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/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,898 | 1,902 | |
/-
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 ... | `C¹`-smooth maps are locally Lipschitz continuous, hence they do not increase the Hausdorff
dimension of sets.
-/
| Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 488 | 490 |
/-
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.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... | | none, some _ => Or.inr rfl
| some a, some b => by simpa [liftOrGet] using h a b
/-- Given an element of `a : Option α`, a default element `b : β` and a function `α → β`, apply this
function to `a` if it comes from `α`, and return `b` otherwise. -/
def casesOn' : Option α → β → (α → β) → β
| none, n, _ => n
| Mathlib/Data/Option/Basic.lean | 231 | 237 |
/-
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... | MeasurableSet.iUnion fun hj => f.mono hj.2 _ ((hu j).measurable hs)
theorem stoppedValue_hitting_mem [ConditionallyCompleteLinearOrder ι] [WellFoundedLT ι]
{u : ι → Ω → β} {s : Set β} {n m : ι} {ω : Ω} (h : ∃ j ∈ Set.Icc n m, u j ω ∈ s) :
stoppedValue u (hitting u s n m) ω ∈ s := by
simp only [stoppedV... | Mathlib/Probability/Process/HittingTime.lean | 231 | 245 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.Restrict
/-! # Mutually singular measures
Two measures `μ`, `ν` are said to be mutually singular (`MeasureTheory.Mea... | theorem zero_right : μ ⟂ₘ 0 :=
⟨∅, MeasurableSet.empty, measure_empty, rfl⟩
| Mathlib/MeasureTheory/Measure/MutuallySingular.lean | 73 | 74 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yakov Pechersky, Eric Wieser
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.enum`
## Deprecation note
Many lemmas in this file have been replaced by the... | Mathlib/Data/List/Enum.lean | 104 | 105 | |
/-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | (h : Equivalence r) : Equivalence (InvImage r f) :=
⟨fun _ ↦ h.1 _, h.symm, h.trans⟩
instance {α β : Type*} {r : α → β → Prop} {x : α × β} [Decidable (r x.1 x.2)] :
Decidable (uncurry r x) :=
‹Decidable _›
instance {α β : Type*} {r : α × β → Prop} {a : α} {b : β} [Decidable (r (a, b))] :
| Mathlib/Logic/Function/Basic.lean | 1,025 | 1,032 |
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir
-/
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Construction of the hyperreal numbers as an ultrapro... | theorem Infinite.not_infinitesimal {x : ℝ*} (h : Infinite x) : ¬Infinitesimal x := fun h' ↦
h'.not_infinite h
theorem InfinitePos.not_infinitesimal {x : ℝ*} (h : InfinitePos x) : ¬Infinitesimal x :=
Infinite.not_infinitesimal (Or.inl h)
| Mathlib/Data/Real/Hyperreal.lean | 396 | 401 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | Mathlib/Data/List/Nodup.lean | 401 | 402 | |
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Control.Monad.Writer
import Mathlib.Control.Lawful
import Batteries.Tactic.Congr
import Batteries.Lean.Except
import Batte... | end
| Mathlib/Control/Monad/Cont.lean | 196 | 197 |
/-
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... |
lemma setLIntegral_withDensity_eq_lintegral_mul₀' {μ : Measure α} {f : α → ℝ≥0∞}
(hf : AEMeasurable f μ) {g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.withDensity f))
{s : Set α} (hs : MeasurableSet s) :
∫⁻ a in s, g a ∂μ.withDensity f = ∫⁻ a in s, (f * g) a ∂μ := by
rw [restrict_withDensity hs, lintegral_withD... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 397 | 426 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
/-!
# The `arctan` function.
Inequalit... | (∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y = (2 * l + 1) * π / 2) :
tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) := by
simpa only [← Complex.ofReal_inj, Complex.ofReal_sub, Complex.ofReal_add, Complex.ofReal_div,
Complex.ofReal_mul, Complex.ofReal_tan] using
@Complex.tan_add (x : ℂ) (y :... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 32 | 38 |
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryThe... | lemma isZero₃_of_isZero₁₂ (h₁ : IsZero T.obj₁) (h₂ : IsZero T.obj₂) : IsZero T.obj₃ :=
isZero₂_of_isZero₁₃ _ (rot_of_distTriang _ hT) h₂ (by
dsimp
simp only [IsZero.iff_id_eq_zero] at h₁ ⊢
| Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 347 | 350 |
/-
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.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# First-... | namespace Cardinal
open FirstOrder FirstOrder.Language
variable {L : Language.{u, v}} (κ : Cardinal.{w}) (T : L.Theory)
/-- A theory is `κ`-categorical if all models of size `κ` are isomorphic. -/
| Mathlib/ModelTheory/Satisfiability.lean | 460 | 466 |
/-
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.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib... | exact hs.not_lt this
theorem setIntegral_Ioi_zero_rpow (s : ℝ) : ∫ x in Ioi (0 : ℝ), x ^ s = 0 :=
MeasureTheory.integral_undef (not_integrableOn_Ioi_rpow s)
theorem integral_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
∫ t : ℝ in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1) := by
have hd : ∀ x ∈ I... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 157 | 179 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable
/-!
# Asymptotic bounds for Jacobi theta functions
The goal of this file is to establish some tech... | lemma f_le_g_nat (k : ℕ) {a t : ℝ} (ha : 0 ≤ a) (ht : 0 < t) (n : ℕ) :
‖f_nat k a t n‖ ≤ g_nat k a t n := by
rw [f_nat, norm_of_nonneg (by positivity)]
refine mul_le_mul_of_nonneg_left ?_ (by positivity)
rw [Real.exp_le_exp, mul_le_mul_right ht,
mul_le_mul_left_of_neg (neg_lt_zero.mpr pi_pos), ← sub_nonne... | Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean | 71 | 80 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... | namespace List
variable {α : Type*}
/-- Auxiliary for `List.splitWrtComposition`. -/
def splitWrtCompositionAux : List α → List ℕ → List (List α)
| Mathlib/Combinatorics/Enumerative/Composition.lean | 686 | 691 |
/-
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.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Combinatorics.Pigeonhole
/-!
# Conserv... | rcases hf.exists_mem_iterate_mem hTm hμT with ⟨x, hxt, m, hm₀, hmt⟩
-- Then `N + m > N`, `x ∈ s`, and `f^[N + m] x = f^[N] (f^[m] x) ∈ s`.
-- This contradicts `x ∈ T ⊆ (⋃ n > N, t n)ᶜ`.
refine hxt.2 <| mem_iUnion₂.2 ⟨N + m, ?_, hxt.1.1, ?_⟩
· simpa [pos_iff_ne_zero]
· simpa only [iterate_add] using hmt.1.2
... | Mathlib/Dynamics/Ergodic/Conservative.lean | 121 | 130 |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | have : Cardinal.mk (Emb F F) = 1 := le_antisymm
(Cardinal.le_one_iff_subsingleton.2 AlgHom.subsingleton)
(Cardinal.one_le_iff_ne_zero.2 <| Cardinal.mk_ne_zero _)
rw [finSepDegree, Nat.card, this, Cardinal.one_toNat]
| Mathlib/FieldTheory/SeparableDegree.lean | 168 | 172 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | theorem disjoint_compl_left : Disjoint aᶜ a :=
| Mathlib/Order/Heyting/Basic.lean | 658 | 658 |
/-
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... |
theorem support_X (H : ¬(1 : R) = 0) : (X : R[X]).support = singleton 1 := by
| Mathlib/Algebra/Polynomial/Basic.lean | 818 | 819 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## Pushouts of Monoids and Groups
This file defin... |
@[simp]
theorem prod_summand_smul {i : ι} (g : G i) (w : NormalWord d) :
| Mathlib/GroupTheory/PushoutI.lean | 531 | 533 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Data.Finset.Grade
import Mathlib.Data.Finset.Sups
import Mathlib.Logic.Function.Iterate
/-!
# Shadows
This file defines shad... | let ⟨t, ht, hts, _⟩ := mem_upShadow_iff_exists_mem_card_add_one.1 hs
⟨t, ht, hts⟩
/-- `t ∈ ∂^k 𝒜` iff `t` is exactly `k` elements more than something in `𝒜`. -/
theorem mem_upShadow_iff_exists_mem_card_add :
s ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ #t + k = #s := by
induction k generalizing 𝒜 s with
| zero =>... | Mathlib/Combinatorics/SetFamily/Shadow.lean | 272 | 280 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
/-!
# The category of functors and natural transf... | theorem id_app (F : C ⥤ D) (X : C) : (𝟙 F : F ⟶ F).app X = 𝟙 (F.obj X) := rfl
| Mathlib/CategoryTheory/Functor/Category.lean | 68 | 68 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | ∃ L L' : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜),
IsTwoBlockDiagonal ((L.map toMatrix).prod * M * (L'.map toMatrix).prod) := by
by_cases H : IsTwoBlockDiagonal M
· refine ⟨List.nil, List.nil, by simpa using H⟩
-- we have already proved this when the last coefficient is nonzero
by_cases hM : M (inr... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 549 | 562 |
/-
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` ... | · intro h t ht
rcases h t ht with ⟨u, u_open, hu⟩
rw [inter_comm, hu]
apply IsOpen.inter u_open hs
· intro h t ht
refine ⟨s ∩ f ⁻¹' t, h t ht, ?_⟩
rw [@inter_comm _ s (f ⁻¹' t), inter_assoc, inter_self]
| Mathlib/Topology/ContinuousOn.lean | 656 | 662 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Connected.Clopen
/-!
# Totally disconnected and totally separated topological spaces
## Main definitions
We define th... | Mathlib/Topology/Connected/TotallyDisconnected.lean | 334 | 339 | |
/-
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.Order.Filter.Partial
import Mathlib.Topology.Neighborhoods
/-!
# Partial functions and topological spaces
In this file we prove properties of `Filter.P... | apply all_mem_nhds_filter
apply Rel.preimage_mono
theorem ptendsto_nhds {f : Y →. X} {l : Filter Y} {x : X} :
PTendsto f l (𝓝 x) ↔ ∀ s, IsOpen s → x ∈ s → f.core s ∈ l :=
| Mathlib/Topology/Partial.lean | 30 | 34 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... | have h4 x : (x + x + 1) / 2 = x := by rw [← two_mul, add_comm]; simp [add_mul_div_left]
cases a <;> cases b <;> simp [h4] <;> split_ifs
<;> simp_all +decide [two_mul]
lemma bit_mod_two_eq_zero_iff (a x) :
bit a x % 2 = 0 ↔ !a := by
simp
lemma bit_mod_two_eq_one_iff (a x) :
bit a x % 2 = 1 ↔ a := by
... | Mathlib/Data/Nat/Bitwise.lean | 83 | 95 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | · exact (hh₂.hasDerivWithinAt.comp x hh.hasDerivWithinAt hs).derivWithin hsx
· simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx]
@[deprecated (since := "2024-10-31")] alias derivWithin.comp := derivWithin_comp
| Mathlib/Analysis/Calculus/Deriv/Comp.lean | 273 | 276 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | Nat.sub_add_cancel (show 1 ≤ n from hpos), sub_add_cancel]
/-- If `n = 0` in `R` and `b` is a unit, then `a * X ^ n + b * X + c` is separable. -/
theorem separable_C_mul_X_pow_add_C_mul_X_add_C
{n : ℕ} (a b c : R) (hn : (n : R) = 0) (hb : IsUnit b) :
(C a * X ^ n + C b * X + C c).Separable := by
| Mathlib/FieldTheory/Separable.lean | 259 | 264 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Algebra.Lie.Quotient
/-!
# The normalizer of Lie submodules and subalgebras.
... |
theorem normalizer_inf : (N₁ ⊓ N₂).normalizer = N₁.normalizer ⊓ N₂.normalizer := by
ext; simp [← forall_and]
| Mathlib/Algebra/Lie/Normalizer.lean | 64 | 67 |
/-
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.CategoryTheory.Sites.Over
/-! Internal hom of sheaves
In this file, given two sheaves `F` and `G` on a site `(C, J)` with values
in a category `A`, we define ... | @[simp]
lemma presheafHom_map_app_op_mk_id {X Y : C} (g : Y ⟶ X)
(α : (presheafHom F G).obj (op X)) :
((presheafHom F G).map g.op α).app (op (Over.mk (𝟙 Y))) =
α.app (op (Over.mk g)) :=
presheafHom_map_app (𝟙 Y) g g (by simp) α
| Mathlib/CategoryTheory/Sites/SheafHom.lean | 67 | 72 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Field.Pi
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.Normed.Group.Pointwise
import Mat... | simp_rw [ball_pi _ hδ, Real.ball_eq_Ioo] at hw
exact ((lt_sub_iff_add_lt.2 <| hyz _).trans (hw _ <| mem_univ _).1).le
theorem IsLowerSet.mem_interior_of_forall_lt (hs : IsLowerSet s) (hx : x ∈ closure s)
(h : ∀ i, y i < x i) : y ∈ interior s := by
cases nonempty_fintype ι
obtain ⟨ε, hε, hxy⟩ := Pi.exists_f... | Mathlib/Analysis/Normed/Order/UpperLower.lean | 93 | 108 |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.Layercake
/-!
# The integral of the real power of a nonnegative function
In thi... | theorem lintegral_rpow_eq_lintegral_meas_le_mul :
∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) := by
have one_lt_p : -1 < p - 1 := by linarith
have obs : ∀ x : ℝ, ∫ t : ℝ in (0)..x, t ^ (p - 1) = x ^ p / p := by
intro x
rw... | Mathlib/Analysis/SpecialFunctions/Pow/Integral.lean | 50 | 72 |
/-
Copyright (c) 2021 Roberto Alvarez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Roberto Alvarez
-/
import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Algebra.Equiv.TransferInstance
import ... | Mathlib/Topology/Homotopy/HomotopyGroup.lean | 535 | 539 | |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... |
@[simp]
theorem star_def : (Star.star : K → K) = conj :=
| Mathlib/Analysis/RCLike/Basic.lean | 358 | 360 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.NumberTheory.LSeries.Basic
/-!
# Linearity of the L-series of `f` as a function of `f`
We show that the `LSer... | lemma LSeries.term_smul_apply (f : ℕ → ℂ) (c s : ℂ) (n : ℕ) :
term (c • f) s n = c * term f s n := by
simp [term_smul]
lemma LSeriesHasSum.smul {f : ℕ → ℂ} (c : ℂ) {s a : ℂ} (hf : LSeriesHasSum f s a) :
| Mathlib/NumberTheory/LSeries/Linearity.lean | 107 | 111 |
/-
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.Adjunction.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Batteries.Tactic.Congr
/-... | theorem ofConeEquiv_symm_apply_desc {D : Type u₄} [Category.{v₄} D] {G : K ⥤ D}
(h : Cone G ≌ Cone F) {c : Cone G} (P : IsLimit c) (s) :
((ofConeEquiv h).symm P).lift s =
(h.counitIso.inv.app s).hom ≫ (h.functor.map (P.liftConeMorphism (h.inverse.obj s))).hom :=
| Mathlib/CategoryTheory/Limits/IsLimit.lean | 228 | 231 |
/-
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.Max
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Multiset.Sort
import Mathlib.Order.RelIso.Set
/-!
# Construct a sorted list f... | -- multiset uses `0` not `∅` for empty sets
if s.card = 0 then "∅" else repr s.1
end Finset
namespace Fin
| Mathlib/Data/Finset/Sort.lean | 271 | 276 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | theorem sin_eq_zero_iff_cos_eq {x : ℝ} : sin x = 0 ↔ cos x = 1 ∨ cos x = -1 := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 505 | 505 |
/-
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.MeasureTheory.Measure.Trim
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
/-!
# Almost everywhere measurable functions
A funct... |
theorem MeasureTheory.Measure.map_mono_of_aemeasurable {f : α → δ} (h : μ ≤ ν)
(hf : AEMeasurable f ν) : μ.map f ≤ ν.map f :=
le_iff.2 fun s hs ↦ by simpa [hf, hs, hf.mono_measure h] using h (f ⁻¹' s)
/-- If the `σ`-algebra of the codomain of a null measurable function is countably generated,
then the function ... | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | 358 | 366 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | theorem sublistsLen_zero (l : List α) : sublistsLen 0 l = [[]] :=
sublistsLenAux_zero _ _ _
@[simp]
theorem sublistsLen_succ_nil (n) : sublistsLen (n + 1) (@nil α) = [] :=
rfl
| Mathlib/Data/List/Sublists.lean | 213 | 218 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 1,266 | 1,268 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,804 | 2,805 | |
/-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Algebra.Divisibility.Units
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural n... | have ⟨c, hc⟩ := h
have : 0 < c := by
apply Nat.pos_of_ne_zero
intro h'
rw [h',mul_zero] at hc
omega
simp [hc, base_pow_mul hp this]
end maxPowDiv
| Mathlib/Data/Nat/MaxPowDiv.lean | 101 | 109 |
/-
Copyright (c) 2024 Ira Fesefeldt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ira Fesefeldt
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
/-!
# Ordinal Approximants for the Fixed points on complete lattices
This file sets up the ordinal-indexed approximation t... | theorem lfpApprox_add_one (h : x ≤ f x) (a : Ordinal) :
lfpApprox f x (a+1) = f (lfpApprox f x a) := by
apply le_antisymm
· conv => left; rw [lfpApprox]
apply sSup_le
simp only [Ordinal.add_one_eq_succ, lt_succ_iff, exists_prop, Set.union_singleton,
Set.mem_insert_iff, Set.mem_setOf_eq, forall_eq_... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 92 | 112 |
/-
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... | section IndexOf
| Mathlib/Data/List/Basic.lean | 521 | 522 |
/-
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.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | d ≤ 3 * c := by omega
| Mathlib/Data/Ordmap/Ordset.lean | 157 | 157 |
/-
Copyright (c) 2021 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.Fintype.Order
import Mathlib.Order.Closure
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Encoding
/-!
# First-Orde... | inj' S T h := by
simp only [SetLike.coe_set_eq, Substructure.mk.injEq] at h
exact h
map_rel_iff' {_ _} := Iff.rfl
| Mathlib/ModelTheory/Substructures.lean | 696 | 699 |
/-
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
-/
import Mathlib.RingTheory.Localization.LocalizationLocalization
/-!
# Localizations of domains as subalgebras of the fraction field.
Given a domain `A` with fra... | rintro ⟨x, rfl⟩
obtain ⟨a, s, rfl⟩ := IsLocalization.mk'_surjective S x
use a, s, s.2
apply IsLocalization.lift_mk', by
rintro ⟨a, s, hs, rfl⟩
use IsLocalization.mk' _ a ⟨s, hs⟩
apply IsLocalization.lift_mk'⟩
instance isLocalization_range_mapToFractionRing (B : Type*) [CommRing B] [Algebra ... | Mathlib/RingTheory/Localization/AsSubring.lean | 49 | 60 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finsupp.Fin
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.MvPolynomial.Rename... | section
-- this speeds up typeclass search in the lemma below
attribute [local instance] IsScalarTower.right
/-- The algebra isomorphism between multivariable polynomials in `Option S₁` and
polynomials with coefficients in `MvPolynomial S₁ R`.
-/
@[simps! -isSimp]
| Mathlib/Algebra/MvPolynomial/Equiv.lean | 339 | 347 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Data.ENat.Basic
/-!
# Trailing degree of univariate polynomials
## Main definitions
* `trailingDegree... |
@[simp]
theorem trailingDegree_one : trailingDegree (1 : R[X]) = (0 : ℕ∞) :=
trailingDegree_C one_ne_zero
@[simp]
theorem trailingDegree_X : trailingDegree (X : R[X]) = 1 :=
trailingDegree_monomial one_ne_zero
| Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 346 | 354 |
/-
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... | Mathlib/Data/Set/Function.lean | 1,449 | 1,457 | |
/-
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... |
/-- A function is `C^n` at a point, for `n : ℕ`, if and only if it is `C^n` on
a neighborhood of this point. -/
| Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 842 | 844 |
/-
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... | @[deprecated (since := "2025-02-17")] alias abs_cpow_of_imp := norm_cpow_of_imp
@[deprecated (since := "2025-02-17")] alias abs_cpow_le := norm_cpow_le
@[deprecated (since := "2025-02-17")] alias abs_cpow_real := norm_cpow_real
@[deprecated (since := "2025-02-17")] alias abs_cpow_inv_nat := norm_cpow_inv_nat
@[deprecat... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 316 | 322 |
/-
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.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | ((ι Q).comp (LinearMap.fst _ _ _) - (Algebra.linearMap R _).comp (LinearMap.snd _ _ _))
haveI f_apply : ∀ x y, f x y = (ι Q x.1 + algebraMap R _ x.2) * (ι Q y.1 - algebraMap R _ y.2) :=
fun x y => by rfl
haveI hc : ∀ (r : R) (x : CliffordAlgebra Q), Commute (algebraMap _ _ r) x := Algebra.commutes
haveI... | Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean | 137 | 141 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Lie.Weights.Killing
import Mathlib.LinearAlgebra.RootSystem.Basic
import Mathlib.LinearAlgebra.RootSystem.Reduced
import Mathlib.LinearAlgebra.RootSy... | toFun := β - β (coroot α) • α
genWeightSpace_ne_bot' := by
by_cases hα : α.IsZero
· simpa [hα.eq] using β.genWeightSpace_ne_bot
rw [sub_eq_neg_add, apply_coroot_eq_cast α β, ← neg_smul, ← Int.cast_neg,
Int.cast_smul_eq_zsmul, rootSpace_zsmul_add_ne_bot_iff α β hα]
omega
lemma reflectRoot_isNo... | Mathlib/Algebra/Lie/Weights/RootSystem.lean | 357 | 366 |
/-
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.Order.Ring.Abs
/-!
# Further lemmas about the integers
The distinction between this file and `Data.Int.Order.Basic` is not particularly clear.
... | Mathlib/Data/Int/Order/Lemmas.lean | 35 | 37 | |
/-
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.Basic
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
import Mathli... | Mathlib/Order/Interval/Set/Pi.lean | 351 | 361 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 1,083 | 1,087 | |
/-
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... | end Num
namespace PosNum
open Num
| Mathlib/Data/Num/Lemmas.lean | 686 | 690 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Eric Wieser, Joachim Breitner
-/
import Mathlib.GroupTheory.FreeGroup.Basic
/-!
# Free groups structures on arbitrary types
This file defines the notion of free basis of a gr... | simp
/-- The canonical basis of the free group over `X`. -/
| Mathlib/GroupTheory/FreeGroup/IsFreeGroup.lean | 80 | 82 |
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | · rfl
· exact Finsupp.not_mem_support_iff.mp h
| Mathlib/Algebra/Polynomial/Laurent.lean | 257 | 258 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algeb... |
/-- A product of weighted homogeneous polynomials is weighted homogeneous, with weighted degree
equal to the sum of the weighted degrees. -/
theorem prod {ι : Type*} (s : Finset ι) (φ : ι → MvPolynomial σ R) (n : ι → M) {w : σ → M} :
| Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 259 | 262 |
/-
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.Limits.Shapes.SplitCoequalizer
import Mathlib.CategoryTheory.Limits.Shapes.SplitEqualizer
import Mathlib.CategoryTheory.Limits.Preserves.Bas... |
end Equalizers
section Coequalizers
| Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean | 105 | 109 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Tactic.ByContra
/-! # Embeddings of comp... |
/-- The opposite embedding in `Embedding c.symm c'.symm` of `e : Embedding c c'`. -/
@[simps]
def op : Embedding c.symm c'.symm where
| Mathlib/Algebra/Homology/Embedding/Basic.lean | 72 | 75 |
/-
Copyright (c) 2019 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Paul Lezeau, Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
/-!
# Minimal... |
variable [IsDomain S] [NoZeroSMulDivisors R S]
variable [IsIntegrallyClosed R]
| Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 61 | 64 |
/-
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.NoZeroSMulDivisors.Basic
import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Positivity.Co... | elim b hb a₁ a₂ h := by
simp only [← hf, ← lt_iff_lt_of_le_iff_le' hf hf, zero, smul] at *
| Mathlib/Algebra/Order/Module/Defs.lean | 1,111 | 1,112 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Order.Group.Multiset
import Mathlib.Data.Setoid.Basic
import Mathlib.Data.Vector.Basic
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Tactic.Ap... | · intro s s'
obtain ⟨b, -⟩ := exists_mem s
rw [eq_replicate_of_subsingleton b s', eq_replicate_of_subsingleton b s]⟩
instance inhabitedSym [Inhabited α] (n : ℕ) : Inhabited (Sym α n) :=
⟨replicate n default⟩
instance inhabitedSym' [Inhabited α] (n : ℕ) : Inhabited (Sym' α n) :=
⟨Quotient.mk' (List... | Mathlib/Data/Sym/Basic.lean | 311 | 319 |
/-
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... | rw [image_iff]
refine h ?_
simp only [Submodule.coe_mk] at hxy
rw [hxy] at hx
rw [← image_iff hx]
simp [hxy]
theorem le_graph_of_le {f g : E →ₗ.[R] F} (h : f ≤ g) : f.graph ≤ g.graph := by
intro x hx
rw [mem_graph_iff] at hx ⊢
obtain ⟨y, hx⟩ := hx
use ⟨y, h.1 y.2⟩
simp only [hx, Submodule.coe_mk,... | Mathlib/LinearAlgebra/LinearPMap.lean | 824 | 845 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... |
@[simp]
theorem sbtw_const_vadd_iff {x y z : P} (v : V) :
Sbtw R (v +ᵥ x) (v +ᵥ y) (v +ᵥ z) ↔ Sbtw R x y z := by
| Mathlib/Analysis/Convex/Between.lean | 205 | 208 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Discrete.Basic
/-!
# Categorical (co)products
This file defines (co)products ... | simpa using w }
instance (priority := 100) hasProduct_unique [Nonempty β] [Subsingleton β] (f : β → C) :
HasProduct f :=
let ⟨_⟩ := nonempty_unique β; HasLimit.mk (limitConeOfUnique f)
| Mathlib/CategoryTheory/Limits/Shapes/Products.lean | 735 | 740 |
/-
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.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
/-!
# Regular conditional probability distribution
We define the regular con... | rw [heq, condDistrib]
refine eq_condKernel_of_measure_eq_compProd _ ?_
convert hκ
exact heq.symm
section Integrability
theorem integrable_toReal_condDistrib (hX : AEMeasurable X μ) (hs : MeasurableSet s) :
Integrable (fun a => (condDistrib Y X μ (X a)).real s) μ := by
| Mathlib/Probability/Kernel/CondDistrib.lean | 134 | 142 |
/-
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.Algebra.Algebra.Basic
import Mathlib.Algebra.Module.Submodule.Pointwise
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.Non... | /-
In this section, we check that in an `R`-algebra `A` over a ring equipped with a topology,
a basis of `R`-submodules which is compatible with the topology on `R` is also a submodule basis
in the sense of `R`-modules (forgetting about the ring structure on `A`) and those two points of
view definitionaly gives the sam... | Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean | 339 | 345 |
/-
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.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.RegularMono... | rw [←
show _ = a from
(Category.comp_id _).symm.trans
((IsKernelPair.id_of_mono f).isLimit.conePointUniqueUpToIso_inv_comp h.isLimit
WalkingCospan.left)]
infer_instance
theorem of_isIso_of_mono [IsIso a] [Mono f] : IsKernelPair f a a := by
| Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean | 199 | 206 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
/-!
# Exterior Algebras
We construct the e... | `cond : ∀ m : M, f m * f m = 0`, this is the canonical lift of `f` to a morphism of `R`-algebras
from `ExteriorAlgebra R M` to `A`.
| Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 97 | 98 |
/-
Copyright (c) 2024 Lawrence Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lawrence Wu
-/
import Mathlib.Analysis.Fourier.Inversion
import Mathlib.Analysis.MellinTransform
/-!
# Mellin inversion formula
We derive the Mellin inversion formula as a consequence o... | /-- The inverse Mellin transform of the Mellin transform applied to `x > 0` is x. -/
theorem mellin_inversion (σ : ℝ) (f : ℝ → E) {x : ℝ} (hx : 0 < x) (hf : MellinConvergent f σ)
(hFf : VerticalIntegrable (mellin f) σ) (hfx : ContinuousAt f x) :
mellinInv σ (mellin f) x = f x := by
let g := fun (u : ℝ) => Rea... | Mathlib/Analysis/MellinInversion.lean | 89 | 121 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.InducedShiftSequence
import Mathlib.CategoryTheory.Shift.Localization
import Mathlib.Algebra.Homology.HomotopyCategory.Shift
import Mathlib.... | rw [id_comp, id_comp]
-- This `erw` is required to bridge the gap between
-- `((shortComplexFunctor C (up ℤ) a').obj K).homology`
-- (the target of the first morphism)
-- and
| Mathlib/Algebra/Homology/HomotopyCategory/ShiftSequence.lean | 93 | 97 |
/-
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.Order.SuccPred.Archimedean
/-!
# Relations on types with a `SuccOrder`
This file contains properties about relations on types with a `SuccOrder`
... |
/-- For `n < m`, `(n, m)` is in the transitive closure of a relation `~` if `i ~ succ i`
for all `i` between `n` and `m`. -/
theorem transGen_of_succ_of_lt (r : α → α → Prop) {n m : α} (h : ∀ i ∈ Ico n m, r i (succ i))
| Mathlib/Order/SuccPred/Relation.lean | 40 | 43 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Determinant
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Ri... | · -- non-terminating case
calc
| Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 271 | 272 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,684 | 1,686 | |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Group.End
import Mathlib.Algebra.Module.NatInt
import Mathlib.Algebra.Order.Archimedean.Basic
/-!
# M... | @[scoped simp]
theorem map_int_add' [AddCommGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b]
(f : F) (n : ℤ) (x : G) : f (↑n + x) = f x + n • b := by
| Mathlib/Algebra/AddConstMap/Basic.lean | 223 | 225 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... | p₁ ≠ p₂ := by
rintro rfl
simp [collinear_pair] at h
| Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 517 | 520 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Topology.Algebra.InfiniteSum.NatInt
import Mathlib... |
@[to_additive]
protected theorem Multipliable.tprod_ne_one_iff (hf : Multipliable f) :
∏' i, f i ≠ 1 ↔ ∃ x, f x ≠ 1 := by
rw [Ne, hf.tprod_eq_one_iff, not_forall]
@[deprecated (since := "2025-04-12")] alias tsum_ne_zero_iff := Summable.tsum_ne_zero_iff
@[to_additive existing, deprecated (since := "2025-04-12")]... | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | 291 | 300 |
/-
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.List.TakeDrop
import Mathlib.Data.List.Induction
/-!
# Prefixes, suffixes, infixes
This file proves properties about
* `List.isPrefix`: `l₁` is ... | Mathlib/Data/List/Infix.lean | 557 | 560 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 464 | 471 | |
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Fintype.BigOperators
import Mat... | Mathlib/Algebra/BigOperators/Fin.lean | 495 | 509 | |
/-
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.Coherent
import Mathlib.Topology.UniformSpace.Equiv
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformAp... |
/-- Let `u₁`, `u₂` be two uniform structures on `γ` and `𝔖₁ 𝔖₂ : Set (Set α)`. If `u₁ ≤ u₂` and
`𝔖₂ ⊆ 𝔖₁` then `𝒱(α, γ, 𝔖₁, u₁) ≤ 𝒱(α, γ, 𝔖₂, u₂)`. -/
protected theorem mono ⦃u₁ u₂ : UniformSpace γ⦄ (hu : u₁ ≤ u₂) ⦃𝔖₁ 𝔖₂ : Set (Set α)⦄
| Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 794 | 797 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... |
theorem OrderEmbedding.wcovBy_of_apply {α β : Type*} [Preorder α] [Preorder β]
(f : α ↪o β) {x y : α} (h : f x ⩿ f y) : x ⩿ y := by
use f.le_iff_le.1 h.1
intro a
| Mathlib/Order/Cover.lean | 122 | 126 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 2,659 | 2,670 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | rfl
variable [AddMonoid M]
/-- The additive equivalence between `(α ⊕ β) →₀ M` and `(α →₀ M) × (β →₀ M)`.
This is the `Finsupp` version of `Equiv.sum_arrow_equiv_prod_arrow`. -/
@[simps! apply symm_apply]
def sumFinsuppAddEquivProdFinsupp {α β : Type*} : (α ⊕ β →₀ M) ≃+ (α →₀ M) × (β →₀ M) :=
{ sumFinsuppEquivPr... | Mathlib/Data/Finsupp/Basic.lean | 1,204 | 1,214 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power basis
This file defines a structure ... | refine sub_eq_zero.mpr ((pb.basis.linearCombination_repr (pb.gen ^ pb.dim)).symm.trans ?_)
rw [Finsupp.linearCombination_apply, Finsupp.sum_fintype] <;>
simp only [pb.coe_basis, zero_smul, eq_self_iff_true, imp_true_iff]
theorem minpolyGen_monic (pb : PowerBasis A S) : Monic (minpolyGen pb) := by
| Mathlib/RingTheory/PowerBasis.lean | 163 | 167 |
/-
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... | let (i, ⟨b', _⟩) ← addAtomQ q($a⁻¹)
pure ⟨b', ExBase.atom i, q(Eq.refl $b')⟩
| Mathlib/Tactic/Ring/Basic.lean | 980 | 981 |
/-
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... | UniformEquicontinuousOn F S ↔ UniformContinuousOn (ofFun ∘ Function.swap F : β → ι →ᵤ α) S := by
rw [UniformContinuousOn, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff]
rfl
| Mathlib/Topology/UniformSpace/Equicontinuity.lean | 529 | 531 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | theorem sameRay_neg_smul_right_iff_of_ne {v : M} {r : R} (hv : v ≠ 0) (hr : r ≠ 0) :
SameRay R (-v) (r • v) ↔ r < 0 := by
simp only [sameRay_neg_smul_right_iff, hv, or_false, hr.le_iff_lt]
| Mathlib/LinearAlgebra/Ray.lean | 483 | 485 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
/-!
# Stalks for presheaved spaces
... | def restrictStalkIso {U : TopCat} (X : PresheafedSpace.{_, _, v} C) {f : U ⟶ (X : TopCat.{v})}
(h : IsOpenEmbedding f) (x : U) : (X.restrict h).presheaf.stalk x ≅ X.presheaf.stalk (f x) :=
haveI := initial_of_adjunction (h.isOpenMap.adjunctionNhds x)
Final.colimitIso (h.isOpenMap.functorNhds x).op ((OpenNhds.in... | Mathlib/Geometry/RingedSpace/Stalks.lean | 56 | 59 |
/-
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... | rw [← lt_div_iff₀ this, div_div_cancel₀ h.ne']
exact lt_max_iff.2 (Or.inl <| lt_add_one _)
| Mathlib/Algebra/Order/Field/Basic.lean | 199 | 200 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | theorem separable_gcd_left {F : Type*} [Field F] [DecidableEq F[X]]
{f : F[X]} (hf : f.Separable) (g : F[X]) :
(EuclideanDomain.gcd f g).Separable :=
| Mathlib/FieldTheory/Separable.lean | 92 | 94 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.Algebra
impo... |
theorem descPochhammer_zero_eval_zero : (descPochhammer R 0).eval 0 = 1 := by simp
@[simp]
| Mathlib/RingTheory/Polynomial/Pochhammer.lean | 284 | 287 |
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