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 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... |
theorem iterate_derivative_at_0_eq_zero_of_lt (n : ℕ) {ν k : ℕ} :
k < ν → (Polynomial.derivative^[k] (bernsteinPolynomial R n ν)).eval 0 = 0 := by
rcases ν with - | ν
· rintro ⟨⟩
| Mathlib/RingTheory/Polynomial/Bernstein.lean | 134 | 138 |
/-
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.Algebra.BigOperators.Finprod
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.Sh... | ∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, ∑ i ∈ I, ρ i x = 1 ∧ support (ρ · x) ⊆ I := by
rcases ρ.exists_finset_nhd' x₀ with ⟨I, H⟩
use I
rwa [nhdsWithin_univ, ← eventually_and] at H
theorem exists_finset_nhd_support_subset {U : ι → Set X} (hso : f.IsSubordinate U)
(ho : ∀ i, IsOpen (U i)) (x : X) :
| Mathlib/Topology/PartitionOfUnity.lean | 289 | 295 |
/-
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... | lemma kstar_def (l : Language α) : l∗ = {x | ∃ L : List (List α), x = L.flatten ∧ ∀ y ∈ L, y ∈ l} :=
| Mathlib/Computability/Language.lean | 104 | 104 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Anne Baanen
-/
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Tactic.Conv
import Mathlib.Util.Qq
/-!
# `ring_nf` tactic
A tactic which ... | (Rat.rawCast (.ofNat n) d : R) = Nat.rawCast n / Nat.rawCast d := by simp
theorem rat_rawCast_neg {R} [DivisionRing R] :
| Mathlib/Tactic/Ring/RingNF.lean | 124 | 125 |
/-
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 | 2,108 | 2,111 | |
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
/-!
# ω-limits
For a function `ϕ : τ → α → β` where `β` is a topological space, we
define the ω-limit under `ϕ` of... |
theorem omegaLimit_eq_iInter_inter {v : Set τ} (hv : v ∈ f) :
ω f ϕ s = ⋂ u : ↥f.sets, closure (image2 ϕ (u ∩ v) s) := by
| Mathlib/Dynamics/OmegaLimit.lean | 183 | 185 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Tactic.FinCases
import Mathlib.Topology.Connected.LocallyConnected
import Mathlib.Topology.Sets.Closeds
/-!
# L... | Mathlib/Topology/LocallyConstant/Basic.lean | 628 | 636 | |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... |
theorem nhdsWithin_eq_map_subtype_coe {s : Set α} {a : α} (h : a ∈ s) :
𝓝[s] a = map ((↑) : s → α) (𝓝 ⟨a, h⟩) :=
| Mathlib/Topology/ContinuousOn.lean | 486 | 488 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.Average
/-!
# Integral average over an interval
In this file we... |
/-- Interval averages are invariant when functions change along discrete sets. -/
theorem intervalAverage_congr_codiscreteWithin {a b : ℝ} {f₁ f₂ : ℝ → ℝ}
| Mathlib/MeasureTheory/Integral/IntervalAverage.lean | 52 | 54 |
/-
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 | 813 | 821 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... | Lagrange.basis s v i = C (nodalWeight s v i) * (nodal s v / (X - C (v i))) := by
simp_rw [Lagrange.basis, basisDivisor, nodalWeight, prod_mul_distrib, map_prod, ←
nodal_erase_eq_nodal_div hi, nodal]
| Mathlib/LinearAlgebra/Lagrange.lean | 585 | 588 |
/-
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.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.GroupTheory.MonoidLocalization.Basic
impo... | ∃ (Λ : ι → ℝ) (Λ₀ : ℝ), (Λ, Λ₀) ≠ 0 ∧ (∑ i, Λ i • f' i) + Λ₀ • φ' = 0 := by
letI := Classical.decEq ι
replace hextr : IsLocalExtrOn φ {x | (fun i => f i x) = fun i => f i x₀} x₀ := by
simpa only [funext_iff] using hextr
rcases hextr.exists_linear_map_of_hasStrictFDerivAt (hasStrictFDerivAt_pi.2 fun i => h... | Mathlib/Analysis/Calculus/LagrangeMultipliers.lean | 108 | 121 |
/-
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.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
/-!
# Filters used in box-based integrals
First ... | theorem toFilter_mono (I : Box ι) {l₁ l₂ : IntegrationParams} (h : l₁ ≤ l₂) :
l₁.toFilter I ≤ l₂.toFilter I :=
iSup_mono fun _ => toFilterDistortion_mono I h le_rfl
@[gcongr, mono]
theorem toFilteriUnion_mono (I : Box ι) {l₁ l₂ : IntegrationParams} (h : l₁ ≤ l₂)
(π₀ : Prepartition I) : l₁.toFilteriUnion I π₀... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 420 | 430 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.GroupTheory.MonoidLocalization.Away
import Mathlib.Algebra.Algebra.Pi
import Mathlib.RingTheory.I... | · rw [hm]
/-- Localizing the localization of `R` at `x` at the image of `y` is the same as localizing
`R` at `x * y`. See `IsLocalization.Away.mul` for the `y * x` version. -/
lemma mul' (T : Type*) [CommSemiring T] [Algebra S T] [Algebra R T] [IsScalarTower R S T] (x y : R)
[IsLocalization.Away x S] [IsLocali... | Mathlib/RingTheory/Localization/Away/Basic.lean | 246 | 262 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Option.Basic
import Batteries.Tactic.Congr
import Mathlib.Data.Set.Basic
import Mathlib.Tactic.Contrapose
/-!
# Partial Equivalences
In this file, ... | theorem trans_bot (f : α ≃. β) : f.trans (⊥ : β ≃. γ) = ⊥ := by
ext; dsimp [PEquiv.trans]; simp
@[simp]
theorem bot_trans (f : β ≃. γ) : (⊥ : α ≃. β).trans f = ⊥ := by
ext; dsimp [PEquiv.trans]; simp
theorem isSome_symm_get (f : α ≃. β) {a : α} (h : isSome (f a)) :
isSome (f.symm (Option.get _ h)) :=
| Mathlib/Data/PEquiv.lean | 274 | 282 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Kim Morrison, Apurva Nakade, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.SetTheory.PGame.Algebra
import Mathl... | ((Equiv.sumEmpty _ _).trans (Equiv.punitProd _)) ?_ ?_ <;>
· rintro (⟨⟨⟩, _⟩ | ⟨⟨⟩, _⟩)
exact ((((PGame.zero_mul (mk _ _ _ _)).add (PGame.one_mul _)).trans (PGame.zero_add _)).sub
| Mathlib/SetTheory/Game/Basic.lean | 645 | 647 |
/-
Copyright (c) 2018 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Sequences in topological spaces
In t... | This property is equivalent to the definition of `FrechetUrysohnSpace`, see
`FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto`. -/
theorem tendsto_nhds_iff_seq_tendsto [FrechetUrysohnSpace X] {f : X → Y} {a : X} {b : Y} :
Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b) := ... | Mathlib/Topology/Sequences.lean | 113 | 116 |
/-
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.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 964 | 966 | |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | refine (add_le_add_right ?_ _).trans hx
simpa only [mul_one] using mul_le_mul_of_nonpos_left ih hx0
theorem geom_sum_alternating_of_lt_neg_one [Ring R] [PartialOrder R] [IsStrictOrderedRing R]
(hx : x + 1 < 0) (hn : 1 < n) :
if Even n then (∑ i ∈ range n, x ^ i) < 0 else 1 < ∑ i ∈ range n, x ^ i :=... | Mathlib/Algebra/GeomSum.lean | 498 | 517 |
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir
-/
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Construction of the hyperreal numbers as an ultrapro... | Mathlib/Data/Real/Hyperreal.lean | 830 | 833 | |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Constructions
import Mathlib.Order.Filter.ListTraverse
import Mathlib.Tactic.AdaptationNote
import Mathlib.Topology.Algebra.Monoid.Defs
/-!
# To... | theorem Filter.Tendsto.cons {α : Type*} {f : α → β} {g : α → List β} {a : Filter α} {b : β}
{l : List β} (hf : Tendsto f a (𝓝 b)) (hg : Tendsto g a (𝓝 l)) :
| Mathlib/Topology/List.lean | 74 | 75 |
/-
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.ConcreteCategory
import Mathlib.Algebra.Category.ModuleCat.Colimits
/-!
# Homology and exactness of short complexes of modules
In... | Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean | 188 | 192 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | [DecidableEq n] (M : Matrix n n 𝕜) :
∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
(L.map toMatrix).prod * M * (L'.map toMatrix).prod = diagonal D := by
suffices ∀ cn, Fintype.card n = cn →
∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
(L.map toMatrix).prod * M * (L'.map... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 639 | 655 |
/-
Copyright (c) 2021 Manuel Candales. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Manuel Candales, Benjamin Davidson
-/
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Geometry.Euclidean.Sphere.Basic
/-!
# Power of a point (intersecting ch... | obtain ⟨k, hk_ne_one, hk⟩ := h₁
let r := (k - 1)⁻¹ * (k + 1)
have hxy : x = r • y := by
rw [← smul_smul, eq_inv_smul_iff₀ (sub_ne_zero.mpr hk_ne_one), ← sub_eq_zero]
calc
(k - 1) • x - (k + 1) • y = k • x - x - (k • y + y) := by
simp_rw [sub_smul, add_smul, one_smul]
_ = k • x - k • y ... | Mathlib/Geometry/Euclidean/Sphere/Power.lean | 40 | 64 |
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.GroupTheory.SpecificGroups.Dihedral
/-!
# Quaternion Groups
We define the (generalised) quat... | simp only [ZMod.val_natCast, ZMod.val_zero, Nat.zero_div, Nat.mod_mul_left_div_self,
Nat.div_self (NeZero.pos n), reduceCtorEq] at h'
· norm_num
/-- In the special case `n = 1`, `Quaternion 1` is a cyclic group (of order `4`). -/
theorem quaternionGroup_one_isCyclic : IsCyclic (QuaternionGroup 1) := by
| Mathlib/GroupTheory/SpecificGroups/Quaternion.lean | 200 | 205 |
/-
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.Geometry.RingedSpace.OpenImmersion
import Mathlib.AlgebraicGeometry.Scheme
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.Categor... | TopologicalSpace.Opens.ext (Set.range_eq_univ.mpr f.homeomorph.surjective)
lemma opensRange_comp_of_isIso {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
[IsIso f] [IsOpenImmersion g] : (f ≫ g).opensRange = g.opensRange := by
| Mathlib/AlgebraicGeometry/OpenImmersion.lean | 110 | 113 |
/-
Copyright (c) 2023 Paul Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Reichert, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Affine.AddTorsorBases
/-!
# Intrinsic frontier and interior
This file defines the intrinsic frontier, interior and closur... | @[simp]
theorem closure_diff_intrinsicFrontier (s : Set P) :
closure s \ intrinsicFrontier 𝕜 s = intrinsicInterior 𝕜 s :=
intrinsicClosure_eq_closure 𝕜 s ▸ intrinsicClosure_diff_intrinsicFrontier s
end NormedAddTorsor
private theorem aux {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] (φ : α ≃ₜ β)
... | Mathlib/Analysis/Convex/Intrinsic.lean | 281 | 291 |
/-
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.Analysis.Convex.Gauge
import Mathlib.Analysis.Normed.Module.Convex
/-!
# "Gauge rescale" homeomorphism between convex sets
Given two convex von Neum... | /-- Given two convex bounded sets in a topological vector space with nonempty interiors,
there exists a homeomorphism of the ambient space
that sends the interior, the closure, and the frontier of one set
to the interior, the closure, and the frontier of the other set.
In particular, if both `s` and `t` are open set o... | Mathlib/Analysis/Convex/GaugeRescale.lean | 149 | 170 |
/-
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.Logic.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... | instance IsWellOrder.gt [Preorder α] [IsWellOrder α (· > ·)] : IsWellOrder (WithTop α) (· > ·) where
instance _root_.WithBot.trichotomous.lt [Preorder α] [h : IsTrichotomous α (· < ·)] :
| Mathlib/Order/WithBot.lean | 934 | 936 |
/-
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.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsing... | WellFoundedGT α ↔ ∀ a : ℕ →o α, ∃ n, ∀ m, n ≤ m → a n = a m :=
wellFoundedGT_iff_monotone_chain_condition'.trans <| by
congrm ∀ a, ∃ n, ∀ m h, ?_
rw [lt_iff_le_and_ne]
simp [a.mono h]
| Mathlib/Order/OrderIsoNat.lean | 228 | 233 |
/-
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... | /-- Binet's formula as a dependent equality. -/
theorem Real.coe_fib_eq : ∀ n, (Nat.fib n : ℝ) = (φ ^ n - ψ ^ n) / √5 := by
rw [← funext_iff, Real.coe_fib_eq']
| Mathlib/Data/Real/GoldenRatio.lean | 202 | 204 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.Module.Card
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.SetTheory.Cardinal.Continuum
import Mathlib.SetTheory.Cardinal.C... | the same cardinality as the whole space. -/
theorem cardinal_eq_of_isOpen
{E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddGroup E] [MulActionWithZero 𝕜 E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E] {s : Set E}
(hs : IsOpen s) (h's : s.Nonempty) : #s = #E := by
rcases h's with ... | Mathlib/Topology/Algebra/Module/Cardinality.lean | 110 | 115 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.PNat.Defs... | q + r = (q.num * r.den + q.den * r.num : ℤ) /. (↑q.den * ↑r.den : ℤ) := by
have hqd : (q.den : ℤ) ≠ 0 := Int.natCast_ne_zero_iff_pos.2 q.den_pos
have hrd : (r.den : ℤ) ≠ 0 := Int.natCast_ne_zero_iff_pos.2 r.den_pos
| Mathlib/Data/Rat/Lemmas.lean | 104 | 106 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... |
end IsBase
section dep_indep
| Mathlib/Data/Matroid/Basic.lean | 518 | 520 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,479 | 1,481 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | @[simp]
theorem det_conj {N : Type*} [AddCommGroup N] [Module A N] (f : M →ₗ[A] M) (e : M ≃ₗ[A] N) :
LinearMap.det ((e : M →ₗ[A] N) ∘ₗ f ∘ₗ (e.symm : N →ₗ[A] M)) = LinearMap.det f := by
| Mathlib/LinearAlgebra/Determinant.lean | 281 | 283 |
/-
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.Algebra.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | (nonnegg_pos_neg.1 xy) (nonnegg_neg_pos.1 zw)
rw [add_def, neg_add_eq_sub]
rwa [Int.subNatNat_eq_coe, Int.subNatNat_eq_coe] at this
theorem Nonneg.add {a b : ℤ√d} (ha : Nonneg a) (hb : Nonneg b) : Nonneg (a + b) := by
rcases nonneg_cases ha with ⟨x, y, rfl | rfl | rfl⟩ <;>
rcases nonneg_cases hb with ⟨... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 569 | 586 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Reduced
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.FieldTheory.Separable
/-!
# Perfect fie... | (X ^ p - C y).roots = p • {(frobeniusEquiv R p).symm y} := by
have H := roots_X_pow_char_pow_sub_C (p := p) (n := 1) (y := y)
rwa [pow_one, iterateFrobeniusEquiv_one] at H
theorem roots_X_pow_char_sub_C_pow {y : R} {m : ℕ} :
((X ^ p - C y) ^ m).roots = (m * p) • {(frobeniusEquiv R p).symm y} := by
have H... | Mathlib/FieldTheory/Perfect.lean | 300 | 306 |
/-
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.Geometry.Euclidean.Angle.Oriented.RightAngle
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Angles in circles and sphere.
This file proves results ... | theorem cospherical_or_collinear_of_two_zsmul_oangle_eq {p₁ p₂ p₃ p₄ : P}
(h : (2 : ℤ) • ∡ p₁ p₂ p₄ = (2 : ℤ) • ∡ p₁ p₃ p₄) :
Cospherical ({p₁, p₂, p₃, p₄} : Set P) ∨ Collinear ℝ ({p₁, p₂, p₃, p₄} : Set P) := by
by_cases hc : Collinear ℝ ({p₁, p₂, p₄} : Set P)
· by_cases he : p₁ = p₄
· rw [he, Set.inser... | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | 365 | 379 |
/-
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... | rcases f; rfl
section CompMeasurable
| Mathlib/MeasureTheory/Function/AEEqFun.lean | 287 | 289 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Int.Order.Basic
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Compare
impor... | of_not_not fun h ↦
let ⟨_, hb⟩ := not_isMin_iff.1 h
| Mathlib/Order/Monotone/Basic.lean | 270 | 271 |
/-
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.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... |
@[simp]
| Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 333 | 334 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | · intro h
rw [← norm_mul_cos_add_sin_mul_I z, h]
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 254 | 255 |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
/-!
# `init` and `tail`
Given a Witt vecto... |
/-- `WittVector.init n x` is polynomial in the coefficients of `x`. -/
| Mathlib/RingTheory/WittVector/InitTail.lean | 221 | 222 |
/-
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 `... | theorem one_div_ne_zero : 1 / p ≠ 0 := ne_of_gt h.one_div_pos
/-- For `r`, instead of `p` -/
theorem pos' : 0 < r := inv_pos.mp <| h.inv_add_inv_eq_inv ▸ add_pos h.inv_pos h.symm.inv_pos
| Mathlib/Data/Real/ConjExponents.lean | 85 | 88 |
/-
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.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
/-!
# Non integrable functions
In this f... | (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
(hfg : deriv f =O[l] g) : ¬IntervalIntegrable g volume a b := by
rw [intervalIntegrable_iff']
exact not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter _ hl hd hf hfg
/-- If `a ≠ b`, `c ∈ [a, b]`, `f` is differentiab... | Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean | 127 | 132 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | rw [← hg.comap_uniformity, ← map_le_iff_le_comap, Filter.map_map, comp_def, comp_def]
theorem IsUniformInducing.isInducing {f : α → β} (h : IsUniformInducing f) : IsInducing f := by
obtain rfl := h.comap_uniformSpace
| Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 104 | 107 |
/-
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.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... | [F.PreservesRightHomologyOf S] : (S.map F).Exact := by
have := h.hasHomology
exact h.map_of_preservesLeftHomologyOf F
variable (S)
lemma exact_map_iff_of_faithful [S.HasHomology]
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 225 | 231 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.DFinsupp.BigOperators
import Mathlib.Data.DFinsupp.Order
/-!
# Equivalence between `Multiset` and `ℕ`-valued finitely supported functions
This defines... | Multiset.equivDFinsupp.symm.injective
| Mathlib/Data/DFinsupp/Multiset.lean | 115 | 116 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Defs
/-!
# Division of `AddMonoidAlgebra` by monomials
This file is most important for when `G = ℕ` (polynomials) or `G = σ →₀ ℕ` (mu... |
theorem of'_mul_modOf (g : G) (x : k[G]) : of' k G g * x %ᵒᶠ g = 0 := by
ext g'
| Mathlib/Algebra/MonoidAlgebra/Division.lean | 139 | 141 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 826 | 829 | |
/-
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.NonUnitalSubalgebra
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Star.Center
import Mathlib.Algebra.Star.SelfAdjoint
/-... | gc := NonUnitalStarAlgebra.gc
le_l_u S := (NonUnitalStarAlgebra.gc (S : Set A) (adjoin R S)).1 <| le_rfl
| Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 710 | 711 |
/-
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.Algebra.Ring.Pointwise.Set
import Mathlib.Order.Filter.AtTopBot.CompleteLattice
import Mathlib.Order.Filter.AtTopBot.G... | /-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]`
with `a < u`. -/
theorem mem_nhdsGE_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α} {s : Set α} :
s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s :=
nhdsGE_basis_Icc.mem_iff
| Mathlib/Topology/Order/LeftRightNhds.lean | 288 | 293 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | contained in it, the dimensions of `K₁` and the intersection of its
orthogonal subspace with `K₂` add to that of `K₂`. -/
theorem finrank_add_inf_finrank_orthogonal {K₁ K₂ : Submodule 𝕜 E}
[FiniteDimensional 𝕜 K₂] (h : K₁ ≤ K₂) :
finrank 𝕜 K₁ + finrank 𝕜 (K₁ᗮ ⊓ K₂ : Submodule 𝕜 E) = finrank 𝕜 K₂ := by
| Mathlib/Analysis/InnerProductSpace/Projection.lean | 1,071 | 1,075 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Fintype.Lattice
import Mathlib.Data.Fintype.Sum
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.MetricSpace.Antilipschitz
... |
theorem comp_continuousOn_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} {s : Set γ} :
| Mathlib/Topology/MetricSpace/Isometry.lean | 151 | 152 |
/-
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 | 720 | 722 | |
/-
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.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.TensorPower.Basic
/-!
# Tensor algebras as direct sums of tensor powers
In this file we show... | end TensorPower
namespace TensorAlgebra
/-- The canonical map from a direct sum of tensor powers to the tensor algebra. -/
| Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean | 68 | 72 |
/-
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.Geometry.Euclidean.Inversion.Basic
import Mathlib.Geometry.Euclidean.PerpBisector
/-!
# Image of a hyperplane under inversion
In this file we prove... |
theorem preimage_inversion_perpBisector (hR : R ≠ 0) (hy : y ≠ c) :
inversion c R ⁻¹' perpBisector c y = sphere (inversion c R y) (R ^ 2 / dist y c) \ {c} := by
rw [← dist_inversion_center, ← preimage_inversion_perpBisector_inversion hR,
| Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean | 56 | 59 |
/-
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, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | · have hft : IntegrableOn f t μ := by rwa [ht.integrableOn_iff hs hf]
calc
∫ x in s, f x ∂μ = ∑' g : G, ∫ x in s ∩ g • t, f x ∂μ := ht.setIntegral_eq_tsum hfs
_ = ∑' g : G, ∫ x in g • t ∩ s, f (g⁻¹ • x) ∂μ := by simp only [hf, inter_comm]
_ = ∫ x in t, f x ∂μ := (hs.setIntegral_eq_tsum' hft).sym... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 416 | 421 |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation.Hausdorff
import Mathlib.Topology.Connected.Clopen
/-!
# Separated maps and locally injective maps ou... |
variable {s : Set A} {g g₁ g₂ : A → E} (sep : IsSeparatedMap p) (inj : IsLocallyInjective p)
include sep inj
| Mathlib/Topology/SeparatedMap.lean | 196 | 199 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
import Mathlib.Tactic.ComputeDegree
/-!
# Division polynomials of Weierst... |
lemma preΨ_ne_zero [Nontrivial R] {n : ℤ} (h : (n : R) ≠ 0) : W.preΨ n ≠ 0 := by
induction n using Int.negInduction with
| nat n => simpa only [preΨ_ofNat] using W.preΨ'_ne_zero <| by exact_mod_cast h
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | 311 | 314 |
/-
Copyright (c) 2021 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.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | have : Tendsto (fun n x ↦ g x + φ n x) atTop (𝓝 (g + f)) :=
tendsto_pi_nhds.2 (fun x ↦ tendsto_const_nhds.add (hφ x))
apply measurable_of_tendsto_metrizable (fun n ↦ ?_) this
exact hg.add_simpleFunc _
| Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 454 | 457 |
/-
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.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... | NFBelow a.1 (repr b)
/-- The `oadd` pseudo-constructor for `NONote` -/
def oadd (e : NONote) (n : ℕ+) (a : NONote) (h : below a e) : NONote :=
| Mathlib/SetTheory/Ordinal/Notation.lean | 1,208 | 1,211 |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | theorem IsLindelof.image_of_continuousOn {f : X → Y} (hs : IsLindelof s) (hf : ContinuousOn f s) :
IsLindelof (f '' s) := by
intro l lne _ ls
have : NeBot (l.comap f ⊓ 𝓟 s) :=
comap_inf_principal_neBot_of_image_mem lne (le_principal_iff.1 ls)
obtain ⟨x, hxs, hx⟩ : ∃ x ∈ s, ClusterPt x (l.comap f ⊓ 𝓟 s) ... | Mathlib/Topology/Compactness/Lindelof.lean | 98 | 110 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.Separation.Basic
/-!
# Topology on the set of filters on a type... | theorem HasBasis.nhds' {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : HasBasis l p s) :
HasBasis (𝓝 l) p fun i => { l' | s i ∈ l' } := by simpa only [Iic_principal] using h.nhds
protected theorem mem_nhds_iff {l : Filter α} {S : Set (Filter α)} :
| Mathlib/Topology/Filter.lean | 95 | 98 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | using Cochain.congr_v (id φ) p p (add_zero p)
@[reassoc]
lemma inl_v_d (i j k : ℤ) (hij : i + (-1) = j) (hik : k + (-1) = i) :
(inl φ).v i j hij ≫ (mappingCone φ).d j i =
φ.f i ≫ (inr φ).f i - F.d i k ≫ (inl φ).v _ _ hik := by
dsimp [mappingCone, inl, inr]
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 234 | 240 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.Pseudo.Basic
import Mathlib.Topology.MetricSpace.Pseudo.Le... | Mathlib/Topology/MetricSpace/Basic.lean | 347 | 357 | |
/-
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.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
/-!
# One-dimensional iterated derivatives
We define the `n`-th de... | (Hdiff : ∀ m : ℕ, (m : ℕ∞) < n → DifferentiableOn 𝕜 (fun x => iteratedDerivWithin m f s x) s) :
ContDiffOn 𝕜 n f s := by
apply contDiffOn_of_continuousOn_differentiableOn
| Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 119 | 121 |
/-
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.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | protected theorem _root_.AEMeasurable.nullMeasurable {f : α → β} (h : AEMeasurable f μ) :
NullMeasurable f μ :=
let ⟨_g, hgm, hg⟩ := h; hgm.nullMeasurable.congr hg.symm
lemma _root_.AEMeasurable.nullMeasurableSet_preimage {f : α → β} {s : Set β}
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,296 | 1,300 |
/-
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... | simp [expNear, range_succ, mul_add, add_left_comm, add_assoc, pow_succ, div_eq_mul_inv,
mul_inv, Nat.factorial]
| Mathlib/Data/Complex/Exponential.lean | 566 | 567 |
/-
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 | 866 | 867 | |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | (Finset.le_sup hi : p i ≤ s.sup p) x
theorem finset_sup_apply_lt {p : ι → Seminorm 𝕜 E} {s : Finset ι} {x : E} {a : ℝ} (ha : 0 < a)
(h : ∀ i, i ∈ s → p i x < a) : s.sup p x < a := by
lift a to ℝ≥0 using ha.le
rw [finset_sup_apply, NNReal.coe_lt_coe, Finset.sup_lt_iff]
· exact h
| Mathlib/Analysis/Seminorm.lean | 382 | 388 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | /-- AddEquiv between (ι →₀ M) and M, when ι has a unique element -/
noncomputable def _root_.AddEquiv.finsuppUnique {ι : Type*} [Unique ι] :
(ι →₀ M) ≃+ M where
| Mathlib/Data/Finsupp/Defs.lean | 504 | 506 |
/-
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.LinearAlgebra.TensorProduct.RightExactness
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.TensorProduct.MvPolynomial
/-!
... | theorem baseChangeAux_surj {σ : Type*} {f : MvPolynomial σ R →ₐ[R] A} (hf : Function.Surjective f) :
Function.Surjective (Algebra.TensorProduct.map (AlgHom.id B B) f) := by
show Function.Surjective (TensorProduct.map (AlgHom.id R B) f)
apply TensorProduct.map_surjective
· exact Function.RightInverse.surjectiv... | Mathlib/RingTheory/FiniteStability.lean | 31 | 36 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | theorem natDegree_scaleRoots (p : R[X]) (s : R) : natDegree (scaleRoots p s) = natDegree p := by
simp only [natDegree, degree_scaleRoots]
theorem monic_scaleRoots_iff {p : R[X]} (s : R) : Monic (scaleRoots p s) ↔ Monic p := by
simp only [Monic, leadingCoeff, natDegree_scaleRoots, coeff_scaleRoots_natDegree]
theor... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 78 | 86 |
/-
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 [← smul_neg, ne_eq, LieSubmodule.eq_bot_iff, not_forall]
exact ⟨_, toEnd_pow_apply_mem hf hx n, prim.pow_toEnd_f_ne_zero_of_eq_nat rfl hn⟩
lemma rootSpace_neg_nsmul_add_chainTop_of_lt (hα : α.IsNonZero) {n : ℕ} (hn : chainLength α β < n) :
rootSpace H (- (n • α) + chainTop α β) = ⊥ := by
by_contra ... | Mathlib/Algebra/Lie/Weights/RootSystem.lean | 108 | 129 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... | (calc
x ⊓ y ⊓ z ⊔ x ⊓ y \ z = x ⊓ (y ⊓ z) ⊔ x ⊓ y \ z := by rw [inf_assoc]
_ = x ⊓ (y ⊓ z ⊔ y \ z) := by rw [← inf_sup_left]
_ = x ⊓ y := by rw [sup_inf_sdiff])
(calc
x ⊓ y ⊓ z ⊓ (x ⊓ y \ z) = x ⊓ x ⊓ (y ⊓ z ⊓ y \ z) := by ac_rfl
_ = ⊥ := by rw [inf_inf_sdiff, inf_bot_eq])
| Mathlib/Order/BooleanAlgebra.lean | 417 | 423 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | simp only [C_apply, ← single_eq_monomial, (Finsupp.ext isEmptyElim (α := σ) : a = 0),
single_eq_same]
rfl
| Mathlib/Algebra/MvPolynomial/Basic.lean | 217 | 220 |
/-
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.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | subcollections of `s` form a topological basis. -/
theorem isTopologicalBasis_of_subbasis {s : Set (Set α)} (hs : t = generateFrom s) :
IsTopologicalBasis ((fun f => ⋂₀ f) '' { f : Set (Set α) | f.Finite ∧ f ⊆ s }) := by
subst t; letI := generateFrom s
refine ⟨?_, ?_, le_antisymm (le_generateFrom ?_) <| generat... | Mathlib/Topology/Bases.lean | 77 | 90 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... |
theorem strictMonoOn_logb_of_base_lt_one : StrictMonoOn (logb b) (Set.Iio 0) := by
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 324 | 325 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... | simp only [ENNReal.coe_toReal]
refine Eq.subst (motive := (∫⁻ x, · x ∂μ < ⊤)) (funext fun x ↦ ?_) this
rw [← neg_mul, Real.rpow_mul (h_one_add x).le]
exact Real.enorm_rpow_of_nonneg (Real.rpow_nonneg (h_one_add x).le _) NNReal.zero_le_coe
refine hl.hasFiniteIntegral.mono' (ae_of_al... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 689 | 717 |
/-
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.SetTheory.Game.State
/-!
# Domineering as a combinatorial game.
We define the game of Domineering, played on a chessboard of arbitrary shape
(possibly ev... | theorem moveRight_card {b : Board} {m : ℤ × ℤ} (h : m ∈ right b) :
Finset.card (moveRight b m) + 2 = Finset.card b := by
dsimp only [moveRight]
rw [Finset.card_erase_of_mem (fst_pred_mem_erase_of_mem_right h)]
rw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)]
exact tsub_add_cancel_of_le (card_o... | Mathlib/SetTheory/Game/Domineering.lean | 101 | 106 |
/-
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... | /-! ### Weak convergence of finite measures with bounded continuous real-valued functions
In this section we characterize the weak convergence of finite measures by the usual (defining)
condition that the integrals of all bounded continuous real-valued functions converge.
-/
| Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 638 | 644 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
import Mathlib.Comp... | Mathlib/Computability/TMToPartrec.lean | 1,878 | 1,880 | |
/-
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
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... |
theorem pred_card_le_card_erase : #s - 1 ≤ #(s.erase a) := by
| Mathlib/Data/Finset/Card.lean | 156 | 157 |
/-
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.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
/-!
# Set family operations
This file defines a few binary operations on `... | lemma sdiff_mem_diffs : a ∈ s → b ∈ t → a \ b ∈ s \\ t := mem_image₂_of_mem
lemma diffs_subset : s₁ ⊆ s₂ → t₁ ⊆ t₂ → s₁ \\ t₁ ⊆ s₂ \\ t₂ := image₂_subset
| Mathlib/Data/Finset/Sups.lean | 531 | 533 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,576 | 1,577 | |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker, Devon Tuma, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probabili... | · exact (integral_map hX aestronglyMeasurable_id).symm
· rw [map_of_not_aemeasurable hX, integral_zero_measure, integral_non_aestronglyMeasurable]
rwa [aestronglyMeasurable_iff_aemeasurable]
end IsUniform
variable {X : Ω → E}
lemma IsUniform.cond {s : Set E} :
| Mathlib/Probability/Distributions/Uniform.lean | 172 | 180 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.RingTheory.Adjoin.Field
import Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra
/-!
# Splitting fields
This file introduces the notion of a splitting... | theorem IsIntegral.mem_intermediateField_of_minpoly_splits {x : L} (int : IsIntegral K x)
{F : IntermediateField K L} (h : Splits (algebraMap K F) (minpoly K x)) : x ∈ F := by
rw [← F.fieldRange_val]; exact int.mem_range_algebraMap_of_minpoly_splits h
| Mathlib/FieldTheory/SplittingField/IsSplittingField.lean | 157 | 160 |
/-
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.Algebra.Rat
import Mathlib.Data.Nat.Cast.Field
import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Definition of well-known power series
In t... | simp [cos, apply_ite f]
end Field
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 218 | 220 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Module.Convex
import Mathlib.Analysis.Normed.Module.Ray
import Mathlib.Analysis.NormedSpace.Pointwise
/-!
# Strictly conv... | refine
StrictConvexSpace.of_strictConvex_unitClosedBall ℝ
((convex_closedBall _ _).strictConvex' fun x hx y hy hne => ?_)
rw [interior_closedBall (0 : E) one_ne_zero, closedBall_diff_ball,
mem_sphere_zero_iff_norm] at hx hy
rcases h x y hx hy hne with ⟨a, b, hab, hlt⟩
use b
rwa [AffineMap.lineMa... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 95 | 106 |
/-
Copyright (c) 2024 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau, Calle Sönne
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.CommSq
/-!
# HomLift
Given a functor `p : 𝒳 ⥤ 𝒮`, this file provides API for... |
/-- If `φ : a ⟶ b` lifts `𝟙 T` and `ψ : b ⟶ c` lifts `f`, then `φ ≫ ψ` lifts `f` -/
lemma comp_lift_id_left' {a b c : 𝒳} (R : 𝒮) (φ : a ⟶ b) [p.IsHomLift (𝟙 R) φ]
| Mathlib/CategoryTheory/FiberedCategory/HomLift.lean | 144 | 146 |
/-
Copyright (c) 2020 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.Field.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.Co... | · exact one_div_pos.2 hgcd
theorem cancel_factors_eq {α} [Field α] {a b ad bd a' b' gcd : α} (ha : ad * a = a')
(hb : bd * b = b') (had : ad ≠ 0) (hbd : bd ≠ 0) (hgcd : gcd ≠ 0) :
(a = b) = (1 / gcd * (bd * a') = 1 / gcd * (ad * b')) := by
rw [← ha, ← hb, ← mul_assoc bd, ← mul_assoc ad, mul_comm bd]
| Mathlib/Tactic/CancelDenoms/Core.lean | 81 | 86 |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... | theorem exists_leftDescent_of_ne_one {w : W} (hw : w ≠ 1) : ∃ i : B, cs.IsLeftDescent w i := by
rcases cs.exists_reduced_word w with ⟨ω, h, rfl⟩
have h₁ : ω ≠ [] := by rintro rfl; simp at hw
rcases List.exists_cons_of_ne_nil h₁ with ⟨i, ω', rfl⟩
| Mathlib/GroupTheory/Coxeter/Length.lean | 291 | 294 |
/-
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
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,380 | 1,388 | |
/-
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.Order.Hom.Bounded
import Mathlib.Topology.Order.Hom.Basic
/-!
# Esakia morphisms
This file defines pseudo-epimorphisms and Esakia morphisms.
We use the ... | Mathlib/Topology/Order/Hom/Esakia.lean | 360 | 362 | |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Hull
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Topology.Bornology.A... | refine hs.smul_mem ?_ ha
rw [norm_smul, norm_inv, ← div_eq_inv_mul]
exact div_le_one_of_le₀ hba (norm_nonneg _)
(a⁻¹ • b) • a • x = b • x := by rw [smul_comm, smul_assoc, smul_inv_smul₀ ha₀]
theorem Balanced.subset_smul (hs : Balanced 𝕜 s) (ha : 1 ≤ ‖a‖) : s ⊆ a • s := by
rw [← @norm_o... | Mathlib/Analysis/LocallyConvex/Basic.lean | 208 | 217 |
/-
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.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## Pushouts of Monoids and Groups
This file defin... | rw [lift_base]
rfl
| Mathlib/GroupTheory/PushoutI.lean | 471 | 473 |
/-
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.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Order.Filter.ENNReal
import Mathlib.Probability.Unif... | obtain ⟨d, cd, hd⟩ : ∃ d, c < d ∧ d < essSup f μ := exists_between hc
let t := {x | d < f x}
have A : 0 < (μ.restrict t) t := by
simp only [Measure.restrict_apply_self]
rw [essSup_eq_sInf] at hd
have : d ∉ {a | μ {x | a < f x} = 0} := not_mem_of_lt_csInf hd (OrderBot.bddBelow _)
exact bot_lt_iff_n... | Mathlib/MeasureTheory/Function/EssSup.lean | 307 | 329 |
/-
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.Data.Int.Interval
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology... |
@[simp]
| Mathlib/Topology/Instances/Int.lean | 63 | 64 |
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