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/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
| Mathlib/Data/ENNReal/Real.lean | 372 | 374 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Algebra.MvPolynomial.Degrees
/-!
# Variables of polynomials
This file establishes man... |
section Mul
theorem vars_mul [DecidableEq σ] (φ ψ : MvPolynomial σ R) : (φ * ψ).vars ⊆ φ.vars ∪ ψ.vars := by
simp_rw [vars_def, ← Multiset.toFinset_add, Multiset.toFinset_subset]
| Mathlib/Algebra/MvPolynomial/Variables.lean | 108 | 112 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sheaf
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "covering pres... | theorem isSheaf_iff_isLimit_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ D) :
Presheaf.IsSheaf (toGrothendieck _ K) P ↔ ∀ ⦃X : C⦄ (R : Presieve X),
R ∈ K.covering X →
Nonempty (IsLimit (P.mapCone (Sieve.generate R).arrows.cocone.op)) := by
simp only [Presheaf.IsSheaf, Presieve.isSheaf_coverage, isLimit_iff_... | Mathlib/CategoryTheory/Sites/Coverage.lean | 416 | 422 |
/-
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... | (∃ᶠ x in f ⊓ 𝓟 s, p x) ↔ ∃ᶠ x in f, x ∈ s ∧ p x := by
simp only [Filter.Frequently, eventually_inf_principal, not_and]
alias ⟨Frequently.of_inf_principal, Frequently.inf_principal⟩ := frequently_inf_principal
| Mathlib/Order/Filter/Basic.lean | 821 | 824 |
/-
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.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... |
@[simp]
| Mathlib/Data/ZMod/Basic.lean | 595 | 596 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.RingTheory.PowerBas... |
end PowerBasis
| Mathlib/LinearAlgebra/Matrix/Charpoly/Minpoly.lean | 83 | 92 |
/-
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.Topology.Order.IsLUB
/-!
# Monotone functions on an order topology
This file contains lemmas about limits and contin... | exists_lt_of_csInf_lt (h_nonempty.image _) hm
filter_upwards [Ioo_mem_nhdsGT xz] with w hw
exact (Mf ⟨hw.1, hw.2.trans zy⟩ ⟨xz, zy⟩ hw.2.le).trans_lt zm
lemma MonotoneOn.tendsto_nhdsLT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [Topolog... | Mathlib/Topology/Order/Monotone.lean | 282 | 292 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | @[simp]
lemma φ_zero : W.φ 0 = 1 := by
rw [φ, ψ_zero, zero_pow two_ne_zero, mul_zero, zero_sub, zero_add, ψ_one, one_mul, zero_sub,
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 502 | 504 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Shift.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Algebra.Grou... | obj X := X.mapObj p
map φ := mapMap φ p
| Mathlib/CategoryTheory/GradedObject.lean | 432 | 433 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,527 | 2,529 | |
/-
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.Topology.GDelta.Basic
/-!
# Baire spaces
A topological space is called a *Baire space*
if a countable intersection of dense open subsets is den... | rintro ⟨t, ts, ho, hd⟩
exact mem_of_superset (residual_of_dense_Gδ ho hd) ts
/-- A property holds on a residual (comeagre) set if and only if it holds on some dense `Gδ` set. -/
theorem eventually_residual {p : X → Prop} :
(∀ᶠ x in residual X, p x) ↔ ∃ t : Set X, IsGδ t ∧ Dense t ∧ ∀ x ∈ t, p x := by
simp on... | Mathlib/Topology/Baire/Lemmas.lean | 74 | 81 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,866 | 2,868 | |
/-
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.Group.Multiset.Basic
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.OrderedMonoid
i... | theorem prod_zero : (0 : PrimeMultiset).prod = 1 := by
exact Multiset.prod_zero
| Mathlib/Data/PNat/Factors.lean | 184 | 186 |
/-
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.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Data.Nat.Cast.WithTop
/-!
# `WithBot ℕ`
Lemmas about the type... | cases m
· simp [WithBot.add_bot]
simp [← WithBot.coe_add, add_eq_zero_iff_of_nonneg]
theorem add_eq_one_iff {n m : WithBot ℕ} : n + m = 1 ↔ n = 0 ∧ m = 1 ∨ n = 1 ∧ m = 0 := by
cases n
| Mathlib/Data/Nat/WithBot.lean | 28 | 33 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Aesop
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Disjointness and complements
This file defines `Disjoint`, `Codisjoint`, and the `IsCompl` predicate.
... | /-- The sublattice of complemented elements. -/
abbrev Complementeds (α : Type*) [Lattice α] [BoundedOrder α] : Type _ :=
{a : α // IsComplemented a}
| Mathlib/Order/Disjoint.lean | 619 | 621 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.SheafOfTypes
import Mathlib.Order.Closure
/-!
# Closed sieves
A natural closure operator on sieves is a closure operator on `Sieve X... | -/
@[simps]
def Functor.closedSieves : Cᵒᵖ ⥤ Type max v u where
obj X := { S : Sieve X.unop // J₁.IsClosed S }
map f S := ⟨S.1.pullback f.unop, J₁.isClosed_pullback f.unop _ S.2⟩
/-- The presheaf of `J`-closed sieves is a `J`-sheaf.
The proof of this is adapted from [MM92], Chapter III, Section 7, Lemma 1.
-/
theo... | Mathlib/CategoryTheory/Sites/Closed.lean | 149 | 159 |
/-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.AdicCompletion.Functoriality
import Mathlib.RingTheory.Filtration
/-!
# Exac... | /-- `AdicCompletion` over a Noetherian ring is exact on finitely generated modules. -/
theorem map_exact : Function.Exact (map I f) (map I g) := by
refine LinearMap.exact_of_comp_eq_zero_of_ker_le_range ?_ (fun y ↦ ?_)
· rw [map_comp, hfg.linearMap_comp_eq_zero, AdicCompletion.map_zero]
· apply AdicCompletion.ind... | Mathlib/RingTheory/AdicCompletion/Exactness.lean | 181 | 200 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | structure IsMaximumClique [Fintype α] (G : SimpleGraph α) (s : Finset α) : Prop where
(isClique : G.IsClique s)
(maximum : ∀ t : Finset α, G.IsClique t → #t ≤ #s)
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 641 | 644 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Group.Subgroup.Map
import Mathlib.Algebra.Module.Submodule.... | def submoduleOfEquivOfLe {p q : Submodule R M} (h : p ≤ q) : p.submoduleOf q ≃ₗ[R] p where
toFun m := ⟨m.1, m.2⟩
| Mathlib/Algebra/Module/Submodule/Map.lean | 268 | 269 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl, Damiano Testa,
Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Data.Ordering.Basic
imp... | theorem one_le_of_le_mul_left [MulRightReflectLE α] {a b : α}
(h : b ≤ a * b) :
1 ≤ a :=
le_of_mul_le_mul_right' (a := b) <| by simpa only [one_mul]
| Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean | 442 | 445 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Module.Basic
/-!
# Group and ring filter bases
A `GroupFilterBasis` is a `FilterBasis` on a group with some properties relating
the ... | Mathlib/Topology/Algebra/FilterBasis.lean | 393 | 413 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Field.Subfield.Defs
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.Normed.Ring.Basic
/-!
# Norm... | Mathlib/Analysis/Normed/Field/Basic.lean | 996 | 997 | |
/-
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... | end omegaLimit
/-!
### ω-limits of flows by a monoid
-/
namespace Flow
variable {τ : Type*} [TopologicalSpace τ] [AddMonoid τ] [ContinuousAdd τ] {α : Type*}
[TopologicalSpace α] (f : Filter τ) (ϕ : Flow τ α) (s : Set α)
open omegaLimit
theorem isInvariant_omegaLimit (hf : ∀ t, Tendsto (t + ·) f f) : IsInvariant ϕ... | Mathlib/Dynamics/OmegaLimit.lean | 296 | 314 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | fun h => h.1.union h.2⟩
| Mathlib/Data/Set/Function.lean | 190 | 191 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | intro x hx
simp only [Perm.mul_apply, Equiv.trans_apply, Equiv.sumCongr_apply]
obtain ⟨n, rfl⟩ := hσ.exists_pow_eq (Classical.choose_spec hσ).1 (mem_support.1 hx)
simp [← Perm.mul_apply, ← pow_succ']
theorem IsCycle.isConj_iff (hσ : IsCycle σ) (hτ : IsCycle τ) :
IsConj σ τ ↔ #σ.support = #τ.support where
... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 648 | 662 |
/-
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.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... |
variable (S M)
| Mathlib/RingTheory/Localization/Basic.lean | 442 | 443 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | @[simp]
| Mathlib/Data/Set/Prod.lean | 104 | 104 |
/-
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... |
open Metric in
theorem IsCompact.exists_thickening_image_subset
[PseudoEMetricSpace α] {β : Type*} [PseudoEMetricSpace β]
{f : α → β} {K : Set α} {U : Set β} (hK : IsCompact K) (ho : IsOpen U)
(hf : ∀ x ∈ K, ContinuousAt f x) (hKU : MapsTo f K U) :
∃ ε > 0, ∃ V ∈ 𝓝ˢ K, thickening ε (f '' V) ⊆ U := by
... | Mathlib/Topology/MetricSpace/Thickening.lean | 641 | 648 |
/-
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 abscissa of (absolute) convergence of the arithmetic function `ζ` is `1`. -/
lemma abscissaOfAbsConv_zeta : abscissaOfAbsConv ↗ζ = 1 := by
rw [abscissaOfAbsConv_congr (g := 1) fun hn ↦ by simp [hn], abscissaOfAbsConv_one]
/-- The L-series of the arithmetic function `ζ` equals the Riemann Zeta Function on its... | Mathlib/NumberTheory/LSeries/Dirichlet.lean | 260 | 268 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | lemma mul_pow_sub_one (hn : n ≠ 0) (a : M) : a * a ^ (n - 1) = a ^ n := by
rw [← pow_succ', Nat.sub_add_cancel <| Nat.one_le_iff_ne_zero.2 hn]
| Mathlib/Algebra/Group/Basic.lean | 153 | 155 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | Mathlib/Data/Int/Bitwise.lean | 541 | 542 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | (fun s : typein ((· < ·) : o.toType → o.toType → Prop) ⁻¹' Set.range f =>
Classical.choose s.prop)
fun s t hst => by
let H := congr_arg f hst
rwa [Classical.choose_spec s.prop, Classical.choose_spec t.prop, typein_inj,
Subtype.coe_inj] at H)
have := ty... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 180 | 213 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | exact congr rfl (funext fun i => ih i ((by simp [Function.comp_apply, hv'])))
/-- A special case of `realize_restrictVar`, included because we can add the `simp` attribute
to it -/
@[simp]
| Mathlib/ModelTheory/Semantics.lean | 130 | 134 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... |
theorem prodMap_image_prod (f : α → β) (g : γ → δ) (s : Set α) (t : Set γ) :
(Prod.map f g) '' (s ×ˢ t) = (f '' s) ×ˢ (g '' t) := by
ext
aesop
theorem insert_prod : insert a s ×ˢ t = Prod.mk a '' t ∪ s ×ˢ t := by
simp only [insert_eq, union_prod, singleton_prod]
theorem prod_insert : s ×ˢ insert b t = (fun... | Mathlib/Data/Set/Prod.lean | 153 | 162 |
/-
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.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | (fun x ↦ let y := n / x; if x * y = n then some (x, y) else none)
variable {n}
@[simp]
| Mathlib/NumberTheory/Divisors.lean | 68 | 72 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,534 | 1,534 | |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
/-!
In this file, we prove that sheafification is compatible w... | slice_lhs 2 3 => rw [plusCompIso_whiskerRight]
rw [Category.assoc, ← J.plusMap_comp, whiskerRight_toPlus_comp_plusCompIso_hom, ←
Category.assoc, whiskerRight_toPlus_comp_plusCompIso_hom]
rfl
| Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean | 110 | 114 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Module.Defs
/-!
# Ordered algebras
This file proves properties of algebras where both rings are ordered... | @[mono] lemma algebraMap_strictMono : StrictMono (algebraMap α β) :=
fun a₁ a₂ ha ↦ by
simpa only [Algebra.algebraMap_eq_smul_one] using smul_lt_smul_of_pos_right ha zero_lt_one
| Mathlib/Algebra/Order/Module/Algebra.lean | 53 | 55 |
/-
Copyright (c) 2020 Jean Lo, Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yury Kudryashov
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Ring.Action.Pointwise.Set
import Mathlib.Topology.Bornology.Basic... | lemma _root_.absorbs_iUnion {ι : Sort*} [Finite ι] {t : ι → Set α} :
Absorbs M s (⋃ i, t i) ↔ ∀ i, Absorbs M s (t i) :=
(finite_range t).absorbs_sUnion.trans forall_mem_range
| Mathlib/Topology/Bornology/Absorbs.lean | 97 | 99 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.IsManifold.ExtChartAt
import Mathlib.Geometry.Manifold.LocalInvariantProperties
/-!
# `C^n` functions betwee... | theorem ContMDiff.contMDiffAt (h : ContMDiff I I' n f) : ContMDiffAt I I' n f x :=
h x
@[deprecated (since := "2024-11-20")] alias Smooth.smoothAt := ContMDiff.contMDiffAt
| Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 239 | 242 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Tactic.AdaptationNote
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Tenso... | /--
Tensor product of `M` over a singleton set is equivalent to `M`
-/
@[simps symm_apply]
def subsingletonEquiv [Subsingleton ι] (i₀ : ι) : (⨂[R] _ : ι, M) ≃ₗ[R] M where
toFun := lift (MultilinearMap.ofSubsingleton R M M i₀ .id)
| Mathlib/LinearAlgebra/PiTensorProduct.lean | 821 | 826 |
/-
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... | (by
rw [← inf_eq_right] at hs
rwa [sup_inf_right, inf_sup_right, sup_comm x, inf_sup_self, inf_comm, sup_comm z,
| Mathlib/Order/BooleanAlgebra.lean | 218 | 220 |
/-
Copyright (c) 2020 Johan Commelin. 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.Truncated
import Mathlib.RingTheory.WittVector.Identities
import Mathlib.NumberTheory.Padics.RingHoms
/-!
# Co... | theorem charP_zmod : CharP (TruncatedWittVector p n (ZMod p)) (p ^ n) :=
charP_of_prime_pow_injective _ _ _ (card_zmod _ _) (eq_of_le_of_cast_pow_eq_zero p n (ZMod p))
| Mathlib/RingTheory/WittVector/Compare.lean | 60 | 61 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 227 | 232 | |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.ZLattice.Basic
import Mathlib.Analysis.InnerProductSpace.ProdL2
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheo... | (fun x ↦ mem_rat_span_latticeBasis K x)
suffices ((latticeBasis K).restrictScalars ℚ).repr.toLinearMap ∘ₗ f =
(integralBasis K).repr.toLinearMap from DFunLike.congr_fun (LinearMap.congr_fun this x) i
refine Basis.ext (integralBasis K) (fun i ↦ ?_)
have : f (integralBasis K i) = ((latticeBasis K).restrictS... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 163 | 177 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Ri... | theorem U_eval_one (n : ℤ) : (U R n).eval 1 = n + 1 := by
induction n using Polynomial.Chebyshev.induct with
| zero => simp
| one => simp; norm_num
| add_two n ih1 ih2 =>
| Mathlib/RingTheory/Polynomial/Chebyshev.lean | 226 | 230 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | left
trans Nat.card ({1} : Set (K ≃ₐ[k] K))
· congr with x
simp only [SetLike.mem_coe, mem_stabilizer_mk_iff, Set.mem_singleton_iff, or_iff_left_iff_imp,
h x, IsEmpty.forall_iff]
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 915 | 919 |
/-
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.Small.Module
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.TensorProduc... | rw [Submodule.fg_top]; exact Module.FinitePresentation.fg_ker _ (hl.comp H)
letI : AddCommGroup (LinearMap.ker (l ∘ₗ π)) := inferInstance
let f : LinearMap.ker (l ∘ₗ π) →ₗ[R] LinearMap.ker l := LinearMap.restrict π (fun x ↦ id)
have e : π ∘ₗ Submodule.subtype _ = Submodule.subtype _ ∘ₗ f := by ext; rfl
have... | Mathlib/Algebra/Module/FinitePresentation.lean | 226 | 273 |
/-
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.Units.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Rat.Defs
i... | simp only [intCast_eq_divInt, natCast_eq_divInt, num_divInt_den] at this ⊢; assumption
| Mathlib/Algebra/Ring/Rat.lean | 91 | 91 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Johan Commelin, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Constructions.Pullbacks
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Abelian/Basic.lean | 798 | 802 | |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
/-!
# Birkhoff average
In this file we define `birkhoffAverage f g n x` to be
$$
\fr... | theorem birkhoffAverage_one (f : α → α) (g : α → M) (x : α) :
birkhoffAverage R f g 1 x = g x := by simp [birkhoffAverage]
| Mathlib/Dynamics/BirkhoffSum/Average.lean | 50 | 51 |
/-
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.Real
/-!
# Power function... | theorem rpow_le_one {x : ℝ≥0} {z : ℝ} (hx2 : x ≤ 1) (hz : 0 ≤ z) : x ^ z ≤ 1 :=
Real.rpow_le_one x.2 hx2 hz
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 340 | 341 |
/-
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.Asymptotics
/-!
# Continu... | ContinuousAt (· ^ b) a :=
Tendsto.comp (@continuousAt_cpow (a, b) ha) (continuousAt_id.prodMk continuousAt_const)
theorem Filter.Tendsto.cpow {l : Filter α} {f g : α → ℂ} {a b : ℂ} (hf : Tendsto f l (𝓝 a))
(hg : Tendsto g l (𝓝 b)) (ha : a ∈ slitPlane) :
Tendsto (fun x => f x ^ g x) l (𝓝 (a ^ b)) :=
... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 84 | 91 |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Basic Translation Lemmas Between Functions Defined for Continued... |
theorem exists_s_b_of_partDen {b : α}
| Mathlib/Algebra/ContinuedFractions/Translations.lean | 58 | 59 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... |
theorem HasFDerivAtFilter.isBigO_sub_rev (hf : HasFDerivAtFilter f f' x L) {C}
(hf' : AntilipschitzWith C f') : (fun x' => x' - x) =O[L] fun x' => f x' - f x :=
have : (fun x' => x' - x) =O[L] fun x' => f' (x' - x) :=
isBigO_iff.2 ⟨C, Eventually.of_forall fun _ => ZeroHomClass.bound_of_antilipschitz f' hf' _... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 780 | 788 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
import Mathlib.CategoryTheory.Limits.Shape... | -- TODO this is another good candidate for a future `UniqueUpToCanonicalIso`.
/-- Two factorisations through monomorphisms satisfying the universal property
must factor through isomorphic objects. -/
| Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 187 | 189 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | ⟨fun h => e.symm_image_image s ▸ e.symm.uniqueDiffOn_image h, e.uniqueDiffOn_image⟩
@[simp]
theorem ContinuousLinearEquiv.uniqueDiffOn_preimage_iff (e : F ≃L[𝕜] E) :
UniqueDiffOn 𝕜 (e ⁻¹' s) ↔ UniqueDiffOn 𝕜 s := by
| Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 495 | 499 |
/-
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.FiberedCategory.HomLift
import Mathlib.CategoryTheory.Bicategory.Strict
import Mathlib.CategoryTheory.Functor.Category
import Ma... | /-- For a based functor `F : 𝒳 ⟶ 𝒴`, and an arrow `φ` in `𝒳`, then `φ` lifts an arrow `f` in `𝒮`
if `F(φ)` does. -/
lemma isHomLift_map [IsHomLift 𝒴.p f (F.map φ)] : IsHomLift 𝒳.p f φ := by
apply of_fac 𝒳.p f φ (F.w_obj a ▸ domain_eq 𝒴.p f (F.map φ))
(F.w_obj b ▸ codomain_eq 𝒴.p f (F.map φ))
simp [con... | Mathlib/CategoryTheory/FiberedCategory/BasedCategory.lean | 117 | 122 |
/-
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.NNReal
/-!
# Limits and a... |
theorem IsTheta.rpow (hr : 0 ≤ r) (hf : 0 ≤ᶠ[l] f) (hg : 0 ≤ᶠ[l] g) (h : f =Θ[l] g) :
(fun x => f x ^ r) =Θ[l] fun x => g x ^ r :=
⟨h.1.rpow hr hg, h.2.rpow hr hf⟩
theorem IsLittleO.rpow (hr : 0 < r) (hg : 0 ≤ᶠ[l] g) (h : f =o[l] g) :
(fun x => f x ^ r) =o[l] fun x => g x ^ r := by
refine .of_isBigOWith f... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 237 | 246 |
/-
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.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.Piecewise
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Semicontinuous maps
A ... | -/
/-! #### Basic dot notation interface for upper semicontinuity -/
| Mathlib/Topology/Semicontinuous.lean | 666 | 671 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 1,954 | 1,956 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Module.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
/-!
# Midpoint of a segment
## Main definitions
* `midpoint R x y`: midp... |
theorem midpoint_vadd_midpoint (v v' : V) (p p' : P) :
midpoint R v v' +ᵥ midpoint R p p' = midpoint R (v +ᵥ p) (v' +ᵥ p') :=
| Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 83 | 85 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Affine
import Mathlib.LinearAlgebra.FreeModule.Norm
import Mathlib.RingTheory.ClassGroup
import Mat... | ClassGroup.mk (XYIdeal' <| (nonsingular_neg ..).mpr h) * ClassGroup.mk (XYIdeal' h) = 1 := by
rw [← map_mul]
exact (ClassGroup.mk_eq_one_of_coe_ideal <| (FractionalIdeal.coeIdeal_mul ..).symm.trans <|
FractionalIdeal.coeIdeal_inj.mpr <| XYIdeal_neg_mul h).mpr ⟨_, XClass_ne_zero W _, rfl⟩
| Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean | 389 | 393 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... | nonsing_inv_mul_cancel_right A B (isUnit_det_of_invertible A)
@[simp]
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 257 | 259 |
/-
Copyright (c) 2021 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Eugster, Eric Wieser
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.FreeAlgebra
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.SimpleRing.B... | refine (CharP.cast_eq_zero_iff A q p).mp ?_
rw [← map_natCast f p, CharP.cast_eq_zero, map_zero]
/-- Given `R →+* A`, where `R` is a domain with `char R > 0`, then `char A = char R`. -/
| Mathlib/Algebra/CharP/Algebra.lean | 34 | 37 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
/-! # Subtypes of cond... | theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf]
| Mathlib/Order/CompleteLatticeIntervals.lean | 97 | 99 |
/-
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 | 1,671 | 1,672 | |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.Complex.Convex
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.Calculus.Deriv.Shift
/-!
# Estimates for the complex ... |
/-- The difference `log (1+z) - z` is bounded by `‖z‖^2/(2*(1-‖z‖))` when `‖z‖ < 1`. -/
lemma norm_log_one_add_sub_self_le {z : ℂ} (hz : ‖z‖ < 1) :
‖log (1 + z) - z‖ ≤ ‖z‖ ^ 2 * (1 - ‖z‖)⁻¹ / 2 := by
convert norm_log_sub_logTaylor_le 1 hz using 2
· simp [logTaylor_succ, logTaylor_zero, sub_eq_add_neg]
| Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | 174 | 179 |
/-
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 ... | include hc
| Mathlib/Algebra/Homology/HomotopyCofiber.lean | 477 | 478 |
/-
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.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
/-!
# Conve... | apply Summable.const_div
simpa [hq, Nat.cast_add, one_div, norm_inv, norm_pow, RCLike.norm_natCast,
Real.summable_nat_pow_inv, iff_true]
using summable_nat_add_iff (f := fun x => ‖1 / (x ^ q : α)‖) k
end shifted
| Mathlib/Analysis/PSeries.lean | 468 | 491 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Algebra.Group.Idempotent
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.PUnit
import Mathlib.Order.Hom.BoundedLattic... | map_top' := f.map_one'
map_bot' := f.map_zero'
| Mathlib/Algebra/Ring/BooleanRing.lean | 311 | 313 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Basics on... | Mathlib/ModelTheory/Basic.lean | 994 | 996 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.Finite.Sum
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Finiteness.Ideal
import Mathlib.RingTheory.Ideal.Quotient.Operations
imp... |
/-- An algebra is finitely presented if and only if it is a quotient of a polynomial ring whose
variables are indexed by a fintype by a finitely generated ideal. -/
theorem iff_quotient_mvPolynomial' :
FinitePresentation R A ↔
∃ (ι : Type*) (_ : Fintype ι) (f : MvPolynomial ι R →ₐ[R] A),
Surjective f... | Mathlib/RingTheory/FinitePresentation.lean | 148 | 156 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | exact nfp_zero_left 0
apply le_antisymm
· apply nfp_le_fp (isNormal_mul_right ha).monotone hba
| Mathlib/SetTheory/Ordinal/FixedPoint.lean | 435 | 437 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | Mathlib/NumberTheory/ArithmeticFunction.lean | 1,337 | 1,345 | |
/-
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.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
/-!
# Hausdorff dimension
The Hausdorff dimension of a set `X` in ... | exact iSup₂_mono fun x hx => ((hC x (hus hx)).mono inter_subset_right).dimH_image_le hr
/-- If `f : X → Y` is Hölder continuous in a neighborhood of every point `x : X` with the same
positive exponent `r` but possibly different coefficients, then the Hausdorff dimension of the range
of `f` is at most the Hausdorff d... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 300 | 304 |
/-
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, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... | M⁻¹.toBlock (fun i => b i < k) fun i => b i < k :=
inv_eq_left_inv <| hM.toBlock_inverse_mul_toBlock_eq_one k
/-- An upper-left subblock of an invertible block-triangular matrix is invertible. -/
def BlockTriangular.invertibleToBlock [LinearOrder α] [Invertible M] (hM : BlockTriangular M b)
(k : α) : Inver... | Mathlib/LinearAlgebra/Matrix/Block.lean | 326 | 343 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | theorem pred_lt_iff : pred a < b ↔ a ≤ b :=
pred_lt_iff_of_not_isMin <| not_isMin a
| Mathlib/Order/SuccPred/Basic.lean | 844 | 846 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Uthaiwat, Oliver Nash
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.RingTheory.I... | protected lemma isNilpotent_iff :
IsNilpotent P ↔ ∀ i, IsNilpotent (coeff P i) := by
refine
⟨P.recOnHorner (by simp) (fun p r hp₀ _ hp hpr i ↦ ?_) (fun p _ hnp hpX i ↦ ?_), fun h ↦ ?_⟩
· rw [← sum_monomial_eq P]
exact isNilpotent_sum (fun i _ ↦ by simpa only [isNilpotent_monomial_iff] using h i)
· hav... | Mathlib/RingTheory/Polynomial/Nilpotent.lean | 71 | 89 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Fourie... | convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this x).tsum_eq.symm using 1
· simpa only [F, coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add]
using (hasSum_apply (summable_of... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 107 | 121 |
/-
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
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
/-!
# Projection of a line onto a closed interval
Given a linearly... | theorem IccExtend_range (f : Icc a b → β) : range (IccExtend h f) = range f := by
simp only [IccExtend, range_comp f, range_projIcc, image_univ]
| Mathlib/Order/Interval/Set/ProjIcc.lean | 179 | 180 |
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Tactic.IntervalCases
/-!
# Cubics and discriminants
This file defines cubic polynomials ... |
theorem map_roots [IsDomain S] : (map φ P).roots = (Polynomial.map φ P.toPoly).roots := by
| Mathlib/Algebra/CubicDiscriminant.lean | 400 | 401 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.CategoryTheory.Action
import Mathlib.Combinatorics.Quiver.Arborescence
import Mathlib.Combinatorics.Quiver.ConnectedComponent
import Mathlib.GroupTheory.FreeGr... | def loopOfHom {a b : G} (p : a ⟶ b) : End (root' T) :=
treeHom T a ≫ p ≫ inv (treeHom T b)
| Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean | 178 | 179 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... | theorem lt_opow_iff_log_lt' {b x c : Ordinal} (hb : 1 < b) (hc : c ≠ 0) : x < b ^ c ↔ log b x < c :=
lt_iff_lt_of_le_iff_le (opow_le_iff_le_log' hb hc)
theorem lt_opow_of_log_lt {b x c : Ordinal} (hb : 1 < b) : log b x < c → x < b ^ c :=
| Mathlib/SetTheory/Ordinal/Exponential.lean | 379 | 382 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingT... |
theorem dvd_map_of_isScalarTower (A K : Type*) {R : Type*} [CommRing A] [Field K] [Ring R]
| Mathlib/FieldTheory/Minpoly/Field.lean | 80 | 81 |
/-
Copyright (c) 2014 Microsoft Corporation. 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.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | obtain - | m := m <;> obtain - | n := n <;>
| Mathlib/Data/Num/Lemmas.lean | 771 | 771 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | Mathlib/Algebra/Polynomial/Basic.lean | 414 | 414 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | theorem infEdist_smul {M} [SMul M α] [IsIsometricSMul M α] (c : M) (x : α) (s : Set α) :
infEdist (c • x) (c • s) = infEdist x s :=
infEdist_image (isometry_smul _ _)
theorem _root_.IsOpen.exists_iUnion_isClosed {U : Set α} (hU : IsOpen U) :
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 196 | 200 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | theorem rotate'_mod (l : List α) (n : ℕ) : l.rotate' (n % l.length) = l.rotate' n :=
calc
| Mathlib/Data/List/Rotate.lean | 88 | 89 |
/-
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... | haveI : HasLimit (E ⋙ F ⋙ G) := h
ext; erw [assoc, limit.post_π, ← G.map_comp, limit.pre_π, assoc, limit.pre_π, limit.post_π]
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 431 | 433 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Ordinals as games
We define the canonical map `Ordinal... | ⟨by contrapose; rw [not_lt]; exact fun h => not_lt_of_le (toPGame_le h), toPGame_lt⟩
| Mathlib/SetTheory/Game/Ordinal.lean | 130 | 131 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,384 | 1,385 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... | @[simp]
theorem isGLB_upperBounds : IsGLB (upperBounds s) a ↔ IsLUB s a :=
| Mathlib/Order/Bounds/Basic.lean | 820 | 821 |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Data.Fintype.Parity
import Mathlib.LinearAl... | (((algebraMap R ℝ (g 0 0) : ℂ) * z + (algebraMap R ℝ (g 0 1) : ℂ)) /
((algebraMap R ℝ (g 1 0) : ℂ) * z + (algebraMap R ℝ (g 1 1) : ℂ)))
(g • z).property :=
rfl
variable (g : GL(2, ℝ)⁺) (z : ℍ)
@[simp]
theorem coe_smul : ↑(g • z) = num g z / denom g z :=
rfl
| Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean | 275 | 284 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... |
theorem Filter.HasBasis.equicontinuousWithinAt_iff_right {p : κ → Prop}
{s : κ → Set (α × α)} {F : ι → X → α} {S : Set X} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
EquicontinuousWithinAt F S x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝[S] x₀, ∀ i, (F i x₀, F i x) ∈ s k := by
rw [equicontinuousWithinAt_iff_continuousWithinAt, Cont... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 625 | 631 |
/-
Copyright (c) 2024 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn,
Mario Carneiro
-/
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Util... | · exact ih (Nat.le_of_succ_le_succ hn)
| Mathlib/Data/List/GetD.lean | 46 | 46 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Peter Nelson
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determina... |
We derive the `det_vandermonde` formula from `det_projVandermonde`,
which is proved using an induction argument involving row operations and division.
To circumvent issues with non-invertible elements while still maintaining the generality of rings,
we first prove it for fields using the private lemma `det_projVanderm... | Mathlib/LinearAlgebra/Vandermonde.lean | 49 | 56 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import... | f.measure (Ioi x) = ofReal (l - f x) := by
rw [← Ici_diff_left, measure_diff _ (nullMeasurableSet_singleton x), measure_singleton,
f.measure_Ici hf, ← ofReal_sub _ (sub_nonneg.mpr <| Monotone.leftLim_le f.mono' le_rfl)]
<;> simp
lemma measure_Ioi_of_tendsto_atTop_atTop (hf : Tendsto f atTop atTop) (x : ℝ... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 455 | 462 |
/-
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.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | instance nhdsLT_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) := nhdsWithin_Iio_neBot (le_refl a)
@[deprecated nhdsLT_neBot (since := "2024-12-22")]
| Mathlib/Topology/Order/DenselyOrdered.lean | 206 | 208 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
import Mathlib.Data.Set.SymmDiff
/-!
# Indicator function
- `Set.indicator (s : Set α) (f ... | simp only [mulIndicator]
split_ifs <;> simp [*]
@[to_additive]
theorem comp_mulIndicator_const (c : M) (f : M → N) (hf : f 1 = 1) :
| Mathlib/Algebra/Group/Indicator.lean | 223 | 227 |
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