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) 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 | 891 | 895 | theorem continuousMul_inf {t₁ t₂ : TopologicalSpace M} (h₁ : @ContinuousMul M t₁ _)
(h₂ : @ContinuousMul M t₂ _) : @ContinuousMul M (t₁ ⊓ t₂) _ := by |
rw [inf_eq_iInf]
refine continuousMul_iInf fun b => ?_
cases b <;> assumption
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 739 | 743 | theorem rpow_lt_one_iff (hx : 0 ≤ x) :
x ^ y < 1 ↔ x = 0 ∧ y ≠ 0 ∨ 1 < x ∧ y < 0 ∨ x < 1 ∧ 0 < y := by |
rcases hx.eq_or_lt with (rfl | hx)
· rcases _root_.em (y = 0) with (rfl | hy) <;> simp [*, lt_irrefl, zero_lt_one]
· simp [rpow_lt_one_iff_of_pos hx, hx.ne.symm]
|
/-
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.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 789 | 790 | theorem add_eq_right {a b : Cardinal} (hb : ℵ₀ ≤ b) (ha : a ≤ b) : a + b = b := by |
rw [add_comm, add_eq_left hb ha]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 1,497 | 1,498 | theorem eventuallyEq_univ {s : Set α} {l : Filter α} : s =ᶠ[l] univ ↔ s ∈ l := by |
simp [eventuallyEq_set]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 411 | 413 | theorem restrict_sUnion_congr {S : Set (Set α)} (hc : S.Countable) :
μ.restrict (⋃₀ S) = ν.restrict (⋃₀ S) ↔ ∀ s ∈ S, μ.restrict s = ν.restrict s := by |
rw [sUnion_eq_biUnion, restrict_biUnion_congr hc]
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunction... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 456 | 460 | theorem integral_inv (h : (0 : ℝ) ∉ [[a, b]]) : ∫ x in a..b, x⁻¹ = log (b / a) := by |
have h' := fun x (hx : x ∈ [[a, b]]) => ne_of_mem_of_not_mem hx h
rw [integral_deriv_eq_sub' _ deriv_log' (fun x hx => differentiableAt_log (h' x hx))
(continuousOn_inv₀.mono <| subset_compl_singleton_iff.mpr h),
log_div (h' b right_mem_uIcc) (h' a left_mem_uIcc)]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathli... | Mathlib/Probability/Martingale/Basic.lean | 509 | 515 | theorem Supermartingale.le_zero_of_predictable [Preorder E] [SigmaFiniteFiltration μ 𝒢]
{f : ℕ → Ω → E} (hfmgle : Supermartingale f 𝒢 μ) (hfadp : Adapted 𝒢 fun n => f (n + 1))
(n : ℕ) : f n ≤ᵐ[μ] f 0 := by |
induction' n with k ih
· rfl
· exact ((Germ.coe_eq.mp <| congr_arg Germ.ofFun <| condexp_of_stronglyMeasurable (𝒢.le _)
(hfadp _) <| hfmgle.integrable _).symm.trans_le (hfmgle.2.1 k (k + 1) k.le_succ)).trans ih
|
/-
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.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Polynomial/Derivative.lean | 478 | 483 | theorem dvd_iterate_derivative_pow (f : R[X]) (n : ℕ) {m : ℕ} (c : R) (hm : m ≠ 0) :
(n : R) ∣ eval c (derivative^[m] (f ^ n)) := by |
obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hm
rw [Function.iterate_succ_apply, derivative_pow, mul_assoc, C_eq_natCast,
iterate_derivative_natCast_mul, eval_mul, eval_natCast]
exact dvd_mul_right _ _
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
#align_... | Mathlib/Geometry/Euclidean/Basic.lean | 390 | 394 | theorem eq_orthogonalProjection_of_eq_subspace {s s' : AffineSubspace ℝ P} [Nonempty s]
[Nonempty s'] [HasOrthogonalProjection s.direction] [HasOrthogonalProjection s'.direction]
(h : s = s') (p : P) : (orthogonalProjection s p : P) = (orthogonalProjection s' p : P) := by |
subst h
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, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 1,451 | 1,457 | theorem Iic_union_Ioc' (h₁ : c < b) : Iic b ∪ Ioc c d = Iic (max b d) := by |
ext1 x
simp_rw [mem_union, mem_Iic, mem_Ioc, le_max_iff]
by_cases hc : c < x
· simp only [hc, true_and] -- Porting note: restore `tauto`
· have hxb : x ≤ b := (le_of_not_gt hc).trans h₁.le
simp only [hxb, true_or] -- Porting note: restore `tauto`
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Dual
/-!
# Matroid Restriction
Given `M : Matroid α` and `R : Set α`, the independent sets of `M` that are contained in `R`
are the independ... | Mathlib/Data/Matroid/Restrict.lean | 421 | 423 | theorem Basis.eq_exchange_of_diff_eq_singleton (hI : M.Basis I X) (hJ : M.Basis J X)
(hIJ : I \ J = {e}) : ∃ f ∈ J \ I, J = insert f I \ {e} := by |
rw [← base_restrict_iff] at hI hJ; exact hI.eq_exchange_of_diff_eq_singleton hJ hIJ
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analys... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 344 | 346 | theorem rpow_inv_natCast_pow (x : ℝ≥0) {n : ℕ} (hn : n ≠ 0) : (x ^ (n⁻¹ : ℝ)) ^ n = x := by |
rw [← NNReal.coe_inj, NNReal.coe_pow, coe_rpow]
exact Real.rpow_inv_natCast_pow x.2 hn
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_spa... | Mathlib/Analysis/NormedSpace/Real.lean | 40 | 43 | theorem inv_norm_smul_mem_closed_unit_ball (x : E) :
‖x‖⁻¹ • x ∈ closedBall (0 : E) 1 := by |
simp only [mem_closedBall_zero_iff, norm_smul, norm_inv, norm_norm, ← div_eq_inv_mul,
div_self_le_one]
|
/-
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 | 1,790 | 1,799 | theorem continuousOn_setToFun_of_dominated (hT : DominatedFinMeasAdditive μ T C) {fs : X → α → E}
{bound : α → ℝ} {s : Set X} (hfs_meas : ∀ x ∈ s, AEStronglyMeasurable (fs x) μ)
(h_bound : ∀ x ∈ s, ∀ᵐ a ∂μ, ‖fs x a‖ ≤ bound a) (bound_integrable : Integrable bound μ)
(h_cont : ∀ᵐ a ∂μ, ContinuousOn (fun x =>... |
intro x hx
refine continuousWithinAt_setToFun_of_dominated hT ?_ ?_ bound_integrable ?_
· filter_upwards [self_mem_nhdsWithin] with x hx using hfs_meas x hx
· filter_upwards [self_mem_nhdsWithin] with x hx using h_bound x hx
· filter_upwards [h_cont] with a ha using ha x hx
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,783 | 1,784 | theorem prod_eq_bot_iff {H : Subgroup G} {K : Subgroup N} : H.prod K = ⊥ ↔ H = ⊥ ∧ K = ⊥ := by |
simpa only [← Subgroup.toSubmonoid_eq] using Submonoid.prod_eq_bot_iff
|
/-
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.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "l... | Mathlib/Algebra/Polynomial/Monic.lean | 318 | 322 | theorem Monic.natDegree_map [Semiring S] [Nontrivial S] {P : R[X]} (hmo : P.Monic) (f : R →+* S) :
(P.map f).natDegree = P.natDegree := by |
refine le_antisymm (natDegree_map_le _ _) (le_natDegree_of_ne_zero ?_)
rw [coeff_map, Monic.coeff_natDegree hmo, RingHom.map_one]
exact one_ne_zero
|
/-
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.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomi... | Mathlib/Algebra/Polynomial/Taylor.lean | 130 | 134 | theorem taylor_injective {R} [CommRing R] (r : R) : Function.Injective (taylor r) := by |
intro f g h
apply_fun taylor (-r) at h
simpa only [taylor_apply, comp_assoc, add_comp, X_comp, C_comp, C_neg, neg_add_cancel_right,
comp_X] using h
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"... | Mathlib/FieldTheory/RatFunc/Defs.lean | 202 | 211 | theorem liftOn_mk {P : Sort v} (p q : K[X]) (f : K[X] → K[X] → P) (f0 : ∀ p, f p 0 = f 0 1)
(H' : ∀ {p q p' q'} (_hq : q ≠ 0) (_hq' : q' ≠ 0), q' * p = q * p' → f p q = f p' q')
(H : ∀ {p q p' q'} (_hq : q ∈ K[X]⁰) (_hq' : q' ∈ K[X]⁰), q' * p = q * p' → f p q = f p' q' :=
fun {p q p' q'} hq hq' h => H' (n... |
by_cases hq : q = 0
· subst hq
simp only [mk_zero, f0, ← Localization.mk_zero 1, Localization.liftOn_mk,
liftOn_ofFractionRing_mk, Submonoid.coe_one]
· simp only [mk_eq_localization_mk _ hq, Localization.liftOn_mk, liftOn_ofFractionRing_mk]
|
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | Mathlib/NumberTheory/Modular.lean | 199 | 236 | theorem tendsto_lcRow0 {cd : Fin 2 → ℤ} (hcd : IsCoprime (cd 0) (cd 1)) :
Tendsto (fun g : { g : SL(2, ℤ) // (↑ₘg) 1 = cd } => lcRow0 cd ↑(↑g : SL(2, ℝ))) cofinite
(cocompact ℝ) := by |
let mB : ℝ → Matrix (Fin 2) (Fin 2) ℝ := fun t => of ![![t, (-(1 : ℤ) : ℝ)], (↑) ∘ cd]
have hmB : Continuous mB := by
refine continuous_matrix ?_
simp only [mB, Fin.forall_fin_two, continuous_const, continuous_id', of_apply, cons_val_zero,
cons_val_one, and_self_iff]
refine Filter.Tendsto.of_tendst... |
/-
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, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Rel
import Mathlib... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 884 | 887 | theorem incidence_other_prop {v : V} {e : Sym2 V} (h : e ∈ G.incidenceSet v) :
G.otherVertexOfIncident h ∈ G.neighborSet v := by |
cases' h with he hv
rwa [← Sym2.other_spec' hv, mem_edgeSet] at he
|
/-
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 | 1,668 | 1,669 | theorem flip_injective_iff₁ [FiniteDimensional K V₁] : Injective B.flip ↔ Surjective B := by |
rw [← dualMap_surjective_iff, ← (evalEquiv K V₁).toEquiv.surjective_comp]; rfl
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTh... | Mathlib/RingTheory/Polynomial/Basic.lean | 271 | 274 | theorem coeffs_one : coeffs (1 : R[X]) ⊆ {1} := by |
classical
simp_rw [coeffs, Finset.image_subset_iff]
simp_all [coeff_one]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 808 | 811 | theorem continuousWithinAt_insert_self {f : α → β} {x : α} {s : Set α} :
ContinuousWithinAt f (insert x s) x ↔ ContinuousWithinAt f s x := by |
simp only [← singleton_union, continuousWithinAt_union, continuousWithinAt_singleton,
true_and_iff]
|
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Image
#... | Mathlib/Data/Complex/Basic.lean | 330 | 330 | theorem I_mul_im (z : ℂ) : (I * z).im = z.re := by | simp
|
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer, Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Geometry.RingedSpace.SheafedSpace
import Mathlib.Topology.Sheaves.Stalks
import Ma... | Mathlib/Geometry/RingedSpace/Basic.lean | 173 | 178 | theorem isUnit_res_basicOpen {U : Opens X} (f : X.presheaf.obj (op U)) :
IsUnit (X.presheaf.map (@homOfLE (Opens X) _ _ _ (X.basicOpen_le f)).op f) := by |
apply isUnit_of_isUnit_germ
rintro ⟨_, ⟨x, (hx : IsUnit _), rfl⟩⟩
convert hx
convert X.presheaf.germ_res_apply _ _ _
|
/-
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 | 146 | 149 | theorem IsAssociatedPrime.annihilator_le (h : IsAssociatedPrime I M) :
(⊤ : Submodule R M).annihilator ≤ I := by |
obtain ⟨hI, x, rfl⟩ := h
exact Submodule.annihilator_mono le_top
|
/-
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.Basic
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.subobject.factor_thru from ... | Mathlib/CategoryTheory/Subobject/FactorThru.lean | 139 | 142 | theorem factorThru_comp_arrow {X Y : C} {P : Subobject Y} (f : X ⟶ P) (h) :
P.factorThru (f ≫ P.arrow) h = f := by |
ext
simp
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Ri... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 261 | 263 | theorem Matrix.mulVecLin_one [DecidableEq n] :
Matrix.mulVecLin (1 : Matrix n n R) = LinearMap.id := by |
ext; simp [Matrix.one_apply, Pi.single_apply]
|
/-
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 | 515 | 540 | theorem fourierCoeffOn_of_hasDeriv_right {a b : ℝ} (hab : a < b) {f f' : ℝ → ℂ}
{n : ℤ} (hn : n ≠ 0)
(hf : ContinuousOn f [[a, b]])
(hff' : ∀ x, x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x)
(hf' : IntervalIntegrable f' volume a b) :
fourierCoeffOn hab f n = 1 / (-2 * π * I * n)... |
rw [← ofReal_sub]
have hT : Fact (0 < b - a) := ⟨by linarith⟩
simp_rw [fourierCoeffOn_eq_integral, smul_eq_mul, real_smul, ofReal_div, ofReal_one]
conv => pattern (occs := 1 2 3) fourier _ _ * _ <;> (rw [mul_comm])
rw [integral_mul_deriv_eq_deriv_mul_of_hasDeriv_right hf
(fun x _ ↦ has_antideriv_at_fouri... |
/-
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 | 2,796 | 2,798 | theorem filter_erase (a : α) (s : Finset α) : filter p (erase s a) = erase (filter p s) a := by |
ext x
simp only [and_assoc, mem_filter, iff_self_iff, mem_erase]
|
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.init_t... | Mathlib/RingTheory/WittVector/InitTail.lean | 154 | 155 | theorem init_add_tail (x : 𝕎 R) (n : ℕ) : init n x + tail n x = x := by |
simp only [init, tail, ← not_lt, select_add_select_not]
|
/-
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 | 421 | 426 | theorem subperm_of_subset (d : Nodup l₁) (H : l₁ ⊆ l₂) : l₁ <+~ l₂ := by |
induction d with
| nil => exact ⟨nil, .nil, nil_sublist _⟩
| cons h _ IH =>
have ⟨H₁, H₂⟩ := forall_mem_cons.1 H
exact cons_subperm_of_not_mem_of_mem (h _ · rfl) H₁ (IH H₂)
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shape... | Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 522 | 526 | theorem image.eq_fac [HasEqualizers C] (h : f = f') :
image.ι f = (image.eqToIso h).hom ≫ image.ι f' := by |
apply image.ext
dsimp [asIso,image.eqToIso, image.eqToHom]
rw [image.lift_fac] -- Porting note: simp did not fire with this it seems
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 126 | 129 | theorem IsOfFinOrder.apply {η : Type*} {Gs : η → Type*} [∀ i, Monoid (Gs i)] {x : ∀ i, Gs i}
(h : IsOfFinOrder x) : ∀ i, IsOfFinOrder (x i) := by |
obtain ⟨n, npos, hn⟩ := h.exists_pow_eq_one
exact fun _ => isOfFinOrder_iff_pow_eq_one.mpr ⟨n, npos, (congr_fun hn.symm _).symm⟩
|
/-
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.RingTheory.Flat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Vanishing
import Mathlib.Algebra.Module.FinitePresentation
/-! # The equational criterion... | Mathlib/RingTheory/Flat/EquationalCriterion.lean | 283 | 295 | theorem exists_factorization_of_isFinitelyPresented [Flat R M] {P : Type u} [AddCommGroup P]
[Module R P] [FinitePresentation R P] (h₁ : P →ₗ[R] M) :
∃ (κ : Type u) (_ : Fintype κ) (h₂ : P →ₗ[R] (κ →₀ R)) (h₃ : (κ →₀ R) →ₗ[R] M),
h₁ = h₃ ∘ₗ h₂ := by |
obtain ⟨L, _, _, K, ϕ, _, _, hK⟩ := FinitePresentation.equiv_quotient R P
haveI : Finite R ↥K := Module.Finite.iff_fg.mpr hK
have : (h₁ ∘ₗ ϕ.symm ∘ₗ K.mkQ) ∘ₗ K.subtype = 0 := by
simp_rw [comp_assoc, (LinearMap.exact_subtype_mkQ K).linearMap_comp_eq_zero, comp_zero]
obtain ⟨κ, hκ, a, y, hay, ha⟩ := exists_... |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Felix Weilacher
-/
import Mathlib.Data.Real.Cardinality
import Mathlib.Topology.MetricSpace.Perfect
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
i... | Mathlib/MeasureTheory/Constructions/Polish.lean | 307 | 317 | theorem _root_.MeasurableSet.analyticSet {α : Type*} [t : TopologicalSpace α] [PolishSpace α]
[MeasurableSpace α] [BorelSpace α] {s : Set α} (hs : MeasurableSet s) : AnalyticSet s := by |
/- For a short proof (avoiding measurable induction), one sees `s` as a closed set for a finer
topology `t'`. It is analytic for this topology. As the identity from `t'` to `t` is continuous
and the image of an analytic set is analytic, it follows that `s` is also analytic for `t`. -/
obtain ⟨t', t't, t'_p... |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-communi... | Mathlib/Algebra/Polynomial/EraseLead.lean | 89 | 92 | theorem eraseLead_ne_zero (f0 : 2 ≤ f.support.card) : eraseLead f ≠ 0 := by |
rw [Ne, ← card_support_eq_zero, eraseLead_support]
exact
(zero_lt_one.trans_le <| (tsub_le_tsub_right f0 1).trans Finset.pred_card_le_card_erase).ne.symm
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 708 | 709 | theorem Ioo_eq_empty_iff [DenselyOrdered α] : Ioo a b = ∅ ↔ ¬a < b := by |
rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioo]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.mul_p from "leanprover-community/mathlib"@"7abfbc92eec87190fba3ed3d5ec58e7c167e7144... | Mathlib/RingTheory/WittVector/MulP.lean | 72 | 80 | theorem bind₁_wittMulN_wittPolynomial (n k : ℕ) :
bind₁ (wittMulN p n) (wittPolynomial p ℤ k) = n * wittPolynomial p ℤ k := by |
induction' n with n ih
· simp [wittMulN, Nat.cast_zero, zero_mul, bind₁_zero_wittPolynomial]
· rw [wittMulN, ← bind₁_bind₁, wittAdd, wittStructureInt_prop]
simp only [AlgHom.map_add, Nat.cast_succ, bind₁_X_right]
rw [add_mul, one_mul, bind₁_rename, bind₁_rename]
simp only [ih, Function.uncurry, Funct... |
/-
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 | 188 | 190 | theorem refl_symm {a : X} : (Path.refl a).symm = Path.refl a := by |
ext
rfl
|
/-
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 | 422 | 423 | theorem tail_interleave (s₁ s₂ : Stream' α) : tail (s₁ ⋈ s₂) = s₂ ⋈ tail s₁ := by |
unfold interleave corecOn; rw [corec_eq]; rfl
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Topology.Bornology.Basic
#align_import topology.bornology.constructions from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621... | Mathlib/Topology/Bornology/Constructions.lean | 170 | 172 | theorem boundedSpace_subtype_iff {p : α → Prop} :
BoundedSpace (Subtype p) ↔ IsBounded { x | p x } := by |
rw [boundedSpace_induced_iff, Subtype.range_coe_subtype]
|
/-
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 | 676 | 681 | theorem Perm.join {l₁ l₂ : List (List α)} (h : l₁ ~ l₂) : l₁.join ~ l₂.join := by |
induction h with
| nil => rfl
| cons _ _ ih => simp only [join_cons, perm_append_left_iff, ih]
| swap => simp only [join_cons, ← append_assoc, perm_append_right_iff]; exact perm_append_comm ..
| trans _ _ ih₁ ih₂ => exact trans ih₁ ih₂
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.M... | Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 78 | 80 | theorem eigenvectorUnitary_mulVec (j : n) :
eigenvectorUnitary hA *ᵥ Pi.single j 1 = ⇑(hA.eigenvectorBasis j) := by |
simp only [mulVec_single, eigenvectorUnitary_apply, mul_one]
|
/-
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 | 65 | 72 | theorem closure_image_mem_nhds {s : Set α} {a : α} (di : DenseInducing i) (hs : s ∈ 𝓝 a) :
closure (i '' s) ∈ 𝓝 (i a) := by |
rw [di.nhds_eq_comap a, ((nhds_basis_opens _).comap _).mem_iff] at hs
rcases hs with ⟨U, ⟨haU, hUo⟩, sub : i ⁻¹' U ⊆ s⟩
refine mem_of_superset (hUo.mem_nhds haU) ?_
calc
U ⊆ closure (i '' (i ⁻¹' U)) := di.dense.subset_closure_image_preimage_of_isOpen hUo
_ ⊆ closure (i '' s) := closure_mono (image_subs... |
/-
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.Init.Data.Quot
import Mathlib.Logic.Relator
import Mathlib.Logic.Unique
import Mathlib.Mathport.Notation
#align_import data.quot from "leanprover-comm... | Mathlib/Data/Quot.lean | 405 | 406 | theorem Quotient.out_equiv_out {s : Setoid α} {x y : Quotient s} : x.out ≈ y.out ↔ x = y := by |
rw [← Quotient.eq_mk_iff_out, Quotient.out_eq]
|
/-
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 | 326 | 334 | theorem affineIndependent_equiv {ι' : Type*} (e : ι ≃ ι') {p : ι' → P} :
AffineIndependent k (p ∘ e) ↔ AffineIndependent k p := by |
refine ⟨?_, AffineIndependent.comp_embedding e.toEmbedding⟩
intro h
have : p = p ∘ e ∘ e.symm.toEmbedding := by
ext
simp
rw [this]
exact h.comp_embedding e.symm.toEmbedding
|
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
/-!
# Cou... | Mathlib/Data/Int/CardIntervalMod.lean | 106 | 109 | theorem Ioc_filter_modEq_card (v : ℕ) : ((Ioc a b).filter (· ≡ v [MOD r])).card =
max (⌊(b - v) / (r : ℚ)⌋ - ⌊(a - v) / (r : ℚ)⌋) 0 := by |
simp_rw [← Ioc_filter_modEq_cast _ _ ▸ card_map _,
Int.Ioc_filter_modEq_card _ _ (cast_lt.mpr hr), Int.cast_natCast]
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 348 | 353 | theorem memℒp_top_const (c : E) : Memℒp (fun _ : α => c) ∞ μ := by |
refine ⟨aestronglyMeasurable_const, ?_⟩
by_cases h : μ = 0
· simp only [h, snorm_measure_zero, ENNReal.zero_lt_top]
· rw [snorm_const _ ENNReal.top_ne_zero h]
simp only [ENNReal.top_toReal, div_zero, ENNReal.rpow_zero, mul_one, ENNReal.coe_lt_top]
|
/-
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, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algeb... | Mathlib/Algebra/Divisibility/Units.lean | 202 | 203 | theorem isRelPrime_mul_unit_left_right : IsRelPrime y (x * z) ↔ IsRelPrime y z := by |
rw [isRelPrime_comm, isRelPrime_mul_unit_left_left hu, isRelPrime_comm]
|
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Hom
import ... | Mathlib/Algebra/Polynomial/GroupRingAction.lean | 71 | 73 | theorem eval_smul' [MulSemiringAction G S] (g : G) (f : S[X]) (x : S) :
f.eval (g • x) = g • (g⁻¹ • f).eval x := by |
rw [← smul_eval_smul, smul_inv_smul]
|
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb... | Mathlib/Data/Real/EReal.lean | 1,118 | 1,120 | theorem bot_mul_of_pos {x : EReal} (h : 0 < x) : ⊥ * x = ⊥ := by |
rw [EReal.mul_comm]
exact mul_bot_of_pos h
|
/-
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.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLat... | Mathlib/Algebra/Tropical/BigOperators.lean | 99 | 103 | theorem trop_sInf_image [ConditionallyCompleteLinearOrder R] (s : Finset S) (f : S → WithTop R) :
trop (sInf (f '' s)) = ∑ i ∈ s, trop (f i) := by |
rcases s.eq_empty_or_nonempty with (rfl | h)
· simp only [Set.image_empty, coe_empty, sum_empty, WithTop.sInf_empty, trop_top]
rw [← inf'_eq_csInf_image _ h, inf'_eq_inf, s.trop_inf]
|
/-
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.Order.Chain
#align_import order.zorn from "leanprover-community/mathlib"@"46a64b5b4268c594af770c44d9e502afc6a515cb"
/-!
# Zorn's lemmas
This file pr... | Mathlib/Order/Zorn.lean | 144 | 149 | theorem zorn_nonempty_Ici₀ (a : α)
(ih : ∀ c ⊆ Ici a, IsChain (· ≤ ·) c → ∀ y ∈ c, ∃ ub, ∀ z ∈ c, z ≤ ub)
(x : α) (hax : a ≤ x) : ∃ m, x ≤ m ∧ ∀ z, m ≤ z → z ≤ m := by |
let ⟨m, _, hxm, hm⟩ := zorn_nonempty_preorder₀ (Ici a) (fun c hca hc y hy ↦ ?_) x hax
· exact ⟨m, hxm, fun z hmz => hm _ (hax.trans <| hxm.trans hmz) hmz⟩
· have ⟨ub, hub⟩ := ih c hca hc y hy; exact ⟨ub, (hca hy).trans (hub y hy), hub⟩
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 274 | 275 | theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by | simp_rw [subset_iInter_iff]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 460 | 489 | theorem exists_pos_set_lintegral_lt_of_measure_lt {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {ε : ℝ≥0∞}
(hε : ε ≠ 0) : ∃ δ > 0, ∀ s, μ s < δ → ∫⁻ x in s, f x ∂μ < ε := by |
rcases exists_between (pos_iff_ne_zero.mpr hε) with ⟨ε₂, hε₂0, hε₂ε⟩
rcases exists_between hε₂0 with ⟨ε₁, hε₁0, hε₁₂⟩
rcases exists_simpleFunc_forall_lintegral_sub_lt_of_pos h hε₁0.ne' with ⟨φ, _, hφ⟩
rcases φ.exists_forall_le with ⟨C, hC⟩
use (ε₂ - ε₁) / C, ENNReal.div_pos_iff.2 ⟨(tsub_pos_iff_lt.2 hε₁₂).ne... |
/-
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 | 474 | 478 | theorem forall_neBot_le_iff {g : Filter α} {p : Filter α → Prop} (hp : Monotone p) :
(∀ f : Filter α, NeBot f → f ≤ g → p f) ↔ ∀ f : Ultrafilter α, ↑f ≤ g → p f := by |
refine ⟨fun H f hf => H f f.neBot hf, ?_⟩
intro H f hf hfg
exact hp (of_le f) (H _ ((of_le f).trans hfg))
|
/-
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.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupActio... | Mathlib/GroupTheory/Index.lean | 270 | 270 | theorem relindex_self : H.relindex H = 1 := by | rw [relindex, subgroupOf_self, index_top]
|
/-
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 | 3,571 | 3,574 | theorem dropSlice_eq (xs : List α) (n m : ℕ) : dropSlice n m xs = xs.take n ++ xs.drop (n + m) := by |
induction n generalizing xs
· cases xs <;> simp [dropSlice]
· cases xs <;> simp [dropSlice, *, Nat.succ_add]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | Mathlib/Order/BooleanAlgebra.lean | 465 | 466 | theorem inf_sdiff_distrib_left (a b c : α) : a ⊓ b \ c = (a ⊓ b) \ (a ⊓ c) := by |
rw [sdiff_inf, sdiff_eq_bot_iff.2 inf_le_left, bot_sup_eq, inf_sdiff_assoc]
|
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.pid from "leanprover-community/mathli... | Mathlib/RingTheory/DedekindDomain/PID.lean | 206 | 243 | theorem IsLocalization.OverPrime.mem_normalizedFactors_of_isPrime [IsDomain S]
{P : Ideal Sₚ} (hP : IsPrime P) (hP0 : P ≠ ⊥) :
P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p) := by |
have non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰ :=
map_le_nonZeroDivisors_of_injective _ (NoZeroSMulDivisors.algebraMap_injective _ _)
p.primeCompl_le_nonZeroDivisors
letI : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
haveI : IsScalarTower R (Localizati... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury G. Kudryashov
-/
import Batteries.Data.Sum.Basic
import Batteries.Logic
/-!
# Disjoint union of types
Theorems about the definitions introduced in `Batteries.Da... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 173 | 177 | theorem LiftRel.mono (hr : ∀ a b, r₁ a b → r₂ a b) (hs : ∀ a b, s₁ a b → s₂ a b)
(h : LiftRel r₁ s₁ x y) : LiftRel r₂ s₂ x y := by |
cases h
· exact LiftRel.inl (hr _ _ ‹_›)
· exact LiftRel.inr (hs _ _ ‹_›)
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
#align_import algebra.star.self_adjoint from "leanpro... | Mathlib/Algebra/Star/SelfAdjoint.lean | 540 | 542 | theorem smul_mem [Monoid R] [DistribMulAction R A] [StarModule R A] (r : R) {x : A}
(h : x ∈ skewAdjoint A) : r • x ∈ skewAdjoint A := by |
rw [mem_iff, star_smul, star_trivial, mem_iff.mp h, smul_neg r]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere.second_inter from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6d... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 175 | 183 | theorem Sphere.sbtw_secondInter {s : Sphere P} {p p' : P} (hp : p ∈ s)
(hp' : dist p' s.center < s.radius) : Sbtw ℝ p p' (s.secondInter p (p' -ᵥ p)) := by |
refine ⟨Sphere.wbtw_secondInter hp hp'.le, ?_, ?_⟩
· rintro rfl
rw [mem_sphere] at hp
simp [hp] at hp'
· rintro h
rw [h, mem_sphere.1 ((Sphere.secondInter_mem _).2 hp)] at hp'
exact lt_irrefl _ hp'
|
/-
Copyright (c) 2021 Manuel Candales. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Manuel Candales, Benjamin Davidson
-/
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere... | Mathlib/Geometry/Euclidean/Sphere/Power.lean | 122 | 129 | theorem mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_zero {a b c d p : P}
(h : Cospherical ({a, b, c, d} : Set P)) (hab : a ≠ b) (hcd : c ≠ d) (hapb : ∠ a p b = 0)
(hcpd : ∠ c p d = 0) : dist a p * dist b p = dist c p * dist d p := by |
obtain ⟨-, k₁, -, hab₁⟩ := angle_eq_zero_iff.mp hapb
obtain ⟨-, k₂, -, hcd₁⟩ := angle_eq_zero_iff.mp hcpd
refine mul_dist_eq_mul_dist_of_cospherical h ⟨k₁, ?_, hab₁⟩ ⟨k₂, ?_, hcd₁⟩ <;> by_contra hnot <;>
simp_all only [Classical.not_not, one_smul]
exacts [hab (vsub_left_cancel hab₁).symm, hcd (vsub_left_ca... |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.M... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 529 | 542 | theorem weightedDegree_eq_zero_iff (hw : NonTorsionWeight w) {m : σ →₀ ℕ} :
weightedDegree w m = 0 ↔ ∀ x : σ, m x = 0 := by |
simp only [weightedDegree, Finsupp.total, LinearMap.toAddMonoidHom_coe, coe_lsum,
LinearMap.coe_smulRight, LinearMap.id_coe, id_eq]
rw [Finsupp.sum, Finset.sum_eq_zero_iff]
apply forall_congr'
intro x
rw [Finsupp.mem_support_iff]
constructor
· intro hx
by_contra hx'
exact absurd (hw _ _ (hx h... |
/-
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 | 511 | 513 | theorem iInf_mul_right' {ι} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} (h : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0)
(h0 : a = 0 → Nonempty ι) : ⨅ i, f i * a = (⨅ i, f i) * a := by |
simpa only [mul_comm a] using iInf_mul_left' h h0
|
/-
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
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 205 | 231 | theorem MeasurableEmbedding.prod_mk {α β γ δ : Type*} {mα : MeasurableSpace α}
{mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {mδ : MeasurableSpace δ} {f : α → β}
{g : γ → δ} (hg : MeasurableEmbedding g) (hf : MeasurableEmbedding f) :
MeasurableEmbedding fun x : γ × α => (g x.1, f x.2) := by |
have h_inj : Function.Injective fun x : γ × α => (g x.fst, f x.snd) := by
intro x y hxy
rw [← @Prod.mk.eta _ _ x, ← @Prod.mk.eta _ _ y]
simp only [Prod.mk.inj_iff] at hxy ⊢
exact ⟨hg.injective hxy.1, hf.injective hxy.2⟩
refine ⟨h_inj, ?_, ?_⟩
· exact (hg.measurable.comp measurable_fst).prod_mk (h... |
/-
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.Order.RelIso.Set
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintyp... | Mathlib/Data/Finset/Sort.lean | 203 | 206 | theorem orderEmbOfFin_last {s : Finset α} {k : ℕ} (h : s.card = k) (hz : 0 < k) :
orderEmbOfFin s h ⟨k - 1, Nat.sub_lt hz (Nat.succ_pos 0)⟩ =
s.max' (card_pos.mp (h.symm ▸ hz)) := by |
simp [orderEmbOfFin_apply, max'_eq_sorted_last, h]
|
/-
Copyright (c) 2021 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.free.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d... | Mathlib/CategoryTheory/Monoidal/Free/Basic.lean | 266 | 296 | theorem Hom.inductionOn {motive : {X Y : F C} → (X ⟶ Y) → Prop} {X Y : F C} (t : X ⟶ Y)
(id : (X : F C) → motive (𝟙 X))
(α_hom : (X Y Z : F C) → motive (α_ X Y Z).hom)
(α_inv : (X Y Z : F C) → motive (α_ X Y Z).inv)
(l_hom : (X : F C) → motive (λ_ X).hom)
(l_inv : (X : F C) → motive (λ_ X).inv)
... |
apply Quotient.inductionOn
intro f
induction f with
| id X => exact id X
| α_hom X Y Z => exact α_hom X Y Z
| α_inv X Y Z => exact α_inv X Y Z
| l_hom X => exact l_hom X
| l_inv X => exact l_inv X
| ρ_hom X => exact ρ_hom X
| ρ_inv X => exact ρ_inv X
| comp f g hf hg => exact comp _ _ (hf ⟦f⟧) (h... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 494 | 501 | theorem tendsto_set_lintegral_zero {ι} {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {l : Filter ι}
{s : ι → Set α} (hl : Tendsto (μ ∘ s) l (𝓝 0)) :
Tendsto (fun i => ∫⁻ x in s i, f x ∂μ) l (𝓝 0) := by |
simp only [ENNReal.nhds_zero, tendsto_iInf, tendsto_principal, mem_Iio,
← pos_iff_ne_zero] at hl ⊢
intro ε ε0
rcases exists_pos_set_lintegral_lt_of_measure_lt h ε0.ne' with ⟨δ, δ0, hδ⟩
exact (hl δ δ0).mono fun i => hδ _
|
/-
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 | 120 | 123 | theorem formPerm_apply_mem_eq_next (hl : Nodup l) (x : α) (hx : x ∈ l) :
formPerm l x = next l x hx := by |
obtain ⟨k, rfl⟩ := get_of_mem hx
rw [next_get _ hl, formPerm_apply_get _ hl]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f70... | Mathlib/Data/PFun.lean | 447 | 447 | theorem preimage_univ : f.preimage Set.univ = f.Dom := by | ext; simp [mem_preimage, mem_dom]
|
/-
Copyright (c) 2019 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.con... | Mathlib/Analysis/Convex/Combination.lean | 70 | 71 | theorem Finset.centerMass_singleton (hw : w i ≠ 0) : ({i} : Finset ι).centerMass w z = z i := by |
rw [centerMass, sum_singleton, sum_singleton, ← mul_smul, inv_mul_cancel hw, one_smul]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 2,844 | 2,846 | theorem mem_seq_iff {f : Filter (α → β)} {g : Filter α} {s : Set β} :
s ∈ f.seq g ↔ ∃ u ∈ f, ∃ t ∈ g, Set.seq u t ⊆ s := by |
simp only [mem_seq_def, seq_subset, exists_prop, iff_self_iff]
|
/-
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 | 3,033 | 3,041 | theorem mem_takeWhile_imp {x : α} (hx : x ∈ takeWhile p l) : p x := by |
induction l with simp [takeWhile] at hx
| cons hd tl IH =>
cases hp : p hd
· simp [hp] at hx
· rw [hp, mem_cons] at hx
rcases hx with (rfl | hx)
· exact hp
· exact IH hx
|
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.DFinsupp.WellFounded
import Mathlib.Data.Finsupp.Lex
#align_import data.finsupp.well_founded from "leanprover-community/mathlib"@"5fd3186f1ec30a75d5f65732e... | Mathlib/Data/Finsupp/WellFounded.lean | 37 | 42 | theorem Lex.acc (x : α →₀ N) (h : ∀ a ∈ x.support, Acc (rᶜ ⊓ (· ≠ ·)) a) :
Acc (Finsupp.Lex r s) x := by |
rw [lex_eq_invImage_dfinsupp_lex]
classical
refine InvImage.accessible toDFinsupp (DFinsupp.Lex.acc (fun _ => hbot) (fun _ => hs) _ ?_)
simpa only [toDFinsupp_support] using h
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFunc... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 595 | 600 | theorem _root_.List.stronglyMeasurable_prod' (l : List (α → M))
(hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod := by |
induction' l with f l ihl; · exact stronglyMeasurable_one
rw [List.forall_mem_cons] at hl
rw [List.prod_cons]
exact hl.1.mul (ihl hl.2)
|
/-
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 | 769 | 770 | theorem insert_image_compl_eq_range (f : α → β) (x : α) : insert (f x) (f '' {x}ᶜ) = range f := by |
rw [← image_insert_eq, insert_eq, union_compl_self, image_univ]
|
/-
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.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"7... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 356 | 357 | theorem degree_mul_C (a0 : a ≠ 0) : (p * C a).degree = p.degree := by |
rw [degree_mul, degree_C a0, add_zero]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Andrew Yang
-/
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.End from "leanprover-community/mathlib"@"85075bccb68ab7fa49fb05db8... | Mathlib/CategoryTheory/Monoidal/End.lean | 167 | 171 | theorem μ_naturalityᵣ {m n n' : M} (g : n ⟶ n') (X : C) :
(F.map g).app ((F.obj m).obj X) ≫ (F.μ m n').app X =
(F.μ m n).app X ≫ (F.map (m ◁ g)).app X := by |
rw [← id_tensorHom, ← μ_naturality₂ F (𝟙 m) g X]
simp
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.ord_connected_component from "leanprover-community/... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 77 | 79 | theorem ordConnectedComponent_inter (s t : Set α) (x : α) :
ordConnectedComponent (s ∩ t) x = ordConnectedComponent s x ∩ ordConnectedComponent t x := by |
simp [ordConnectedComponent, setOf_and]
|
/-
Copyright (c) 2022 Daniel Roca González. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Roca González
-/
import Mathlib.Analysis.InnerProductSpace.Dual
#align_import analysis.inner_product_space.lax_milgram from "leanprover-community/mathlib"@"46b633fd842bef... | Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean | 65 | 71 | theorem antilipschitz (coercive : IsCoercive B) : ∃ C : ℝ≥0, 0 < C ∧ AntilipschitzWith C B♯ := by |
rcases coercive.bounded_below with ⟨C, C_pos, below_bound⟩
refine ⟨C⁻¹.toNNReal, Real.toNNReal_pos.mpr (inv_pos.mpr C_pos), ?_⟩
refine ContinuousLinearMap.antilipschitz_of_bound B♯ ?_
simp_rw [Real.coe_toNNReal', max_eq_left_of_lt (inv_pos.mpr C_pos), ←
inv_mul_le_iff (inv_pos.mpr C_pos)]
simpa using bel... |
/-
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 | 1,109 | 1,113 | theorem lift_prod {ι : Type u} (c : ι → Cardinal.{v}) :
lift.{w} (prod c) = prod fun i => lift.{w} (c i) := by |
lift c to ι → Type v using fun _ => trivial
simp only [← mk_pi, ← mk_uLift]
exact mk_congr (Equiv.ulift.trans <| Equiv.piCongrRight fun i => Equiv.ulift.symm)
|
/-
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.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
/-!
# Asco... | Mathlib/Topology/UniformSpace/Ascoli.lean | 212 | 240 | theorem EquicontinuousOn.comap_uniformOnFun_eq {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
(UniformOnFun.uniformSpace X α 𝔖).comap F =
(Pi.uniformSpace _).comap ((⋃₀ 𝔖).restrict ∘ F) := by |
-- Recall that the uniform structure on `X →ᵤ[𝔖] α` is the one induced by all the maps
-- `K.restrict : (X →ᵤ[𝔖] α) → (K →ᵤ α)` for `K ∈ 𝔖`. Its pullback along `F`, which is
-- the LHS of our goal, is thus the uniform structure induced by the maps
-- `K.restrict ∘ F : ι → (K →ᵤ α)` for `K ∈ 𝔖`.
have H1 :... |
/-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.tower from "leanprover-community/mathlib"@"bb168510e... | Mathlib/RingTheory/MvPolynomial/Tower.lean | 35 | 37 | theorem aeval_map_algebraMap (x : σ → B) (p : MvPolynomial σ R) :
aeval x (map (algebraMap R A) p) = aeval x p := by |
rw [aeval_def, aeval_def, eval₂_map, IsScalarTower.algebraMap_eq R A B]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 750 | 751 | theorem ae_restrict_of_ae_eq_of_ae_restrict {s t} (hst : s =ᵐ[μ] t) {p : α → Prop} :
(∀ᵐ x ∂μ.restrict s, p x) → ∀ᵐ x ∂μ.restrict t, p x := by | simp [Measure.restrict_congr_set hst]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Module.Defs
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.FreeGroup.Basic
#align_import gr... | Mathlib/GroupTheory/FreeAbelianGroup.lean | 129 | 135 | theorem map_hom {α β γ} [AddCommGroup β] [AddCommGroup γ] (a : FreeAbelianGroup α) (f : α → β)
(g : β →+ γ) : g (lift f a) = lift (g ∘ f) a := by |
show (g.comp (lift f)) a = lift (g ∘ f) a
apply lift.unique
intro a
show g ((lift f) (of a)) = g (f a)
simp only [(· ∘ ·), lift.of]
|
/-
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.Convex.Topology
#align_import topology.algebra.module.locally_convex from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea5... | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | 83 | 89 | theorem LocallyConvexSpace.ofBasisZero {ι : Type*} (b : ι → Set E) (p : ι → Prop)
(hbasis : (𝓝 0).HasBasis p b) (hconvex : ∀ i, p i → Convex 𝕜 (b i)) :
LocallyConvexSpace 𝕜 E := by |
refine LocallyConvexSpace.ofBases 𝕜 E (fun (x : E) (i : ι) => (x + ·) '' b i) (fun _ => p)
(fun x => ?_) fun x i hi => (hconvex i hi).translate x
rw [← map_add_left_nhds_zero]
exact hbasis.map _
|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 827 | 830 | theorem closedBall_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).closedBall x r = f ⁻¹' p.closedBall (f x) r := by |
ext
simp_rw [closedBall, mem_preimage, comp_apply, Set.mem_setOf_eq, map_sub]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 34 | 35 | theorem aeval_U (x : A) (n : ℤ) : aeval x (U R n) = (U A n).eval x := by |
rw [aeval_def, eval₂_eq_eval_map, map_U]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 1,306 | 1,307 | theorem isBigOWith_principal {s : Set α} : IsBigOWith c (𝓟 s) f g ↔ ∀ x ∈ s, ‖f x‖ ≤ c * ‖g x‖ := by |
rw [IsBigOWith_def, eventually_principal]
|
/-
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 | 379 | 380 | theorem comp_whiskerLeft_symm (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h') :
f ◁ g ◁ η = (α_ f g h).inv ≫ (f ≫ g) ◁ η ≫ (α_ f g h').hom := by | simp
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
import Mathlib.Order.WellFounded
#align_import order.succ_pred.basic from "l... | Mathlib/Order/SuccPred/Basic.lean | 290 | 295 | theorem succ_le_succ (h : a ≤ b) : succ a ≤ succ b := by |
by_cases hb : IsMax b
· by_cases hba : b ≤ a
· exact (hb <| hba.trans <| le_succ _).trans (le_succ _)
· exact succ_le_of_lt ((h.lt_of_not_le hba).trans_le <| le_succ b)
· rwa [succ_le_iff_of_not_isMax fun ha => hb <| ha.mono h, lt_succ_iff_of_not_isMax hb]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 1,757 | 1,758 | theorem strictMono_restrict [Preorder α] [Preorder β] {f : α → β} {s : Set α} :
StrictMono (s.restrict f) ↔ StrictMonoOn f s := by | simp [Set.restrict, StrictMono, StrictMonoOn]
|
/-
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.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-communit... | Mathlib/AlgebraicGeometry/Gluing.lean | 371 | 388 | theorem fromGlued_injective : Function.Injective 𝒰.fromGlued.1.base := by |
intro x y h
obtain ⟨i, x, rfl⟩ := 𝒰.gluedCover.ι_jointly_surjective x
obtain ⟨j, y, rfl⟩ := 𝒰.gluedCover.ι_jointly_surjective y
erw [← comp_apply, ← comp_apply] at h -- now `erw` after #13170
simp_rw [← SheafedSpace.comp_base, ← LocallyRingedSpace.comp_val] at h
erw [ι_fromGlued, ι_fromGlued] at h
let ... |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, E. W. Ayers
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.L... | Mathlib/CategoryTheory/Sites/Sieves.lean | 522 | 522 | theorem pullback_id : S.pullback (𝟙 _) = S := by | simp [Sieve.ext_iff]
|
/-
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.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 1,177 | 1,186 | theorem Machine.map_respects (g₁ : PointedMap Λ Λ') (g₂ : Λ' → Λ) {S} (ss : Supports M S)
(f₂₁ : Function.RightInverse f₁ f₂) (g₂₁ : ∀ q ∈ S, g₂ (g₁ q) = q) :
Respects (step M) (step (M.map f₁ f₂ g₁ g₂)) fun a b ↦ a.q ∈ S ∧ Cfg.map f₁ g₁ a = b := by |
intro c _ ⟨cs, rfl⟩
cases e : step M c
· rw [← M.map_step f₁ f₂ g₁ g₂ f₂₁ g₂₁ _ cs, e]
rfl
· refine ⟨_, ⟨step_supports M ss e cs, rfl⟩, TransGen.single ?_⟩
rw [← M.map_step f₁ f₂ g₁ g₂ f₂₁ g₂₁ _ cs, e]
rfl
|
/-
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 | 228 | 229 | theorem Coprime.gcd_mul_right_cancel (m : ℕ+) {n k : ℕ+} :
k.Coprime n → (m * k).gcd n = m.gcd n := by | rw [mul_comm]; apply Coprime.gcd_mul_left_cancel
|
/-
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, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 809 | 809 | theorem div_mul_comm : a / b * c = c / b * a := by | simp
|
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