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) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.NoncommRing
#align_import algebra.algebra.spectrum from "leanprover-c... | Mathlib/Algebra/Algebra/Spectrum.lean | 116 | 117 | theorem zero_mem_iff {a : A} : (0 : R) ∈ σ a ↔ ¬IsUnit a := by |
rw [mem_iff, map_zero, zero_sub, IsUnit.neg_iff]
|
/-
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.Convex.Topology
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.calculus.... | Mathlib/Analysis/Calculus/TangentCone.lean | 321 | 330 | theorem UniqueDiffWithinAt.univ_pi (ι : Type*) [Finite ι] (E : ι → Type*)
[∀ i, NormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)] (s : ∀ i, Set (E i)) (x : ∀ i, E i)
(h : ∀ i, UniqueDiffWithinAt 𝕜 (s i) (x i)) : UniqueDiffWithinAt 𝕜 (Set.pi univ s) x := by |
classical
simp only [uniqueDiffWithinAt_iff, closure_pi_set] at h ⊢
refine ⟨(dense_pi univ fun i _ => (h i).1).mono ?_, fun i _ => (h i).2⟩
norm_cast
simp only [← Submodule.iSup_map_single, iSup_le_iff, LinearMap.map_span, Submodule.span_le,
← mapsTo']
exact fun i => (mapsTo_tangentCone_pi fun j _ => (... |
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.PEmpty
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.... | Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean | 540 | 543 | theorem IsInitial.mono_from [InitialMonoClass C] {I} {X : C} (hI : IsInitial I) (f : I ⟶ X) :
Mono f := by |
rw [hI.hom_ext f (hI.to X)]
apply InitialMonoClass.isInitial_mono_from
|
/-
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.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0... | Mathlib/Geometry/Euclidean/MongePoint.lean | 775 | 798 | theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
{t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
s = insert t.orthocenter (Set.range t.points) := by |
rcases ho with ⟨t₀, ht₀o, ht₀s⟩
rw [ht₀s] at ht
rcases exists_of_range_subset_orthocentricSystem ht₀o ht t.independent.injective with
(⟨i₁, i₂, i₃, j₂, j₃, h₁₂, h₁₃, h₂₃, h₁₂₃, h₁, hj₂₃, h₂, h₃⟩ | hs)
· obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ... |
/-
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.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 209 | 215 | theorem restrict_ofFunction (s : Set α) (hm : Monotone m) :
restrict s (OuterMeasure.ofFunction m m_empty) =
OuterMeasure.ofFunction (fun t => m (t ∩ s)) (by simp; simp [m_empty]) := by |
rw [restrict]
simp only [inter_comm _ s, LinearMap.comp_apply]
rw [comap_ofFunction _ (Or.inl hm)]
simp only [map_ofFunction Subtype.coe_injective, Subtype.image_preimage_coe]
|
/-
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.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_prod... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 61 | 66 | theorem OrthonormalBasis.volume_parallelepiped (b : OrthonormalBasis ι ℝ F) :
volume (parallelepiped b) = 1 := by |
haveI : Fact (finrank ℝ F = finrank ℝ F) := ⟨rfl⟩
let o := (stdOrthonormalBasis ℝ F).toBasis.orientation
rw [← o.measure_eq_volume]
exact o.measure_orthonormalBasis b
|
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | Mathlib/Algebra/Ring/Ext.lean | 224 | 229 | theorem toNonUnitalSemiring_injective :
Function.Injective (@toNonUnitalSemiring R) := by |
intro _ _ h
ext x y
· exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
|
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johanes Hölzl, Patrick Massot, Yury Kudryashov, Kevin Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@... | Mathlib/Order/Filter/Prod.lean | 404 | 406 | theorem prod_inf_prod {f₁ f₂ : Filter α} {g₁ g₂ : Filter β} :
(f₁ ×ˢ g₁) ⊓ (f₂ ×ˢ g₂) = (f₁ ⊓ f₂) ×ˢ (g₁ ⊓ g₂) := by |
simp only [SProd.sprod, Filter.prod, comap_inf, inf_comm, inf_assoc, inf_left_comm]
|
/-
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, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 817 | 818 | theorem nhds_le_of_le {f} (h : x ∈ s) (o : IsOpen s) (sf : 𝓟 s ≤ f) : 𝓝 x ≤ f := by |
rw [nhds_def]; exact iInf₂_le_of_le s ⟨h, o⟩ sf
|
/-
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.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import ... | Mathlib/Data/Set/Basic.lean | 2,423 | 2,425 | theorem Injective.nonempty_apply_iff {f : Set α → Set β} (hf : Injective f) (h2 : f ∅ = ∅)
{s : Set α} : (f s).Nonempty ↔ s.Nonempty := by |
rw [nonempty_iff_ne_empty, ← h2, nonempty_iff_ne_empty, hf.ne_iff]
|
/-
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 | 2,890 | 2,894 | theorem filter_ne [DecidableEq β] (s : Finset β) (b : β) :
(s.filter fun a => b ≠ a) = s.erase b := by |
ext
simp only [mem_filter, mem_erase, Ne, decide_not, Bool.not_eq_true', decide_eq_false_iff_not]
tauto
|
/-
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 | 269 | 270 | theorem sub_int_iff : LiouvilleWith p (x - m) ↔ LiouvilleWith p x := by |
rw [← Rat.cast_intCast, sub_rat_iff]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 354 | 359 | theorem norm_inner_le_norm (x y : F) : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ :=
nonneg_le_nonneg_of_sq_le_sq (mul_nonneg (sqrt_nonneg _) (sqrt_nonneg _)) <|
calc
‖⟪x, y⟫‖ * ‖⟪x, y⟫‖ = ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ := by | rw [norm_inner_symm]
_ ≤ re ⟪x, x⟫ * re ⟪y, y⟫ := inner_mul_inner_self_le x y
_ = ‖x‖ * ‖y‖ * (‖x‖ * ‖y‖) := by simp only [inner_self_eq_norm_mul_norm]; ring
|
/-
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.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDe... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 1,099 | 1,109 | theorem map_extend_nhdsWithin_eq_image {y : M} (hy : y ∈ f.source) :
map (f.extend I) (𝓝[s] y) = 𝓝[f.extend I '' ((f.extend I).source ∩ s)] f.extend I y := by |
set e := f.extend I
calc
map e (𝓝[s] y) = map e (𝓝[e.source ∩ s] y) :=
congr_arg (map e) (nhdsWithin_inter_of_mem (extend_source_mem_nhdsWithin f I hy)).symm
_ = 𝓝[e '' (e.source ∩ s)] e y :=
((f.extend I).leftInvOn.mono inter_subset_left).map_nhdsWithin_eq
((f.extend I).left_inv <| ... |
/-
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.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Algebra.SMulWithZero
import Mathlib.Order.Hom.Basic
imp... | Mathlib/Algebra/Tropical/Basic.lean | 338 | 339 | theorem add_eq_left_iff {x y : Tropical R} : x + y = x ↔ x ≤ y := by |
rw [trop_add_def, trop_eq_iff_eq_untrop, ← untrop_le_iff, min_eq_left_iff]
|
/-
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 | 764 | 768 | theorem nats_eq : nats = cons 0 (map succ nats) := by |
apply Stream'.ext; intro n
cases n
· rfl
rw [get_succ]; rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 96 | 100 | theorem measure_diff_null (ht : μ t = 0) : μ (s \ t) = μ s :=
(measure_mono diff_subset).antisymm <| calc
μ s ≤ μ (s ∩ t) + μ (s \ t) := measure_le_inter_add_diff _ _ _
_ ≤ μ t + μ (s \ t) := by | gcongr; apply inter_subset_right
_ = μ (s \ t) := by simp [ht]
|
/-
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 | 100 | 102 | theorem Filter.Eventually.intCast_atBot [StrictOrderedRing R] [Archimedean R] {p : R → Prop}
(h : ∀ᶠ (x:R) in atBot, p x) : ∀ᶠ (n:ℤ) in atBot, p n := by |
rw [← Int.comap_cast_atBot (R := R)]; exact h.comap _
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Nonarchimedean.Basic
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Algebra.Module.Submodule.Pointwise
#align_import topo... | Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean | 390 | 398 | theorem RingFilterBasis.submodulesBasisIsBasis (BR : RingFilterBasis R) {B : ι → Submodule R M}
(hB : BR.SubmodulesBasis B) : @_root_.SubmodulesBasis ι R _ M _ _ BR.topology B :=
let _ := BR.topology -- Porting note: failed to synthesize instance `TopologicalSpace R`
{ inter := hB.inter
smul := by |
letI := BR.topology
intro m i
rcases hB.smul m i with ⟨V, V_in, hV⟩
exact mem_of_superset (BR.toAddGroupFilterBasis.mem_nhds_zero V_in) hV }
|
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Topology.Algebra.Order.Liminf... | Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 292 | 297 | theorem superpolynomialDecay_param_pow_mul_iff (hk : Tendsto k l atTop) (n : ℕ) :
SuperpolynomialDecay l k (k ^ n * f) ↔ SuperpolynomialDecay l k f := by |
induction' n with n hn
· simp
· simpa [pow_succ, ← mul_comm k, mul_assoc,
superpolynomialDecay_param_mul_iff (k ^ n * f) hk] using hn
|
/-
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 | 601 | 602 | theorem subsingleton_reverse_iff {s : Cycle α} : s.reverse.Subsingleton ↔ s.Subsingleton := by |
simp [length_subsingleton_iff]
|
/-
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.Set.Finite
import Mathlib.Data.Countable.Basic
import Mathlib.Logic.Equiv.List
import Mathlib.Data.Set.Subsingleton
#align_import data.set.counta... | Mathlib/Data/Set/Countable.lean | 222 | 226 | theorem countable_iUnion {t : ι → Set α} [Countable ι] (ht : ∀ i, (t i).Countable) :
(⋃ i, t i).Countable := by |
have := fun i ↦ (ht i).to_subtype
rw [iUnion_eq_range_psigma]
apply countable_range
|
/-
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.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.f... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 358 | 370 | theorem inv_of_red_of_ne {x1 b1 x2 b2} (H1 : (x1, b1) ≠ (x2, b2))
(H2 : Red ((x1, b1) :: L₁) ((x2, b2) :: L₂)) : Red L₁ ((x1, not b1) :: (x2, b2) :: L₂) := by |
have : Red ((x1, b1) :: L₁) ([(x2, b2)] ++ L₂) := H2
rcases to_append_iff.1 this with ⟨_ | ⟨p, L₃⟩, L₄, eq, h₁, h₂⟩
· simp [nil_iff] at h₁
· cases eq
show Red (L₃ ++ L₄) ([(x1, not b1), (x2, b2)] ++ L₂)
apply append_append _ h₂
have h₁ : Red ((x1, not b1) :: (x1, b1) :: L₃) [(x1, not b1), (x2, b2)]... |
/-
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, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 501 | 504 | theorem toReal_laverage {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
(⨍⁻ x, f x ∂μ).toReal = ⨍ x, (f x).toReal ∂μ := by |
rw [average_eq, laverage_eq, smul_eq_mul, toReal_div, div_eq_inv_mul, ←
integral_toReal hf (hf'.mono fun _ => lt_top_iff_ne_top.2)]
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.CommSq
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.C... | Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | 998 | 1,008 | theorem Functor.map_isPullback [PreservesLimit (cospan h i) F] (s : IsPullback f g h i) :
IsPullback (F.map f) (F.map g) (F.map h) (F.map i) := by |
-- This is made slightly awkward because `C` and `D` have different universes,
-- and so the relevant `WalkingCospan` diagrams live in different universes too!
refine
IsPullback.of_isLimit' (F.map_commSq s.toCommSq)
(IsLimit.equivOfNatIsoOfIso (cospanCompIso F h i) _ _ (WalkingCospan.ext ?_ ?_ ?_)
... |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.TypeTags
import Mathlib.Algebra.Ring.Hom.Defs
import M... | Mathlib/Data/Nat/Cast/Basic.lean | 159 | 164 | theorem ext_nat'' [MonoidWithZeroHomClass F ℕ A] (f g : F) (h_pos : ∀ {n : ℕ}, 0 < n → f n = g n) :
f = g := by |
apply DFunLike.ext
rintro (_ | n)
· simp [map_zero f, map_zero g]
· exact h_pos n.succ_pos
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | Mathlib/Analysis/Convex/Segment.lean | 137 | 147 | theorem insert_endpoints_openSegment (x y : E) :
insert x (insert y (openSegment 𝕜 x y)) = [x -[𝕜] y] := by |
simp only [subset_antisymm_iff, insert_subset_iff, left_mem_segment, right_mem_segment,
openSegment_subset_segment, true_and_iff]
rintro z ⟨a, b, ha, hb, hab, rfl⟩
refine hb.eq_or_gt.imp ?_ fun hb' => ha.eq_or_gt.imp ?_ fun ha' => ?_
· rintro rfl
rw [← add_zero a, hab, one_smul, zero_smul, add_zero]
... |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 1,700 | 1,701 | theorem dvd_iff_mod_eq_zero {m n : Num} : m ∣ n ↔ n % m = 0 := by |
rw [← dvd_to_nat, Nat.dvd_iff_mod_eq_zero, ← to_nat_inj, mod_to_nat]; rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 316 | 319 | theorem SeparationQuotient.completeSpace_iff :
CompleteSpace (SeparationQuotient α) ↔ CompleteSpace α := by |
rw [completeSpace_iff_isComplete_univ, ← range_mk,
← completeSpace_iff_isComplete_range uniformInducing_mk]
|
/-
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 | 726 | 726 | theorem sdiff_compl : x \ yᶜ = x ⊓ y := by | rw [sdiff_eq, compl_compl]
|
/-
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 | 796 | 796 | theorem image.map_ι : image.map sq ≫ image.ι g.hom = image.ι f.hom ≫ sq.right := by | simp
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,608 | 1,612 | theorem norm_multiset_prod_le (m : Multiset E) : ‖m.prod‖ ≤ (m.map fun x => ‖x‖).sum := by |
rw [← Multiplicative.ofAdd_le, ofAdd_multiset_prod, Multiset.map_map]
refine Multiset.le_prod_of_submultiplicative (Multiplicative.ofAdd ∘ norm) ?_ (fun x y => ?_) _
· simp only [comp_apply, norm_one', ofAdd_zero]
· exact norm_mul_le' x y
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriente... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 572 | 573 | theorem oangle_sub_eq_oangle_sub_rev_of_norm_eq {x y : V} (h : ‖x‖ = ‖y‖) :
o.oangle x (x - y) = o.oangle (y - x) y := by | simp [oangle, h]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Topology.Order.Monot... | Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 379 | 382 | theorem disjoint_withBotCoe {I J : WithBot (Box ι)} :
Disjoint (I : Set (ι → ℝ)) J ↔ Disjoint I J := by |
simp only [disjoint_iff_inf_le, ← withBotCoe_subset_iff, coe_inf]
rfl
|
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Adam Topaz, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors
import M... | Mathlib/Algebra/FreeAlgebra.lean | 423 | 426 | theorem lift_unique (f : X → A) (g : FreeAlgebra R X →ₐ[R] A) :
(g : FreeAlgebra R X → A) ∘ ι R = f ↔ g = lift R f := by |
rw [← (lift R).symm_apply_eq, lift]
rfl
|
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Ev... | Mathlib/Algebra/Polynomial/Smeval.lean | 290 | 294 | theorem aeval_eq_smeval {R : Type*} [CommSemiring R] {S : Type*} [Semiring S] [Algebra R S]
(x : S) (p : R[X]) : aeval x p = p.smeval x := by |
rw [aeval_def, eval₂_def, Algebra.algebraMap_eq_smul_one', smeval_def]
simp only [Algebra.smul_mul_assoc, one_mul]
exact rfl
|
/-
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.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 95 | 97 | theorem trinomial_leadingCoeff (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).leadingCoeff = w := by |
rw [leadingCoeff, trinomial_natDegree hkm hmn hw, trinomial_leading_coeff' hkm hmn]
|
/-
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, Yury Kudryashov
-/
import Mathlib.Topology.ContinuousOn
import Mathlib.Order.Minimal
/-!
# Irreducibility in topological spaces
## Main definitions
* ... | Mathlib/Topology/Irreducible.lean | 296 | 301 | theorem subset_closure_inter_of_isPreirreducible_of_isOpen {S U : Set X} (hS : IsPreirreducible S)
(hU : IsOpen U) (h : (S ∩ U).Nonempty) : S ⊆ closure (S ∩ U) := by |
by_contra h'
obtain ⟨x, h₁, h₂, h₃⟩ :=
hS _ (closure (S ∩ U))ᶜ hU isClosed_closure.isOpen_compl h (inter_compl_nonempty_iff.mpr h')
exact h₃ (subset_closure ⟨h₁, h₂⟩)
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f6856... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 453 | 457 | theorem tangentMapWithin_subset {p : TangentBundle I M} (st : s ⊆ t)
(hs : UniqueMDiffWithinAt I s p.1) (h : MDifferentiableWithinAt I I' f t p.1) :
tangentMapWithin I I' f s p = tangentMapWithin I I' f t p := by |
simp only [tangentMapWithin, mfld_simps]
rw [mfderivWithin_subset st hs h]
|
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.Layercake
#align_import measure_theory.integral.layercake from "leanprover-commu... | Mathlib/Analysis/SpecialFunctions/Pow/Integral.lean | 86 | 94 | theorem lintegral_rpow_eq_lintegral_meas_lt_mul :
∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (t ^ (p - 1)) := by |
rw [lintegral_rpow_eq_lintegral_meas_le_mul μ f_nn f_mble p_pos]
apply congr_arg fun z => ENNReal.ofReal p * z
apply lintegral_congr_ae
filter_upwards [meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f]
with t ht
rw [ht]
|
/-
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.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 284 | 287 | theorem orderEmbOfFin_boundaries :
c.boundaries.orderEmbOfFin c.card_boundaries_eq_succ_length = c.boundary := by |
refine (Finset.orderEmbOfFin_unique' _ ?_).symm
exact fun i => (Finset.mem_map' _).2 (Finset.mem_univ _)
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Scott Morrison
-/
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Simple
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.CategoryTheory.Endomorp... | Mathlib/CategoryTheory/Preadditive/Schur.lean | 193 | 201 | theorem finrank_hom_simple_simple_eq_zero_iff (X Y : C) [FiniteDimensional 𝕜 (X ⟶ X)]
[FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X] [Simple Y] :
finrank 𝕜 (X ⟶ Y) = 0 ↔ IsEmpty (X ≅ Y) := by |
rw [← not_nonempty_iff, ← not_congr (finrank_hom_simple_simple_eq_one_iff 𝕜 X Y)]
refine ⟨fun h => by rw [h]; simp, fun h => ?_⟩
have := finrank_hom_simple_simple_le_one 𝕜 X Y
interval_cases finrank 𝕜 (X ⟶ Y)
· rfl
· exact False.elim (h rfl)
|
/-
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.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
/-!
# Convex join
... | Mathlib/Analysis/Convex/Join.lean | 36 | 37 | theorem mem_convexJoin : x ∈ convexJoin 𝕜 s t ↔ ∃ a ∈ s, ∃ b ∈ t, x ∈ segment 𝕜 a b := by |
simp [convexJoin]
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.char_zero.quotient from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
/-!
# Lem... | Mathlib/Algebra/CharZero/Quotient.lean | 42 | 47 | theorem nsmul_mem_zmultiples_iff_exists_sub_div {r : R} {n : ℕ} (hn : n ≠ 0) :
n • r ∈ AddSubgroup.zmultiples p ↔
∃ k : Fin n, r - (k : ℕ) • (p / n : R) ∈ AddSubgroup.zmultiples p := by |
rw [← natCast_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (Int.natCast_ne_zero.mpr hn),
Int.cast_natCast]
rfl
|
/-
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 | 275 | 281 | theorem valuation_of_mk' {r : R} {s : nonZeroDivisors R} :
v.valuation (IsLocalization.mk' K r s) = v.intValuation r / v.intValuation s := by |
erw [valuation_def, (IsLocalization.toLocalizationMap (nonZeroDivisors R) K).lift_mk',
div_eq_mul_inv, mul_eq_mul_left_iff]
left
rw [Units.val_inv_eq_inv_val, inv_inj]
rfl
|
/-
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 | 330 | 331 | theorem equivMapDomain_trans (f : α ≃ β) (g : β ≃ γ) (l : α →₀ M) :
equivMapDomain (f.trans g) l = equivMapDomain g (equivMapDomain f l) := by | ext x; rfl
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Equiv.Fin
import Mathlib.ModelTheory.Langu... | Mathlib/ModelTheory/Syntax.lean | 284 | 290 | theorem comp_onTerm {L'' : Language} (φ : L' →ᴸ L'') (ψ : L →ᴸ L') :
((φ.comp ψ).onTerm : L.Term α → L''.Term α) = φ.onTerm ∘ ψ.onTerm := by |
ext t
induction' t with _ _ _ _ ih
· rfl
· simp_rw [onTerm, ih]
rfl
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu, Anne Baanen
-/
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.Module.LocalizedModule
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Lo... | Mathlib/RingTheory/Localization/Module.lean | 56 | 71 | theorem LinearIndependent.of_isLocalizedModule {ι : Type*} {v : ι → M}
(hv : LinearIndependent R v) : LinearIndependent Rₛ (f ∘ v) := by |
rw [linearIndependent_iff'] at hv ⊢
intro t g hg i hi
choose! a g' hg' using IsLocalization.exist_integer_multiples S t g
have h0 : f (∑ i ∈ t, g' i • v i) = 0 := by
apply_fun ((a : R) • ·) at hg
rw [smul_zero, Finset.smul_sum] at hg
rw [map_sum, ← hg]
refine Finset.sum_congr rfl fun i hi => ?_... |
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 235 | 238 | theorem of_mono {X Y : C} (f : X ⟶ Y) [Mono f] (i : IsZero Y) : IsZero X := by |
have hf := i.eq_zero_of_tgt f
subst hf
exact IsZero.of_mono_zero X Y
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 999 | 1,007 | theorem range_equiv_smaller_toFun_bijective (o : Ordinal) :
Function.Bijective (range_equiv_smaller_toFun C o) := by |
dsimp (config := { unfoldPartialApp := true }) [range_equiv_smaller_toFun]
refine ⟨fun a b hab ↦ ?_, fun ⟨a, b, hb⟩ ↦ ?_⟩
· ext1
simp only [Subtype.mk.injEq] at hab
exact injective_πs C o hab
· use ⟨b, hb.1⟩
simpa only [Subtype.mk.injEq] using hb.2
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanp... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 347 | 350 | theorem cthickening_closedBall (hε : 0 ≤ ε) (hδ : 0 ≤ δ) (x : E) :
cthickening ε (closedBall x δ) = closedBall x (ε + δ) := by |
rw [← cthickening_singleton _ hδ, cthickening_cthickening hε hδ,
cthickening_singleton _ (add_nonneg hε hδ)]
|
/-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.Pad... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 173 | 220 | theorem sub_one_pow_totient_lt_cyclotomic_eval {n : ℕ} {q : ℝ} (hn' : 2 ≤ n) (hq' : 1 < q) :
(q - 1) ^ totient n < (cyclotomic n ℝ).eval q := by |
have hn : 0 < n := pos_of_gt hn'
have hq := zero_lt_one.trans hq'
have hfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖ := by
intro ζ' hζ'
rw [mem_primitiveRoots hn] at hζ'
convert norm_sub_norm_le (↑q) ζ'
· rw [Complex.norm_real, Real.norm_of_nonneg hq.le]
· rw [hζ'.norm'_eq_one hn.ne']
... |
/-
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.DirectLimit
import Mathlib.Algebra.CharP.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Construction
#al... | Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean | 85 | 94 | theorem spanEval_ne_top : spanEval k ≠ ⊤ := by |
rw [Ideal.ne_top_iff_one, spanEval, Ideal.span, ← Set.image_univ,
Finsupp.mem_span_image_iff_total]
rintro ⟨v, _, hv⟩
replace hv := congr_arg (toSplittingField k v.support) hv
rw [AlgHom.map_one, Finsupp.total_apply, Finsupp.sum, AlgHom.map_sum, Finset.sum_eq_zero] at hv
· exact zero_ne_one hv
intro j ... |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 772 | 776 | theorem ncard_le_ncard_of_injOn {t : Set β} (f : α → β) (hf : ∀ a ∈ s, f a ∈ t) (f_inj : InjOn f s)
(ht : t.Finite := by | toFinite_tac) :
s.ncard ≤ t.ncard := by
have hle := encard_le_encard_of_injOn hf f_inj
to_encard_tac; rwa [ht.cast_ncard_eq, (ht.finite_of_encard_le hle).cast_ncard_eq]
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... | Mathlib/RingTheory/Kaehler.lean | 656 | 663 | theorem KaehlerDifferential.kerTotal_map' (h : Function.Surjective (algebraMap A B)) :
(KaehlerDifferential.kerTotal R A ⊔
Submodule.span A (Set.range fun x ↦ .single (algebraMap R A x) 1)).map finsupp_map =
(KaehlerDifferential.kerTotal R B).restrictScalars _ := by |
rw [Submodule.map_sup, ← kerTotal_map R R A B h, Submodule.map_span, ← Set.range_comp]
congr
refine congr_arg Set.range ?_
ext; simp [IsScalarTower.algebraMap_eq R A B]
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.FinitelyGenerated
import Mathlib.ModelTheory.DirectLimit
import Mathlib.ModelTheory.Bundled
#align_import model_theory.fraisse from "leanp... | Mathlib/ModelTheory/Fraisse.lean | 169 | 182 | theorem age.countable_quotient [h : Countable M] : (Quotient.mk' '' L.age M).Countable := by |
classical
refine (congr_arg _ (Set.ext <| Quotient.forall.2 fun N => ?_)).mp
(countable_range fun s : Finset M => ⟦⟨closure L (s : Set M), inferInstance⟩⟧)
constructor
· rintro ⟨s, hs⟩
use Bundled.of (closure L (s : Set M))
exact ⟨⟨(fg_iff_structure_fg _).1 (fg_closure s.finite_toSet), ⟨Substructur... |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Utensil Song
-/
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
import Mathlib.LinearAlge... | Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean | 285 | 287 | theorem ι_mul_ι_mul_of_isOrtho (x : CliffordAlgebra Q) {a b : M} (h : Q.IsOrtho a b) :
ι Q a * (ι Q b * x) = -(ι Q b * (ι Q a * x)) := by |
rw [← mul_assoc, ι_mul_ι_comm_of_isOrtho h, neg_mul, mul_assoc]
|
/-
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.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Analysis.Complex.CauchyIntegral
#align_impo... | Mathlib/Analysis/Complex/RemovableSingularity.lean | 34 | 43 | theorem analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt {f : ℂ → E} {c : ℂ}
(hd : ∀ᶠ z in 𝓝[≠] c, DifferentiableAt ℂ f z) (hc : ContinuousAt f c) : AnalyticAt ℂ f c := by |
rcases (nhdsWithin_hasBasis nhds_basis_closedBall _).mem_iff.1 hd with ⟨R, hR0, hRs⟩
lift R to ℝ≥0 using hR0.le
replace hc : ContinuousOn f (closedBall c R) := by
refine fun z hz => ContinuousAt.continuousWithinAt ?_
rcases eq_or_ne z c with (rfl | hne)
exacts [hc, (hRs ⟨hz, hne⟩).continuousAt]
exa... |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 926 | 927 | theorem dotProduct_smul [SMulCommClass R α α] (x : R) (v w : m → α) :
v ⬝ᵥ x • w = x • (v ⬝ᵥ w) := by | simp [dotProduct, Finset.smul_sum, mul_smul_comm]
|
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Junyan Xu
-/
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.Algebra.Category.GroupCat.Injective
import Mathli... | Mathlib/Algebra/Module/CharacterModule.lean | 212 | 214 | theorem _root_.rTensor_injective_iff_lcomp_surjective {f : A →ₗ[R] A'} :
Function.Injective (f.rTensor B) ↔ Function.Surjective (f.lcomp R <| CharacterModule B) := by |
simp [← dual_rTensor_conj_homEquiv, dual_surjective_iff_injective]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 477 | 481 | theorem _root_.List.foldr_inf_eq_inf_toFinset [DecidableEq α] (l : List α) :
l.foldr (· ⊓ ·) ⊤ = l.toFinset.inf id := by |
rw [← coe_fold_r, ← Multiset.fold_dedup_idem, inf_def, ← List.toFinset_coe, toFinset_val,
Multiset.map_id]
rfl
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Grou... | Mathlib/RingTheory/Coprime/Basic.lean | 163 | 164 | theorem IsCoprime.mul_right_iff : IsCoprime x (y * z) ↔ IsCoprime x y ∧ IsCoprime x z := by |
rw [isCoprime_comm, IsCoprime.mul_left_iff, isCoprime_comm, @isCoprime_comm _ _ z]
|
/-
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
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.Spec... | Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 111 | 114 | theorem HasStrictDerivAt.log (hf : HasStrictDerivAt f f' x) (hx : f x ≠ 0) :
HasStrictDerivAt (fun y => log (f y)) (f' / f x) x := by |
rw [div_eq_inv_mul]
exact (hasStrictDerivAt_log hx).comp x hf
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 100 | 101 | theorem Q_f_0_eq (q : ℕ) : ((Q q).f 0 : X _[0] ⟶ X _[0]) = 0 := by |
simp only [HomologicalComplex.sub_f_apply, HomologicalComplex.id_f, Q, P_f_0_eq, sub_self]
|
/-
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.Geometry.RingedSpace.PresheafedSpace
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.Topology.Sheaves.Limits
import Mathlib.Categor... | Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean | 300 | 310 | theorem desc_fac (F : J ⥤ PresheafedSpace.{_, _, v} C) (s : Cocone F) (j : J) :
(colimitCocone F).ι.app j ≫ desc F s = s.ι.app j := by |
ext U
· simp [desc]
· -- Porting note: the original proof is just `ext; dsimp [desc, descCApp]; simpa`,
-- but this has to be expanded a bit
rw [NatTrans.comp_app, PresheafedSpace.comp_c_app, whiskerRight_app]
dsimp [desc, descCApp]
simp only [eqToHom_app, op_obj, Opens.map_comp_obj, eqToHom_map,... |
/-
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.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "lean... | Mathlib/Analysis/Convex/Side.lean | 785 | 800 | theorem setOf_sSameSide_eq_image2 {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
{ y | s.SSameSide x y } = Set.image2 (fun (t : R) q => t • (x -ᵥ p) +ᵥ q) (Set.Ioi 0) s := by |
ext y
simp_rw [Set.mem_setOf, Set.mem_image2, Set.mem_Ioi]
constructor
· rw [sSameSide_iff_exists_left hp]
rintro ⟨-, hy, p₂, hp₂, h | h | ⟨r₁, r₂, hr₁, hr₂, h⟩⟩
· rw [vsub_eq_zero_iff_eq] at h
exact False.elim (hx (h.symm ▸ hp))
· rw [vsub_eq_zero_iff_eq] at h
exact False.elim (hy (h.s... |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.normalized from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f... | Mathlib/AlgebraicTopology/DoldKan/Normalized.lean | 77 | 78 | theorem PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap (X : SimplicialObject A) :
PInftyToNormalizedMooreComplex X ≫ inclusionOfMooreComplexMap X = PInfty := by | aesop_cat
|
/-
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 | 780 | 783 | theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ := by |
simp only [iUnion_and, @iUnion_comm _ ι]
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Multiset.Basic
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Tactic.ApplyFun
#align_import data.sym.basic from "lean... | Mathlib/Data/Sym/Basic.lean | 675 | 680 | theorem decode_encode [DecidableEq α] (s : Sym (Option α) n.succ) : decode (encode s) = s := by |
by_cases h : none ∈ s
· simp [h]
· simp only [decode, h, not_false_iff, encode_of_not_none_mem, Embedding.some_apply, map_map,
comp_apply, Option.some_get]
convert s.attach_map_coe
|
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
#align_import dynamics.omega_limit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
... | Mathlib/Dynamics/OmegaLimit.lean | 142 | 144 | theorem mem_omegaLimit_iff_frequently₂ (y : β) :
y ∈ ω f ϕ s ↔ ∀ n ∈ 𝓝 y, ∃ᶠ t in f, (ϕ t '' s ∩ n).Nonempty := by |
simp_rw [mem_omegaLimit_iff_frequently, image_inter_nonempty_iff]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.GaloisConnection
#align_import order.modular_lattice from "leanpr... | Mathlib/Order/ModularLattice.lean | 402 | 407 | theorem disjoint_sup_left_of_disjoint_sup_right [Lattice α] [OrderBot α]
[IsModularLattice α] (h : Disjoint b c) (hsup : Disjoint a (b ⊔ c)) :
Disjoint (a ⊔ b) c := by |
rw [disjoint_comm, sup_comm]
apply Disjoint.disjoint_sup_right_of_disjoint_sup_left h.symm
rwa [sup_comm, disjoint_comm] at hsup
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
imp... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 90 | 101 | theorem cos_eq_cos_iff {x y : ℂ} : cos x = cos y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x :=
calc
cos x = cos y ↔ cos x - cos y = 0 := sub_eq_zero.symm
_ ↔ -2 * sin ((x + y) / 2) * sin ((x - y) / 2) = 0 := by | rw [cos_sub_cos]
_ ↔ sin ((x + y) / 2) = 0 ∨ sin ((x - y) / 2) = 0 := by simp [(by norm_num : (2 : ℂ) ≠ 0)]
_ ↔ sin ((x - y) / 2) = 0 ∨ sin ((x + y) / 2) = 0 := or_comm
_ ↔ (∃ k : ℤ, y = 2 * k * π + x) ∨ ∃ k : ℤ, y = 2 * k * π - x := by
apply or_congr <;>
field_simp [sin_eq_zero_iff, (by norm... |
/-
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.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.Measur... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,937 | 1,943 | theorem of_measurable_inverse_on_range {g : range f → α} (hf₁ : Measurable f)
(hf₂ : MeasurableSet (range f)) (hg : Measurable g) (H : LeftInverse g (rangeFactorization f)) :
MeasurableEmbedding f := by |
set e : α ≃ᵐ range f :=
⟨⟨rangeFactorization f, g, H, H.rightInverse_of_surjective surjective_onto_range⟩,
hf₁.subtype_mk, hg⟩
exact (MeasurableEmbedding.subtype_coe hf₂).comp e.measurableEmbedding
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.P... | Mathlib/NumberTheory/Bernoulli.lean | 137 | 150 | theorem sum_bernoulli' (n : ℕ) : (∑ k ∈ range n, (n.choose k : ℚ) * bernoulli' k) = n := by |
cases' n with n
· simp
suffices
((n + 1 : ℚ) * ∑ k ∈ range n, ↑(n.choose k) / (n - k + 1) * bernoulli' k) =
∑ x ∈ range n, ↑(n.succ.choose x) * bernoulli' x by
rw_mod_cast [sum_range_succ, bernoulli'_def, ← this, choose_succ_self_right]
ring
simp_rw [mul_sum, ← mul_assoc]
refine sum_congr r... |
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 131 | 133 | theorem inv_pos_le_iff_one_le_mul (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ b * a := by |
rw [inv_eq_one_div]
exact div_le_iff ha
|
/-
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 | 733 | 734 | theorem coe_eq_singleton {s : Finset α} {a : α} : (s : Set α) = {a} ↔ s = {a} := by |
rw [← coe_singleton, coe_inj]
|
/-
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.Asymptotics
#align_import... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 177 | 181 | theorem continuousAt_const_rpow' {a b : ℝ} (h : b ≠ 0) : ContinuousAt (a ^ ·) b := by |
simp only [rpow_def]
refine Complex.continuous_re.continuousAt.comp ?_
refine (continuousAt_const_cpow' ?_).comp Complex.continuous_ofReal.continuousAt
norm_cast
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 378 | 378 | theorem zero_of_to_zero {X : C} (f : X ⟶ 0) : f = 0 := by | ext
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Anal... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 615 | 620 | theorem hasSum_cauchyPowerSeries_integral {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ}
(hf : CircleIntegrable f c R) (hw : abs w < R) :
HasSum (fun n => cauchyPowerSeries f c R n fun _ => w)
((2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - (c + w))⁻¹ • f z) := by |
simp only [cauchyPowerSeries_apply]
exact (hasSum_two_pi_I_cauchyPowerSeries_integral hf hw).const_smul _
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Ba... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 462 | 465 | theorem adjugate_conjTranspose [StarRing α] (A : Matrix n n α) : A.adjugateᴴ = adjugate Aᴴ := by |
dsimp only [conjTranspose]
have : Aᵀ.adjugate.map star = adjugate (Aᵀ.map star) := (starRingEnd α).map_adjugate Aᵀ
rw [A.adjugate_transpose, this]
|
/-
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.MeasurableIntegral
#align_import probability.kernel.composition from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7c... | Mathlib/Probability/Kernel/Composition.lean | 515 | 523 | theorem compProd_eq_sum_compProd_right (κ : kernel α β) (η : kernel (α × β) γ)
[IsSFiniteKernel η] : κ ⊗ₖ η = kernel.sum fun n => κ ⊗ₖ seq η n := by |
by_cases hκ : IsSFiniteKernel κ
swap
· simp_rw [compProd_of_not_isSFiniteKernel_left _ _ hκ]
simp
rw [compProd_eq_sum_compProd]
simp_rw [compProd_eq_sum_compProd_left κ _]
rw [kernel.sum_comm]
|
/-
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.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprove... | Mathlib/GroupTheory/Perm/List.lean | 218 | 231 | theorem formPerm_apply_get (xs : List α) (h : Nodup xs) (i : Fin xs.length) :
formPerm xs (xs.get i) =
xs.get ⟨((i.val + 1) % xs.length), (Nat.mod_lt _ (i.val.zero_le.trans_lt i.isLt))⟩ := by |
let ⟨n, hn⟩ := i
cases' xs with x xs
· simp at hn
· have : n ≤ xs.length := by
refine Nat.le_of_lt_succ ?_
simpa using hn
rcases this.eq_or_lt with (rfl | hn')
· simp
· rw [formPerm_apply_lt_get (x :: xs) h _ (Nat.succ_lt_succ hn')]
congr
rw [Nat.mod_eq_of_lt]; simpa [Nat.su... |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theor... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 232 | 233 | theorem normal_mul (N H : Subgroup G) [N.Normal] : (↑(N ⊔ H) : Set G) = N * H := by |
rw [← set_mul_normal_comm, sup_comm, mul_normal]
|
/-
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.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import n... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 258 | 262 | theorem subOneIntegralPowerBasis_gen [IsCyclotomicExtension {p ^ k} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) :
hζ.subOneIntegralPowerBasis.gen =
⟨ζ - 1, Subalgebra.sub_mem _ (hζ.isIntegral (p ^ k).pos) (Subalgebra.one_mem _)⟩ := by |
simp [subOneIntegralPowerBasis]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlg... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 181 | 183 | theorem dist_midpoint_right (p₁ p₂ : P) :
dist (midpoint 𝕜 p₁ p₂) p₂ = ‖(2 : 𝕜)‖⁻¹ * dist p₁ p₂ := by |
rw [midpoint_comm, dist_midpoint_left, dist_comm]
|
/-
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.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearA... | Mathlib/Algebra/Quaternion.lean | 731 | 731 | theorem star_coe : star (x : ℍ[R,c₁,c₂]) = x := by | ext <;> simp
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Ba... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 119 | 122 | theorem cramer_row_self (i : n) (h : ∀ j, b j = A j i) : A.cramer b = Pi.single i A.det := by |
rw [← transpose_transpose A, det_transpose]
convert cramer_transpose_row_self Aᵀ i
exact funext h
|
/-
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.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.... | Mathlib/Order/Interval/Set/OrderIso.lean | 30 | 32 | theorem preimage_Ici (e : α ≃o β) (b : β) : e ⁻¹' Ici b = Ici (e.symm b) := by |
ext x
simp [← e.le_iff_le]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 1,047 | 1,055 | theorem IsCompact.mem_closure_iff_exists_inseparable {K : Set X} (hK : IsCompact K) :
y ∈ closure K ↔ ∃ x ∈ K, Inseparable x y := by |
refine ⟨fun hy ↦ ?_, fun ⟨x, hxK, hxy⟩ ↦
(hxy.mem_closed_iff isClosed_closure).1 <| subset_closure hxK⟩
contrapose! hy
have : Disjoint (𝓝 y) (𝓝ˢ K) := hK.disjoint_nhdsSet_right.2 fun x hx ↦
(disjoint_nhds_nhds_iff_not_inseparable.2 (hy x hx)).symm
simpa only [disjoint_iff, not_mem_closure_iff_nhdsWit... |
/-
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 | 107 | 107 | theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by | rw [sup_comm, sup_inf_sdiff]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import ... | Mathlib/Data/List/Chain.lean | 414 | 424 | theorem Chain.induction (p : α → Prop) (l : List α) (h : Chain r a l)
(hb : getLast (a :: l) (cons_ne_nil _ _) = b) (carries : ∀ ⦃x y : α⦄, r x y → p y → p x)
(final : p b) : ∀ i ∈ a :: l, p i := by |
induction' l with _ _ l_ih generalizing a
· cases hb
simpa using final
· rw [chain_cons] at h
simp only [mem_cons]
rintro _ (rfl | H)
· apply carries h.1 (l_ih h.2 hb _ (mem_cons.2 (Or.inl rfl)))
· apply l_ih h.2 hb _ (mem_cons.2 H)
|
/-
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.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 255 | 255 | theorem not_coprime_zero_zero : ¬Coprime 0 0 := by | simp
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matri... | Mathlib/Data/Matrix/Notation.lean | 436 | 438 | theorem natCast_fin_two (n : ℕ) : (n : Matrix (Fin 2) (Fin 2) α) = !![↑n, 0; 0, ↑n] := by |
ext i j
fin_cases i <;> fin_cases j <;> rfl
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 755 | 757 | theorem preimage_const_mul_Ioc_of_neg (a b : α) {c : α} (h : c < 0) :
(c * ·) ⁻¹' Ioc a b = Ico (b / c) (a / c) := by |
simpa only [mul_comm] using preimage_mul_const_Ioc_of_neg a b h
|
/-
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 | 421 | 427 | theorem supermartingale_of_setIntegral_succ_le [IsFiniteMeasure μ] {f : ℕ → Ω → ℝ}
(hadp : Adapted 𝒢 f) (hint : ∀ i, Integrable (f i) μ)
(hf : ∀ i, ∀ s : Set Ω, MeasurableSet[𝒢 i] s → ∫ ω in s, f (i + 1) ω ∂μ ≤ ∫ ω in s, f i ω ∂μ) :
Supermartingale f 𝒢 μ := by |
rw [← neg_neg f]
refine (submartingale_of_setIntegral_le_succ hadp.neg (fun i => (hint i).neg) ?_).neg
simpa only [integral_neg, Pi.neg_apply, neg_le_neg_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, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 111 | 112 | theorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by |
rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]
|
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topolo... | Mathlib/Topology/MetricSpace/Lipschitz.lean | 41 | 44 | theorem lipschitzWith_iff_dist_le_mul [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0}
{f : α → β} : LipschitzWith K f ↔ ∀ x y, dist (f x) (f y) ≤ K * dist x y := by |
simp only [LipschitzWith, edist_nndist, dist_nndist]
norm_cast
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align... | Mathlib/Data/Real/Sqrt.lean | 97 | 98 | theorem sqrt_mul (x y : ℝ≥0) : sqrt (x * y) = sqrt x * sqrt y := by |
rw [sqrt_eq_iff_eq_sq, mul_pow, sq_sqrt, sq_sqrt]
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Best, Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.Data.PNat.Prime
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.A... | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 504 | 519 | theorem norm_pow_sub_one_two {k : ℕ} (hζ : IsPrimitiveRoot ζ (2 ^ (k + 1)))
[IsCyclotomicExtension {(2 : ℕ+) ^ (k + 1)} K L]
(hirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)) :
norm K (ζ ^ 2 ^ k - 1) = (-2 : K) ^ 2 ^ k := by |
have := hζ.pow_of_dvd (fun h => two_ne_zero (pow_eq_zero h)) (pow_dvd_pow 2 (le_succ k))
rw [Nat.pow_div (le_succ k) zero_lt_two, Nat.succ_sub (le_refl k), Nat.sub_self, pow_one] at this
have H : (-1 : L) - (1 : L) = algebraMap K L (-2) := by
simp only [map_neg, map_ofNat]
ring
-- Porting note: `simpa us... |
/-
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 | 1,424 | 1,426 | theorem absorbent_ball (hx : ‖x‖ < r) : Absorbent 𝕜 (Metric.ball x r) := by |
rw [← ball_normSeminorm 𝕜]
exact (normSeminorm _ _).absorbent_ball hx
|
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