Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 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.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | Mathlib/Algebra/Polynomial/Roots.lean | 819 | 823 | |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import M... | theorem Metric.cobounded_eq_cocompact [ProperSpace α] : cobounded α = cocompact α := by
nontriviality α; inhabit α
exact cobounded_le_cocompact.antisymm <| (hasBasis_cobounded_compl_closedBall default).ge_iff.2
fun _ _ ↦ (isCompact_closedBall _ _).compl_mem_cocompact
theorem tendsto_dist_right_cocompact_atTop ... | Mathlib/Topology/MetricSpace/Bounded.lean | 531 | 537 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | generalize hx : Nat.digitChar (n % b) = x
simp only [hx, hnb, if_false] at ih
simp only [hnb, if_false]
specialize ih (n / b) c (x :: tl)
rw [← ih]
| Mathlib/Data/Nat/Digits.lean | 766 | 770 |
/-
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... | theorem pow_card (a : K) : a ^ q = a := by
by_cases h : a = 0; · rw [h]; apply zero_pow Fintype.card_ne_zero
rw [← Nat.succ_pred_eq_of_pos Fintype.card_pos, pow_succ, Nat.pred_eq_sub_one,
pow_card_sub_one_eq_one a h, one_mul]
| Mathlib/FieldTheory/Finite/Basic.lean | 225 | 228 |
/-
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.Algebra.Group.PUnit
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence... | section Group
variable {G H : Type*} [Group G] [Group H]
| Mathlib/GroupTheory/Coprod/Basic.lean | 601 | 604 |
/-
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... | exact h 2 one_lt_two
refine le_of_forall_gt_imp_ge_of_dense fun x hbx => ?_
convert h (x / b) ((one_lt_div hb).mpr hbx)
rw [mul_div_cancel₀ _ hb.ne']
| Mathlib/Algebra/Order/Field/Basic.lean | 278 | 281 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
/-!
# The Giry monad
Let X be a measurable space. The collection of all measures on X again
forms a measura... | Measure.ofMeasurable_apply s hs
theorem le_join_apply (m : Measure (Measure α)) (s : Set α) : ∫⁻ μ, μ s ∂m ≤ join m s := by
| Mathlib/MeasureTheory/Measure/GiryMonad.lean | 123 | 125 |
/-
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... |
/-- The disjoint union of finsets is a sum -/
| Mathlib/Data/Finset/Basic.lean | 629 | 630 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | · rw [← cons_self_properDivisors hn, Finset.sum_cons, add_comm]
/-- `n : ℕ` is perfect if and only the sum of the proper divisors of `n` is `n` and `n`
| Mathlib/NumberTheory/Divisors.lean | 377 | 379 |
/-
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... | section
| Mathlib/Tactic/Ring/Basic.lean | 974 | 975 |
/-
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 | 2,643 | 2,658 | |
/-
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... | apply Nat.lt_succ_self
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,083 | 1,084 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | map φ hφ f = algebraMap _ _ (φ f.num) / algebraMap _ _ (φ f.denom) := by
rw [← num_div_denom f, map_apply_div_ne_zero, num_div_denom f]
| Mathlib/FieldTheory/RatFunc/Basic.lean | 979 | 980 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.IndepAxioms
/-!
# Matroid Duality
For a matroid `M` on ground set `E`, the collection of complements of the bases of `M` is the
collection o... | theorem eq_dual_iff_dual_eq {M₁ M₂ : Matroid α} : M₁ = M₂✶ ↔ M₁✶ = M₂ :=
dual_involutive.eq_iff.symm
| Mathlib/Data/Matroid/Dual.lean | 170 | 171 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | simpa only [polynomial_eq] using Cubic.monic_of_b_eq_one'
lemma irreducible_polynomial [IsDomain R] : Irreducible W'.polynomial := by
by_contra h
rcases (monic_polynomial.not_irreducible_iff_exists_add_mul_eq_coeff natDegree_polynomial).mp h
with ⟨f, g, h0, h1⟩
simp only [polynomial_eq, Cubic.coeff_eq_c, C... | Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 161 | 172 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... | have h : k ≠ j := fun h ↦ (h ▸ hj) (Finset.mem_insert_self _ _)
rw [Finset.sum_insert hk, add_comm ((c k) • M k), ← add_assoc, det_updateRow_add,
det_updateRow_smul, det_updateRow_eq_zero h, mul_zero, add_zero, h_ind]
exact fun h ↦ hj (Finset.mem_insert_of_mem h)
/-- If we replace a row of a ... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 422 | 431 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | -- Porting note: proofs were `tidy` or `simp`
toFun x := x.1.2
| Mathlib/LinearAlgebra/Prod.lean | 554 | 555 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.ModelTheory... | Mathlib/ModelTheory/Syntax.lean | 1,141 | 1,143 | |
/-
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 ... |
See `PowerBasis.liftEquiv` for a bundled equiv sending `⟨y, hy⟩` to the algebra map.
-/
noncomputable def lift (pb : PowerBasis A S) (y : S') (hy : aeval y (minpoly A pb.gen) = 0) :
S →ₐ[A] S' :=
{ pb.basis.constr A fun i => y ^ (i : ℕ) with
| Mathlib/RingTheory/PowerBasis.lean | 276 | 281 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
/-!
# Homomorphisms of semirings and... | @[simp]
theorem cancel_left {g : β →ₙ+* γ} {f₁ f₂ : α →ₙ+* β} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
| Mathlib/Algebra/Ring/Hom/Defs.lean | 279 | 281 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
/-!
# Continu... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 499 | 508 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.CategoryTheory.Linear.Yoneda
import Mathlib.Algebra.Category.ModuleCat.Monoi... | (MonoidalClosed.pre f).app P = ofHom (homLinearEquiv.symm.toLinearMap ∘ₗ
LinearMap.lcomp _ _ f.hom ∘ₗ homLinearEquiv.toLinearMap) :=
rfl
| Mathlib/Algebra/Category/ModuleCat/Monoidal/Closed.lean | 88 | 91 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.Limits
import Mathlib.Al... | (s : (structureSheaf R).1.obj (op U)) (p : V) :
(comap f U V hUV s).1 p =
Localization.localRingHom (PrimeSpectrum.comap f p.1).asIdeal _ f rfl
| Mathlib/AlgebraicGeometry/StructureSheaf.lean | 1,022 | 1,024 |
/-
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... | refine .iUnion fun n ↦ .image ?_ ?_
· have : IsSeparable {f : Fin n → R × M | ∀ (i : Fin n), f i ∈ Set.univ ×ˢ s} := by
apply isSeparable_pi (fun i ↦ .prod (.of_separableSpace Set.univ) hs)
rwa [Set.univ_prod] at this
· apply continuous_finset_sum _ (fun i _ ↦ ?_)
exact (continuous_fst.comp (continu... | Mathlib/Topology/Algebra/Module/Basic.lean | 114 | 125 |
/-
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.Linear.Basic
import Mathlib.Algebra.Homology.ComplexShapeSigns
import Mathlib.Algebra.Homology.HomologicalBicomplex
import Mathlib.Algebra.Module.... |
namespace totalAux
@[reassoc (attr := simp)]
lemma ιMapObj_D₁ (i₁₂ i₁₂' : I₁₂) (i : I₁ × I₂) (h : ComplexShape.π c₁ c₂ c₁₂ i = i₁₂) :
| Mathlib/Algebra/Homology/TotalComplex.lean | 138 | 142 |
/-
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... | induction n with
| zero => simp
| succ n ih => simp [pow_succ, pow_mul, ih, pow_card]
| Mathlib/FieldTheory/Finite/Basic.lean | 231 | 234 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Truncated
import Mathlib.RingTheory.WittVector.Identities
import Mathlib.NumberTheory.Padics.RingHoms
/-!
# Co... | ```text
ZMod (p^n) ----------------------------> ZMod (p^m)
| |
| Mathlib/RingTheory/WittVector/Compare.lean | 79 | 81 |
/-
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... | rw [le_iff_eq_or_lt] at hza hzb
rcases hza with hza | hza; · exact hle_a _ hza.symm
rcases hzb with hzb | hzb; · exact hle_b _ hzb
wlog hC₀ : 0 < C generalizing C
· refine le_of_forall_gt_imp_ge_of_dense fun C' hC' => this (fun w hw => ?_) (fun w hw => ?_) ?_
· exact (hle_a _ hw).trans hC'.le
· exact ... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 115 | 221 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
/-!
# Ordered Subtraction
This file proves l... | Contravariant.AddLECancellable.tsub_eq_of_eq_add_rev h
@[simp]
theorem add_tsub_cancel_right (a b : α) : a + b - b = a :=
| Mathlib/Algebra/Order/Sub/Defs.lean | 324 | 327 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying
-/
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
/-!
# Bilinear form
This file defines the conversio... | Mathlib/LinearAlgebra/Matrix/BilinearForm.lean | 411 | 413 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,559 | 1,565 | |
/-
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 ... | rintro i x ⟨x, rfl⟩
use lp.single 2 i x
exact hV.linearIsometry_apply_single x
exact hV.linearIsometry.isometry.isUniformInducing.isComplete_range.isClosed
end OrthogonalFamily
section IsHilbertSum
variable (𝕜 G)
variable [CompleteSpace E] (V : ∀ i, G i →ₗᵢ[𝕜] E) (F : ι → Submodule 𝕜 E)
/--... | Mathlib/Analysis/InnerProductSpace/l2Space.lean | 244 | 262 |
/-
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... | · rw [f0n, Nat.bitwise_zero_left]
cases g false true <;> rfl
| Mathlib/Data/Num/Lemmas.lean | 774 | 775 |
/-
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 | 3,326 | 3,327 | |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
/-!
# Sesquilinear maps
... |
variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
(B : LinearMap.BilinForm R M) (f : M → M)
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 468 | 470 |
/-
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... | -- attribute. Also, the linter does not complain about that attribute.
@[simp]
| Mathlib/MeasureTheory/Measure/MutuallySingular.lean | 68 | 69 |
/-
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... |
section Abelian
variable [Abelian C]
/-- Given a morphism of short complexes `φ : S₁ ⟶ S₂` in an abelian category, if `S₁.f`
and `S₁.g` are zero (e.g. when `S₁` is of the form `0 ⟶ S₁.X₂ ⟶ 0`) and `S₂.f = 0`
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 827 | 833 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | · rw [hI]; apply Submodule.fg_bot
have hI : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI
exact I.fg_of_isUnit (IsFractionRing.injective A (FractionRing A)) (h.isUnit hI)
theorem integrallyClosed : IsIntegrallyClosed A := by
-- It suffices to show that for integral `x`,
| Mathlib/RingTheory/DedekindDomain/Ideal.lean | 283 | 288 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Andrew Yang
-/
import Mathlib.CategoryTheory.Monoidal.Functor
/-!
# Endofunctors as a monoidal category.
We give the monoidal category structure on `C ⥤ C`,
and show that... | (η F).app ((F.obj n).obj X) = (μ F n (𝟙_ M)).app X ≫ (F.map (ρ_ n).hom).app X := by
rw [map_rightUnitor]
dsimp
simp only [Category.comp_id, μ_δ_app_assoc]
@[reassoc]
| Mathlib/CategoryTheory/Monoidal/End.lean | 226 | 231 |
/-
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... | Mathlib/LinearAlgebra/LinearPMap.lean | 1,037 | 1,041 | |
/-
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.Analysis.Convolution
import Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension
import Mathlib.Analysis.Calculus.BumpFunction.Normed
import Math... | φ.support_normed_eq.subset φ.nonneg_normed φ.integral_normed hmg hg
/-- `(φ i ⋆ g i) (k i)` tends to `z₀` as `i` tends to some filter `l` if
* `φ` is a sequence of normed bump functions
such that `(φ i).rOut` tends to `0` as `i` tends to `l`;
| Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean | 75 | 79 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... | instance (priority := 100) _root_.PrincipalIdealRing.isNoetherianRing [IsPrincipalIdealRing R] :
IsNoetherianRing R where
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 114 | 115 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra... | protected theorem map_neg {F : Type*} [Ring β] [FunLike F α β] [RingHomClass F α β]
(f : F) (u : αˣ) : map (f : α →* β) (-u) = -map (f : α →* β) u :=
| Mathlib/Algebra/Ring/Units.lean | 82 | 83 |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Pigeonhole
... |
theorem mem_eraseLast {s : CompositionSeries X} {x : X} (h : 0 < s.length) :
x ∈ s.eraseLast ↔ x ≠ s.last ∧ x ∈ s := by
simp only [RelSeries.mem_def, eraseLast]
constructor
· rintro ⟨i, rfl⟩
have hi : (i : ℕ) < s.length := by
conv_rhs => rw [← Nat.add_one_sub_one s.length, Nat.succ_sub h]
| Mathlib/Order/JordanHolder.lean | 208 | 215 |
/-
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... | Mathlib/LinearAlgebra/Ray.lean | 401 | 401 | |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
/-!
# Characteris... |
section CayleyHamilton
| Mathlib/LinearAlgebra/Charpoly/Basic.lean | 60 | 61 |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | lemma ih24_neg : IH24 x₁ x₂ y → IH24 (-x₂) (-x₁) y ∧ IH24 x₁ x₂ (-y) := by
simp_rw [IH24, ← P24_neg_right, isOption_neg]
refine fun h ↦ ⟨fun z ↦ ⟨?_, ?_, ?_⟩,
fun z ↦ ⟨(@h z).1, (@h z).2.1, P24_neg_right.2 ∘ (@h <| -z).2.2⟩⟩
all_goals
rw [P24_neg_left]
simp only [neg_neg]
first | exact (@h <| -z).... | Mathlib/SetTheory/Surreal/Multiplication.lean | 323 | 331 |
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | set S : ℕ → Set X := fun n => {x ∈ s | (↑n)⁻¹ ≤ infEdist x t}
have Ssep (n) : Metric.AreSeparated (S n) t :=
⟨n⁻¹, ENNReal.inv_ne_zero.2 (ENNReal.natCast_ne_top _),
fun x hx y hy ↦ hx.2.trans <| infEdist_le_edist_of_mem hy⟩
have Ssep' : ∀ n, Metric.AreSeparated (S n) (s ∩ t) := fun n =>
(Ssep n).mon... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 159 | 226 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
/-!
# Filters used in box-based integrals
First ... | end IntegrationParams
end BoxIntegral
| Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 522 | 527 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 1,342 | 1,344 | |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | (h_mul : local_hMul[R, inst₁] = local_hMul[R, inst₂]) :
inst₁ = inst₂ := by
have h : inst₁.toNonUnitalNonAssocSemiring = inst₂.toNonUnitalNonAssocSemiring := by
| Mathlib/Algebra/Ring/Ext.lean | 133 | 135 |
/-
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... | A ⊗ₖ (1 : Matrix n n α) = blockDiagonal fun _ => A :=
(kronecker_diagonal _ _).trans <| congr_arg _ <| funext fun _ => Matrix.ext fun _ _ => mul_one _
theorem one_kronecker [MulZeroOneClass α] [DecidableEq l] (B : Matrix m n α) :
(1 : Matrix l l α) ⊗ₖ B =
Matrix.reindex (Equiv.prodComm _ _) (Equiv.prod... | Mathlib/Data/Matrix/Kronecker.lean | 350 | 356 |
/-
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... | theorem subtypeDomain_sum {s : Finset ι} {h : ι → α →₀ M} :
(∑ c ∈ s, h c).subtypeDomain p = ∑ c ∈ s, (h c).subtypeDomain p :=
map_sum subtypeDomainAddMonoidHom _ s
theorem subtypeDomain_finsupp_sum [Zero N] {s : β →₀ N} {h : β → N → α →₀ M} :
(s.sum h).subtypeDomain p = s.sum fun c d => (h c d).subtypeDomai... | Mathlib/Data/Finsupp/Basic.lean | 989 | 996 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 998 | 999 | |
/-
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.Group.Action.Pointwise.Finset
import Mathlib.Algebra.Ring.Nat
/-!
# e-transforms
e-transforms are a family of transformations of pairs of finite ... |
@[to_additive (attr := simp)]
theorem mulETransformLeft_inv : mulETransformLeft e⁻¹ x = (mulETransformRight e x.swap).swap := by
| Mathlib/Combinatorics/Additive/ETransform.lean | 158 | 160 |
/-
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.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
/-!
# Met... | div_mul_eq_mul_div]
· gcongr
exact EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist (a : ℂ) b c (conj (b : ℂ))
| Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 96 | 98 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs
import Mathlib.Tactic.Linter.Deprecate... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 414 | 415 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Basic
/-!
# Split a box along one or more hyperplanes
## Main definitions
A hyperplane `{x : ι → ℝ | x i = a}` spli... | rw [← iUnion_eq_empty, iUnion_compl, h.iUnion_eq, diff_self]
@[simp]
theorem compl_top : (⊤ : Prepartition I).compl = ⊥ :=
(isPartitionTop I).compl_eq_bot
| Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 346 | 350 |
/-
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.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | rw [← C_mul_X_pow_eq_monomial, aroots_C_mul_X_pow ha]
variable (R S) in
@[simp]
| Mathlib/Algebra/Polynomial/Roots.lean | 473 | 476 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes, Kevin Buzzard
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units.Basic
/-!
# Monoid homomorphisms and units
This file allows to... | (Units.liftRight f fun x => (hf x).unit) fun _ => rfl
| Mathlib/Algebra/Group/Units/Hom.lean | 198 | 199 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma
import Mathlib.Combinatorics.Simp... |
private lemma card_bound (hP₁ : P.IsEquipartition) (hP₃ : #P.parts ≤ bound (ε / 8) ⌈4/ε⌉₊)
(hX : s ∈ P.parts) : card α / (2 * bound (ε / 8) ⌈4 / ε⌉₊ : ℝ) ≤ #s := by
cases isEmpty_or_nonempty α
· simp [Fintype.card_eq_zero]
have := Finset.Nonempty.card_pos ⟨_, hX⟩
calc
_ ≤ card α / (2 * #P.parts : ℝ) :=... | Mathlib/Combinatorics/SimpleGraph/Triangle/Removal.lean | 56 | 65 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Adam Topaz, Johan Commelin, Jakob von Raumer
-/
import Mathlib.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTh... |
theorem exact_iff_isIso_imageToKernel : S.Exact ↔ IsIso (imageToKernel S.f S.g S.zero) := by
rw [S.exact_iff_epi_imageToKernel]
constructor
· intro
apply isIso_of_mono_of_epi
· intro
infer_instance
/-- In an abelian category, a short complex `S` is exact
iff `imageSubobject S.f = kernelSubobject S.g`.... | Mathlib/CategoryTheory/Abelian/Exact.lean | 66 | 81 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Na... | theorem norm_iteratedFDeriv_clm_apply {f : E → F →L[𝕜] G} {g : E → F} {N : WithTop ℕ∞} {n : ℕ}
(hf : ContDiff 𝕜 N f) (hg : ContDiff 𝕜 N g) (x : E) (hn : n ≤ N) :
‖iteratedFDeriv 𝕜 n (fun y : E => (f y) (g y)) x‖ ≤ ∑ i ∈ Finset.range (n + 1),
↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv ... | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | 537 | 548 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Theory.Model.realize_of_mem φ h
| Mathlib/ModelTheory/Semantics.lean | 685 | 686 |
/-
Copyright (c) 2022 Yaël Dillies, George Shakan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, George Shakan
-/
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Tactic.FieldSimp
import Mathli... |
/-- **Ruzsa's triangle inequality**. Multiplication version. -/
@[to_additive "**Ruzsa's triangle inequality**. Addition version."]
theorem ruzsa_triangle_inequality_mul_mul_mul (A B C : Finset G) :
| Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 166 | 169 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | theorem enorm'_le_iff_norm_le {x : E} {y : F} : ‖x‖ₑ ≤ ‖y‖ₑ ↔ ‖x‖ ≤ ‖y‖ := by
simp only [← ofReal_norm']
refine ⟨fun h ↦ ?_, fun h ↦ by gcongr⟩
| Mathlib/Analysis/Normed/Group/Basic.lean | 714 | 716 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... |
@[simp] theorem drop_tail' {i : ℕ} {s : Stream' α} : drop i (tail s) = s.drop (i + 1) := rfl
| Mathlib/Data/Stream/Init.lean | 65 | 66 |
/-
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.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 776 | 779 | |
/-
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.Idempotents.Karoubi
/-!
# Extension of functors to the idempotent completion
In this file, we construct an extension `functorExtension₁`
of func... | rw [← φ.naturality, ← assoc, ← F.map_comp]
conv_lhs => rw [← id_comp (φ.app P), ← F.map_id]
congr
apply decompId
namespace FunctorExtension₁
| Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean | 35 | 40 |
/-
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.... | | h => by
rw [← @ofReal_inj K, im_eq_conj_sub, h, sub_self, mul_zero, zero_div,
ofReal_zero]
| Mathlib/Analysis/RCLike/Basic.lean | 338 | 340 |
/-
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.Data.Finsupp.Basic
import Mathlib.Data.List.AList
/-!
# Connections between `Finsupp` and `AList`
## Main definitions
* `Fin... | (hx : x ≠ 0) : l.lookupFinsupp a = x ↔ x ∈ l.lookup a := by
rw [lookupFinsupp_apply]
rcases lookup a l with - | m <;> simp [hx.symm]
theorem lookupFinsupp_eq_zero_iff [DecidableEq α] {l : AList fun _x : α => M} {a : α} :
| Mathlib/Data/Finsupp/AList.lean | 82 | 86 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... |
section val
| Mathlib/Order/Interval/Finset/Fin.lean | 130 | 131 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... |
theorem logb_mul (hx : x ≠ 0) (hy : y ≠ 0) : logb b (x * y) = logb b x + logb b y := by
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 72 | 73 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... | variable (E) in
theorem martingale_zero (ℱ : Filtration ι m0) (μ : Measure Ω) : Martingale (0 : ι → Ω → E) ℱ μ :=
⟨adapted_zero E ℱ, fun i j _ => by rw [Pi.zero_apply, condExp_zero]; simp⟩
namespace Martingale
| Mathlib/Probability/Martingale/Basic.lean | 75 | 79 |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
/-!
# A predicate on adjoining root... | -- simp only [map_add, Finsupp.comapDomain_add_of_injective Fin.val_injective, toFinsupp_add]
map_smul' := fun c x => by
rw [map_smul, toFinsupp_smul, Finsupp.comapDomain_smul_of_injective Fin.val_injective,
RingHom.id_apply] }
-- Porting note: the original simp proof with the same l... | Mathlib/RingTheory/IsAdjoinRoot.lean | 394 | 399 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.Analysis.Calculus.ContDiff.CPolynomial
import... | lemma fderiv_fourierIntegral
[MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] {μ : Measure V}
(hf : Integrable f μ) (hf' : Integrable (fun v : V ↦ ‖v‖ * ‖f v‖) μ) :
fderiv ℝ (fourierIntegral 𝐞 μ L.toLinearMap₂ f) =
fourierIntegral 𝐞 μ L.toLinearMap₂ (fourierSMulRight L f) := by
ext w... | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 235 | 241 |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
impo... | theorem div_add_div_same (a b c : K) : a / c + b / c = (a + b) / c :=
| Mathlib/Algebra/Field/Basic.lean | 29 | 29 |
/-
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.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.FunctorCategory.EpiMono
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.Categ... | Mathlib/CategoryTheory/Sites/Subsheaf.lean | 301 | 309 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | (S ⧸ I.map (algebraMap R S)) ≃ₐ[R]
Polynomial (R ⧸ I) ⧸
Ideal.span ({(minpoly R pb.gen).map (Ideal.Quotient.mk I)} : Set (Polynomial (R ⧸ I))) :=
(ofRingEquiv
(show ∀ x,
(Ideal.quotientEquiv _ (Ideal.map (AdjoinRoot.of (minpoly R pb.gen)) I)
| Mathlib/RingTheory/AdjoinRoot.lean | 795 | 800 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 1,841 | 1,845 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Basic
import Mathlib.CategoryTheory.Preadditive.Basic
/-!
# Factoring through subobjects
The predicate `h : P.Fact... |
end Subobject
| Mathlib/CategoryTheory/Subobject/FactorThru.lean | 195 | 197 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,442 | 1,446 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Iterate
impor... | (h : ∀ k < n, c * u k < u (k + 1)) : c ^ n * u 0 < u n := by
apply (monotone_mul_left_of_nonneg hc).seq_pos_lt_seq_of_le_of_lt hn _ _ h
· simp
· simp [_root_.pow_succ', mul_assoc, le_refl]
| Mathlib/Analysis/SpecificLimits/Basic.lean | 208 | 211 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Data.List.Iterate
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Tactic.Group
/-!
# ... | (nodup_of_pairwise_disjoint_cycles h₁l₂ h₂l₂)).mpr
fun σ => ?_
by_cases hσ : σ.IsCycle
· obtain _ := not_forall.mp (mt ext hσ.ne_one)
rw [mem_list_cycles_iff h₁l₁ h₂l₁, mem_list_cycles_iff h₁l₂ h₂l₂, h₀]
· exact iff_of_false (mt (h₁l₁ σ) hσ) (mt (h₁l₂ σ) hσ)
/-- Factors a permutat... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 461 | 482 |
/-
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... | protected theorem zero_lt_one : (0 : ℝ) < 1 := by
convert ratCast_lt.2 zero_lt_one <;> simp [← ofCauchy_ratCast, ofCauchy_one, ofCauchy_zero]
| Mathlib/Data/Real/Basic.lean | 338 | 339 |
/-
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.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | filter_upwards [Measure.rnDeriv_le_one_of_le (hκη a)] with x hx_le_one
simp_rw [rnDerivAux]
split_ifs with hα
· refine ENNReal.toReal_le_of_le_ofReal zero_le_one ?_
simp only [Pi.one_apply, ENNReal.ofReal_one]
exact hx_le_one
· have := hαγ.countableOrCountablyGenerated.resolve_left hα
exact densit... | Mathlib/Probability/Kernel/RadonNikodym.lean | 101 | 110 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.... | simp only [ofMkLE, ofLEMk, ofLE, ofMkLEMk, ← Functor.map_comp underlying,
assoc, Iso.hom_inv_id_assoc]
congr 1
| Mathlib/CategoryTheory/Subobject/Basic.lean | 392 | 395 |
/-
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.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Data.Set.Monotone
import Mathlib.Order.Interval.Set... | /-! ### `natAbs` -/
| Mathlib/Data/Int/Lemmas.lean | 36 | 37 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Angles in circles and sphere.
This file proves results ... | multiple (by half the inverse of the tangent of the angle at one of those points) of a `π / 2`
rotation of the vector between the other two points, plus the midpoint of those points. -/
theorem inv_tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center {s : Sphere P}
{p₁ p₂ p₃ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ ... | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | 161 | 171 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe,
InnerProductGeometry.tan_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)]
/-- The tangent of an angle in a right-angled triangle as a ratio of sides. -/
theorem tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle ... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 118 | 122 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... | __ := Prop.instHeytingAlgebra
__ := GeneralizedHeytingAlgebra.toDistribLattice
| Mathlib/Order/BooleanAlgebra.lean | 712 | 713 |
/-
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.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,306 | 1,315 | |
/-
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.Order.Floor.Div
import Mathlib.Data.Nat.Factorization.Defs
/-!
# Roots of natural numbers, rounded up and down
This file defines the flooring and... | @[simp] lemma ceilRoot_zero_right (n : ℕ) : ceilRoot n 0 = 0 := by simp [ceilRoot]
| Mathlib/Data/Nat/Factorization/Root.lean | 121 | 121 |
/-
Copyright (c) 2019 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.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | (prod.braiding _ _).hom_inv_id
/-- The braiding isomorphism is symmetric. -/
@[reassoc]
| Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 878 | 881 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Algebra.Module.Stro... |
/-- The norm of the `0` operator is `0`. -/
theorem opNorm_zero : ‖(0 : E →SL[σ₁₂] F)‖ = 0 :=
| Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean | 173 | 175 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | rcases hd i k with ⟨j, hji, hjk⟩
use j
rw [← measure_diff hjk (h _) (this _ hjk)]
gcongr
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 518 | 521 |
/-
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... | (hw : ∀ i g, ⟨i, g⟩ ∈ w.toList → g ∈ d.set i)
(hφw : ∀ j g, ⟨j, g⟩ ∈ (CoprodI.of (φ i h) • w).toList → g ∈ d.set j) :
h = 1 := by
simp only [← (d.compl _).equiv_snd_eq_self_iff_mem (one_mem _)] at hw hφw
have hhead : ((d.compl i).equiv (Word.equivPair i w).head).2 =
(Word.equivPair i w).head := by... | Mathlib/GroupTheory/PushoutI.lean | 331 | 343 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | monomial a₁ b₁ = monomial a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ ∨ b₁ = 0 ∧ b₂ = 0 :=
Finsupp.single_eq_single_iff _ _ _ _
| Mathlib/Algebra/MvPolynomial/Basic.lean | 338 | 340 |
/-
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 | 473 | 482 | |
/-
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.HausdorffDistance
import Mathlib.Topology.Sets.Compacts
/-!
# Closed subsets
... | · refine isCompact_iff_totallyBounded_isComplete.2 ⟨?_, s.isClosed.isComplete⟩
refine totallyBounded_iff.2 fun ε (εpos : 0 < ε) => ?_
-- we have to show that s is covered by finitely many eballs of radius ε
-- pick a nonempty compact set t at distance at most ε/2 of s
rcases mem_closure_iff.1 hs (ε / ... | Mathlib/Topology/MetricSpace/Closeds.lean | 256 | 291 |
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