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) 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.Order.Star.Basic
import Mathlib.Data.Matrix.RowCol
/-!
# Dot product of two vectors
This file conta... |
@[deprecated (since := "2024-12-12")]
protected alias Matrix.dotProduct_eq_zero_iff := dotProduct_eq_zero_iff
| Mathlib/LinearAlgebra/Matrix/DotProduct.lean | 54 | 56 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | end ToSubring
variable (T : Subring R)
/-- Given a polynomial whose coefficients are in some subring, return
the corresponding polynomial whose coefficients are in the ambient ring. -/
def ofSubring (p : T[X]) : R[X] :=
∑ i ∈ p.support, monomial i (p.coeff i : R)
| Mathlib/RingTheory/Polynomial/Basic.lean | 433 | 440 |
/-
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... | theorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → M) :
(∏ k ∈ Ico a (b + 1), f k) = (∏ k ∈ Ico a b, f k) * f b := by
rw [Nat.Ico_succ_right_eq_insert_Ico hab, prod_insert right_not_mem_Ico, mul_comm]
| Mathlib/Algebra/BigOperators/Intervals.lean | 111 | 113 |
/-
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.Group.Measure
/-!
# Lebesgue Integration on Groups
We develop properties of integrals with a group as domain.
This file contains pr... | (f : G → ℝ≥0∞) (g : G) : (∫⁻ x, f (g / x) ∂μ) = ∫⁻ x, f x ∂μ := by
simp_rw [div_eq_mul_inv, lintegral_inv_eq_self (f <| g * ·), lintegral_mul_left_eq_self]
| Mathlib/MeasureTheory/Group/LIntegral.lean | 71 | 73 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Algebra.Module.Stro... | rw [← mem_closure_zero_iff_norm, ← specializes_iff_mem_closure, ← map_zero f] at *
exact hx.map hf
section
| Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean | 54 | 57 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | (bind_assoc := fun f g k => funext fun a => (f a).bind_assoc (fun b => g b a) fun b => k b a)
| Mathlib/Data/PFun.lean | 201 | 202 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,681 | 1,683 | |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.Analysis.Normed.Group.NullSubmodule
/-!
# The Fréche... | lemma nullSubmodule_le_ker_toDualMap_left : nullSubmodule 𝕜 E ≤ ker (toDualMap 𝕜 E) :=
fun _ hx ↦ ContinuousLinearMap.ext <| fun y ↦ inner_eq_zero_of_left y hx
end NullSubmodule
end Seminormed
section Normed
variable [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
| Mathlib/Analysis/InnerProductSpace/Dual.lean | 81 | 90 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Basic
/-!
# Homomorphisms of `R`-algebras
This file defines bundled homomorphisms of `R`-algebras.
## Main definitions
* `Al... |
/-- This is a special case of a more general instance that we define in a later file. -/
instance subsingleton_id : Subsingleton (R →ₐ[R] A) :=
⟨fun f g => AlgHom.ext fun _ => (f.commutes _).trans (g.commutes _).symm⟩
| Mathlib/Algebra/Algebra/Hom.lean | 395 | 398 |
/-
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.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | image2 f (s.image g) t = (image2 f' s t).image g' :=
(image_image2_distrib_left fun a b => (h_left_comm a b).symm).symm
| Mathlib/Data/Set/NAry.lean | 244 | 245 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 613 | 614 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Data.Set.Image
/-!
# Directed indexed families and sets
This file defines directed indexed families and directed sets. An indexed famil... |
lemma DirectedOn.is_bot_of_is_min {s : Set α} (hd : DirectedOn (· ≥ ·) s)
{m} (hm : m ∈ s) (hmin : ∀ a ∈ s, a ≤ m → m ≤ a) : ∀ a ∈ s, m ≤ a := fun a as =>
let ⟨x, xs, xm, xa⟩ := hd m hm a as
(hmin x xs xm).trans xa
lemma DirectedOn.is_top_of_is_max {s : Set α} (hd : DirectedOn (· ≤ ·) s)
{m} (hm : m ∈ s) ... | Mathlib/Order/Directed.lean | 234 | 243 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.ConeCategory
/-!
... | @[reassoc]
theorem snd_app_right (a) : K.ι.app (WalkingMultispan.left a) = I.snd a ≫ K.π _ := by
rw [← K.w (WalkingMultispan.Hom.snd a)]
| Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | 553 | 555 |
/-
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.Basis.Submodule
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix... |
/-- A matrix whose columns form a basis `b'`, expressed w.r.t. a basis `b`, is invertible. -/
def Basis.invertibleToMatrix [DecidableEq ι] [Fintype ι] (b b' : Basis ι R₂ M₂) :
Invertible (b.toMatrix b') :=
⟨b'.toMatrix b, Basis.toMatrix_mul_toMatrix_flip _ _, Basis.toMatrix_mul_toMatrix_flip _ _⟩
@[simp]
| Mathlib/LinearAlgebra/Matrix/Basis.lean | 257 | 263 |
/-
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.Module.Submodule.Ker
import Mathlib.Algebra.Module.Submodul... | theorem range_eq_bot {f : M →ₛₗ[τ₁₂] M₂} : range f = ⊥ ↔ f = 0 := by
rw [← range_le_bot_iff, le_bot_iff]
| Mathlib/Algebra/Module/Submodule/Range.lean | 184 | 185 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | obtain ⟨ε, ⟨ε_pos, ε_lt⟩⟩ := exists_real_pos_lt_infEdist_of_not_mem_closure h
filter_upwards [eventually_lt_nhds ε_pos] with δ hδ
simp only [thickening, mem_setOf_eq, not_lt]
exact (ENNReal.ofReal_le_ofReal hδ.le).trans ε_lt.le
/-- The (open) thickening equals the preimage of an open interval under `EMetric.in... | Mathlib/Topology/MetricSpace/Thickening.lean | 59 | 66 |
/-
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.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Notation.Prod
import Mathlib.Data.Set.Image
/-!
# Support of a func... | @[to_additive (attr := simp)]
theorem mulSupport_eq_empty_iff {f : α → M} : mulSupport f = ∅ ↔ f = 1 := by
rw [← subset_empty_iff, mulSupport_subset_iff', funext_iff]
simp
| Mathlib/Algebra/Group/Support.lean | 113 | 116 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Prime.Basic
import Ma... | 4 ∣ n ∨ ∃ p, Prime p ∧ p ∣ n ∧ Odd p := by
obtain ⟨_ | _ | k, rfl⟩ | ⟨p, hp, hdvd, hodd⟩ := n.eq_two_pow_or_exists_odd_prime_and_dvd
· contradiction
· contradiction
· simp [Nat.pow_succ, mul_assoc]
| Mathlib/Data/Nat/Factors.lean | 305 | 309 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | Mathlib/Topology/Connected/Basic.lean | 1,322 | 1,334 | |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | have := (hc.hasFDerivAt.clm_apply hu.hasFDerivAt).hasDerivAt
rwa [add_apply, comp_apply, flip_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this
theorem derivWithin_clm_apply (hc : DifferentiableWithinAt 𝕜 c s x)
| Mathlib/Analysis/Calculus/Deriv/Mul.lean | 480 | 484 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | rw [← h a, ofSubtype_apply_of_mem (p := (· ∈ s)) _ ha, subtypePerm_apply]
· rintro ⟨hc, h⟩ a
| Mathlib/GroupTheory/Perm/Support.lean | 363 | 364 |
/-
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.CharP.Defs
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... | @[simp]
theorem constantCoeff_X : constantCoeff R X = 0 :=
MvPowerSeries.coeff_zero_X _
| Mathlib/RingTheory/PowerSeries/Basic.lean | 376 | 379 |
/-
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.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | (e.continuousAt_symm (e.map_source hx)).continuousWithinAt
(e.continuousAt hx).continuousWithinAt
theorem map_nhdsWithin_preimage_eq {x} (hx : x ∈ e.source) (s : Set Y) :
map e (𝓝[e ⁻¹' s] x) = 𝓝[s] e x := by
| Mathlib/Topology/PartialHomeomorph.lean | 343 | 347 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,704 | 1,711 | |
/-
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.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
/-!
# Ackermann function
In this file, we def... | Mathlib/Computability/Ackermann.lean | 70 | 70 | |
/-
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 [← SetLike.coe_set_eq]
apply le_antisymm
· intro x hx
simp only [SetLike.mem_coe, LinearMap.mem_range]
rw [Ideal.map, ← submodule_span_eq] at hx
refine Submodule.span_induction ?_ ?_ ?_ ?_ hx
· intro x
simp only [includeRight_apply, Set.mem_image, SetLike.mem_coe]
rintro ⟨y, hy, rfl... | Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 512 | 578 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | Mathlib/LinearAlgebra/Ray.lean | 738 | 742 | |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Cardinal
import Mathlib.SetTheory.Cardi... | ext n
cases h : f (n + 1) <;> simp [h, _root_.pow_succ']
| Mathlib/Data/Real/Cardinality.lean | 82 | 83 |
/-
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.Algebra.Polynomial.Identities
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.Topology.A... | ∀ {ε}, ε > 0 → ∃ N : ℕ, ∀ {n}, n ≥ N → ‖F.derivative.eval a‖ * T ^ 2 ^ n < ε := fun hε ↦
eventually_atTop.1 <| (bound' hnorm).eventually <| gt_mem_nhds hε
private theorem bound'_sq :
Tendsto (fun n : ℕ => ‖F.derivative.eval a‖ ^ 2 * T ^ 2 ^ n) atTop (𝓝 0) := by
rw [← mul_zero ‖F.derivative.eval a‖, sq]
... | Mathlib/NumberTheory/Padics/Hensel.lean | 299 | 319 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space.Integrable
/-!
# Uniform integrability
This file contains the def... | ENNReal.rpow_le_rpow_iff (ENNReal.toReal_pos hp hp_top), ENNReal.ofReal_div_of_pos hM]
· simpa only [ENNReal.ofReal_eq_zero, not_le, Ne]
section
variable {f : α → β}
/-- Auxiliary lemma for `MeasureTheory.MemLp.eLpNorm_indicator_le`. -/
theorem MemLp.eLpNorm_indicator_le' (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) (h... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 346 | 373 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Ralf Stephan
-/
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.Squarefree
/-!
# Smooth numbers
For `s : Finset ℕ` we define the set `Nat.factoredNum... | /-- Membership in `Nat.smoothNumbers n` is decidable. -/
instance (n : ℕ) : DecidablePred (· ∈ smoothNumbers n) :=
inferInstanceAs <| DecidablePred fun x ↦ x ∈ {m | m ≠ 0 ∧ ∀ p ∈ primeFactorsList m, p < n}
/-- A number that divides an `n`-smooth number is itself `n`-smooth. -/
| Mathlib/NumberTheory/SmoothNumbers.lean | 291 | 295 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Data.Set.UnionLift
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.RingTheory.NonUnitalSubring.Basic
... | theorem mem_inf {S T : NonUnitalSubalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T :=
Iff.rfl
| Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean | 763 | 764 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 1,062 | 1,064 | |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Ess... |
/-- The limit cone obtained from a right adjoint of the constant functor. -/
@[simps]
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 536 | 538 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... |
theorem sqrt_ne_zero (h : 0 ≤ x) : √x ≠ 0 ↔ x ≠ 0 := by rw [not_iff_not, sqrt_eq_zero h]
| Mathlib/Data/Real/Sqrt.lean | 242 | 244 |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.UpperLower.Basic
/-!
# Young diagrams
A You... | · rfl
· exact List.pairwise_iff_get.mp hw _ _ h
| Mathlib/Combinatorics/Young/YoungDiagram.lean | 420 | 421 |
/-
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 ... |
theorem unpaired' {f : ℕ → ℕ → ℕ} : Nat.Primrec (Nat.unpaired f) ↔ Primrec₂ f :=
| Mathlib/Computability/Primrec.lean | 372 | 373 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen,
Kim Morrison, Chris Hughes, Anne Baanen, Junyan Xu
-/
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimensi... | refine not_le_of_gt (span_lt_top_of_card_lt_finrank
(show (b '' (Set.univ \ {i})).toFinset.card < finrank K V from ?_)) ?_
· calc
(b '' (Set.univ \ {i})).toFinset.card = ((Set.univ \ {i}).toFinset.image b).card := by
rw [Set.toFinset_card, Fintype.card_ofFinset]
_ ≤ (Set.univ \ {... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 123 | 166 |
/-
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... | instance isSplitEpi_prod_snd [HasZeroMorphisms C] {X Y : C} [HasLimit (pair X Y)] :
IsSplitEpi (prod.snd : X ⨯ Y ⟶ Y) :=
IsSplitEpi.mk' { section_ := prod.lift 0 (𝟙 Y) }
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 574 | 576 |
/-
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.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathli... | rintro ⟨a, ha, b, hb, rfl⟩
exact s.mul_mem ha hb
@[to_additive]
theorem closure_mul_le (S T : Set M) : closure (S * T) ≤ closure S ⊔ closure T :=
| Mathlib/Algebra/Group/Submonoid/Pointwise.lean | 72 | 76 |
/-
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... | Continuous fun a ↦ Fin.tail (f a) :=
continuous_iff_continuousAt.2 fun _ ↦ hf.continuousAt.finTail
end Fin
theorem isOpen_set_pi {i : Set ι} {s : ∀ a, Set (π a)} (hi : i.Finite)
| Mathlib/Topology/Constructions.lean | 850 | 855 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Martin Dvorak
-/
import Mathlib.Algebra.Order.Kleene
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.DeriveFintype
/-!
# Languages... | constructor
· rintro ⟨a, ha, b, ⟨S, rfl, rfl, hS⟩, rfl⟩
exact ⟨a :: S, rfl, rfl, forall_mem_cons.2 ⟨ha, hS⟩⟩
· rintro ⟨_ | ⟨a, S⟩, rfl, hn, hS⟩ <;> cases hn
rw [forall_mem_cons] at hS
exact ⟨a, hS.1, _, ⟨S, rfl, rfl, hS.2⟩, rfl⟩
theorem kstar_eq_iSup_pow (l : Language α) : l∗ = ⨆ i : ℕ, l... | Mathlib/Computability/Language.lean | 234 | 249 |
/-
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.RingTheory.Trace.Basic
import Mathlib.FieldTheory.Finite.GaloisField
/-!
# The trace map for finite fields
We state the fact that the trace map from a ... | ∃ b : F, Algebra.trace (ZMod (ringChar F)) F (a * b) ≠ 0 := by
haveI : Fact (ringChar F).Prime := ⟨CharP.char_is_prime F _⟩
have htr := traceForm_nondegenerate (ZMod (ringChar F)) F a
simp_rw [Algebra.traceForm_apply] at htr
by_contra! hf
exact ha (htr hf)
end FiniteField
| Mathlib/FieldTheory/Finite/Trace.lean | 25 | 32 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | Ico a b ∪ Ico b c = Ico a c := by
rw [← coe_inj, coe_union, coe_Ico, coe_Ico, coe_Ico, Set.Ico_union_Ico_eq_Ico hab hbc]
@[simp]
| Mathlib/Order/Interval/Finset/Basic.lean | 822 | 825 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
/-!
# A computable model of ZFA without infinity
In this file we define finite hereditary list... | | ⟨_, l⟩ => l.toList
/-- Predicate stating that a ZFA list is proper. -/
def IsList (l : Lists α) : Prop :=
l.1
/-- Converts a `List` of ZFA lists to a ZFA list. -/
| Mathlib/SetTheory/Lists.lean | 201 | 207 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | theorem idealFactorsFunOfQuotHom_comp {f : R ⧸ I →+* A ⧸ J} {g : A ⧸ J →+* B ⧸ L}
(hf : Function.Surjective f) (hg : Function.Surjective g) :
(idealFactorsFunOfQuotHom hg).comp (idealFactorsFunOfQuotHom hf) =
idealFactorsFunOfQuotHom (show Function.Surjective (g.comp f) from hg.comp hf) := by
refine Ord... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 1,052 | 1,065 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 1,723 | 1,730 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Filter.Tendsto
import Mathlib.Order.Filter.Bases.Finite
/-!
# (Co)product of a family of f... |
theorem mem_of_pi_mem_pi [∀ i, NeBot (f i)] {I : Set ι} (h : I.pi s ∈ pi f) {i : ι} (hi : i ∈ I) :
s i ∈ f i := by
classical
| Mathlib/Order/Filter/Pi.lean | 74 | 77 |
/-
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.MvPolynomial.Variables
/-!
# Multivariate polynomials over a ring
Many results about polynomials hold when th... | by_contra! hc
have h := support_sub σ f g hm
simp only [mem_support_iff, Ne, coeff_sub, sub_eq_zero] at hm
rcases Finset.mem_union.1 h with cf | cg
· exact hm (hf m cf hc)
· exact hm (hg m cg hc)
end DegreeOf
section TotalDegree
@[simp]
theorem totalDegree_neg (a : MvPolynomial σ R) : (-a).totalDegree = ... | Mathlib/Algebra/MvPolynomial/CommRing.lean | 183 | 195 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 477 | 479 | |
/-
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 | 1,333 | 1,336 | |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Utensil Song
-/
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
import Mathlib.LinearAlge... |
See `CliffordAlgebra.equivOfIsometry` for the case when `f` is a `QuadraticForm.IsometryEquiv`. -/
def map (f : Q₁ →qᵢ Q₂) :
| Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean | 285 | 287 |
/-
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, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | /-- Extension for the `positivity` tactic: exponentiation by a real number is positive (namely 1)
when the exponent is zero. The other cases are done in `evalRpow`. -/
@[positivity (_ : ℝ) ^ (0 : ℝ)]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 356 | 358 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | rw [mul_smul, hx, hy, mul_assoc]
/-- Each element of the free product corresponds to a unique reduced word. -/
| Mathlib/GroupTheory/CoprodI.lean | 595 | 597 |
/-
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.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | rw [mul_assoc, content_C_mul, content_C_mul, mul_comm p.primPart, mul_assoc, content_C_mul,
content_C_mul, h, mul_one, content_primPart, content_primPart, mul_one, mul_one]
rw [← normalize_content, normalize_eq_one, isUnit_iff_dvd_one,
content_eq_gcd_leadingCoeff_content_eraseLead, leadingCoeff_... | Mathlib/RingTheory/Polynomial/Content.lean | 346 | 353 |
/-
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.LpSeminorm.Trim
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.StronglyMeasura... | rw [Set.indicator_support]
exact hfm.ae_eq_mk.symm
have hf'_meas : StronglyMeasurable[m] f' := hfm.stronglyMeasurable_mk.indicator (hs_f.diff hs_g)
have hf'_Lp : MemLp f' p μ := hf.ae_eq hff'
let g' := (s_g \ s_f).indicator (hgm.mk g)
have hgg' : g =ᵐ[μ] g' := by
have : s_g \ s_f =ᵐ[μ] s_g := by
... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 572 | 614 |
/-
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 ... | Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean | 405 | 405 | |
/-
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... | theorem circumcenter_eq_point (s : Simplex ℝ P 0) (i : Fin 1) : s.circumcenter = s.points i := by
have h := s.circumcenter_mem_affineSpan
have : Unique (Fin 1) := ⟨⟨0, by decide⟩, fun a => by simp only [Fin.eq_zero]⟩
simp only [Set.range_unique, AffineSubspace.mem_affineSpan_singleton] at h
rw [h]
| Mathlib/Geometry/Euclidean/Circumcenter.lean | 307 | 311 |
/-
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... | lemma map_succ_succ (i j : ℕ) (hi : i + 1 < n + 1 + 1) (hj : j + 1 < n + 1 + 1)
(hij : i + 1 ≤ j + 1) :
map F f ⟨i + 1, hi⟩ ⟨j + 1, hj⟩ hij = F.map' i j := rfl
| Mathlib/CategoryTheory/ComposableArrows.lean | 330 | 332 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.Iso
/-!
# Natural isomorphisms
For the most ... | theorem naturality_2 (α : F ≅ G) (f : X ⟶ Y) : α.hom.app X ≫ G.map f ≫ α.inv.app Y = F.map f := by
simp
| Mathlib/CategoryTheory/NatIso.lean | 176 | 177 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Order.Group.Multiset
import Mathlib.Data.Setoid.Basic
import Mathlib.Data.Vector.Basic
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Tactic.Ap... | coe_injective <| Multiset.map_injective hf <| coe_inj.2 h
| Mathlib/Data/Sym/Basic.lean | 384 | 385 |
/-
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.Connected.Basic
/-!
# Locally connected topological spaces
A topological space is **locally connected** if each neighborhood filter admi... | exact forall_congr' fun x => Filter.hasBasis_self.symm
theorem locallyConnectedSpace_of_connected_bases {ι : Type*} (b : α → ι → Set α) (p : α → ι → Prop)
(hbasis : ∀ x, (𝓝 x).HasBasis (p x) (b x))
(hconnected : ∀ x i, p x i → IsPreconnected (b x i)) : LocallyConnectedSpace α := by
| Mathlib/Topology/Connected/LocallyConnected.lean | 118 | 122 |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.GroupAction.Defs
import Mathlib.GroupTheory... |
/-- Stabilizers in monoid SubMulAction coincide with stabilizers in the ambient space -/
@[to_additive]
theorem stabilizer_of_subMul.submonoid {p : SubMulAction R M} (m : p) :
MulAction.stabilizerSubmonoid R m = MulAction.stabilizerSubmonoid R (m : M) := by
| Mathlib/GroupTheory/GroupAction/SubMulAction.lean | 382 | 386 |
/-
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.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... | rw [withDensity_apply _ hs, withDensity_apply _ hs, restrict_withDensity hs,
lintegral_withDensity_eq_lintegral_mul₀ hf.restrict hg.restrict]
theorem withDensity_mul (μ : Measure α) {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) :
μ.withDensity (f * g) = (μ.withDensity f).withDensity g :=
| Mathlib/MeasureTheory/Measure/WithDensity.lean | 491 | 495 |
/-
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 | 2,136 | 2,139 | |
/-
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.HomologicalComplexBiprod
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
/-! The homotopy ... | @[reassoc]
lemma sndX_inrX (i : ι) (hi : ¬ c.Rel i (c.next i)) :
sndX φ i ≫ inrX φ i = 𝟙 _ := by
dsimp [sndX, inrX]
simp only [dif_neg hi, Iso.hom_inv_id]
| Mathlib/Algebra/Homology/HomotopyCofiber.lean | 101 | 105 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 528 | 536 | |
/-
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.Data.List.Lemmas
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Li... | induction' is with t is ih generalizing is'; · simp
simp only [foldr_permutationsAux2, ih, map_flatMap, cons_append, permutationsAux_cons, map_append,
reverse_cons, append_assoc, singleton_append]
congr 2
funext _
rw [map_permutationsAux2]
simp +singlePass only [← permutationsAux2_comp_append]
| Mathlib/Data/List/Permutation.lean | 235 | 241 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Exte... | (of (φ.symm b : G) : HNNExtension G A B φ) = t⁻¹ * of (b : G) * t := by
rw [mul_assoc, of_mul_t]; simp
| Mathlib/GroupTheory/HNNExtension.lean | 81 | 83 |
/-
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... | · simp only [ha, if_true]
· simp only [ha, if_false]
split_ifs <;> simp only [neg_eq_zero, one_ne_zero, not_false_iff]
@[simp]
theorem quadraticCharFun_zero : quadraticCharFun F 0 = 0 := by
| Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 66 | 71 |
/-
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.Algebra.Polynomial.Identities
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.Topology.A... | Tendsto (fun i => ‖F.eval (newton_cau_seq hnorm i)‖) atTop (𝓝 0) :=
squeeze_zero (fun _ => norm_nonneg _) (newton_seq_norm_le hnorm) (bound'_sq hnorm)
private theorem newton_seq_dist_tendsto' :
Tendsto (fun n => ‖newton_cau_seq hnorm n - a‖) atTop (𝓝 ‖soln - a‖) :=
(continuous_norm.tendsto _).comp ((newt... | Mathlib/NumberTheory/Padics/Hensel.lean | 328 | 344 |
/-
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.RightHomology
/-!
# Homology of short complexes
In this file, we shall define the homology of short complexes `S`, i.e. diagrams... | (by rw [← cancel_mono h₂.ι, assoc, RightHomologyData.liftH_ι,
LeftHomologyData.f'_i_assoc, RightHomologyData.wp, zero_comp]) := by
simp only [← cancel_mono h₂.ι, ← cancel_epi h₁.π, π_leftRightHomologyComparison'_ι,
LeftHomologyData.π_descH_assoc, RightHomologyData.liftH_ι]
end
| Mathlib/Algebra/Homology/ShortComplex/Homology.lean | 570 | 576 |
/-
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.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | independent.
This is the affine version of `LinearIndependent.units_smul`. -/
theorem AffineIndependent.units_lineMap {p : ι → P} (hp : AffineIndependent k p) (j : ι)
(w : ι → Units k) : AffineIndependent k fun i => AffineMap.lineMap (p j) (p i) (w i : k) := by
| Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 258 | 262 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.Order.Disjointed
/-!
# Indu... | theorem subset_generatePiSystem_self (S : Set (Set α)) : S ⊆ generatePiSystem S := fun _ =>
generatePiSystem.base
| Mathlib/MeasureTheory/PiSystem.lean | 215 | 217 |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | theorem exists_seq_of_forall_finset_exists' {α : Type*} (P : α → Prop) (r : α → α → Prop)
| Mathlib/Data/Fintype/Basic.lean | 282 | 282 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... | rw [← attachFin_Iio, map_valEmbedding_attachFin]
| Mathlib/Order/Interval/Finset/Fin.lean | 210 | 211 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
import Mathlib.MeasureTheory.Measure.Prod
impo... | theorem testAgainstNN_mono (μ : FiniteMeasure Ω) {f g : Ω →ᵇ ℝ≥0} (f_le_g : (f : Ω → ℝ≥0) ≤ g) :
μ.testAgainstNN f ≤ μ.testAgainstNN g := by
simp only [← ENNReal.coe_le_coe, testAgainstNN_coe_eq]
gcongr
apply f_le_g
| Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 350 | 354 |
/-
Copyright (c) 2023 Bhavik Mehta, Rishi Mehta, Linus Sommer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Rishi Mehta, Linus Sommer
-/
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Combinatorics.SimpleG... | variable {p : G.Walk a a}
lemma IsHamiltonianCycle.isCycle (hp : p.IsHamiltonianCycle) : p.IsCycle :=
hp.toIsCycle
lemma IsHamiltonianCycle.map {H : SimpleGraph β} (f : G →g H) (hf : Bijective f)
(hp : p.IsHamiltonianCycle) : (p.map f).IsHamiltonianCycle where
toIsCycle := hp.isCycle.map hf.injective
isHami... | Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean | 73 | 85 |
/-
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... | @[simp]
protected theorem coe_toBasis_repr (b : OrthonormalBasis ι 𝕜 E) :
b.toBasis.equivFun = b.repr.toLinearEquiv :=
Basis.equivFun_ofEquivFun _
@[simp]
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 407 | 412 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Bilin... | simp only [Basis.map_apply, LinearEquiv.coe_mk]
exact his' _ ⟨_, rfl⟩
variable [Algebra.IsSeparable K L]
/-- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is
integrally closed and Noetherian, the integral closure `C` of `A` in `L` is
Noetherian over `A`. -/
theorem IsIntegralClosure.isNoe... | Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean | 145 | 167 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | ⟨ae_eq_zero_of_eLpNorm'_eq_zero (le_of_lt hq0_lt) hf, eLpNorm'_eq_zero_of_ae_zero hq0_lt⟩
variable {ε : Type*} [ENorm ε] in
theorem enorm_ae_le_eLpNormEssSup {_ : MeasurableSpace α} (f : α → ε) (μ : Measure α) :
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 987 | 990 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.MeasureTheory.Integral.ExpDecay
/-!
# The Gamma function
This file defines the `Γ` functio... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 613 | 639 | |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Field.UnitBall
/-!
# The circle
This file defines `circle` to be the m... | instance instSMulCommClass_left [SMul ℂ β] [SMul α β] [SMulCommClass ℂ α β] :
SMulCommClass Circle α β := Submonoid.smulCommClass_left _
instance instSMulCommClass_right [SMul ℂ β] [SMul α β] [SMulCommClass α ℂ β] :
SMulCommClass α Circle β := Submonoid.smulCommClass_right _
| Mathlib/Analysis/Complex/Circle.lean | 160 | 164 |
/-
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.ObjectProperty.ClosedUnderIsomorphisms
import Mathlib.CategoryTheory.MorphismProperty.Composition
import Mathlib.CategoryTheory.Localization.Adjun... | lemma W_isoClosure : W P.isoClosure = W P := by
ext X Y f
constructor
· intro hf Z hZ
exact hf _ (P.le_isoClosure _ hZ)
· rintro hf Z ⟨Z', hZ', ⟨e⟩⟩
constructor
· intro g₁ g₂ eq
rw [← cancel_mono e.hom]
apply (hf _ hZ').1
simp only [reassoc_of% eq]
· intro g
obtain ⟨a, h⟩... | Mathlib/CategoryTheory/Localization/Bousfield.lean | 61 | 74 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,583 | 2,587 | |
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space.AEEqFun
import Mathlib.MeasureTheory.Function.LpSpace.Complete
import Mathlib.MeasureThe... | end Induction
section CoeToLp
variable [Fact (1 ≤ p)]
| Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 648 | 653 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... |
@[simp]
theorem image₂_right [DecidableEq β] (h : s.Nonempty) : image₂ (fun _ y => y) s t = t :=
coe_injective <| by
push_cast
| Mathlib/Data/Finset/NAry.lean | 298 | 302 |
/-
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.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.LSeries.Convolution
import Mathlib.NumberTheory.LSeries.Deriv
import Mathlib.NumberThe... | /-- The L-series of the twist of the von Mangoldt function `Λ` by a Dirichlet character `χ` at `s`
equals the negative logarithmic derivative of the L-series of `χ` when `re s > 1`. -/
lemma LSeries_twist_vonMangoldt_eq {N : ℕ} (χ : DirichletCharacter ℂ N) {s : ℂ} (hs : 1 < s.re) :
L (↗χ * ↗Λ) s = - deriv (L ↗χ) s ... | Mathlib/NumberTheory/LSeries/Dirichlet.lean | 366 | 381 |
/-
Copyright (c) 2020 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.Data.Finite.Sum
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.CardCommute
import Mathlib.Grou... | · simp [← r_zero]
/-- If `0 < n`, then `r 1` has order `n`.
-/
@[simp]
theorem orderOf_r_one : orderOf (r 1 : DihedralGroup n) = n := by
rcases eq_zero_or_neZero n with (rfl | hn)
· rw [orderOf_eq_zero_iff']
intro n hn
rw [r_one_pow, one_def]
apply mt r.inj
simpa using hn.ne'
· apply (Nat.le_of... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 170 | 184 |
/-
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.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | outer measures is nonempty. -/
theorem restrict_sInf_eq_sInf_restrict (m : Set (OuterMeasure α)) {s : Set α} (hm : m.Nonempty) :
restrict s (sInf m) = sInf (restrict s '' m) := by
simp only [sInf_eq_iInf, restrict_biInf, hm, iInf_image]
end sInfGen
| Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 458 | 463 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Lie.Weights.Killing
import Mathlib.LinearAlgebra.RootSystem.Basic
import Mathlib.LinearAlgebra.RootSystem.Reduced
import Mathlib.LinearAlgebra.RootSy... | simp only [Int.cast_sub, Int.cast_natCast, Int.cast_mul, Int.cast_ofNat, eq_sub_iff_add_eq',
this, mul_comm (2 : K)]
lemma rootSpace_neg_nsmul_add_chainTop_of_le {n : ℕ} (hn : n ≤ chainLength α β) :
rootSpace H (- (n • α) + chainTop α β) ≠ ⊥ := by
by_cases hα : α.IsZero
· simpa only [hα.eq, smul_zero, ne... | Mathlib/Algebra/Lie/Weights/RootSystem.lean | 95 | 106 |
/-
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 | 1,578 | 1,580 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | rw [← add_left_cancel_iff, div_add_mod, ← natCast_div, ← natCast_mul, ← Nat.cast_add,
Nat.div_add_mod]
@[simp]
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,087 | 1,090 |
/-
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... |
@[continuity, fun_prop]
theorem continuous_quot_lift {f : X → Y} (hr : ∀ a b, r a b → f a = f b) (h : Continuous f) :
Continuous (Quot.lift f hr : Quot r → Y) :=
| Mathlib/Topology/Constructions.lean | 549 | 552 |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.DistribMulAction
import Mathlib.GroupTheory.OreLocalization.Basic
import Mathlib.Algebra.GroupWi... | Mathlib/RingTheory/OreLocalization/Basic.lean | 556 | 561 | |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
import Mathlib.SetTheory.Nimber.Basic
/-!
# Nim and the Sprague-Grundy theore... | Mathlib/SetTheory/Game/Nim.lean | 405 | 409 | |
/-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
import M... | Mathlib/MeasureTheory/Function/AEEqFun.lean | 927 | 928 | |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri
-/
import Mathlib.Geometry.Manifold.ContMDiffMap
import Mathlib.Geometry.Manifold.MFDeriv.Basic
/-!
# `C^n` monoid
A `C^n` monoid is a monoid that is also a `C^n` m... | ⟨(g * ·), contMDiff_mul_left⟩
/-- Right multiplication by `g`. It is meant to mimic the usual notation in Lie groups.
Used mostly through the notation `𝑹`.
| Mathlib/Geometry/Manifold/Algebra/Monoid.lean | 209 | 212 |
/-
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.LpSeminorm.Trim
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.StronglyMeasura... |
/-- If `f` belongs to `lpMeasSubgroup F m p μ`, then the measurable function it is almost
everywhere equal to (given by `AEMeasurable.mk`) belongs to `ℒp` for the measure `μ.trim hm`. -/
| Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 241 | 243 |
/-
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.FieldTheory.Finiteness
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.... | exact LinearMap.ker_le_ker_comp _ _
theorem embeddingPiTangent_injective_mfderiv (x : M) (hx : x ∈ s) :
Injective (mfderiv I 𝓘(ℝ, ι → E × ℝ) f.embeddingPiTangent x) :=
LinearMap.ker_eq_bot.1 (f.embeddingPiTangent_ker_mfderiv x hx)
/-- Baby version of the **Whitney weak embedding theorem**: if `M` admits a fi... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 101 | 107 |
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