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) 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.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 796 | 796 | theorem tensor_right_iff {X Y : C} (f g : X ⟶ Y) : f ⊗ 𝟙 (𝟙_ C) = g ⊗ 𝟙 (𝟙_ C) ↔ f = g := by | simp
|
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
/-!
# Reflections, inversions, and inversion sequences
Throughout this file, `B` is a type and... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 271 | 278 | theorem getD_rightInvSeq_mul_self (ω : List B) (j : ℕ) :
((ris ω).getD j 1) * ((ris ω).getD j 1) = 1 := by |
simp [getD_rightInvSeq, mul_assoc]
rcases em (j < ω.length) with hj | nhj
· rw [get?_eq_get hj]
simp [← mul_assoc]
· rw [get?_eq_none.mpr (by omega)]
simp
|
/-
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.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
im... | Mathlib/Algebra/BigOperators/Finsupp.lean | 101 | 104 | theorem prod_ite_eq [DecidableEq α] (f : α →₀ M) (a : α) (b : α → M → N) :
(f.prod fun x v => ite (a = x) (b x v) 1) = ite (a ∈ f.support) (b a (f a)) 1 := by |
dsimp [Finsupp.prod]
rw [f.support.prod_ite_eq]
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathli... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 243 | 247 | theorem degree_compl [Fintype (Gᶜ.neighborSet v)] [Fintype V] :
Gᶜ.degree v = Fintype.card V - 1 - G.degree v := by |
classical
rw [← card_neighborSet_union_compl_neighborSet G v, Set.toFinset_union]
simp [card_union_of_disjoint (Set.disjoint_toFinset.mpr (compl_neighborSet_disjoint G v))]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#a... | Mathlib/Data/Rat/Floor.lean | 96 | 101 | theorem Nat.coprime_sub_mul_floor_rat_div_of_coprime {n d : ℕ} (n_coprime_d : n.Coprime d) :
((n : ℤ) - d * ⌊(n : ℚ) / d⌋).natAbs.Coprime d := by |
have : (n : ℤ) % d = n - d * ⌊(n : ℚ) / d⌋ := Int.mod_nat_eq_sub_mul_floor_rat_div
rw [← this]
have : d.Coprime n := n_coprime_d.symm
rwa [Nat.Coprime, Nat.gcd_rec] at this
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 684 | 703 | theorem LinearIndependent.maximal_iff {ι : Type w} {R : Type u} [Ring R] [Nontrivial R] {M : Type v}
[AddCommGroup M] [Module R M] {v : ι → M} (i : LinearIndependent R v) :
i.Maximal ↔
∀ (κ : Type v) (w : κ → M) (_i' : LinearIndependent R w) (j : ι → κ) (_h : w ∘ j = v),
Surjective j := by |
constructor
· rintro p κ w i' j rfl
specialize p (range w) i'.coe_range (range_comp_subset_range _ _)
rw [range_comp, ← image_univ (f := w)] at p
exact range_iff_surjective.mp (image_injective.mpr i'.injective p)
· intro p w i' h
specialize
p w ((↑) : w → M) i' (fun i => ⟨v i, range_subset_... |
/-
Copyright (c) 2018 Mitchell Rowett. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Rowett, Scott Morrison
-/
import Mathlib.Algebra.Quotient
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Group... | Mathlib/GroupTheory/Coset.lean | 250 | 261 | theorem leftCoset_eq_iff {x y : α} : x • (s : Set α) = y • s ↔ x⁻¹ * y ∈ s := by |
rw [Set.ext_iff]
simp_rw [mem_leftCoset_iff, SetLike.mem_coe]
constructor
· intro h
apply (h y).mpr
rw [mul_left_inv]
exact s.one_mem
· intro h z
rw [← mul_inv_cancel_right x⁻¹ y]
rw [mul_assoc]
exact s.mul_mem_cancel_left h
|
/-
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 | 864 | 868 | theorem adjoin_finset_isCompactElement (S : Finset E) :
IsCompactElement (adjoin F S : IntermediateField F E) := by |
rw [← biSup_adjoin_simple]
simp_rw [Finset.mem_coe, ← Finset.sup_eq_iSup]
exact isCompactElement_finsetSup S fun x _ => adjoin_simple_isCompactElement x
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.s... | Mathlib/Data/Stream/Init.lean | 615 | 622 | theorem append_take_drop : ∀ (n : Nat) (s : Stream' α),
appendStream' (take n s) (drop n s) = s := by |
intro n
induction' n with n' ih
· intro s
rfl
· intro s
rw [take_succ, drop_succ, cons_append_stream, ih (tail s), Stream'.eta]
|
/-
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 | 856 | 858 | theorem integral_comp_div_sub (hc : c ≠ 0) (d) :
(∫ x in a..b, f (x / c - d)) = c • ∫ x in a / c - d..b / c - d, f x := by |
simpa only [div_eq_inv_mul, inv_inv] using integral_comp_mul_sub f (inv_ne_zero hc) d
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0... | Mathlib/Data/Int/Log.lean | 93 | 96 | theorem log_of_right_le_zero (b : ℕ) {r : R} (hr : r ≤ 0) : log b r = 0 := by |
rw [log_of_right_le_one _ (hr.trans zero_le_one),
Nat.clog_of_right_le_one ((Nat.ceil_eq_zero.mpr <| inv_nonpos.2 hr).trans_le zero_le_one),
Int.ofNat_zero, neg_zero]
|
/-
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 | 1,232 | 1,236 | theorem exists_path_through_family {n : ℕ} (p : Fin (n + 1) → X) :
∃ γ : Path (p 0) (p n), ∀ i, p i ∈ range γ := by |
have : IsPathConnected (univ : Set X) := pathConnectedSpace_iff_univ.mp (by infer_instance)
rcases this.exists_path_through_family p fun _i => True.intro with ⟨γ, -, h⟩
exact ⟨γ, h⟩
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e... | Mathlib/CategoryTheory/Subobject/Limits.lean | 158 | 160 | theorem kernelSubobjectMap_arrow (sq : Arrow.mk f ⟶ Arrow.mk f') :
kernelSubobjectMap sq ≫ (kernelSubobject f').arrow = (kernelSubobject f).arrow ≫ sq.left := by |
simp [kernelSubobjectMap]
|
/-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.Fin.VecNotation
import ... | Mathlib/Data/Nat/Choose/Multinomial.lean | 80 | 85 | theorem multinomial_insert_one [DecidableEq α] (h : a ∉ s) (h₁ : f a = 1) :
multinomial (insert a s) f = (s.sum f).succ * multinomial s f := by |
simp only [multinomial, one_mul, factorial]
rw [Finset.sum_insert h, Finset.prod_insert h, h₁, add_comm, ← succ_eq_add_one, factorial_succ]
simp only [factorial_one, one_mul, Function.comp_apply, factorial, mul_one, ← one_eq_succ_zero]
rw [Nat.mul_div_assoc _ (prod_factorial_dvd_factorial_sum _ _)]
|
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 820 | 821 | theorem coeFn_nonneg (f : Lp.simpleFunc G p μ) : (0 : α → G) ≤ᵐ[μ] f ↔ 0 ≤ f := by |
rw [← Subtype.coe_le_coe, Lp.coeFn_nonneg, AddSubmonoid.coe_zero]
|
/-
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 | 540 | 562 | theorem support_extend_domain (f : α ≃ Subtype p) {g : Perm α} :
support (g.extendDomain f) = g.support.map f.asEmbedding := by |
ext b
simp only [exists_prop, Function.Embedding.coeFn_mk, toEmbedding_apply, mem_map, Ne,
Function.Embedding.trans_apply, mem_support]
by_cases pb : p b
· rw [extendDomain_apply_subtype _ _ pb]
constructor
· rintro h
refine ⟨f.symm ⟨b, pb⟩, ?_, by simp⟩
contrapose! h
simp [h]
... |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.indicator_function from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b8... | Mathlib/Order/Filter/IndicatorFunction.lean | 76 | 79 | theorem Antitone.mulIndicator_eventuallyEq_iInter {ι} [Preorder ι] [One β] (s : ι → Set α)
(hs : Antitone s) (f : α → β) (a : α) :
(fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋂ i, s i) f a := by |
classical exact hs.piecewise_eventually_eq_iInter f 1 a
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot
-/
import Mathlib.Algebra.Algebra.Subalgebra.Operations
import Mathlib.Algebra.Ring.Fin
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.ideal.q... | Mathlib/RingTheory/Ideal/QuotientOperations.lean | 49 | 56 | theorem lift_injective_of_ker_le_ideal (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hI : ker f ≤ I) : Function.Injective (Ideal.Quotient.lift I f H) := by |
rw [RingHom.injective_iff_ker_eq_bot, RingHom.ker_eq_bot_iff_eq_zero]
intro u hu
obtain ⟨v, rfl⟩ := Ideal.Quotient.mk_surjective u
rw [Ideal.Quotient.lift_mk] at hu
rw [Ideal.Quotient.eq_zero_iff_mem]
exact hI ((RingHom.mem_ker f).mpr hu)
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
[`data.finset.sym`@`98e83c3d541c77cdb7da20d79611a780ff8e7d90`..`02ba8949f486ebecf93fe7460f1ed0564b5e442c`](https://leanprover-community.github.io/mathlib-port-status/file/d... | Mathlib/Data/Finset/Sym.lean | 51 | 53 | theorem mem_sym2_iff {m : Sym2 α} : m ∈ s.sym2 ↔ ∀ a ∈ m, a ∈ s := by |
rw [mem_mk, sym2_val, Multiset.mem_sym2_iff]
simp only [mem_val]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 1,020 | 1,021 | theorem braid_natural [HasBinaryProducts C] {W X Y Z : C} (f : X ⟶ Y) (g : Z ⟶ W) :
prod.map f g ≫ (prod.braiding _ _).hom = (prod.braiding _ _).hom ≫ prod.map g f := by | simp
|
/-
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.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTh... | Mathlib/MeasureTheory/Function/LpSpace.lean | 1,626 | 1,655 | theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
(hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
(h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by |
by_cases hp_top : p = ∞
· simp_rw [hp_top] at *
have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
simp_rw [ae_all_iff]
exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
simp_rw [snorm_exponent_top, snormEssSup] at h_cau
refine h_cau_ae... |
/-
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 | 218 | 221 | theorem ite_sub_ite {α} [Sub α] (P : Prop) [Decidable P] (a b c d : α) :
((if P then a else b) - if P then c else d) = if P then a - c else b - d := by |
split
repeat rfl
|
/-
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.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | Mathlib/Topology/Algebra/Monoid.lean | 789 | 792 | theorem continuousOn_multiset_prod {f : ι → X → M} (s : Multiset ι) {t : Set X} :
(∀ i ∈ s, ContinuousOn (f i) t) → ContinuousOn (fun a => (s.map fun i => f i a).prod) t := by |
rcases s with ⟨l⟩
simpa using continuousOn_list_prod l
|
/-
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.Lebesgue.Complex
import Mathlib.MeasureTheory.Integral.DivergenceTheorem
import Mathlib.MeasureTheory.Integral.CircleIntegral
i... | Mathlib/Analysis/Complex/CauchyIntegral.lean | 633 | 636 | theorem analyticOn_univ_iff_differentiable {f : ℂ → E} :
AnalyticOn ℂ f univ ↔ Differentiable ℂ f := by |
simp only [← differentiableOn_univ]
exact analyticOn_iff_differentiableOn isOpen_univ
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homo... | Mathlib/Algebra/Homology/Homotopy.lean | 115 | 121 | theorem dNext_nat (C D : ChainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
dNext i f = C.d i (i - 1) ≫ f (i - 1) i := by |
dsimp [dNext]
cases i
· simp only [shape, ChainComplex.next_nat_zero, ComplexShape.down_Rel, Nat.one_ne_zero,
not_false_iff, zero_comp]
· congr <;> simp
|
/-
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 | 1,340 | 1,344 | theorem support_rotate {u v : V} (c : G.Walk v v) (h : u ∈ c.support) :
(c.rotate h).support.tail ~r c.support.tail := by |
simp only [rotate, tail_support_append]
apply List.IsRotated.trans List.isRotated_append
rw [← tail_support_append, take_spec]
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analy... | Mathlib/Analysis/NormedSpace/Dual.lean | 254 | 260 | theorem isBounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
IsBounded (polar 𝕜 s) := by |
obtain ⟨a, ha⟩ : ∃ a : 𝕜, 1 < ‖a‖ := NormedField.exists_one_lt_norm 𝕜
obtain ⟨r, r_pos, r_ball⟩ : ∃ r : ℝ, 0 < r ∧ ball 0 r ⊆ s := Metric.mem_nhds_iff.1 s_nhd
exact isBounded_closedBall.subset
(((dualPairing 𝕜 E).flip.polar_antitone r_ball).trans <|
polar_ball_subset_closedBall_div ha r_pos)
|
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace
#a... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 189 | 234 | theorem TendstoInMeasure.exists_seq_tendsto_ae (hfg : TendstoInMeasure μ f atTop g) :
∃ ns : ℕ → ℕ, StrictMono ns ∧ ∀ᵐ x ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) := by |
/- Since `f` tends to `g` in measure, it has a subsequence `k ↦ f (ns k)` such that
`μ {|f (ns k) - g| ≥ 2⁻ᵏ} ≤ 2⁻ᵏ` for all `k`. Defining
`s := ⋂ k, ⋃ i ≥ k, {|f (ns k) - g| ≥ 2⁻ᵏ}`, we see that `μ s = 0` by the
first Borel-Cantelli lemma.
On the other hand, as `s` is precisely the set for which `f... |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
impor... | Mathlib/Analysis/Convolution.lean | 216 | 235 | theorem _root_.BddAbove.convolutionExistsAt' {x₀ : G} {s : Set G}
(hbg : BddAbove ((fun i => ‖g i‖) '' ((fun t => -t + x₀) ⁻¹' s))) (hs : MeasurableSet s)
(h2s : (support fun t => L (f t) (g (x₀ - t))) ⊆ s) (hf : IntegrableOn f s μ)
(hmg : AEStronglyMeasurable g <| map (fun t => x₀ - t) (μ.restrict s)) :
... |
rw [ConvolutionExistsAt]
rw [← integrableOn_iff_integrable_of_support_subset h2s]
set s' := (fun t => -t + x₀) ⁻¹' s
have : ∀ᵐ t : G ∂μ.restrict s,
‖L (f t) (g (x₀ - t))‖ ≤ s.indicator (fun t => ‖L‖ * ‖f t‖ * ⨆ i : s', ‖g i‖) t := by
filter_upwards
refine le_indicator (fun t ht => ?_) fun t ht =>... |
/-
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.BigOperators.Group.List
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.I... | Mathlib/Data/Vector/Basic.lean | 357 | 359 | theorem scanl_singleton (v : Vector α 1) : scanl f b v = b ::ᵥ f b v.head ::ᵥ nil := by |
rw [← cons_head_tail v]
simp only [scanl_cons, scanl_nil, head_cons, singleton_tail]
|
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Mathport.Rename
#align_import init.data.list.instances from "leanprover-community/lean"@"9af482290ef68e8aaa5ead01aa7b09b7be7019fd"
/-!
# De... | Mathlib/Init/Data/List/Instances.lean | 30 | 32 | theorem map_eq_bind {α β} (f : α → β) (l : List α) : map f l = l.bind fun x => [f x] := by |
simp only [← map_singleton]
rw [← bind_singleton' l, bind_map, bind_singleton']
|
/-
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
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Init.Data.Fin.Basic
#align_import data.list.nodup from "leanprover-... | Mathlib/Data/List/Nodup.lean | 336 | 338 | theorem nodup_join {L : List (List α)} :
Nodup (join L) ↔ (∀ l ∈ L, Nodup l) ∧ Pairwise Disjoint L := by |
simp only [Nodup, pairwise_join, disjoint_left.symm, forall_mem_ne]
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.PInfty
#align_import algebraic_topology.dold_kan.decomposition from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190... | Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean | 150 | 155 | theorem preComp_φ : (f.preComp g).φ = g.app (op [n + 1]) ≫ f.φ := by |
unfold φ preComp
simp only [PInfty_f, comp_add]
congr 1
· simp only [P_f_naturality_assoc]
· simp only [comp_sum, P_f_naturality_assoc, SimplicialObject.δ_naturality_assoc]
|
/-
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.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 482 | 483 | theorem associator_naturality_right (X Y : C) {Z Z' : C} (f : Z ⟶ Z') :
(X ⊗ Y) ◁ f ≫ (α_ X Y Z').hom = (α_ X Y Z).hom ≫ X ◁ Y ◁ f := by | simp
|
/-
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 | 315 | 323 | theorem Perm.rec_heq {β : List α → Sort _} {f : ∀ a l, β l → β (a :: l)} {b : β []} {l l' : List α}
(hl : l ~ l') (f_congr : ∀ {a l l' b b'}, l ~ l' → HEq b b' → HEq (f a l b) (f a l' b'))
(f_swap : ∀ {a a' l b}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))) :
HEq (@List.rec α β b f l) (@List.re... |
induction hl with
| nil => rfl
| cons a h ih => exact f_congr h ih
| swap a a' l => exact f_swap
| trans _h₁ _h₂ ih₁ ih₂ => exact ih₁.trans ih₂
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
/-... | Mathlib/Topology/MetricSpace/Infsep.lean | 532 | 534 | theorem infsep_zero_iff_subsingleton_of_finite [Finite s] : s.infsep = 0 ↔ s.Subsingleton := by |
rw [infsep_zero, einfsep_eq_top_iff, or_iff_right_iff_imp]
exact fun H => (einfsep_pos_of_finite.ne' H).elim
|
/-
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 | 187 | 190 | theorem rangeIsAffineOpenOfOpenImmersion {X Y : Scheme} [IsAffine X] (f : X ⟶ Y)
[H : IsOpenImmersion f] : IsAffineOpen (Scheme.Hom.opensRange f) := by |
refine isAffineOfIso (IsOpenImmersion.isoOfRangeEq f (Y.ofRestrict _) ?_).inv
exact Subtype.range_val.symm
|
/-
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.ContMDiff.Product
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
/-! ## Equivalence of smoothness wit... | Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean | 51 | 55 | theorem contMDiffWithinAt_iff_contDiffWithinAt {f : E → E'} {s : Set E} {x : E} :
ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') n f s x ↔ ContDiffWithinAt 𝕜 n f s x := by |
simp (config := { contextual := true }) only [ContMDiffWithinAt, liftPropWithinAt_iff',
ContDiffWithinAtProp, iff_def, mfld_simps]
exact ContDiffWithinAt.continuousWithinAt
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
Coinductive formalization of unbounded computations.
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "le... | Mathlib/Data/Seq/Computation.lean | 587 | 588 | theorem results_thinkN_pure (a : α) (n) : Results (thinkN (pure a) n) a n := by |
have := results_thinkN n (results_pure a); rwa [Nat.zero_add] at this
|
/-
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 | 836 | 838 | theorem Ioo_filter_lt (a b c : α) : (Ioo a b).filter (· < c) = Ioo a (min b c) := by |
ext
simp [and_assoc]
|
/-
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.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 88 | 92 | theorem IsPositive.conj_adjoint {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F) :
(S ∘L T ∘L S†).IsPositive := by |
refine ⟨hT.isSelfAdjoint.conj_adjoint S, fun x => ?_⟩
rw [reApplyInnerSelf, comp_apply, ← adjoint_inner_right]
exact hT.inner_nonneg_left _
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.AlgebraicTopology.DoldKan.Notations
#align_import algebraic_topology.dold_kan.homotopies from "leanprover-community/mat... | Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean | 86 | 90 | theorem cs_down_0_not_rel_left (j : ℕ) : ¬c.Rel 0 j := by |
intro hj
dsimp at hj
apply Nat.not_succ_le_zero j
rw [Nat.succ_eq_add_one, hj]
|
/-
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.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 3,134 | 3,147 | theorem toFinset_eq_singleton_iff (s : Multiset α) (a : α) :
s.toFinset = {a} ↔ card s ≠ 0 ∧ s = card s • {a} := by |
refine ⟨fun H ↦ ⟨fun h ↦ ?_, ext' fun x ↦ ?_⟩, fun H ↦ ?_⟩
· rw [card_eq_zero.1 h, toFinset_zero] at H
exact Finset.singleton_ne_empty _ H.symm
· rw [count_nsmul, count_singleton]
by_cases hx : x = a
· simp_rw [hx, ite_true, mul_one, count_eq_card]
intro y hy
rw [← mem_toFinset, H, Finset... |
/-
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
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 309 | 311 | theorem image_empty (f : α → β) : f '' ∅ = ∅ := by |
ext
simp
|
/-
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 | 1,124 | 1,127 | theorem IsPrime.prod_mem_iff_exists_mem {I : Ideal R} (hI : I.IsPrime) (s : Finset R) :
s.prod (fun x ↦ x) ∈ I ↔ ∃ p ∈ s, p ∈ I := by |
rw [Finset.prod_eq_multiset_prod, Multiset.map_id']
exact hI.multiset_prod_mem_iff_exists_mem s.val
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.Bin... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 214 | 217 | theorem inr_pushoutZeroZeroIso_hom (X Y : C) [HasBinaryCoproduct X Y] :
pushout.inr ≫ (pushoutZeroZeroIso X Y).hom = coprod.inr := by |
dsimp [pushoutZeroZeroIso]
simp
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.... | Mathlib/RingTheory/Localization/Submodule.lean | 138 | 162 | theorem mem_span_iff {N : Type*} [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]
{x : N} {a : Set N} :
x ∈ Submodule.span S a ↔ ∃ y ∈ Submodule.span R a, ∃ z : M, x = mk' S 1 z • y := by |
constructor
· intro h
refine Submodule.span_induction h ?_ ?_ ?_ ?_
· rintro x hx
exact ⟨x, Submodule.subset_span hx, 1, by rw [mk'_one, _root_.map_one, one_smul]⟩
· exact ⟨0, Submodule.zero_mem _, 1, by rw [mk'_one, _root_.map_one, one_smul]⟩
· rintro _ _ ⟨y, hy, z, rfl⟩ ⟨y', hy', z', rfl⟩
... |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Nilpotent.lean | 617 | 633 | theorem nilpotencyClass_quotient_center [hH : IsNilpotent G] :
Group.nilpotencyClass (G ⧸ center G) = Group.nilpotencyClass G - 1 := by |
generalize hn : Group.nilpotencyClass G = n
rcases n with (rfl | n)
· simp [nilpotencyClass_zero_iff_subsingleton] at *
exact Quotient.instSubsingletonQuotient (leftRel (center G))
· suffices Group.nilpotencyClass (G ⧸ center G) = n by simpa
apply le_antisymm
· apply upperCentralSeries_eq_top_iff_n... |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathl... | Mathlib/NumberTheory/ZetaValues.lean | 103 | 117 | theorem bernoulliFourierCoeff_recurrence (k : ℕ) {n : ℤ} (hn : n ≠ 0) :
bernoulliFourierCoeff k n =
1 / (-2 * π * I * n) * (ite (k = 1) 1 0 - k * bernoulliFourierCoeff (k - 1) n) := by |
unfold bernoulliFourierCoeff
rw [fourierCoeffOn_of_hasDerivAt zero_lt_one hn
(fun x _ => (hasDerivAt_bernoulliFun k x).ofReal_comp)
((continuous_ofReal.comp <|
continuous_const.mul <| Polynomial.continuous _).intervalIntegrable
_ _)]
simp_rw [ofReal_one, ofReal_zero, sub_zero, one... |
/-
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.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 912 | 913 | theorem modifyHead_modifyHead (l : List α) (f g : α → α) :
(l.modifyHead f).modifyHead g = l.modifyHead (g ∘ f) := by | cases l <;> simp
|
/-
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 | 95 | 98 | theorem mem_prod_top {s : Set (α × β)} :
s ∈ f ×ˢ (⊤ : Filter β) ↔ { a | ∀ b, (a, b) ∈ s } ∈ f := by |
rw [← principal_univ, mem_prod_principal]
simp only [mem_univ, forall_true_left]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#alig... | Mathlib/RingTheory/Ideal/Basic.lean | 733 | 736 | theorem span_singleton_neg (x : α) : (span {-x} : Ideal α) = span {x} := by |
ext
simp only [mem_span_singleton']
exact ⟨fun ⟨y, h⟩ => ⟨-y, h ▸ neg_mul_comm y x⟩, fun ⟨y, h⟩ => ⟨-y, h ▸ neg_mul_neg y x⟩⟩
|
/-
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.Combinatorics.SimpleGraph.Prod
import Mathlib.Data.Fin.SuccPred
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Relation
import Mathlib.Tact... | Mathlib/Combinatorics/SimpleGraph/Hasse.lean | 108 | 111 | theorem pathGraph_adj {n : ℕ} {u v : Fin n} :
(pathGraph n).Adj u v ↔ u.val + 1 = v.val ∨ v.val + 1 = u.val := by |
simp only [pathGraph, hasse]
simp_rw [← Fin.coe_covBy_iff, Nat.covBy_iff_succ_eq]
|
/-
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.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,489 | 1,500 | theorem GoodProducts.head!_eq_o_of_maxProducts [Inhabited I] (l : ↑(MaxProducts C ho)) :
l.val.val.head! = term I ho := by |
rw [eq_comm, ← ord_term ho]
have hm := l.prop.2
have := Products.prop_of_isGood_of_contained C _ l.prop.1 hsC l.val.val.head!
(List.head!_mem_self (List.ne_nil_of_mem hm))
simp only [Order.lt_succ_iff] at this
refine eq_of_le_of_not_lt this (not_lt.mpr ?_)
have h : ord I (term I ho) ≤ ord I l.val.val.h... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathlib.GroupTheory.GroupAction.Ring
im... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 116 | 119 | theorem det_eq_elem_of_subsingleton [Subsingleton n] (A : Matrix n n R) (k : n) :
det A = A k k := by |
have := uniqueOfSubsingleton k
convert det_unique A
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yaël Dillies
-/
import Mathlib.Topology.Sets.Opens
#align_import topology.sets.closeds from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd298... | Mathlib/Topology/Sets/Closeds.lean | 172 | 173 | theorem coe_iInf {ι} (s : ι → Closeds α) : ((⨅ i, s i : Closeds α) : Set α) = ⋂ i, s i := by |
ext; simp
|
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.MeasureTheory.Group.FundamentalDomain
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Rin... | Mathlib/Algebra/Module/Zlattice/Basic.lean | 109 | 112 | theorem repr_ceil_apply (m : E) (i : ι) : b.repr (ceil b m) i = ⌈b.repr m i⌉ := by |
classical simp only [ceil, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,
Finset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,
Finset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, map_sum]
|
/-
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 | 766 | 767 | theorem edges_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :
(p.concat h).edges = p.edges.concat s(v, w) := by | simp [edges]
|
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir
-/
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
#align_import data.real.hyperreal from "leanprover-communi... | Mathlib/Data/Real/Hyperreal.lean | 388 | 389 | theorem isSt_iff_abs_sub_lt_delta {x : ℝ*} {r : ℝ} : IsSt x r ↔ ∀ δ : ℝ, 0 < δ → |x - ↑r| < δ := by |
simp only [abs_sub_lt_iff, sub_lt_iff_lt_add, IsSt, and_comm, add_comm]
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
/-!
# Unbundled semirin... | Mathlib/Deprecated/Ring.lean | 107 | 110 | theorem map_neg (hf : IsRingHom f) : f (-x) = -f x :=
calc
f (-x) = f (-x + x) - f x := by | rw [hf.map_add]; simp
_ = -f x := by simp [hf.map_zero]
|
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.Algebra.Group.Fin
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.circulant from "leanprover-community/mathlib"@"3e068... | Mathlib/LinearAlgebra/Matrix/Circulant.lean | 81 | 82 | theorem transpose_circulant [AddGroup n] (v : n → α) :
(circulant v)ᵀ = circulant fun i => v (-i) := by | ext; simp
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.m... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 214 | 219 | theorem Commute.zpow_right {A B : M} (h : Commute A B) (m : ℤ) : Commute A (B ^ m) := by |
rcases nonsing_inv_cancel_or_zero B with (⟨hB, _⟩ | hB)
· refine SemiconjBy.zpow_right ?_ ?_ h _ <;> exact isUnit_det_of_left_inverse hB
· cases m
· simpa using h.pow_right _
· simp [← inv_pow', hB]
|
/-
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.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 565 | 565 | theorem mk_bool : #Bool = 2 := by | simp
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
/-!
# Map... | Mathlib/RingTheory/Ideal/Maps.lean | 616 | 617 | theorem comap_ker (f : S →+* R) (g : T →+* S) : f.ker.comap g = ker (f.comp g) := by |
rw [RingHom.ker_eq_comap_bot, Ideal.comap_comap, RingHom.ker_eq_comap_bot]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
import Mat... | Mathlib/NumberTheory/ArithmeticFunction.lean | 1,201 | 1,202 | theorem coe_zeta_mul_coe_moebius [Ring R] : (ζ * μ : ArithmeticFunction R) = 1 := by |
rw [← coe_coe, ← intCoe_mul, coe_zeta_mul_moebius, intCoe_one]
|
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
#align_import analysis.special_functions.trigonometric.bounds from "leanprover-commu... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean | 201 | 222 | theorem cos_lt_one_div_sqrt_sq_add_one {x : ℝ} (hx1 : -(3 * π / 2) ≤ x) (hx2 : x ≤ 3 * π / 2)
(hx3 : x ≠ 0) : cos x < (1 / √(x ^ 2 + 1) : ℝ) := by |
suffices ∀ {y : ℝ}, 0 < y → y ≤ 3 * π / 2 → cos y < 1 / sqrt (y ^ 2 + 1) by
rcases lt_or_lt_iff_ne.mpr hx3.symm with ⟨h⟩
· exact this h hx2
· convert this (by linarith : 0 < -x) (by linarith) using 1
· rw [cos_neg]
· rw [neg_sq]
intro y hy1 hy2
have hy3 : ↑0 < y ^ 2 + 1 := by linarith [sq... |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 492 | 495 | theorem isTwoCoboundary_of_twoCoboundaries (f : twoCoboundaries (Rep.ofDistribMulAction k G A)) :
IsTwoCoboundary (A := A) f.1.1 := by |
rcases mem_range_of_mem_twoCoboundaries f.2 with ⟨x, hx⟩
exact ⟨x, fun g h => Function.funext_iff.1 hx (g, h)⟩
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.List.Sigma
import Mathlib.Data.Int.Range
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.Testing.SlimCheck.Sample... | Mathlib/Testing/SlimCheck/Functions.lean | 312 | 332 | theorem List.applyId_zip_eq [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs)
(h₁ : xs.length = ys.length) (x y : α) (i : ℕ) (h₂ : xs.get? i = some x) :
List.applyId.{u} (xs.zip ys) x = y ↔ ys.get? i = some y := by |
induction xs generalizing ys i with
| nil => cases h₂
| cons x' xs xs_ih =>
cases i
· injection h₂ with h₀; subst h₀
cases ys
· cases h₁
· -- Porting note: `open List` no longer makes `zip_cons_cons` visible
simp only [List.applyId, Prod.toSigma, Option.getD_some, List.get?, Lis... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 548 | 551 | theorem comap_eval_surjective : (Submodule.comap (eval K V)).Surjective := by |
apply Submodule.comap_surjective_of_injective
rw [← LinearMap.ker_eq_bot]
exact eval_ker K V
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimen... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 677 | 679 | theorem Complex.coe_orthonormalBasisOneI :
(Complex.orthonormalBasisOneI : Fin 2 → ℂ) = ![1, I] := by |
simp [Complex.orthonormalBasisOneI]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from ... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 165 | 170 | theorem support_formPerm [Fintype α] (s : Cycle α) (h : Nodup s) (hn : Nontrivial s) :
support (formPerm s h) = s.toFinset := by |
induction' s using Quot.inductionOn with s
refine support_formPerm_of_nodup s h ?_
rintro _ rfl
simpa [Nat.succ_le_succ_iff] using length_nontrivial hn
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 778 | 782 | theorem support_sdiff_support_subset_support_add [DecidableEq σ] (p q : MvPolynomial σ R) :
p.support \ q.support ⊆ (p + q).support := by |
intro m hm
simp only [Classical.not_not, mem_support_iff, Finset.mem_sdiff, Ne] at hm
simp [hm.2, hm.1]
|
/-
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 | 388 | 390 | theorem ratCast_lt {x y : ℚ} : (x : ℝ) < (y : ℝ) ↔ x < y := by |
erw [mk_lt]
exact const_lt
|
/-
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 | 608 | 608 | theorem mem_support_nil_iff {u v : V} : u ∈ (nil : G.Walk v v).support ↔ u = v := by | simp
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
/-!
# Matroids
A `Matroid` is a structure that combinatorially abstracts
the ... | Mathlib/Data/Matroid/Basic.lean | 535 | 536 | theorem Dep.nonempty (hD : M.Dep D) : D.Nonempty := by |
rw [nonempty_iff_ne_empty]; rintro rfl; exact hD.not_indep M.empty_indep
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison
-/
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the language o... | Mathlib/Tactic/Abel.lean | 136 | 138 | theorem term_add_const {α} [AddCommMonoid α] (n x a k a') (h : a + k = a') :
@term α _ n x a + k = term n x a' := by |
simp [h.symm, term, add_assoc]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d4510... | Mathlib/Data/Fintype/Basic.lean | 319 | 322 | theorem univ_filter_mem_range (f : α → β) [Fintype β] [DecidablePred fun y => y ∈ Set.range f]
[DecidableEq β] : (Finset.univ.filter fun y => y ∈ Set.range f) = Finset.univ.image f := by |
letI : DecidablePred (fun y => ∃ x, f x = y) := by simpa using ‹_›
exact univ_filter_exists f
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 1,105 | 1,108 | theorem smul {R : Type*} [Semiring R] [DistribMulAction R M] [ContinuousConstSMul R M] {r : R}
{v : VectorMeasure α M} {w : VectorMeasure α N} (h : v ≪ᵥ w) : r • v ≪ᵥ w := by |
intro s hs
rw [smul_apply, h hs, smul_zero]
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 531 | 550 | theorem realize_toPrenexImp {φ ψ : L.BoundedFormula α n} (hφ : IsPrenex φ) (hψ : IsPrenex ψ)
{v : α → M} {xs : Fin n → M} : (φ.toPrenexImp ψ).Realize v xs ↔ (φ.imp ψ).Realize v xs := by |
revert ψ
induction' hφ with _ _ hφ _ _ _hφ ih _ _ _hφ ih <;> intro ψ hψ
· rw [hφ.toPrenexImp]
exact realize_toPrenexImpRight hφ hψ
· unfold toPrenexImp
rw [realize_ex]
refine _root_.trans (exists_congr fun _ => ih hψ.liftAt) ?_
simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.ZornAtoms
#align_import order.filter.ultrafilter from "leanprover-community... | Mathlib/Order/Filter/Ultrafilter.lean | 317 | 320 | theorem eq_pure_of_finite_mem (h : s.Finite) (h' : s ∈ f) : ∃ x ∈ s, f = pure x := by |
rw [← biUnion_of_singleton s] at h'
rcases (Ultrafilter.finite_biUnion_mem_iff h).mp h' with ⟨a, has, haf⟩
exact ⟨a, has, eq_of_le (Filter.le_pure_iff.2 haf)⟩
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 703 | 704 | theorem differentiable_const_sub_iff (c : F) :
(Differentiable 𝕜 fun y => c - f y) ↔ Differentiable 𝕜 f := by | simp [sub_eq_add_neg]
|
/-
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.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order... | Mathlib/Order/CompleteLattice.lean | 1,657 | 1,659 | theorem iInf_nat_gt_zero_eq (f : ℕ → α) : ⨅ i > 0, f i = ⨅ i, f (i + 1) := by |
rw [← iInf_range, Nat.range_succ]
simp
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
/-... | Mathlib/Topology/MetricSpace/Bounded.lean | 552 | 572 | theorem _root_.IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α}
(h0 : IsComplete (s 0)) (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n))
(h : ∀ N, (⋂ n ≤ N, s n).Nonempty) (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) :
(⋂ n, s n).Nonempty := by |
let u N := (h N).some
have I : ∀ n N, n ≤ N → u N ∈ s n := by
intro n N hn
apply mem_of_subset_of_mem _ (h N).choose_spec
intro x hx
simp only [mem_iInter] at hx
exact hx n hn
have : CauchySeq u := by
apply cauchySeq_of_le_tendsto_0 _ _ h'
intro m n N hm hn
exact dist_le_diam_of_m... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.... | Mathlib/RingTheory/TensorProduct/Basic.lean | 83 | 86 | theorem baseChange_add : (f + g).baseChange A = f.baseChange A + g.baseChange A := by |
ext
-- Porting note: added `-baseChange_tmul`
simp [baseChange_eq_ltensor, -baseChange_tmul]
|
/-
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 | 280 | 281 | theorem preimage_sub_const_Icc : (fun x => x - a) ⁻¹' Icc b c = Icc (b + a) (c + a) := by |
simp [sub_eq_add_neg]
|
/-
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.UnionFind.Basic
namespace Batteries.UnionFind
@[simp] theorem arr_empty : empty.arr = #[] := rfl
@[simp] theorem parent_empty : empty.parent a... | .lake/packages/batteries/Batteries/Data/UnionFind/Lemmas.lean | 64 | 97 | theorem root_link {self : UnionFind} {x y : Fin self.size}
(xroot : self.parent x = x) (yroot : self.parent y = y) :
∃ r, (r = x ∨ r = y) ∧ ∀ i,
(link self x y yroot).rootD i =
if self.rootD i = x ∨ self.rootD i = y then r.1 else self.rootD i := by |
if h : x.1 = y then
refine ⟨x, .inl rfl, fun i => ?_⟩
rw [rootD_ext (m2 := self) (fun _ => by rw [parent_link, if_pos h])]
split <;> [obtain _ | _ := ‹_› <;> simp [*]; rfl]
else
have {x y : Fin self.size}
(xroot : self.parent x = x) (yroot : self.parent y = y) {m : UnionFind}
(hm : ∀ i, m... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | Mathlib/RingTheory/AdjoinRoot.lean | 737 | 742 | theorem quotMapOfEquivQuotMapCMapSpanMk_symm_mk (x : AdjoinRoot f) :
(quotMapOfEquivQuotMapCMapSpanMk I f).symm
(Ideal.Quotient.mk ((I.map (C : R →+* R[X])).map (Ideal.Quotient.mk (span {f}))) x) =
Ideal.Quotient.mk (I.map (of f)) x := by |
rw [quotMapOfEquivQuotMapCMapSpanMk, Ideal.quotEquivOfEq_symm]
exact Ideal.quotEquivOfEq_mk _ _
|
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Order.Group.PosPart
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.Order.Lattice
#align_import analysis.normed.ord... | Mathlib/Analysis/Normed/Order/Lattice.lean | 119 | 130 | theorem norm_inf_sub_inf_le_add_norm (a b c d : α) : ‖a ⊓ b - c ⊓ d‖ ≤ ‖a - c‖ + ‖b - d‖ := by |
rw [← norm_abs_eq_norm (a - c), ← norm_abs_eq_norm (b - d)]
refine le_trans (solid ?_) (norm_add_le |a - c| |b - d|)
rw [abs_of_nonneg (add_nonneg (abs_nonneg (a - c)) (abs_nonneg (b - d)))]
calc
|a ⊓ b - c ⊓ d| = |a ⊓ b - c ⊓ b + (c ⊓ b - c ⊓ d)| := by rw [sub_add_sub_cancel]
_ ≤ |a ⊓ b - c ⊓ b| + |c ... |
/-
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.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | Mathlib/Topology/Algebra/Monoid.lean | 663 | 668 | theorem Filter.tendsto_cocompact_mul_left {a b : M} (ha : b * a = 1) :
Filter.Tendsto (fun x : M => a * x) (Filter.cocompact M) (Filter.cocompact M) := by |
refine Filter.Tendsto.of_tendsto_comp ?_ (Filter.comap_cocompact_le (continuous_mul_left b))
convert Filter.tendsto_id
ext x
simp [← mul_assoc, ha]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@... | Mathlib/Data/List/Intervals.lean | 213 | 216 | theorem filter_lt_of_succ_bot {n m : ℕ} (hnm : n < m) :
((Ico n m).filter fun x => x < n + 1) = [n] := by |
have r : min m (n + 1) = n + 1 := (@inf_eq_right _ _ m (n + 1)).mpr hnm
simp [filter_lt n m (n + 1), r]
|
/-
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 | 1,108 | 1,110 | theorem toSet_union (x y : ZFSet.{u}) : (x ∪ y).toSet = x.toSet ∪ y.toSet := by |
change (⋃₀ {x, y}).toSet = _
simp
|
/-
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 | 62 | 68 | theorem linearIndependent_leftExact : LinearIndependent R u := by |
rw [linearIndependent_sum]
refine ⟨?_, LinearIndependent.of_comp S.g hw, disjoint_span_sum hS hw huv⟩
rw [huv, LinearMap.linearIndependent_iff S.f]; swap
· rw [LinearMap.ker_eq_bot, ← mono_iff_injective]
infer_instance
exact hv
|
/-
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 | 1,616 | 1,625 | theorem dvd_of_mem_factors {a p : Associates α} (hm : p ∈ factors a) :
p ∣ a := by |
rcases eq_or_ne a 0 with rfl | ha0
· exact dvd_zero p
obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha0
rw [← Associates.factors_prod a]
rw [← ha', factors_mk a0 nza] at hm ⊢
rw [prod_coe]
apply Multiset.dvd_prod; apply Multiset.mem_map.mpr
exact ⟨⟨p, irreducible_of_mem_factorSet hm⟩, mem_factorSet_some... |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 232 | 237 | theorem factorization_pow (n k : ℕ) : factorization (n ^ k) = k • n.factorization := by |
induction' k with k ih; · simp
rcases eq_or_ne n 0 with (rfl | hn)
· simp
rw [Nat.pow_succ, mul_comm, factorization_mul hn (pow_ne_zero _ hn), ih,
add_smul, one_smul, add_comm]
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
/-!
# Matroids
A `Matroid` is a structure that combinatorially abstracts
the ... | Mathlib/Data/Matroid/Basic.lean | 946 | 950 | theorem Base.base_of_basis_superset (hB : M.Base B) (hBX : B ⊆ X) (hIX : M.Basis I X) :
M.Base I := by |
by_contra h
obtain ⟨e,heBI,he⟩ := hIX.indep.exists_insert_of_not_base h hB
exact heBI.2 (hIX.mem_of_insert_indep (hBX heBI.1) he)
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 90 | 91 | theorem natDegree_scaleRoots (p : R[X]) (s : R) : natDegree (scaleRoots p s) = natDegree p := by |
simp only [natDegree, degree_scaleRoots]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 409 | 424 | theorem setToSimpleFunc_add_left' (T T' T'' : Set α → E →L[ℝ] F)
(h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) {f : α →ₛ E}
(hf : Integrable f μ) : setToSimpleFunc T'' f = setToSimpleFunc T f + setToSimpleFunc T' f := by |
simp_rw [setToSimpleFunc_eq_sum_filter]
suffices
∀ x ∈ filter (fun x : E => x ≠ 0) f.range, T'' (f ⁻¹' {x}) = T (f ⁻¹' {x}) + T' (f ⁻¹' {x}) by
rw [← sum_add_distrib]
refine Finset.sum_congr rfl fun x hx => ?_
rw [this x hx]
push_cast
rw [Pi.add_apply]
intro x hx
refine
h_add (f ⁻¹'... |
/-
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.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.projection from "leanprover-community/mathlib"@"6d584f1709bedb... | Mathlib/LinearAlgebra/Projection.lean | 139 | 143 | theorem prodEquivOfIsCompl_symm_apply_snd_eq_zero (h : IsCompl p q) {x : E} :
((prodEquivOfIsCompl p q h).symm x).2 = 0 ↔ x ∈ p := by |
conv_rhs => rw [← (prodEquivOfIsCompl p q h).apply_symm_apply x]
rw [coe_prodEquivOfIsCompl', Submodule.add_mem_iff_right _ (Submodule.coe_mem _),
mem_left_iff_eq_zero_of_disjoint h.disjoint]
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.OpenImmersion
/-!
# Restriction of Schemes and Morphisms
## Main definition
- `AlgebraicGeometry.Scheme.restrict`: The restriction of a s... | Mathlib/AlgebraicGeometry/Restrict.lean | 279 | 287 | theorem isPullback_morphismRestrict {X Y : Scheme.{u}} (f : X ⟶ Y) (U : Opens Y) :
IsPullback (f ∣_ U) (Scheme.ιOpens (f ⁻¹ᵁ U)) (Scheme.ιOpens U) f := by |
delta morphismRestrict
rw [← Category.id_comp f]
refine
(IsPullback.of_horiz_isIso ⟨?_⟩).paste_horiz
(IsPullback.of_hasPullback f (Y.ofRestrict U.openEmbedding)).flip
-- Porting note: changed `rw` to `erw`
erw [pullbackRestrictIsoRestrict_inv_fst]; rw [Category.comp_id]
|
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-commun... | Mathlib/Algebra/CharP/ExpChar.lean | 162 | 164 | theorem ExpChar.exists [Ring R] [IsDomain R] : ∃ q, ExpChar R q := by |
obtain _ | ⟨p, ⟨hp⟩, _⟩ := CharP.exists' R
exacts [⟨1, .zero⟩, ⟨p, .prime hp⟩]
|
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