Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
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.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Ma... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 145 | 154 | theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
(hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y := by |
by_cases h : ∃ x₀, Summable fun n => f n x₀
· rcases h with ⟨x₀, hf0⟩
intro x
exact (hasFDerivAt_tsum hu hf hf' hf0 x).differentiableAt
· push_neg at h
have : (fun x => ∑' n, f n x) = 0 := by ext1 x; exact tsum_eq_zero_of_not_summable (h x)
rw [this]
exact differentiable_const 0
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Basic
#align_import analysis.normed.group.hom from "leanprover-community/mathlib"@"3c4225288b55380a90df078ebae0991080b12393"
/-... | Mathlib/Analysis/Normed/Group/Hom.lean | 760 | 762 | theorem ker_zero : (0 : NormedAddGroupHom V₁ V₂).ker = ⊤ := by |
ext
simp [mem_ker]
|
/-
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.Order.Module.Defs
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.order from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5... | Mathlib/Data/DFinsupp/Order.lean | 313 | 316 | theorem support_inf : (f ⊓ g).support = f.support ∩ g.support := by |
ext
simp only [inf_apply, mem_support_iff, Ne, Finset.mem_inter]
simp only [inf_eq_min, ← nonpos_iff_eq_zero, min_le_iff, not_or]
|
/-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Filtered.Final
/-!
# Finally small categories
A category given by `(J : Type u) [Category.{v} J]` is `w`-finally small if there exists a... | Mathlib/CategoryTheory/Limits/FinallySmall.lean | 132 | 136 | theorem FinallySmall.exists_small_weakly_terminal_set [FinallySmall.{w} J] :
∃ (s : Set J) (_ : Small.{w} s), ∀ i, ∃ j ∈ s, Nonempty (i ⟶ j) := by |
refine ⟨Set.range (fromFinalModel J).obj, inferInstance, fun i => ?_⟩
obtain ⟨f⟩ : Nonempty (StructuredArrow i (fromFinalModel J)) := IsConnected.is_nonempty
exact ⟨(fromFinalModel J).obj f.right, Set.mem_range_self _, ⟨f.hom⟩⟩
|
/-
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.Algebra.Module.BigOperators
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Int.ModEq
import Mat... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 365 | 390 | theorem mem_list_cycles_iff {α : Type*} [Finite α] {l : List (Perm α)}
(h1 : ∀ σ : Perm α, σ ∈ l → σ.IsCycle) (h2 : l.Pairwise Disjoint) {σ : Perm α} :
σ ∈ l ↔ σ.IsCycle ∧ ∀ a, σ a ≠ a → σ a = l.prod a := by |
suffices σ.IsCycle → (σ ∈ l ↔ ∀ a, σ a ≠ a → σ a = l.prod a) by
exact ⟨fun hσ => ⟨h1 σ hσ, (this (h1 σ hσ)).mp hσ⟩, fun hσ => (this hσ.1).mpr hσ.2⟩
intro h3
classical
cases nonempty_fintype α
constructor
· intro h a ha
exact eq_on_support_mem_disjoint h h2 _ (mem_support.mpr ha)
· intro... |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.Se... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 149 | 150 | theorem mul_star_self_ne_zero_iff (x : E) : x * x⋆ ≠ 0 ↔ x ≠ 0 := by |
simp only [Ne, mul_star_self_eq_zero_iff]
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space
#align_import measure_theory.function.uniform_integrable from "lea... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 760 | 775 | theorem uniformIntegrable_finite [Finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
(hf : ∀ i, Memℒp (f i) p μ) : UniformIntegrable f p μ := by |
cases nonempty_fintype ι
refine ⟨fun n => (hf n).1, unifIntegrable_finite hp_one hp_top hf, ?_⟩
by_cases hι : Nonempty ι
· choose _ hf using hf
set C := (Finset.univ.image fun i : ι => snorm (f i) p μ).max'
⟨snorm (f hι.some) p μ, Finset.mem_image.2 ⟨hι.some, Finset.mem_univ _, rfl⟩⟩
refine ⟨C.to... |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover... | Mathlib/Topology/Connected/PathConnected.lean | 1,208 | 1,209 | theorem pathConnectedSpace_iff_eq : PathConnectedSpace X ↔ ∃ x : X, pathComponent x = univ := by |
simp [pathConnectedSpace_iff_univ, isPathConnected_iff_eq]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 456 | 477 | theorem norm_eq_zero_iff {f : lp E p} : ‖f‖ = 0 ↔ f = 0 := by |
refine ⟨fun h => ?_, by rintro rfl; exact norm_zero⟩
rcases p.trichotomy with (rfl | rfl | hp)
· ext i
have : { i : α | ¬f i = 0 } = ∅ := by simpa [lp.norm_eq_card_dsupport f] using h
have : (¬f i = 0) = False := congr_fun this i
tauto
· cases' isEmpty_or_nonempty α with _i _i
· simp [eq_iff_tr... |
/-
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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 950 | 956 | theorem leadingCoeff_add_of_degree_eq (h : degree p = degree q)
(hlc : leadingCoeff p + leadingCoeff q ≠ 0) :
leadingCoeff (p + q) = leadingCoeff p + leadingCoeff q := by |
have : natDegree (p + q) = natDegree p := by
apply natDegree_eq_of_degree_eq
rw [degree_add_eq_of_leadingCoeff_add_ne_zero hlc, h, max_self]
simp only [leadingCoeff, this, natDegree_eq_of_degree_eq h, coeff_add]
|
/-
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.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 1,421 | 1,428 | theorem Ideal.squarefree_span_singleton {a : R} :
Squarefree (span {a}) ↔ Squarefree a := by |
refine ⟨fun h x hx ↦ ?_, fun h I hI ↦ ?_⟩
· rw [← span_singleton_dvd_span_singleton_iff_dvd, ← span_singleton_mul_span_singleton] at hx
simpa using h _ hx
· rw [← span_singleton_generator I, span_singleton_mul_span_singleton,
span_singleton_dvd_span_singleton_iff_dvd] at hI
exact isUnit_iff.mpr <| ... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 2,303 | 2,304 | theorem countP_add (s t) : countP p (s + t) = countP p s + countP p t := by |
simp [countP_eq_card_filter]
|
/-
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.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 744 | 744 | theorem coe_eq_univ : (s : Set α) = univ ↔ s = ⊤ := by | simp [SetLike.ext'_iff]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Chris Hughes, Michael Howes
-/
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Semiconj.Units
#align_import algebra.group.conj from "leanprover-community... | Mathlib/Algebra/Group/Conj.lean | 110 | 113 | theorem conj_pow {i : ℕ} {a b : α} : (a * b * a⁻¹) ^ i = a * b ^ i * a⁻¹ := by |
induction' i with i hi
· simp
· simp [pow_succ, hi]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 1,732 | 1,748 | theorem map_injective_of_injective {f : G →g G'} (hinj : Function.Injective f) (u v : V) :
Function.Injective (Walk.map f : G.Walk u v → G'.Walk (f u) (f v)) := by |
intro p p' h
induction p with
| nil =>
cases p'
· rfl
· simp at h
| cons _ _ ih =>
cases p' with
| nil => simp at h
| cons _ _ =>
simp only [map_cons, cons.injEq] at h
cases hinj h.1
simp only [cons.injEq, heq_iff_eq, true_and_iff]
apply ih
simpa using h.2
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprove... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 53 | 55 | theorem terminatedAt_iff_part_denom_none :
g.TerminatedAt n ↔ g.partialDenominators.get? n = none := by |
rw [terminatedAt_iff_s_none, part_denom_none_iff_s_none]
|
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.RingTheory.Valuation.Integers
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localizat... | Mathlib/RingTheory/Valuation/ValuationRing.lean | 303 | 307 | theorem unique_irreducible [ValuationRing R] ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
Associated p q := by |
have := dvd_total p q
rw [Irreducible.dvd_comm hp hq, or_self_iff] at this
exact associated_of_dvd_dvd (Irreducible.dvd_symm hq hp this) this
|
/-
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.DivergenceTheorem
import Mathlib.Analysis.BoxIntegral.Integrability
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.... | Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean | 143 | 245 | theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (Fin (n + 1)))
(f : ℝⁿ⁺¹ → Eⁿ⁺¹)
(f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹)
(s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))
(Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x)
(Hi : IntegrableOn (∑ i, f' · (e i) i) (B... |
/- Choose a monotone sequence `J k` of subboxes that cover the interior of `I` and prove that
these boxes satisfy the assumptions of the previous lemma. -/
rcases I.exists_seq_mono_tendsto with ⟨J, hJ_sub, hJl, hJu⟩
have hJ_sub' : ∀ k, Box.Icc (J k) ⊆ Box.Icc I := fun k => (hJ_sub k).trans I.Ioo_subset_Icc
... |
/-
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.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 308 | 309 | theorem logb_le_logb_of_base_lt_one (h : 0 < x) (h₁ : 0 < y) : logb b x ≤ logb b y ↔ y ≤ x := by |
rw [logb, logb, div_le_div_right_of_neg (log_neg b_pos b_lt_one), log_le_log_iff h₁ h]
|
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Zinkevich, Vincent Beffara
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.Probability.Independence.Basic
#align_import probability.integration from "lean... | Mathlib/Probability/Integration.lean | 45 | 73 | theorem lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator {Mf mΩ : MeasurableSpace Ω}
{μ : Measure Ω} (hMf : Mf ≤ mΩ) (c : ℝ≥0∞) {T : Set Ω} (h_meas_T : MeasurableSet T)
(h_ind : IndepSets {s | MeasurableSet[Mf] s} {T} μ) (h_meas_f : Measurable[Mf] f) :
(∫⁻ ω, f ω * T.indicator (fun _ => c) ω ∂μ)... |
revert f
have h_mul_indicator : ∀ g, Measurable g → Measurable fun a => g a * T.indicator (fun _ => c) a :=
fun g h_mg => h_mg.mul (measurable_const.indicator h_meas_T)
apply @Measurable.ennreal_induction _ Mf
· intro c' s' h_meas_s'
simp_rw [← inter_indicator_mul]
rw [lintegral_indicator _ (Measur... |
/-
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.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 87 | 92 | theorem inv_anti_mono (hI : I ≠ 0) (hJ : J ≠ 0) (hIJ : I ≤ J) : J⁻¹ ≤ I⁻¹ := by |
-- Porting note: in Lean3, introducing `x` would just give `x ∈ J⁻¹ → x ∈ I⁻¹`, but
-- in Lean4, it goes all the way down to the subtypes
intro x
simp only [val_eq_coe, mem_coe, mem_inv_iff hJ, mem_inv_iff hI]
exact fun h y hy => h y (hIJ hy)
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
/-!
# Darts in graphs
A `Dart` or half-edge or bond in a graph is an ordered pair of adjacent vertices, regarded as an
orie... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 33 | 34 | theorem Dart.ext_iff (d₁ d₂ : G.Dart) : d₁ = d₂ ↔ d₁.toProd = d₂.toProd := by |
cases d₁; cases d₂; simp
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 218 | 221 | theorem exists_prime_factors (a : α) :
a ≠ 0 → ∃ f : Multiset α, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a := by |
simp_rw [← UniqueFactorizationMonoid.irreducible_iff_prime]
apply WfDvdMonoid.exists_factors a
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-com... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 477 | 479 | theorem sheafifyMap_id (P : Cᵒᵖ ⥤ D) : J.sheafifyMap (𝟙 P) = 𝟙 (J.sheafify P) := by |
dsimp [sheafifyMap, sheafify]
simp
|
/-
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.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 424 | 429 | theorem support_prod_le (l : List (Perm α)) : l.prod.support ≤ (l.map support).foldr (· ⊔ ·) ⊥ := by |
induction' l with hd tl hl
· 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
|
/-
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.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 1,568 | 1,585 | theorem divMod_to_nat_aux {n d : PosNum} {q r : Num} (h₁ : (r : ℕ) + d * _root_.bit0 (q : ℕ) = n)
(h₂ : (r : ℕ) < 2 * d) :
((divModAux d q r).2 + d * (divModAux d q r).1 : ℕ) = ↑n ∧ ((divModAux d q r).2 : ℕ) < d := by |
unfold divModAux
have : ∀ {r₂}, Num.ofZNum' (Num.sub' r (Num.pos d)) = some r₂ ↔ (r : ℕ) = r₂ + d := by
intro r₂
apply Num.mem_ofZNum'.trans
rw [← ZNum.to_int_inj, Num.cast_toZNum, Num.cast_sub', sub_eq_iff_eq_add, ← Int.natCast_inj]
simp
cases' e : Num.ofZNum' (Num.sub' r (Num.pos d)) with r₂ <;... |
/-
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.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,787 | 1,796 | theorem controlled_prod_of_mem_closure_range {j : E →* F} {b : F}
(hb : b ∈ closure (j.range : Set F)) {f : ℕ → ℝ} (b_pos : ∀ n, 0 < f n) :
∃ a : ℕ → E,
Tendsto (fun n => ∏ i ∈ range (n + 1), j (a i)) atTop (𝓝 b) ∧
‖j (a 0) / b‖ < f 0 ∧ ∀ n, 0 < n → ‖j (a n)‖ < f n := by |
obtain ⟨v, sum_v, v_in, hv₀, hv_pos⟩ := controlled_prod_of_mem_closure hb b_pos
choose g hg using v_in
exact
⟨g, by simpa [← hg] using sum_v, by simpa [hg 0] using hv₀,
fun n hn => by simpa [hg] using hv_pos n hn⟩
|
/-
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 Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,375 | 1,376 | theorem range_subset {m n : Nat} : range m ⊆ range n ↔ m ≤ n := by |
simp only [range_eq_range', range'_subset_right, lt_succ_self]
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Array.Lemmas
import Batteries.Tactic.Lint.Misc
namespace Batteries
/-- Union-find node type -/
structure UFNode where
/-- Parent of node -/
... | .lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | 328 | 336 | theorem parentD_findAux_lt {self : UnionFind} {x : Fin self.size} (h : i < self.size) :
parentD (findAux self x).s i < self.size := by |
if h' : (self.arr.get x).parent = x then
rw [findAux_s, if_pos h']; apply self.parentD_lt h
else
rw [parentD_findAux]; split <;> [simp [rootD_lt]; skip]
have := Nat.sub_lt_sub_left (self.lt_rankMax x) (self.rank'_lt _ ‹_›)
apply parentD_findAux_lt h
termination_by self.rankMax - self.rank x
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
/-!
# Compl... | Mathlib/GroupTheory/Complement.lean | 144 | 153 | theorem isComplement_univ_left : IsComplement univ S ↔ ∃ g : G, S = {g} := by |
refine
⟨fun h => Set.exists_eq_singleton_iff_nonempty_subsingleton.mpr ⟨?_, fun a ha b hb => ?_⟩, ?_⟩
· obtain ⟨a, _⟩ := h.2 1
exact ⟨a.2.1, a.2.2⟩
· have : (⟨⟨_, mem_top a⁻¹⟩, ⟨a, ha⟩⟩ : (⊤ : Set G) × S) = ⟨⟨_, mem_top b⁻¹⟩, ⟨b, hb⟩⟩ :=
h.1 ((inv_mul_self a).trans (inv_mul_self b).symm)
exact ... |
/-
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.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
#align_import logic.equiv.set from "leanprover-... | Mathlib/Logic/Equiv/Set.lean | 155 | 158 | theorem prod_assoc_symm_preimage {α β γ} {s : Set α} {t : Set β} {u : Set γ} :
(Equiv.prodAssoc α β γ).symm ⁻¹' (s ×ˢ t) ×ˢ u = s ×ˢ t ×ˢ u := by |
ext
simp [and_assoc]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 1,429 | 1,437 | theorem continuous_of_le_add_edist {f : α → ℝ≥0∞} (C : ℝ≥0∞) (hC : C ≠ ∞)
(h : ∀ x y, f x ≤ f y + C * edist x y) : Continuous f := by |
refine continuous_iff_continuousAt.2 fun x => ENNReal.tendsto_nhds_of_Icc fun ε ε0 => ?_
rcases ENNReal.exists_nnreal_pos_mul_lt hC ε0.ne' with ⟨δ, δ0, hδ⟩
rw [mul_comm] at hδ
filter_upwards [EMetric.closedBall_mem_nhds x (ENNReal.coe_pos.2 δ0)] with y hy
refine ⟨tsub_le_iff_right.2 <| (h x y).trans ?_, (h y... |
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 1,435 | 1,437 | theorem algEquivOfEq_apply_root {p q : K[X]} (hp : p ≠ 0) (h_eq : p = q) :
algEquivOfEq hp h_eq (root p) = root q := by |
rw [← coe_algHom, algEquivOfEq_toAlgHom, liftHom_root]
|
/-
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.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#ali... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 579 | 595 | theorem isPairSelfAdjoint_equiv (e : M₁ ≃ₗ[R] M) (f : Module.End R M) :
IsPairSelfAdjoint B F f ↔
IsPairSelfAdjoint (B.compl₁₂ ↑e ↑e) (F.compl₁₂ ↑e ↑e) (e.symm.conj f) := by |
have hₗ :
(F.compl₁₂ (↑e : M₁ →ₗ[R] M) (↑e : M₁ →ₗ[R] M)).comp (e.symm.conj f) =
(F.comp f).compl₁₂ (↑e : M₁ →ₗ[R] M) (↑e : M₁ →ₗ[R] M) := by
ext
simp only [LinearEquiv.symm_conj_apply, coe_comp, LinearEquiv.coe_coe, compl₁₂_apply,
LinearEquiv.apply_symm_apply, Function.comp_apply]
have hᵣ ... |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | Mathlib/SetTheory/Game/Nim.lean | 73 | 75 | theorem moveLeft_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) := inferInstance
HEq (nim o).moveLeft fun i : o.out.α => nim (typein (· < ·) i) := by | rw [nim_def]; rfl
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | Mathlib/Analysis/Fourier/AddCircle.lean | 419 | 422 | theorem hasSum_fourier_series_L2 (f : Lp ℂ 2 <| @haarAddCircle T hT) :
HasSum (fun i => fourierCoeff f i • fourierLp 2 i) f := by |
simp_rw [← fourierBasis_repr]; rw [← coe_fourierBasis]
exact HilbertBasis.hasSum_repr fourierBasis f
|
/-
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.Star.Basic
import Mathlib.Algebra.Order.CauSeq.Completion
#align_import data.real.basic from "leanprover-community/mathlib"@... | Mathlib/Data/Real/Basic.lean | 332 | 334 | theorem mk_pos {f : CauSeq ℚ abs} : 0 < mk f ↔ Pos f := by |
rw [← mk_zero, mk_lt]
exact iff_of_eq (congr_arg Pos (sub_zero f))
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 118 | 119 | theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by |
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
|
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.RingTheory.Prime
import Mathlib.RingTheory.Polynomial.Content
import Mathlib.RingTheory.Ideal.Quotient
#align_import ... | Mathlib/RingTheory/EisensteinCriterion.lean | 52 | 61 | theorem le_natDegree_of_map_eq_mul_X_pow {n : ℕ} {P : Ideal R} (hP : P.IsPrime) {q : R[X]}
{c : Polynomial (R ⧸ P)} (hq : map (mk P) q = c * X ^ n) (hc0 : c.degree = 0) :
n ≤ q.natDegree :=
Nat.cast_le.1
(calc
↑n = degree (q.map (mk P)) := by |
rw [hq, degree_mul, hc0, zero_add, degree_pow, degree_X, nsmul_one]
_ ≤ degree q := degree_map_le _ _
_ ≤ natDegree q := degree_le_natDegree
)
|
/-
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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 978 | 980 | theorem eval₂_map [Semiring T] (g : S →+* T) (x : T) :
(p.map f).eval₂ g x = p.eval₂ (g.comp f) x := by |
rw [eval₂_eq_eval_map, eval₂_eq_eval_map, map_map]
|
/-
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.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.RingTheory.Loca... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 616 | 644 | theorem of_affine_open_cover {X : Scheme} (V : X.affineOpens) (S : Set X.affineOpens)
{P : X.affineOpens → Prop}
(hP₁ : ∀ (U : X.affineOpens) (f : X.presheaf.obj <| op U.1), P U → P (X.affineBasicOpen f))
(hP₂ :
∀ (U : X.affineOpens) (s : Finset (X.presheaf.obj <| op U))
(_ : Ideal.span (s : S... |
classical
have : ∀ (x : V.1), ∃ f : X.presheaf.obj <| op V.1,
↑x ∈ X.basicOpen f ∧ P (X.affineBasicOpen f) := by
intro x
obtain ⟨W, hW⟩ := Set.mem_iUnion.mp (by simpa only [← hS] using Set.mem_univ (x : X))
obtain ⟨f, g, e, hf⟩ := exists_basicOpen_le_affine_inter V.prop W.1.prop x ⟨x.prop, hW⟩
... |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Separation
import Mathlib.Topology.Bases
#align_import topology.dense_embedding from "leanprover-community/mathl... | Mathlib/Topology/DenseEmbedding.lean | 117 | 124 | theorem tendsto_comap_nhds_nhds {d : δ} {a : α} (di : DenseInducing i)
(H : Tendsto h (𝓝 d) (𝓝 (i a))) (comm : h ∘ g = i ∘ f) : Tendsto f (comap g (𝓝 d)) (𝓝 a) := by |
have lim1 : map g (comap g (𝓝 d)) ≤ 𝓝 d := map_comap_le
replace lim1 : map h (map g (comap g (𝓝 d))) ≤ map h (𝓝 d) := map_mono lim1
rw [Filter.map_map, comm, ← Filter.map_map, map_le_iff_le_comap] at lim1
have lim2 : comap i (map h (𝓝 d)) ≤ comap i (𝓝 (i a)) := comap_mono H
rw [← di.nhds_eq_comap] at l... |
/-
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.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
/-!
# Exact sequences with free modules
This file proves resu... | Mathlib/Algebra/Category/ModuleCat/Free.lean | 166 | 174 | theorem free_shortExact_finrank_add {n p : ℕ} [Module.Free R S.X₁] [Module.Free R S.X₃]
[Module.Finite R S.X₁] [Module.Finite R S.X₃]
(hN : FiniteDimensional.finrank R S.X₁ = n)
(hP : FiniteDimensional.finrank R S.X₃ = p)
[StrongRankCondition R] :
FiniteDimensional.finrank R S.X₂ = n + p := by |
apply FiniteDimensional.finrank_eq_of_rank_eq
rw [free_shortExact_rank_add hS', ← hN, ← hP]
simp only [Nat.cast_add, finrank_eq_rank]
|
/-
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.Hom.Bounded
import Mathlib.Order.SymmDiff
#align_import order.hom.lattice from "leanprover-community/mathlib"@"7581030920af3dcb241d1df0e36f6ec8289dd... | Mathlib/Order/Hom/Lattice.lean | 295 | 296 | theorem Codisjoint.map (h : Codisjoint a b) : Codisjoint (f a) (f b) := by |
rw [codisjoint_iff, ← map_sup, h.eq_top, map_top]
|
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 225 | 266 | theorem hasDerivAt_ofReal_cpow {x : ℝ} (hx : x ≠ 0) {r : ℂ} (hr : r ≠ -1) :
HasDerivAt (fun y : ℝ => (y : ℂ) ^ (r + 1) / (r + 1)) (x ^ r) x := by |
rw [Ne, ← add_eq_zero_iff_eq_neg, ← Ne] at hr
rcases lt_or_gt_of_ne hx.symm with (hx | hx)
· -- easy case : `0 < x`
-- Porting note: proof used to be
-- convert (((hasDerivAt_id (x : ℂ)).cpow_const _).div_const (r + 1)).comp_ofReal using 1
-- · rw [add_sub_cancel, id.def, mul_one, mul_comm, mul_div_c... |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Algebra.Lie.Quotient
#align_import algebra.lie.normalizer from "leanprover-com... | Mathlib/Algebra/Lie/Normalizer.lean | 134 | 135 | theorem mem_normalizer_iff' (x : L) : x ∈ H.normalizer ↔ ∀ y : L, y ∈ H → ⁅y, x⁆ ∈ H := by |
rw [Subtype.forall']; rfl
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 235 | 246 | theorem peval_polyOfInterest' (n : ℕ) (x y : 𝕎 k) :
peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i] =
(x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) -
x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1) := by |
rw [peval_polyOfInterest]
have : (p : k) = 0 := CharP.cast_eq_zero k p
simp only [this, Nat.cast_pow, ne_eq, add_eq_zero, and_false, zero_pow, zero_mul, add_zero,
not_false_eq_true]
have sum_zero_pow_mul_pow_p (y : 𝕎 k) : ∑ x ∈ range (n + 1 + 1),
(0 : k) ^ x * y.coeff x ^ p ^ (n + 1 - x) = y.coeff 0... |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_imp... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 63 | 64 | theorem unitors_equal : (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom := by |
coherence
|
/-
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.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 430 | 441 | theorem exists_seq_norm_le_one_le_norm_sub' {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
(h : ¬FiniteDimensional 𝕜 E) :
∃ f : ℕ → E, (∀ n, ‖f n‖ ≤ R) ∧ Pairwise fun m n => 1 ≤ ‖f m - f n‖ := by |
have : IsSymm E fun x y : E => 1 ≤ ‖x - y‖ := by
constructor
intro x y hxy
rw [← norm_neg]
simpa
apply
exists_seq_of_forall_finset_exists' (fun x : E => ‖x‖ ≤ R) fun (x : E) (y : E) => 1 ≤ ‖x - y‖
rintro s -
exact exists_norm_le_le_norm_sub_of_finset hc hR h s
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Simon Hudon
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.PEmpty
#align_i... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean | 242 | 246 | theorem tensor_id (X₁ X₂ : C) : tensorHom ℬ (𝟙 X₁) (𝟙 X₂) = 𝟙 (tensorObj ℬ X₁ X₂) := by |
apply IsLimit.hom_ext (ℬ _ _).isLimit;
rintro ⟨⟨⟩⟩ <;>
· dsimp [tensorHom]
simp
|
/-
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
-/
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Com... | Mathlib/Topology/Algebra/UniformGroup.lean | 369 | 373 | theorem Filter.HasBasis.uniformity_of_nhds_one_inv_mul_swapped {ι} {p : ι → Prop} {U : ι → Set α}
(h : (𝓝 (1 : α)).HasBasis p U) :
(𝓤 α).HasBasis p fun i => { x : α × α | x.2⁻¹ * x.1 ∈ U i } := by |
rw [uniformity_eq_comap_inv_mul_nhds_one_swapped]
exact h.comap _
|
/-
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.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Tactic.GCongr.Core
#align_import algebra.order... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 724 | 725 | theorem mul_le_of_le_one_left [MulPosMono α] (hb : 0 ≤ b) (h : a ≤ 1) : a * b ≤ b := by |
simpa only [one_mul] using mul_le_mul_of_nonneg_right h hb
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanpro... | Mathlib/RingTheory/Ideal/Operations.lean | 153 | 159 | theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop}
(smul : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn))
(add : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by |
refine Exists.elim ?_ fun (h : x ∈ I • N) (H : p x h) => H
exact
smul_induction_on hx (fun a ha x hx => ⟨_, smul _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ =>
⟨_, add _ _ _ _ hx hy⟩
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 375 | 378 | theorem filter_lt_le_eq_Ioc [DecidablePred fun j => a < j ∧ j ≤ b] :
(univ.filter fun j => a < j ∧ j ≤ b) = Ioc a b := by |
ext
simp
|
/-
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.Topology.Sets.Closeds
#align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Noetheri... | Mathlib/Topology/NoetherianSpace.lean | 223 | 264 | theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [NoetherianSpace α]
(Z : Set α) (H : Z ∈ irreducibleComponents α) :
∃ o : Set α, IsOpen o ∧ o ≠ ∅ ∧ o ≤ Z := by |
classical
let ι : Set (Set α) := irreducibleComponents α \ {Z}
have hι : ι.Finite := NoetherianSpace.finite_irreducibleComponents.subset Set.diff_subset
have hι' : Finite ι := by rwa [Set.finite_coe_iff]
let U := Z \ ⋃ (x : ι), x
have hU0 : U ≠ ∅ := fun r ↦ by
obtain ⟨Z', hZ'⟩ := isIrreducible_iff_sU... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 72 | 81 | theorem affineIndependent_iff_of_fintype [Fintype ι] (p : ι → P) :
AffineIndependent k p ↔
∀ w : ι → k, ∑ i, w i = 0 → Finset.univ.weightedVSub p w = (0 : V) → ∀ i, w i = 0 := by |
constructor
· exact fun h w hw hs i => h Finset.univ w hw hs i (Finset.mem_univ _)
· intro h s w hw hs i hi
rw [Finset.weightedVSub_indicator_subset _ _ (Finset.subset_univ s)] at hs
rw [← Finset.sum_indicator_subset _ (Finset.subset_univ s)] at hw
replace h := h ((↑s : Set ι).indicator w) hw hs i
... |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 320 | 322 | theorem cyclotomic.monic (n : ℕ) (R : Type*) [Ring R] : (cyclotomic n R).Monic := by |
rw [← map_cyclotomic_int]
exact (int_cyclotomic_spec n).2.2.map _
|
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johanes Hölzl, Patrick Massot, Yury Kudryashov, Kevin Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@... | Mathlib/Order/Filter/Prod.lean | 519 | 520 | theorem coprod_bot (l : Filter α) : l.coprod (⊥ : Filter β) = comap Prod.fst l := by |
simp [Filter.coprod]
|
/-
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.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
impor... | Mathlib/Order/SupIndep.lean | 440 | 442 | theorem Independent.injective (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Injective t := by |
suffices univ = {i | t i ≠ ⊥} by rw [injective_iff_injOn_univ, this]; exact ht.injOn
aesop
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,977 | 1,980 | theorem sub_vecMul [Fintype m] (A : Matrix m n α) (x y : m → α) :
(x - y) ᵥ* A = x ᵥ* A - y ᵥ* A := by |
ext
apply sub_dotProduct
|
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
/-!
# Purely inseparable extension and relative perfect closure
This file contains basic... | Mathlib/FieldTheory/PurelyInseparable.lean | 982 | 985 | theorem sepDegree_mul_sepDegree_of_isAlgebraic (K : Type v) [Field K] [Algebra F K]
[Algebra E K] [IsScalarTower F E K] [Algebra.IsAlgebraic F E] :
sepDegree F E * sepDegree E K = sepDegree F K := by |
simpa only [Cardinal.lift_id] using lift_sepDegree_mul_lift_sepDegree_of_isAlgebraic F E K
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 345 | 347 | theorem comp_add_left (hf : IntervalIntegrable f volume a b) (c : ℝ) :
IntervalIntegrable (fun x => f (c + x)) volume (a - c) (b - c) := by |
simpa only [add_comm] using IntervalIntegrable.comp_add_right hf c
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_ze... | Mathlib/Topology/Algebra/GroupWithZero.lean | 219 | 221 | theorem Continuous.div₀ (hf : Continuous f) (hg : Continuous g) (h₀ : ∀ x, g x ≠ 0) :
Continuous (fun x => f x / g x) := by |
simpa only [div_eq_mul_inv] using hf.mul (hg.inv₀ h₀)
|
/-
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.ModelTheory.Syntax
import Mathlib.ModelTheory.Semantics
import Mathlib.Algebra.Ring.Equiv
/-!
# First Order Language of Rings
This file defines the fir... | Mathlib/ModelTheory/Algebra/Ring/Basic.lean | 138 | 140 | theorem card_ring : card Language.ring = 5 := by |
have : Fintype.card Language.ring.Symbols = 5 := rfl
simp [Language.card, this]
|
/-
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.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Noetherian
#align_... | Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 83 | 103 | theorem exists_le_isAssociatedPrime_of_isNoetherianRing [H : IsNoetherianRing R] (x : M)
(hx : x ≠ 0) : ∃ P : Ideal R, IsAssociatedPrime P M ∧ (R ∙ x).annihilator ≤ P := by |
have : (R ∙ x).annihilator ≠ ⊤ := by
rwa [Ne, Ideal.eq_top_iff_one, Submodule.mem_annihilator_span_singleton, one_smul]
obtain ⟨P, ⟨l, h₁, y, rfl⟩, h₃⟩ :=
set_has_maximal_iff_noetherian.mpr H
{ P | (R ∙ x).annihilator ≤ P ∧ P ≠ ⊤ ∧ ∃ y : M, P = (R ∙ y).annihilator }
⟨(R ∙ x).annihilator, rfl.le... |
/-
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.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 333 | 336 | theorem ker_prodMap (f : M →ₗ[R] M₂) (g : M₃ →ₗ[R] M₄) :
ker (LinearMap.prodMap f g) = Submodule.prod (ker f) (ker g) := by |
dsimp only [ker]
rw [← prodMap_comap_prod, Submodule.prod_bot]
|
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.Associated
import Mathlib.NumberTheory.Divisors
#align_import algebra.is_prime_pow from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0... | Mathlib/Algebra/IsPrimePow.lean | 84 | 91 | theorem isPrimePow_nat_iff_bounded (n : ℕ) :
IsPrimePow n ↔ ∃ p : ℕ, p ≤ n ∧ ∃ k : ℕ, k ≤ n ∧ p.Prime ∧ 0 < k ∧ p ^ k = n := by |
rw [isPrimePow_nat_iff]
refine Iff.symm ⟨fun ⟨p, _, k, _, hp, hk, hn⟩ => ⟨p, k, hp, hk, hn⟩, ?_⟩
rintro ⟨p, k, hp, hk, rfl⟩
refine ⟨p, ?_, k, (Nat.lt_pow_self hp.one_lt _).le, hp, hk, rfl⟩
conv => { lhs; rw [← (pow_one p)] }
exact Nat.pow_le_pow_right hp.one_lt.le hk
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Andrew Yang
-/
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.RingTheory.GradedAlgeb... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 393 | 419 | theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q := by |
revert c
refine DirectSum.Decomposition.inductionOn 𝒜 ?_ ?_ ?_
· rw [zero_smul]; exact carrier.zero_mem f_deg hm _
· rintro n ⟨a, ha⟩ i
simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
-- Porting note: having trouble with Mul instance
let product : A⁰_ f :=
Mul.mul (H... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
imp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 353 | 354 | theorem braiding_tensorUnit_right (X : C) : (β_ X (𝟙_ C)).hom = (ρ_ X).hom ≫ (λ_ X).inv := by |
simp [← rightUnitor_inv_braiding]
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 1,055 | 1,055 | theorem lift_comp : (f.lift hg).comp f.toMap = g := by | ext; exact f.lift_eq hg _
|
/-
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.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.MulA... | Mathlib/Topology/Algebra/Module/Basic.lean | 222 | 234 | theorem LinearMap.continuous_on_pi {ι : Type*} {R : Type*} {M : Type*} [Finite ι] [Semiring R]
[TopologicalSpace R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [ContinuousAdd M]
[ContinuousSMul R M] (f : (ι → R) →ₗ[R] M) : Continuous f := by |
cases nonempty_fintype ι
classical
-- for the proof, write `f` in the standard basis, and use that each coordinate is a continuous
-- function.
have : (f : (ι → R) → M) = fun x => ∑ i : ι, x i • f fun j => if i = j then 1 else 0 := by
ext x
exact f.pi_apply_eq_sum_univ x
rw [this]
r... |
/-
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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 522 | 528 | theorem coeff_zero_eq_eval_zero (p : R[X]) : coeff p 0 = p.eval 0 :=
calc
coeff p 0 = coeff p 0 * 0 ^ 0 := by | simp
_ = p.eval 0 := by
symm
rw [eval_eq_sum]
exact Finset.sum_eq_single _ (fun b _ hb => by simp [zero_pow hb]) (by simp)
|
/-
Copyright (c) 2023 Paul Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Reichert, Yaël Dillies
-/
import Mathlib.Analysis.NormedSpace.AddTorsorBases
#align_import analysis.convex.intrinsic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be51... | Mathlib/Analysis/Convex/Intrinsic.lean | 147 | 149 | theorem intrinsicClosure_singleton (x : P) : intrinsicClosure 𝕜 ({x} : Set P) = {x} := by |
simpa only [intrinsicClosure, preimage_coe_affineSpan_singleton, closure_univ, image_univ,
Subtype.range_coe] using coe_affineSpan_singleton _ _ _
|
/-
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, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 197 | 202 | theorem roots_C (x : R) : (C x).roots = 0 := by |
classical exact
if H : x = 0 then by rw [H, C_0, roots_zero]
else
Multiset.ext.mpr fun r => (by
rw [count_roots, count_zero, rootMultiplicity_eq_zero (not_isRoot_C _ _ H)])
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "lean... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 54 | 58 | theorem IicSnd_apply (r : ℝ) {s : Set α} (hs : MeasurableSet s) :
ρ.IicSnd r s = ρ (s ×ˢ Iic r) := by |
rw [IicSnd, fst_apply hs,
restrict_apply' (MeasurableSet.univ.prod (measurableSet_Iic : MeasurableSet (Iic r))), ←
prod_univ, prod_inter_prod, inter_univ, univ_inter]
|
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
/-!
# B... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 485 | 485 | theorem unitors_inv_equal : (λ_ (𝟙 a)).inv = (ρ_ (𝟙 a)).inv := by | simp [Iso.inv_eq_inv]
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 251 | 254 | theorem equicontinuousAt_finite [Finite ι] {F : ι → X → α} {x₀ : X} :
EquicontinuousAt F x₀ ↔ ∀ i, ContinuousAt (F i) x₀ := by |
simp [EquicontinuousAt, ContinuousAt, (nhds_basis_uniformity' (𝓤 α).basis_sets).tendsto_right_iff,
UniformSpace.ball, @forall_swap _ ι]
|
/-
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.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import... | Mathlib/Data/Real/Irrational.lean | 432 | 435 | theorem div_int (h : Irrational x) {m : ℤ} (hm : m ≠ 0) : Irrational (x / m) := by |
rw [← cast_intCast]
refine h.div_rat ?_
rwa [Int.cast_ne_zero]
|
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Order.Hom.CompleteLattice
import Mathlib.Topology.Homeomorph
import Mathlib.Topology.Order.Lattice
#align_import topology.order.lower_topology... | Mathlib/Topology/Order/LowerUpperTopology.lean | 218 | 219 | theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen { t | ∃ a, (Ici a)ᶜ = t } s := by |
rw [topology_eq α]; rfl
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.FieldTheory.Finite.Trace
import Mathlib.Algebra.Group.AddChar
import Mathlib.Data.ZMod.Units
import... | Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 208 | 213 | theorem zmodChar_primitive_of_primitive_root (n : ℕ+) {ζ : C} (h : IsPrimitiveRoot ζ n) :
IsPrimitive (zmodChar n ((IsPrimitiveRoot.iff_def ζ n).mp h).left) := by |
apply zmod_char_primitive_of_eq_one_only_at_zero
intro a ha
rw [zmodChar_apply, ← pow_zero ζ] at ha
exact (ZMod.val_eq_zero a).mp (IsPrimitiveRoot.pow_inj h (ZMod.val_lt a) n.pos ha)
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 378 | 378 | theorem image_neg_Iic : Neg.neg '' Iic a = Ici (-a) := by | simp
|
/-
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.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Anal... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 384 | 399 | theorem y_eq_zero_of_not_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (hx : x ∉ ball (0 : E) (1 + D)) :
y D x = 0 := by |
change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 0
have B : ∀ y, y ∈ ball x D → φ y = 0 := by
intro y hy
simp only [φ, indicator, mem_closedBall_zero_iff, ite_eq_right_iff, one_ne_zero]
intro h'y
have C : ball y D ⊆ ball 0 (1 + D) := by
apply ball_subset_ball'
rw [← dist_zero_right] at h'y
lina... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 176 | 178 | theorem sup_product_right (s : Finset β) (t : Finset γ) (f : β × γ → α) :
(s ×ˢ t).sup f = t.sup fun i' => s.sup fun i => f ⟨i, i'⟩ := by |
rw [sup_product_left, Finset.sup_comm]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f0242... | Mathlib/Data/PNat/Prime.lean | 302 | 305 | theorem Coprime.gcd_mul (k : ℕ+) {m n : ℕ+} (h : m.Coprime n) :
k.gcd (m * n) = k.gcd m * k.gcd n := by |
rw [← coprime_coe] at h; apply eq
simp only [gcd_coe, mul_coe]; apply Nat.Coprime.gcd_mul k h
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.LinearAlgebra.Dual
#align_import analysi... | Mathlib/Analysis/Calculus/LagrangeMultipliers.lean | 108 | 121 | theorem IsLocalExtrOn.exists_multipliers_of_hasStrictFDerivAt {ι : Type*} [Fintype ι]
{f : ι → E → ℝ} {f' : ι → E →L[ℝ] ℝ} (hextr : IsLocalExtrOn φ {x | ∀ i, f i x = f i x₀} x₀)
(hf' : ∀ i, HasStrictFDerivAt (f i) (f' i) x₀) (hφ' : HasStrictFDerivAt φ φ' x₀) :
∃ (Λ : ι → ℝ) (Λ₀ : ℝ), (Λ, Λ₀) ≠ 0 ∧ (∑ i, Λ i... |
letI := Classical.decEq ι
replace hextr : IsLocalExtrOn φ {x | (fun i => f i x) = fun i => f i x₀} x₀ := by
simpa only [Function.funext_iff] using hextr
rcases hextr.exists_linear_map_of_hasStrictFDerivAt (hasStrictFDerivAt_pi.2 fun i => hf' i)
hφ' with
⟨Λ, Λ₀, h0, hsum⟩
rcases (LinearEquiv.piRin... |
/-
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.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... | Mathlib/Algebra/Ring/Defs.lean | 416 | 416 | theorem mul_sub_one (a b : α) : a * (b - 1) = a * b - a := by | rw [mul_sub, mul_one]
|
/-
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.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,409 | 1,411 | theorem ContDiffOn.sub {s : Set E} {f g : E → F} (hf : ContDiffOn 𝕜 n f s)
(hg : ContDiffOn 𝕜 n g s) : ContDiffOn 𝕜 n (fun x => f x - g x) s := by |
simpa only [sub_eq_add_neg] using hf.add hg.neg
|
/-
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.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import n... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 286 | 288 | theorem normAtPlace_apply_isReal {w : InfinitePlace K} (hw : IsReal w) (x : E K):
normAtPlace w x = ‖x.1 ⟨w, hw⟩‖ := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, dif_pos]
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Eq... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 819 | 826 | theorem whiskerLeft_id_bimod {X Y Z : Mon_ C} {M : Bimod X Y} {N : Bimod Y Z} :
whiskerLeft M (𝟙 N) = 𝟙 (M.tensorBimod N) := by |
ext
apply Limits.coequalizer.hom_ext
dsimp only [tensorBimod_X, whiskerLeft_hom, id_hom']
simp only [MonoidalCategory.whiskerLeft_id, ι_colimMap, parallelPair_obj_one,
parallelPairHom_app_one, Category.id_comp]
erw [Category.comp_id]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import ... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 182 | 185 | theorem toComplex_comp_ofComplex : toComplex.comp ofComplex = AlgHom.id ℝ ℂ := by |
ext1
dsimp only [AlgHom.comp_apply, Subtype.coe_mk, AlgHom.id_apply]
rw [ofComplex_I, toComplex_ι, one_smul]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.AEMeasurable
#align_import dynamics.ergodic.measure_preserving from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409... | Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 182 | 186 | theorem exists_mem_iterate_mem [IsFiniteMeasure μ] (hf : MeasurePreserving f μ μ)
(hs : MeasurableSet s) (hs' : μ s ≠ 0) : ∃ x ∈ s, ∃ m ≠ 0, f^[m] x ∈ s := by |
rcases ENNReal.exists_nat_mul_gt hs' (measure_ne_top μ (Set.univ : Set α)) with ⟨N, hN⟩
rcases hf.exists_mem_iterate_mem_of_volume_lt_mul_volume hs hN with ⟨x, hx, m, hm, hmx⟩
exact ⟨x, hx, m, hm.1.ne', hmx⟩
|
/-
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
#align_import topology.fiber_... | Mathlib/Topology/FiberBundle/Basic.lean | 668 | 670 | theorem localTrivAt_apply (p : Z.TotalSpace) : (Z.localTrivAt p.1) p = ⟨p.1, p.2⟩ := by |
rw [localTrivAt, localTriv_apply, coordChange_self]
exact Z.mem_baseSet_at p.1
|
/-
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.Order.Monotone.Odd
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Anal... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | 35 | 41 | theorem hasStrictDerivAt_sin (x : ℂ) : HasStrictDerivAt sin (cos x) x := by |
simp only [cos, div_eq_mul_inv]
convert ((((hasStrictDerivAt_id x).neg.mul_const I).cexp.sub
((hasStrictDerivAt_id x).mul_const I).cexp).mul_const I).mul_const (2 : ℂ)⁻¹ using 1
simp only [Function.comp, id]
rw [sub_mul, mul_assoc, mul_assoc, I_mul_I, neg_one_mul, neg_neg, mul_one, one_mul, mul_assoc,
... |
/-
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.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "lean... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,216 | 1,219 | theorem contDiffOn_top_iff_fderiv_of_isOpen (hs : IsOpen s) :
ContDiffOn 𝕜 ∞ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ∞ (fun y => fderiv 𝕜 f y) s := by |
rw [contDiffOn_top_iff_fderivWithin hs.uniqueDiffOn]
exact Iff.rfl.and <| contDiffOn_congr fun x hx ↦ fderivWithin_of_isOpen hs hx
|
/-
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, Mario Carneiro
-/
import Batteries.Tactic.Alias
import Batteries.Data.List.Init.Attach
import Batteries.Data.List.Pairwise
-- Adaptation note: ... | .lake/packages/batteries/Batteries/Data/List/Perm.lean | 607 | 612 | theorem perm_insert_swap (x y : α) (l : List α) :
List.insert x (List.insert y l) ~ List.insert y (List.insert x l) := by |
by_cases xl : x ∈ l <;> by_cases yl : y ∈ l <;> simp [xl, yl]
if xy : x = y then simp [xy] else
simp [List.insert, xl, yl, xy, Ne.symm xy]
constructor
|
/-
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.Set.Lattice
import Mathlib.Logic.Small.Basic
import Mathlib.Logic.Function.OfArity
import Mathlib.Order.WellFounded
#align_import set_theory.zfc.... | Mathlib/SetTheory/ZFC/Basic.lean | 806 | 810 | theorem nonempty_mk_iff {x : PSet} : (mk x).Nonempty ↔ x.Nonempty := by |
refine ⟨?_, fun ⟨a, h⟩ => ⟨mk a, h⟩⟩
rintro ⟨a, h⟩
induction a using Quotient.inductionOn
exact ⟨_, h⟩
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.Antichain
import Mathlib.Order.SetNotation
#align_import data.set.intervals.ord_connect... | Mathlib/Order/Interval/Set/OrdConnected.lean | 227 | 230 | theorem ordConnected_singleton {α : Type*} [PartialOrder α] {a : α} :
OrdConnected ({a} : Set α) := by |
rw [← Icc_self]
exact ordConnected_Icc
|
/-
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.Fourier.AddCircle
import Mathlib.Ana... | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 209 | 233 | theorem hasFDerivAt_fourierIntegral
[MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] {μ : Measure V}
(hf : Integrable f μ) (hf' : Integrable (fun v : V ↦ ‖v‖ * ‖f v‖) μ) (w : W) :
HasFDerivAt (fourierIntegral 𝐞 μ L.toLinearMap₂ f)
(fourierIntegral 𝐞 μ L.toLinearMap₂ (fourierSMulRight L... |
let F : W → V → E := fun w' v ↦ 𝐞 (-L v w') • f v
let F' : W → V → W →L[ℝ] E := fun w' v ↦ 𝐞 (-L v w') • fourierSMulRight L f v
let B : V → ℝ := fun v ↦ 2 * π * ‖L‖ * ‖v‖ * ‖f v‖
have h0 (w' : W) : Integrable (F w') μ :=
(fourierIntegral_convergent_iff continuous_fourierChar
(by apply L.continuous₂... |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Functor
#align_import control.traversa... | Mathlib/Control/Traversable/Instances.lean | 31 | 32 | theorem Option.id_traverse {α} (x : Option α) : Option.traverse (pure : α → Id α) x = x := by |
cases x <;> rfl
|
/-
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.Init.Data.Nat.Notation
import Mathlib.Control.Functor
import Mathlib.Data.SProd... | Mathlib/Data/List/Defs.lean | 569 | 571 | theorem iterateTR_loop_eq (f : α → α) (a : α) (n : ℕ) (l : List α) :
iterateTR.loop f a n l = reverse l ++ iterate f a n := by |
induction n generalizing a l <;> simp [*]
|
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