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.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_import number_theory.liouville.liouville_with from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Filter Metric Real Set
open sc... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 203 | 203 | theorem rat_add_iff : LiouvilleWith p (r + x) ↔ LiouvilleWith p x := by | rw [add_comm, add_rat_iff]
|
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section ConditionallyCompleteLinearOrder
variable [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [Top... | Mathlib/Topology/Order/Monotone.lean | 92 | 96 | theorem Antitone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iSup g)) (Af : Antitone f)
(bdd : BddAbove (range g) := by | bddDefault) : f (⨆ i, g i) = ⨅ i, f (g i) := by
rw [iSup, Antitone.map_sSup_of_continuousAt' Cf Af (range_nonempty g) bdd, ← range_comp, iInf]
rfl
|
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.add from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal F... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 327 | 329 | theorem fderivWithin_const_add (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) :
fderivWithin 𝕜 (fun y => c + f y) s x = fderivWithin 𝕜 f s x := by |
simpa only [add_comm] using fderivWithin_add_const hxs c
|
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 139 | 147 | theorem int_valuation_le_pow_iff_dvd (r : R) (n : ℕ) :
v.intValuationDef r ≤ Multiplicative.ofAdd (-(n : ℤ)) ↔ v.asIdeal ^ n ∣ Ideal.span {r} := by |
rw [intValuationDef]
split_ifs with hr
· simp_rw [hr, Ideal.dvd_span_singleton, zero_le', Submodule.zero_mem]
· rw [WithZero.coe_le_coe, ofAdd_le, neg_le_neg_iff, Int.ofNat_le, Ideal.dvd_span_singleton, ←
Associates.le_singleton_iff,
Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hr)
... |
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vandermonde
import Mathlib.Tactic.FieldSimp
#align_import data.polynomial.hasse_deriv from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 127 | 129 | theorem hasseDeriv_C (r : R) (hk : 0 < k) : hasseDeriv k (C r) = 0 := by |
rw [← monomial_zero_left, hasseDeriv_monomial, Nat.choose_eq_zero_of_lt hk, Nat.cast_zero,
zero_mul, monomial_zero_right]
|
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 583 | 587 | theorem tendsto_measure_iUnion' {α ι : Type*} [MeasurableSpace α] {μ : Measure α} [Countable ι]
[Preorder ι] [IsDirected ι (· ≤ ·)] {f : ι → Set α} :
Tendsto (fun i ↦ μ (Accumulate f i)) atTop (𝓝 (μ (⋃ i, f i))) := by |
rw [measure_iUnion_eq_iSup']
exact tendsto_atTop_iSup fun i j hij ↦ by gcongr
|
import Mathlib.Data.List.Sigma
#align_import data.list.alist from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v w
open List
variable {α : Type u} {β : α → Type v}
structure AList (β : α → Type v) : Type max u v where
entries : List (Sigma β)
nodupKeys : entri... | Mathlib/Data/List/AList.lean | 183 | 190 | theorem keys_subset_keys_of_entries_subset_entries
{s₁ s₂ : AList β} (h : s₁.entries ⊆ s₂.entries) : s₁.keys ⊆ s₂.keys := by |
intro k hk
letI : DecidableEq α := Classical.decEq α
have := h (mem_lookup_iff.1 (Option.get_mem (lookup_isSome.2 hk)))
rw [← mem_lookup_iff, Option.mem_def] at this
rw [← mem_keys, ← lookup_isSome, this]
exact Option.isSome_some
|
import Mathlib.Topology.Bornology.Basic
#align_import topology.bornology.constructions from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
open Set Filter Bornology Function
open Filter
variable {α β ι : Type*} {π : ι → Type*} [Bornology α] [Bornology β]
[∀ i, Bornology (π i)]
inst... | Mathlib/Topology/Bornology/Constructions.lean | 126 | 131 | theorem isBounded_pi : IsBounded (pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ i, IsBounded (S i) := by |
by_cases hne : ∃ i, S i = ∅
· simp [hne, univ_pi_eq_empty_iff.2 hne]
· simp only [hne, false_or_iff]
simp only [not_exists, ← Ne.eq_def, ← nonempty_iff_ne_empty, ← univ_pi_nonempty_iff] at hne
exact isBounded_pi_of_nonempty hne
|
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.AdaptationNote
#align_import data.subtype from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
#align_import init.data.subtype.basic from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
open Function
namesp... | Mathlib/Data/Subtype.lean | 169 | 171 | theorem restrict_apply {α} {β : α → Type*} (f : ∀ x, β x) (p : α → Prop) (x : Subtype p) :
restrict p f x = f x.1 := by |
rfl
|
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Nat.Lattice
#align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2"
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
noncomputable def dist (u v : V)... | Mathlib/Combinatorics/SimpleGraph/Metric.lean | 99 | 102 | theorem nonempty_of_pos_dist {u v : V} (h : 0 < G.dist u v) :
(Set.univ : Set (G.Walk u v)).Nonempty := by |
simpa [Set.range_nonempty_iff_nonempty, Set.nonempty_iff_univ_nonempty] using
Nat.nonempty_of_pos_sInf h
|
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 163 | 176 | theorem ContinuousLinearMap.continuous_det : Continuous fun f : E →L[𝕜] E => f.det := by |
change Continuous fun f : E →L[𝕜] E => LinearMap.det (f : E →ₗ[𝕜] E)
-- Porting note: this could be easier with `det_cases`
by_cases h : ∃ s : Finset E, Nonempty (Basis (↥s) 𝕜 E)
· rcases h with ⟨s, ⟨b⟩⟩
haveI : FiniteDimensional 𝕜 E := FiniteDimensional.of_fintype_basis b
simp_rw [LinearMap.det_eq... |
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 | 81 | 86 | theorem continuousAt_zpow {𝕜 : Type*} [NontriviallyNormedField 𝕜] {m : ℤ} {x : 𝕜} :
ContinuousAt (fun x ↦ x ^ m) x ↔ x ≠ 0 ∨ 0 ≤ m := by |
refine ⟨?_, continuousAt_zpow₀ _ _⟩
contrapose!; rintro ⟨rfl, hm⟩ hc
exact not_tendsto_atTop_of_tendsto_nhds (hc.tendsto.mono_left nhdsWithin_le_nhds).norm
(tendsto_norm_zpow_nhdsWithin_0_atTop hm)
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.IndicatorConstPointwise
#align_import measure_theory.constructions.borel_space.metrizable from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Filter MeasureT... | Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean | 31 | 47 | theorem measurable_of_tendsto_metrizable' {ι} {f : ι → α → β} {g : α → β} (u : Filter ι) [NeBot u]
[IsCountablyGenerated u] (hf : ∀ i, Measurable (f i)) (lim : Tendsto f u (𝓝 g)) :
Measurable g := by |
letI : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
apply measurable_of_isClosed'
intro s h1s h2s h3s
have : Measurable fun x => infNndist (g x) s := by
suffices Tendsto (fun i x => infNndist (f i x) s) u (𝓝 fun x => infNndist (g x) s) from
NNReal.measurable_of_tendsto' u (fun i => (hf... |
import Batteries.Data.List.Lemmas
import Batteries.Tactic.Classical
import Mathlib.Tactic.TypeStar
import Mathlib.Mathport.Rename
#align_import data.list.tfae from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
namespace List
def TFAE (l : List Prop) : Prop :=
∀ x ∈ l, ∀ y ∈ l, x ↔ ... | Mathlib/Data/List/TFAE.lean | 110 | 115 | theorem exists_tfae {α : Type*} (l : List (α → Prop)) (H : ∀ a : α, (l.map (fun p ↦ p a)).TFAE) :
(l.map (fun p ↦ ∃ a, p a)).TFAE := by |
simp only [TFAE, List.forall_mem_map_iff]
intros p₁ hp₁ p₂ hp₂
exact exists_congr fun a ↦ H a (p₁ a) (mem_map_of_mem (fun p ↦ p a) hp₁)
(p₂ a) (mem_map_of_mem (fun p ↦ p a) hp₂)
|
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 60 | 73 | theorem definable_iff_exists_formula_sum :
A.Definable L s ↔ ∃ φ : L.Formula (A ⊕ α), s = {v | φ.Realize (Sum.elim (↑) v)} := by |
rw [Definable, Equiv.exists_congr_left (BoundedFormula.constantsVarsEquiv)]
refine exists_congr (fun φ => iff_iff_eq.2 (congr_arg (s = ·) ?_))
ext
simp only [Formula.Realize, BoundedFormula.constantsVarsEquiv, constantsOn, mk₂_Relations,
BoundedFormula.mapTermRelEquiv_symm_apply, mem_setOf_eq]
refine Bou... |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {𝕜 E : Type*}
variable [NormedField 𝕜]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 237 | 239 | theorem disjoint_closedBall_ball_iff (hδ : 0 ≤ δ) (hε : 0 < ε) :
Disjoint (closedBall x δ) (ball y ε) ↔ δ + ε ≤ dist x y := by |
rw [disjoint_comm, disjoint_ball_closedBall_iff hε hδ, add_comm, dist_comm]
|
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : ℕ} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 72 | 75 | theorem modPart_lt_p : modPart p r < p := by |
convert Int.emod_lt _ _
· simp
· exact mod_cast hp_prime.1.ne_zero
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 933 | 934 | theorem coe_abs_toReal_of_sign_nonneg {θ : Angle} (h : 0 ≤ θ.sign) : ↑|θ.toReal| = θ := by |
rw [abs_eq_self.2 (toReal_nonneg_iff_sign_nonneg.2 h), coe_toReal]
|
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform_space.cauchy from "leanprover-community/mathlib"@"22131150f88a2d125713ffa0f4693e3355b1eb49"
universe u v
open scoped Classical
open Filter TopologicalSpace Set Uni... | Mathlib/Topology/UniformSpace/Cauchy.lean | 316 | 324 | theorem cauchySeq_shift {u : ℕ → α} (k : ℕ) : CauchySeq (fun n ↦ u (n + k)) ↔ CauchySeq u := by |
constructor <;> intro h
· rw [cauchySeq_iff] at h ⊢
intro V mV
obtain ⟨N, h⟩ := h V mV
use N + k
intro a ha b hb
convert h (a - k) (Nat.le_sub_of_add_le ha) (b - k) (Nat.le_sub_of_add_le hb) <;> omega
· exact h.comp_tendsto (tendsto_add_atTop_nat k)
|
import Mathlib.Topology.Gluing
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits
#align_import algebraic_geometry.presheafed_space.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean... | Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean | 251 | 274 | theorem ι_image_preimage_eq (i j : D.J) (U : Opens (D.U i).carrier) :
(Opens.map (𝖣.ι j).base).obj ((D.ι_openEmbedding i).isOpenMap.functor.obj U) =
(D.f_open j i).openFunctor.obj
((Opens.map (𝖣.t j i).base).obj ((Opens.map (𝖣.f i j).base).obj U)) := by |
ext1
dsimp only [Opens.map_coe, IsOpenMap.functor_obj_coe]
rw [← show _ = (𝖣.ι i).base from 𝖣.ι_gluedIso_inv (PresheafedSpace.forget _) i, ←
show _ = (𝖣.ι j).base from 𝖣.ι_gluedIso_inv (PresheafedSpace.forget _) j]
-- Porting note (#11224): change `rw` to `erw` on `coe_comp`
erw [coe_comp, coe_comp, ... |
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.basic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical Topology Filter ENNReal ... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 772 | 775 | theorem HasDerivAt.le_of_lip' {f : 𝕜 → F} {f' : F} {x₀ : 𝕜} (hf : HasDerivAt f f' x₀)
{C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) :
‖f'‖ ≤ C := by |
simpa using HasFDerivAt.le_of_lip' hf.hasFDerivAt hC₀ hlip
|
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover-community/mathlib"@"ed90a7d327c3a5caf65a6faf7e8a0d63c4605df7"
open IsLocalization Pointwise nonZeroDivisors
namespace FractionalIdeal
open Set Submodule
variable... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 123 | 124 | theorem map_map_symm (g : P ≃ₐ[R] P') : (I.map (g : P →ₐ[R] P')).map (g.symm : P' →ₐ[R] P) = I := by |
rw [← map_comp, g.symm_comp, map_id]
|
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Baire.LocallyCompactRegular
import Mathlib.Topology.Instances.Irrational
#align_import number_theory.liouville.residual from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
open scope... | Mathlib/NumberTheory/Liouville/Residual.lean | 59 | 72 | theorem eventually_residual_liouville : ∀ᶠ x in residual ℝ, Liouville x := by |
rw [Filter.Eventually, setOf_liouville_eq_irrational_inter_iInter_iUnion]
refine eventually_residual_irrational.and ?_
refine residual_of_dense_Gδ ?_ (Rat.denseEmbedding_coe_real.dense.mono ?_)
· exact .iInter fun n => IsOpen.isGδ <|
isOpen_iUnion fun a => isOpen_iUnion fun b => isOpen_iUnion fun _hb... |
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Order.Archimedean
import Mathlib.GroupTheory.Coset
#align_import algebra.periodic from "leanprover-community/mathlib"@"3041... | Mathlib/Algebra/Periodic.lean | 333 | 336 | theorem Periodic.map_vadd_multiples [AddCommMonoid α] (hf : Periodic f c)
(a : AddSubmonoid.multiples c) (x : α) : f (a +ᵥ x) = f x := by |
rcases a with ⟨_, m, rfl⟩
simp [AddSubmonoid.vadd_def, add_comm _ x, hf.nsmul m x]
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
#align_import linear_algebra.matrix.hermitian from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
namespace Matrix
variable {α β : Type*} {m n : Type*} {A : Matrix n n α}
open scoped Matrix
local ... | Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 56 | 57 | theorem IsHermitian.ext {A : Matrix n n α} : (∀ i j, star (A j i) = A i j) → A.IsHermitian := by |
intro h; ext i j; exact h i j
|
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd"
namespace Polynomial
open Polynomial Finsupp Finset
open... | Mathlib/Algebra/Polynomial/Reverse.lean | 133 | 135 | theorem reflect_add (f g : R[X]) (N : ℕ) : reflect N (f + g) = reflect N f + reflect N g := by |
ext
simp only [coeff_add, coeff_reflect]
|
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 | 79 | 81 | theorem convexJoin_union_left (s₁ s₂ t : Set E) :
convexJoin 𝕜 (s₁ ∪ s₂) t = convexJoin 𝕜 s₁ t ∪ convexJoin 𝕜 s₂ t := by |
simp_rw [convexJoin, mem_union, iUnion_or, iUnion_union_distrib]
|
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 363 | 364 | theorem ι_gluedIso_inv (i : D.J) : (D.mapGlueData F).ι i ≫ (D.gluedIso F).inv = F.map (D.ι i) := by |
rw [Iso.comp_inv_eq, ι_gluedIso_hom]
|
import Mathlib.Logic.Relation
import Mathlib.Order.GaloisConnection
#align_import data.setoid.basic from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
variable {α : Type*} {β : Type*}
def Setoid.Rel (r : Setoid α) : α → α → Prop :=
@Setoid.r _ r
#align setoid.rel Setoid.Rel
instanc... | Mathlib/Data/Setoid/Basic.lean | 319 | 323 | theorem lift_unique {r : Setoid α} {f : α → β} (H : r ≤ ker f) (g : Quotient r → β)
(Hg : f = g ∘ Quotient.mk'') : Quotient.lift f H = g := by |
ext ⟨x⟩
erw [Quotient.lift_mk f H, Hg]
rfl
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,520 | 1,525 | theorem set_lintegral_dirac {a : α} (f : α → ℝ≥0∞) (s : Set α) [MeasurableSingletonClass α]
[Decidable (a ∈ s)] : ∫⁻ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 := by |
rw [restrict_dirac]
split_ifs
· exact lintegral_dirac _ _
· exact lintegral_zero_measure _
|
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic.Monotonicity
#align_import number_theory.pell_matiyasevic from "leanprover-community/mathlib"@"795b501869b9f... | Mathlib/NumberTheory/PellMatiyasevic.lean | 565 | 568 | theorem x_sub_y_dvd_pow_lem (y2 y1 y0 yn1 yn0 xn1 xn0 ay a2 : ℤ) :
(a2 * yn1 - yn0) * ay + y2 - (a2 * xn1 - xn0) =
y2 - a2 * y1 + y0 + a2 * (yn1 * ay + y1 - xn1) - (yn0 * ay + y0 - xn0) := by |
ring
|
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
#align_import topology.vector_bundle.constructions from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Cl... | Mathlib/Topology/VectorBundle/Constructions.lean | 50 | 55 | theorem trivialization.coordChangeL (b : B) :
(trivialization B F).coordChangeL 𝕜 (trivialization B F) b =
ContinuousLinearEquiv.refl 𝕜 F := by |
ext v
rw [Trivialization.coordChangeL_apply']
exacts [rfl, ⟨mem_univ _, mem_univ _⟩]
|
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[Topolo... | Mathlib/Topology/Inseparable.lean | 267 | 269 | theorem not_inseparable_iff_exists_open :
¬(x ~ᵢ y) ↔ ∃ s : Set X, IsOpen s ∧ Xor' (x ∈ s) (y ∈ s) := by |
simp [inseparable_iff_forall_open, ← xor_iff_not_iff]
|
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 276 | 276 | theorem top_div_of_ne_top (h : a ≠ ∞) : ∞ / a = ∞ := by | simp [top_div, h]
|
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Data.ZMod.Quotient
import Mathlib.RingTheory.DedekindDomain.AdicValuation
#align_import ring_theory.dedekind_domain.selmer_group from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
set_option quotPrecheck false
local notation K "... | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | 120 | 131 | theorem valuation_of_unit_eq (x : Rˣ) :
v.valuationOfNeZero (Units.map (algebraMap R K : R →* K) x) = 1 := by |
rw [← WithZero.coe_inj, valuationOfNeZero_eq, Units.coe_map, eq_iff_le_not_lt]
constructor
· exact v.valuation_le_one x
· cases' x with x _ hx _
change ¬v.valuation (algebraMap R K x) < 1
apply_fun v.intValuation at hx
rw [map_one, map_mul] at hx
rw [not_lt, ← hx, ← mul_one <| v.valuation _, va... |
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.ApplyFun
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import category_theory.subobject.basic from "leanprover-community/mathlib"@"70fd9563a21e7b... | Mathlib/CategoryTheory/Subobject/Basic.lean | 399 | 402 | theorem ofLEMk_comp_ofMkLE {B A : C} (X : Subobject B) (f : A ⟶ B) [Mono f] (Y : Subobject B)
(h₁ : X ≤ mk f) (h₂ : mk f ≤ Y) : ofLEMk X f h₁ ≫ ofMkLE f Y h₂ = ofLE X Y (h₁.trans h₂) := by |
simp only [ofMkLE, ofLEMk, ofLE, ← Functor.map_comp underlying, assoc, Iso.hom_inv_id_assoc]
congr 1
|
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
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
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 72 | 74 | theorem mul_self : t * t = 1 := by |
rcases ht with ⟨w, i, rfl⟩
simp
|
import Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
#align_import geometry.manifold.bump_function from "leanprover-community/mathlib"@"b018406ad2f2a73223a3a9e198ccae61e6f05318"
universe uE uF uH uM
va... | Mathlib/Geometry/Manifold/BumpFunction.lean | 334 | 344 | theorem smooth_smul {G} [NormedAddCommGroup G] [NormedSpace ℝ G] {g : M → G}
(hg : SmoothOn I 𝓘(ℝ, G) g (chartAt H c).source) : Smooth I 𝓘(ℝ, G) fun x => f x • g x := by |
refine contMDiff_of_tsupport fun x hx => ?_
have : x ∈ (chartAt H c).source :=
-- Porting note: was a more readable `calc`
-- calc
-- x ∈ tsupport fun x => f x • g x := hx
-- _ ⊆ tsupport f := tsupport_smul_subset_left _ _
-- _ ⊆ (chart_at _ c).source := f.tsupport_subset_chartAt_source
f.tsupp... |
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 | 172 | 174 | theorem map₂_map_left (m : γ → β → δ) (n : α → γ) :
map₂ m (f.map n) g = map₂ (fun a b => m (n a) b) f g := by |
rw [← map_prod_eq_map₂, ← map_prod_eq_map₂, ← @map_id _ g, prod_map_map_eq, map_map, map_id]; rfl
|
import Mathlib.Algebra.Associated
import Mathlib.NumberTheory.Divisors
#align_import algebra.is_prime_pow from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
def IsPrimePow : Prop :=
∃ (p : R) (k : ℕ), Prime p ∧ 0 < k ∧ p... | Mathlib/Algebra/IsPrimePow.lean | 76 | 77 | theorem isPrimePow_nat_iff (n : ℕ) : IsPrimePow n ↔ ∃ p k : ℕ, Nat.Prime p ∧ 0 < k ∧ p ^ k = n := by |
simp only [isPrimePow_def, Nat.prime_iff]
|
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 76 | 80 | theorem edgeSet_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeSet =
G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s) := by |
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
|
import Mathlib.Order.Filter.Bases
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.filter_basis from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Set TopologicalSpace Function
open Topology Filter Pointwise
universe u
class GroupFilterBasis (... | Mathlib/Topology/Algebra/FilterBasis.lean | 197 | 200 | theorem nhds_hasBasis (B : GroupFilterBasis G) (x₀ : G) :
HasBasis (@nhds G B.topology x₀) (fun V : Set G ↦ V ∈ B) fun V ↦ (fun y ↦ x₀ * y) '' V := by |
rw [B.nhds_eq]
apply B.hasBasis
|
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Atoms
def Order.radical (α : Type*) [Preorder α] [OrderTop α] [InfSet α] : α :=
⨅ a ∈ {H | IsCoatom H}, a
variable {α : Type*} [CompleteLattice α]
lemma Order.radical_le_coatom {a : α} (h : IsCoatom a) : radical α ≤ a := biInf_le _ h
variable {β : Typ... | Mathlib/Order/Radical.lean | 30 | 36 | theorem OrderIso.map_radical (f : α ≃o β) : f (Order.radical α) = Order.radical β := by |
unfold Order.radical
simp only [OrderIso.map_iInf]
fapply Equiv.iInf_congr
· exact f.toEquiv
· intros
simp
|
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 165 | 172 | theorem WithDensityᵥEq.congr_ae {f g : α → E} (h : f =ᵐ[μ] g) :
μ.withDensityᵥ f = μ.withDensityᵥ g := by |
by_cases hf : Integrable f μ
· ext i hi
rw [withDensityᵥ_apply hf hi, withDensityᵥ_apply (hf.congr h) hi]
exact integral_congr_ae (ae_restrict_of_ae h)
· have hg : ¬Integrable g μ := by intro hg; exact hf (hg.congr h.symm)
rw [withDensityᵥ, withDensityᵥ, dif_neg hf, dif_neg hg]
|
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 785 | 834 | theorem tendsto_addHaar_inter_smul_one_of_density_one_aux (s : Set E) (hs : MeasurableSet s)
(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) (h''t : μ t ≠ ∞) :
Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x... |
have I : ∀ u v, μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v →
μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u := by
intro u v uzero utop vmeas
simp_rw [div_eq_mul_inv]
rw [← ENNReal.sub_mul]; swap
· simp only [uzero, ENNReal.inv_eq_top, imp_true_iff, Ne, not_false_iff]
congr 1
apply
ENNReal.... |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
section Fintype
variable {α β : Type*} [Fintype α] [DecidableEq β] (e : Equiv.Perm α) (f : α ↪ β)
... | Mathlib/Logic/Equiv/Fintype.lean | 54 | 57 | theorem Function.Embedding.toEquivRange_eq_ofInjective :
f.toEquivRange = Equiv.ofInjective f f.injective := by |
ext
simp
|
import Mathlib.CategoryTheory.Limits.KanExtension
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.Topology.Sheaves.Init
import Mathlib.Data.Set.Subsingleton
#align_import topology.sheaves.presheaf from "leanprover-community/mathlib"@"5dc6092d09e5e4891068652... | Mathlib/Topology/Sheaves/Presheaf.lean | 251 | 253 | theorem id_hom_app' (U) (p) : (id ℱ).hom.app (op ⟨U, p⟩) = ℱ.map (𝟙 (op ⟨U, p⟩)) := by |
dsimp [id]
simp
|
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Pi
#align_import order.filter.cofinite from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Function
variable {ι α β : Type*} {l : Filter α}
namespace Filter
def cofinite : Filter α :=
comk Set.Finite finite_e... | Mathlib/Order/Filter/Cofinite.lean | 143 | 146 | theorem countable_compl_ker [l.IsCountablyGenerated] (h : cofinite ≤ l) : Set.Countable l.kerᶜ := by |
rcases exists_antitone_basis l with ⟨s, hs⟩
simp only [hs.ker, iInter_true, compl_iInter]
exact countable_iUnion fun n ↦ Set.Finite.countable <| h <| hs.mem _
|
import Mathlib.MeasureTheory.Integral.IntegrableOn
#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace Bornology
open scoped Topology Interval ENNReal
variabl... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 83 | 100 | theorem LocallyIntegrableOn.exists_countable_integrableOn [SecondCountableTopology X]
(hf : LocallyIntegrableOn f s μ) : ∃ T : Set (Set X), T.Countable ∧
(∀ u ∈ T, IsOpen u) ∧ (s ⊆ ⋃ u ∈ T, u) ∧ (∀ u ∈ T, IntegrableOn f (u ∩ s) μ) := by |
have : ∀ x : s, ∃ u, IsOpen u ∧ x.1 ∈ u ∧ IntegrableOn f (u ∩ s) μ := by
rintro ⟨x, hx⟩
rcases hf x hx with ⟨t, ht, h't⟩
rcases mem_nhdsWithin.1 ht with ⟨u, u_open, x_mem, u_sub⟩
exact ⟨u, u_open, x_mem, h't.mono_set u_sub⟩
choose u u_open xu hu using this
obtain ⟨T, T_count, hT⟩ : ∃ T : Set s, T... |
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Data.Set.MulAntidiagonal
#align_import data.finset.mul_antidiagonal from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
namespace Set
open Pointwise
variable {α : Type*} {s t : Set α}
@[to_additive]
theorem IsPWO.mul [OrderedCanc... | Mathlib/Data/Finset/MulAntidiagonal.lean | 40 | 45 | theorem IsWF.min_mul (hs : s.IsWF) (ht : t.IsWF) (hsn : s.Nonempty) (htn : t.Nonempty) :
(hs.mul ht).min (hsn.mul htn) = hs.min hsn * ht.min htn := by |
refine le_antisymm (IsWF.min_le _ _ (mem_mul.2 ⟨_, hs.min_mem _, _, ht.min_mem _, rfl⟩)) ?_
rw [IsWF.le_min_iff]
rintro _ ⟨x, hx, y, hy, rfl⟩
exact mul_le_mul' (hs.min_le _ hx) (ht.min_le _ hy)
|
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.Data.Nat.PartENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function
variable {α : Type u}
namespace Cardinal
noncomputable def toPartENat : Cardinal →+o PartEN... | Mathlib/SetTheory/Cardinal/PartENat.lean | 104 | 105 | theorem toPartENat_lift (c : Cardinal.{v}) : toPartENat (lift.{u, v} c) = toPartENat c := by |
simp only [← partENatOfENat_toENat, toENat_lift]
|
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 | 231 | 232 | theorem rpow_one_div_le_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x ^ (1 / z) ≤ y ↔ x ≤ y ^ z := by |
rw [← rpow_le_rpow_iff hz, rpow_self_rpow_inv hz.ne']
|
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Init.Data.Ordering.Lemmas
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.NormNum
#align_import set_theory.ordinal.notation from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
set_option linter.uppercaseLean3 ... | Mathlib/SetTheory/Ordinal/Notation.lean | 157 | 159 | theorem omega_le_oadd (e n a) : ω ^ repr e ≤ repr (oadd e n a) := by |
refine le_trans ?_ (le_add_right _ _)
simpa using (Ordinal.mul_le_mul_iff_left <| opow_pos (repr e) omega_pos).2 (natCast_le.2 n.2)
|
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 | 145 | 150 | theorem Submartingale.stoppedValue_leastGE_snorm_le' [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hr : 0 ≤ r) (hf0 : f 0 = 0) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) (i : ℕ) :
snorm (stoppedValue f (leastGE f r i)) 1 μ ≤
ENNReal.toNNReal (2 * μ Set.univ * ENNReal.ofReal (r + R)) := by |
refine (hf.stoppedValue_leastGE_snorm_le hr hf0 hbdd i).trans ?_
simp [ENNReal.coe_toNNReal (measure_ne_top μ _), ENNReal.coe_toNNReal]
|
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 327 | 340 | theorem T_mul (m n : ℤ) : T R (m * n) = (T R m).comp (T R n) := by |
induction m using Polynomial.Chebyshev.induct with
| zero => simp
| one => simp
| add_two m ih1 ih2 =>
have h₁ := mul_T R ((m + 1) * n) n
have h₂ := congr_arg (comp · (T R n)) <| T_add_two R m
simp only [sub_comp, mul_comp, ofNat_comp, X_comp] at h₂
linear_combination (norm := ring_nf) -ih2 - h... |
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.Tactic.PPWithUniv
import Mathlib.Data.Set.Defs
#align_import category_theory.types from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace CategoryTheory
-- morphism levels be... | Mathlib/CategoryTheory/Types.lean | 157 | 157 | theorem map_id_apply (a : F.obj X) : (F.map (𝟙 X)) a = a := by | simp [types_id]
|
import Mathlib.CategoryTheory.Comma.Basic
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.Logic.Small.Set
#align_import category_theory.structured_arrow from "leanprover-community/mathlib"@"8a318021995877a44630c898d... | Mathlib/CategoryTheory/Comma/StructuredArrow.lean | 102 | 105 | theorem eqToHom_right {X Y : StructuredArrow S T} (h : X = Y) :
(eqToHom h).right = eqToHom (by rw [h]) := by |
subst h
simp only [eqToHom_refl, id_right]
|
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.instances from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
section OrderedSemiring
variable [OrderedSe... | Mathlib/Algebra/Order/Interval/Set/Instances.lean | 201 | 203 | theorem coe_eq_zero [Nontrivial α] {x : Ico (0 : α) 1} : (x : α) = 0 ↔ x = 0 := by |
symm
exact Subtype.ext_iff
|
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
open Finset Function Sym
universe u
variab... | Mathlib/Data/Sym/Sym2.lean | 73 | 74 | theorem Rel.trans {x y z : α × α} (a : Rel α x y) (b : Rel α y z) : Rel α x z := by |
aesop (rule_sets := [Sym2])
|
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 272 | 274 | theorem Chain'.rel_head? {x l} (h : Chain' R (x :: l)) ⦃y⦄ (hy : y ∈ head? l) : R x y := by |
rw [← cons_head?_tail hy] at h
exact h.rel_head
|
import Mathlib.Order.Partition.Equipartition
#align_import combinatorics.simple_graph.regularity.equitabilise from "leanprover-community/mathlib"@"bf7ef0e83e5b7e6c1169e97f055e58a2e4e9d52d"
open Finset Nat
namespace Finpartition
variable {α : Type*} [DecidableEq α] {s t : Finset α} {m n a b : ℕ} {P : Finpartitio... | Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean | 42 | 139 | theorem equitabilise_aux (hs : a * m + b * (m + 1) = s.card) :
∃ Q : Finpartition s,
(∀ x : Finset α, x ∈ Q.parts → x.card = m ∨ x.card = m + 1) ∧
(∀ x, x ∈ P.parts → (x \ (Q.parts.filter fun y => y ⊆ x).biUnion id).card ≤ m) ∧
(Q.parts.filter fun i => card i = m + 1).card = b := by |
-- Get rid of the easy case `m = 0`
obtain rfl | m_pos := m.eq_zero_or_pos
· refine ⟨⊥, by simp, ?_, by simpa [Finset.filter_true_of_mem] using hs.symm⟩
simp only [le_zero_iff, card_eq_zero, mem_biUnion, exists_prop, mem_filter, id, and_assoc,
sdiff_eq_empty_iff_subset, subset_iff]
exact fun x hx a... |
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
... | Mathlib/Data/Finset/Prod.lean | 137 | 139 | theorem product_biUnion [DecidableEq γ] (s : Finset α) (t : Finset β) (f : α × β → Finset γ) :
(s ×ˢ t).biUnion f = s.biUnion fun a => t.biUnion fun b => f (a, b) := by |
classical simp_rw [product_eq_biUnion, biUnion_biUnion, image_biUnion]
|
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
-- Make sure we don't import algebra
assert_not_exists Monoid
universe u
open Nat
namespace List
variable {α : Type u} {β γ δ ε : Type*}
#align list.zip_with_cons_cons Li... | Mathlib/Data/List/Zip.lean | 112 | 112 | theorem unzip_right (l : List (α × β)) : (unzip l).2 = l.map Prod.snd := by | simp only [unzip_eq_map]
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory.measure.content from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
universe u v w
noncomputable section
open Set Topologic... | Mathlib/MeasureTheory/Measure/Content.lean | 324 | 327 | theorem is_mul_left_invariant_outerMeasure [Group G] [TopologicalGroup G]
(h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
(A : Set G) : μ.outerMeasure ((g * ·) ⁻¹' A) = μ.outerMeasure A := by |
convert μ.outerMeasure_preimage (Homeomorph.mulLeft g) (fun K => h g) A
|
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 303 | 309 | theorem sum_geometric_two_le (n : ℕ) : (∑ i ∈ range n, (1 / (2 : ℝ)) ^ i) ≤ 2 := by |
have : ∀ i, 0 ≤ (1 / (2 : ℝ)) ^ i := by
intro i
apply pow_nonneg
norm_num
convert sum_le_tsum (range n) (fun i _ ↦ this i) summable_geometric_two
exact tsum_geometric_two.symm
|
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "�... | Mathlib/Analysis/RCLike/Basic.lean | 541 | 542 | theorem inv_re (z : K) : re z⁻¹ = re z / normSq z := by |
rw [inv_def, normSq_eq_def', mul_comm, re_ofReal_mul, conj_re, div_eq_inv_mul]
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
open Nat
namespace List
@[simp]
theo... | Mathlib/Data/List/Sublists.lean | 76 | 78 | theorem sublists'_cons (a : α) (l : List α) :
sublists' (a :: l) = sublists' l ++ map (cons a) (sublists' l) := by |
simp [sublists'_eq_sublists'Aux, foldr_cons, sublists'Aux_eq_map]
|
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
#align_import analysis.special_functions.trigonometric.complex from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable secti... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 75 | 76 | theorem tan_ne_zero_iff {θ : ℂ} : tan θ ≠ 0 ↔ ∀ k : ℤ, (k * π / 2 : ℂ) ≠ θ := by |
rw [← not_exists, not_iff_not, tan_eq_zero_iff]
|
import Batteries.Data.HashMap.Basic
import Batteries.Data.Array.Lemmas
import Batteries.Data.Nat.Lemmas
namespace Batteries.HashMap
namespace Imp
attribute [-simp] Bool.not_eq_true
namespace Buckets
@[ext] protected theorem ext : ∀ {b₁ b₂ : Buckets α β}, b₁.1.data = b₂.1.data → b₁ = b₂
| ⟨⟨_⟩, _⟩, ⟨⟨_⟩, _⟩, rfl ... | .lake/packages/batteries/Batteries/Data/HashMap/WF.lean | 38 | 40 | theorem mk_size (h) : (mk n h : Buckets α β).size = 0 := by |
simp only [mk, mkArray, size_eq]; clear h
induction n <;> simp [*]
|
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 445 | 446 | theorem Ico_subset_Iio_self : Ico a b ⊆ Iio b := by |
simpa [← coe_subset] using Set.Ico_subset_Iio_self
|
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Set
open Filter
namespace Filter
variable {α β γ δ : Type*} {ι : Sort*}
section Prod
variable {s : Set α} {t : Set β} {f : Filter α} {g : Filter β}
protected ... | Mathlib/Order/Filter/Prod.lean | 308 | 313 | theorem prod_inj {f₁ f₂ : Filter α} {g₁ g₂ : Filter β} [NeBot f₁] [NeBot g₁] :
f₁ ×ˢ g₁ = f₂ ×ˢ g₂ ↔ f₁ = f₂ ∧ g₁ = g₂ := by |
refine ⟨fun h => ?_, fun h => h.1 ▸ h.2 ▸ rfl⟩
have hle : f₁ ≤ f₂ ∧ g₁ ≤ g₂ := prod_le_prod.1 h.le
haveI := neBot_of_le hle.1; haveI := neBot_of_le hle.2
exact ⟨hle.1.antisymm <| (prod_le_prod.1 h.ge).1, hle.2.antisymm <| (prod_le_prod.1 h.ge).2⟩
|
import Mathlib.Order.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
open Function OrderDual
universe u
variable {ι α β : Type*}
section
variable (α β)
instance Prod.instHImp [HImp α] [HImp β] : HImp (α × β) :=
⟨fun a b => (a.1 ... | Mathlib/Order/Heyting/Basic.lean | 645 | 646 | theorem Disjoint.sup_sdiff_cancel_right (h : Disjoint a b) : (a ⊔ b) \ b = a := by |
rw [sup_sdiff, sdiff_self, sup_bot_eq, h.sdiff_eq_left]
|
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Fintype.Prod
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Option
#align_import group_theory.perm.option from "leanprover-community/mathlib"@"c3019c79074b0619edb4b27553a91b2e82242395"
open Equiv
@[simp]
theo... | Mathlib/GroupTheory/Perm/Option.lean | 80 | 81 | theorem Equiv.Perm.decomposeOption_symm_sign {α : Type*} [DecidableEq α] [Fintype α] (e : Perm α) :
Perm.sign (Equiv.Perm.decomposeOption.symm (none, e)) = Perm.sign e := by | simp
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 828 | 843 | theorem lintegral_add_mul_meas_add_le_le_lintegral {f g : α → ℝ≥0∞} (hle : f ≤ᵐ[μ] g)
(hg : AEMeasurable g μ) (ε : ℝ≥0∞) :
∫⁻ a, f a ∂μ + ε * μ { x | f x + ε ≤ g x } ≤ ∫⁻ a, g a ∂μ := by |
rcases exists_measurable_le_lintegral_eq μ f with ⟨φ, hφm, hφ_le, hφ_eq⟩
calc
∫⁻ x, f x ∂μ + ε * μ { x | f x + ε ≤ g x } = ∫⁻ x, φ x ∂μ + ε * μ { x | f x + ε ≤ g x } := by
rw [hφ_eq]
_ ≤ ∫⁻ x, φ x ∂μ + ε * μ { x | φ x + ε ≤ g x } := by
gcongr
exact fun x => (add_le_add_right (hφ_le _) _).... |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Positivity.Core
#align_import data.nat.factorial.double_factorial from "leanprover-community/mathlib"@"7daeaf3072304c498b653628add84a88d0e78767"
open Nat
namespace Nat
@[sim... | Mathlib/Data/Nat/Factorial/DoubleFactorial.lean | 48 | 48 | theorem doubleFactorial_add_one (n : ℕ) : (n + 1)‼ = (n + 1) * (n - 1)‼ := by | cases n <;> rfl
|
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Topological... | Mathlib/Probability/Martingale/Upcrossing.lean | 186 | 189 | theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by |
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· simp only [upperCrossingTime_succ, hitting_le]
|
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 81 | 86 | theorem condCount_isProbabilityMeasure {s : Set Ω} (hs : s.Finite) (hs' : s.Nonempty) :
IsProbabilityMeasure (condCount s) :=
{ measure_univ := by |
rw [condCount, cond_apply _ hs.measurableSet, Set.inter_univ, ENNReal.inv_mul_cancel]
· exact fun h => hs'.ne_empty <| Measure.empty_of_count_eq_zero h
· exact (Measure.count_apply_lt_top.2 hs).ne }
|
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 168 | 170 | theorem Ico_succ_succ : Ico a.succ b.succ = Ioc a b := by |
ext x
rw [mem_Ico, mem_Ioc, succ_le_iff, Nat.lt_succ_iff]
|
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 | 307 | 311 | theorem nfpBFamily_le_apply (ho : o ≠ 0) (H : ∀ i hi, IsNormal (f i hi)) {a b} :
(∃ i hi, nfpBFamily.{u, v} o f a ≤ f i hi b) ↔ nfpBFamily.{u, v} o f a ≤ b := by |
rw [← not_iff_not]
push_neg
exact apply_lt_nfpBFamily_iff.{u, v} ho H
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 182 | 184 | theorem fib_two_mul_add_one (n : ℕ) : fib (2 * n + 1) = fib (n + 1) ^ 2 + fib n ^ 2 := by |
rw [two_mul, fib_add]
ring
|
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.OpenPos
#align_import measure_theory.constructions.prod.basic from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb32... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 73 | 77 | theorem IsPiSystem.prod {C : Set (Set α)} {D : Set (Set β)} (hC : IsPiSystem C)
(hD : IsPiSystem D) : IsPiSystem (image2 (· ×ˢ ·) C D) := by |
rintro _ ⟨s₁, hs₁, t₁, ht₁, rfl⟩ _ ⟨s₂, hs₂, t₂, ht₂, rfl⟩ hst
rw [prod_inter_prod] at hst ⊢; rw [prod_nonempty_iff] at hst
exact mem_image2_of_mem (hC _ hs₁ _ hs₂ hst.1) (hD _ ht₁ _ ht₂ hst.2)
|
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.manifold.cont_mdiff from "leanprover-community/mathlib"@"e5ab837fc252451f3eb9124ae6e7b6f57455e7b9"
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scope... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 706 | 711 | theorem contMDiffWithinAt_iff_nat :
ContMDiffWithinAt I I' n f s x ↔ ∀ m : ℕ, (m : ℕ∞) ≤ n → ContMDiffWithinAt I I' m f s x := by |
refine ⟨fun h m hm => h.of_le hm, fun h => ?_⟩
cases' n with n
· exact contMDiffWithinAt_top.2 fun n => h n le_top
· exact h n le_rfl
|
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 2,353 | 2,356 | theorem ite_eq_of_subset_right (t : Set α) {s₁ s₂ : Set α} (h : s₂ ⊆ s₁) :
t.ite s₁ s₂ = (s₁ ∩ t) ∪ s₂ := by |
ext x
by_cases hx : x ∈ t <;> simp [*, Set.ite, or_iff_left_of_imp (@h x)]
|
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 | 291 | 296 | theorem ae_withDensity_iff' {p : α → Prop} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
(∀ᵐ x ∂μ.withDensity f, p x) ↔ ∀ᵐ x ∂μ, f x ≠ 0 → p x := by |
rw [ae_iff, ae_iff, withDensity_apply_eq_zero' hf, iff_iff_eq]
congr
ext x
simp only [exists_prop, mem_inter_iff, iff_self_iff, mem_setOf_eq, not_forall]
|
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 | 429 | 431 | theorem image_const_sub_Icc : (fun x => a - x) '' Icc b c = Icc (a - c) (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.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerS... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 206 | 208 | theorem map_exp : map (f : A →+* A') (exp A) = exp A' := by |
ext
simp
|
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 368 | 369 | theorem not_subset : ¬s ⊆ t ↔ ∃ a ∈ s, a ∉ t := by |
simp only [subset_def, not_forall, exists_prop]
|
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 172 | 176 | theorem lcm_cycleType (σ : Perm α) : σ.cycleType.lcm = orderOf σ := by |
induction σ using cycle_induction_on with
| base_one => simp
| base_cycles σ hσ => simp [hσ.cycleType, hσ.orderOf]
| induction_disjoint σ τ hd _ hσ hτ => simp [hd.cycleType, hd.orderOf, lcm_eq_nat_lcm, hσ, hτ]
|
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
| Mathlib/NumberTheory/Pell.lean | 83 | 85 | theorem is_pell_solution_iff_mem_unitary {d : ℤ} {a : ℤ√d} :
a.re ^ 2 - d * a.im ^ 2 = 1 ↔ a ∈ unitary (ℤ√d) := by |
rw [← norm_eq_one_iff_mem_unitary, norm_def, sq, sq, ← mul_assoc]
|
import Mathlib.Init.Data.Ordering.Basic
import Mathlib.Order.Synonym
#align_import order.compare from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {α β : Type*}
def cmpLE {α} [LE α] [@DecidableRel α (· ≤ ·)] (x y : α) : Ordering :=
if x ≤ y then if y ≤ x then Ordering.eq ... | Mathlib/Order/Compare.lean | 141 | 142 | theorem swap_orElse (o₁ o₂) : (orElse o₁ o₂).swap = orElse o₁.swap o₂.swap := by |
cases o₁ <;> rfl
|
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 | 453 | 455 | theorem lt_base_pow_length_digits {b m : ℕ} (hb : 1 < b) : m < b ^ (digits b m).length := by |
rcases b with (_ | _ | b) <;> try simp_all
exact lt_base_pow_length_digits'
|
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_theory.abelian.opposite from "leanprover-community/mathlib"@"a5ff45a1c92c278b03b52459a620cfd9c49ebc80"
noncomputable section
namespace CategoryTheor... | Mathlib/CategoryTheory/Abelian/Opposite.lean | 164 | 172 | theorem image_ι_op_comp_imageUnopOp_hom :
(image.ι g.unop).op ≫ (imageUnopOp g).hom = factorThruImage g := by |
simp only [imageUnopOp, Iso.trans, Iso.symm, Iso.op, cokernelOpOp_inv, cokernelEpiComp_hom,
cokernelCompIsIso_hom, Abelian.coimageIsoImage'_hom, ← Category.assoc, ← op_comp]
simp only [Category.assoc, Abelian.imageIsoImage_hom_comp_image_ι, kernel.lift_ι,
Quiver.Hom.op_unop, cokernelIsoOfEq_hom_comp_desc_a... |
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 | 49 | 54 | theorem Nat.card_eq (α : Type*) :
Nat.card α = if h : Finite α then @Fintype.card α (Fintype.ofFinite α) else 0 := by |
cases finite_or_infinite α
· letI := Fintype.ofFinite α
simp only [*, Nat.card_eq_fintype_card, dif_pos]
· simp only [*, card_eq_zero_of_infinite, not_finite_iff_infinite.mpr, dite_false]
|
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.OpenPos
#align_import measure_theory.constructions.prod.basic from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb32... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 341 | 344 | theorem prod_apply {s : Set (α × β)} (hs : MeasurableSet s) :
μ.prod ν s = ∫⁻ x, ν (Prod.mk x ⁻¹' s) ∂μ := by |
simp_rw [Measure.prod, bind_apply hs (Measurable.map_prod_mk_left (ν := ν)),
map_apply measurable_prod_mk_left hs]
|
import Mathlib.MeasureTheory.OuterMeasure.Basic
open Filter Set
open scoped ENNReal
namespace MeasureTheory
variable {α β F : Type*} [FunLike F (Set α) ℝ≥0∞] [OuterMeasureClass F α] {μ : F} {s t : Set α}
def ae (μ : F) : Filter α :=
.ofCountableUnion (μ · = 0) (fun _S hSc ↦ (measure_sUnion_null_iff hSc).2) fu... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 231 | 233 | theorem inter_ae_eq_left_of_ae_eq_univ (h : t =ᵐ[μ] univ) : (s ∩ t : Set α) =ᵐ[μ] s := by |
convert ae_eq_set_inter (ae_eq_refl s) h
rw [inter_univ]
|
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
noncomputable section
open scoped Classical
open Filter
open Function Set Filter
namespace BoxIntegral
variable {ι M : Type*} {... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 126 | 127 | theorem splitUpper_eq_self : I.splitUpper i x = I ↔ x ≤ I.lower i := by |
simp [splitUpper, update_eq_iff]
|
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.theta from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter
open Topology
namespace Asymptotics
set_option linter.uppercaseLean3 false -- is_Theta
v... | Mathlib/Analysis/Asymptotics/Theta.lean | 316 | 318 | theorem isTheta_const_mul_right {c : 𝕜} {g : α → 𝕜} (hc : c ≠ 0) :
(f =Θ[l] fun x ↦ c * g x) ↔ f =Θ[l] g := by |
simpa only [← smul_eq_mul] using isTheta_const_smul_right hc
|
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.Polynomial.Pochhammer
section Multichoose
open Function Polynomial
class BinomialRing (R : Type*) [AddCommMonoid R] [Pow R ℕ] where
nsmul_right_injective (n : ℕ) (h : n ≠ 0) : Injective (n • · : R →... | Mathlib/RingTheory/Binomial.lean | 129 | 138 | theorem ascPochhammer_smeval_neg_eq_descPochhammer (r : R) (k : ℕ) :
(ascPochhammer ℕ k).smeval (-r) = (-1)^k * (descPochhammer ℤ k).smeval r := by |
induction k with
| zero => simp only [ascPochhammer_zero, descPochhammer_zero, smeval_one, npow_zero, one_mul]
| succ k ih =>
simp only [ascPochhammer_succ_right, smeval_mul, ih, descPochhammer_succ_right, sub_eq_add_neg]
have h : (X + (k : ℕ[X])).smeval (-r) = - (X + (-k : ℤ[X])).smeval r := by
si... |
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Order.Monotone
#align_import topology.algebra.order.left_right_lim from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Set Filter
open Topology
section
variable {α β : Type*} [LinearOrder α] [TopologicalSpace β]
n... | Mathlib/Topology/Order/LeftRightLim.lean | 75 | 78 | theorem leftLim_eq_of_eq_bot [hα : TopologicalSpace α] [h'α : OrderTopology α] (f : α → β) {a : α}
(h : 𝓝[<] a = ⊥) : leftLim f a = f a := by |
rw [h'α.topology_eq_generate_intervals] at h
simp [leftLim, ite_eq_left_iff, h]
|
import Mathlib.Topology.MetricSpace.Lipschitz
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
#align_import topology.metric_space.holder from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
variable {X Y Z : Type*}
open Filter Set
open NNReal ENNReal Topology
section Emetric
... | Mathlib/Topology/MetricSpace/Holder.lean | 120 | 126 | theorem comp {Cg rg : ℝ≥0} {g : Y → Z} {t : Set Y} (hg : HolderOnWith Cg rg g t) {Cf rf : ℝ≥0}
{f : X → Y} (hf : HolderOnWith Cf rf f s) (hst : MapsTo f s t) :
HolderOnWith (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) s := by |
intro x hx y hy
rw [ENNReal.coe_mul, mul_comm rg, NNReal.coe_mul, ENNReal.rpow_mul, mul_assoc,
← ENNReal.coe_rpow_of_nonneg _ rg.coe_nonneg, ← ENNReal.mul_rpow_of_nonneg _ _ rg.coe_nonneg]
exact hg.edist_le_of_le (hst hx) (hst hy) (hf.edist_le hx hy)
|
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