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/-
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.Order.SuccPred.Archimedean
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Successor and predecessor limits
We define the pre... | Mathlib/Order/SuccPred/Limit.lean | 111 | 111 | |
/-
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.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... | · exact hs
| Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 164 | 164 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | theorem one_div_le_one_div_of_le (ha : 0 < a) (h : a ≤ b) : 1 / b ≤ 1 / a := by
| Mathlib/Algebra/Order/Field/Basic.lean | 110 | 110 |
/-
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.Data.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... | variable [Lattice α] {a b c : α}
theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
inf_le_left.trans le_sup_left
| Mathlib/Order/Lattice.lean | 521 | 525 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... |
@[simp] lemma graph_inj {f g : α → β} : f.graph = g.graph ↔ f = g := graph_injective.eq_iff
| Mathlib/Data/Rel.lean | 331 | 332 |
/-
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.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
/-!
# Extending an alternating map to the exterior algebra
## Main de... | (liftAlternating (R := R) (M := M) (N := N) f).compAlternatingMap (ιMulti R n) = f n :=
AlternatingMap.ext <| liftAlternating_apply_ιMulti f
@[simp]
theorem liftAlternating_comp (g : N →ₗ[R] N') (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) :
(liftAlternating (R := R) (M := M) (N := N') fun i => g.compAlternatingMap (f i)) ... | Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean | 102 | 114 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | Mathlib/Algebra/Ring/Parity.lean | 156 | 156 | |
/-
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... | exact not_cliqueFree_of_top_embedding <| f.comp (topEmbeddingOfNotCliqueFree h)
@[simp] theorem cliqueFree_map_iff {f : α ↪ β} [Nonempty α] :
(G.map f).CliqueFree n ↔ G.CliqueFree n := by
obtain (hle | hlt) := le_or_lt n 1
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 376 | 380 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | def sum : α :=
@prod (Multiplicative _) _ x
variable {x y}
| Mathlib/GroupTheory/FreeGroup/Basic.lean | 787 | 790 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
/-!
# Density of si... | `F n x ∈ s`. -/
noncomputable def approxOn (f : β → α) (hf : Measurable f) (s : Set α) (y₀ : α) (h₀ : y₀ ∈ s)
[SeparableSpace s] (n : ℕ) : β →ₛ α :=
haveI : Nonempty s := ⟨⟨y₀, h₀⟩⟩
comp (nearestPt (fun k => Nat.casesOn k y₀ ((↑) ∘ denseSeq s) : ℕ → α) n) f hf
| Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | 116 | 121 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Order.Interval.Finset.Fin
import Mathlib.Data.Vector.Basic
/-!
# The structure of `Fintype (Fin n)`
This file contains some basic results about the `Fintyp... | hxs, List.count_cons, add_comm (ite (x = a) 1 0), beq_iff_eq]
end Fin
| Mathlib/Data/Fintype/Fin.lean | 55 | 58 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | @[simp]
theorem one_add_of_omega0_le {o} (h : ω ≤ o) : 1 + o = o :=
mod_cast natCast_add_of_omega0_le h 1
open Ordinal
theorem isLimit_iff_omega0_dvd {a : Ordinal} : IsLimit a ↔ a ≠ 0 ∧ ω ∣ a := by
refine ⟨fun l => ⟨l.ne_zero, ⟨a / ω, le_antisymm ?_ (mul_div_le _ _)⟩⟩, fun h => ?_⟩
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,156 | 1,163 |
/-
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 | 828 | 829 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... |
theorem sub (hf : Martingale f ℱ μ) (hg : Martingale g ℱ μ) : Martingale (f - g) ℱ μ := by
rw [sub_eq_add_neg]; exact hf.add hg.neg
theorem smul (c : ℝ) (hf : Martingale f ℱ μ) : Martingale (c • f) ℱ μ := by
| Mathlib/Probability/Martingale/Basic.lean | 109 | 113 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... | (mulSupport fun v : HeightOneSpectrum R => v.maxPowDividing I).Finite :=
haveI h_subset : {v : HeightOneSpectrum R | v.maxPowDividing I ≠ 1} ⊆
{v : HeightOneSpectrum R |
((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ) ≠ 0} := by
intro v hv h_zero
have hv' : v.maxPowDividing I... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 97 | 107 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... |
theorem dvd_ordProj_of_dvd {n p : ℕ} (hn : n ≠ 0) (pp : p.Prime) (h : p ∣ n) : p ∣ ordProj[p] n :=
dvd_pow_self p (Prime.factorization_pos_of_dvd pp hn h).ne'
@[deprecated (since := "2024-10-24")] alias dvd_ord_proj_of_dvd := dvd_ordProj_of_dvd
theorem not_dvd_ordCompl {n p : ℕ} (hp : Prime p) (hn : n ≠ 0) : ¬p ∣ ... | Mathlib/Data/Nat/Factorization/Basic.lean | 219 | 227 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.Const
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.Yoneda
import Mat... |
/-- There is a morphism from an extended cone to the original cone. -/
@[simps]
def extend (s : Cone F) {X : C} (f : X ⟶ s.pt) : s.extend f ⟶ s where
| Mathlib/CategoryTheory/Limits/Cones.lean | 304 | 307 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | /-- The property that `f 0 = 0` in terms of the graph. -/
theorem graph_fst_eq_zero_snd (f : E →ₗ.[R] F) {x : E} {x' : F} (h : (x, x') ∈ f.graph)
| Mathlib/LinearAlgebra/LinearPMap.lean | 767 | 768 |
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
/-!
# Projective Spaces
This file contains the defin... | conv_lhs => rw [← v.mk_rep]
rfl
| Mathlib/LinearAlgebra/Projectivization/Basic.lean | 136 | 138 |
/-
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 isLittleO_exp_neg_mul_rpow_atTop {a : ℝ} (ha : 0 < a) (b : ℝ) :
IsLittleO atTop (fun x : ℝ => exp (-a * x)) fun x : ℝ => x ^ b := by
apply isLittleO_of_tendsto'
| Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 317 | 319 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Analysis.Normed.Group.Basic
/-!
# Hamming spaces
The Hamming metric counts the number of places two members of a (finite) Pi type
differ. The Hamming n... | theorem hammingDist_eq_zero {x y : ∀ i, β i} : hammingDist x y = 0 ↔ x = y :=
⟨eq_of_hammingDist_eq_zero, fun H => by
rw [H]
| Mathlib/InformationTheory/Hamming.lean | 86 | 88 |
/-
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... | (unitOfCoprime x h : ZMod n) = x :=
rfl
theorem val_coe_unit_coprime {n : ℕ} (u : (ZMod n)ˣ) : Nat.Coprime (u : ZMod n).val n := by
| Mathlib/Data/ZMod/Basic.lean | 772 | 775 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,288 | 1,294 | |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Sums of collections of Finsupp, ... | theorem Finset.mem_sup_support_iff [Zero M] {s : Finset (ι →₀ M)} {x : ι} :
x ∈ s.sup Finsupp.support ↔ ∃ f ∈ s, x ∈ f.support :=
Multiset.mem_sup_map_support_iff
open scoped Function -- required for scoped `on` notation
| Mathlib/Data/Finsupp/BigOperators.lean | 69 | 73 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Countable.Defs
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.PNat.Equiv
import Mathlib.... | def equiv : α ≃ ULower α :=
| Mathlib/Logic/Encodable/Basic.lean | 420 | 420 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | (e₁.prodCongr e₂ : M × M₃ →ₗ[R] M₂ × M₄) = (e₁ : M →ₗ[R] M₂).prodMap (e₂ : M₃ →ₗ[R] M₄) :=
rfl
| Mathlib/LinearAlgebra/Prod.lean | 729 | 731 |
/-
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.Additive
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.Algebra.Homology.ShortComplex.Preadditive
import Mathlib.Tactic.Linar... |
/-- The canonical isomorphism `K.X j ≅ K.opCycles j` when the differential to `j` is zero. -/
| Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | 537 | 538 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | exact_mod_cast Nat.one_le_of_lt h.lt]
exact Real.log_natCast_le_rpow_div n hε
end Complex
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 962 | 965 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Alge... |
theorem mul_fin_three [AddCommMonoid α] [Mul α]
(a₁₁ a₁₂ a₁₃ a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃ b₁₁ b₁₂ b₁₃ b₂₁ b₂₂ b₂₃ b₃₁ b₃₂ b₃₃ : α) :
!![a₁₁, a₁₂, a₁₃;
a₂₁, a₂₂, a₂₃;
a₃₁, a₃₂, a₃₃] * !![b₁₁, b₁₂, b₁₃;
| Mathlib/Data/Matrix/Notation.lean | 467 | 472 |
/-
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.AlgebraicGeometry.Cover.MorphismProperty
/-!
# Open covers of schemes
This file provides the basic API for open covers of schemes.
## Main definition
- `A... | Mathlib/AlgebraicGeometry/Cover/Open.lean | 434 | 441 | |
/-
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.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | (show (r : ℝ≥0∞) < p.radius from h.symm ▸ ENNReal.coe_lt_top)
refine .of_norm_bounded
(fun n ↦ (C : ℝ) * a ^ n) ((summable_geometric_of_lt_one ha.1.le ha.2).mul_left _) fun n ↦ ?_
specialize hp n
rwa [Real.norm_of_nonneg (mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg n))]
| Mathlib/Analysis/Analytic/Basic.lean | 283 | 287 |
/-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.LinearAlgebra.LinearIndependent.Defs
import Mathlib.LinearAlgebra.Multilinear.Basis
/-!
# Alte... | @[simp]
theorem curryLeft_compLinearMap {n : ℕ} (g : M₂'' →ₗ[R'] M'')
(f : M'' [⋀^Fin n.succ]→ₗ[R'] N'') (m : M₂'') :
(f.compLinearMap g).curryLeft m = (f.curryLeft (g m)).compLinearMap g :=
ext fun v => congr_arg f <| funext <| by
refine Fin.cases ?_ ?_
| Mathlib/LinearAlgebra/Alternating/Basic.lean | 940 | 945 |
/-
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.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | end ContinuousAdd
section IsTopologicalAddGroup
| Mathlib/Analysis/LocallyConvex/Bounded.lean | 320 | 322 |
/-
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... | (calc
(∑' b : α, (s.count b : ℝ≥0∞) / (Multiset.card s))
= (Multiset.card s : ℝ≥0∞)⁻¹ * ∑' b, (s.count b : ℝ≥0∞) := by
| Mathlib/Probability/Distributions/Uniform.lean | 328 | 330 |
/-
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... |
theorem init_neg (x : 𝕎 R) (n : ℕ) : init n (-x) = init n (-init n x) := by
| Mathlib/RingTheory/WittVector/InitTail.lean | 201 | 202 |
/-
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 continuousOn_iff :
ContinuousOn f s ↔
∀ x ∈ s, ∀ t : Set β, IsOpen t → f x ∈ t → ∃ u, IsOpen u ∧ x ∈ u ∧ u ∩ s ⊆ f ⁻¹' t := by
simp only [ContinuousOn, ContinuousWithinAt, tendsto_nhds, mem_nhdsWithin]
| Mathlib/Topology/ContinuousOn.lean | 577 | 581 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | end Sylow
end InfiniteSylow
open Equiv Equiv.Perm Finset Function List QuotientGroup
universe u
variable {G : Type u} [Group G]
theorem QuotientGroup.card_preimage_mk (s : Subgroup G) (t : Set (G ⧸ s)) :
Nat.card (QuotientGroup.mk ⁻¹' t) = Nat.card s * Nat.card t := by
rw [← Nat.card_prod, Nat.card_congr (pr... | Mathlib/GroupTheory/Sylow.lean | 499 | 519 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.HahnSeries.Multiplication
/-!
# Summable families of Hahn Series
We introduce a notion of formal summability for families of Hahn series, a... | (a : α × β) :
∑ x ∈ VAddAntidiagonal (s a.1).isPWO_support' (t a.2).isPWO_support' g, (s a.1).coeff x.1 •
(t a.2).coeff x.2 = ∑ x ∈ VAddAntidiagonal s.isPWO_iUnion_support' t.isPWO_iUnion_support' g,
| Mathlib/RingTheory/HahnSeries/Summable.lean | 363 | 365 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Data.Set.Sigma
import Mathlib.Order.CompleteLattice.Finset
/-!
# Finite sets in a ... | Mathlib/Data/Finset/Sigma.lean | 60 | 60 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | -/
partial def evalPow₁ {a : Q($α)} {b : Q(ℕ)} (va : ExSum sα a) (vb : ExProd sℕ b) :
| Mathlib/Tactic/Ring/Basic.lean | 847 | 848 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | def _root_.Set.fintypeOfMemBounds {s : Set α} [DecidablePred (· ∈ s)] (ha : a ∈ lowerBounds s)
(hb : b ∈ upperBounds s) : Fintype s :=
| Mathlib/Order/Interval/Finset/Basic.lean | 259 | 260 |
/-
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... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 938 | 945 | |
/-
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.Topology.Instances.Int
import Mathlib.Data.Nat.Lattice
/-!
# Topology on the natural numbers
The structure of a metric space on `ℕ` i... |
instance : ProperSpace ℕ :=
⟨fun x r => by
rw [closedBall_eq_Icc]
exact (Set.finite_Icc _ _).isCompact⟩
instance : NoncompactSpace ℕ :=
noncompactSpace_of_neBot <| by simp [Filter.atTop_neBot]
| Mathlib/Topology/Instances/Nat.lean | 55 | 63 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | Mathlib/Algebra/Lie/Submodule.lean | 1,250 | 1,255 | |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Stopping
import Mathlib.Tactic.AdaptationNote
/-!
# Hitting time
Given a stochastic process, the hitting time provides th... | simp only [hitting]
split_ifs with h
· obtain ⟨j, hj₁, hj₂⟩ := h
change j ∈ {i | u i ω ∈ s} at hj₂
exact (csInf_le (BddBelow.inter_of_left bddBelow_Icc) (Set.mem_inter hj₁ hj₂)).trans hj₁.2
· exact le_rfl
| Mathlib/Probability/Process/HittingTime.lean | 78 | 84 |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Cast.Field
import Mathlib.NumberTheory.ArithmeticFunction
/-!
# The von Mangoldt Function
In ... | theorem vonMangoldt_le_log : ∀ {n : ℕ}, Λ n ≤ Real.log (n : ℝ)
| 0 => by simp
| n + 1 => by
rw [← vonMangoldt_sum]
exact single_le_sum (by exact fun _ _ => vonMangoldt_nonneg)
(mem_divisors_self _ n.succ_ne_zero)
end ArithmeticFunction
| Mathlib/NumberTheory/VonMangoldt.lean | 144 | 160 |
/-
Copyright (c) 2023 Mohanad ahmed. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mohanad Ahmed
-/
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.SemiringInverse
/-! # Block Matrices from Rows and Columns
This file provides the basic definiti... | @[simp]
lemma fromRows_mulVec [Fintype n] (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) (v : n → R) :
fromRows A₁ A₂ *ᵥ v = Sum.elim (A₁ *ᵥ v) (A₂ *ᵥ v) := by
| Mathlib/Data/Matrix/ColumnRowPartitioned.lean | 203 | 205 |
/-
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... | M.copy E M.IsBase Indep hE (fun _ ↦ Iff.rfl) h
/-- create a copy of `M : Matroid α` with a base predicate and ground set defeq
| Mathlib/Data/Matroid/Basic.lean | 797 | 799 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
/-!
# Contents
In this file... | have : (↑(L' ⊔ M') : Set G) ⊆ U' := by
simp only [Compacts.coe_sup, union_subset_iff, hL'U', true_and]
exact hM'.trans diff_subset
rw [μ.outerMeasure_of_isOpen (↑U') U'.2]
refine le_trans (ge_of_eq ?_) (μ.le_innerContent _ _ this)
exact μ.sup_disjoint L' M' (subset_diff.1 hM').2.symm isClosed_closure is... | Mathlib/MeasureTheory/Measure/Content.lean | 352 | 392 |
/-
Copyright (c) 2022 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# The filter of small sets
This file defines the ... | @[simp]
theorem smallSets_top : (⊤ : Filter α).smallSets = ⊤ := by
| Mathlib/Order/Filter/SmallSets.lean | 116 | 117 |
/-
Copyright (c) 2021 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Jeremy Tan
-/
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
/-!
# Strongly regular graphs
## Main definitio... | theorem IsSRGWith.top :
(⊤ : SimpleGraph V).IsSRGWith (Fintype.card V) (Fintype.card V - 1) (Fintype.card V - 2) μ where
card := rfl
regular := IsRegularOfDegree.top
of_adj _ _ := card_commonNeighbors_top
of_not_adj v w h h' := (h' ((top_adj v w).2 h)).elim
theorem IsSRGWith.card_neighborFinset_union_eq {v... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 73 | 80 |
/-
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.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Nat
/-!
# Lemmas about squ... | rw [← not_iff_not, hp.dvd_iff_one_le_factorization hn.ne_zero, not_le, lt_one_iff,
hp.dvd_iff_one_le_factorization hm.ne_zero, not_le, lt_one_iff] at h₁
have h₂ := hn.natFactorization_le_one p
have h₃ := hm.natFactorization_le_one p
omega
rw [factorization_eq_zero_of_non_prime _ hp, factorizatio... | Mathlib/Data/Nat/Squarefree.lean | 76 | 91 |
/-
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... | Mathlib/Data/Num/Lemmas.lean | 1,327 | 1,328 | |
/-
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... | #α = #(Set.univ : Set α) := mk_univ.symm
_ ≤ #l.toFinset := mk_le_mk_of_subset fun x _ => List.mem_toFinset.mpr (h x)
_ = l.toFinset.card := Cardinal.mk_coe_finset
_ ≤ l.length := Nat.cast_le.mpr (List.toFinset_card_le l)
theorem three_le {α : Type*} (h : 3 ≤ #α) (x : α) (y : α) : ∃ z : α, z ≠ x ∧ z ≠ ... | Mathlib/SetTheory/Cardinal/Basic.lean | 935 | 951 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.NumberTheory.Padics.PadicVal.Basic
/-!
# p-adic norm
This file defines the `p`-adic norm on `ℚ`.
The `p`-adic valuation on `ℚ` is the difference o... | · simp_all
theorem sum_le {α : Type*} {F : α → ℚ} {t : ℚ} {s : Finset α} :
| Mathlib/NumberTheory/Padics/PadicNorm.lean | 288 | 290 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L1
This file contains ... | @[deprecated (since := "2025-01-21")] noncomputable alias condexpIndL1 := condExpIndL1
variable {hm : m ≤ m0} [SigmaFinite (μ.trim hm)]
| Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 185 | 187 |
/-
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.Set.Image
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Sets in sigma types
This file defines `Set.sigma`, the indexed sum of sets.
-/
namespace Set... |
theorem _root_.biInf_sigma' (s : Set ι) (t : ∀ i, Set (α i)) (f : ∀ i, α i → β) :
⨅ (i ∈ s) (j ∈ t i), f i j = ⨅ ij ∈ s.sigma t, f ij.fst ij.snd :=
| Mathlib/Data/Set/Sigma.lean | 131 | 133 |
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Junyan Xu, Jack McKoen
-/
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Localization.AsSubring
import Mathlib.Algebra.Ring.Subring.Pointwise
impor... | at the expense of a more complicated right hand side.
-/
| Mathlib/RingTheory/Valuation/ValuationSubring.lean | 520 | 521 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunc... | intro x
nlinarith
have h₂ : ContDiff ℝ ω fun w => (4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v := by
refine (contDiff_const.smul contDiff_id).add ?_
exact (h₀.sub contDiff_const).smul contDiff_const
| Mathlib/Geometry/Manifold/Instances/Sphere.lean | 163 | 167 |
/-
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.Finset.Basic
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
/-!
# The fold operation for a commutative associative operation ... | Mathlib/Data/Finset/Fold.lean | 235 | 240 | |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | · rw [snoc_castSucc]
exact (cast_heq _ _).symm
| Mathlib/Data/Fin/Tuple/Basic.lean | 887 | 888 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber, Yaël Dillies, Kin Yau James Wong
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Topo... |
theorem compl_cylinder (s : Finset ι) (S : Set (∀ i : s, α i)) :
(cylinder s S)ᶜ = cylinder s (Sᶜ) := by
| Mathlib/MeasureTheory/Constructions/Cylinders.lean | 209 | 211 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Order.Field
impo... |
variable (G)
| Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 74 | 76 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.RightHomology
/-!
# Homology of short complexes
In this file, we shall define the homology of short complexes `S`, i.e. diagrams... | · rw [← cancel_epi S₂.homologyπ, reassoc_of% hz, homologyπ_naturality,
IsIso.inv_hom_id_assoc, comp_id]
lemma isIso_homologyMap_of_isIso_opcyclesMap_of_mono {φ : S₁ ⟶ S₂}
[S₁.HasHomology] [S₂.HasHomology] (h₁ : IsIso (opcyclesMap φ)) (h₂ : Mono φ.τ₃) :
IsIso (homologyMap φ) := by
have h : (S₂.homolog... | Mathlib/Algebra/Homology/ShortComplex/Homology.lean | 1,085 | 1,097 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | alias ⟨indepSet.congr, _⟩ := indepSet_congr
lemma iIndepFun_congr {β : ι → Type*} {m : ∀ x : ι, MeasurableSpace (β x)}
{f : ∀ x : ι, Ω → β x} (h : κ =ᵐ[μ] η) : iIndepFun f κ μ ↔ iIndepFun f η μ :=
iIndep_congr h
| Mathlib/Probability/Independence/Kernel.lean | 193 | 197 |
/-
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... | private theorem sdiff_le' : x \ y ≤ x :=
calc
x \ y ≤ x ⊓ y ⊔ x \ y := le_sup_right
_ = x := sup_inf_sdiff x y
-- Use `sdiff_sup_self`
| Mathlib/Order/BooleanAlgebra.lean | 127 | 132 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | theorem neg_rotation_neg_pi_div_two (x : V) :
-o.rotation (-π / 2 : ℝ) x = o.rotation (π / 2 : ℝ) x := by
rw [neg_rotation, ← Real.Angle.coe_add, neg_div, ← sub_eq_add_neg, sub_half]
| Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 173 | 175 |
/-
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
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... |
theorem tmul_neg (m : M) (n : N) : m ⊗ₜ (-n) = -m ⊗ₜ[R] n :=
(mk R M N _).map_neg _
| Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,297 | 1,299 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Topology.Algebra.Monoid
/-!
# Topological group with zero
I... | (continuousAt_zpow₀ _ _ (Or.inl hx)).continuousWithinAt
| Mathlib/Topology/Algebra/GroupWithZero.lean | 323 | 324 |
/-
Copyright (c) 2023 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.Complex.PhragmenLindelof
/-!
# Hadamard three-lines Theorem
In this file we present a ... | lemma sSupNormIm_nonneg (x : ℝ) : 0 ≤ sSupNormIm f x := by
apply Real.sSup_nonneg
rintro y ⟨z1, _, hz2⟩
simp only [← hz2, comp, norm_nonneg]
| Mathlib/Analysis/Complex/Hadamard.lean | 96 | 99 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... |
theorem prod_withDensity_right {g : β → ℝ≥0∞} (hg : Measurable g) :
μ.prod (ν.withDensity g) = (μ.prod ν).withDensity (fun z ↦ g z.2) :=
| Mathlib/MeasureTheory/Measure/WithDensity.lean | 660 | 662 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | instance list : Primcodable (List α) :=
⟨letI H := @Primcodable.prim (List ℕ) _
have : Primrec₂ fun (a : α) (o : Option (List ℕ)) => o.map (List.cons (encode a)) :=
option_map snd <| (list_cons' H).comp ((@Primrec.encode α _).comp (fst.comp fst)) snd
have :
| Mathlib/Computability/Primrec.lean | 806 | 810 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
/-!
# Filtered categories
A category is filtered if every finite diagram admits a cocone.
We give a... | · exact ⟨leftToMax _ _⟩
· exact ⟨(w' (Finset.mem_of_mem_insert_of_ne mY h)).some ≫ rightToMax _ _⟩
variable (O : Finset C) (H : Finset (Σ' (X Y : C) (_ : X ∈ O) (_ : Y ∈ O), X ⟶ Y))
/-- Given any `Finset` of objects `{X, ...}` and
| Mathlib/CategoryTheory/Filtered/Basic.lean | 214 | 219 |
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Complement
/-!
# Semidirect product
This file defines semidirect products of groups, and the canonical maps in and out of the
semidirect prod... | right_inv _ := by simp
| Mathlib/GroupTheory/SemidirectProduct.lean | 296 | 296 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Monic
/-!
# Lemmas for the interaction between polynomials and `∑` and `∏`.
Recall that `∑` and `∏` are notation for ... | nontriviality R
rw [natDegree_multiset_prod']
simp_rw [Ne, Multiset.prod_eq_zero_iff, Multiset.mem_map, leadingCoeff_eq_zero]
rintro ⟨_, h, rfl⟩
contradiction
/-- The degree of a product of polynomials is equal to
| Mathlib/Algebra/Polynomial/BigOperators.lean | 329 | 335 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Algebra.Group.End
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Common
/-!
#... | Mathlib/GroupTheory/Perm/Basic.lean | 428 | 432 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... |
end Sylow
/-- A generalization of **Sylow's first theorem**.
Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
Exists.elim
(zorn_le_nonempty₀ { Q : Subgroup G | IsPGroup p Q }
(fun c hc1 hc2 Q hQ ... | Mathlib/GroupTheory/Sylow.lean | 148 | 162 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Complex.Convex
import Mathlib.D... | · have h_exp : ∀ x, ContinuousAt (fun x => exp (- x)) x := fun x => continuousAt_neg.rexp
rw [← Ioc_union_Ioi_eq_Ioi zero_le_one, integrableOn_union]
constructor
· rw [← integrableOn_Icc_iff_integrableOn_Ioc]
refine IntegrableOn.mul_continuousOn ?_ ?_ isCompact_Icc
· refine (intervalIntegrable... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 63 | 89 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.Separation.Regular
/-!
# Dense embeddings
This file defines three properties of f... | have hV₂x : V₂ ∈ 𝓝 x := IsOpen.mem_nhds V₂_op x_in₂
apply V'_closed.mem_of_tendsto x_in₁
use V₂
tauto
theorem continuous_extend [T3Space γ] {f : α → γ} (di : IsDenseInducing i)
(hf : ∀ b, ∃ c, Tendsto f (comap i (𝓝 b)) (𝓝 c)) : Continuous (di.extend f) :=
continuous_iff_continuousAt.mpr fun _ => di.co... | Mathlib/Topology/DenseEmbedding.lean | 192 | 212 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
/-!
# Locally integrable functions
A function is called *locally integrable* (`MeasureTheory.LocallyIntegrabl... | (integrable_zero X E μ).locallyIntegrable
theorem locallyIntegrableOn_zero : LocallyIntegrableOn (fun _ ↦ (0 : E)) s μ :=
locallyIntegrable_zero.locallyIntegrableOn s
theorem LocallyIntegrable.indicator (hf : LocallyIntegrable f μ) {s : Set X}
| Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 269 | 274 |
/-
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.Algebra.GroupWithZero.Pointwise.Set.Basic
import Mathlib.Algebra.Ring.Pointwise.Set
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topol... | ext fun _ => rfl
| Mathlib/Topology/MetricSpace/IsometricSMul.lean | 143 | 144 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | exact continuousInv_sInf (Set.forall_mem_range.mpr h')
@[to_additive]
theorem continuousInv_inf {t₁ t₂ : TopologicalSpace G} (h₁ : @ContinuousInv G t₁ _)
| Mathlib/Topology/Algebra/Group/Basic.lean | 322 | 325 |
/-
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.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin.Rotate
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutatio... | · exact ⟨_, _, hlt', by simp [hlt.le]⟩
exact ⟨_, _, hle.lt_of_ne (by simp [hlt.ne]), by simp [hlt.le]⟩
| Mathlib/GroupTheory/Perm/Fin.lean | 315 | 316 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.RingTheory.Artinian.Module
/-!
# Lie subalgebras
This file defines Lie subalgebras of a Lie algebra and provides basic rel... | zero_add := bot_sup_eq
add_zero := sup_bot_eq
add_comm := sup_comm
nsmul := nsmulRec
instance : IsOrderedAddMonoid (LieSubalgebra R L) where
add_le_add_left _ _ := sup_le_sup_left
| Mathlib/Algebra/Lie/Subalgebra.lean | 496 | 502 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | @[simp]
theorem biproduct.isoProduct_hom {f : J → C} [HasBiproduct f] :
| Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 508 | 509 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,464 | 1,466 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... | image₂ (fun a b i ↦ f _ (a i) (b i)) (piFinset s) (piFinset t) := by
ext; simp only [mem_piFinset, mem_image₂, Classical.skolem, forall_and, funext_iff]
end Fintype
| Mathlib/Data/Finset/NAry.lean | 589 | 592 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... | rw [hurwitzZetaEven_def_of_ne_or_ne, div_eq_mul_inv]
simpa [sub_eq_zero, eq_comm (a := s)] using hs'
rw [this, completedHurwitzZetaEven_one_sub, inv_Gammaℝ_one_sub hs, cosZeta,
Function.update_of_ne (by simpa using hs 0), ← Gammaℂ]
generalize Gammaℂ s * cos (π * s / 2) = A -- speeds up ring_nf call
ri... | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 759 | 767 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,290 | 1,326 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Hull
/-!
# Convex join
This file defines the convex join of two sets. The convex join of `s` and `t` is the union of the
segments with on... | convexJoin 𝕜 (segment 𝕜 a b) (segment 𝕜 c d) = convexHull 𝕜 {a, b, c, d} := by
simp_rw [← convexHull_pair, convexHull_insert (insert_nonempty _ _),
convexHull_insert (singleton_nonempty _), convexJoin_assoc,
convexHull_singleton]
theorem convexJoin_segment_singleton (a b c : E) :
convexJoin 𝕜 (s... | Mathlib/Analysis/Convex/Join.lean | 177 | 187 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
/-!
# Quadratic characters of finite fields
Further facts relying ... | obtain ⟨a, ha⟩ := quadraticChar_exists_neg_one' hF
refine ne_one_iff.mpr ⟨a, ?_⟩
simpa only [ringHomComp_apply, ha, eq_intCast, Int.cast_neg, Int.cast_one, χ] using
Ring.neg_one_ne_one_of_char_ne_two hF'
have h := Char.card_pow_card hχ₁ ((quadraticChar_isQuadratic F).comp _) h hF'
rw [← quadraticC... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 79 | 92 |
/-
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
/... | theorem mem_ball_self (hr : 0 < r) : x ∈ ball p x r := by simp [hr]
theorem mem_closedBall_self (hr : 0 ≤ r) : x ∈ closedBall p x r := by simp [hr]
theorem mem_ball_zero : y ∈ ball p 0 r ↔ p y < r := by rw [mem_ball, sub_zero]
theorem mem_closedBall_zero : y ∈ closedBall p 0 r ↔ p y ≤ r := by rw [mem_closedBall, sub... | Mathlib/Analysis/Seminorm.lean | 621 | 627 |
/-
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.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | @[simp]
theorem toWithTop_top' {h : Decidable (⊤ : PartENat).Dom} : toWithTop ⊤ = ⊤ := by
| Mathlib/Data/Nat/PartENat.lean | 504 | 505 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,504 | 2,509 | |
/-
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 ... | noncomputable def desc (α : Cochain F K (-1)) (β : G ⟶ K)
(eq : δ (-1) 0 α = Cochain.ofHom (φ ≫ β)) : mappingCone φ ⟶ K :=
Cocycle.homOf (descCocycle φ α (Cocycle.ofHom β) (neg_add_cancel 1) (by simp [eq]))
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 358 | 361 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Data.Sum.Order
import Mathlib.Order.RelIso.Set
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order... |
@[simp]
| Mathlib/Order/InitialSeg.lean | 369 | 370 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Fintype.Order
import Mathlib.ModelTheory.FinitelyGenerated
import Mathlib.ModelTheory.Quotients
import... | intro S hS x _
let out := Quotient.out (s := DirectLimit.setoid G f)
refine hS (out x).1 ⟨(out x).2, ?_, ?_⟩
· rw [(Classical.choose_spec (h (out x).1).out).2]
trivial
| Mathlib/ModelTheory/DirectLimit.lean | 414 | 418 |
/-
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.ChartedSpace
/-!
# Local properties invariant under a groupoid
We study properties of a triple `(g, s, x)` ... | (h : LiftProp P g) : LiftPropOn P g s := by
rw [← liftPropOn_univ] at h
exact liftPropOn_mono mono h (subset_univ _)
theorem liftPropAt_of_mem_maximalAtlas [HasGroupoid M G] (hG : G.LocalInvariantProp G Q)
(hQ : ∀ y, Q id univ y) (he : e ∈ maximalAtlas M G) (hx : x ∈ e.source) : LiftPropAt Q e x := by
si... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 448 | 454 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Ab
import Mathlib.Algebra.Homology.ShortComplex.ExactFunctor
import Mathlib.Algebra.Homology.ShortComplex.SnakeLemma
import Mathlib... | ∃ (x₁ : (forget₂ C Ab).obj S.X₁), ((forget₂ C Ab).map S.f) x₁ = x₂ := by
rw [S.exact_iff_exact_map_forget₂, ab_exact_iff]
rfl
| Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean | 87 | 90 |
/-
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 Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Int.Notation
import Mathlib.Data.Nat.BinaryRec
import Mathlib.L... | class CancelMonoid (M : Type u) extends LeftCancelMonoid M, RightCancelMonoid M
/-- Commutative version of `AddCancelMonoid`. -/
class AddCancelCommMonoid (M : Type u) extends AddCommMonoid M, AddLeftCancelMonoid M
| Mathlib/Algebra/Group/Defs.lean | 702 | 705 |
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