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 |
|---|---|---|---|---|---|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlg... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 169 | 171 | theorem transpose_adjMatrix [Zero α] [One α] : (G.adjMatrix α)ᵀ = G.adjMatrix α := by |
ext
simp [adj_comm]
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.MeasureTheory.Integral.Periodic
import Mathlib.Data.ZMod.Quotient
#align_import measure_theory.group.add_circle from "leanprover-community/mathlib"@"f2ce608... | Mathlib/MeasureTheory/Group/AddCircle.lean | 54 | 92 | theorem isAddFundamentalDomain_of_ae_ball (I : Set <| AddCircle T) (u x : AddCircle T)
(hu : IsOfFinAddOrder u) (hI : I =ᵐ[volume] ball x (T / (2 * addOrderOf u))) :
IsAddFundamentalDomain (AddSubgroup.zmultiples u) I := by |
set G := AddSubgroup.zmultiples u
set n := addOrderOf u
set B := ball x (T / (2 * n))
have hn : 1 ≤ (n : ℝ) := by norm_cast; linarith [hu.addOrderOf_pos]
refine IsAddFundamentalDomain.mk_of_measure_univ_le ?_ ?_ ?_ ?_
· -- `NullMeasurableSet I volume`
exact measurableSet_ball.nullMeasurableSet.congr hI... |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import... | Mathlib/RingTheory/Nilpotent/Lemmas.lean | 66 | 68 | theorem nilpotent_iff_mem_prime : IsNilpotent x ↔ ∀ J : Ideal R, J.IsPrime → x ∈ J := by |
rw [← mem_nilradical, nilradical_eq_sInf, Submodule.mem_sInf]
rfl
|
/-
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.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
#align_import algebra.big_operators.nat_antidiagonal from "leanprover-com... | Mathlib/Algebra/BigOperators/NatAntidiagonal.lean | 35 | 38 | theorem prod_antidiagonal_swap {n : ℕ} {f : ℕ × ℕ → M} :
∏ p ∈ antidiagonal n, f p.swap = ∏ p ∈ antidiagonal n, f p := by |
conv_lhs => rw [← map_swap_antidiagonal, Finset.prod_map]
rfl
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.... | Mathlib/Logic/Basic.lean | 990 | 994 | theorem ExistsUnique.intro₂ {α : Sort*} {p : α → Sort*} [∀ x, Subsingleton (p x)]
{q : ∀ (x : α) (_ : p x), Prop} (w : α) (hp : p w) (hq : q w hp)
(H : ∀ (y) (hy : p y), q y hy → y = w) : ∃! x, ∃! hx : p x, q x hx := by |
simp only [exists_unique_iff_exists]
exact ExistsUnique.intro w ⟨hp, hq⟩ fun y ⟨hyp, hyq⟩ ↦ H y hyp hyq
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
/-!
... | Mathlib/Data/QPF/Univariate/Basic.lean | 678 | 690 | theorem liftp_iff_of_isUniform (h : q.IsUniform) {α : Type u} (x : F α) (p : α → Prop) :
Liftp p x ↔ ∀ u ∈ supp x, p u := by |
rw [liftp_iff, ← abs_repr x]
cases' repr x with a f; constructor
· rintro ⟨a', f', abseq, hf⟩ u
rw [supp_eq_of_isUniform h, h _ _ _ _ abseq]
rintro ⟨i, _, hi⟩
rw [← hi]
apply hf
intro h'
refine ⟨a, f, rfl, fun i => h' _ ?_⟩
rw [supp_eq_of_isUniform h]
exact ⟨i, mem_univ i, rfl⟩
|
/-
Copyright (c) 2021 Bhavik Mehta, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Order.Antichain
import Mathlib.Order.Interval.Finset.Nat
#alig... | Mathlib/Data/Finset/Slice.lean | 70 | 72 | theorem sized_iUnion₂ {f : ∀ i, κ i → Set (Finset α)} :
(⋃ (i) (j), f i j).Sized r ↔ ∀ i j, (f i j).Sized r := by |
simp only [Set.sized_iUnion]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.tagged from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97... | Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean | 156 | 165 | theorem forall_biUnionTagged (p : (ι → ℝ) → Box ι → Prop) (π : Prepartition I)
(πi : ∀ J, TaggedPrepartition J) :
(∀ J ∈ π.biUnionTagged πi, p ((π.biUnionTagged πi).tag J) J) ↔
∀ J ∈ π, ∀ J' ∈ πi J, p ((πi J).tag J') J' := by |
simp only [mem_biUnionTagged]
refine ⟨fun H J hJ J' hJ' => ?_, fun H J' ⟨J, hJ, hJ'⟩ => ?_⟩
· rw [← π.tag_biUnionTagged hJ hJ']
exact H J' ⟨J, hJ, hJ'⟩
· rw [π.tag_biUnionTagged hJ hJ']
exact H J hJ J' hJ'
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.Catego... | Mathlib/CategoryTheory/Equivalence.lean | 405 | 407 | theorem cancel_unit_right_assoc {W X X' Y : C} (f : W ⟶ X) (g : X ⟶ Y) (f' : W ⟶ X') (g' : X' ⟶ Y) :
f ≫ g ≫ e.unit.app Y = f' ≫ g' ≫ e.unit.app Y ↔ f ≫ g = f' ≫ g' := by |
simp only [← Category.assoc, cancel_mono]
|
/-
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 | 50 | 52 | theorem equalizerSubobject_arrow :
(equalizerSubobjectIso f g).hom ≫ equalizer.ι f g = (equalizerSubobject f g).arrow := by |
simp [equalizerSubobjectIso]
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valu... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 303 | 307 | theorem valuation_exists_uniformizer : ∃ π : K, v.valuation π = Multiplicative.ofAdd (-1 : ℤ) := by |
obtain ⟨r, hr⟩ := v.int_valuation_exists_uniformizer
use algebraMap R K r
rw [valuation_def, Valuation.extendToLocalization_apply_map_apply]
exact hr
|
/-
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 | 126 | 128 | theorem factorThru_self {X : C} (P : Subobject X) (h) : P.factorThru P.arrow h = 𝟙 (P : C) := by |
ext
simp
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-... | Mathlib/Order/LiminfLimsup.lean | 186 | 189 | theorem IsBoundedUnder.bddAbove_range [Preorder β] [IsDirected β (· ≤ ·)] {f : ℕ → β}
(hf : IsBoundedUnder (· ≤ ·) atTop f) : BddAbove (range f) := by |
rw [← Nat.cofinite_eq_atTop] at hf
exact hf.bddAbove_range_of_cofinite
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison
-/
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib... | Mathlib/SetTheory/Game/PGame.lean | 1,423 | 1,424 | theorem neg_fuzzy_iff {x y : PGame} : -x ‖ y ↔ x ‖ -y := by |
rw [← neg_neg y, neg_fuzzy_neg_iff, neg_neg]
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | Mathlib/Probability/Martingale/Upcrossing.lean | 556 | 572 | theorem upcrossingsBefore_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ} (hN₁ : N ≤ N₁)
(hN₁' : f N₁ ω < a) (hN₂ : N₁ ≤ N₂) (hN₂' : b < f N₂ ω) :
upcrossingsBefore a b f N ω < upcrossingsBefore a b f (N₂ + 1) ω := by |
refine lt_of_lt_of_le (Nat.lt_succ_self _) (le_csSup (upperCrossingTime_lt_bddAbove hab) ?_)
rw [Set.mem_setOf_eq, upperCrossingTime_succ_eq, hitting_lt_iff _ le_rfl]
refine ⟨N₂, ⟨?_, Nat.lt_succ_self _⟩, hN₂'.le⟩
rw [lowerCrossingTime, hitting_le_iff_of_lt _ (Nat.lt_succ_self _)]
refine ⟨N₁, ⟨le_trans ?_ hN... |
/-
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
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Regular
import Mathlib.Tactic.Common
#align_import algebra.gcd_monoid.basic from "leanp... | Mathlib/Algebra/GCDMonoid/Basic.lean | 172 | 181 | theorem normalize_eq_normalize {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
normalize a = normalize b := by |
nontriviality α
rcases associated_of_dvd_dvd hab hba with ⟨u, rfl⟩
refine by_cases (by rintro rfl; simp only [zero_mul]) fun ha : a ≠ 0 => ?_
suffices a * ↑(normUnit a) = a * ↑u * ↑(normUnit a) * ↑u⁻¹ by
simpa only [normalize_apply, mul_assoc, normUnit_mul ha u.ne_zero, normUnit_coe_units]
calc
a * ↑... |
/-
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 | 949 | 950 | theorem adjoin_intCast (n : ℤ) : F⟮(n : E)⟯ = ⊥ := by |
exact adjoin_simple_eq_bot_iff.mpr (intCast_mem ⊥ n)
|
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra... | Mathlib/Algebra/BigOperators/Finprod.lean | 585 | 586 | theorem finprod_eq_one_of_forall_eq_one {f : α → M} (h : ∀ x, f x = 1) : ∏ᶠ i, f i = 1 := by |
simp (config := { contextual := true }) [h]
|
/-
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.List.Basic
#align_import data.list.count from "leanprover-community/mathl... | Mathlib/Data/List/Count.lean | 90 | 93 | theorem count_cons' (a b : α) (l : List α) :
count a (b :: l) = count a l + if a = b then 1 else 0 := by |
simp only [count, beq_iff_eq, countP_cons, Nat.add_right_inj]
simp only [eq_comm]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 342 | 344 | theorem nthRoots_two_eq_zero_iff {r : R} : nthRoots 2 r = 0 ↔ ¬IsSquare r := by |
simp_rw [isSquare_iff_exists_sq, eq_zero_iff_forall_not_mem, mem_nthRoots (by norm_num : 0 < 2),
← not_exists, eq_comm]
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 295 | 299 | theorem addHaar_preimage_linearEquiv (f : E ≃ₗ[ℝ] E) (s : Set E) :
μ (f ⁻¹' s) = ENNReal.ofReal |LinearMap.det (f.symm : E →ₗ[ℝ] E)| * μ s := by |
have A : LinearMap.det (f : E →ₗ[ℝ] E) ≠ 0 := (LinearEquiv.isUnit_det' f).ne_zero
convert addHaar_preimage_linearMap μ A s
simp only [LinearEquiv.det_coe_symm]
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
#align_import analysis.calculus.deriv.slope from "leanprover-community/... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 215 | 220 | theorem HasDerivWithinAt.limsup_slope_norm_le (hf : HasDerivWithinAt f f' s x) (hr : ‖f'‖ < r) :
∀ᶠ z in 𝓝[s] x, ‖z - x‖⁻¹ * (‖f z‖ - ‖f x‖) < r := by |
apply (hf.limsup_norm_slope_le hr).mono
intro z hz
refine lt_of_le_of_lt (mul_le_mul_of_nonneg_left (norm_sub_norm_le _ _) ?_) hz
exact inv_nonneg.2 (norm_nonneg _)
|
/-
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 | 773 | 777 | theorem indicatorConstLp_sub {c' : E} :
indicatorConstLp p hs hμs c - indicatorConstLp p hs hμs c' =
indicatorConstLp p hs hμs (c - c') := by |
simp_rw [indicatorConstLp, ← Memℒp.toLp_sub, indicator_sub]
rfl
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c46... | Mathlib/Algebra/Homology/ComplexShape.lean | 154 | 158 | theorem next_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.next i = j := by |
apply c.next_eq _ h
rw [next]
rw [dif_pos]
exact Exists.choose_spec ⟨j, h⟩
|
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Quasispectrum
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
import Mathlib.Analysis.Complex.Liouville
import Mathlib.Analysis.Complex.P... | Mathlib/Analysis/NormedSpace/Spectrum.lean | 263 | 265 | theorem eventually_isUnit_resolvent (a : A) : ∀ᶠ z in cobounded 𝕜, IsUnit (resolvent a z) := by |
rw [atTop_basis_Ioi.cobounded_of_norm.eventually_iff]
exact ⟨‖a‖ * ‖(1 : A)‖, trivial, fun _ ↦ isUnit_resolvent.mp ∘ mem_resolventSet_of_norm_lt_mul⟩
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
/-!
# Measure with a given density with respect to another measure
For a measure `μ` on `α` and a fun... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 489 | 505 | theorem lintegral_withDensity_eq_lintegral_mul_non_measurable₀ (μ : Measure α) {f : α → ℝ≥0∞}
(hf : AEMeasurable f μ) (h'f : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) :
∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by |
let f' := hf.mk f
calc
∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, g a ∂μ.withDensity f' := by
rw [withDensity_congr_ae hf.ae_eq_mk]
_ = ∫⁻ a, (f' * g) a ∂μ := by
apply lintegral_withDensity_eq_lintegral_mul_non_measurable _ hf.measurable_mk
filter_upwards [h'f, hf.ae_eq_mk]
intro x hx h'x
... |
/-
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 | 325 | 328 | theorem glued_cover_cocycle_snd (x y z : 𝒰.J) :
gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y ≫ pullback.snd =
pullback.snd := by |
apply pullback.hom_ext <;> simp [pullback.condition]
|
/-
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.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 776 | 777 | theorem toIocDiv_eq_sub (a b : α) : toIocDiv hp a b = toIocDiv hp 0 (b - a) := by |
rw [toIocDiv_sub_eq_toIocDiv_add, zero_add]
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 1,474 | 1,479 | theorem eq_of_root {x y : L} (hx : IsAlgebraic K x)
(h_ev : (Polynomial.aeval y) (minpoly K x) = 0) : minpoly K y = minpoly K x := by |
have hy : IsAlgebraic K y := ⟨minpoly K x, ne_zero hx.isIntegral, h_ev⟩
exact Polynomial.eq_of_monic_of_associated (monic hy.isIntegral) (monic hx.isIntegral)
(Irreducible.associated_of_dvd (irreducible hy.isIntegral)
(irreducible hx.isIntegral) (dvd K y h_ev))
|
/-
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.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
/-!
# Cycles of a li... | Mathlib/Data/List/Cycle.lean | 94 | 106 | theorem nextOr_mem {xs : List α} {x d : α} (hd : d ∈ xs) : nextOr xs x d ∈ xs := by |
revert hd
suffices ∀ xs' : List α, (∀ x ∈ xs, x ∈ xs') → d ∈ xs' → nextOr xs x d ∈ xs' by
exact this xs fun _ => id
intro xs' hxs' hd
induction' xs with y ys ih
· exact hd
cases' ys with z zs
· exact hd
rw [nextOr]
split_ifs with h
· exact hxs' _ (mem_cons_of_mem _ (mem_cons_self _ _))
· exac... |
/-
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.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 65 | 76 | theorem eval_iterate_derivative_rootMultiplicity {p : R[X]} {t : R} :
(derivative^[p.rootMultiplicity t] p).eval t =
(p.rootMultiplicity t).factorial • (p /ₘ (X - C t) ^ p.rootMultiplicity t).eval t := by |
set m := p.rootMultiplicity t with hm
conv_lhs => rw [← p.pow_mul_divByMonic_rootMultiplicity_eq t, ← hm]
rw [iterate_derivative_mul, eval_finset_sum, sum_eq_single_of_mem _ (mem_range.mpr m.succ_pos)]
· rw [m.choose_zero_right, one_smul, eval_mul, m.sub_zero, iterate_derivative_X_sub_pow_self,
eval_natC... |
/-
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 Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lat... | Mathlib/Data/List/Perm.lean | 523 | 528 | theorem bind_append_perm (l : List α) (f g : α → List β) :
l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x := by |
induction' l with a l IH <;> simp
refine (Perm.trans ?_ (IH.append_left _)).append_left _
rw [← append_assoc, ← append_assoc]
exact perm_append_comm.append_right _
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 326 | 328 | theorem dual_node4L (l : Ordnode α) (x : α) (m : Ordnode α) (y : α) (r : Ordnode α) :
dual (node4L l x m y r) = node4R (dual r) y (dual m) x (dual l) := by |
cases m <;> simp [node4L, node4R, node3R, dual_node3L, dual_node', add_comm]
|
/-
Copyright (c) 2021 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, David Loeffler
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Numbe... | Mathlib/NumberTheory/BernoulliPolynomials.lean | 57 | 63 | theorem bernoulli_def (n : ℕ) : bernoulli n =
∑ i ∈ range (n + 1), Polynomial.monomial i (_root_.bernoulli (n - i) * choose n i) := by |
rw [← sum_range_reflect, add_succ_sub_one, add_zero, bernoulli]
apply sum_congr rfl
rintro x hx
rw [mem_range_succ_iff] at hx
rw [choose_symm hx, tsub_tsub_cancel_of_le hx]
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
#align_import number_theory.modular_forms.jacob... | Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 58 | 72 | theorem norm_exp_mul_sq_le {τ : ℂ} (hτ : 0 < τ.im) (n : ℤ) :
‖cexp (π * I * (n : ℂ) ^ 2 * τ)‖ ≤ rexp (-π * τ.im) ^ n.natAbs := by |
let y := rexp (-π * τ.im)
have h : y < 1 := exp_lt_one_iff.mpr (mul_neg_of_neg_of_pos (neg_lt_zero.mpr pi_pos) hτ)
refine (le_of_eq ?_).trans (?_ : y ^ n ^ 2 ≤ _)
· rw [Complex.norm_eq_abs, Complex.abs_exp]
have : (π * I * n ^ 2 * τ : ℂ).re = -π * τ.im * (n : ℝ) ^ 2 := by
rw [(by push_cast; ring : (π... |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | Mathlib/Algebra/Polynomial/Laurent.lean | 537 | 539 | theorem degree_C {a : R} (a0 : a ≠ 0) : (C a).degree = 0 := by |
rw [← mul_one (C a), ← T_zero]
exact degree_C_mul_T 0 a a0
|
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Extensions of Groups
This file defines the HNN extension of a group `G`, `HNN... | Mathlib/GroupTheory/HNNExtension.lean | 494 | 496 | theorem of_smul_eq_smul (g : G) (w : NormalWord d) :
(of g : HNNExtension G A B φ) • w = g • w := by |
simp [instHSMul, SMul.smul, MulAction.toEndHom]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 352 | 353 | theorem commute_int_add (n : ℤ) : Function.Commute f (n + ·) := by |
simpa only [add_comm _ (n : ℝ)] using f.commute_add_int n
|
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Complement
import Mathlib.GroupTheory.Sylow
import Mathlib.GroupTheory.Subgroup.Center
#align_import group_theory.transfer from "leanpro... | Mathlib/GroupTheory/Transfer.lean | 222 | 228 | theorem transferSylow_eq_pow_aux (g : G) (hg : g ∈ P) (k : ℕ) (g₀ : G)
(h : g₀⁻¹ * g ^ k * g₀ ∈ P) : g₀⁻¹ * g ^ k * g₀ = g ^ k := by |
haveI : (P : Subgroup G).IsCommutative :=
⟨⟨fun a b => Subtype.ext (hP (le_normalizer b.2) a a.2)⟩⟩
replace hg := (P : Subgroup G).pow_mem hg k
obtain ⟨n, hn, h⟩ := P.conj_eq_normalizer_conj_of_mem (g ^ k) g₀ hg h
exact h.trans (Commute.inv_mul_cancel (hP hn (g ^ k) hg).symm)
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Group.Units
import Ma... | Mathlib/Algebra/Order/Ring/Defs.lean | 956 | 957 | theorem bit0_lt_bit0 : bit0 a < bit0 b ↔ a < b := by |
rw [bit0, bit0, ← two_mul, ← two_mul, mul_lt_mul_left (zero_lt_two : 0 < (2 : α))]
|
/-
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.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
import Mathlib.MeasureTheory.Decomposition.Jordan
import Mathlib.MeasureThe... | Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean | 131 | 145 | theorem singularPart_mutuallySingular (s : SignedMeasure α) (μ : Measure α) :
s.toJordanDecomposition.posPart.singularPart μ ⟂ₘ
s.toJordanDecomposition.negPart.singularPart μ := by |
by_cases hl : s.HaveLebesgueDecomposition μ
· obtain ⟨i, hi, hpos, hneg⟩ := s.toJordanDecomposition.mutuallySingular
rw [s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add μ] at hpos
rw [s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add μ] at hneg
rw [add_apply, add_eq_zero_iff] ... |
/-
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
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 733 | 735 | theorem nhdsWithin_eq_comap_uniformity_of_mem {x : α} {T : Set α} (hx : x ∈ T) (S : Set α) :
𝓝[S] x = (𝓤 α ⊓ 𝓟 (T ×ˢ S)).comap (Prod.mk x) := by |
simp [nhdsWithin, nhds_eq_comap_uniformity, hx]
|
/-
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 | 547 | 561 | theorem integral_cos_mul_complex {z : ℂ} (hz : z ≠ 0) (a b : ℝ) :
(∫ x in a..b, Complex.cos (z * x)) = Complex.sin (z * b) / z - Complex.sin (z * a) / z := by |
apply integral_eq_sub_of_hasDerivAt
swap
· apply Continuous.intervalIntegrable
exact Complex.continuous_cos.comp (continuous_const.mul Complex.continuous_ofReal)
intro x _
have a := Complex.hasDerivAt_sin (↑x * z)
have b : HasDerivAt (fun y => y * z : ℂ → ℂ) z ↑x := hasDerivAt_mul_const _
have c : Ha... |
/-
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 | 670 | 673 | theorem fromEdgeSet_union (s t : Set (Sym2 V)) :
fromEdgeSet (s ∪ t) = fromEdgeSet s ⊔ fromEdgeSet t := by |
ext v w
simp [Set.mem_union, or_and_right]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_impo... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 216 | 216 | theorem int_add_iff : LiouvilleWith p (m + x) ↔ LiouvilleWith p x := by | rw [add_comm, add_int_iff]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
#align_import analysis.normed_space.multilinear from "leanprover-community/mathlib"@"f40476639bac089693a4... | Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean | 61 | 70 | theorem ContinuousLinearMap.norm_map_tail_le
(f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G) (m : ∀ i, Ei i) :
‖f (m 0) (tail m)‖ ≤ ‖f‖ * ∏ i, ‖m i‖ :=
calc
‖f (m 0) (tail m)‖ ≤ ‖f (m 0)‖ * ∏ i, ‖(tail m) i‖ := (f (m 0)).le_opNorm _
_ ≤ ‖f‖ * ‖m 0‖ * ∏ i, ‖tail m i‖ := mul... | ring
_ = ‖f‖ * ∏ i, ‖m i‖ := by
rw [prod_univ_succ]
rfl
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.Order.CompleteBooleanAlgebra
#align_import category_theory.morphism_propert... | Mathlib/CategoryTheory/MorphismProperty/Basic.lean | 313 | 313 | theorem isomorphisms.iff : (isomorphisms C) f ↔ IsIso f := by | rfl
|
/-
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.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
impo... | Mathlib/RingTheory/Ideal/Cotangent.lean | 63 | 65 | theorem map_toCotangent_ker : I.toCotangent.ker.map I.subtype = I ^ 2 := by |
rw [Ideal.toCotangent, Submodule.ker_mkQ, pow_two, Submodule.map_smul'' I ⊤ (Submodule.subtype I),
Algebra.id.smul_eq_mul, Submodule.map_subtype_top]
|
/-
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 | 317 | 322 | theorem support_sum [DecidableEq β] [Zero M] [AddCommMonoid N] {f : α →₀ M} {g : α → M → β →₀ N} :
(f.sum g).support ⊆ f.support.biUnion fun a => (g a (f a)).support := by |
have : ∀ c, (f.sum fun a b => g a b c) ≠ 0 → ∃ a, f a ≠ 0 ∧ ¬(g a (f a)) c = 0 := fun a₁ h =>
let ⟨a, ha, ne⟩ := Finset.exists_ne_zero_of_sum_ne_zero h
⟨a, mem_support_iff.mp ha, ne⟩
simpa only [Finset.subset_iff, mem_support_iff, Finset.mem_biUnion, sum_apply, exists_prop]
|
/-
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.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topol... | Mathlib/Topology/EMetricSpace/Basic.lean | 376 | 381 | theorem tendstoLocallyUniformly_iff {ι : Type*} [TopologicalSpace β] {F : ι → β → α} {f : β → α}
{p : Filter ι} :
TendstoLocallyUniformly F f p ↔
∀ ε > 0, ∀ x : β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, edist (f y) (F n y) < ε := by |
simp only [← tendstoLocallyUniformlyOn_univ, tendstoLocallyUniformlyOn_iff, mem_univ,
forall_const, exists_prop, nhdsWithin_univ]
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint f... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 223 | 240 | theorem norm_adjoint_comp_self (A : E →L[𝕜] F) :
‖ContinuousLinearMap.adjoint A ∘L A‖ = ‖A‖ * ‖A‖ := by |
refine le_antisymm ?_ ?_
· calc
‖A† ∘L A‖ ≤ ‖A†‖ * ‖A‖ := opNorm_comp_le _ _
_ = ‖A‖ * ‖A‖ := by rw [LinearIsometryEquiv.norm_map]
· rw [← sq, ← Real.sqrt_le_sqrt_iff (norm_nonneg _), Real.sqrt_sq (norm_nonneg _)]
refine opNorm_le_bound _ (Real.sqrt_nonneg _) fun x => ?_
have :=
calc
... |
/-
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.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 319 | 327 | theorem liftPropWithinAt_indep_chart_source [HasGroupoid M G] (he : e ∈ G.maximalAtlas M)
(xe : x ∈ e.source) :
LiftPropWithinAt P g s x ↔ LiftPropWithinAt P (g ∘ e.symm) (e.symm ⁻¹' s) (e x) := by |
rw [liftPropWithinAt_self_source, liftPropWithinAt_iff',
e.symm.continuousWithinAt_iff_continuousWithinAt_comp_right xe, e.symm_symm]
refine and_congr Iff.rfl ?_
rw [Function.comp_apply, e.left_inv xe, ← Function.comp.assoc,
hG.liftPropWithinAt_indep_chart_source_aux (chartAt _ (g x) ∘ g) (chart_mem_maxi... |
/-
Copyright (c) 2023 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Sébastien Gouëzel, Jireh Loreaux
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
/-!
# `L^p` distance on products of two metric space... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 240 | 243 | theorem prod_dist_eq_sup (f g : WithLp ∞ (α × β)) :
dist f g = dist f.fst g.fst ⊔ dist f.snd g.snd := by |
dsimp [dist]
exact if_neg ENNReal.top_ne_zero
|
/-
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 | 1,370 | 1,370 | theorem iInf_emptyset {f : β → α} : ⨅ x ∈ (∅ : Set β), f x = ⊤ := by | simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 492 | 494 | theorem span_inl_union_inr {s : Set M} {t : Set M₂} :
span R (inl R M M₂ '' s ∪ inr R M M₂ '' t) = (span R s).prod (span R t) := by |
rw [span_union, prod_eq_sup_map, ← span_image, ← span_image]
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-com... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 237 | 261 | theorem eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.Cover X} (x : Meq P S) (y : Meq P T) :
mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by |
constructor
· intro h
obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq.{u} _ _ _ h
use W.unop, h1.unop, h2.unop
ext I
apply_fun Multiequalizer.ι (W.unop.index P) I at hh
convert hh
all_goals
dsimp [diagram]
erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_a... |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
#align_import number_theory.legendre_symbol.mul_characte... | Mathlib/NumberTheory/MulChar/Basic.lean | 274 | 274 | theorem one_apply_coe (a : Rˣ) : (1 : MulChar R R') a = 1 := by | classical exact dif_pos a.isUnit
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"... | Mathlib/Probability/Kernel/Basic.lean | 731 | 735 | theorem setIntegral_piecewise {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(a : α) (g : β → E) (t : Set β) :
∫ b in t, g b ∂piecewise hs κ η a =
if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a := by |
simp_rw [piecewise_apply]; split_ifs <;> rfl
|
/-
Copyright (c) 2023 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
#align_import combinatorics.quiver.single_obj from "leanprover-community/mat... | Mathlib/Combinatorics/Quiver/SingleObj.lean | 139 | 142 | theorem pathToList_listToPath (l : List α) : pathToList (listToPath l) = l := by |
induction' l with a l ih
· rfl
· change a :: pathToList (listToPath l) = a :: l; rw [ih]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 334 | 335 | theorem equivMapDomain_trans' (f : α ≃ β) (g : β ≃ γ) :
@equivMapDomain _ _ M _ (f.trans g) = equivMapDomain g ∘ equivMapDomain f := by | ext x; rfl
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e... | Mathlib/Topology/MetricSpace/Thickening.lean | 158 | 160 | theorem thickening_singleton (δ : ℝ) (x : X) : thickening δ ({x} : Set X) = ball x δ := by |
ext
simp [mem_thickening_iff]
|
/-
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 | 1,729 | 1,730 | theorem mul_lt_aleph0_iff_of_ne_zero {a b : Cardinal} (ha : a ≠ 0) (hb : b ≠ 0) :
a * b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ := by | simp [mul_lt_aleph0_iff, ha, hb]
|
/-
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 | 707 | 709 | theorem integral_sin_pow_succ_le : (∫ x in (0)..π, sin x ^ (n + 1)) ≤ ∫ x in (0)..π, sin x ^ n := by |
let H x h := pow_le_pow_of_le_one (sin_nonneg_of_mem_Icc h) (sin_le_one x) (n.le_add_right 1)
refine integral_mono_on pi_pos.le ?_ ?_ H <;> exact (continuous_sin.pow _).intervalIntegrable 0 π
|
/-
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, Sander Dahmen, Scott Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_im... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 271 | 276 | theorem rank_pi [Finite η] : Module.rank R (∀ i, φ i) =
Cardinal.sum fun i => Module.rank R (φ i) := by |
cases nonempty_fintype η
let B i := chooseBasis R (φ i)
let b : Basis _ R (∀ i, φ i) := Pi.basis fun i => B i
simp [← b.mk_eq_rank'', fun i => (B i).mk_eq_rank'']
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# Heyting algebr... | Mathlib/Order/Heyting/Basic.lean | 1,044 | 1,045 | theorem codisjoint_hnot_hnot_left_iff : Codisjoint (￢￢a) b ↔ Codisjoint a b := by |
simp_rw [← hnot_le_iff_codisjoint_right, hnot_hnot_hnot]
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.NumberTheory.NumberField.Discriminant
#align_import number_theory.cyclotomic.discriminant from... | Mathlib/NumberTheory/Cyclotomic/Discriminant.lean | 193 | 201 | theorem discr_prime_pow_eq_unit_mul_pow [IsCyclotomicExtension {p ^ k} K L]
[hp : Fact (p : ℕ).Prime] (hζ : IsPrimitiveRoot ζ ↑(p ^ k))
(hirr : Irreducible (cyclotomic (↑(p ^ k) : ℕ) K)) :
∃ (u : ℤˣ) (n : ℕ), discr K (hζ.powerBasis K).basis = u * p ^ n := by |
rw [discr_prime_pow hζ hirr]
by_cases heven : Even ((p ^ k : ℕ).totient / 2)
· exact ⟨1, (p : ℕ) ^ (k - 1) * ((p - 1) * k - 1), by rw [heven.neg_one_pow]; norm_num⟩
· exact ⟨-1, (p : ℕ) ^ (k - 1) * ((p - 1) * k - 1), by
rw [(odd_iff_not_even.2 heven).neg_one_pow]; norm_num⟩
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... | Mathlib/Analysis/BoundedVariation.lean | 572 | 585 | theorem comp_eq_of_antitoneOn (f : α → E) {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t) :
eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t) := by |
apply le_antisymm (comp_le_of_antitoneOn f φ hφ (mapsTo_image φ t))
cases isEmpty_or_nonempty β
· convert zero_le (_ : ℝ≥0∞)
exact eVariationOn.subsingleton f <| (subsingleton_of_subsingleton.image _).anti
(surjOn_image φ t)
let ψ := φ.invFunOn t
have ψφs : EqOn (φ ∘ ψ) id (φ '' t) := (surjOn_image... |
/-
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 | 571 | 577 | theorem factors_pow_count_prod [DecidableEq α] {x : α} (hx : x ≠ 0) :
(∏ p ∈ (factors x).toFinset, p ^ (factors x).count p) ~ᵤ x :=
calc
_ = prod (∑ a ∈ toFinset (factors x), count a (factors x) • {a}) := by |
simp only [prod_sum, prod_nsmul, prod_singleton]
_ = prod (factors x) := by rw [toFinset_sum_count_nsmul_eq (factors x)]
_ ~ᵤ x := factors_prod hx
|
/-
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 | 568 | 583 | theorem ord_compl_dvd_ord_compl_of_dvd {a b : ℕ} (hab : a ∣ b) (p : ℕ) :
ord_compl[p] a ∣ ord_compl[p] b := by |
rcases em' p.Prime with (pp | pp)
· simp [pp, hab]
rcases eq_or_ne b 0 with (rfl | hb0)
· simp
rcases eq_or_ne a 0 with (rfl | ha0)
· cases hb0 (zero_dvd_iff.1 hab)
have ha := (Nat.div_pos (ord_proj_le p ha0) (ord_proj_pos a p)).ne'
have hb := (Nat.div_pos (ord_proj_le p hb0) (ord_proj_pos b p)).ne'
... |
/-
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 | 212 | 229 | theorem IsChain.exists_maxChain (hc : IsChain r c) : ∃ M, @IsMaxChain _ r M ∧ c ⊆ M := by |
-- Porting note: the first three lines replace the following two lines in mathlib3.
-- The mathlib3 `obtain` supports holes for proof obligations, this is not yet implemented in 4.
-- obtain ⟨M, ⟨_, hM₀⟩, hM₁, hM₂⟩ :=
-- zorn_subset_nonempty { s | c ⊆ s ∧ IsChain r s } _ c ⟨Subset.rfl, hc⟩
have H := zorn_s... |
/-
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.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Nonarchimedean.Basic
/-!
# Infinite sums and products in nonarchimedean abelian groups
... | Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean | 31 | 48 | theorem cauchySeq_prod_of_tendsto_cofinite_one {f : α → G} (hf : Tendsto f cofinite (𝓝 1)) :
CauchySeq (fun s ↦ ∏ i ∈ s, f i) := by |
/- Let `U` be a neighborhood of `1`. It suffices to show that there exists `s : Finset α` such
that for any `t : Finset α` disjoint from `s`, we have `∏ i ∈ t, f i ∈ U`. -/
apply cauchySeq_finset_iff_prod_vanishing.mpr
intro U hU
-- Since `G` is nonarchimedean, `U` contains an open subgroup `V`.
rcases is_... |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padic... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 838 | 842 | theorem norm_p : ‖(p : ℚ_[p])‖ = (p : ℝ)⁻¹ := by |
rw [← @Rat.cast_natCast ℝ _ p]
rw [← @Rat.cast_natCast ℚ_[p] _ p]
simp [hp.1.ne_zero, hp.1.ne_one, norm, padicNorm, padicValRat, padicValInt, zpow_neg,
-Rat.cast_natCast]
|
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import d... | Mathlib/Data/PNat/Factors.lean | 321 | 330 | theorem factorMultiset_le_iff {m n : ℕ+} : factorMultiset m ≤ factorMultiset n ↔ m ∣ n := by |
constructor
· intro h
rw [← prod_factorMultiset m, ← prod_factorMultiset m]
apply Dvd.intro (n.factorMultiset - m.factorMultiset).prod
rw [← PrimeMultiset.prod_add, PrimeMultiset.factorMultiset_prod, add_tsub_cancel_of_le h,
prod_factorMultiset]
· intro h
rw [← mul_div_exact h, factorMultis... |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin... | Mathlib/RingTheory/IsAdjoinRoot.lean | 547 | 562 | theorem coeff_root_pow (h : IsAdjoinRootMonic S f) {n} (hn : n < natDegree f) :
h.coeff (h.root ^ n) = Pi.single n 1 := by |
ext i
rw [coeff_apply]
split_ifs with hi
· calc
h.basis.repr (h.root ^ n) ⟨i, _⟩ = h.basis.repr (h.basis ⟨n, hn⟩) ⟨i, hi⟩ := by
rw [h.basis_apply, Fin.val_mk]
_ = Pi.single (f := fun _ => R) ((⟨n, hn⟩ : Fin _) : ℕ) (1 : (fun _ => R) n)
↑(⟨i, _⟩ : Fin _) := by
rw [h.basis.rep... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 933 | 942 | theorem tendsto_comp_of_locally_uniform_limit_within (h : ContinuousWithinAt f s x)
(hg : Tendsto g p (𝓝[s] x))
(hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u) :
Tendsto (fun n => F n (g n)) p (𝓝 (f x)) := by |
refine Uniform.tendsto_nhds_right.2 fun u₀ hu₀ => ?_
obtain ⟨u₁, h₁, u₁₀⟩ : ∃ u ∈ 𝓤 β, u ○ u ⊆ u₀ := comp_mem_uniformity_sets hu₀
rcases hunif u₁ h₁ with ⟨s, sx, hs⟩
have A : ∀ᶠ n in p, g n ∈ s := hg sx
have B : ∀ᶠ n in p, (f x, f (g n)) ∈ u₁ := hg (Uniform.continuousWithinAt_iff'_right.1 h h₁)
exact B.mp... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "lean... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,269 | 1,289 | theorem hasFTaylorSeriesUpToOn_univ_iff :
HasFTaylorSeriesUpToOn n f p univ ↔ HasFTaylorSeriesUpTo n f p := by |
constructor
· intro H
constructor
· exact fun x => H.zero_eq x (mem_univ x)
· intro m hm x
rw [← hasFDerivWithinAt_univ]
exact H.fderivWithin m hm x (mem_univ x)
· intro m hm
rw [continuous_iff_continuousOn_univ]
exact H.cont m hm
· intro H
constructor
· exact fun ... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "lea... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 54 | 63 | theorem snorm_add_le {f g : α → E} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
(hp1 : 1 ≤ p) : snorm (f + g) p μ ≤ snorm f p μ + snorm g p μ := by |
by_cases hp0 : p = 0
· simp [hp0]
by_cases hp_top : p = ∞
· simp [hp_top, snormEssSup_add_le]
have hp1_real : 1 ≤ p.toReal := by
rwa [← ENNReal.one_toReal, ENNReal.toReal_le_toReal ENNReal.one_ne_top hp_top]
repeat rw [snorm_eq_snorm' hp0 hp_top]
exact snorm'_add_le hf hg hp1_real
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib... | Mathlib/Data/List/Sublists.lean | 472 | 486 | theorem range_bind_sublistsLen_perm (l : List α) :
((List.range (l.length + 1)).bind fun n => sublistsLen n l) ~ sublists' l := by |
induction' l with h tl l_ih
· simp [range_succ]
· simp_rw [range_succ_eq_map, length, cons_bind, map_bind, sublistsLen_succ_cons, sublists'_cons,
List.sublistsLen_zero, List.singleton_append]
refine ((bind_append_perm (range (tl.length + 1)) _ _).symm.cons _).trans ?_
simp_rw [← List.bind_map, ← co... |
/-
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.MeasureTheory.Constructions.HaarToSphere
import Mathlib.MeasureTheory.Integral.Gamma
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Spe... | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | 297 | 315 | theorem Complex.volume_sum_rpow_le [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
volume {x : ι → ℂ | (∑ i, ‖x i‖ ^ p) ^ (1 / p) ≤ r} = (.ofReal r) ^ (2 * card ι) *
.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * card ι / p + 1)) := by |
have h₁ : 0 < p := by linarith
-- We collect facts about `Lp` norms that will be used in `measure_lt_one_eq_integral_div_gamma`
have eq_norm := fun x : ι → ℂ => (PiLp.norm_eq_sum (p := .ofReal p) (f := x)
((toReal_ofReal (le_of_lt h₁)).symm ▸ h₁))
simp_rw [toReal_ofReal (le_of_lt h₁)] at eq_norm
have : F... |
/-
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 | 656 | 657 | theorem ssubset_toFinset {s : Finset α} [Fintype t] : s ⊂ t.toFinset ↔ ↑s ⊂ t := by |
rw [← Finset.coe_ssubset, coe_toFinset]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean fr... | Mathlib/Order/Filter/Archimedean.lean | 296 | 299 | theorem Tendsto.atBot_zsmul_const {f : α → ℤ} (hr : 0 < r) (hf : Tendsto f l atBot) :
Tendsto (fun x => f x • r) l atBot := by |
simp only [← tendsto_neg_atTop_iff, ← neg_zsmul] at hf ⊢
exact hf.atTop_zsmul_const hr
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.or... | Mathlib/Algebra/Order/Group/MinMax.lean | 75 | 76 | theorem max_div_div_left' (a b c : α) : max (a / b) (a / c) = a / min b c := by |
simp only [div_eq_mul_inv, max_mul_mul_left, max_inv_inv']
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 396 | 396 | theorem add_neg {X Y : C} (a b : X ⟶ Y) : a + -b = a - b := by | rw [add_def, neg_neg]
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 378 | 380 | theorem Filter.EventuallyEq.lineDeriv_eq (h : f₁ =ᶠ[𝓝 x] f) :
lineDeriv 𝕜 f₁ x v = lineDeriv 𝕜 f x v := by |
rw [← lineDerivWithin_univ, ← lineDerivWithin_univ, h.lineDerivWithin_eq_nhds]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 730 | 732 | theorem submodule_span_le_lieSpan : Submodule.span R s ≤ lieSpan R L s := by |
rw [Submodule.span_le]
apply subset_lieSpan
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import linear_algebra.clifford_algeb... | Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean | 35 | 37 | theorem one_le_evenOdd_zero : 1 ≤ evenOdd Q 0 := by |
refine le_trans ?_ (le_iSup _ ⟨0, Nat.cast_zero⟩)
exact (pow_zero _).ge
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Star.Spectrum
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Analysis.NormedSpace.Algebra
import Mathlib.Topology.Cont... | Mathlib/Analysis/NormedSpace/Star/GelfandDuality.lean | 275 | 281 | theorem gelfandStarTransform_naturality {A B : Type*} [NormedCommRing A] [NormedAlgebra ℂ A]
[CompleteSpace A] [StarRing A] [CstarRing A] [StarModule ℂ A] [NormedCommRing B]
[NormedAlgebra ℂ B] [CompleteSpace B] [StarRing B] [CstarRing B] [StarModule ℂ B]
(φ : A →⋆ₐ[ℂ] B) :
(gelfandStarTransform B : _ →... |
rfl
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Filter.Bases
import Mathlib.Order.Filter.Ultrafilter
/-!
# Subsingleton filters
We say that a filter `l` is a *subsingleton* if there exists a... | Mathlib/Order/Filter/Subsingleton.lean | 70 | 71 | theorem subsingleton_iff_exists_singleton_mem [Nonempty α] : l.Subsingleton ↔ ∃ a, {a} ∈ l := by |
simp only [subsingleton_iff_exists_le_pure, le_pure_iff]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... | Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 471 | 472 | theorem biprod.lift_eq {T : C} {f : T ⟶ X} {g : T ⟶ Y} :
biprod.lift f g = f ≫ biprod.inl + g ≫ biprod.inr := by | ext <;> simp [add_comp]
|
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 1,234 | 1,238 | theorem Ideal.IsPrime.mem_pow_mul (I : Ideal R) [hI : I.IsPrime] {a b : R} {n : ℕ}
(h : a * b ∈ I ^ n) : a ∈ I ^ n ∨ b ∈ I := by |
rw [mul_comm] at h
rw [or_comm]
exact Ideal.IsPrime.mul_mem_pow _ 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 | 1,358 | 1,360 | theorem lintegral_inter_add_diff {B : Set α} (f : α → ℝ≥0∞) (A : Set α) (hB : MeasurableSet B) :
∫⁻ x in A ∩ B, f x ∂μ + ∫⁻ x in A \ B, f x ∂μ = ∫⁻ x in A, f x ∂μ := by |
rw [← lintegral_add_measure, restrict_inter_add_diff _ hB]
|
/-
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, Sander Dahmen, Scott Morrison
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeMod... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 417 | 420 | theorem FiniteDimensional.finrank_pos_iff [NoZeroSMulDivisors R M] :
0 < finrank R M ↔ Nontrivial M := by |
rw [← rank_pos_iff_nontrivial (R := R), ← finrank_eq_rank]
norm_cast
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/math... | Mathlib/Probability/Kernel/CondDistrib.lean | 298 | 316 | theorem _root_.MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk
{Ω F} {mΩ : MeasurableSpace Ω} (X : Ω → β) {μ : Measure Ω} [TopologicalSpace F] {f : Ω → F}
(hf : AEStronglyMeasurable f μ) :
AEStronglyMeasurable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) := by |
refine ⟨fun x => hf.mk f x.2, hf.stronglyMeasurable_mk.comp_measurable measurable_snd, ?_⟩
suffices h : Measure.QuasiMeasurePreserving Prod.snd (μ.map fun ω ↦ (X ω, ω)) μ from
Measure.QuasiMeasurePreserving.ae_eq h hf.ae_eq_mk
refine ⟨measurable_snd, Measure.AbsolutelyContinuous.mk fun s hs hμs => ?_⟩
rw [... |
/-
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 | 260 | 262 | theorem ofMatroid_le_iff {M M' : Matroid α} :
Matroidᵣ.ofMatroid M ≤ Matroidᵣ.ofMatroid M' ↔ M ≤r M' := by |
simp
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedS... | Mathlib/Analysis/RCLike/Basic.lean | 774 | 775 | theorem re_eq_norm_of_mul_conj (x : K) : re (x * conj x) = ‖x * conj x‖ := by |
rw [mul_conj, ← ofReal_pow]; simp [-ofReal_pow]
|
/-
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.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Analysis/ODE/Gronwall.lean | 96 | 101 | theorem gronwallBound_continuous_ε (δ K x : ℝ) : Continuous fun ε => gronwallBound δ K ε x := by |
by_cases hK : K = 0
· simp only [gronwallBound_K0, hK]
exact continuous_const.add (continuous_id.mul continuous_const)
· simp only [gronwallBound_of_K_ne_0 hK]
exact continuous_const.add ((continuous_id.mul continuous_const).mul continuous_const)
|
/-
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.Order.Hom.CompleteLattice
import Mathlib.Topology.Bases
import Mathlib.Topology.Homeomorph
import Mathlib.Topology.Co... | Mathlib/Topology/Sets/Opens.lean | 278 | 279 | theorem ne_bot_iff_nonempty (U : Opens α) : U ≠ ⊥ ↔ Set.Nonempty (U : Set α) := by |
rw [Ne, ← not_nonempty_iff_eq_bot, not_not]
|
/-
Copyright (c) 2023 Bulhwi Cha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bulhwi Cha, Mario Carneiro
-/
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqF... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 228 | 231 | theorem prev_of_valid (cs : List Char) (c : Char) (cs' : List Char) :
prev ⟨cs ++ c :: cs'⟩ ⟨utf8Len cs + csize c⟩ = ⟨utf8Len cs⟩ := by |
simp [prev]; refine (if_neg (Pos.ne_of_gt add_csize_pos)).trans ?_
rw [utf8PrevAux_of_valid] <;> simp
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-communit... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 398 | 402 | theorem toOrdinal_cast_nat (n : ℕ) : toOrdinal n = n := by |
induction' n with n hn
· rfl
· change (toOrdinal n) ♯ 1 = n + 1
rw [hn]; exact nadd_one n
|
/-
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 | 1,133 | 1,139 | theorem comp_eval₂Hom [CommSemiring S₂] (f : R →+* S₁) (g : σ → S₁) (φ : S₁ →+* S₂) :
φ.comp (eval₂Hom f g) = eval₂Hom (φ.comp f) fun i => φ (g i) := by |
apply MvPolynomial.ringHom_ext
· intro r
rw [RingHom.comp_apply, eval₂Hom_C, eval₂Hom_C, RingHom.comp_apply]
· intro i
rw [RingHom.comp_apply, eval₂Hom_X', eval₂Hom_X']
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.