topology/MetricSpace/PiNat.lean | 134 | 147 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Hilbert sum of ... | refine IsHilbertSum.mkInternal _ K.orthogonalFamily_self ?_
refine le_trans ?_ (Submodule.le_topologicalClosure _)
rw [iSup_bool_eq, cond, cond]
refine Codisjoint.top_le ?_
exact Submodule.isCompl_orthogonal_of_completeSpace.codisjoint
end IsHilbertSum
| Mathlib/Analysis/InnerProductSpace/l2Space.lean | 354 | 361 |
/-
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.AlgebraicGeometry.Morphisms.UnderlyingMap
import Mathlib.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
/-!
# Quasi-compact morphisms
A morp... | refine Set.Subset.trans ?_ (Set.subset_iUnion₂ j hj)
exact Set.Subset.rfl
instance : HasAffineProperty @QuasiCompact (fun X _ _ _ ↦ CompactSpace X) where
| Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 116 | 119 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Alge... |
variable (succ' : ∀ {X₀ X₁ : V} (f : X₁ ⟶ X₀), Σ' (X₂ : V) (d : X₂ ⟶ X₁), d ≫ f = 0)
| Mathlib/Algebra/Homology/HomologicalComplex.lean | 722 | 724 |
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | simp only [vadd_vsub, smul_smul, vsub_vadd_eq_vsub_sub, smul_sub, ← sub_smul]
ring_nf
theorem Wbtw.sameRay_vsub_left {x y z : P} (h : Wbtw R x y z) : SameRay R (y -ᵥ x) (z -ᵥ x) := by
| Mathlib/Analysis/Convex/Between.lean | 504 | 507 |
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Constructi... | haveI := Algebra.isSeparable_tower_top_of_isSeparable F F⟮α⟯ E
rw [IntermediateField.finrank_top, ← AlgHom.card_of_splits _ _ A, Fintype.card_eq_one_iff]
· exact ⟨{ __ := φ, commutes' := fun _ ↦ rfl }, fun ψ ↦ AlgHom.restrictScalars_injective F <|
Eq.symm <| h _ (ψ.commutes <| AdjoinSimple.gen F α).symm⟩
... | Mathlib/FieldTheory/PrimitiveElement.lean | 394 | 401 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.FiniteStability
import Mathlib.RingTheory.Localization.InvSubmonoid
import Mathlib.RingTheory.RingHom.Finite
/-!
# The meta properties of finite... | open scoped Classical in
/-- Let `S` be an `R`-algebra, `M` a submonoid of `R`, and `S' = M⁻¹S`.
If the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
then there exists some `m : M` such that `m • x` falls in the
adjoin of `IsLocalization.finsetIntegerMultiple _ s` over `R`.
-/
theorem IsLo... | Mathlib/RingTheory/RingHom/FiniteType.lean | 104 | 112 |
/-
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... | map F f 0 ⟨j + 2, hj⟩ (by simp) = f ≫ F.map' 0 (j+1) := rfl
| Mathlib/CategoryTheory/ComposableArrows.lean | 327 | 328 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | /-- The determinant of an endomorphism given a basis.
See `LinearMap.det` for a version that populates the basis non-computably.
Although the `Trunc (Basis ι A M)` parameter makes it slightly more convenient to switch bases,
| Mathlib/LinearAlgebra/Determinant.lean | 115 | 119 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | have hfg' : f + g ≈ f := by
change LimZero (g - 0) at hg
show LimZero (f + g - f); · simpa only [add_sub_cancel_left, sub_zero] using hg
have hcfg : (f + g).norm = f.norm := norm_equiv hfg'
have hcl : g.norm = 0 := (norm_zero_iff g).2 hg
have : max f.norm g.norm = f.n... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 360 | 366 |
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Topology.Category.TopCat.Limits.Konig
/-!
# Cofiltered systems
This file de... | (h : ∀ j : J, ∃ (i : _) (f : i ⟶ j), (range <| F.map f).Finite) : F.IsMittagLeffler := by
intro j
obtain ⟨i, hi, hf⟩ := h j
| Mathlib/CategoryTheory/CofilteredSystem.lean | 233 | 235 |
/-
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.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
/-!
# Preadditive monoidal categories
A monoidal category is `M... | ext
simp? [leftDistributor_hom, Preadditive.sum_comp, Preadditive.comp_sum_assoc, biproduct.ι_π,
comp_dite] says
simp only [leftDistributor_hom, Category.assoc, Preadditive.sum_comp, biproduct.ι_π, comp_dite,
comp_zero, Finset.sum_dite_eq', Finset.mem_univ, ↓reduceIte, eqToHom_refl, Category.comp_id... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 309 | 315 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.Pointwise
/-!
# Submonoid of inverses
Given a submonoid `N` of a monoid `M`, we define the submonoid `N.leftInv` as the submonoid ... | theorem unit_mem_leftInv (x : Mˣ) (hx : (x : M) ∈ S) : ((x⁻¹ :) : M) ∈ S.leftInv :=
⟨⟨x, hx⟩, x.inv_val⟩
@[to_additive]
| Mathlib/GroupTheory/Submonoid/Inverses.lean | 73 | 76 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 488 | 500 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | [SMulCommClass G R S] [SMulCommClass G S S] (g : G) (x) :
leftMulMatrix (g • b) x = leftMulMatrix b x := by
ext
simp_rw [leftMulMatrix_apply, LinearMap.toMatrix_apply, coe_lmul_eq_mul, LinearMap.mul_apply',
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 846 | 849 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... | ext
simp_rw [← coeff_coe]
congr
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 839 | 841 |
/-
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 | 1,475 | 1,486 | |
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir
-/
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Construction of the hyperreal numbers as an ultrapro... | theorem infinitePos_mul_infinitePos {x y : ℝ*} :
InfinitePos x → InfinitePos y → InfinitePos (x * y) := fun hx hy =>
| Mathlib/Data/Real/Hyperreal.lean | 701 | 702 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | (monomial s a).degreeOf i = s i := by
classical rw [degreeOf_def, degrees_monomial_eq _ _ ha, Finsupp.count_toMultiset]
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 259 | 261 |
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Topology.Category.TopCat.Limits.Basic
/-!
# Topological Kőnig's lemma
A topological version of Kőnig's lemma i... | · rintro u hu f hf
have : ιA f ∈ A.2.image ιA ⊔ B.2.image ιB := by
apply Finset.mem_union_left
rw [Finset.mem_image]
exact ⟨f, hf, rfl⟩
exact hu this
· rintro u hu f hf
have : ιB f ∈ A.2.image ιA ⊔ B.2.image ιB := by
apply Finset.mem_union_right
rw [Finset.mem_image]
... | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | 83 | 103 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... |
open scoped Classical in
@[simp]
theorem denom_div (p : K[X]) {q : K[X]} (hq : q ≠ 0) :
| Mathlib/FieldTheory/RatFunc/Basic.lean | 832 | 835 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,487 | 3,499 | |
/-
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.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotent
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.... | rw [IsScalarTower.algebraMap_apply R A, RingHom.toFun_eq_coe, Ideal.Quotient.algebraMap_eq,
Ideal.Quotient.lift_mk]
exact f.map_algebraMap r
theorem _root_.AlgHom.ker_kerSquareLift (f : A →ₐ[R] B) :
| Mathlib/RingTheory/Ideal/Cotangent.lean | 183 | 187 |
/-
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.CategoryTheory.GradedObject.Associator
import Mathlib.CategoryTheory.GradedObject.Single
/-!
# The left and right unitors
Given a bifunctor `F : C ⥤ D ⥤ D`, an ... | lemma mapBifunctor_triangle
(triangle : ∀ (X₁ : C₁) (X₃ : C₃), ((associator.hom.app X₁).app X₂).app X₃ ≫
(G.obj X₁).map (e₂.hom.app X₃) = (G.map (e₁.hom.app X₁)).app X₃) :
(mapBifunctorAssociator associator τ.ρ₁₂ τ.ρ₂₃ X₁ ((single₀ I₂).obj X₂) X₃).hom ≫
mapBifunctorMapMap G π (𝟙 X₁) (mapBifunctorLeftUn... | Mathlib/CategoryTheory/GradedObject/Unitor.lean | 326 | 351 |
/-
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.Constructions
import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps, which move a matroid on one ty... | lemma comapOn_preimage_eq (N : Matroid β) (f : α → β) : N.comapOn (f ⁻¹' N.E) f = N.comap f := by
rw [comapOn, restrict_eq_self_iff]; rfl
| Mathlib/Data/Matroid/Map.lean | 254 | 255 |
/-
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.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
/-!
# Lie algeb... |
variable {R L} in
| Mathlib/Algebra/Lie/OfAssociative.lean | 217 | 218 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 154 | 154 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.FunLike.Basic
import Mathlib.Logic.Embedding.Basic
import Mathlib.Order.RelClasses
/-!
# Relation homomorphisms, embeddings, isomorphisms
This f... | theorem ofMonotone_coe [IsTrichotomous α r] [IsAsymm β s] (f : α → β) (H) :
(@ofMonotone _ _ r s _ _ f H : α → β) = f :=
rfl
/-- A relation embedding from an empty type. -/
def ofIsEmpty (r : α → α → Prop) (s : β → β → Prop) [IsEmpty α] : r ↪r s :=
⟨Embedding.ofIsEmpty, @fun a => isEmptyElim a⟩
/-- `Sum.inl` ... | Mathlib/Order/RelIso/Basic.lean | 435 | 444 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... | exact Pi.mulSingle_apply_commute x i j
convert (noncommProd_congr (insert_erase (mem_univ i)).symm _ _).trans _
· intro j
exact Pi.mulSingle j (x j) i
· intro j _; dsimp
· rw [noncommProd_insert_of_not_mem _ _ _ _ (not_mem_erase _ _),
noncommProd_eq_pow_card (univ.erase i), one_pow, mul_one]
·... | Mathlib/Data/Finset/NoncommProd.lean | 411 | 424 |
/-
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... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,646 | 1,653 | |
/-
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,
Kim Morrison
-/
import Mathlib.Data.List.Basic
/-!
# Lattice structure of lists
This files pro... | theorem count_bagInter {a : α} :
| Mathlib/Data/List/Lattice.lean | 195 | 195 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... | variable [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂]
variable [Fintype n] [Fintype n']
variable (b₁ : Basis n R M₁) (b₂ : Basis n' R M₂)
variable (J J₂ : Matrix n n R) (J' : Matrix n' n' R)
variable (A : Matrix n' n R) (A' : Matrix n n' R)
variable (A₁ A₂ : Matrix n n R)
| Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 481 | 487 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Extreme
import Mathlib.Analysis.Convex.Function
import Mathlib.Topology.Algebra.Module.LinearMap
import Mathlib... | Mathlib/Analysis/Convex/Exposed.lean | 253 | 263 | |
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | simp [cliqueFinset_eq_image, map_eq_image]
| Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 217 | 218 |
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.Submodule.Equiv
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.Algebra.Module.Rat
im... | theorem lie_skew : -⁅y, x⁆ = ⁅x, y⁆ := by
| Mathlib/Algebra/Lie/Basic.lean | 176 | 176 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 1,184 | 1,192 | |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.Portmanteau
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | dist μ ν := levyProkhorovDist μ.toMeasure ν.toMeasure
dist_self _ := levyProkhorovDist_self _
dist_comm _ _ := levyProkhorovDist_comm _ _
dist_triangle _ _ _ := levyProkhorovDist_triangle _ _ _
edist_dist μ ν := by simp [← ENNReal.ofReal_coe_nnreal]
lemma LevyProkhorov.dist_def (μ ν : LevyProkhorov (Probabil... | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | 256 | 270 |
/-
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
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | (subset_image_diff f s t)
open scoped symmDiff in
theorem image_symmDiff (hf : Injective f) (s t : Set α) : f '' s ∆ t = (f '' s) ∆ (f '' t) := by
| Mathlib/Data/Set/Image.lean | 379 | 382 |
/-
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... | Mathlib/Data/List/Basic.lean | 2,220 | 2,221 | |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.Sigma
/-!
# Association Lists
This file defines association lists. An association list is a list where every element consists o... | simp only [AList.entries_insert, List.kerase_cons_eq, and_self_iff, AList.singleton_entries,
heq_iff_eq, eq_self_iff_true]
| Mathlib/Data/List/AList.lean | 304 | 306 |
/-
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 | 1,389 | 1,396 | |
/-
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.Analysis.Convex.Hull
/-!
# Convex join
This file defines the convex join of two sets. The convex join of `s` and `t` is the union of the
segments with on... | Mathlib/Analysis/Convex/Join.lean | 208 | 212 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Iterate
impor... |
theorem tsum_geometric_of_lt_one {r : ℝ} (h₁ : 0 ≤ r) (h₂ : r < 1) : ∑' n : ℕ, r ^ n = (1 - r)⁻¹ :=
(hasSum_geometric_of_lt_one h₁ h₂).tsum_eq
theorem hasSum_geometric_two : HasSum (fun n : ℕ ↦ ((1 : ℝ) / 2) ^ n) 2 := by
convert hasSum_geometric_of_lt_one _ _ <;> norm_num
| Mathlib/Analysis/SpecificLimits/Basic.lean | 303 | 309 |
/-
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.LineDeriv.Measurable
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar... | let L : E →L[ℝ] ℝ :=
LinearMap.toContinuousLinearMap (B.constr ℝ (fun i ↦ lineDeriv ℝ f x (B i)))
refine ⟨L, fun v hv ↦ ?_⟩
have J : L v = lineDeriv ℝ f x v := by convert (hx v hv).symm <;> simp [L, B.sum_repr v]
simpa [J] using (h'x v hv).hasLineDerivAt
omit [MeasurableSpace E] in
/-- If a Lipschitz funct... | Mathlib/Analysis/Calculus/Rademacher.lean | 257 | 307 |
/-
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.Algebra.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... | theorem const_pos {x : α} : Pos (const x) ↔ 0 < x :=
⟨fun ⟨_, K0, _, h⟩ => lt_of_lt_of_le K0 (h _ le_rfl), fun h => ⟨x, h, 0, fun _ _ => le_rfl⟩⟩
theorem add_pos {f g : CauSeq α abs} : Pos f → Pos g → Pos (f + g)
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 583 | 586 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
/-!
# Bilinear form
This file defines ort... | exact b₁ n x (b₁ x n (b₁ n x (hx₂ n hn)))
refine IsCompl.of_eq this (eq_top_of_finrank_eq <| (finrank_le _).antisymm ?_)
conv_rhs => rw [← add_zero (finrank K _)]
rw [← finrank_bot K V, ← this, finrank_sup_add_finrank_inf_eq,
finrank_add_finrank_orthogonal b₁]
exact le_self_add
/-- A subspace is comple... | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 358 | 367 |
/-
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... | "Given a family of sets `t : ι → Set α`, a finite set `I` in the index type such that
all sets `t i`, `i ∈ I`, are finite, if all `t i`, `i ∈ I`, are pairwise disjoint, then the
sum of `f a` over `a ∈ ⋃ i ∈ I, t i` is equal to the sum over `i ∈ I` of the sums of `f a`
| Mathlib/Algebra/BigOperators/Finprod.lean | 922 | 924 |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.NormNum.Ineq
/-!
# `norm_num` extension for integer div/mod and divides
This file adds support f... | theorem isInt_dvd_false : {a b : ℤ} → {a' b' : ℤ} →
IsInt a a' → IsInt b b' → Int.emod b' a' != 0 → ¬a ∣ b
| _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, e => mt Int.emod_eq_zero_of_dvd (by simpa using e)
| Mathlib/Tactic/NormNum/DivMod.lean | 151 | 153 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Logic.Function.Iterate
import Mathlib.Logic.Relation
/-!
# R... | {r : {a // p a} → {a // p a} → Prop} :
(l.attachWith p h).Chain' r ↔ l.Chain' fun a b ↦ ∃ ha hb, r ⟨a, ha⟩ ⟨b, hb⟩ := by
induction l with
| nil => rfl
| cons a l IH =>
rw [attachWith_cons, chain'_cons', chain'_cons', IH, and_congr_left]
| Mathlib/Data/List/Chain.lean | 337 | 342 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L1
This file contains ... | by_cases hx_mem : x ∈ s <;> simp [hx_mem]
@[deprecated (since := "2025-01-21")] alias condexpInd_of_measurable := condExpInd_of_measurable
| Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 388 | 391 |
/-
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 | 1,002 | 1,003 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.AEMeasurable
import Mathlib.Order.Filter.EventuallyConst
/-!
# Measure preserving maps
We say that `f : α → β` is a measure p... |
/-- An alias of `MeasureTheory.MeasurePreserving.comp` with a convenient defeq and argument order
for `MeasurableEquiv` -/
| Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 92 | 94 |
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