Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.FixedPoints
import Mathlib.GroupTheory.Perm.Support
open Equiv List MulAction Pointwise Set Subgroup
variable {G α : Type*} [Group G] [MulAction G α] [DecidableEq α]
theorem finite_compl_fixedBy_closure_iff {S : Set G} :
(∀ g ∈ closure S, ... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 127 | 129 | theorem closure_of_isSwap_of_isPretransitive [Finite α] {S : Set (Perm α)} (hS : ∀ σ ∈ S, σ.IsSwap)
[MulAction.IsPretransitive (Subgroup.closure S) α] : Subgroup.closure S = ⊤ := by |
simp [eq_top_iff', mem_closure_isSwap hS, orbit_eq_univ, Set.toFinite]
|
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 585 | 588 | theorem kahler_map {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
[hF : Fact (finrank ℝ F = 2)] (φ : E ≃ₗᵢ[ℝ] F) (x y : F) :
(Orientation.map (Fin 2) φ.toLinearEquiv o).kahler x y = o.kahler (φ.symm x) (φ.symm y) := by |
simp [kahler_apply_apply, areaForm_map]
|
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 53 | 57 | theorem coeff_smul [SMulZeroClass S R] (r : S) (p : R[X]) (n : ℕ) :
coeff (r • p) n = r • coeff p n := by |
rcases p with ⟨⟩
simp_rw [← ofFinsupp_smul, coeff]
exact Finsupp.smul_apply _ _ _
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,518 | 1,522 | theorem bsup_not_succ_of_ne_bsup {o : Ordinal.{u}} {f : ∀ a < o, Ordinal.{max u v}}
(hf : ∀ {i : Ordinal} (h : i < o), f i h ≠ bsup.{_, v} o f) (a) :
a < bsup.{_, v} o f → succ a < bsup.{_, v} o f := by |
rw [← sup_eq_bsup] at *
exact sup_not_succ_of_ne_sup fun i => hf _
|
import Mathlib.Algebra.DirectLimit
import Mathlib.Algebra.CharP.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Construction
#align_import field_theory.is_alg_closed.algebraic_closure from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
univ... | Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean | 272 | 292 | theorem Step.isIntegral (n) : ∀ z : Step k n, IsIntegral k z := by |
induction' n with a h
· intro z
exact isIntegral_algebraMap
· intro z
change RingHom.IsIntegralElem _ _
revert z
change RingHom.IsIntegral _
unfold algebraMap
unfold Algebra.toRingHom
unfold algebra
unfold RingHom.toAlgebra
unfold RingHom.toAlgebra'
simp only
rw [toSte... |
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Martingale.OptionalStopping
import Mathlib.Probability.Martingale.Centering
#align_import probability.martingale.borel_cantelli from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open Filter
open scoped NNRea... | Mathlib/Probability/Martingale/BorelCantelli.lean | 101 | 115 | theorem Submartingale.stoppedValue_leastGE [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) (r : ℝ) :
Submartingale (fun i => stoppedValue f (leastGE f r i)) ℱ μ := by |
rw [submartingale_iff_expected_stoppedValue_mono]
· intro σ π hσ hπ hσ_le_π hπ_bdd
obtain ⟨n, hπ_le_n⟩ := hπ_bdd
simp_rw [stoppedValue_stoppedValue_leastGE f σ r fun i => (hσ_le_π i).trans (hπ_le_n i)]
simp_rw [stoppedValue_stoppedValue_leastGE f π r hπ_le_n]
refine hf.expected_stoppedValue_mono ?_... |
import Mathlib.Algebra.Homology.ImageToKernel
#align_import algebra.homology.exact from "leanprover-community/mathlib"@"3feb151caefe53df080ca6ca67a0c6685cfd1b82"
universe v v₂ u u₂
open CategoryTheory CategoryTheory.Limits
variable {V : Type u} [Category.{v} V]
variable [HasImages V]
namespace CategoryTheory
... | Mathlib/Algebra/Homology/Exact.lean | 140 | 144 | theorem comp_eq_zero_of_image_eq_kernel {A B C : V} (f : A ⟶ B) (g : B ⟶ C)
(p : imageSubobject f = kernelSubobject g) : f ≫ g = 0 := by |
suffices Subobject.arrow (imageSubobject f) ≫ g = 0 by
rw [← imageSubobject_arrow_comp f, Category.assoc, this, comp_zero]
rw [p, kernelSubobject_arrow_comp]
|
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 92 | 93 | theorem nodup_range (n : ℕ) : Nodup (range n) := by |
simp (config := {decide := true}) only [range_eq_range', nodup_range']
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.add from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
variable... | Mathlib/Analysis/Calculus/Deriv/Add.lean | 208 | 210 | theorem derivWithin.neg (hxs : UniqueDiffWithinAt 𝕜 s x) :
derivWithin (fun y => -f y) s x = -derivWithin f s x := by |
simp only [derivWithin, fderivWithin_neg hxs, ContinuousLinearMap.neg_apply]
|
import Mathlib.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
#align_import algebraic_geometry.Scheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"
-- Explicit universe annotations were used in this file to improv... | Mathlib/AlgebraicGeometry/Scheme.lean | 144 | 146 | theorem comp_val_base_apply {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
(f ≫ g).val.base x = g.val.base (f.val.base x) := by |
simp
|
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 57 | 60 | theorem isPrimePow_iff_card_primeFactors_eq_one {n : ℕ} :
IsPrimePow n ↔ n.primeFactors.card = 1 := by |
simp_rw [isPrimePow_iff_factorization_eq_single, ← Nat.support_factorization,
Finsupp.card_support_eq_one', pos_iff_ne_zero]
|
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 1,085 | 1,107 | theorem isPreconnected_iff_subset_of_disjoint_closed :
IsPreconnected s ↔
∀ u v, IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v := by |
constructor <;> intro h
· intro u v hu hv hs huv
rw [isPreconnected_closed_iff] at h
specialize h u v hu hv hs
contrapose! huv
simp [not_subset] at huv
rcases huv with ⟨⟨x, hxs, hxu⟩, ⟨y, hys, hyv⟩⟩
have hxv : x ∈ v := or_iff_not_imp_left.mp (hs hxs) hxu
have hyu : y ∈ u := or_iff_not_i... |
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 829 | 830 | theorem mul_ne_zero {a b : Ordinal} : a ≠ 0 → b ≠ 0 → a * b ≠ 0 := by |
simpa only [Ordinal.pos_iff_ne_zero] using mul_pos
|
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 100 | 117 | theorem coeff_expand {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
(expand R p f).coeff n = if p ∣ n then f.coeff (n / p) else 0 := by |
simp only [expand_eq_sum]
simp_rw [coeff_sum, ← pow_mul, C_mul_X_pow_eq_monomial, coeff_monomial, sum]
split_ifs with h
· rw [Finset.sum_eq_single (n / p), Nat.mul_div_cancel' h, if_pos rfl]
· intro b _ hb2
rw [if_neg]
intro hb3
apply hb2
rw [← hb3, Nat.mul_div_cancel_left b hp]
... |
import Mathlib.Topology.Separation
import Mathlib.Algebra.BigOperators.Finprod
#align_import topology.algebra.infinite_sum.basic from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
noncomputable section
open Filter Function
open scoped Topology
variable {α β γ : Type*}
section HasP... | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | 174 | 175 | theorem HasProd.unique {a₁ a₂ : α} [T2Space α] : HasProd f a₁ → HasProd f a₂ → a₁ = a₂ := by |
classical exact tendsto_nhds_unique
|
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 96 | 100 | theorem epi_of_nonzero_to_simple [HasEqualizers C] {X Y : C} [Simple Y] {f : X ⟶ Y} [HasImage f]
(w : f ≠ 0) : Epi f := by |
rw [← image.fac f]
haveI : IsIso (image.ι f) := isIso_of_mono_of_nonzero fun h => w (eq_zero_of_image_eq_zero h)
apply epi_comp
|
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 210 | 216 | theorem integrable_condexp : Integrable (μ[f|m]) μ := by |
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]; exact integrable_zero _ _ _
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]; exact integrable_zero _ _ _
haveI : SigmaFinite (μ.trim hm) := hμm
exact (integrable_condexpL1 f).congr (condexp_ae_eq_condexpL1 hm f).s... |
import Mathlib.Data.List.Basic
import Mathlib.Data.Sigma.Basic
#align_import data.list.prod_sigma from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
variable {α β : Type*}
namespace List
@[simp]
theorem nil_product (l : List β) : (@nil α) ×ˢ l = [] :=
rfl
#align list.nil_product... | Mathlib/Data/List/ProdSigma.lean | 89 | 93 | theorem length_sigma' (l₁ : List α) (l₂ : ∀ a, List (σ a)) :
length (l₁.sigma l₂) = Nat.sum (l₁.map fun a ↦ length (l₂ a)) := by |
induction' l₁ with x l₁ IH
· rfl
· simp only [map, sigma_cons, length_append, length_map, IH, Nat.sum_cons]
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-community/mathlib"@"991ff3b5269848f6dd942ae8e9dd3c946035dc8b"
noncomputable section
universe u
open List
namespace Ordinal
@[elab_as_elim]
noncomputabl... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 150 | 158 | theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :
x ∈ CNF b o → x.2 < b := by |
refine CNFRec b ?_ (fun o ho IH ↦ ?_) o
· simp only [CNF_zero, not_mem_nil, IsEmpty.forall_iff]
· rw [CNF_ne_zero ho]
intro h
cases' (mem_cons.mp h) with h h
· rw [h]; simpa only using div_opow_log_lt o hb
· exact IH h
|
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory
open CategoryTheory.Limits Opposite
universe w v u
var... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 529 | 533 | theorem isIso_toSheafify {P : Cᵒᵖ ⥤ D} (hP : Presheaf.IsSheaf J P) : IsIso (J.toSheafify P) := by |
dsimp [toSheafify]
haveI := isIso_toPlus_of_isSheaf J P hP
change (IsIso (toPlus J P ≫ (J.plusFunctor D).map (toPlus J P)))
infer_instance
|
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 203 | 207 | theorem fold_union_empty_singleton [DecidableEq α] (s : Finset α) :
Finset.fold (· ∪ ·) ∅ singleton s = s := by |
induction' s using Finset.induction_on with a s has ih
· simp only [fold_empty]
· rw [fold_insert has, ih, insert_eq]
|
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Init.Data.Fin.Basic
#align_import data.list.nodup from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
universe u v
open Nat Function
variable {α : Type u} {β : Type v} {l l₁ l₂ : List α} {r : α → α ... | Mathlib/Data/List/Nodup.lean | 384 | 387 | theorem Nodup.union [DecidableEq α] (l₁ : List α) (h : Nodup l₂) : (l₁ ∪ l₂).Nodup := by |
induction' l₁ with a l₁ ih generalizing l₂
· exact h
· exact (ih h).insert
|
import Mathlib.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function Set
namespace Cardinal
variable {α : Type u} {c d : Cardinal.{u}}
noncomputable def toNat : Cardinal →*₀ ℕ :=
ENat.toNat.com... | Mathlib/SetTheory/Cardinal/ToNat.lean | 164 | 165 | theorem toNat_lift (c : Cardinal.{v}) : toNat (lift.{u, v} c) = toNat c := by |
simp only [← toNat_toENat, toENat_lift]
|
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Init.Data.List.Instances
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.list.basic... | Mathlib/Data/List/Basic.lean | 1,439 | 1,444 | theorem modifyNthTail_modifyNthTail_le {f g : List α → List α} (m n : ℕ) (l : List α)
(h : n ≤ m) :
(l.modifyNthTail f n).modifyNthTail g m =
l.modifyNthTail (fun l => (f l).modifyNthTail g (m - n)) n := by |
rcases Nat.exists_eq_add_of_le h with ⟨m, rfl⟩
rw [Nat.add_comm, modifyNthTail_modifyNthTail, Nat.add_sub_cancel]
|
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
universe u v y
variable {R : Type u} {S : Typ... | Mathlib/Algebra/Polynomial/Monic.lean | 196 | 200 | theorem natDegree_mul_comm (hp : p.Monic) (q : R[X]) : (p * q).natDegree = (q * p).natDegree := by |
by_cases h : q = 0
· simp [h]
rw [hp.natDegree_mul' h, Polynomial.natDegree_mul', add_comm]
simpa [hp.leadingCoeff, leadingCoeff_ne_zero]
|
import Mathlib.Combinatorics.SimpleGraph.Regularity.Bound
import Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise
import Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform
#align_import combinatorics.simple_graph.regularity.chunk from "leanprover-community/mathlib"@"bf7ef0e83e5b7e6c1169e97f055e58a2e4e9d52d... | Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean | 526 | 537 | theorem edgeDensity_chunk_uniform [Nonempty α] (hPα : P.parts.card * 16 ^ P.parts.card ≤ card α)
(hPε : ↑100 ≤ ↑4 ^ P.parts.card * ε ^ 5) (hU : U ∈ P.parts) (hV : V ∈ P.parts) :
(G.edgeDensity U V : ℝ) ^ 2 - ε ^ 5 / ↑25 ≤
(∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts,
(G.edgeDensity ... |
apply (edgeDensity_chunk_aux (hP := hP) hPα hPε hU hV).trans
have key : ↑16 ^ P.parts.card =
(((chunk hP G ε hU).parts ×ˢ (chunk hP G ε hV).parts).card : ℝ) := by
rw [card_product, cast_mul, card_chunk (m_pos hPα).ne', card_chunk (m_pos hPα).ne', ←
cast_mul, ← mul_pow]; norm_cast
simp_rw [key]
... |
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.measure.haar.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set Inv Function Topological... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 377 | 390 | theorem nonempty_iInter_clPrehaar (K₀ : PositiveCompacts G) :
(haarProduct (K₀ : Set G) ∩ ⋂ V : OpenNhdsOf (1 : G), clPrehaar K₀ V).Nonempty := by |
have : IsCompact (haarProduct (K₀ : Set G)) := by
apply isCompact_univ_pi; intro K; apply isCompact_Icc
refine this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => ?_
let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
have h1V₀ : IsOpen V₀ := isOpen_biInter_finset <| by rintro ⟨⟨... |
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.computation.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
open Generali... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 240 | 244 | theorem get?_of_eq_some_of_succ_get?_intFractPair_stream {ifp_succ_n : IntFractPair K}
(stream_succ_nth_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n) :
(of v).s.get? n = some ⟨1, ifp_succ_n.b⟩ := by |
unfold of IntFractPair.seq1
simp [Stream'.Seq.map_tail, Stream'.Seq.get?_tail, Stream'.Seq.map_get?, stream_succ_nth_eq]
|
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Coset
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Hom
#align_import group_theory.group_action.... | Mathlib/GroupTheory/GroupAction/Quotient.lean | 108 | 109 | theorem Quotient.mk_smul_out' [QuotientAction β H] (b : β) (q : α ⧸ H) :
QuotientGroup.mk (b • q.out') = b • q := by | rw [← Quotient.smul_mk, QuotientGroup.out_eq']
|
import Mathlib.Order.Filter.Partial
import Mathlib.Topology.Basic
#align_import topology.partial from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter
open Topology
variable {X Y : Type*} [TopologicalSpace X]
theorem rtendsto_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
... | Mathlib/Topology/Partial.lean | 61 | 83 | theorem pcontinuous_iff' {f : X →. Y} :
PContinuous f ↔ ∀ {x y} (h : y ∈ f x), PTendsto' f (𝓝 x) (𝓝 y) := by |
constructor
· intro h x y h'
simp only [ptendsto'_def, mem_nhds_iff]
rintro s ⟨t, tsubs, opent, yt⟩
exact ⟨f.preimage t, PFun.preimage_mono _ tsubs, h _ opent, ⟨y, yt, h'⟩⟩
intro hf s os
rw [isOpen_iff_nhds]
rintro x ⟨y, ys, fxy⟩ t
rw [mem_principal]
intro (h : f.preimage s ⊆ t)
change t ∈ ... |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.fixed_point from "leanprover-community/mathlib"@"0dd4319a17376eda5763cd0a7e0d35bbaaa50e83"
noncomputable section
universe u v
open Function Order
namespace Ordinal
section
variable {ι ... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 713 | 721 | theorem deriv_mul_eq_opow_omega_mul {a : Ordinal.{u}} (ha : 0 < a) (b) :
deriv (a * ·) b = (a^omega) * b := by |
revert b
rw [← funext_iff,
IsNormal.eq_iff_zero_and_succ (deriv_isNormal _) (mul_isNormal (opow_pos omega ha))]
refine ⟨?_, fun c h => ?_⟩
· dsimp only; rw [deriv_zero, nfp_mul_zero, mul_zero]
· rw [deriv_succ, h]
exact nfp_mul_opow_omega_add c ha zero_lt_one (one_le_iff_pos.2 (opow_pos _ ha))
|
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 151 | 154 | theorem prod_univ_six [CommMonoid β] (f : Fin 6 → β) :
∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 := by |
rw [prod_univ_castSucc, prod_univ_five]
rfl
|
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 144 | 147 | theorem isFiniteMeasure_withDensity {f : α → ℝ≥0∞} (hf : ∫⁻ a, f a ∂μ ≠ ∞) :
IsFiniteMeasure (μ.withDensity f) :=
{ measure_univ_lt_top := by |
rwa [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ, lt_top_iff_ne_top] }
|
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb370b"
assert_not_exists MonoidWithZero
universe u
variable {m n : ℕ}
def finZeroEquiv : Fin 0 ≃ Empty :=
Equiv.equivEmpty _
#align fin_... | Mathlib/Logic/Equiv/Fin.lean | 359 | 360 | theorem finAddFlip_apply_castAdd (k : Fin m) (n : ℕ) :
finAddFlip (Fin.castAdd n k) = Fin.natAdd n k := by | simp [finAddFlip]
|
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
suppress_compilation
open Bornology Metric Set Real
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special cases about linear (as o... | Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean | 70 | 86 | theorem tendsto_of_tendsto_pointwise_of_cauchySeq {f : ℕ → E' →SL[σ₁₂] F} {g : E' →SL[σ₁₂] F}
(hg : Tendsto (fun n x => f n x) atTop (𝓝 g)) (hf : CauchySeq f) : Tendsto f atTop (𝓝 g) := by |
/- Since `f` is a Cauchy sequence, there exists `b → 0` such that `‖f n - f m‖ ≤ b N` for any
`m, n ≥ N`. -/
rcases cauchySeq_iff_le_tendsto_0.1 hf with ⟨b, hb₀, hfb, hb_lim⟩
-- Since `b → 0`, it suffices to show that `‖f n x - g x‖ ≤ b n * ‖x‖` for all `n` and `x`.
suffices ∀ n x, ‖f n x - g x‖ ≤ b n * ‖x... |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.congruence_subgroups from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 156 | 172 | theorem Gamma1_to_Gamma0_mem (N : ℕ) (A : Gamma0 N) : A ∈ Gamma1' N ↔
((↑ₘA 0 0 : ℤ) : ZMod N) = 1 ∧ ((↑ₘA 1 1 : ℤ) : ZMod N) = 1 ∧ ((↑ₘA 1 0 : ℤ) : ZMod N) = 0 := by |
constructor
· intro ha
have hA := A.property
rw [Gamma0_mem] at hA
have adet := Gamma0_det N A
rw [Matrix.det_fin_two] at adet
simp only [Gamma0Map, coe_matrix_coe, Int.coe_castRingHom, map_apply, Gamma1_mem',
MonoidHom.coe_mk, OneHom.coe_mk, Int.cast_sub, Int.cast_mul] at *
rw [hA, h... |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 157 | 160 | theorem mem_powers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : ℕ} [hp : Fact p.Prime]
(h : Fintype.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ Submonoid.powers g := by |
rw [mem_powers_iff_mem_zpowers]
exact mem_zpowers_of_prime_card h hg
|
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 145 | 146 | theorem laverage_congr {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ⨍⁻ x, f x ∂μ = ⨍⁻ x, g x ∂μ := by |
simp only [laverage_eq, lintegral_congr_ae h]
|
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ι : Sort*} {𝕜 E : Type*}
section OrderedSemiring
variable (𝕜) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t s₁ s₂ t₁ t₂ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 61 | 61 | theorem convexJoin_empty_right (s : Set E) : convexJoin 𝕜 s ∅ = ∅ := by | simp [convexJoin]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 227 | 230 | theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by |
by_cases h : x = 0
· simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re]
rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)]
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Bits
import Mathlib.Data.Nat.Log
import Mathlib.Data.List.Indexes
import Mathlib.Data.List.Palindrome
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
impo... | Mathlib/Data/Nat/Digits.lean | 648 | 655 | theorem ofDigits_modEq' (b b' : ℕ) (k : ℕ) (h : b ≡ b' [MOD k]) (L : List ℕ) :
ofDigits b L ≡ ofDigits b' L [MOD k] := by |
induction' L with d L ih
· rfl
· dsimp [ofDigits]
dsimp [Nat.ModEq] at *
conv_lhs => rw [Nat.add_mod, Nat.mul_mod, h, ih]
conv_rhs => rw [Nat.add_mod, Nat.mul_mod]
|
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Group.Pointwise
#align_import measure_theory.group.fundamental_domain from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
open scoped ENNReal Pointwise Topology NNRea... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 319 | 321 | theorem measure_eq_tsum (h : IsFundamentalDomain G s μ) (t : Set α) :
μ t = ∑' g : G, μ (g • t ∩ s) := by |
simpa only [set_lintegral_one] using h.set_lintegral_eq_tsum' (fun _ => 1) t
|
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 118 | 122 | theorem sin_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.sin (o.oangle (x + y) y) = ‖x‖ / ‖x + y‖ := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).sin_oangle_add_right_of_oangle_eq_pi_div_two h
|
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 393 | 408 | theorem angle_eq_zero_iff_ne_and_wbtw {p₁ p₂ p₃ : P} :
∠ p₁ p₂ p₃ = 0 ↔ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁ := by |
constructor
· rw [angle, angle_eq_zero_iff]
rintro ⟨hp₁p₂, r, hr0, hp₃p₂⟩
rcases le_or_lt 1 r with (hr1 | hr1)
· refine Or.inl ⟨vsub_ne_zero.1 hp₁p₂, r⁻¹, ⟨(inv_pos.2 hr0).le, inv_le_one hr1⟩, ?_⟩
rw [AffineMap.lineMap_apply, hp₃p₂, smul_smul, inv_mul_cancel hr0.ne.symm, one_smul,
vsub_va... |
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.Probability.Independence.Basic
#align_import probability.integration from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
noncomputable section
open Set MeasureTheory
open scoped ENNReal MeasureTheory
variable {Ω : Type*... | Mathlib/Probability/Integration.lean | 180 | 197 | theorem IndepFun.integrable_right_of_integrable_mul {β : Type*} [MeasurableSpace β] {X Y : Ω → β}
[NormedDivisionRing β] [BorelSpace β] (hXY : IndepFun X Y μ) (h'XY : Integrable (X * Y) μ)
(hX : AEStronglyMeasurable X μ) (hY : AEStronglyMeasurable Y μ) (h'X : ¬X =ᵐ[μ] 0) :
Integrable Y μ := by |
refine ⟨hY, ?_⟩
have I : (∫⁻ ω, ‖X ω‖₊ ∂μ) ≠ 0 := fun H ↦ by
have I : (fun ω => ‖X ω‖₊ : Ω → ℝ≥0∞) =ᵐ[μ] 0 := (lintegral_eq_zero_iff' hX.ennnorm).1 H
apply h'X
filter_upwards [I] with ω hω
simpa using hω
refine lt_top_iff_ne_top.2 fun H => ?_
have J : IndepFun (fun ω => ‖X ω‖₊ : Ω → ℝ≥0∞) (fun ... |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.Hall.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.SetTheory.Cardinal.Finite
#align_import combinatorics.configuration from "leanprover-community/mathlib"@"d2d8742b0c21426362a9dacebc6005db895ca963"
open Finset
nam... | Mathlib/Combinatorics/Configuration.lean | 225 | 245 | theorem HasLines.card_le [HasLines P L] [Fintype P] [Fintype L] :
Fintype.card P ≤ Fintype.card L := by |
classical
by_contra hc₂
obtain ⟨f, hf₁, hf₂⟩ := Nondegenerate.exists_injective_of_card_le (le_of_not_le hc₂)
have :=
calc
∑ p, lineCount L p = ∑ l, pointCount P l := sum_lineCount_eq_sum_pointCount P L
_ ≤ ∑ l, lineCount L (f l) :=
(Finset.sum_le_sum fun l _ => HasLines.pointCount_le_li... |
import Mathlib.Algebra.Homology.ImageToKernel
#align_import algebra.homology.exact from "leanprover-community/mathlib"@"3feb151caefe53df080ca6ca67a0c6685cfd1b82"
universe v v₂ u u₂
open CategoryTheory CategoryTheory.Limits
variable {V : Type u} [Category.{v} V]
variable [HasImages V]
namespace CategoryTheory
... | Mathlib/Algebra/Homology/Exact.lean | 282 | 285 | theorem comp_eq_zero_of_exact (h : Exact f g) {X Y : V} {ι : X ⟶ B} (hι : ι ≫ g = 0) {π : B ⟶ Y}
(hπ : f ≫ π = 0) : ι ≫ π = 0 := by |
rw [← kernel.lift_ι _ _ hι, ← cokernel.π_desc _ _ hπ, Category.assoc,
kernel_comp_cokernel_assoc _ _ h, zero_comp, comp_zero]
|
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import algebra.quaternion from "leanprover-community/mathlib"@"cf7a7252c19... | Mathlib/Algebra/Quaternion.lean | 614 | 616 | theorem rank_eq_four [StrongRankCondition R] : Module.rank R ℍ[R,c₁,c₂] = 4 := by |
rw [rank_eq_card_basis (basisOneIJK c₁ c₂), Fintype.card_fin]
norm_num
|
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 158 | 162 | theorem polynomialQuotientEquivQuotientPolynomial_map_mk (I : Ideal R) (f : R[X]) :
I.polynomialQuotientEquivQuotientPolynomial (f.map <| Quotient.mk I) =
Quotient.mk (map C I : Ideal R[X]) f := by |
apply (polynomialQuotientEquivQuotientPolynomial I).symm.injective
rw [RingEquiv.symm_apply_apply, polynomialQuotientEquivQuotientPolynomial_symm_mk]
|
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 516 | 517 | theorem KaehlerDifferential.kerTotal_mkQ_single_algebraMap_one (x) : (x𝖣1) = 0 := by |
rw [← (algebraMap R S).map_one, KaehlerDifferential.kerTotal_mkQ_single_algebraMap]
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 437 | 439 | theorem ext_iff_of_iUnion_eq_univ [Countable ι] {s : ι → Set α} (hs : ⋃ i, s i = univ) :
μ = ν ↔ ∀ i, μ.restrict (s i) = ν.restrict (s i) := by |
rw [← restrict_iUnion_congr, hs, restrict_univ, restrict_univ]
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 1,110 | 1,115 | theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}
(h : ∀ x ∈ s, ∃ t, IsOpen t ∧ x ∈ t ∧ ContinuousOn f (s ∩ t)) : ContinuousOn f s := by |
intro x xs
rcases h x xs with ⟨t, open_t, xt, ct⟩
have := ct x ⟨xs, xt⟩
rwa [ContinuousWithinAt, ← nhdsWithin_restrict _ xt open_t] at this
|
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 90 | 100 | theorem map_permutationsAux2' {α' β'} (g : α → α') (g' : β → β') (t : α) (ts ys : List α)
(r : List β) (f : List α → β) (f' : List α' → β') (H : ∀ a, g' (f a) = f' (map g a)) :
map g' (permutationsAux2 t ts r ys f).2 =
(permutationsAux2 (g t) (map g ts) (map g' r) (map g ys) f').2 := by |
induction' ys with ys_hd _ ys_ih generalizing f f'
· simp
· simp only [map, permutationsAux2_snd_cons, cons_append, cons.injEq]
rw [ys_ih, permutationsAux2_fst]
· refine ⟨?_, rfl⟩
simp only [← map_cons, ← map_append]; apply H
· intro a; apply H
|
import Mathlib.Init.Order.Defs
import Mathlib.Logic.Nontrivial.Defs
import Mathlib.Tactic.Attr.Register
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Subtype
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Unique
#align_import logic.nontrivial from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc95... | Mathlib/Logic/Nontrivial/Basic.lean | 41 | 43 | theorem Subtype.nontrivial_iff_exists_ne (p : α → Prop) (x : Subtype p) :
Nontrivial (Subtype p) ↔ ∃ (y : α) (_ : p y), y ≠ x := by |
simp only [_root_.nontrivial_iff_exists_ne x, Subtype.exists, Ne, Subtype.ext_iff]
|
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈... | Mathlib/Topology/Basic.lean | 711 | 716 | theorem frontier_inter_subset (s t : Set X) :
frontier (s ∩ t) ⊆ frontier s ∩ closure t ∪ closure s ∩ frontier t := by |
simp only [frontier_eq_closure_inter_closure, compl_inter, closure_union]
refine (inter_subset_inter_left _ (closure_inter_subset_inter_closure s t)).trans_eq ?_
simp only [inter_union_distrib_left, union_inter_distrib_right, inter_assoc,
inter_comm (closure t)]
|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
noncomputable section
open scoped Classical
variable {α β γ : Type*}
def Finite.equivFin (α : Type*) [Finite α] : α ≃ Fin (Nat.card α) := by
have := (Finite.... | Mathlib/Data/Finite/Card.lean | 109 | 113 | theorem card_le_of_surjective [Finite α] (f : α → β) (hf : Function.Surjective f) :
Nat.card β ≤ Nat.card α := by |
haveI := Fintype.ofFinite α
haveI := Fintype.ofSurjective f hf
simpa only [Nat.card_eq_fintype_card, ge_iff_le] using Fintype.card_le_of_surjective f hf
|
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib.Logic.Small.Defs
import Mathlib.Order.GameAdd
#align_import set_theory.game.pgame from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
set_option autoImplicit true
names... | Mathlib/SetTheory/Game/PGame.lean | 821 | 821 | theorem equiv_of_eq {x y : PGame} (h : x = y) : x ≈ y := by | subst h; rfl
|
import Mathlib.MeasureTheory.Integral.Periodic
import Mathlib.Data.ZMod.Quotient
#align_import measure_theory.group.add_circle from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter MeasureTheory MeasureTheory.Measure Metric
open scoped MeasureTheory Pointwise Top... | Mathlib/MeasureTheory/Group/AddCircle.lean | 95 | 104 | theorem volume_of_add_preimage_eq (s I : Set <| AddCircle T) (u x : AddCircle T)
(hu : IsOfFinAddOrder u) (hs : (u +ᵥ s : Set <| AddCircle T) =ᵐ[volume] s)
(hI : I =ᵐ[volume] ball x (T / (2 * addOrderOf u))) :
volume s = addOrderOf u • volume (s ∩ I) := by |
let G := AddSubgroup.zmultiples u
haveI : Fintype G := @Fintype.ofFinite _ hu.finite_zmultiples.to_subtype
have hsG : ∀ g : G, (g +ᵥ s : Set <| AddCircle T) =ᵐ[volume] s := by
rintro ⟨y, hy⟩; exact (vadd_ae_eq_self_of_mem_zmultiples hs hy : _)
rw [(isAddFundamentalDomain_of_ae_ball I u x hu hI).measure_eq_... |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Data.ZMod.Algebra
#align_import ring_theory.polynomial.cyclotomic.expand from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"
namespace Polynomial
@[simp]
theorem cyclotomic_expand_eq_cyclotomic_mul {p n : ℕ} (hp : Na... | Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean | 166 | 179 | theorem isRoot_cyclotomic_prime_pow_mul_iff_of_charP {m k p : ℕ} {R : Type*} [CommRing R]
[IsDomain R] [hp : Fact (Nat.Prime p)] [hchar : CharP R p] {μ : R} [NeZero (m : R)] :
(Polynomial.cyclotomic (p ^ k * m) R).IsRoot μ ↔ IsPrimitiveRoot μ m := by |
rcases k.eq_zero_or_pos with (rfl | hk)
· rw [pow_zero, one_mul, isRoot_cyclotomic_iff]
refine ⟨fun h => ?_, fun h => ?_⟩
· rw [IsRoot.def, cyclotomic_mul_prime_pow_eq R (NeZero.not_char_dvd R p m) hk, eval_pow]
at h
replace h := pow_eq_zero h
rwa [← IsRoot.def, isRoot_cyclotomic_iff] at h
· rw... |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import geometry.euclidean.angle.unoriented.basic from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
assert_not_exists HasFDerivAt
assert_not_exists ConformalAt
noncom... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 155 | 158 | theorem angle_smul_right_of_pos (x y : V) {r : ℝ} (hr : 0 < r) : angle x (r • y) = angle x y := by |
unfold angle
rw [inner_smul_right, norm_smul, Real.norm_eq_abs, abs_of_nonneg (le_of_lt hr), ← mul_assoc,
mul_comm _ r, mul_assoc, mul_div_mul_left _ _ (ne_of_gt hr)]
|
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 291 | 294 | theorem exists_rightDescent_of_ne_one {w : W} (hw : w ≠ 1) : ∃ i : B, cs.IsRightDescent w i := by |
simp only [← isLeftDescent_inv_iff]
apply exists_leftDescent_of_ne_one
simpa
|
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k V₁ P₁ V₂ P₂ : Type*} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁]
[Module k V₂] [AddTorsor V₁ P₁] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 33 | 36 | theorem AffineSubspace.nonempty_map {E : AffineSubspace k P₁} [Ene : Nonempty E] {φ : P₁ →ᵃ[k] P₂} :
Nonempty (E.map φ) := by |
obtain ⟨x, hx⟩ := id Ene
exact ⟨⟨φ x, AffineSubspace.mem_map.mpr ⟨x, hx, rfl⟩⟩⟩
|
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 233 | 233 | theorem right_le_or : ∀ x y : Bool, y ≤ (x || y) := by | decide
|
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
| Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 211 | 217 | theorem riemannZeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : ℂ) ^ (2 * k - 1)
* (π : ℂ) ^ (2 * k) * bernoulli (2 * k) / (2 * k)! := by |
convert congr_arg ((↑) : ℝ → ℂ) (hasSum_zeta_nat hk).tsum_eq
· rw [← Nat.cast_two, ← Nat.cast_mul, zeta_nat_eq_tsum_of_gt_one (by omega)]
simp only [push_cast]
· norm_cast
|
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
#align_import dynamics.omega_limit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter Topology
section omegaLimit
variable {τ : Type*} {α : Type*} {β : Type*} {ι : Type*}
def omegaLimit [Topol... | Mathlib/Dynamics/OmegaLimit.lean | 210 | 215 | theorem omegaLimit_subset_closure_fw_image {u : Set τ} (hu : u ∈ f) :
ω f ϕ s ⊆ closure (image2 ϕ u s) := by |
rw [omegaLimit_eq_iInter]
intro _ hx
rw [mem_iInter] at hx
exact hx ⟨u, hu⟩
|
import Mathlib.Order.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
open Function Set
open Filter
namespace Filter
variable {α α' β β' γ γ' δ δ' ε ε' : Type*} {m : α → β → γ} {f f₁ f₂ : Filter α}
{g g₁ g₂ : Filter β} {h h₁ h₂ : Filt... | Mathlib/Order/Filter/NAry.lean | 204 | 209 | theorem map₂_assoc {m : δ → γ → ε} {n : α → β → δ} {m' : α → ε' → ε} {n' : β → γ → ε'}
{h : Filter γ} (h_assoc : ∀ a b c, m (n a b) c = m' a (n' b c)) :
map₂ m (map₂ n f g) h = map₂ m' f (map₂ n' g h) := by |
rw [← map_prod_eq_map₂ n, ← map_prod_eq_map₂ n', map₂_map_left, map₂_map_right,
← map_prod_eq_map₂, ← map_prod_eq_map₂, ← prod_assoc, map_map]
simp only [h_assoc, Function.comp, Equiv.prodAssoc_apply]
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingleton
#align_import combinatorics.composition from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open List
variable {n : ℕ}
... | Mathlib/Combinatorics/Enumerative/Composition.lean | 204 | 204 | theorem sizeUpTo_zero : c.sizeUpTo 0 = 0 := by | simp [sizeUpTo]
|
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 423 | 425 | theorem image_const_sub_Iio : (fun x => a - x) '' Iio b = Ioi (a - b) := by |
have := image_comp (fun x => a + x) fun x => -x; dsimp [Function.comp_def] at this
simp [sub_eq_add_neg, this, add_comm]
|
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Order.Filter.ModEq
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.List.TFAE
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.specific_lim... | Mathlib/Analysis/SpecificLimits/Normed.lean | 562 | 564 | theorem geom_series_mul_shift (x : R) (h : ‖x‖ < 1) :
x * ∑' i : ℕ, x ^ i = ∑' i : ℕ, x ^ (i + 1) := by |
simp_rw [← (NormedRing.summable_geometric_of_norm_lt_one _ h).tsum_mul_left, ← _root_.pow_succ']
|
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 138 | 140 | theorem setLaverage_eq' (f : α → ℝ≥0∞) (s : Set α) :
⨍⁻ x in s, f x ∂μ = ∫⁻ x, f x ∂(μ s)⁻¹ • μ.restrict s := by |
simp only [laverage_eq', restrict_apply_univ]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
var... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 162 | 164 | theorem list_prod_map_rpow' {ι} (l : List ι) (f : ι → ℝ≥0) (r : ℝ) :
(l.map (f · ^ r)).prod = (l.map f).prod ^ r := by |
rw [← list_prod_map_rpow, List.map_map]; rfl
|
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
#align_import algebra.indicator_function from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
assert_not_exists MonoidWithZero
open Function
variable {α β ι M N : Type*}
namespace Set
section One
variable [On... | Mathlib/Algebra/Group/Indicator.lean | 318 | 320 | theorem mulIndicator_preimage_of_not_mem (s : Set α) (f : α → M) {t : Set M} (ht : (1 : M) ∉ t) :
mulIndicator s f ⁻¹' t = f ⁻¹' t ∩ s := by |
simp [mulIndicator_preimage, Pi.one_def, Set.preimage_const_of_not_mem ht]
|
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 76 | 98 | theorem exists_root_sum_quadratic [Fintype R] {f g : R[X]} (hf2 : degree f = 2) (hg2 : degree g = 2)
(hR : Fintype.card R % 2 = 1) : ∃ a b, f.eval a + g.eval b = 0 :=
letI := Classical.decEq R
suffices ¬Disjoint (univ.image fun x : R => eval x f)
(univ.image fun x : R => eval x (-g)) by
simp only [disjo... |
rw [card_union_of_disjoint hd];
simp [natDegree_eq_of_degree_eq_some hf2, natDegree_eq_of_degree_eq_some hg2, mul_add]
|
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
open Function Set
open scoped ENNReal Classical
noncomputable section
variable {α β δ : Type*} [MeasurableSpace α] [MeasurableSpace β] {s : Set α} ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 97 | 98 | theorem sum_smul_dirac [Countable α] [MeasurableSingletonClass α] (μ : Measure α) :
(sum fun a => μ {a} • dirac a) = μ := by | simpa using (map_eq_sum μ id measurable_id).symm
|
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Set Fin Filter Function
open scoped NNReal Topology
section Real
variab... | Mathlib/Analysis/Calculus/ContDiff/RCLike.lean | 151 | 157 | theorem ContDiff.lipschitzWith_of_hasCompactSupport {f : E' → F'} {n : ℕ∞}
(hf : HasCompactSupport f) (h'f : ContDiff 𝕂 n f) (hn : 1 ≤ n) :
∃ C, LipschitzWith C f := by |
obtain ⟨C, hC⟩ := (hf.fderiv 𝕂).exists_bound_of_continuous (h'f.continuous_fderiv hn)
refine ⟨⟨max C 0, le_max_right _ _⟩, ?_⟩
apply lipschitzWith_of_nnnorm_fderiv_le (h'f.differentiable hn) (fun x ↦ ?_)
simp [← NNReal.coe_le_coe, hC x]
|
import Mathlib.Algebra.Ring.Equiv
#align_import algebra.ring.comp_typeclasses from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
variable {R₁ : Type*} {R₂ : Type*} {R₃ : Type*}
variable [Semiring R₁] [Semiring R₂] [Semiring R₃]
-- This at first seems not very useful. However we need ... | Mathlib/Algebra/Ring/CompTypeclasses.lean | 106 | 108 | theorem comp_apply_eq₂ {x : R₂} : σ (σ' x) = x := by |
rw [← RingHom.comp_apply, comp_eq₂]
simp
|
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 171 | 180 | theorem nnnorm_diagonal [DecidableEq n] (v : n → α) : ‖diagonal v‖₊ = ‖v‖₊ := by |
simp_rw [nnnorm_def, Pi.nnnorm_def]
congr 1 with i : 1
refine le_antisymm (Finset.sup_le fun j hj => ?_) ?_
· obtain rfl | hij := eq_or_ne i j
· rw [diagonal_apply_eq]
· rw [diagonal_apply_ne _ hij, nnnorm_zero]
exact zero_le _
· refine Eq.trans_le ?_ (Finset.le_sup (Finset.mem_univ i))
rw ... |
import Mathlib.CategoryTheory.Limits.Filtered
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.DiscreteCategory
#align_import category_theory.limits.opposites from "leanprover-community/mathlib"@"ac3ae212f394f508df43e37aa09372... | Mathlib/CategoryTheory/Limits/Opposites.lean | 528 | 533 | theorem opProductIsoCoproduct_inv_comp_lift {X : C} (π : (a : α) → X ⟶ Z a) :
(opProductIsoCoproduct Z).inv ≫ (Pi.lift π).op = Sigma.desc (fun a ↦ (π a).op) := by |
convert opProductIsoCoproduct'_inv_comp_lift (productIsProduct Z)
(coproductIsCoproduct (op <| Z ·)) (Fan.mk _ π)
· ext; simp [Pi.lift, productIsProduct]
· ext; simp [Sigma.desc, coproductIsCoproduct]
|
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 271 | 276 | theorem norm_le_norm_mul_left (x : ℤ[i]) {y : ℤ[i]} (hy : y ≠ 0) :
(norm x).natAbs ≤ (norm (x * y)).natAbs := by |
rw [Zsqrtd.norm_mul, Int.natAbs_mul]
exact le_mul_of_one_le_right (Nat.zero_le _) (Int.ofNat_le.1 (by
rw [abs_natCast_norm]
exact Int.add_one_le_of_lt (norm_pos.2 hy)))
|
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.Logic.Equiv.Basic
#align_import category_theory.limits.has_limits from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
noncomputable sec... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 272 | 276 | theorem limit.lift_map {F G : J ⥤ C} [HasLimit F] [HasLimit G] (c : Cone F) (α : F ⟶ G) :
limit.lift F c ≫ limMap α = limit.lift G ((Cones.postcompose α).obj c) := by |
ext
rw [assoc, limMap_π, limit.lift_π_assoc, limit.lift_π]
rfl
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Sort
#align_import data.polynomial.basic from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
set_option linter.uppercaseLean3 false
noncomputable section
structure ... | Mathlib/Algebra/Polynomial/Basic.lean | 271 | 272 | theorem toFinsupp_eq_one {a : R[X]} : a.toFinsupp = 1 ↔ a = 1 := by |
rw [← toFinsupp_one, toFinsupp_inj]
|
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 65 | 69 | theorem not_countably_generated_nhds_infty_opc : ¬IsCountablyGenerated (𝓝 (∞ : ℚ∞)) := by |
intro
have : IsCountablyGenerated (comap (OnePoint.some : ℚ → ℚ∞) (𝓝 ∞)) := by infer_instance
rw [OnePoint.comap_coe_nhds_infty, coclosedCompact_eq_cocompact] at this
exact not_countably_generated_cocompact this
|
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 322 | 369 | theorem tendsto_lintegral_norm_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ}
(F_measurable : ∀ n, AEStronglyMeasurable (F n) μ)
(bound_hasFiniteIntegral : HasFiniteIntegral bound μ)
(h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
(h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f... |
have f_measurable : AEStronglyMeasurable f μ :=
aestronglyMeasurable_of_tendsto_ae _ F_measurable h_lim
let b a := 2 * ENNReal.ofReal (bound a)
/- `‖F n a‖ ≤ bound a` and `F n a --> f a` implies `‖f a‖ ≤ bound a`, and thus by the
triangle inequality, have `‖F n a - f a‖ ≤ 2 * (bound a)`. -/
have hb : ∀... |
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section TopologicalGroup
variable [CommGroup α] [TopologicalSpace α] [TopologicalGroup α]
variable {f g : β → α} {a a₁... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 175 | 178 | theorem tprod_eq_mul_tprod_ite [DecidableEq β] (hf : Multipliable f) (b : β) :
∏' n, f n = f b * ∏' n, ite (n = b) 1 (f n) := by |
rw [(hasProd_ite_div_hasProd hf.hasProd b).tprod_eq]
exact (mul_div_cancel _ _).symm
|
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 314 | 315 | theorem imageSubobject_arrow' :
(imageSubobjectIso f).inv ≫ (imageSubobject f).arrow = image.ι f := by | simp [imageSubobjectIso]
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 403 | 405 | theorem mem_closure_pi {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
{s : ∀ i, Set (α i)} {x : ∀ i, α i} : x ∈ closure (pi I s) ↔ ∀ i ∈ I, x i ∈ closure (s i) := by |
simp only [mem_closure_iff_nhdsWithin_neBot, nhdsWithin_pi_neBot]
|
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder α] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 143 | 145 | theorem Ico_disjoint_Ico : Disjoint (Ico a₁ a₂) (Ico b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁ := by |
simp_rw [Set.disjoint_iff_inter_eq_empty, Ico_inter_Ico, Ico_eq_empty_iff, inf_eq_min, sup_eq_max,
not_lt]
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 316 | 319 | theorem mem_direction_iff_eq_vsub_left {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (v : V) :
v ∈ s.direction ↔ ∃ p2 ∈ s, v = p -ᵥ p2 := by |
rw [← SetLike.mem_coe, coe_direction_eq_vsub_set_left hp]
exact ⟨fun ⟨p2, hp2, hv⟩ => ⟨p2, hp2, hv.symm⟩, fun ⟨p2, hp2, hv⟩ => ⟨p2, hp2, hv.symm⟩⟩
|
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ι : Type*} {R : Type*} {M₁ M₂ N₁ N₂ : Type*} {Mᵢ Nᵢ : ι → Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 313 | 325 | theorem anisotropic_of_pi [Fintype ι] {R} [OrderedCommRing R] [∀ i, Module R (Mᵢ i)]
{Q : ∀ i, QuadraticForm R (Mᵢ i)} (h : (pi Q).Anisotropic) : ∀ i, (Q i).Anisotropic := by |
simp_rw [Anisotropic, pi_apply, Function.funext_iff, Pi.zero_apply] at h
intro i x hx
classical
have := h (Pi.single i x) ?_ i
· rw [Pi.single_eq_same] at this
exact this
apply Finset.sum_eq_zero
intro j _
by_cases hji : j = i
· subst hji; rw [Pi.single_eq_same, hx]
· rw [Pi.single_eq_of_ne hji... |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 368 | 371 | theorem le_natTrailingDegree (hp : p ≠ 0) (hn : ∀ m < n, p.coeff m = 0) :
n ≤ p.natTrailingDegree := by |
rw [natTrailingDegree_eq_support_min' hp]
exact Finset.le_min' _ _ _ fun m hm => not_lt.1 fun hmn => mem_support_iff.1 hm <| hn _ hmn
|
import Mathlib.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Noetherian
#align_import ring_theory.ideal.associated_prime from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [... | Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 132 | 142 | theorem biUnion_associatedPrimes_eq_zero_divisors [IsNoetherianRing R] :
⋃ p ∈ associatedPrimes R M, p = { r : R | ∃ x : M, x ≠ 0 ∧ r • x = 0 } := by |
simp_rw [← Submodule.mem_annihilator_span_singleton]
refine subset_antisymm (Set.iUnion₂_subset ?_) ?_
· rintro _ ⟨h, x, ⟨⟩⟩ r h'
refine ⟨x, ne_of_eq_of_ne (one_smul R x).symm ?_, h'⟩
refine mt (Submodule.mem_annihilator_span_singleton _ _).mpr ?_
exact (Ideal.ne_top_iff_one _).mp h.ne_top
· intro ... |
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTh... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 72 | 73 | theorem cyclotomic'_zero (R : Type*) [CommRing R] [IsDomain R] : cyclotomic' 0 R = 1 := by |
simp only [cyclotomic', Finset.prod_empty, primitiveRoots_zero]
|
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
suppress_compilation
variable (𝕜 A : Type*) [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A]
variable [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A]
open ContinuousLinearMap
namespace Unitizati... | Mathlib/Analysis/NormedSpace/Unitization.lean | 185 | 190 | theorem uniformity_eq_aux :
𝓤[instUniformSpaceProd.comap <| addEquiv 𝕜 A] = 𝓤 (Unitization 𝕜 A) := by |
have key : UniformInducing (addEquiv 𝕜 A) :=
antilipschitzWith_addEquiv.uniformInducing lipschitzWith_addEquiv.uniformContinuous
rw [← key.comap_uniformity]
rfl
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 639 | 640 | theorem single_eq_of_ne {i i' b} (h : i ≠ i') : (single i b : Π₀ i, β i) i' = 0 := by |
simp only [single_apply, dif_neg h]
|
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
universe u
variable {α : Type u}
def invertibleNeg [Mul α] [One α] [HasDistribNeg α] (a : α) [Invertible a] : Invertible... | Mathlib/Algebra/Ring/Invertible.lean | 54 | 62 | theorem Ring.inverse_add_inverse [Semiring α] {a b : α} (h : IsUnit a ↔ IsUnit b) :
Ring.inverse a + Ring.inverse b = Ring.inverse a * (a + b) * Ring.inverse b := by |
by_cases ha : IsUnit a
· have hb := h.mp ha
obtain ⟨ia⟩ := ha.nonempty_invertible
obtain ⟨ib⟩ := hb.nonempty_invertible
simp_rw [inverse_invertible, invOf_add_invOf]
· have hb := h.not.mp ha
simp [inverse_non_unit, ha, hb]
|
import Mathlib.CategoryTheory.Limits.Filtered
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.DiscreteCategory
#align_import category_theory.limits.opposites from "leanprover-community/mathlib"@"ac3ae212f394f508df43e37aa09372... | Mathlib/CategoryTheory/Limits/Opposites.lean | 671 | 672 | theorem unop_inr {X Y Z : Cᵒᵖ} {f : X ⟶ Z} {g : Y ⟶ Z} (c : PullbackCone f g) :
c.unop.inr = c.snd.unop := by | aesop_cat
|
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 156 | 158 | theorem Ici_eq_finset_subtype : Ici a = (Icc (a : ℕ) n).fin n := by |
ext
simp
|
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 183 | 184 | theorem compRel_assoc {r s t : Set (α × α)} : r ○ s ○ t = r ○ (s ○ t) := by |
ext ⟨a, b⟩; simp only [mem_compRel]; tauto
|
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 110 | 111 | theorem frontier_sphere (x : E) {r : ℝ} (hr : r ≠ 0) : frontier (sphere x r) = sphere x r := by |
rw [isClosed_sphere.frontier_eq, interior_sphere x hr, diff_empty]
|
import Mathlib.Control.Monad.Basic
import Mathlib.Control.Monad.Writer
import Mathlib.Init.Control.Lawful
#align_import control.monad.cont from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
universe u v w u₀ u₁ v₀ v₁
structure MonadCont.Label (α : Type w) (m : Type u → Type v) (β : Typ... | Mathlib/Control/Monad/Cont.lean | 128 | 130 | theorem ExceptT.goto_mkLabel {α β ε : Type _} (x : Label (Except.{u, u} ε α) m β) (i : α) :
goto (ExceptT.mkLabel x) i = ExceptT.mk (Except.ok <$> goto x (Except.ok i)) := by |
cases x; rfl
|
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