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) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
/-!
# The sheaf condition for a presieve
We define what it means fo... | (t₂ : x₂.Compatible) : x₁.restrict (le_generate R) = x₂.restrict (le_generate R) → x₁ = x₂ :=
fun h => by
rw [← extend_restrict t₁, ← extend_restrict t₂]
-- Porting note: congr fails to make progress
apply congr_arg
exact h
| Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 246 | 252 |
/-
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, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.GCongr.CoreAttrs
import Mathlib.Tactic.Common
import Mathlib.Tactic.... |
theorem factorial_inj' (h : 1 < n ∨ 1 < m) : n ! = m ! ↔ n = m := by
obtain hn|hm := h
· exact factorial_inj hn
· rw [eq_comm, factorial_inj hm, eq_comm]
theorem self_le_factorial : ∀ n : ℕ, n ≤ n !
| 0 => Nat.zero_le _
| k + 1 => Nat.le_mul_of_pos_right _ (Nat.one_le_of_lt k.factorial_pos)
| Mathlib/Data/Nat/Factorial/Basic.lean | 121 | 129 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Basic
/... | @[simp]
theorem expand_zero (f : R[X]) : expand R 0 f = C (eval 1 f) := by simp [expand]
@[simp]
| Mathlib/Algebra/Polynomial/Expand.lean | 69 | 72 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | · intro h i j hi hj hij
rw [Ne, ← Option.some_inj, ← getElem?_eq_getElem, ← getElem?_eq_getElem]
exact h i j hij hj
set_option linter.deprecated false in
| Mathlib/Data/List/Nodup.lean | 103 | 107 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,484 | 2,488 | |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | | nil =>
exfalso
exact not_mem_nil h
| Mathlib/Data/List/Basic.lean | 571 | 573 |
/-
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... | _ = (x ⊔ y \ z) ⊓ y := by rw [sup_inf_right, sup_inf_sdiff]
_ = x ⊓ y ⊔ y \ z := by rw [inf_sup_right, inf_sdiff_left]
| Mathlib/Order/BooleanAlgebra.lean | 315 | 316 |
/-
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... |
end GroupWithZero
protected theorem Absorbs.restrict_scalars
| Mathlib/Topology/Bornology/Absorbs.lean | 249 | 252 |
/-
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.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | exact (ncard_le_ncard_of_injOn f hf h' ht).not_lt hc
theorem le_ncard_of_inj_on_range {n : ℕ} (f : ℕ → α) (hf : ∀ i < n, f i ∈ s)
(f_inj : ∀ i < n, ∀ j < n, f i = f j → i = j) (hs : s.Finite := by toFinite_tac) :
n ≤ s.ncard := by
| Mathlib/Data/Set/Card.lean | 787 | 791 |
/-
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... | Tendsto (logb b) (𝓝[>] 0) atTop :=
tendsto_log_nhdsGT_zero.atBot_mul_const_of_neg (inv_lt_zero.2 (log_neg hb₀ hb))
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 441 | 443 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,129 | 1,132 | |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 335 | 337 | |
/-
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 | 3,190 | 3,192 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
# Basic properties of the manifold Fréchet ... | theorem mdifferentiableWithinAt_iff_target :
MDifferentiableWithinAt I I' f s x ↔
ContinuousWithinAt f s x ∧
MDifferentiableWithinAt I 𝓘(𝕜, E') (extChartAt I' (f x) ∘ f) s x := by
simp_rw [MDifferentiableWithinAt, liftPropWithinAt_iff', ← and_assoc]
| Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 211 | 215 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Topology.S... | rw [nhds_nhdsAdjoint, sup_of_le_right]
exact le_iInf₂ fun γ hγ ↦ le_principal_iff.2 <| zero_lt_iff.2 hγ
| Mathlib/Topology/Algebra/WithZeroTopology.lean | 47 | 49 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Data.Set.UnionLift
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.RingTheory.NonUnitalSubring.Basic
... | ((⨅ i, S i).map f : NonUnitalSubalgebra R B) = ⨅ i, (S i).map f := by
apply SetLike.coe_injective
| Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean | 803 | 804 |
/-
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.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
/-!
# Identically distributed random variables
Two random variable... | have : ∀ k, (fun x ↦ Set.indicator {x | C ≤ ‖f k x‖} (fun a ↦ ‖f k a‖) x) = F ∘ f k := by
intro k
| Mathlib/Probability/IdentDistrib.lean | 311 | 312 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | simp [nd.1]
· simp [h, ih nd.2]
@[simp]
theorem dlookup_kerase (a) {l : List (Sigma β)} (nd : l.NodupKeys) :
dlookup a (kerase a l) = none :=
dlookup_eq_none.mpr (not_mem_keys_kerase a nd)
@[simp]
theorem dlookup_kerase_ne {a a'} {l : List (Sigma β)} (h : a ≠ a') :
dlookup a (kerase a' l) = dlooku... | Mathlib/Data/List/Sigma.lean | 464 | 476 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Ess... | rfl
@[simp]
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 405 | 407 |
/-
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.Algebra.MvPolynomial.Monad
/-!
## Expand multivariate polynomials
Given a multivariate polynomial `φ`, one may replace every occurre... | theorem map_expand (f : R →+* S) (p : ℕ) (φ : MvPolynomial σ R) :
map f (expand p φ) = expand p (map f φ) := by simp [expand, map_bind₁]
@[simp]
theorem rename_expand (f : σ → τ) (p : ℕ) (φ : MvPolynomial σ R) :
| Mathlib/Algebra/MvPolynomial/Expand.lean | 64 | 68 |
/-
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.Fin.Fin2
import Mathlib.Data.PFun
import Mathlib.Data.Vector3
import Mathlib.NumberTheory.PellMatiyasevic
/-!
# Diophantine functions and Matiyas... | dsimp
-- Porting note: `congr` used to be enough here
suffices v ∘ (id ⊗ elim0) = v ⊗ fun i ↦ f i v by rw [this]
ext x; obtain _ | _ | _ := x; rfl
| succ n, S, d, f =>
f.consElim fun f fl => by
simp only [vectorAllP_cons, and_imp]
exact fun df dfl =>
have : Dioph ... | Mathlib/NumberTheory/Dioph.lean | 465 | 473 |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
import Mathlib.CategoryTheory.LiftingProperties.Adjunction
/-!
# Preservati... | F.map_injective ((cancel_mono (F.map f)).1 (by rw [← F.map_comp, hgh, F.map_comp]))⟩
instance (priority := 100) reflectsEpimorphisms_of_faithful (F : C ⥤ D) [Faithful F] :
ReflectsEpimorphisms F where
reflects {X} {Y} f _ :=
⟨fun {Z} g h hgh =>
F.map_injective ((cancel_epi (F.map f)).1 (by rw [← ... | Mathlib/CategoryTheory/Functor/EpiMono.lean | 175 | 182 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
exact Exists.elim this fun R hR => by
rw [← Sylow.ext (H.mp hR)]
| Mathlib/GroupTheory/Sylow.lean | 289 | 291 |
/-
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.Language
import Mathlib.Tactic.AdaptationNote
/-!
# Regular Expressions
This file contains the formal definition for regular expressions and ... | rw [← h] at hQ
exact hQ
· left
rw [List.cons_append, List.cons_eq_cons] at h
refine ⟨t, u, h.2, ?_, hQ⟩
rw [rmatch] at hP
convert hP
exact h.1
· rw [ih]
constructor <;> rintro ⟨t, u, h, hP, hQ⟩
· exact ⟨a :: t, u, by tauto⟩
... | Mathlib/Computability/RegularExpressions.lean | 231 | 242 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.Algebra.Group.Submonoid.Finsupp
import Mathlib.Order.Filter.AtTopBot.Defs
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Gra... | rcases c1 with ⟨i1, ⟨n1, hn1⟩, ⟨d1, hd1⟩, h1⟩
rcases c2 with ⟨i2, ⟨n2, hn2⟩, ⟨d2, hd2⟩, h2⟩
dsimp only [Subtype.coe_mk] at *
subst hdeg hnum hden
congr
instance : Neg (NumDenSameDeg 𝒜 x) where
| Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | 109 | 115 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.CompleteLattice.Lemmas
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection.Basic
/-... | lemma himp_le_iff : a ⇨ b ≤ c ↔ ∀ d, d ⊓ a ≤ b → d ≤ c := by simp [himp_eq_sSup]
| Mathlib/Order/CompleteBooleanAlgebra.lean | 434 | 435 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.Algebra.Group.Submonoid.Finsupp
import Mathlib.Order.Filter.AtTopBot.Defs
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Gra... | -/
def map (g : A →+* B)
(comap_le : P ≤ Q.comap g) (hg : ∀ i, ∀ a ∈ 𝒜 i, g a ∈ ℬ i) :
HomogeneousLocalization 𝒜 P →+* HomogeneousLocalization ℬ Q where
toFun := Quotient.map'
(fun x ↦ ⟨x.1, ⟨_, hg _ _ x.2.2⟩, ⟨_, hg _ _ x.3.2⟩, comap_le x.4⟩)
fun x y (e : x.embedding = y.embedding) ↦ by
apply... | Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | 599 | 612 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
/-!
# Limits and a... |
/-- `x ^ s = o(exp x)` as `x → ∞` for any real `s`. -/
theorem isLittleO_rpow_exp_atTop (s : ℝ) : (fun x : ℝ => x ^ s) =o[atTop] exp := by
| Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 311 | 313 |
/-
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.Data.List.Basic
/-!
# Double universal quantification on a list
This file provides an API for `List.Forall₂` (definition in `Data.Lis... | fun h =>
match u, h with
| Mathlib/Data/List/Forall2.lean | 73 | 74 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | theorem Right.sign_neg [AddRightStrictMono α] (a : α) :
sign (-a) = -sign a := by
simp_rw [sign_apply, Right.neg_pos_iff, Right.neg_neg_iff]
| Mathlib/Data/Sign.lean | 439 | 441 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | -- NB. Changing this lemma to use ‖‖ₑ makes it false (only => still holds);
-- unlike a nnnorm, the enorm can be ∞.
lemma eLpNormEssSup_lt_top_iff_isBoundedUnder :
eLpNormEssSup f μ < ⊤ ↔ IsBoundedUnder (· ≤ ·) (ae μ) fun x ↦ ‖f x‖₊ where
mp h := ⟨(eLpNormEssSup f μ).toNNReal, by
simp_rw [← ENNReal.coe_le_coe... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 1,020 | 1,026 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Countable.Defs
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.PNat.Equiv
import Mathlib.... | @[simp 1100]
theorem decode_nat (n : ℕ) : decode n = some n :=
rfl
| Mathlib/Logic/Encodable/Basic.lean | 113 | 116 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Data.Rat.Sqrt
imp... | @[deprecated (since := "2025-04-01")] alias of_add_int := of_add_intCast
theorem intCast_add (h : Irrational x) (m : ℤ) : Irrational (m + x) := by
rw [← cast_intCast]
exact h.ratCast_add m
| Mathlib/Data/Real/Irrational.lean | 239 | 243 |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathl... | f.genEigenspace μ k = ⨆ l : {l : ℕ // l ≤ k}, f.genEigenspace μ l := by
simp_rw [genEigenspace_nat, genEigenspace, OrderHom.coe_mk, iSup_subtype]
| Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 105 | 106 |
/-
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 | 1,687 | 1,693 | |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Lim... | Function.Surjective (F.map f) := by
have : Epi f := epi_of_nonempty_of_isConnected F f
exact surjective_on_fiber_of_epi F f
/-- If `X : ι → C` is a finite family of objects with non-empty fiber, then
also `∏ᶜ X` has non-empty fiber. -/
instance nonempty_fiber_pi_of_nonempty_of_finite {ι : Type*} [Finite ι] (X ... | Mathlib/CategoryTheory/Galois/Basic.lean | 331 | 338 |
/-
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.Lie.Derivation.Killing
import Mathlib.Algebra.Lie.Killing
import Mathlib.Algebra.Lie.Sl2
import Mathlib.Algebra.Lie.Weights.Chain
import Mathlib.Line... | simpa using hx
lemma disjoint_ker_weight_corootSpace (α : Weight K H L) :
Disjoint α.ker (corootSpace α) := by
| Mathlib/Algebra/Lie/Weights/Killing.lean | 369 | 372 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Yaël Dillies, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Ring.Unbundled.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Alge... | @[deprecated "Use `[CommSemiring R] [PartialOrder R] [IsOrderedRing R]` instead."
(since := "2025-04-10")]
structure OrderedCommSemiring (R : Type u) extends OrderedSemiring R, CommSemiring R where
mul_le_mul_of_nonneg_right a b c ha hc :=
-- parentheses ensure this generates an `optParam` rather than an `autoP... | Mathlib/Algebra/Order/Ring/Defs.lean | 250 | 254 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 2,165 | 2,166 | |
/-
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 | 993 | 994 | |
/-
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.Data.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 1,638 | 1,642 | |
/-
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.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... | theorem Convex.mem_smul_of_zero_mem (h : Convex 𝕜 s) {x : E} (zero_mem : (0 : E) ∈ s) (hx : x ∈ s)
{t : 𝕜} (ht : 1 ≤ t) : x ∈ t • s := by
rw [mem_smul_set_iff_inv_smul_mem₀ (zero_lt_one.trans_le ht).ne']
exact h.smul_mem_of_zero_mem zero_mem hx
| Mathlib/Analysis/Convex/Basic.lean | 521 | 524 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Johan Commelin, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Constructions.Pullbacks
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.CategoryT... |
attribute [local instance] nonPreadditiveAbelian
/-- In an abelian category, an epi is the cokernel of its kernel. More precisely:
| Mathlib/CategoryTheory/Abelian/Basic.lean | 426 | 429 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | · simp
· rw [List.prod_cons, List.map_cons, List.foldr_cons]
refine (support_mul_le hd tl.prod).trans ?_
exact sup_le_sup le_rfl hl
theorem support_zpow_le (σ : Perm α) (n : ℤ) : (σ ^ n).support ≤ σ.support := fun _ h1 =>
mem_support.mpr fun h2 => mem_support.mp h1 (zpow_apply_eq_self_of_apply_eq_self h2... | Mathlib/GroupTheory/Perm/Support.lean | 450 | 463 |
/-
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... |
instance {_ : MeasurableSpace α} : InfSet (Measure α) :=
⟨fun m => (sInf (toOuterMeasure '' m)).toMeasure <| sInf_caratheodory⟩
theorem sInf_apply (hs : MeasurableSet s) : sInf m s = sInf (toOuterMeasure '' m) s :=
toMeasure_apply _ _ hs
private theorem measure_sInf_le (h : μ ∈ m) : sInf m ≤ μ :=
have : sInf (... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 991 | 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.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Order.Mono... | AntivaryOn (f₁ * f₂) g s := fun _i hi _j hj hij ↦ mul_le_mul' (h₁ hi hj hij) (h₂ hi hj hij)
| Mathlib/Algebra/Order/Monovary.lean | 50 | 51 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | theorem log_inv_eq_ite (x : ℂ) : log x⁻¹ = if x.arg = π then -conj (log x) else -log x := by
| Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 113 | 113 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | lemma iSup_genWeightSpaceOf_eq_top [IsTriangularizable R L M] (x : L) :
⨆ (φ : R), genWeightSpaceOf M φ x = ⊤ := by
rw [← LieSubmodule.toSubmodule_inj, LieSubmodule.iSup_toSubmodule,
LieSubmodule.top_toSubmodule]
dsimp [genWeightSpaceOf]
exact IsTriangularizable.maxGenEigenspace_eq_top x
open LinearMap M... | Mathlib/Algebra/Lie/Weights/Basic.lean | 730 | 739 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | variable (K)
theorem convexBodySumFun_continuous :
Continuous (convexBodySumFun : mixedSpace K → ℝ) := by
refine continuous_finset_sum Finset.univ fun w ↦ ?_
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 306 | 310 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... |
theorem support_prod_of_pairwise_disjoint (l : List (Perm α)) (h : l.Pairwise Disjoint) :
l.prod.support = (l.map support).foldr (· ⊔ ·) ⊥ := by
induction' l with hd tl hl
· simp
· rw [List.pairwise_cons] at h
| Mathlib/GroupTheory/Perm/Support.lean | 422 | 427 |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... | {f : Sigma β → α} (hs : s.SupIndep fun i => (g i).sup fun b => f ⟨i, b⟩)
(hg : ∀ i ∈ s, (g i).SupIndep fun b => f ⟨i, b⟩) : (s.sigma g).SupIndep f := by
rintro t ht ⟨i, b⟩ hi hit
rw [Finset.disjoint_sup_right]
rintro ⟨j, c⟩ hj
have hbc := (ne_of_mem_of_not_mem hj hit).symm
replace hj := ht hj
rw [me... | Mathlib/Order/SupIndep.lean | 186 | 194 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Algebra.Module.WeakBilin
/-!
# Weak Dual in Topolog... | def toSeminormFamily (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : SeminormFamily 𝕜 E F := fun y =>
(B.flip y).toSeminorm
@[simp]
| Mathlib/Analysis/LocallyConvex/WeakDual.lean | 73 | 76 |
/-
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.Algebra.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... | rw [mem_Icc, mem_Ico, lt_iff_le_pred h]
| Mathlib/Order/Interval/Finset/Nat.lean | 120 | 121 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 1,640 | 1,644 | |
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlge... | (i : ι) (j : κ) :
(Basis.tensorProduct b c).repr (m ⊗ₜ n) (i, j) = c.repr n j • b.repr m i := by
simp [Basis.tensorProduct, mul_comm]
| Mathlib/LinearAlgebra/TensorProduct/Basis.lean | 50 | 53 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | exact ((le_of_add_le_right (mem_antidiagonal.mp hx).le).trans_lt <| not_le.mp h).ne
theorem coeff_X_pow_mul' (p : R[X]) (n d : ℕ) :
(X ^ n * p).coeff d = ite (n ≤ d) (p.coeff (d - n)) 0 := by
rw [(commute_X_pow p n).eq, coeff_mul_X_pow']
| Mathlib/Algebra/Polynomial/Coeff.lean | 245 | 250 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | by_cases hx0 : f x = 0
· simp [hx0, IH]
have hy0 : f y ≠ 0 := by
contrapose! H
simpa [H, degree_eq_bot] using hx0
refine absurd H (h ?_ ?_ fun H => hx ?_)
· simp [hx0]
· simp [hy, hy0]
· exact H.symm ▸ hy
· rw [← IH, sup_eq_left.mpr H.le, degree_add_eq_left_... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 199 | 222 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | dlookup a l₁ = some b ∨ a ∉ l₁.keys ∧ dlookup a l₂ = some b :=
mem_dlookup_kunion
theorem mem_dlookup_kunion_middle {a} {b : β a} {l₁ l₂ l₃ : List (Sigma β)}
(h₁ : b ∈ dlookup a (kunion l₁ l₃)) (h₂ : a ∉ keys l₂) :
b ∈ dlookup a (kunion (kunion l₁ l₂) l₃) :=
| Mathlib/Data/List/Sigma.lean | 724 | 729 |
/-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
/-!
# Hermite polynomials
This file defines `Polynomial.hermite n`, the `n`... | | succ k =>
rw [coeff_hermite_succ_succ, ih, ih, mul_zero, sub_zero]
| Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 125 | 126 |
/-
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.ExactSequence
import Mathlib.Algebra.Homology.ShortComplex.Limits
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The snake lemma
The ... | @[reassoc (attr := simp)]
lemma lift_φ₂ {A : C} (a : A ⟶ S.L₁.X₂) (b : A ⟶ S.L₀.X₃) (h : a ≫ S.L₁.g = b ≫ S.v₀₁.τ₃) :
pullback.lift a b h ≫ S.φ₂ = a ≫ S.v₁₂.τ₂ := by
simp [φ₂]
| Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean | 226 | 229 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... | Mathlib/Data/Finset/NAry.lean | 666 | 668 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.NormNum
/-!
# Some facts about finite rings
-/
open Finset ZMod
section Ring
variable {R : Type*} [Ring R] [Fint... | lemma Finset.univ_of_card_le_three (h : Fintype.card R ≤ 3) :
(univ : Finset R) = {0, 1, -1} := by
refine (eq_of_subset_of_card_le (subset_univ _) ?_).symm
rcases lt_or_eq_of_le h with h | h
· apply card_le_card
rw [Finset.univ_of_card_le_two (Nat.lt_succ_iff.1 h)]
intro a ha
simp only [mem_insert... | Mathlib/RingTheory/Fintype.lean | 28 | 54 |
/-
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... | mem_nonzeroDivisors_of_coeff_mem _ h
end nonZeroDivisors
theorem natDegree_pos_of_monic_of_aeval_eq_zero [Nontrivial R] [Semiring S] [Algebra R S]
[FaithfulSMul R S] {p : R[X]} (hp : p.Monic) {x : S} (hx : aeval x p = 0) :
0 < p.natDegree :=
natDegree_pos_of_aeval_root (Monic.ne_zero hp) hx
| Mathlib/Algebra/Polynomial/RingDivision.lean | 129 | 136 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... |
-- TODO: eventually, deprecate and remove the nnnorm version
theorem eLpNorm'_le_nnreal_smul_eLpNorm'_of_ae_le_mul' {f : α → ε} {g : α → ε'} {c : ℝ≥0}
(h : ∀ᵐ x ∂μ, ‖f x‖ₑ ≤ c * ‖g x‖ₑ) {p : ℝ} (hp : 0 < p) :
eLpNorm' f p μ ≤ c • eLpNorm' g p μ := by
simp_rw [eLpNorm'_eq_lintegral_enorm]
rw [← ENNReal.rpow... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 1,150 | 1,156 |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kyle Miller, Eric Wieser
-/
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Int.GCD
import Mathlib.Tactic.NormNum
/-! # `norm_num` extensions for GC... |
theorem nat_lcm_helper (x y d m : ℕ) (hd : Nat.gcd x y = d)
(d0 : Nat.beq d 0 = false)
(dm : x * y = d * m) : Nat.lcm x y = m :=
mul_right_injective₀ (Nat.ne_of_beq_eq_false d0) <| by
dsimp only
| Mathlib/Tactic/NormNum/GCD.lean | 57 | 62 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Principal ordinals
We define principal or indecomposable ordinals, and we prove the standa... | theorem principal_mul_one : Principal (· * ·) 1 := by
rw [principal_one_iff]
exact zero_mul _
theorem principal_mul_two : Principal (· * ·) 2 := by
intro a b ha hb
rw [← succ_one, lt_succ_iff] at *
convert mul_le_mul' ha hb
exact (mul_one 1).symm
theorem principal_mul_of_le_two (ho : o ≤ 2) : Principal (·... | Mathlib/SetTheory/Ordinal/Principal.lean | 248 | 263 |
/-
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.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Left Homology of short complexes
Given a short complex `S : Shor... | LeftHomologyData.liftK_i _ k hk
@[reassoc]
| Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean | 942 | 944 |
/-
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... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,494 | 1,501 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 313 | 313 | |
/-
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.Dynamics.Newton
import Mathlib.LinearAlgebra.Semisimple
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
/-!
# Jordan-Chevalley-Dunford decomposition
... |
theorem exists_isNilpotent_isSemisimple_of_separable_of_dvd_pow {P : K[X]} {k : ℕ}
(sep : P.Separable) (nil : minpoly K f ∣ P ^ k) :
∃ᵉ (n ∈ adjoin K {f}) (s ∈ adjoin K {f}), IsNilpotent n ∧ IsSemisimple s ∧ f = n + s := by
set ff : adjoin K {f} := ⟨f, self_mem_adjoin_singleton K f⟩
set P' := derivative P
... | Mathlib/LinearAlgebra/JordanChevalley.lean | 42 | 65 |
/-
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, Heather Macbeth
-/
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
/-!
# Fiber bundles
Mathemat... |
variable (b : B) (a : F)
theorem open_source' (i : ι) : IsOpen (Z.localTrivAsPartialEquiv i).source := by
apply TopologicalSpace.GenerateOpen.basic
simp only [exists_prop, mem_iUnion, mem_singleton_iff]
refine ⟨i, Z.baseSet i ×ˢ univ, (Z.isOpen_baseSet i).prod isOpen_univ, ?_⟩
ext p
simp only [localTrivAsPa... | Mathlib/Topology/FiberBundle/Basic.lean | 528 | 540 |
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.Topology.Sheaves.Forget
import Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections
import Mathlib.CategoryTheory.Limits.Types.Shapes
/-!
#... | · rw [eqToHom_op, eqToHom_op, eqToHom_trans, eqToHom_refl, F.1.map_id,
ConcreteCategory.id_apply]
· exact gl'_spec i
@[ext]
theorem eq_of_locally_eq (s t : ToType (F.1.obj (op (iSup U))))
(h : ∀ i, F.1.map (Opens.leSupr U i).op s = F.1.map (Opens.leSupr U i).op t) : s = t := by
let sf : ∀ i : ι, ... | Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean | 198 | 211 |
/-
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... | private theorem measure_le_sInf (h : ∀ μ' ∈ m, μ ≤ μ') : μ ≤ sInf m :=
have : μ.toOuterMeasure ≤ sInf (toOuterMeasure '' m) :=
le_sInf <| forall_mem_image.2 fun _ hμ ↦ toOuterMeasure_le.2 <| h _ hμ
le_iff.2 fun s hs => by rw [sInf_apply hs]; exact this s
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,002 | 1,005 |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.NumberTheory.ModularForms.SlashActions
/-!
# Slash invariant forms
This file defines functions that are invariant under a `SlashAction` which forms t... | rw [SlashAction.add_slash, slash_action_eqn f γ hγ, slash_action_eqn g γ hγ] }⟩
@[simp]
| Mathlib/NumberTheory/ModularForms/SlashInvariantForms.lean | 100 | 102 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Martingale.Basic
/-!
# Centering lemma for stochastic processes
Any `ℕ`-indexed stochastic process which is adapted and integrable can be wri... | theorem martingalePart_bdd_difference {R : ℝ≥0} {f : ℕ → Ω → ℝ} (ℱ : Filtration ℕ m0)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, ∀ i, |martingalePart f ℱ μ (i + 1) ω - martingalePart f ℱ μ i ω| ≤ ↑(2 * R) := by
filter_upwards [hbdd, predictablePart_bdd_difference ℱ hbdd] with ω hω₁ hω₂ i
si... | Mathlib/Probability/Martingale/Centering.lean | 156 | 162 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Riccardo Brasca
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.RingTheory.Ideal.Quotient.Operations
import Mathl... | lemma nhds_one_hasBasis : (𝓝 (1 : M ⧸ S)).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {x | ‖x‖ < ε} := by
have : ∀ ε : ℝ, mk '' ball (1 : M) ε = {x : M ⧸ S | ‖x‖ < ε} := by
| Mathlib/Analysis/Normed/Group/Quotient.lean | 146 | 147 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-! # Positive Definite Matrices
This file defi... | whenPPOption getPPNotation <|
whenNotPPOption getPPAnalysisSkip <|
withOverApp 7 <|
withOptionAtCurrPos `pp.analysis.skip true do
let e ← getExpr
guard <| e.isAppOfArity ``Matrix.PosSemidef.sqrt 7
let optionsPerPos ← withNaryArg 6 do
return (← read).optionsPerPos.setBool (← getPos) `pp.proofs.... | Mathlib/LinearAlgebra/Matrix/PosDef.lean | 185 | 219 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | f /ₘ (X - C a) + (X - C a) * derivative (f /ₘ (X - C a)) = derivative f := by
have key := by apply congrArg derivative <| X_sub_C_mul_divByMonic_eq_sub_modByMonic f a
simpa only [derivative_mul, derivative_sub, derivative_X, derivative_C, sub_zero, one_mul,
modByMonic_X_sub_C_eq_C_eval] using key
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 598 | 602 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Bounded
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topology.MetricSpace.Thickening
/-!
# P... |
@[to_additive]
theorem ball_mul_singleton : ball x δ * {y} = ball (x * y) δ := by
| Mathlib/Analysis/Normed/Group/Pointwise.lean | 102 | 104 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 2,103 | 2,116 | |
/-
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... | instance OrderDual.instCoheytingAlgebra : CoheytingAlgebra αᵒᵈ where
| Mathlib/Order/Heyting/Basic.lean | 754 | 754 |
/-
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.Order.Filter.Curry
import Mathlib.Data.Set.Countable
/-!
# Filters with countable intersection property
In this file we define `CountableInterFilte... | ⋃ i, s i ≤ᶠ[l] ⋃ i, t i :=
(eventually_countable_forall.2 h).mono fun _ hst hs => mem_iUnion.2 <| (mem_iUnion.1 hs).imp hst
theorem EventuallyEq.countable_iUnion [Countable ι] {s t : ι → Set α} (h : ∀ i, s i =ᶠ[l] t i) :
⋃ i, s i =ᶠ[l] ⋃ i, t i :=
| Mathlib/Order/Filter/CountableInter.lean | 71 | 75 |
/-
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.CategoryTheory.FinCategory.AsType
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import ... | (@ULiftHom.category (ULift J) (@uliftCategory J hJ)) :=
@ULiftHomULiftCategory.equiv J hJ
apply @hasLimitsOfShape_of_equivalence (ULiftHom (ULift J))
(@ULiftHom.category (ULift J) (@uliftCategory J hJ)) C _ J hJ
(@Equivalence.symm J hJ (ULiftHom (ULift J))
(@ULiftHo... | Mathlib/CategoryTheory/Limits/Shapes/FiniteLimits.lean | 71 | 82 |
/-
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta, Doga Can Sertbas
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Real.Archimedea... | /-- The Schnirelmann density of `A` containing `0` is `1` if and only if `A` is the naturals. -/
lemma schnirelmannDensity_eq_one_iff_of_zero_mem (hA : 0 ∈ A) :
schnirelmannDensity A = 1 ↔ A = Set.univ := by
rw [schnirelmannDensity_eq_one_iff]
constructor
· refine fun h => Set.eq_univ_of_forall fun x => ?_
... | Mathlib/Combinatorics/Schnirelmann.lean | 128 | 138 |
/-
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.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Module.LinearMap.Prod
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Analysis... | obtain rfl | hb := hb.eq_or_lt
· rw [add_zero] at hab
rwa [hab, one_smul, zero_smul, add_zero]
exact h hy ha hb hab
theorem starConvex_iff_forall_ne_pos (hx : x ∈ s) :
StarConvex 𝕜 x s ↔
∀ ⦃y⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s := by
refine ⟨fun h y hy _... | Mathlib/Analysis/Convex/Star.lean | 165 | 175 |
/-
Copyright (c) 2020 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Arthur Paulino, Kyle Miller, Pim Otte
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph
import Mathlib.Com... |
theorem IsMatching.toEdge.surjective (h : M.IsMatching) : Surjective h.toEdge := by
rintro ⟨⟨x, y⟩, he⟩
exact ⟨⟨x, M.edge_vert he⟩, h.toEdge_eq_of_adj _ he⟩
| Mathlib/Combinatorics/SimpleGraph/Matching.lean | 77 | 80 |
/-
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... | @[deprecated (since := "2025-04-22")] alias norm_eq_sqrt_inner := norm_eq_sqrt_re_inner
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 347 | 348 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.FunLike.Equiv
import Mathlib.Data.Quot
import Mathlib.Data.Subtype
import Mathlib.Logic.U... | Mathlib/Logic/Equiv/Defs.lean | 899 | 900 | |
/-
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.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.Cast.Order.Ring
/-!
# Order properties of cast of integers
This... | lemma cast_le_neg_one_of_neg (h : a < 0) : (a : R) ≤ -1 := by
rw [← Int.cast_one, ← Int.cast_neg, cast_le]
| Mathlib/Algebra/Order/Ring/Cast.lean | 82 | 83 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | exact h.mono (subset_univ _)
theorem Continuous.continuousWithinAt (h : Continuous f) :
ContinuousWithinAt f s x :=
| Mathlib/Topology/ContinuousOn.lean | 818 | 821 |
/-
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.BooleanAlgebra
import Mathlib.Tactic.Common
/-!
# Co-Heyting boundary
The boundary of an element of a co-Heyting algebra is the intersection of its... | Mathlib/Order/Heyting/Boundary.lean | 140 | 143 | |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.Data.Matrix.ConjTranspose
/-! # Hermitian matrices
Th... | conjTranspose_natCast _
theorem IsHermitian.pow [Fintype n] [DecidableEq n] {A : Matrix n n α} (h : A.IsHermitian) (k : ℕ) :
| Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 215 | 217 |
/-
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.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | apply Cochain.comp_smul
@[simp]
protected lemma comp_id {n : ℤ} (z₁ : Cochain F G n) :
z₁.comp (Cochain.ofHom (𝟙 G)) (add_zero n) = z₁ := by
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 373 | 377 |
/-
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.Sphere.Basic
/-!
# Second intersection of a sphere and a line
This file defines and proves basic results about the second intersection... |
/-- A point on a line through a point on a sphere equals that point or `secondInter`. -/
theorem Sphere.eq_or_eq_secondInter_of_mem_mk'_span_singleton_iff_mem {s : Sphere P} {p : P}
(hp : p ∈ s) {v : V} {p' : P} (hp' : p' ∈ AffineSubspace.mk' p (ℝ ∙ v)) :
p' = p ∨ p' = s.secondInter p v ↔ p' ∈ s := by
refine... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 70 | 78 |
/-
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.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Currying and uncurrying continuous multilinear maps
... | (fun f => by
simp only [curryRight, LinearEquiv.coe_mk]
exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _)
(fun f => by
| Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean | 290 | 293 |
/-
Copyright (c) 2021 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.GroupTheory.MonoidLocalization.Away
impo... | _ < powHalf n := powHalf_succ_lt_powHalf n
· rcases n with - | n
| Mathlib/SetTheory/Surreal/Dyadic.lean | 116 | 117 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Center
import Mathlib.Algebra.GroupWithZero.Units.Basic
/-!
# Center of a group with zero
-/
assert_not_exists RelIso Finset R... | end GroupWithZero
end Set
| Mathlib/Algebra/GroupWithZero/Center.lean | 56 | 62 |
/-
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
/-!
# Tagged partitions
A tagged (pre)partition is a (pre)partition `π` enriched with a tagged point for each ... | is defined to be the `π.tag` of the box of the partition `π` that includes `J`.
Note that usually the result is not a Henstock partition. -/
@[simps -fullyApplied tag]
def biUnionPrepartition (π : TaggedPrepartition I) (πi : ∀ J : Box ι, Prepartition J) :
TaggedPrepartition I where
toPrepartition := π.toPreparti... | Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean | 156 | 165 |
/-
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... | | succ n IH =>
rw [iteratedDerivWithin_succ, Function.iterate_succ']
exact derivWithin_congr (fun y hy => IH) IH
/-- The `n+1`-th iterated derivative within a set with unique derivatives can be obtained by
taking the `n`-th derivative of the derivative. -/
| Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 196 | 201 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | Mathlib/Data/List/Rotate.lean | 684 | 687 |
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