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 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlg... | Mathlib/Algebra/Lie/Nilpotent.lean | 564 | 585 | theorem Function.Surjective.lieModule_lcs_map_eq (k : ℕ) :
(lowerCentralSeries R L M k : Submodule R M).map g = lowerCentralSeries R L₂ M₂ k := by |
induction' k with k ih
· simpa [LinearMap.range_eq_top]
· suffices
g '' {m | ∃ (x : L) (n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, n⁆ = m} =
{m | ∃ (x : L₂) (n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, g n⁆ = m} by
simp only [← LieSubmodule.mem_coeSubmodule] at this
-- Porting note: ... |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Option.Basic
import Mathlib.Data.Set.Basic
#align_import data.pequiv from "leanprover-community/mathlib"@"7c3269ca3fa4c0c19e4d127cd7151edbdbf99ed4"
... | Mathlib/Data/PEquiv.lean | 169 | 170 | theorem refl_trans (f : α ≃. β) : (PEquiv.refl α).trans f = f := by |
ext; dsimp [PEquiv.trans]; rfl
|
/-
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.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 238 | 241 | theorem HasDerivWithinAt.norm_sq {f : ℝ → F} {f' : F} {s : Set ℝ} {x : ℝ}
(hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (‖f ·‖ ^ 2) (2 * Inner.inner (f x) f') s x := by |
simpa using hf.hasFDerivWithinAt.norm_sq.hasDerivWithinAt
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.cont_diff from "lea... | Mathlib/Analysis/Calculus/ContDiff/RCLike.lean | 43 | 49 | theorem ContDiffAt.hasStrictFDerivAt' {f : E' → F'} {f' : E' →L[𝕂] F'} {x : E'}
(hf : ContDiffAt 𝕂 n f x) (hf' : HasFDerivAt f f' x) (hn : 1 ≤ n) :
HasStrictFDerivAt f f' x := by |
rcases hf 1 hn with ⟨u, H, p, hp⟩
simp only [nhdsWithin_univ, mem_univ, insert_eq_of_mem] at H
have := hp.hasStrictFDerivAt le_rfl H
rwa [hf'.unique this.hasFDerivAt]
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "le... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 336 | 365 | theorem IsCompactOperator.continuous {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator f) :
Continuous f := by |
letI : UniformSpace M₂ := TopologicalAddGroup.toUniformSpace _
haveI : UniformAddGroup M₂ := comm_topologicalAddGroup_is_uniform
-- Since `f` is linear, we only need to show that it is continuous at zero.
-- Let `U` be a neighborhood of `0` in `M₂`.
refine continuous_of_continuousAt_zero f fun U hU => ?_
r... |
/-
Copyright (c) 2018 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.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 337 | 338 | theorem Fork.app_one_eq_ι_comp_left (s : Fork f g) : s.π.app one = s.ι ≫ f := by |
rw [← s.app_zero_eq_ι, ← s.w left, parallelPair_map_left]
|
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most one bas... | Mathlib/Data/Matroid/Constructions.lean | 217 | 219 | theorem uniqueBaseOn_inter_basis (hI : I ⊆ E) (hX : X ⊆ E) :
(uniqueBaseOn I E).Basis (X ∩ I) X := by |
rw [uniqueBaseOn_basis_iff hI hX]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 384 | 390 | theorem eq_aleph_of_eq_card_ord {o : Ordinal} (ho : o.card.ord = o) (ho' : ω ≤ o) :
∃ a, (aleph a).ord = o := by |
cases' eq_aleph'_of_eq_card_ord ho with a ha
use a - ω
unfold aleph
rwa [Ordinal.add_sub_cancel_of_le]
rwa [← aleph0_le_aleph', ← ord_le_ord, ha, ord_aleph0]
|
/-
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 | 185 | 190 | theorem basis_restrict_iff' : (M ↾ R).Basis I X ↔ M.Basis I (X ∩ M.E) ∧ X ⊆ R := by |
rw [basis_iff_basis'_subset_ground, basis'_restrict_iff, restrict_ground_eq, and_congr_left_iff,
← basis'_iff_basis_inter_ground]
intro hXR
rw [inter_eq_self_of_subset_left hXR, and_iff_left_iff_imp]
exact fun h ↦ h.subset.trans hXR
|
/-
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.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 675 | 681 | theorem prod_bij' (i : ∀ a ∈ s, κ) (j : ∀ a ∈ t, ι) (hi : ∀ a ha, i a ha ∈ t)
(hj : ∀ a ha, j a ha ∈ s) (left_inv : ∀ a ha, j (i a ha) (hi a ha) = a)
(right_inv : ∀ a ha, i (j a ha) (hj a ha) = a) (h : ∀ a ha, f a = g (i a ha)) :
∏ x ∈ s, f x = ∏ x ∈ t, g x := by |
refine prod_bij i hi (fun a1 h1 a2 h2 eq ↦ ?_) (fun b hb ↦ ⟨_, hj b hb, right_inv b hb⟩) h
rw [← left_inv a1 h1, ← left_inv a2 h2]
simp only [eq]
|
/-
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 | 219 | 222 | theorem prod_add (u v : PrimeMultiset) : (u + v).prod = u.prod * v.prod := by |
change (coePNatMonoidHom (u + v)).prod = _
rw [coePNatMonoidHom.map_add]
exact Multiset.prod_add _ _
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 84 | 89 | theorem t_id (i : 𝒰.J) : t 𝒰 f g i i = 𝟙 _ := by |
apply pullback.hom_ext <;> rw [Category.id_comp]
· apply pullback.hom_ext
· rw [← cancel_mono (𝒰.map i)]; simp only [pullback.condition, Category.assoc, t_fst_fst]
· simp only [Category.assoc, t_fst_snd]
· rw [← cancel_mono (𝒰.map i)]; simp only [pullback.condition, t_snd, Category.assoc]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 52 | 54 | theorem eval₂_congr {R S : Type*} [Semiring R] [Semiring S] {f g : R →+* S} {s t : S}
{φ ψ : R[X]} : f = g → s = t → φ = ψ → eval₂ f s φ = eval₂ g t ψ := by |
rintro rfl rfl rfl; rfl
|
/-
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.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#a... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 185 | 203 | theorem condexpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {f : Lp ℝ 2 μ}
(hf : f =ᵐ[μ.restrict s] 0) : condexpL2 ℝ ℝ hm f =ᵐ[μ.restrict s] (0 : α → ℝ) := by |
suffices h_nnnorm_eq_zero : ∫⁻ x in s, ‖(condexpL2 ℝ ℝ hm f : α → ℝ) x‖₊ ∂μ = 0 by
rw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero
· refine h_nnnorm_eq_zero.mono fun x hx => ?_
dsimp only at hx
rw [Pi.zero_apply] at hx ⊢
· rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
· refine Measurab... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 290 | 325 | theorem norm_eq_iInf_iff_real_inner_eq_zero (K : Submodule ℝ F) {u : F} {v : F} (hv : v ∈ K) :
(‖u - v‖ = ⨅ w : (↑K : Set F), ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0 :=
Iff.intro
(by
intro h
have h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 := by |
rwa [norm_eq_iInf_iff_real_inner_le_zero] at h
exacts [K.convex, hv]
intro w hw
have le : ⟪u - v, w⟫_ℝ ≤ 0 := by
let w' := w + v
have : w' ∈ K := Submodule.add_mem _ hw hv
have h₁ := h w' this
have h₂ : w' - v = w := by
simp only [w', add_neg_cancel... |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417... | Mathlib/Algebra/Polynomial/Lifts.lean | 73 | 75 | theorem lifts_iff_coeff_lifts (p : S[X]) : p ∈ lifts f ↔ ∀ n : ℕ, p.coeff n ∈ Set.range f := by |
rw [lifts_iff_ringHom_rangeS, mem_map_rangeS f]
rfl
|
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset... | Mathlib/GroupTheory/Exponent.lean | 587 | 598 | theorem card_dvd_exponent_pow_rank : Nat.card G ∣ Monoid.exponent G ^ Group.rank G := by |
obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G
rw [← hS1, ← Fintype.card_coe, ← Finset.card_univ, ← Finset.prod_const]
let f : (∀ g : S, zpowers (g : G)) →* G := noncommPiCoprod fun s t _ x y _ _ => mul_comm x _
have hf : Function.Surjective f := by
rw [← MonoidHom.range_top_iff_surjective, eq_top_iff, ← hS2, c... |
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 88 | 92 | theorem degrees_monomial (s : σ →₀ ℕ) (a : R) : degrees (monomial s a) ≤ toMultiset s := by |
classical
refine (supDegree_single s a).trans_le ?_
split_ifs
exacts [bot_le, le_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
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
#alig... | Mathlib/Analysis/PSeries.lean | 173 | 181 | theorem tsum_condensed_le (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) :
(∑' k : ℕ, 2 ^ k * f (2 ^ k)) ≤ f 1 + 2 * ∑' k, f k := by |
rw [ENNReal.tsum_eq_iSup_nat' (tendsto_atTop_mono Nat.le_succ tendsto_id), two_mul, ← two_nsmul]
refine
iSup_le fun n =>
le_trans ?_
(add_le_add_left
(nsmul_le_nsmul_right (ENNReal.sum_le_tsum <| Finset.Ico 2 (2 ^ n + 1)) _) _)
simpa using Finset.sum_condensed_le hf n
|
/-
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.Probability.Independence.Kernel
#align_import probability.independence.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29... | Mathlib/Probability/Independence/Basic.lean | 634 | 647 | theorem indepFun_iff_map_prod_eq_prod_map_map {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
[IsFiniteMeasure μ] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
IndepFun f g μ ↔ μ.map (fun ω ↦ (f ω, g ω)) = (μ.map f).prod (μ.map g) := by |
rw [indepFun_iff_measure_inter_preimage_eq_mul]
have h₀ {s : Set β} {t : Set β'} (hs : MeasurableSet s) (ht : MeasurableSet t) :
μ (f ⁻¹' s) * μ (g ⁻¹' t) = μ.map f s * μ.map g t ∧
μ (f ⁻¹' s ∩ g ⁻¹' t) = μ.map (fun ω ↦ (f ω, g ω)) (s ×ˢ t) :=
⟨by rw [Measure.map_apply_of_aemeasurable hf hs, Measur... |
/-
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.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
/-!
# N-ary maps of fil... | Mathlib/Order/Filter/NAry.lean | 177 | 179 | theorem map₂_map_right (m : α → γ → δ) (n : β → γ) :
map₂ m f (g.map n) = map₂ (fun a b => m a (n b)) f g := by |
rw [map₂_swap, map₂_map_left, map₂_swap]
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 154 | 155 | theorem edgeDensity_empty_left (t : Finset β) : edgeDensity r ∅ t = 0 := by |
rw [edgeDensity, Finset.card_empty, Nat.cast_zero, zero_mul, div_zero]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Thomas Read, Andrew Yang, Dagur Asgeirsson, Joël Riou
-/
import Mathlib.CategoryTheory.Adjunction.Basic
/-!
# Uniqueness of adjoints
This file shows that adjoints are uni... | Mathlib/CategoryTheory/Adjunction/Unique.lean | 117 | 119 | theorem homEquiv_leftAdjointUniq_hom_app {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G)
(x : C) : adj1.homEquiv _ _ ((leftAdjointUniq adj1 adj2).hom.app x) = adj2.unit.app x := by |
simp [leftAdjointUniq]
|
/-
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 | 620 | 633 | theorem comp_comp_symm_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, SymmetricRel t ∧ t ○ t ○ t ⊆ s := by |
rcases comp_symm_mem_uniformity_sets hs with ⟨w, w_in, _, w_sub⟩
rcases comp_symm_mem_uniformity_sets w_in with ⟨t, t_in, t_symm, t_sub⟩
use t, t_in, t_symm
have : t ⊆ t ○ t := subset_comp_self_of_mem_uniformity t_in
-- Porting note: Needed the following `have`s to make `mono` work
have ht := Subset.refl t... |
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.cha... | Mathlib/RepresentationTheory/Character.lean | 99 | 101 | theorem char_linHom (V W : FdRep k G) (g : G) :
(of (linHom V.ρ W.ρ)).character g = V.character g⁻¹ * W.character g := by |
rw [← char_iso (dualTensorIsoLinHom _ _), char_tensor, Pi.mul_apply, char_dual]
|
/-
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 | 320 | 321 | theorem measure_sum_seq (κ : kernel α β) [h : IsSFiniteKernel κ] (a : α) :
(Measure.sum fun n => seq κ n a) = κ a := by | rw [← kernel.sum_apply, kernel_sum_seq κ]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 2,062 | 2,066 | theorem filter_cons {a : α} (s : Multiset α) :
filter p (a ::ₘ s) = (if p a then {a} else 0) + filter p s := by |
split_ifs with h
· rw [filter_cons_of_pos _ h, singleton_add]
· rw [filter_cons_of_neg _ h, zero_add]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Simon Hudon
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.PEmpty
#align_i... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean | 297 | 306 | theorem associator_naturality {X₁ X₂ X₃ Y₁ Y₂ Y₃ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃ : X₃ ⟶ Y₃) :
tensorHom ℬ (tensorHom ℬ f₁ f₂) f₃ ≫ (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom =
(BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫ tensorHom ℬ f₁ (tensorHom ℬ f₂ f₃) := by |
dsimp [tensorHom]
apply IsLimit.hom_ext (ℬ _ _).isLimit; rintro ⟨⟨⟩⟩
· simp
· apply IsLimit.hom_ext (ℬ _ _).isLimit
rintro ⟨⟨⟩⟩
· simp
· simp
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,281 | 1,282 | theorem closure_singleton_one : closure ({1} : Set G) = ⊥ := by |
simp [eq_bot_iff_forall, mem_closure_singleton]
|
/-
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.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 257 | 261 | theorem angle_sub_le_pi_div_two_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
angle x (x - y) ≤ π / 2 := by |
rw [← neg_eq_zero, ← inner_neg_right] at h
rw [sub_eq_add_neg]
exact angle_add_le_pi_div_two_of_inner_eq_zero h
|
/-
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 | 781 | 785 | theorem dvd_of_mem_normalizedFactors {a p : α} (H : p ∈ normalizedFactors a) : p ∣ a := by |
by_cases hcases : a = 0
· rw [hcases]
exact dvd_zero p
· exact dvd_trans (Multiset.dvd_prod H) (Associated.dvd (normalizedFactors_prod hcases))
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "lea... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 161 | 167 | theorem leftDistributor_inv {J : Type} [Fintype J] (X : C) (f : J → C) :
(leftDistributor X f).inv = ∑ j : J, biproduct.π _ j ≫ (X ◁ biproduct.ι f j) := by |
ext
dsimp [leftDistributor, Functor.mapBiproduct, Functor.mapBicone]
simp only [Preadditive.comp_sum, biproduct.ι_π_assoc, dite_comp, zero_comp,
Finset.sum_dite_eq, Finset.mem_univ, ite_true, eqToHom_refl, Category.id_comp,
biproduct.ι_desc]
|
/-
Copyright (c) 2021 Jordan Brown, Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jordan Brown, Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bra... | Mathlib/GroupTheory/Commutator.lean | 225 | 240 | theorem commutator_pi_pi_of_finite {η : Type*} [Finite η] {Gs : η → Type*} [∀ i, Group (Gs i)]
(H K : ∀ i, Subgroup (Gs i)) : ⁅Subgroup.pi Set.univ H, Subgroup.pi Set.univ K⁆ =
Subgroup.pi Set.univ fun i => ⁅H i, K i⁆ := by |
classical
apply le_antisymm (commutator_pi_pi_le H K)
rw [pi_le_iff]
intro i hi
rw [map_commutator]
apply commutator_mono <;>
· rw [le_pi_iff]
intro j _hj
rintro _ ⟨_, ⟨x, hx, rfl⟩, rfl⟩
by_cases h : j = i
· subst h
simpa using hx
· simp [h,... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Data.Set... | Mathlib/GroupTheory/GroupAction/Basic.lean | 429 | 447 | theorem quotient_preimage_image_eq_union_mul (U : Set α) :
letI := orbitRel G α
Quotient.mk' ⁻¹' (Quotient.mk' '' U) = ⋃ g : G, (g • ·) '' U := by |
letI := orbitRel G α
set f : α → Quotient (MulAction.orbitRel G α) := Quotient.mk'
ext a
constructor
· rintro ⟨b, hb, hab⟩
obtain ⟨g, rfl⟩ := Quotient.exact hab
rw [Set.mem_iUnion]
exact ⟨g⁻¹, g • a, hb, inv_smul_smul g a⟩
· intro hx
rw [Set.mem_iUnion] at hx
obtain ⟨g, u, hu₁, hu₂⟩ := ... |
/-
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 | 1,148 | 1,148 | theorem Valid'.node4L_lemma₂ {b c d : ℕ} (mr₂ : b + c + 1 ≤ 3 * d) : c ≤ 3 * d := by | omega
|
/-
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.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theo... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 299 | 301 | theorem decompId_i_naturality {P Q : Karoubi C} (f : P ⟶ Q) :
f ≫ decompId_i Q = decompId_i P ≫ (by exact Hom.mk f.f (by simp)) := by |
aesop_cat
|
/-
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 | 869 | 883 | theorem eventually_nonempty_inter_smul_of_density_one (s : Set E) (x : E)
(h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)) (t : Set E)
(ht : MeasurableSet t) (h't : μ t ≠ 0) :
∀ᶠ r in 𝓝[>] (0 : ℝ), (s ∩ ({x} + r • t)).Nonempty := by |
obtain ⟨t', t'_meas, t't, t'pos, t'top⟩ : ∃ t', MeasurableSet t' ∧ t' ⊆ t ∧ 0 < μ t' ∧ μ t' < ⊤ :=
exists_subset_measure_lt_top ht h't.bot_lt
filter_upwards [(tendsto_order.1
(tendsto_addHaar_inter_smul_one_of_density_one μ s x h t' t'_meas t'pos.ne' t'top.ne)).1
0 zero_lt_one]
intro r hr
h... |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 947 | 949 | theorem le_insertNth_iff {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)} {q : ∀ j, α j} :
q ≤ i.insertNth x p ↔ q i ≤ x ∧ (fun j ↦ q (i.succAbove j)) ≤ p := by |
simp [Pi.le_def, forall_iff_succAbove i]
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.EqToHom
#align_import ca... | Mathlib/CategoryTheory/Comma/Basic.lean | 173 | 176 | theorem eqToHom_right (X Y : Comma L R) (H : X = Y) :
CommaMorphism.right (eqToHom H) = eqToHom (by cases H; rfl) := by |
cases H
rfl
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 312 | 318 | theorem append_right_nil {α : Type*} (u : Fin m → α) (v : Fin n → α) (hv : n = 0) :
append u v = u ∘ Fin.cast (by rw [hv, Nat.add_zero]) := by |
refine funext (Fin.addCases (fun l => ?_) fun r => ?_)
· rw [append_left, Function.comp_apply]
refine congr_arg u (Fin.ext ?_)
simp
· exact (Fin.cast hv r).elim0
|
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
/-!
This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single
element to the back of a vector.... | Mathlib/Data/Vector/Snoc.lean | 42 | 45 | theorem reverse_cons : reverse (x ::ᵥ xs) = (reverse xs).snoc x := by |
cases xs
simp only [reverse, cons, toList_mk, List.reverse_cons, snoc]
congr
|
/-
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 | 138 | 143 | theorem range_deriv_subset_closure_span_image
(f : 𝕜 → F) {t : Set 𝕜} (h : Dense t) :
range (deriv f) ⊆ closure (Submodule.span 𝕜 (f '' t)) := by |
rw [← derivWithin_univ]
apply range_derivWithin_subset_closure_span_image
simp [dense_iff_closure_eq.1 h]
|
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Chris Hughes, Mantas Bakšys
-/
import Mathlib.Data.List.Basic
import Mathlib.Order.MinMax
import Mathlib.Order.WithBot
#align_import data.list.min_max from "leanprover-communi... | Mathlib/Data/List/MinMax.lean | 139 | 142 | theorem argmax_concat (f : α → β) (a : α) (l : List α) :
argmax f (l ++ [a]) =
Option.casesOn (argmax f l) (some a) fun c => if f c < f a then some a else some c := by |
rw [argmax, argmax]; simp [argAux]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,556 | 1,557 | theorem bsup_one (f : ∀ a < (1 : Ordinal), Ordinal) : bsup 1 f = f 0 zero_lt_one := by |
simp_rw [← sup_eq_bsup, sup_unique, familyOfBFamily, familyOfBFamily', typein_one_out]
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
#align_import algebraic_geometry.open_immersion.basic from ... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 582 | 585 | theorem lift_fac (H : Set.range g.base ⊆ Set.range f.base) : lift f g H ≫ f = g := by |
-- Porting note: this instance was automatic
letI := pullback_snd_isIso_of_range_subset _ _ H
erw [Category.assoc]; rw [IsIso.inv_comp_eq]; exact pullback.condition
|
/-
Copyright (c) 2020 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subsemiring.Basic
#align_import ring_theory.subring.basic from "leanprover-community/math... | Mathlib/Algebra/Ring/Subring/Basic.lean | 1,107 | 1,114 | theorem mem_iSup_of_directed {ι} [hι : Nonempty ι] {S : ι → Subring R} (hS : Directed (· ≤ ·) S)
{x : R} : (x ∈ ⨆ i, S i) ↔ ∃ i, x ∈ S i := by |
refine ⟨?_, fun ⟨i, hi⟩ ↦ le_iSup S i hi⟩
let U : Subring R :=
Subring.mk' (⋃ i, (S i : Set R)) (⨆ i, (S i).toSubmonoid) (⨆ i, (S i).toAddSubgroup)
(Submonoid.coe_iSup_of_directed hS) (AddSubgroup.coe_iSup_of_directed hS)
suffices ⨆ i, S i ≤ U by simpa [U] using @this x
exact iSup_le fun i x hx ↦ Set... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 594 | 603 | theorem continuousOn_clm_apply {X : Type*} [TopologicalSpace X] [FiniteDimensional 𝕜 E]
{f : X → E →L[𝕜] F} {s : Set X} : ContinuousOn f s ↔ ∀ y, ContinuousOn (fun x => f x y) s := by |
refine ⟨fun h y => (ContinuousLinearMap.apply 𝕜 F y).continuous.comp_continuousOn h, fun h => ?_⟩
let d := finrank 𝕜 E
have hd : d = finrank 𝕜 (Fin d → 𝕜) := (finrank_fin_fun 𝕜).symm
let e₁ : E ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd
let e₂ : (E →L[𝕜] F) ≃L[𝕜] Fin d → F :=
(e₁.ar... |
/-
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.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
impor... | Mathlib/Dynamics/PeriodicPts.lean | 673 | 678 | theorem zpow_smul_eq_iff_period_dvd {j : ℤ} {g : G} {a : α} :
g ^ j • a = a ↔ (period g a : ℤ) ∣ j := by |
rcases j with n | n
· rw [Int.ofNat_eq_coe, zpow_natCast, Int.natCast_dvd_natCast, pow_smul_eq_iff_period_dvd]
· rw [Int.negSucc_coe, zpow_neg, zpow_natCast, inv_smul_eq_iff, eq_comm, dvd_neg,
Int.natCast_dvd_natCast, pow_smul_eq_iff_period_dvd]
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,078 | 1,081 | theorem op_smul_eq_mul_diagonal [Fintype n] [DecidableEq n] (M : Matrix m n α) (a : α) :
MulOpposite.op a • M = M * (diagonal fun _ : n => a) := by |
ext
simp
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.N... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 240 | 241 | theorem _root_.GroupWithZero.eq_zero_or_unit (a : G₀) : a = 0 ∨ ∃ u : G₀ˣ, a = u := by |
simpa using em _
|
/-
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 | 365 | 376 | theorem formPerm_apply_mem_eq_self_iff (hl : Nodup l) (x : α) (hx : x ∈ l) :
formPerm l x = x ↔ length l ≤ 1 := by |
obtain ⟨⟨k, hk⟩, rfl⟩ := get_of_mem hx
rw [formPerm_apply_get _ hl ⟨k, hk⟩, hl.get_inj_iff, Fin.mk.inj_iff]
simp only [Fin.val_mk]
cases hn : l.length
· exact absurd k.zero_le (hk.trans_le hn.le).not_le
· rw [hn] at hk
rcases (Nat.le_of_lt_succ hk).eq_or_lt with hk' | hk'
· simp [← hk', Nat.succ_le... |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Patrick Stevens, Thomas Browning
-/
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
#align_import... | Mathlib/Data/Nat/Choose/Factorization.lean | 93 | 97 | theorem factorization_centralBinom_of_two_mul_self_lt_three_mul (n_big : 2 < n) (p_le_n : p ≤ n)
(big : 2 * n < 3 * p) : (centralBinom n).factorization p = 0 := by |
refine factorization_choose_of_lt_three_mul ?_ p_le_n (p_le_n.trans ?_) big
· omega
· rw [two_mul, add_tsub_cancel_left]
|
/-
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 | 176 | 178 | theorem spectralRadius_le_nnnorm [NormOneClass A] (a : A) : spectralRadius 𝕜 a ≤ ‖a‖₊ := by |
refine iSup₂_le fun k hk => ?_
exact mod_cast norm_le_norm_of_mem hk
|
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Logic.Function.Basic
#align_import group_theory.semidirect_product from "lean... | Mathlib/GroupTheory/SemidirectProduct.lean | 304 | 304 | theorem map_comp_inr : (map f₁ f₂ h).comp inr = inr.comp f₂ := by | ext <;> simp [map]
|
/-
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.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
#align_import data.sum.interval from "leanprover-community/mathlib... | Mathlib/Data/Sum/Interval.lean | 43 | 57 | theorem mem_sumLift₂ :
c ∈ sumLift₂ f g a b ↔
(∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨
∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g a₂ b₂ := by |
constructor
· cases' a with a a <;> cases' b with b b
· rw [sumLift₂, mem_map]
rintro ⟨c, hc, rfl⟩
exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩
· refine fun h ↦ (not_mem_empty _ h).elim
· refine fun h ↦ (not_mem_empty _ h).elim
· rw [sumLift₂, mem_map]
rintro ⟨c, hc, rfl⟩
exact... |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathli... | Mathlib/CategoryTheory/Subobject/Basic.lean | 638 | 641 | theorem pullback_map_self [HasPullbacks C] (f : X ⟶ Y) [Mono f] (g : Subobject X) :
(pullback f).obj ((map f).obj g) = g := by |
revert g
exact Quotient.ind (fun g' => Quotient.sound ⟨(MonoOver.pullbackMapSelf f).app _⟩)
|
/-
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.Join
#align_import data.list.permutation from "leanprover-community/... | Mathlib/Data/List/Permutation.lean | 140 | 146 | theorem permutations'Aux_eq_permutationsAux2 (t : α) (ts : List α) :
permutations'Aux t ts = (permutationsAux2 t [] [ts ++ [t]] ts id).2 := by |
induction' ts with a ts ih; · rfl
simp only [permutations'Aux, ih, cons_append, permutationsAux2_snd_cons, append_nil, id_eq,
cons.injEq, true_and]
simp (config := { singlePass := true }) only [← permutationsAux2_append]
simp [map_permutationsAux2]
|
/-
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 | 402 | 404 | theorem lift_symm_apply (F : FreeAlgebra R X →ₐ[R] A) : (lift R).symm F = F ∘ ι R := by |
rw [lift]
rfl
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 1,402 | 1,403 | theorem coe_sSup {s : Set α} (hb : BddAbove s) : ↑(sSup s) = (⨆ a ∈ s, ↑a : WithTop α) := by |
rw [coe_sSup' hb, sSup_image]
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Nonneg.Ring
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Int.Lemmas
#align_import data.rat.nnrat fr... | Mathlib/Data/NNRat/Defs.lean | 142 | 142 | theorem coe_eq_zero : (q : ℚ) = 0 ↔ q = 0 := by | norm_cast
|
/-
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 | 1,592 | 1,592 | theorem max'_singleton (a : α) : ({a} : Finset α).max' (singleton_nonempty _) = a := by | simp [max']
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 769 | 769 | theorem mk'_spec' (x) (y : S) : f.toMap y * f.mk' x y = f.toMap x := by | rw [mul_comm, mk'_spec]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury G. Kudryashov, Scott Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.BigOperators.Finsupp
impor... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 1,739 | 1,747 | theorem induction_on [AddMonoid G] {p : k[G] → Prop} (f : k[G])
(hM : ∀ g, p (of k G (Multiplicative.ofAdd g)))
(hadd : ∀ f g : k[G], p f → p g → p (f + g))
(hsmul : ∀ (r : k) (f), p f → p (r • f)) : p f := by |
refine Finsupp.induction_linear f ?_ (fun f g hf hg => hadd f g hf hg) fun g r => ?_
· simpa using hsmul 0 (of k G (Multiplicative.ofAdd 0)) (hM 0)
· convert hsmul r (of k G (Multiplicative.ofAdd g)) (hM g)
-- Porting note: Was `simp only`.
rw [of_apply, toAdd_ofAdd, smul_single', mul_one]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 1,404 | 1,407 | theorem linearIndependent_fin_succ {n} {v : Fin (n + 1) → V} :
LinearIndependent K v ↔
LinearIndependent K (Fin.tail v) ∧ v 0 ∉ Submodule.span K (range <| Fin.tail v) := by |
rw [← linearIndependent_fin_cons, Fin.cons_self_tail]
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne, Adam Topaz
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.Topology.Separation
import Mathlib.Topology.LocallyConstant.Basic
#align_import topology.discrete_quotient f... | Mathlib/Topology/DiscreteQuotient.lean | 213 | 215 | theorem ofLE_refl : ofLE (le_refl A) = id := by |
ext ⟨⟩
rfl
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | Mathlib/Topology/Inseparable.lean | 218 | 225 | theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by |
have : ∀ U, IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U := fun U hU x hx ↦ (hspec x).mem_open hU hx
rw [continuous_def]
intro U hU
rw [piecewise_preimage, ite_eq_of_subset_right _ (this U hU)]
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
#align_import data.polynomial.splits from "leanpro... | Mathlib/Algebra/Polynomial/Splits.lean | 447 | 449 | theorem splits_of_algHom {f : R[X]} (h : Splits (algebraMap R K) f) (e : K →ₐ[R] L) :
Splits (algebraMap R L) f := by |
rw [← e.comp_algebraMap_of_tower R]; exact splits_comp_of_splits _ _ h
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot
-/
import Mathlib.Algebra.Algebra.Subalgebra.Operations
import Mathlib.Algebra.Ring.Fin
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.ideal.q... | Mathlib/RingTheory/Ideal/QuotientOperations.lean | 791 | 801 | theorem quotQuotEquivComm_symm : (quotQuotEquivComm I J).symm = quotQuotEquivComm J I := by |
/- Porting note: this proof used to just be rfl but currently rfl opens up a bottomless pit
of processor cycles. Synthesizing instances does not seem to be an issue.
-/
change (((quotQuotEquivQuotSup I J).trans (quotEquivOfEq (sup_comm ..))).trans
(quotQuotEquivQuotSup J I).symm).symm =
((quotQuotEq... |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
/-!
# Some finiteness results of tensor product
This file contains some finiteness results of... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 65 | 75 | theorem exists_finsupp_left (x : M ⊗[R] N) :
∃ S : M →₀ N, x = S.sum fun m n ↦ m ⊗ₜ[R] n := by |
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨Finsupp.single x y, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
use Sx + Sy
rw [hx, hy]
exact (Finsupp.sum_add_index' (by simp) TensorProduct.tmul_add).symm
|
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Scott Morrison, Adam Topaz
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.C... | Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 227 | 234 | theorem widePullback_ext {B : C} {ι : Type w} {X : ι → C} (f : ∀ j : ι, X j ⟶ B)
[HasWidePullback B X f] [PreservesLimit (wideCospan B X f) (forget C)]
(x y : ↑(widePullback B X f)) (h₀ : base f x = base f y) (h : ∀ j, π f j x = π f j y) :
x = y := by |
apply Concrete.limit_ext
rintro (_ | j)
· exact h₀
· apply h
|
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Floris van Doorn
-/
import Mathlib.Data.PNat.Basic
#align_import data.pnat.find from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-!
#... | Mathlib/Data/PNat/Find.lean | 86 | 87 | theorem find_le_iff (n : ℕ+) : PNat.find h ≤ n ↔ ∃ m ≤ n, p m := by |
simp only [exists_prop, ← lt_add_one_iff, find_lt_iff]
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Yury Kudryashov
-/
import Mathlib.Geometry.Manifold.ContMDiffMap
import Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
#align_import geometry.manifold.diffeo... | Mathlib/Geometry/Manifold/Diffeomorph.lean | 447 | 450 | theorem uniqueMDiffOn_image_aux (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) {s : Set M}
(hs : UniqueMDiffOn I s) : UniqueMDiffOn J (h '' s) := by |
convert hs.uniqueMDiffOn_preimage (h.toPartialHomeomorph_mdifferentiable hn)
simp [h.image_eq_preimage]
|
/-
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 | 273 | 295 | theorem Complex.volume_sum_rpow_lt [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 : ∀ x : ι → ℂ, 0 ≤ ∑ i, ‖x i‖ ^ p := by
refine fun _ => Finset.sum_nonneg' ?_
exact fun i => (fun _ => rpow_nonneg (norm_nonneg _) _) _
have h₂ : ∀ x : ι → ℂ, 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p) := fun x => rpow_nonneg (h₁ x) _
obtain hr | hr := le_or_lt r 0
· have : {x : ι → ℂ | (∑ i, ‖x i‖ ^ p... |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.F... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 352 | 358 | theorem exists_fixed_point_of_prime {p n : ℕ} [hp : Fact p.Prime] (hα : ¬p ∣ Fintype.card α)
{σ : Perm α} (hσ : σ ^ p ^ n = 1) : ∃ a : α, σ a = a := by |
classical
contrapose! hα
simp_rw [← mem_support, ← Finset.eq_univ_iff_forall] at hα
exact Nat.modEq_zero_iff_dvd.1 ((congr_arg _ (Finset.card_eq_zero.2 (compl_eq_bot.2 hα))).mp
(card_compl_support_modEq hσ).symm)
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis
impo... | Mathlib/RingTheory/Adjoin/Basic.lean | 209 | 211 | theorem adjoin_toSubmodule_le {s : Set A} {t : Submodule R A} :
Subalgebra.toSubmodule (adjoin R s) ≤ t ↔ ↑(Submonoid.closure s) ⊆ (t : Set A) := by |
rw [adjoin_eq_span, span_le]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 1,358 | 1,364 | theorem RingHom.ext_zmod {n : ℕ} {R : Type*} [Semiring R] (f g : ZMod n →+* R) : f = g := by |
ext a
obtain ⟨k, rfl⟩ := ZMod.intCast_surjective a
let φ : ℤ →+* R := f.comp (Int.castRingHom (ZMod n))
let ψ : ℤ →+* R := g.comp (Int.castRingHom (ZMod n))
show φ k = ψ k
rw [φ.ext_int ψ]
|
/-
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.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "lea... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 258 | 267 | theorem exists_pos_forall_lt_edist (hs : IsCompact s) (ht : IsClosed t) (hst : Disjoint s t) :
∃ r : ℝ≥0, 0 < r ∧ ∀ x ∈ s, ∀ y ∈ t, (r : ℝ≥0∞) < edist x y := by |
rcases s.eq_empty_or_nonempty with (rfl | hne)
· use 1
simp
obtain ⟨x, hx, h⟩ := hs.exists_isMinOn hne continuous_infEdist.continuousOn
have : 0 < infEdist x t :=
pos_iff_ne_zero.2 fun H => hst.le_bot ⟨hx, (mem_iff_infEdist_zero_of_closed ht).mpr H⟩
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 this wit... |
/-
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 | 306 | 312 | theorem conj_orthogonalProjection {T : E →L[𝕜] E} (hT : IsSelfAdjoint T) (U : Submodule 𝕜 E)
[CompleteSpace U] :
IsSelfAdjoint
(U.subtypeL ∘L orthogonalProjection U ∘L T ∘L U.subtypeL ∘L orthogonalProjection U) := by |
rw [← ContinuousLinearMap.comp_assoc]
nth_rw 1 [← (orthogonalProjection_isSelfAdjoint U).adjoint_eq]
exact hT.adjoint_conj _
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
/-... | Mathlib/Topology/MetricSpace/Infsep.lean | 145 | 148 | theorem le_einfsep_pair : edist x y ⊓ edist y x ≤ ({x, y} : Set α).einfsep := by |
simp_rw [le_einfsep_iff, inf_le_iff, mem_insert_iff, mem_singleton_iff]
rintro a (rfl | rfl) b (rfl | rfl) hab <;> (try simp only [le_refl, true_or, or_true]) <;>
contradiction
|
/-
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 | 178 | 179 | theorem rank_finsupp_self (ι : Type w) : Module.rank R (ι →₀ R) = Cardinal.lift.{u} #ι := by |
simp [rank_finsupp]
|
/-
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
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.LinearAlgebra.Span
#align_import linear_algebra.quotient from... | Mathlib/LinearAlgebra/Quotient.lean | 385 | 385 | theorem ker_mkQ : ker p.mkQ = p := by | ext; simp
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
i... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 395 | 405 | theorem measurable_fderiv : Measurable (fderiv 𝕜 f) := by |
refine measurable_of_isClosed fun s hs => ?_
have :
fderiv 𝕜 f ⁻¹' s =
{ x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ s } ∪
{ x | ¬DifferentiableAt 𝕜 f x } ∩ { _x | (0 : E →L[𝕜] F) ∈ s } :=
Set.ext fun x => mem_preimage.trans fderiv_mem_iff
rw [this]
exact
(measurableSet_of_differ... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 1,411 | 1,413 | theorem frequently_and_distrib_left {f : Filter α} {p : Prop} {q : α → Prop} :
(∃ᶠ x in f, p ∧ q x) ↔ p ∧ ∃ᶠ x in f, q x := by |
simp only [Filter.Frequently, not_and, eventually_imp_distrib_left, Classical.not_imp]
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.AlgebraicGeometry.Scheme
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
#align_import algebraic_geomet... | Mathlib/AlgebraicGeometry/OpenImmersion.lean | 269 | 286 | theorem affineBasisCover_is_basis (X : Scheme.{u}) :
TopologicalSpace.IsTopologicalBasis
{x : Set X |
∃ a : X.affineBasisCover.J, x = Set.range (X.affineBasisCover.map a).1.base} := by |
apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
· rintro _ ⟨a, rfl⟩
exact IsOpenImmersion.isOpen_range (X.affineBasisCover.map a)
· rintro a U haU hU
rcases X.affineCover.Covers a with ⟨x, e⟩
let U' := (X.affineCover.map (X.affineCover.f a)).1.base ⁻¹' U
have hxU' : x ∈ U' := by rw [←... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... | Mathlib/RingTheory/Localization/Basic.lean | 838 | 858 | theorem isLocalization_of_base_ringEquiv [IsLocalization M S] (h : R ≃+* P) :
haveI := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra
IsLocalization (M.map h.toMonoidHom) S := by |
letI : Algebra P S := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra
constructor
· rintro ⟨_, ⟨y, hy, rfl⟩⟩
convert IsLocalization.map_units S ⟨y, hy⟩
dsimp only [RingHom.algebraMap_toAlgebra, RingHom.comp_apply]
exact congr_arg _ (h.symm_apply_apply _)
· intro y
obtain ⟨⟨x, s⟩, e⟩ := IsLoc... |
/-
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.Topology.EMetricSpace.Basic
#align_import topology.metric_space.metric_separated from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb66... | Mathlib/Topology/MetricSpace/MetricSeparated.lean | 115 | 117 | theorem finite_iUnion_right_iff {ι : Type*} {I : Set ι} (hI : I.Finite) {s : Set X}
{t : ι → Set X} : IsMetricSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, IsMetricSeparated s (t i) := by |
simpa only [@comm _ _ s] using finite_iUnion_left_iff hI
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 2,366 | 2,371 | theorem addBottom_map (L : ListBlank (∀ k, Option (Γ k))) :
(addBottom L).map ⟨Prod.snd, by rfl⟩ = L := by |
simp only [addBottom, ListBlank.map_cons]
convert ListBlank.cons_head_tail L
generalize ListBlank.tail L = L'
refine L'.induction_on fun l ↦ ?_; simp
|
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analy... | Mathlib/Analysis/InnerProductSpace/l2Space.lean | 164 | 171 | theorem inner_single_left (i : ι) (a : G i) (f : lp G 2) : ⟪lp.single 2 i a, f⟫ = ⟪a, f i⟫ := by |
refine (hasSum_inner (lp.single 2 i a) f).unique ?_
convert hasSum_ite_eq i ⟪a, f i⟫ using 1
ext j
rw [lp.single_apply]
split_ifs with h
· subst h; rfl
· simp
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d... | Mathlib/Order/Interval/Set/Disjoint.lean | 201 | 205 | theorem IsGLB.biUnion_Ioi_eq (h : IsGLB s a) : ⋃ x ∈ s, Ioi x = Ioi a := by |
refine (iUnion₂_subset fun x hx => ?_).antisymm fun x hx => ?_
· exact Ioi_subset_Ioi (h.1 hx)
· rcases h.exists_between hx with ⟨y, hys, _, hyx⟩
exact mem_biUnion hys hyx
|
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Tactic.Common
#align_import data.vector from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d... | Mathlib/Data/Vector/Defs.lean | 259 | 263 | theorem toList_append {n m : ℕ} (v : Vector α n) (w : Vector α m) :
toList (append v w) = toList v ++ toList w := by |
cases v
cases w
rfl
|
/-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Algebra.Field.Subfield
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.uniform_field ... | Mathlib/Topology/Algebra/UniformField.lean | 72 | 93 | theorem continuous_hatInv [CompletableTopField K] {x : hat K} (h : x ≠ 0) :
ContinuousAt hatInv x := by |
refine denseInducing_coe.continuousAt_extend ?_
apply mem_of_superset (compl_singleton_mem_nhds h)
intro y y_ne
rw [mem_compl_singleton_iff] at y_ne
apply CompleteSpace.complete
have : (fun (x : K) => (↑x⁻¹: hat K)) =
((fun (y : K) => (↑y: hat K))∘(fun (x : K) => (x⁻¹ : K))) := by
unfold Function... |
/-
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 | 289 | 295 | theorem FiniteDimensional.finrank_pi_fintype
{ι : Type v} [Fintype ι] {M : ι → Type w} [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] [∀ i : ι, Module.Finite R (M i)] :
finrank R (∀ i, M i) = ∑ i, finrank R (M i) := by |
letI := nontrivial_of_invariantBasisNumber R
simp only [finrank, fun i => rank_eq_card_chooseBasisIndex R (M i), rank_pi, ← mk_sigma,
mk_toNat_eq_card, Fintype.card_sigma]
|
/-
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, Alistair Tucker, Wen Yang
-/
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
... | Mathlib/Topology/Order/IntermediateValue.lean | 686 | 711 | theorem Continuous.strictMonoOn_of_inj_rigidity {f : α → δ}
(hf_c : Continuous f) (hf_i : Injective f) {a b : α} (hab : a < b)
(hf_mono : StrictMonoOn f (Icc a b)) : StrictMono f := by |
intro x y hxy
let s := min a x
let t := max b y
have hsa : s ≤ a := min_le_left a x
have hbt : b ≤ t := le_max_left b y
have hst : s ≤ t := hsa.trans $ hbt.trans' hab.le
have hf_mono_st : StrictMonoOn f (Icc s t) ∨ StrictAntiOn f (Icc s t) := by
letI := Icc.completeLinearOrder hst
have := Continu... |
/-
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 | 162 | 166 | theorem normUnit_mul_normUnit (a : α) : normUnit (a * normUnit a) = 1 := by |
nontriviality α using Subsingleton.elim a 0
obtain rfl | h := eq_or_ne a 0
· rw [normUnit_zero, zero_mul, normUnit_zero]
· rw [normUnit_mul h (Units.ne_zero _), normUnit_coe_units, mul_inv_eq_one]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 1,642 | 1,647 | theorem IsLittleO.mul_isBigO {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (h₁ : f₁ =o[l] g₁) (h₂ : f₂ =O[l] g₂) :
(fun x => f₁ x * f₂ x) =o[l] fun x => g₁ x * g₂ x := by |
simp only [IsLittleO_def] at *
intro c cpos
rcases h₂.exists_pos with ⟨c', c'pos, hc'⟩
exact ((h₁ (div_pos cpos c'pos)).mul hc').congr_const (div_mul_cancel₀ _ (ne_of_gt c'pos))
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Star.Center
/-!
# Non-unital Star Subalgebras
In this... | Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 722 | 724 | theorem mem_sup_right {S T : NonUnitalStarSubalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T := by |
rw [← SetLike.le_def]
exact le_sup_right
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 99 | 101 | theorem affine_cancel_left_isIso {P : AffineTargetMorphismProperty} (hP : P.toProperty.RespectsIso)
{X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] [IsAffine Z] : P (f ≫ g) ↔ P g := by |
rw [← P.toProperty_apply, ← P.toProperty_apply, hP.cancel_left_isIso]
|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-com... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 258 | 263 | theorem ι_range_map_reverse :
(ι Q).range.map (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q)
= LinearMap.range (ι Q) := by |
rw [reverse, reverseOp, Submodule.map_comp, ι_range_map_lift, LinearMap.range_comp,
← Submodule.map_comp]
exact Submodule.map_id _
|
/-
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 | 2,099 | 2,100 | theorem iSup_option_toFinset (o : Option α) (f : α → β) : ⨆ x ∈ o.toFinset, f x = ⨆ x ∈ o, f x := by |
simp
|
/-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.AlgebraicTopology.ExtraDegeneracy
import Mathlib.CategoryTheory.Abelian.Ext
import Mathlib.R... | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | 176 | 197 | theorem diagonalSucc_hom_single (f : Gⁿ⁺¹) (a : k) :
(diagonalSucc k G n).hom.hom (single f a) =
single (f 0) 1 ⊗ₜ single (fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) a := by |
/- Porting note (#11039): broken proof was
dsimp only [diagonalSucc]
simpa only [Iso.trans_hom, Iso.symm_hom, Action.comp_hom, ModuleCat.comp_def,
LinearMap.comp_apply, Functor.mapIso_hom,
linearization_map_hom_single (actionDiagonalSucc G n).hom f a, asIso_inv,
linearization_μ_inv_hom, actionDiagonalS... |
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