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/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
/-!
# Composition of analytic functions
In this fi... |
/-- The norm of `f.compAlongComposition p c` is controlled by the product of
the norms of the relevant bits of `f` and `p`. -/
theorem compAlongComposition_bound {n : ℕ} (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n)
(f : F [×c.length]→L[𝕜] G) (v : Fin n → E) :
‖f.compAlongComposition p c v‖ ≤ (‖f‖ ... | Mathlib/Analysis/Analytic/Composition.lean | 284 | 291 |
/-
Copyright (c) 2022 Niels Voss. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Niels Voss
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
/-!
# Fermat Pseudoprimes
In this file we define... | · intro h₁
apply Nat.modEq_of_dvd
exact mod_cast h₁
· intro h₁
exact mod_cast Nat.ModEq.dvd h₁
/-- If `n` is a Fermat pseudoprime to base `b`, then `n` is coprime with `b`, assuming that `b` is
positive.
This lemma is a small wrapper based on `coprime_of_probablePrime`
-/
| Mathlib/NumberTheory/FermatPsp.lean | 102 | 112 |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | (∫ x in a..b, cos x ^ (n + 2)) =
(cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a) / (n + 2) +
(n + 1) / (n + 2) * ∫ x in a..b, cos x ^ n := by
field_simp
convert eq_sub_iff_add_eq.mp (integral_cos_pow_aux n) using 1
ring
@[simp]
theorem integral_cos_sq : ∫ x in a..b, cos x ^ 2 = (cos b * sin... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 717 | 738 |
/-
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, Chris Hughes, Daniel Weber
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.GroupWithZero.Associated
import Mathlib.Algebra.Ring.Divisibility.Basic
import Math... | norm_cast
@[deprecated (since := "2024-11-30")]
alias multiplicity.Finite.not_pow_dvd_of_multiplicity_lt :=
FiniteMultiplicity.not_pow_dvd_of_multiplicity_lt
theorem multiplicity_pos_of_dvd (hdiv : a ∣ b) : 0 < multiplicity a b := by
| Mathlib/RingTheory/Multiplicity.lean | 279 | 285 |
/-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Antidiag.Pi
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
im... | variable {α R : Type*} [DecidableEq α]
section Semiring
variable [Semiring R]
open scoped Function -- required for scoped `on` notation
-- TODO: Can we prove one of the following two from the other one?
/-- The **multinomial theorem**. -/
lemma sum_pow_eq_sum_piAntidiag_of_commute (s : Finset α) (f : α → R)
| Mathlib/Data/Nat/Choose/Multinomial.lean | 203 | 212 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... |
section CommRing
| Mathlib/NumberTheory/ArithmeticFunction.lean | 1,093 | 1,094 |
/-
Copyright (c) 2022 Mantas Bakšys. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Set.Image
import Mathlib.Da... |
/-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
| Mathlib/Algebra/Order/Rearrangement.lean | 319 | 321 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | (hf : QuasiMeasurePreserving f μ ν) (h : g =ᵐ[ν] g') : g ∘ f =ᵐ[μ] g' ∘ f :=
ae_eq_comp' hf.aemeasurable h hf.absolutelyContinuous
theorem ae_eq_comp {f : α → β} {g g' : β → δ} (hf : AEMeasurable f μ) (h : g =ᵐ[μ.map f] g') :
| Mathlib/MeasureTheory/Measure/Restrict.lean | 624 | 627 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.Complex.Convex
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.Calculus.Deriv.Shift
/-!
# Estimates for the complex ... | lemma continuousOn_one_add_mul_inv {z : ℂ} (hz : 1 + z ∈ slitPlane) :
ContinuousOn (fun t : ℝ ↦ (1 + t • z)⁻¹) (Set.Icc 0 1) :=
ContinuousOn.inv₀ (by fun_prop)
(fun _ ht ↦ slitPlane_ne_zero <| StarConvex.add_smul_mem starConvex_one_slitPlane hz ht.1 ht.2)
| Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | 31 | 34 |
/-
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.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
impor... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 369 | 380 | |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Adjoint of operators on Hilbert spaces
Given ... | theorem _root_.orthogonalProjection_isSelfAdjoint (U : Submodule 𝕜 E) [CompleteSpace U] :
IsSelfAdjoint (U.subtypeL ∘L U.orthogonalProjection) :=
U.orthogonalProjection_isSymmetric.isSelfAdjoint
theorem conj_orthogonalProjection {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) (U : Submodule 𝕜 E)
| Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 274 | 278 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.MeasurableSpace.Defs
/-!
# σ-algebra of sets invariant under a self-map
In this file we define `MeasurableSpace.invariants (f : α → α)... | theorem le_invariants_iterate (f : α → α) (n : ℕ) :
invariants f ≤ invariants (f^[n]) := by
induction n with
| zero => simp [invariants_le]
| succ n ihn => exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _)
| Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean | 50 | 54 |
/-
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.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... | /-- Special case of the correspondence theorem for isomorphic rings -/
def relIsoOfBijective : Ideal S ≃o Ideal R where
toFun := comap f
invFun := map f
left_inv := map_comap_of_surjective _ hf.2
right_inv J :=
le_antisymm
(fun _ h ↦ have ⟨y, hy, eq⟩ := (mem_map_iff_of_surjective _ hf.2).mp h; hf.1 eq... | Mathlib/RingTheory/Ideal/Maps.lean | 455 | 467 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.CauSeq.Completion
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Rat.Cast.Defs
/-!
# Real numbers from Cauc... | le_antisymm a b := by
induction a using Real.ind_mk
induction b using Real.ind_mk
| Mathlib/Data/Real/Basic.lean | 327 | 329 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.SetSemiring
import Mathlib.MeasureTheory.OuterMeasure.Induced
/-!
# Additive Contents
An additive content `m` on a set ... | m s ≤ m t := by
have h := sum_addContent_le_of_subset (m := m) hC (I := {s}) ?_ ?_ ht ?_
· simpa only [sum_singleton] using h
· rwa [singleton_subset_set_iff]
· simp only [coe_singleton, pairwiseDisjoint_singleton]
· simp [hst]
| Mathlib/MeasureTheory/Measure/AddContent.lean | 152 | 157 |
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
/-!
# ω-limits
For a function `ϕ : τ → α → β` where `β` is a topological space, we
define the ω-limit under `ϕ` of... | ϕ' t (ga x) ∈ image2 ϕ' u s' := mem_image2_of_mem ht.1 (hs hx)
_ = gb (ϕ t x) := ht.2 hx |>.symm
theorem mapsTo_omegaLimit {α' β' : Type*} [TopologicalSpace β'] {f : Filter τ} {ϕ : τ → α → β}
{ϕ' : τ → α' → β'} {ga : α → α'} {s' : Set α'} (hs : MapsTo ga s s') {gb : β → β'}
(hg : ∀ t x, gb (ϕ t x) = ϕ'... | Mathlib/Dynamics/OmegaLimit.lean | 89 | 98 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Gabriel Ebner, Yury Kudryashov
-/
import Mathlib.Data.Set.Accumulate
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Int... | · assumption
· rintro ⟨H, H'⟩
rw [sInf_def (⟨_, H⟩ : s.Nonempty), find_eq_iff]
exact ⟨H, fun n hnk hns ↦ H' <| hs n k (Nat.lt_succ_iff.mp hnk) hns⟩
/-- This instance is necessary, otherwise the lattice operations would be derived via
`ConditionallyCompleteLinearOrderBot` and marked as noncomputable. -/
i... | Mathlib/Data/Nat/Lattice.lean | 110 | 120 |
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.... | (selfAdjointEquiv z : ℂ) = z := by
simpa [selfAdjointEquiv_symm_apply]
using (congr_arg Subtype.val <| Complex.selfAdjointEquiv.left_inv z)
| Mathlib/Data/Complex/Module.lean | 474 | 476 |
/-
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.Exact
import Mathlib.CategoryTheory.ComposableArrows
/-!
# Exact sequences
A sequence of `n` composable arrows `S : ComposableArr... | toIsComplex := S.isComplex₀
-- See https://github.com/leanprover/lean4/issues/2862
exact i hi := by simp at hi
| Mathlib/Algebra/Homology/ExactSequence.lean | 188 | 190 |
/-
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 Codisjoint.himp_eq_right (h : Codisjoint a b) : b ⇨ a = a := by
conv_rhs => rw [← @top_himp _ _ a]
rw [← h.eq_top, sup_himp_self_left]
| Mathlib/Order/Heyting/Basic.lean | 326 | 328 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic
/-!
# Minimal polynomials
This file defines the minimal polynomial of an element `x` of an `A... |
section CommRing
variable [CommRing A]
section Ring
variable [Ring B] [Algebra A B]
| Mathlib/FieldTheory/Minpoly/Basic.lean | 170 | 177 |
/-
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 | 360 | 360 | |
/-
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... | Mathlib/Data/Nat/Factorization/Basic.lean | 802 | 808 | |
/-
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.FieldTheory.SplittingField.Construction
import Mathlib.RingTheory.Localization.Integral
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
impo... | g ∈ lifts (algebraMap (integralClosure R K) K) := by
have := mem_lift_of_splits_of_roots_mem_range (integralClosure R g.SplittingField)
((splits_id_iff_splits _).2 <| SplittingField.splits g) (hg.map _) fun a ha =>
(SetLike.ext_iff.mp (integralClosure R g.SplittingField).range_algebraMap _).mpr <|
... | Mathlib/RingTheory/Polynomial/GaussLemma.lean | 54 | 70 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 531 | 532 | |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.MorphismProperty.Composition
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
import Mathlib.CategoryTheory.Category.Quiv
/-!
# Const... | toFun := W.Q.obj
invFun X := X.as.obj
left_inv _ := rfl
right_inv := by
rintro ⟨⟨X⟩⟩
rfl
/-- A `MorphismProperty` in `W.Localization` is satisfied by all
morphisms in the localized category if it contains the image of the
morphisms in the original category, the inverses of the morphisms
in `W` and if i... | Mathlib/CategoryTheory/Localization/Construction.lean | 192 | 209 |
/-
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.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | hrm <|
this.radical.symm ▸ (sInf_le ⟨hIm, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr
theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] :
(⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn
| Mathlib/RingTheory/Ideal/Operations.lean | 856 | 860 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.FieldTheory.Perfect
import Mathlib.LinearAlgebra.AnnihilatingPolynomial
import Mathlib.RingTheory.Artinian.Instances
im... | @[simp] lemma isSemisimple_neg : (-f).IsSemisimple ↔ f.IsSemisimple := by
simp [isSemisimple_iff, mem_invtSubmodule]
variable (f) in
| Mathlib/LinearAlgebra/Semisimple.lean | 114 | 117 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | open Lean Meta Qq
/-- Extension for the `positivity` tactic: `ENNReal.ofReal`. -/
| Mathlib/Data/ENNReal/Real.lean | 462 | 464 |
/-
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... | theorem le_trailingDegree_C : (0 : ℕ∞) ≤ trailingDegree (C a) :=
le_trailingDegree_monomial
| Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 183 | 185 |
/-
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... | | zero => simp
| one => simp
| Mathlib/Algebra/Ring/Parity.lean | 371 | 372 |
/-
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 ... | (pair right ((@Primcodable.prim β).comp left)))).comp
(pair right ((@Primcodable.prim α).comp left))).of_eq
fun n => by
simp only [Nat.unpaired, Nat.unpair_pair, decode_prod_val]
cases @decode α _ n.unpair.1 <;> simp
| Mathlib/Computability/Primrec.lean | 297 | 301 |
/-
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.... | (homologyFunctorIso C (ComplexShape.up ℤ) a').symm
lemma shiftIso_hom_app (n a a' : ℤ) (ha' : n + a = a') (K : CochainComplex C ℤ) :
(shiftIso C n a a' ha').hom.app K =
ShortComplex.homologyMap ((shiftShortComplexFunctorIso C n a a' ha').hom.app K) := by
| Mathlib/Algebra/Homology/HomotopyCategory/ShiftSequence.lean | 87 | 91 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Peter Nelson
-/
import Mathlib.Order.Antichain
/-!
# Minimality and Maximality
This file proves basic facts about minimality and maximality
of an element with respect to ... | theorem Minimal.not_prop_of_lt (h : Minimal P x) (hlt : y < x) : ¬ P y :=
(minimal_iff_forall_lt.1 h).2 hlt
theorem Maximal.not_prop_of_gt (h : Maximal P x) (hlt : x < y) : ¬ P y :=
(maximal_iff_forall_gt.1 h).2 hlt
theorem Minimal.not_lt (h : Minimal P x) (hy : P y) : ¬ (y < x) :=
| Mathlib/Order/Minimal.lean | 173 | 179 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... | exact Arrow.isoMk S.opcyclesIsoCokernel.symm (Iso.refl _) (by aesop_cat)
lemma QuasiIso.exact_iff {S₁ S₂ : ShortComplex C} (φ : S₁ ⟶ S₂)
[S₁.HasHomology] [S₂.HasHomology] [QuasiIso φ] : S₁.Exact ↔ S₂.Exact := by
simp only [exact_iff_isZero_homology]
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 398 | 402 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Topology.Category.CompHausLike.Limits
/-!
# Explicit limits and colimits
This file applies... | Mathlib/Topology/Category/Profinite/Limits.lean | 213 | 217 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 881 | 882 | |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Basic
/-!
# Intervals as multisets
This file defines intervals as multisets.
## Main declarations
In a `LocallyFiniteOrder`,
* `M... | def Ici (a : α) : Multiset α := (Finset.Ici a).val
| Mathlib/Order/Interval/Multiset.lean | 73 | 73 |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... |
theorem length_simple_mul_ne (w : W) (i : B) : ℓ (s i * w) ≠ ℓ w := by
convert cs.length_mul_simple_ne w⁻¹ i using 1
· convert cs.length_inv ?_ using 2
simp
· simp
theorem length_mul_simple (w : W) (i : B) :
ℓ (w * s i) = ℓ w + 1 ∨ ℓ (w * s i) + 1 = ℓ w := by
rcases Nat.lt_or_gt_of_ne (cs.length_mul_s... | Mathlib/GroupTheory/Coxeter/Length.lean | 177 | 191 |
/-
Copyright (c) 2021 Filippo A. E. Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Linear... | det (A.map fun a => f a 1) ^ Fintype.card n * det (B.map fun b => f 1 b) ^ Fintype.card m :=
calc
det (kroneckerMapBilinear f A B) =
det (kroneckerMapBilinear f A 1 * kroneckerMapBilinear f 1 B) := by
rw [← kroneckerMapBilinear_mul_mul f h_comm, Matrix.mul_one, Matrix.one_mul]
_ = det (blo... | Mathlib/Data/Matrix/Kronecker.lean | 225 | 230 |
/-
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 ... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 530 | 531 | |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | have := embedding_mk_eq φ
rwa [ComplexEmbedding.isReal_iff.mp h, or_self] at this
theorem isReal_iff {w : InfinitePlace K} :
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 372 | 375 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | Mathlib/Analysis/Convex/Segment.lean | 650 | 654 | |
/-
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... | theorem isSuccPrelimit_iff_isSuccLimit [NoMinOrder α] : IsSuccPrelimit a ↔ IsSuccLimit a :=
isSuccPrelimit_iff_isSuccLimit_of_not_isMin (not_isMin a)
| Mathlib/Order/SuccPred/Limit.lean | 84 | 86 |
/-
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.Control.EquivFunctor
import Mathlib.Data.Option.Basic
import Mathlib.Data.Subtype
import Mathlib.Logic.Equiv.Defs
/-!
# Equivalences for `Option α`
We def... | @[simps]
| Mathlib/Logic/Equiv/Option.lean | 276 | 276 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.Order.Lattice
import Mathlib.Order.PartialSups
/-!
# Continuity of `partialSups`
In this file we prove that `partialSups` of a sequence of... | protected lemma ContinuousAt.partialSups (hf : ∀ k ≤ n, ContinuousAt (f k) x) :
ContinuousAt (partialSups f n) x := by
simpa only [← Pi.partialSups_apply] using ContinuousAt.partialSups_apply hf
| Mathlib/Topology/Order/PartialSups.lean | 43 | 45 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
/-!
# Nondeterministic Finite Automata
This file contains the definition of a Nondeterministic Finite... |
@[simp]
| Mathlib/Computability/NFA.lean | 52 | 53 |
/-
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.Analysis.Complex.AbsMax
import Mathlib.Analysis.Asymptotics.SuperpolynomialDecay
/-!
# Phragmen-Lindelöf principle
In this file we prove several ve... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 850 | 864 | |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | rw [mul_comm]
exact (Nat.pred_mul_geom_sum_le a b n).trans tsub_le_self
theorem Nat.geom_sum_Ico_le {b : ℕ} (hb : 2 ≤ b) (a n : ℕ) :
∑ i ∈ Ico 1 n, a / b ^ i ≤ a / (b - 1) := by
rcases n with - | n
· rw [Ico_eq_empty_of_le (zero_le_one' ℕ), sum_empty]
exact Nat.zero_le _
rw [← add_le_add_iff_left a]
... | Mathlib/Algebra/GeomSum.lean | 444 | 459 |
/-
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.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | /-- In the characteristic zero case, the weights in the centroid
indexed by a `Fintype` sum to 1 if the set is nonempty. -/
theorem sum_centroidWeightsIndicator_eq_one_of_nonempty [CharZero k] [Fintype ι] (h : s.Nonempty) :
∑ i, s.centroidWeightsIndicator k i = 1 := by
rw [sum_centroidWeightsIndicator]
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 838 | 842 |
/-
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... |
end Lattice
| Mathlib/Order/Interval/Finset/Basic.lean | 1,010 | 1,012 |
/-
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... | 𝓐 ×ˢ (⊤ : Filter α) :=
forall₂_congr fun U _hU => mem_prod_top.symm
_ ↔ lowerAdjoint 𝓐 ≤ 𝓕 := Iff.rfl
variable [UniformSpace β]
/-- Core of the uniform structure of uniform convergence. -/
protected def uniformCore : UniformSpace.Core (α →ᵤ β) :=
UniformSpace.Core.mkOfBasis (UniformFun.basi... | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 263 | 278 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 930 | 931 | |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Mario Carneiro, Reid Barton, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Func... | ℱ.map (eqToHom (by simp [Opens.map, Set.preimage_preimage])) ≫
H₂.app (op ((Opens.map H₁.inv).obj (unop U))) := by
simp [toPushforwardOfIso, Adjunction.homEquiv_unit]
| Mathlib/Topology/Sheaves/Presheaf.lean | 238 | 241 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... |
/-- If `p` is a prime natural number and `x` is an integer number, then there exist natural numbers
`a ≤ p / 2` and `b ≤ p / 2` such that `a ^ 2 + b ^ 2 ≡ x [ZMOD p]`. This is a version of
| Mathlib/FieldTheory/Finite/Basic.lean | 488 | 490 |
/-
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 | 390 | 392 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | variable {α β : Type*} {m : OuterMeasure α}
/-- If `s : ι → Set α` is a sequence of sets, `S = ⋃ n, s n`, and `m (S \ s n)` tends to zero along
some nontrivial filter (usually `atTop` on `ι = ℕ`), then `m S = ⨆ n, m (s n)`. -/
theorem iUnion_of_tendsto_zero {ι} (m : OuterMeasure α) {s : ι → Set α} (l : Filter ι) [NeBo... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 162 | 167 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Module.ULift
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Tactic.Ring
/-!
# The characteristic predicate of tensor product
## Main ... | convert_to Function.Bijective (LinearMap.id : M ⊗[R] N →ₗ[R] M ⊗[R] N) using 2
· apply TensorProduct.ext'
simp
· exact Function.bijective_id
variable {R M N}
| Mathlib/RingTheory/IsTensorProduct.lean | 60 | 65 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | · rintro ⟨a, ha, rfl⟩
exact ⟨subset_insert _ _, (card_insert_of_not_mem ha).symm⟩
· rintro ⟨hst, h⟩
obtain ⟨a, ha⟩ : ∃ a, t \ s = {a} :=
| Mathlib/Data/Finset/Card.lean | 599 | 602 |
/-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Analytic.CPolynomialDef
/-! # Properties of continuously polynomial functions
We expand the API a... | Mathlib/Analysis/Analytic/CPolynomial.lean | 256 | 263 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.InfiniteSum.Defs
imp... | rw [tprod_eq_finprod (s.finite_toSet.subset hf), finprod_eq_prod_of_mulSupport_subset _ hf]
@[to_additive]
theorem tprod_eq_prod {s : Finset β} (hf : ∀ b ∉ s, f b = 1) :
∏' b, f b = ∏ b ∈ s, f b :=
tprod_eq_prod' <| mulSupport_subset_iff'.2 hf
| Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 376 | 381 |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
import Mathlib.Algebra.Star.Rat
/-!
# Self-adjoint, sk... | @[simp, norm_cast]
theorem val_smul [Monoid R] [DistribMulAction R A] [StarModule R A] (r : R) (x : skewAdjoint A) :
| Mathlib/Algebra/Star/SelfAdjoint.lean | 534 | 535 |
/-
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.Order.RelClasses
import Mathlib.Data.List.Basic
/-!
# Lexicographic ordering of lists.
The lexicographic order on `List α` is defined by `L < M` iff
... | Mathlib/Data/List/Lex.lean | 177 | 179 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... | lemma lt_unbot_iff (hy : y ≠ ⊥) : a < unbot y hy ↔ a < y := by lift y to α using id hy; simp
lemma unbot_lt_iff (hx : x ≠ ⊥) : unbot x hx < b ↔ x < b := by lift x to α using id hx; simp
lemma unbotD_lt_iff (hx : x = ⊥ → a < b) : x.unbotD a < b ↔ x < b := by cases x <;> simp [hx]
| Mathlib/Order/WithBot.lean | 262 | 264 |
/-
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 | 3,159 | 3,162 | |
/-
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... | rwa [Nat.mul_sub_right_distrib, one_mul, ← offDiag_card] at this
have := hP.card_part_le_average_add_one hU
refine Nat.mul_le_mul ((Nat.sub_le_sub_right this 1).trans ?_) this
simp only [Nat.add_succ_sub_one, add_zero, card_univ, le_rfl]
lemma IsEquipartition.card_biUnion_offDiag_le (hε : 0 < ε) (hP : P.IsEq... | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 327 | 332 |
/-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel_right, neg_neg,
mul_one, neg_add_cancel, mul_zero, zero_add] at this
conv_lhs at this => arg 1; intro x; rw [add_comm, ← sub_eq_add_neg, mul_comm]
exact this
theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaI... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 93 | 102 |
/-
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,
Kim Morrison
-/
import Mathlib.Data.List.Basic
/-!
# Lattice structure of lists
This files pro... |
@[simp]
| Mathlib/Data/List/Lattice.lean | 109 | 110 |
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.WittVector.DiscreteValuationRing
/-!
#... | nthRemainder p n bs (truncateFun (n + 1) a₂) := by
rw [← sub_eq_zero]
have := succNthVal_spec p n a₁ a₂ bs ha₁ ha₂
simp only [Polynomial.map_add, Polynomial.eval_X, Polynomial.map_pow, Polynomial.eval_C,
Polynomial.eval_pow, succNthDefiningPoly, Polynomial.eval_mul, Polynomial.eval_add,
Polynomial... | Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean | 123 | 138 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.ConjTranspose
/-!
# Row and column matrices
This file provides results about row and column matrices.
## Main definitions
* `Mat... | theorem updateRow_apply [DecidableEq m] {i' : m} :
updateRow M i b i' j = if i' = i then b j else M i' j := by
by_cases h : i' = i
· rw [h, updateRow_self, if_pos rfl]
| Mathlib/Data/Matrix/RowCol.lean | 261 | 264 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Order.Group.Pointwise.Bounds
import Mathlib.Data.Real.Basic
import Mathlib.Ord... | le_csSup s a hs ha := (Real.isLUB_sSup ⟨a, ha⟩ hs).1 ha
csSup_le s a hs ha := (Real.isLUB_sSup hs ⟨a, ha⟩).2 ha
csInf_le s a hs ha := (Real.isGLB_sInf ⟨a, ha⟩ hs).1 ha
le_csInf s a hs ha := (Real.isGLB_sInf hs ⟨a, ha⟩).2 ha
| Mathlib/Data/Real/Archimedean.lean | 138 | 141 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... | refine Submartingale.tendsto_eLpNorm_one_limitProcess ?_ ?_
· exact (martingale_densityProcess hκν a hs).submartingale
· refine uniformIntegrable_of le_rfl ENNReal.one_ne_top ?_ ?_
· exact fun n ↦ (measurable_densityProcess_right κ ν n a hs).aestronglyMeasurable
· refine fun ε _ ↦ ⟨2, fun n ↦ le_of_eq_of_... | Mathlib/Probability/Kernel/Disintegration/Density.lean | 400 | 409 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... | mul_nonsing_inv_cancel_left A B (isUnit_det_of_invertible A)
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 253 | 254 |
/-
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.CondexpL1
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` ... | swap; · simp_rw [condExp_of_not_sigmaFinite hm hμm]; simp
refine (condExp_ae_eq_condExpL1 hm _).trans ?_
rw [condExpL1_add hf hg]
exact (coeFn_add _ _).trans
((condExp_ae_eq_condExpL1 hm _).symm.add (condExp_ae_eq_condExpL1 hm _).symm)
| Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 327 | 332 |
/-
Copyright (c) 2021 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.Filtration
import Mathlib.Topology.Instances.Discrete
/-!
# Adapted and progressively measurable processes
This file defi... | specialize h_tendsto x.fst
rw [tendsto_nhds] at h_tendsto ⊢
exact fun s hs h_mem => h_tendsto {g | g x.snd ∈ s} (hs.preimage (continuous_apply x.snd)) h_mem
| Mathlib/Probability/Process/Adapted.lean | 164 | 167 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
/-!
# Conve... | ∑ k ∈ range (n + 1), (u (k + 1) - u k) • f (u k) ≤
(u 1 - u 0) • f (u 0) + C • ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k := by
rw [sum_range_succ', add_comm]
gcongr
suffices ∑ k ∈ range n, (u (k + 2) - u (k + 1)) • f (u (k + 1)) ≤
C • ∑ k ∈ range n, ((u (k + 1) - u k) • f (u (k + 1))) by
refine this.trans ... | Mathlib/Analysis/PSeries.lean | 107 | 125 |
/-
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.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... | lemma card_range_of_injective {f : α → β} (hf : Injective f) :
Nat.card (range f) = Nat.card α := by
rw [← Nat.card_preimage_of_injective hf le_rfl]
simp
| Mathlib/SetTheory/Cardinal/Finite.lean | 167 | 170 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 484 | 484 | |
/-
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... | <;> simp only [PosNum.testBit]
· rfl
| Mathlib/Data/Num/Lemmas.lean | 859 | 860 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the langua... | simp [term, h]
theorem term_atom_pfg {α} [AddCommGroup α] (x x' : α) (h : x = x') : x = termg 1 x' 0 := by
simp [termg, h]
| Mathlib/Tactic/Abel.lean | 274 | 276 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jung Tao Cheng, Christian Merten, Andrew Yang
-/
import Mathlib.LinearAlgebra.TensorProduct.RightExactness
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Gene... | def ofFinitePresentation [FinitePresentation R S] : Presentation.{0, 0} R S :=
letI H := FinitePresentation.out (R := R) (A := S)
letI n : ℕ := H.choose
letI f : MvPolynomial (Fin n) R →ₐ[R] S := H.choose_spec.choose
haveI hf : Function.Surjective f := H.choose_spec.choose_spec.1
haveI hf' : (RingHom.ker f).F... | Mathlib/RingTheory/Presentation.lean | 132 | 148 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Hilbert sum of ... | -- Note : this should be `b.repr` composed with an identification of `lp (fun i : ι => 𝕜) p` with
-- `PiLp p (fun i : ι => 𝕜)` (in this case with `p = 2`), but we don't have this yet (July 2022).
/-- An orthonormal basis is a Hilbert basis. -/
protected def _root_.OrthonormalBasis.toHilbertBasis [Fintype ι] (b : Orth... | Mathlib/Analysis/InnerProductSpace/l2Space.lean | 534 | 537 |
/-
Copyright (c) 2022 Filippo A. E. Nuccio Mortarino Majno di Capriglio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Junyan Xu
-/
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Homotopy.Basic
import Mathlib.Topology.Path
/-!
# H-s... | instance (x : X) : HSpace (Path x x) where
hmul := ⟨fun ρ => ρ.1.trans ρ.2, continuous_trans⟩
e := refl x
hmul_e_e := refl_trans_refl
eHmul :=
{ toHomotopy :=
⟨⟨fun p : I × Path x x ↦ delayReflLeft p.1 p.2, continuous_delayReflLeft⟩,
delayReflLeft_zero, delayReflLeft_one⟩
| Mathlib/Topology/Homotopy/HSpaces.lean | 236 | 243 |
/-
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 only [eq_self_iff_true, true_and]
rfl⟩⟩
one_mul a :=
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 607 | 609 |
/-
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.Filter.Prod
/-!
# N-ary maps of filter
This file defines the binary and ternary maps of filters. This is mostly useful to define pointwise
operatio... | map₂ m (f.map n) g = map₂ (fun a b => m (n a) b) f g := by
rw [← map_prod_eq_map₂, ← map_prod_eq_map₂, ← @map_id _ g, prod_map_map_eq, map_map, map_id]; rfl
| Mathlib/Order/Filter/NAry.lean | 142 | 143 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.Order.Group.Multiset
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
/-!
# GCD... | end gcd
end Multiset
| Mathlib/Algebra/GCDMonoid/Multiset.lean | 213 | 215 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | theorem inv_zetaUnit : ((zetaUnit⁻¹ : (ArithmeticFunction R)ˣ) : ArithmeticFunction R) = μ :=
rfl
| Mathlib/NumberTheory/ArithmeticFunction.lean | 1,112 | 1,114 |
/-
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
... | · rw [← Category.assoc, congr_point α x ((α ≫ β).base x) h_eq.symm, Category.assoc]
erw [← stalkMap.comp β α (α.base x)]
rw [congr_hom _ _ (IsIso.inv_hom_id α), stalkMap.id, eqToHom_trans_assoc, eqToHom_refl,
Category.id_comp]
· rw [Category.assoc, ← stalkMap.comp, congr_hom _ _ (IsIso.hom_i... | Mathlib/Geometry/RingedSpace/Stalks.lean | 171 | 177 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | Mathlib/Topology/Constructions.lean | 1,513 | 1,524 | |
/-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Logic.Small.Set
import Mathlib.CategoryTheory.Filtered.Final
/-!
# Finally small categories
A category given by `(J : Type u) [Category.{v} J]` is `w`-... |
variable (J : Type u) [Category.{v} J]
/-- The converse is true if `J` is filtered, see `finallySmall_of_small_weakly_terminal_set`. -/
theorem FinallySmall.exists_small_weakly_terminal_set [FinallySmall.{w} J] :
| Mathlib/CategoryTheory/Limits/FinallySmall.lean | 132 | 136 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
/-!
# One-dimensional iterated derivatives
We define the `n`-th de... | advantage of avoiding the discussion of continuity in the proof (and for `n = ∞` this is optimal).
-/
theorem contDiffOn_of_differentiableOn_deriv {n : ℕ∞}
(h : ∀ m : ℕ, (m : ℕ∞) ≤ n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s) :
ContDiffOn 𝕜 n f s := by
apply contDiffOn_of_differentiableOn
simpa o... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 128 | 134 |
/-
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... |
/-- This version of `rpow_le_rpow_of_exponent_ge` allows `x = 0` but requires `0 ≤ z`.
See also `rpow_le_rpow_of_exponent_ge_of_imp` for the most general version. -/
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 717 | 719 |
/-
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... | /-- Auxiliary lemmas for the "monotonicity on a fundamental interval implies monotonicity" lemmas.
We formulate it for any relation so that the proof works both for `Monotone` and `StrictMono`. -/
protected theorem rel_map_of_Icc [AddCommGroup G] [LinearOrder G] [IsOrderedAddMonoid G]
[Archimedean G] [AddGroup H]
... | Mathlib/Algebra/AddConstMap/Basic.lean | 237 | 276 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | conv_rhs => rw [hf]
rw [seqOfForallFinsetExistsAux]
apply
| Mathlib/Data/Fintype/Basic.lean | 268 | 270 |
/-
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.HomotopyCategory.Shift
/-! Shifting cochains
Let `C` be a preadditive category. Gi... | (a * n' + (a * (a - 1)) / 2).negOnePow • γ.shift a := by
ext p q hpq
simp only [rightShift_v _ a n hn' p q hpq (q + a) (by omega),
leftShift_v _ a n' hn' p (q + a) (by omega) (p + a) (by omega), units_smul_v, shift_v']
dsimp
rw [id_comp, comp_id]
/-- The left and right shift of cochains commute only ... | Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | 479 | 486 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Nailin Guan, Yi Song, Xuchun Li
-/
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.RingTheory.Ideal.Defs
import Mathlib.Topology.Algebra.Group.Quotient
i... |
@[deprecated (since := "2025-03-11")]
alias _root_.OpenAddSubgroup.coe_sum := OpenAddSubgroup.coe_prod
@[to_additive (attr := simp, norm_cast) toAddSubgroup_prod]
theorem toSubgroup_prod (U : OpenSubgroup G) (V : OpenSubgroup H) :
(U.prod V : Subgroup (G × H)) = (U : Subgroup G).prod V :=
rfl
| Mathlib/Topology/Algebra/OpenSubgroup.lean | 168 | 176 |
/-
Copyright (c) 2020 Jean Lo, Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yury Kudryashov
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Ring.Action.Pointwise.Set
import Mathlib.Topology.Bornology.Basic... | lemma Filter.mem_absorbing : s ∈ absorbing G₀ u ↔ Absorbs G₀ s u := .rfl
lemma Set.Finite.absorbs_sInter (hS : S.Finite) :
| Mathlib/Topology/Bornology/Absorbs.lean | 165 | 167 |
/-
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.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,304 | 1,322 | |
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Control.Functor
import Mathlib.Control.Basic
/-!
# `applicative` instances
This file provides `Applicative` instances for... | funext x y
rw [commutative_map]
congr
| Mathlib/Control/Applicative.lean | 123 | 126 |
/-
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.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import Mathli... |
theorem mem_chartAt_source_of_eq_one {i : ι} {x : M} (h : fs i x = 1) :
x ∈ (chartAt H (fs.c i)).source :=
(fs i).support_subset_source <| by simp [h]
theorem mem_extChartAt_source_of_eq_one {i : ι} {x : M} (h : fs i x = 1) :
x ∈ (extChartAt I (fs.c i)).source := by
rw [extChartAt_source]; exact fs.mem_ch... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 410 | 430 |
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