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) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Order.Disjoint
import Mathlib.Order.RelIso.Basic
import Mathlib.Tactic.Monotonicity.Attr
/-!
# Order homomorphisms
This file defines order homomorphi... | Mathlib/Order/Hom/Basic.lean | 1,445 | 1,449 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... |
@[to_additive (attr := simp) enorm_norm]
lemma enorm_norm' (x : E) : ‖‖x‖‖ₑ = ‖x‖ₑ := by simp [enorm]
lemma enorm_enorm {ε : Type*} [ENorm ε] (x : ε) : ‖‖x‖ₑ‖ₑ = ‖x‖ₑ := by simp [enorm]
| Mathlib/Analysis/Normed/Group/Basic.lean | 1,227 | 1,231 |
/-
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
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 287 | 291 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheor... | rfl
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 373 | 373 |
/-
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.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 537 | 541 | |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | rw [sub_eq_iff_eq_add, ← mul_inv_cancel_left₀ two_ne_zero π, mul_assoc, ← mul_add,
mul_right_inj' (two_ne_zero' ℝ), ← eq_sub_iff_add_eq', mul_inv_cancel_left₀ two_ne_zero π,
inv_mul_eq_div, mul_comm] at h
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 688 | 690 |
/-
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.Order.Interval.Finset.Fin
import Mathlib.Data.Vector.Basic
/-!
# The structure of `Fintype (Fin n)`
This file contains some basic results about the `Fintyp... | coe_injective <| by ext; simp [lt_def]
| Mathlib/Data/Fintype/Fin.lean | 36 | 37 |
/-
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.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | nontriviality R
rintro rfl
cases n with
| zero =>
| Mathlib/Algebra/Polynomial/Roots.lean | 359 | 362 |
/-
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
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 300 | 305 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tact... | @[simp]
theorem prod_range_add_one_eq_factorial : ∀ n : ℕ, (∏ x ∈ range n, (x + 1)) = n !
| 0 => rfl
| n + 1 => by simp [factorial, Finset.range_succ, prod_range_add_one_eq_factorial n]
section GaussSum
| Mathlib/Algebra/BigOperators/Intervals.lean | 243 | 249 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | · rw [← h, ← mul_smul, coe_zeta_mul_moebius, one_smul]
· rw [ArithmeticFunction.ext_iff]
apply forall_congr'
intro n
cases n with
| zero => simp
| Mathlib/NumberTheory/ArithmeticFunction.lean | 1,138 | 1,143 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.CauSeq.Completion
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Rat.Cast.Defs
/-!
# Real numbers from Cauc... | Mathlib/Data/Real/Basic.lean | 311 | 311 | |
/-
Copyright (c) 2023 Mohanad ahmed. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mohanad Ahmed
-/
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.SemiringInverse
/-! # Block Matrices from Rows and Columns
This file provides the basic definiti... | ext i (j | j) <;> simp
@[deprecated (since := "2024-12-11")] alias fromColumns_neg := fromCols_neg
end Neg
| Mathlib/Data/Matrix/ColumnRowPartitioned.lean | 175 | 180 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... |
variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] {x : α} {s t u : Set α} {Φ : α → β}
/-- The Hausdorff edistance of a set to itself vanishes. -/
@[simp]
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 251 | 255 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.EraseLead
/-!
# Denominators of evaluation of polynomials ... | rw [Int.cast_ne_zero]
exact b0.ne.symm)
obtain Fa := congr_arg abs hF
rw [eq_one_div_of_mul_eq_one_left bu, eq_intCast, eq_intCast, abs_mul] at Fa
rw [abs_of_pos (pow_pos (Int.cast_pos.mpr b0) _ : 0 < (b : K) ^ _), one_div, eq_intCast] at Fa
rw [div_eq_mul_inv, ← Fa, ← Int.cast_abs, ← Int.cast_o... | Mathlib/Algebra/Polynomial/DenomsClearable.lean | 90 | 107 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Joseph Myers
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
/-!
# Bounds on specific values of the exponential
-/
namesp... | iterate 13 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
| Mathlib/Data/Complex/ExponentialBounds.lean | 20 | 25 |
/-
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, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Group.Subgroup.Map
import Mathlib.Algebra.Module.Submodule.... | ↑(f.submoduleMap p x) = f x := rfl
theorem submoduleMap_surjective (f : M →ₗ[R] M₁) (p : Submodule R M) :
| Mathlib/Algebra/Module/Submodule/Map.lean | 661 | 663 |
/-
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.... | @[to_additive add_one_zsmul]
| Mathlib/Algebra/Group/Basic.lean | 819 | 819 |
/-
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.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
/-!
# Integer powers of square matrices
In this file, we defi... | rw [neg_add, neg_add_cancel_right, zpow_neg h, zpow_natCast]
_ = (A * A ^ n)⁻¹ * A := by
rw [mul_inv_rev, Matrix.mul_assoc, nonsing_inv_mul _ h, Matrix.mul_one]
_ = A ^ (-(n + 1 : ℤ)) * A := by
rw [zpow_neg h, ← Int.natCast_succ, zpow_natCast, pow_succ']
| Mathlib/LinearAlgebra/Matrix/ZPow.lean | 137 | 142 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas
import Mathlib.Data.Fintype.BigOperators
import Mathli... | ((orbitEquivQuotientStabilizer α b).symm a : β) = a • b :=
rfl
@[to_additive (attr := simp)]
theorem stabilizer_quotient {G} [Group G] (H : Subgroup G) :
MulAction.stabilizer G ((1 : G) : G ⧸ H) = H := by
ext
| Mathlib/GroupTheory/GroupAction/Quotient.lean | 180 | 186 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology
import Mathlib.Topology.Sheaves.LocalPredicate
import Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalizatio... | end SectionSubring
section
open SectionSubring
variable {𝒜}
/-- The functions satisfying `isLocallyFraction` form a subring of all dependent functions
`Π x : U, HomogeneousLocalization 𝒜 x`. -/
def sectionsSubring (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) :
Subring (∀ x : U.unop, at x.1) where
carrier := ... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean | 139 | 152 |
/-
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.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Algebra.Order.Monoid.Units
/-!
# Ordered monoid an... |
@[to_additive]
| Mathlib/Algebra/Order/Hom/Monoid.lean | 590 | 591 |
/-
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.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-!
# Open immersions of structured spaces
We say that a m... | instance comp {X Y Z : SheafedSpace C} (f : X ⟶ Y) (g : Y ⟶ Z) [SheafedSpace.IsOpenImmersion f]
[SheafedSpace.IsOpenImmersion g] : SheafedSpace.IsOpenImmersion (f ≫ g) :=
| Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 606 | 607 |
/-
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.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... | (hf : Orientation.map (Fin 2) f.toLinearEquiv o = Complex.orientation) (x y : E) :
| Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 557 | 557 |
/-
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, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
/-!
# p-adic integers
This f... | Mathlib/NumberTheory/Padics/PadicIntegers.lean | 487 | 487 | |
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
/-!
# Monad Operations for Probability Mass Functions
This file constructs two operations on `PMF` ... | simpa only [ENNReal.coe_inj.symm, bind_apply, ENNReal.tsum_mul_left.symm,
ENNReal.tsum_mul_right.symm, mul_assoc, mul_left_comm, mul_comm] using ENNReal.tsum_comm
theorem bind_comm (p : PMF α) (q : PMF β) (f : α → β → PMF γ) :
| Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 140 | 143 |
/-
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... | _ = ite (b = b) (f b) 1 * ∏' x, update f b 1 x := by
congr
exact tprod_eq_mulSingle b fun b' hb' ↦ if_neg hb'
_ = f b * ∏' x, ite (x = b) 1 (f x) := by
simp only [update, eq_self_iff_true, if_true, eq_rec_constant, dite_eq_ite]
| Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 594 | 598 |
/-
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.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 1,336 | 1,352 | |
/-
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.LeftHomology
import Mathlib.CategoryTheory.Limits.Opposites
/-!
# Right Homology of short complexes
In this file, we define the ... | @[simp]
lemma ofZeros_g' (hf : S.f = 0) (hg : S.g = 0) :
(ofZeros S hf hg).g' = 0 := by
rw [← cancel_epi ((ofZeros S hf hg).p), comp_zero, p_g', hg]
| Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean | 205 | 208 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... |
@[simp]
theorem eraseLead_X_pow (n : ℕ) : eraseLead (X ^ n : R[X]) = 0 := by
rw [X_pow_eq_monomial, eraseLead_monomial]
| Mathlib/Algebra/Polynomial/EraseLead.lean | 141 | 144 |
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
/-!
# Projective seminorm on the tensor of a finite family of normed s... | projectiveSeminormAux (p + q) ≤ projectiveSeminormAux p + projectiveSeminormAux q := by
simp [projectiveSeminormAux]
theorem projectiveSeminormAux_smul (p : FreeAddMonoid (𝕜 × Π i, E i)) (a : 𝕜) :
projectiveSeminormAux (p.map (fun (y : 𝕜 × Π i, E i) ↦ (a * y.1, y.2))) =
‖a‖ * projectiveSeminormAux p :... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 66 | 71 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.Algebra... |
@[to_additive]
theorem Multipliable.vanishing (hf : Multipliable f) ⦃e : Set G⦄ (he : e ∈ 𝓝 (1 : G)) :
∃ s : Finset α, ∀ t, Disjoint t s → (∏ k ∈ t, f k) ∈ e := by
classical
letI : UniformSpace G := IsTopologicalGroup.toUniformSpace G
have : IsUniformGroup G := isUniformGroup_of_commGroup
exact cauchySeq_... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 347 | 356 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... |
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [FiniteDimensional 𝕜 V]
variable {S : Submodule 𝕜 V} {L : S →ₗᵢ[𝕜] V}
open Module
/-- Let `S` be a subspace of a finite-dimensional complex inner product space `V`. A linear
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 941 | 947 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Lattice
/-!
# Characteris... | end Nontrivial
| Mathlib/Algebra/CharP/Defs.lean | 244 | 245 |
/-
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.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | simp only [normalize_apply, normUnit_X, Units.val_one, mul_one]
end NormalizationMonoid
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 236 | 239 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Equiv.Opposite
import Mathlib.Algebra.Group.TypeTags.Basic
/-!
# Squares and even elements
This ... | Mathlib/Algebra/Group/Even.lean | 164 | 165 | |
/-
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, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... |
@[simp]
theorem comap_snd : q.comap (snd R M M₂) = prod ⊤ q := by ext ⟨x, y⟩; simp
| Mathlib/LinearAlgebra/Prod.lean | 491 | 493 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Order.Mono... | @[simp] lemma antivary_inv_left₀ (hf : StrongLT 0 f) : Antivary f⁻¹ g ↔ Monovary f g :=
forall₃_congr fun _i _j _ ↦ inv_le_inv₀ (hf _) (hf _)
| Mathlib/Algebra/Order/Monovary.lean | 268 | 269 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... |
theorem image_prodMk_subset_prod {f : α → β} {g : α → γ} {s : Set α} :
(fun x => (f x, g x)) '' s ⊆ (f '' s) ×ˢ (g '' s) := by
rintro _ ⟨x, hx, rfl⟩
| Mathlib/Data/Set/Prod.lean | 277 | 280 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | [Countable ι] (hτ : IsStoppingTime f τ) (i : ι) : MeasurableSet[f i] {ω | i ≤ τ ω} :=
hτ.measurableSet_ge_of_countable_range (Set.to_countable _) i
end IsStoppingTime
end CountableStoppingTime
| Mathlib/Probability/Process/Stopping.lean | 128 | 134 |
/-
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... | if h1 : j ≤ m then
letI l : A⁰_ f := HomogeneousLocalization.mk
⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
letI r : A⁰_ f := (HomogeneousLocalization.mk
⟨m * i, ⟨proj 𝒜 i b ^ m, by rw [← smul_eq_mul]... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 337 | 387 |
/-
Copyright (c) 2023 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.TensorProduct... | rw [← TensorProduct.tmul_smul, ← TensorProduct.tmul_sub]
apply le_comap_range_lTensor f
rw [exact_iff] at hfg
simp only [← hfg, mem_ker, map_sub, map_smul, hgh _, sub_self] }
lemma lTensor.inverse_of_rightInverse_apply
{h : P → N} (hgh : Function.RightInverse h g) (y : Q ⊗[R] N) :
(lTen... | Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 210 | 224 |
/-
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.Data.Finsupp.Lex
import Mathlib.Algebra.MvPolynomial.Degrees
/-!
# Variables of polynomials
This file establishes man... | section Map
variable [CommSemiring S] (f : R →+* S)
variable (p)
theorem vars_map : (map f p).vars ⊆ p.vars := by
classical simp [vars_def, Multiset.subset_of_le degrees_map_le]
variable {f}
theorem vars_map_of_injective (hf : Injective f) : (map f p).vars = p.vars := by
simp [vars, degrees_map_of_injective _ h... | Mathlib/Algebra/MvPolynomial/Variables.lean | 192 | 207 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Alge... | theorem vec3_dotProduct' {a₀ a₁ a₂ b₀ b₁ b₂ : α} :
![a₀, a₁, a₂] ⬝ᵥ ![b₀, b₁, b₂] = a₀ * b₀ + a₁ * b₁ + a₂ * b₂ := by
rw [cons_dotProduct_cons, cons_dotProduct_cons, cons_dotProduct_cons, dotProduct_empty,
| Mathlib/Data/Matrix/Notation.lean | 514 | 516 |
/-
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.Control.Functor
import Mathlib.Tactic.Common
/-!
# Functors with two arguments
This file defines bifunctors.
A bifunctor is a function `F : Type* → Type* ... |
end Bifunctor
| Mathlib/Control/Bifunctor.lean | 92 | 93 |
/-
Copyright (c) 2022 Justin Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justin Thomas
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.Polynomial.Module.AEval
/-!
# Annihilating Ideal
Given ... | apply Polynomial.leadingCoeff_eq_zero.not.mpr
apply (mul_ne_zero_iff.mp h).1
/-- The annihilating ideal generator is a member of the annihilating ideal. -/
theorem annIdealGenerator_mem (a : A) : annIdealGenerator 𝕜 a ∈ annIdeal 𝕜 a :=
Ideal.mul_mem_right _ _ (Submodule.IsPrincipal.generator_mem _)
theore... | Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean | 99 | 108 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.MeasureTheory.Measure.Stieltjes
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Fu... | variable {f : α → ℚ → ℝ} [MeasurableSpace α] (hf : IsMeasurableRatCDF f)
include hf
lemma IsMeasurableRatCDF.stieltjesFunctionAux_eq (a : α) (r : ℚ) :
IsMeasurableRatCDF.stieltjesFunctionAux f a r = f a r := by
rw [← hf.iInf_rat_gt_eq a r, IsMeasurableRatCDF.stieltjesFunctionAux]
refine Equiv.iInf_congr ?_ ?_
| Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | 286 | 292 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.DirectSum.Basic
/-!
# Additively-graded multiplicative structures on `⨁ i, A i`
This module provides a set of h... | /-- A heavily unfolded version of the definition of multiplication -/
theorem mul_eq_sum_support_ghas_mul [∀ (i : ι) (x : A i), Decidable (x ≠ 0)] (a a' : ⨁ i, A i) :
a * a' =
| Mathlib/Algebra/DirectSum/Ring.lean | 302 | 304 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Action.Opposite
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.GroupWithZero.U... | section
| Mathlib/Algebra/Star/Basic.lean | 289 | 290 |
/-
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.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... | cosZeta a (1 - s) = 2 * (2 * π) ^ (-s) * Gamma s * cos (π * s / 2) * hurwitzZetaEven a s := by
rw [← Gammaℂ]
have : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (Gammaℝ (1 - s))⁻¹ := by
rw [cosZeta, Function.update_of_ne, div_eq_mul_inv]
simpa [sub_eq_zero] using (hs 0).symm
rw [this, completedCos... | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 769 | 776 |
/-
Copyright (c) 2022 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Fabian Kruse, Nikolas Kuhn
-/
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
/-!
# Stirling's formula
Thi... | log (stirlingSeq n) = Real.log n ! - 1 / 2 * Real.log (2 * n) - n * log (n / exp 1) := by
cases n
| Mathlib/Analysis/SpecialFunctions/Stirling.lean | 61 | 62 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | using Möbius inversion. -/
| Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 412 | 412 |
/-
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.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... |
/-- Replacing the third point by one on the same line does not change twice the oriented angle. -/
theorem _root_.Collinear.two_zsmul_oangle_eq_right {p₁ p₂ p₃ p₃' : P}
(h : Collinear ℝ ({p₃, p₂, p₃'} : Set P)) (hp₃p₂ : p₃ ≠ p₂) (hp₃'p₂ : p₃' ≠ p₂) :
(2 : ℤ) • ∡ p₁ p₂ p₃ = (2 : ℤ) • ∡ p₁ p₂ p₃' := by
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 555 | 559 |
/-
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, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | Mathlib/Topology/Constructions.lean | 1,615 | 1,622 | |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.NumberTheory.LSeries.Basic
/-!
# Linearity of the L-series of `f` as a function of `f`
We show that the `LSer... | simpa [LSeriesHasSum, term_smul] using hf.const_smul c
lemma LSeriesSummable.smul {f : ℕ → ℂ} (c : ℂ) {s : ℂ} (hf : LSeriesSummable f s) :
| Mathlib/NumberTheory/LSeries/Linearity.lean | 113 | 115 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Dynamics.Newton
import Mathlib.LinearAlgebra.Semisimple
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
/-!
# Jordan-Chevalley-Dunford decomposition
... | over a perfect field may be written as a sum of nilpotent and semisimple endomorphisms. Moreover
these nilpotent and semisimple components are polynomial expressions in the original endomorphism.
-/
theorem exists_isNilpotent_isSemisimple [PerfectField K] :
∃ᵉ (n ∈ adjoin K {f}) (s ∈ adjoin K {f}), IsNilpotent n ∧ ... | Mathlib/LinearAlgebra/JordanChevalley.lean | 71 | 75 |
/-
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.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | @[deprecated (since := "2024-10-24")] alias ord_proj_le := ordProj_le
| Mathlib/Data/Nat/Factorization/Basic.lean | 116 | 116 |
/-
Copyright (c) 2023 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Kernel.Composition.Comp
/-!
# Invariance of measures along a kernel
We say that a measure `μ` is invariant with respect to a kernel `κ` if its ... | Pi.add_apply, Measure.bind_apply hs (Kernel.aemeasurable _),
Measure.bind_apply hs (Kernel.aemeasurable _)]
@[deprecated "Use comp_smul in Composition/MeasureComp" (since := "2025-02-28")]
theorem bind_smul (κ : Kernel α β) (μ : Measure α) (r : ℝ≥0∞) : (r • μ).bind κ = r • μ.bind κ := by
| Mathlib/Probability/Kernel/Invariance.lean | 43 | 47 |
/-
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.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... | apply ext_inner_right ℝ
intro w
rw [Orientation.inner_rightAngleRotation_left]
| Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 527 | 529 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.Algebra.Category.Grp.Limits
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.CategoryTheory.Galois.Decomposition
import Mathlib.Catego... | (autGaloisSystem F ⋙ forget _).sections
noncomputable instance : Group (AutGalois F) :=
inferInstanceAs <| Group (autGaloisSystem F ⋙ forget _).sections
/-- The canonical projection from `AutGalois F` to the `C`-automorphism group of each
| Mathlib/CategoryTheory/Galois/Prorepresentability.lean | 217 | 222 |
/-
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.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | (hz : z ∈ s) : s.WSameSide x y :=
h.symm.wSameSide₃₂ hz
| Mathlib/Analysis/Convex/Side.lean | 321 | 323 |
/-
Copyright (c) 2020 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.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.Limits
import Mathlib.Al... | theorem toOpen_res (U V : Opens (PrimeSpectrum.Top R)) (i : V ⟶ U) :
toOpen R U ≫ (structureSheaf R).1.map i.op = toOpen R V :=
rfl
@[simp]
| Mathlib/AlgebraicGeometry/StructureSheaf.lean | 386 | 390 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | lemma coe_Δ' : W.Δ' = W.Δ :=
rfl
/-- The j-invariant `j` of an elliptic curve, which is invariant under isomorphisms over `R`.
| Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 372 | 375 |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Prime.Pow
import Mathlib.NumberTheory.Divisors
/-!
# Prime powers a... | · simp_all
refine ⟨_, _, (Nat.minFac_prime hn).prime, ?_, h⟩
simp [pos_iff_ne_zero, ← Finsupp.mem_support_iff, Nat.support_factorization, hn',
Nat.minFac_prime hn, Nat.minFac_dvd]
theorem isPrimePow_iff_minFac_pow_factorization_eq {n : ℕ} (hn : n ≠ 1) :
IsPrimePow n ↔ n.minFac ^ n.factorization n.minFac ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 27 | 33 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | rw [← h_eq_add]
exact singularPart_eq_zero_iff_apply_eq_zero κ η a
lemma withDensity_rnDeriv_eq_zero_iff_measure_eq_zero (κ η : Kernel α γ)
[IsFiniteKernel κ] [IsFiniteKernel η] (a : α) :
withDensity η (rnDeriv κ η) a = 0 ↔ κ a (mutuallySingularSetSlice κ η a)ᶜ = 0 := by
have h_eq_add := rnDeriv_add_sing... | Mathlib/Probability/Kernel/RadonNikodym.lean | 444 | 454 |
/-
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... | lemma rightHomologyMap'_congr (h : Homotopy φ₁ φ₂) (h₁ : S₁.RightHomologyData)
(h₂ : S₂.RightHomologyData) : rightHomologyMap' φ₁ h₁ h₂ = rightHomologyMap' φ₂ h₁ h₂ := by
rw [h.eq_add_nullHomotopic, rightHomologyMap'_add, rightHomologyMap'_nullHomotopic, add_zero]
| Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean | 651 | 653 |
/-
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 | 1,477 | 1,487 | |
/-
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.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
/-!
# Contents
In this file... | Mathlib/MeasureTheory/Measure/Content.lean | 455 | 472 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Opposites
import Mathlib.Tactic.Spread
/-!
# Definitions of group actions
This file de... | Mathlib/Algebra/Group/Action/Defs.lean | 655 | 657 | |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Data.Fin.VecNotati... | Mathlib/Data/Real/GoldenRatio.lean | 84 | 84 | |
/-
Copyright (c) 2023 Yaël Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Or... | rw [coe_union]
exact Set.union_subset hts hus)
(by rintro s₁ s₂ hs h₂ _ ⟨t, ht, hts, rfl⟩; exact h₂.finsetSup'_mem ht fun i hi ↦ hs <| hts hi)
| Mathlib/Order/SupClosed.lean | 291 | 294 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
/-!
# The derivative of the scalar restriction of a linear map
For detailed document... | exact ⟨g', hs.eq (hg'.restrictScalars 𝕜) hf.hasFDerivWithinAt⟩
· rintro ⟨f', hf'⟩
exact ⟨f', hf.hasFDerivWithinAt.of_restrictScalars 𝕜 hf'⟩
| Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 99 | 102 |
/-
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, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,419 | 1,420 | |
/-
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.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | UniformCauchySeqOn (fun (i : ι × ι') a => (F i.fst a, F' i.snd a)) (p ×ˢ p') s :=
(congr_arg _ s.inter_self).mp ((h.prodMap h').comp fun a => (a, a))
theorem UniformCauchySeqOn.prod' {β' : Type*} [UniformSpace β'] {F' : ι → α → β'}
| Mathlib/Topology/UniformSpace/UniformConvergence.lean | 481 | 484 |
/-
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.Data.Finset.Grade
import Mathlib.Data.Finset.Powerset
import Mathlib.Order.Interval.Finset.Basic
/-!
# Intervals of finsets as finsets
This file provides... | theorem card_Iio_finset : (Iio s).card = 2 ^ s.card - 1 := by
rw [Iio_eq_ssubsets, ssubsets, card_erase_of_mem (mem_powerset_self _), card_powerset]
| Mathlib/Data/Finset/Interval.lean | 115 | 116 |
/-
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.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | ⟨{ (⊤ : Submodule R M) with lie_mem := fun {x m} _ ↦ mem_univ ⁅x, m⁆ }⟩
@[simp]
theorem top_coe : ((⊤ : LieSubmodule R L M) : Set M) = univ :=
rfl
| Mathlib/Algebra/Lie/Submodule.lean | 293 | 297 |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.CDF
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
/-! # Gamma distributions over ℝ
Define the gamm... |
/-- The gamma pdf is nonnegative -/
lemma gammaPDFReal_nonneg {a r : ℝ} (ha : 0 < a) (hr : 0 < r) (x : ℝ) :
0 ≤ gammaPDFReal a r x := by
unfold gammaPDFReal
| Mathlib/Probability/Distributions/Gamma.lean | 89 | 93 |
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 731 | 732 | |
/-
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.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... |
@[simp] lemma ncard_graphOn (s : Set α) (f : α → β) : (s.graphOn f).ncard = s.ncard := by
rw [← ncard_image_of_injOn fst_injOn_graph, image_fst_graphOn]
| Mathlib/Data/Set/Card.lean | 839 | 842 |
/-
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.RingTheory.DedekindDomain.Ideal
/-!
# The ideal class group
This file defines the ideal class group `ClassGroup R` of fractional ideals of `R`
inside its f... | coe_toPrincipalIdeal, coe_mapEquiv, MulEquiv.refl_apply]
refine ⟨fun ⟨x, hx⟩ => ⟨⟨x, by rw [← hx, coe_spanSingleton]⟩⟩, ?_⟩
intro hI
obtain ⟨x, hx⟩ := @Submodule.IsPrincipal.principal _ _ _ _ _ _ hI
| Mathlib/RingTheory/ClassGroup.lean | 323 | 326 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | end LinearOrderedField
/-!
#### Segments in an ordered space
Relates `segment`, `openSegment` and `Set.Icc`, `Set.Ico`, `Set.Ioc`, `Set.Ioo`
-/
| Mathlib/Analysis/Convex/Segment.lean | 402 | 408 |
/-
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.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 1,355 | 1,363 | |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
/-!
# Separable closure
This file contains basics about the (relative) separable closure of a fie... | @[simp]
theorem sepDegree_self : sepDegree F F = 1 := by
| Mathlib/FieldTheory/SeparableClosure.lean | 317 | 318 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.Seminorm
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Equicontinuit... | Eq.subset (p i).ball_zero_eq_preimage_ball.symm⟩
/-- If a family of seminorms is continuous, then their basis sets are neighborhoods of zero. -/
lemma basisSets_mem_nhds {𝕜 E ι : Type*} [NormedField 𝕜]
[AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] (p : SeminormFamily 𝕜 E ι)
(hp : ∀ i, Contin... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 190 | 207 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
import Mathlib.Logic.Function.ULift
/-!
# The Yoneda emb... |
variable (F : Cᵒᵖ ⥤ Type v) [hF : F.IsRepresentable]
| Mathlib/CategoryTheory/Yoneda.lean | 340 | 342 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,670 | 1,674 | |
/-
Copyright (c) 2023 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.MeasureTheory.MeasurableSpace.Embedding
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSepa... | (h : x ≠ y) : ∃ s, MeasurableSet s ∧ x ∈ s ∧ y ∉ s := by
contrapose! h
exact separatesPoints_def h
| Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 144 | 147 |
/-
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.NumberTheory.ModularForms.JacobiTheta.TwoVariable
/-!
# Asymptotic bounds for Jacobi theta functions
The goal of this file is to establish some tech... | lemma summable_f_nat (k : ℕ) (a : ℝ) {t : ℝ} (ht : 0 < t) : Summable (f_nat k a t) := by
have : Summable fun n : ℕ ↦ n ^ k * exp (-π * (n + a) ^ 2 * t) := by
refine (((summable_pow_mul_jacobiTheta₂_term_bound (|a| * t) ht k).mul_right
(rexp (-π * a ^ 2 * t))).comp_injective Nat.cast_injective).of_norm_bound... | Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean | 85 | 103 |
/-
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.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (σ-algebras)
* A fami... | constructor
· refine fun hS ↦ (Measure.pi_eq fun h hm ↦ ?_).symm
rw [← (h₀ hm).1, ← (h₀ hm).2]
simpa [hm] using hS Finset.univ (sets := h)
· intro h S s hs
specialize h₀ (s := fun i ↦ if i ∈ S then s i else univ)
fun i ↦ by beta_reduce; split_ifs with hiS <;> simp [hiS, hs]
simp only [apply_... | Mathlib/Probability/Independence/Basic.lean | 634 | 647 |
/-
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.Data.Set.Finite.Lemmas
import Mathlib.ModelTheory.Substructures
/-!
# Finitely Generated First-Order Structures
This file defines what it means for a... | theorem fg_closure {s : Set M} (hs : s.Finite) : FG (closure L s) :=
⟨hs.toFinset, by rw [hs.coe_toFinset]⟩
| Mathlib/ModelTheory/FinitelyGenerated.lean | 67 | 68 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... | (setIntegral_condExp (𝒢.le i) (hint _) hs).symm
rw [this]
exact setIntegral_mono_ae (hint i).integrableOn integrable_condExp.integrableOn (hf i)
| Mathlib/Probability/Martingale/Basic.lean | 388 | 391 |
/-
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, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,003 | 2,004 | |
/-
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.MeasureTheory.Integral.IntegrableOn
/-!
# Locally integrable functions
A function is called *locally integrable* (`MeasureTheory.LocallyIntegrabl... | LocallyIntegrable (-f) μ := fun x ↦ (hf x).neg
protected theorem LocallyIntegrable.smul {𝕜 : Type*} [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 E]
[IsBoundedSMul 𝕜 E] (hf : LocallyIntegrable f μ) (c : 𝕜) :
LocallyIntegrable (c • f) μ := fun x ↦ (hf x).smul c
| Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 301 | 305 |
/-
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.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... |
theorem exists_min_root [LinearOrder R] (p : R[X]) (hp : p ≠ 0) : ∃ x₀, ∀ x, p.IsRoot x → x₀ ≤ x :=
Set.exists_lower_bound_image _ _ <| finite_setOf_isRoot hp
| Mathlib/Algebra/Polynomial/Roots.lean | 136 | 139 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
## Basic properties of `C^n` functions between manifolds
In this file, we show that stan... | theorem ContMDiffWithinAt.comp_of_eq {t : Set M'} {g : M' → M''} {x : M} {y : M'}
(hg : ContMDiffWithinAt I' I'' n g t y) (hf : ContMDiffWithinAt I I' n f s x)
(st : MapsTo f s t) (hx : f x = y) : ContMDiffWithinAt I I'' n (g ∘ f) s x := by
subst hx; exact hg.comp x hf st
| Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 81 | 84 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... |
lemma setIntegral_densityProcess_of_mem (hκν : fst κ ≤ ν) [hν : IsFiniteKernel ν]
(n : ℕ) (a : α) {s : Set β} (hs : MeasurableSet s) {u : Set γ}
(hu : u ∈ countablePartition γ n) :
∫ x in u, densityProcess κ ν n a x s ∂(ν a) = (κ a).real (u ×ˢ s) := by
have : IsFiniteKernel κ := isFiniteKernel_of_isFinit... | Mathlib/Probability/Kernel/Disintegration/Density.lean | 197 | 236 |
/-
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.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | modEq_zero_iff_dvd.1 <| by
have xm := (xy_modEq_yn a1 n k).right; rw [← ke] at xm
exact (xm.of_dvd <| by simp [_root_.pow_succ]).symm.trans h.modEq_zero_nat
rw [ke]
| Mathlib/NumberTheory/PellMatiyasevic.lean | 430 | 433 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Birkhoff sums
In this file we define `birkhoffSum f g n x` ... | theorem birkhoffSum_add (f : α → α) (g : α → M) (m n : ℕ) (x : α) :
birkhoffSum f g (m + n) x = birkhoffSum f g m x + birkhoffSum f g n (f^[m] x) := by
simp_rw [birkhoffSum, sum_range_add, add_comm m, iterate_add_apply]
| Mathlib/Dynamics/BirkhoffSum/Basic.lean | 51 | 53 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Holder
/-!
# Real conjugate exponents
This file defines Hölder triple and Hölder conjugate exponents in `ℝ` and `... | lemma HolderConjugate.conjExponent (h : 1 < p) : p.HolderConjugate (conjExponent p) :=
(holderConjugate_iff_eq_conjExponent h).2 rfl
lemma holderConjugate_one_div (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) :
(1 / a).HolderConjugate (1 / b) := by simpa using HolderConjugate.inv_inv ha hb hab
| Mathlib/Data/Real/ConjExponents.lean | 197 | 202 |
/-
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 | 898 | 900 |
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