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.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 316 | 321 | theorem isZero_zero [HasZeroObject V] : IsZero (zero : HomologicalComplex V c) := by |
refine ⟨fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩, fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩⟩
all_goals
ext
dsimp [zero]
apply Subsingleton.elim
|
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 | 238 | 246 | theorem snd_invApp_t_app (i j k : D.J) (U : Opens (pullback (D.f i j) (D.f i k)).carrier) :
(π₂⁻¹ i, j, k) U ≫ (D.t k i).c.app _ =
(D.t' k i j).c.app _ ≫
(π₁⁻¹ k, j, i) (unop _) ≫
(D.V (k, i)).presheaf.map (eqToHom (D.snd_invApp_t_app' i j k U).choose.symm) := by |
have e := (D.snd_invApp_t_app' i j k U).choose_spec
replace e := reassoc_of% e
rw [← e]
simp [eqToHom_map]
|
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import measure_theory.function.egorov from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open MeasureTheory NNReal ENNReal Topology
namespace MeasureTheory
open Set Filt... | Mathlib/MeasureTheory/Function/Egorov.lean | 59 | 70 | theorem measure_inter_notConvergentSeq_eq_zero [SemilatticeSup ι] [Nonempty ι]
(hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :
μ (s ∩ ⋂ j, notConvergentSeq f g n j) = 0 := by |
simp_rw [Metric.tendsto_atTop, ae_iff] at hfg
rw [← nonpos_iff_eq_zero, ← hfg]
refine measure_mono fun x => ?_
simp only [Set.mem_inter_iff, Set.mem_iInter, ge_iff_le, mem_notConvergentSeq_iff]
push_neg
rintro ⟨hmem, hx⟩
refine ⟨hmem, 1 / (n + 1 : ℝ), Nat.one_div_pos_of_nat, fun N => ?_⟩
obtain ⟨n, hn₁... |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 69 | 70 | theorem apply_ne_one_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
¬A i j = 1 ↔ A i j = 0 := by | obtain h | h := h.zero_or_one i j <;> simp [h]
|
import Mathlib.CategoryTheory.Adjunction.Reflective
import Mathlib.Topology.StoneCech
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.CategoryTheory.Elementwise
#align_import topol... | Mathlib/Topology/Category/CompHaus/Basic.lean | 123 | 135 | theorem isIso_of_bijective {X Y : CompHaus.{u}} (f : X ⟶ Y) (bij : Function.Bijective f) :
IsIso f := by |
let E := Equiv.ofBijective _ bij
have hE : Continuous E.symm := by
rw [continuous_iff_isClosed]
intro S hS
rw [← E.image_eq_preimage]
exact isClosedMap f S hS
refine ⟨⟨⟨E.symm, hE⟩, ?_, ?_⟩⟩
· ext x
apply E.symm_apply_apply
· ext x
apply E.apply_symm_apply
|
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
#align_import ring_theory.etale from ... | Mathlib/RingTheory/Smooth/Basic.lean | 306 | 320 | theorem of_isLocalization : FormallySmooth R Rₘ := by |
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
convert (IsLocalization.map_units Rₘ x).map f
simp only [Ideal.Quotient.mk_algebraMap, AlgHom.commutes]
let this : Rₘ →ₐ[R] Q :=
{ IsLocalization.lift t... |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#align_import measure_theory.function.conditional_expectation.condexp_L2 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 126 | 137 | theorem condexpL2_indicator_of_measurable (hm : m ≤ m0) (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞)
(c : E) :
(condexpL2 E 𝕜 hm (indicatorConstLp 2 (hm s hs) hμs c) : α →₂[μ] E) =
indicatorConstLp 2 (hm s hs) hμs c := by |
rw [condexpL2]
haveI : Fact (m ≤ m0) := ⟨hm⟩
have h_mem : indicatorConstLp 2 (hm s hs) hμs c ∈ lpMeas E 𝕜 m 2 μ :=
mem_lpMeas_indicatorConstLp hm hs hμs
let ind := (⟨indicatorConstLp 2 (hm s hs) hμs c, h_mem⟩ : lpMeas E 𝕜 m 2 μ)
have h_coe_ind : (ind : α →₂[μ] E) = indicatorConstLp 2 (hm s hs) hμs c :=... |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 145 | 146 | theorem isCodetecting_unop_iff {𝒢 : Set Cᵒᵖ} : IsCodetecting 𝒢.unop ↔ IsDetecting 𝒢 := by |
rw [← isDetecting_op_iff, Set.unop_op]
|
import Mathlib.CategoryTheory.Sites.Sheaf
#align_import category_theory.sites.plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory.GrothendieckTopology
open CategoryTheory
open CategoryTheory.Limits
open Opposite
universe w v u
variable {C : Type u} [Ca... | Mathlib/CategoryTheory/Sites/Plus.lean | 273 | 287 | theorem isIso_toPlus_of_isSheaf (hP : Presheaf.IsSheaf J P) : IsIso (J.toPlus P) := by |
rw [Presheaf.isSheaf_iff_multiequalizer] at hP
suffices ∀ X, IsIso ((J.toPlus P).app X) from NatIso.isIso_of_isIso_app _
intro X
suffices IsIso (colimit.ι (J.diagram P X.unop) (op ⊤)) from IsIso.comp_isIso
suffices ∀ (S T : (J.Cover X.unop)ᵒᵖ) (f : S ⟶ T), IsIso ((J.diagram P X.unop).map f) from
isIso_ι_... |
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 | 353 | 353 | theorem conj_neg_I : conj (-I) = (I : K) := by | rw [map_neg, conj_I, neg_neg]
|
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 | 271 | 275 | theorem isLeftDescent_inv_iff {w : W} {i : B} :
cs.IsLeftDescent w⁻¹ i ↔ cs.IsRightDescent w i := by |
unfold IsLeftDescent IsRightDescent
nth_rw 1 [← length_inv]
simp
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 267 | 271 | theorem log_lt_sub_one_of_pos (hx1 : 0 < x) (hx2 : x ≠ 1) : log x < x - 1 := by |
have h : log x ≠ 0 := by
rwa [← log_one, log_injOn_pos.ne_iff hx1]
exact mem_Ioi.mpr zero_lt_one
linarith [add_one_lt_exp h, exp_log hx1]
|
import Mathlib.Data.List.Chain
#align_import data.list.destutter from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
variable {α : Type*} (l : List α) (R : α → α → Prop) [DecidableRel R] {a b : α}
namespace List
@[simp]
theorem destutter'_nil : destutter' R a [] = [a] :=
rfl
#align ... | Mathlib/Data/List/Destutter.lean | 60 | 61 | theorem destutter'_singleton : [b].destutter' R a = if R a b then [a, b] else [a] := by |
split_ifs with h <;> simp! [h]
|
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 | 107 | 109 | theorem comap_prod (f : α → β × γ) (b : Filter β) (c : Filter γ) :
comap f (b ×ˢ c) = comap (Prod.fst ∘ f) b ⊓ comap (Prod.snd ∘ f) c := by |
erw [comap_inf, Filter.comap_comap, Filter.comap_comap]
|
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Topology.Category.TopCat.Limits.Basic
#align_import topology.category.Top.limits.konig from "leanprover-community/mathlib"@"dbdf71cee7bb20367cb7e37279c08b0c218cf967"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 ... | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | 130 | 146 | theorem nonempty_limitCone_of_compact_t2_cofiltered_system (F : J ⥤ TopCat.{max v u})
[IsCofilteredOrEmpty J]
[∀ j : J, Nonempty (F.obj j)] [∀ j : J, CompactSpace (F.obj j)] [∀ j : J, T2Space (F.obj j)] :
Nonempty (TopCat.limitCone F).pt := by |
classical
obtain ⟨u, hu⟩ :=
IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed (fun G => partialSections F _)
(partialSections.directed F) (fun G => partialSections.nonempty F _)
(fun G => IsClosed.isCompact (partialSections.closed F _)) fun G =>
partialSections.closed F _
us... |
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 | 160 | 175 | theorem addHaar_eq_zero_of_disjoint_translates {E : Type*} [NormedAddCommGroup E]
[NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E)
[IsAddHaarMeasure μ] {s : Set E} (u : ℕ → E) (hu : IsBounded (range u))
(hs : Pairwise (Disjoint on fun n => {u n} + s)) (h's : Measu... |
suffices H : ∀ R, μ (s ∩ closedBall 0 R) = 0 by
apply le_antisymm _ (zero_le _)
calc
μ s ≤ ∑' n : ℕ, μ (s ∩ closedBall 0 n) := by
conv_lhs => rw [← iUnion_inter_closedBall_nat s 0]
exact measure_iUnion_le _
_ = 0 := by simp only [H, tsum_zero]
intro R
apply addHaar_eq_zero_of_... |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 111 | 114 | theorem encard_union_eq (h : Disjoint s t) : (s ∪ t).encard = s.encard + t.encard := by |
classical
have e := (Equiv.Set.union (by rwa [subset_empty_iff, ← disjoint_iff_inter_eq_empty])).symm
simp [encard, ← PartENat.card_congr e, PartENat.card_sum, PartENat.withTopEquiv]
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.LinearAlgebra.AffineSpace.Restrict
import Mathlib.Tactic.FailIfNoProgress
#align_import analysis.normed_space.affine_isomet... | Mathlib/Analysis/NormedSpace/AffineIsometry.lean | 86 | 88 | theorem toAffineMap_injective : Injective (toAffineMap : (P →ᵃⁱ[𝕜] P₂) → P →ᵃ[𝕜] P₂) := by |
rintro ⟨f, _⟩ ⟨g, _⟩ rfl
rfl
|
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 530 | 533 | theorem prod_divisorsAntidiagonal' {M : Type*} [CommMonoid M] (f : ℕ → ℕ → M) {n : ℕ} :
∏ i ∈ n.divisorsAntidiagonal, f i.1 i.2 = ∏ i ∈ n.divisors, f (n / i) i := by |
rw [← map_swap_divisorsAntidiagonal, Finset.prod_map]
exact prod_divisorsAntidiagonal fun i j => f j i
|
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
universe u... | Mathlib/Analysis/MeanInequalities.lean | 180 | 183 | theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i := by |
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
|
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-community/mathlib"@"b90e72c7eebbe8de7c8293a80208ea2ba135c834"
universe u
open SetTheory PGame
open scoped NaturalOps PGame
namespace Ordinal
noncomputable def toPGame : Ordin... | Mathlib/SetTheory/Game/Ordinal.lean | 53 | 54 | theorem toPGame_leftMoves (o : Ordinal) : o.toPGame.LeftMoves = o.out.α := by |
rw [toPGame, LeftMoves]
|
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
def padicNorm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-padicValRat p q)
#align padic_n... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 288 | 290 | theorem int_lt_one_iff (m : ℤ) : padicNorm p m < 1 ↔ (p : ℤ) ∣ m := by |
rw [← not_iff_not, ← int_eq_one_iff, eq_iff_le_not_lt]
simp only [padicNorm.of_int, true_and_iff]
|
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mathlib"@"132328c4dd48da87adca5d408ca54f315282b719"
universe u v w w₁
namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieR... | Mathlib/Algebra/Lie/Character.lean | 52 | 60 | theorem lieCharacter_apply_of_mem_derived (χ : LieCharacter R L) {x : L}
(h : x ∈ derivedSeries R L 1) : χ x = 0 := by |
rw [derivedSeries_def, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_zero, ←
LieSubmodule.mem_coeSubmodule, LieSubmodule.lieIdeal_oper_eq_linear_span] at h
refine Submodule.span_induction h ?_ ?_ ?_ ?_
· rintro y ⟨⟨z, hz⟩, ⟨⟨w, hw⟩, rfl⟩⟩; apply lieCharacter_apply_lie
· exact χ.map_zero
· intro y z hy ... |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 44 | 47 | theorem inv_gold : φ⁻¹ = -ψ := by |
have : 1 + √5 ≠ 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
|
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 462 | 468 | theorem LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right (g : G ≃ₗᵢ[𝕜] E) (f : E → F)
(hs : UniqueDiffOn 𝕜 s) {x : G} (hx : g x ∈ s) (i : ℕ) :
‖iteratedFDerivWithin 𝕜 i (f ∘ g) (g ⁻¹' s) x‖ = ‖iteratedFDerivWithin 𝕜 i f s (g x)‖ := by |
have : iteratedFDerivWithin 𝕜 i (f ∘ g) (g ⁻¹' s) x =
(iteratedFDerivWithin 𝕜 i f s (g x)).compContinuousLinearMap fun _ => g :=
g.toContinuousLinearEquiv.iteratedFDerivWithin_comp_right f hs hx i
rw [this, ContinuousMultilinearMap.norm_compContinuous_linearIsometryEquiv]
|
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 520 | 521 | theorem toIcoMod_sub' (a b : α) : toIcoMod hp (a - p) b = toIcoMod hp a b - p := by |
simpa only [one_zsmul] using toIcoMod_sub_zsmul' hp a b 1
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 148 | 151 | theorem A_mono (L : E →L[𝕜] F) (r : ℝ) {ε δ : ℝ} (h : ε ≤ δ) : A f L r ε ⊆ A f L r δ := by |
rintro x ⟨r', r'r, hr'⟩
refine ⟨r', r'r, fun y hy z hz => (hr' y hy z hz).trans_le (mul_le_mul_of_nonneg_right h ?_)⟩
linarith [mem_ball.1 hy, r'r.2, @dist_nonneg _ _ y x]
|
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.FreeAlgebra
import Mathlib.RingTheory.Localization.FractionRing
#align_import algebra.char_p.algebra from "leanprover-community/mathlib"@"96782a2d6dcded92116d8ac9ae48efb41d46a27c"
theorem charP_of_injective_ringHom {R A : Type*} [NonAssocSemiring R] [NonAs... | Mathlib/Algebra/CharP/Algebra.lean | 121 | 123 | theorem Algebra.ringChar_eq : ringChar K = ringChar L := by |
rw [ringChar.eq_iff, Algebra.charP_iff K L]
apply ringChar.charP
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sum
#align_import combinatorics.hales_jewett from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
open scoped Classical
universe u v
namespace ... | Mathlib/Combinatorics/HalesJewett.lean | 197 | 200 | theorem horizontal_apply {α ι ι'} (l : Line α ι) (v : ι' → α) (x : α) :
l.horizontal v x = Sum.elim (l x) v := by |
funext i
cases i <;> rfl
|
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Algebra.Module.Submodule.LinearMap
open Function Pointwise Set
variable {R : Type*} {R₁ : Type*} {R₂ : Type*} {R₃ : Type*}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*} {M₃ : Type*}
namespace Submodule
section AddCommMonoid
variable [Semiring R] [... | Mathlib/Algebra/Module/Submodule/Map.lean | 170 | 173 | theorem map_equivMapOfInjective_symm_apply (f : F) (i : Injective f) (p : Submodule R M)
(x : p.map f) : f ((equivMapOfInjective f i p).symm x) = x := by |
rw [← LinearEquiv.apply_symm_apply (equivMapOfInjective f i p) x, coe_equivMapOfInjective_apply,
i.eq_iff, LinearEquiv.apply_symm_apply]
|
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 | 116 | 120 | theorem fold_union_inter [DecidableEq α] {s₁ s₂ : Finset α} {b₁ b₂ : β} :
((s₁ ∪ s₂).fold op b₁ f * (s₁ ∩ s₂).fold op b₂ f) = s₁.fold op b₂ f * s₂.fold op b₁ f := by |
unfold fold
rw [← fold_add op, ← Multiset.map_add, union_val, inter_val, union_add_inter, Multiset.map_add,
hc.comm, fold_add]
|
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Conj
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.adjunction.reflective from "leanprover-community/mathlib"@"239d882c4fb58361ee8b3b39fb2091320edef10a"
universe v₁ v₂ v₃ u₁ u₂ u₃
noncomputable s... | Mathlib/CategoryTheory/Adjunction/Reflective.lean | 87 | 89 | theorem Functor.essImage.unit_isIso [Reflective i] {A : C} (h : A ∈ i.essImage) :
IsIso ((reflectorAdjunction i).unit.app A) := by |
rwa [isIso_unit_app_iff_mem_essImage]
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
#align_import number_theory.zsqrtd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
@[ext]
struct... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 284 | 284 | theorem intCast_im (n : ℤ) : (n : ℤ√d).im = 0 := by | cases n <;> rfl
|
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Topology.Algebra.InfiniteSum.NatInt
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Order.MonotoneConvergence
#align_import topology.algebra.infinite_sum.order from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
... | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | 160 | 163 | theorem one_le_tprod (h : ∀ i, 1 ≤ g i) : 1 ≤ ∏' i, g i := by |
by_cases hg : Multipliable g
· exact hg.hasProd.one_le h
· rw [tprod_eq_one_of_not_multipliable hg]
|
import Mathlib.LinearAlgebra.TensorProduct.Graded.External
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.GroupTheory.GroupAction.Ring
suppress_compilation
open scoped TensorProduct
variable {R ι A B : Type*}
variable [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι]
variable [CommRing R] [R... | Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean | 133 | 135 | theorem auxEquiv_one : auxEquiv R 𝒜 ℬ 1 = 1 := by |
rw [← of_one, Algebra.TensorProduct.one_def, auxEquiv_tmul 𝒜 ℬ, DirectSum.decompose_one,
DirectSum.decompose_one, Algebra.TensorProduct.one_def]
|
import Mathlib.RingTheory.Finiteness
import Mathlib.Logic.Equiv.TransferInstance
universe u v w
open Function
variable (R : Type u) [Semiring R]
@[mk_iff]
class OrzechProperty : Prop where
injective_of_surjective_of_submodule' : ∀ {M : Type u} [AddCommMonoid M] [Module R M]
[Module.Finite R M] {N : Submod... | Mathlib/RingTheory/OrzechProperty.lean | 69 | 82 | theorem injective_of_surjective_of_injective
{N : Type w} [AddCommMonoid N] [Module R N]
(i f : N →ₗ[R] M) (hi : Injective i) (hf : Surjective f) : Injective f := by |
obtain ⟨n, g, hg⟩ := Module.Finite.exists_fin' R M
haveI := small_of_surjective hg
letI := Equiv.addCommMonoid (equivShrink M).symm
letI := Equiv.module R (equivShrink M).symm
let j : Shrink.{u} M ≃ₗ[R] M := Equiv.linearEquiv R (equivShrink M).symm
haveI := Module.Finite.equiv j.symm
let i' := j.symm.toL... |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 490 | 493 | theorem not_dvd_ord_compl {n p : ℕ} (hp : Prime p) (hn : n ≠ 0) : ¬p ∣ ord_compl[p] n := by |
rw [Nat.Prime.dvd_iff_one_le_factorization hp (ord_compl_pos p hn).ne']
rw [Nat.factorization_div (Nat.ord_proj_dvd n p)]
simp [hp.factorization]
|
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 | 437 | 438 | theorem Icc_subset_Iic_self : Icc a b ⊆ Iic b := by |
simpa [← coe_subset] using Set.Icc_subset_Iic_self
|
import Mathlib.LinearAlgebra.Dual
open Function Module
variable (R M N : Type*) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
structure PerfectPairing :=
toLin : M →ₗ[R] N →ₗ[R] R
bijectiveLeft : Bijective toLin
bijectiveRight : Bijective toLin.flip
attribute [nolint docBlame] P... | Mathlib/LinearAlgebra/PerfectPairing.lean | 102 | 105 | theorem toDualRight_symm_comp_toDualLeft :
p.toDualRight.symm.dualMap ∘ₗ (p.toDualLeft : M →ₗ[R] Dual R N) = Dual.eval R M := by |
ext1 x
exact p.toDualRight_symm_toDualLeft x
|
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 141 | 146 | theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by |
simp_rw [sub_eq_neg_add]
rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
|
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
def HomRel (C) [Quiver C] :=
∀ ⦃X Y : C⦄, (X ⟶ Y) → (X ⟶ Y) → Prop
#align hom_rel HomRel
-- Porting Note: `deriving I... | Mathlib/CategoryTheory/Quotient.lean | 65 | 66 | theorem CompClosure.of {a b : C} (m₁ m₂ : a ⟶ b) (h : r m₁ m₂) : CompClosure r m₁ m₂ := by |
simpa using CompClosure.intro (𝟙 _) m₁ m₂ (𝟙 _) h
|
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 63 | 67 | theorem IsMatching.toEdge_eq_of_adj {M : Subgraph G} (h : M.IsMatching) {v w : V} (hv : v ∈ M.verts)
(hvw : M.Adj v w) : h.toEdge ⟨v, hv⟩ = ⟨s(v, w), hvw⟩ := by |
simp only [IsMatching.toEdge, Subtype.mk_eq_mk]
congr
exact ((h (M.edge_vert hvw)).choose_spec.2 w hvw).symm
|
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.PSeries
#align_import number_theory.zeta_values from "leanprover-community/mathlib"@"f0c8bf9245297... | Mathlib/NumberTheory/ZetaValues.lean | 80 | 87 | theorem integral_bernoulliFun_eq_zero {k : ℕ} (hk : k ≠ 0) :
∫ x : ℝ in (0)..1, bernoulliFun k x = 0 := by |
rw [integral_eq_sub_of_hasDerivAt (fun x _ => antideriv_bernoulliFun k x)
((Polynomial.continuous _).intervalIntegrable _ _)]
rw [bernoulliFun_eval_one]
split_ifs with h
· exfalso; exact hk (Nat.succ_inj'.mp h)
· simp
|
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import number_theory.cyclotomic.rat from "leanprover-community/mathlib"@"b353176c24d96c23f0ce1cc63efc3f55019702d9"
universe u
open Algebra IsCyclotomicExtensio... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 65 | 69 | theorem discr_prime_pow_eq_unit_mul_pow' [IsCyclotomicExtension {p ^ k} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) :
∃ (u : ℤˣ) (n : ℕ), discr ℚ (hζ.subOnePowerBasis ℚ).basis = u * p ^ n := by |
rw [hζ.discr_zeta_eq_discr_zeta_sub_one.symm]
exact discr_prime_pow_eq_unit_mul_pow hζ (cyclotomic.irreducible_rat (p ^ k).pos)
|
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.PartENat
#align_import set_theory.cardinal.finite from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
set_option autoImplicit true
open Cardinal Function
noncomputable section
variable {α β : Typ... | Mathlib/SetTheory/Cardinal/Finite.lean | 144 | 146 | theorem card_of_subsingleton (a : α) [Subsingleton α] : Nat.card α = 1 := by |
letI := Fintype.ofSubsingleton a
rw [card_eq_fintype_card, Fintype.card_ofSubsingleton a]
|
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 677 | 683 | theorem hasBasis_coclosedCompact :
(Filter.coclosedCompact X).HasBasis (fun s => IsClosed s ∧ IsCompact s) compl := by |
simp only [Filter.coclosedCompact, iInf_and']
refine hasBasis_biInf_principal' ?_ ⟨∅, isClosed_empty, isCompact_empty⟩
rintro s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩
exact ⟨s ∪ t, ⟨⟨hs₁.union ht₁, hs₂.union ht₂⟩, compl_subset_compl.2 subset_union_left,
compl_subset_compl.2 subset_union_right⟩⟩
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.Matrix.AbsoluteValue
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
import Mathlib.RingTheory.ClassGroup
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.Ri... | Mathlib/NumberTheory/ClassNumber/Finite.lean | 251 | 268 | theorem exists_mem_finset_approx' [Algebra.IsAlgebraic R L] (a : S) {b : S} (hb : b ≠ 0) :
∃ q : S,
∃ r ∈ finsetApprox bS adm, abv (Algebra.norm R (r • a - q * b)) < abv (Algebra.norm R b) := by |
have inj : Function.Injective (algebraMap R L) := by
rw [IsScalarTower.algebraMap_eq R S L]
exact (IsIntegralClosure.algebraMap_injective S R L).comp bS.algebraMap_injective
obtain ⟨a', b', hb', h⟩ := IsIntegralClosure.exists_smul_eq_mul inj a hb
obtain ⟨q, r, hr, hqr⟩ := exists_mem_finsetApprox bS adm a... |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 204 | 208 | theorem HasFTaylorSeriesUpToOn.congr (h : HasFTaylorSeriesUpToOn n f p s)
(h₁ : ∀ x ∈ s, f₁ x = f x) : HasFTaylorSeriesUpToOn n f₁ p s := by |
refine ⟨fun x hx => ?_, h.fderivWithin, h.cont⟩
rw [h₁ x hx]
exact h.zero_eq x hx
|
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Tactic.FinCases
import Mathlib.Topology.Sets.Closeds
#align_import topology.locally_constant.basic from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
variable {X Y Z α : Type*} [TopologicalSpace X]
open Set Filter
open Top... | Mathlib/Topology/LocallyConstant/Basic.lean | 152 | 159 | theorem apply_eq_of_isPreconnected {f : X → Y} (hf : IsLocallyConstant f) {s : Set X}
(hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by |
let U := f ⁻¹' {f y}
suffices x ∉ Uᶜ from Classical.not_not.1 this
intro hxV
specialize hs U Uᶜ (hf {f y}) (hf {f y}ᶜ) _ ⟨y, ⟨hy, rfl⟩⟩ ⟨x, ⟨hx, hxV⟩⟩
· simp only [union_compl_self, subset_univ]
· simp only [inter_empty, Set.not_nonempty_empty, inter_compl_self] at hs
|
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 44 | 45 | theorem tsub_le_tsub_iff_right (h : c ≤ b) : a - c ≤ b - c ↔ a ≤ b := by |
rw [tsub_le_iff_right, tsub_add_cancel_of_le h]
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
section Image
variable {f : α → β} {s t : Set... | Mathlib/Data/Set/Image.lean | 335 | 337 | theorem image_singleton {f : α → β} {a : α} : f '' {a} = {f a} := by |
ext
simp [image, eq_comm]
|
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 | 448 | 455 | theorem nonempty_inter_of_measure_lt_add {m : MeasurableSpace α} (μ : Measure α) {s t u : Set α}
(ht : MeasurableSet t) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ u < μ s + μ t) :
(s ∩ t).Nonempty := by |
rw [← Set.not_disjoint_iff_nonempty_inter]
contrapose! h
calc
μ s + μ t = μ (s ∪ t) := (measure_union h ht).symm
_ ≤ μ u := measure_mono (union_subset h's h't)
|
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 | 220 | 221 | theorem add_nat_iff : LiouvilleWith p (x + n) ↔ LiouvilleWith p x := by |
rw [← Rat.cast_natCast n, add_rat_iff]
|
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
open Nat
namespace Nat
def choose : ℕ → ℕ → ℕ
| _, 0 => 1
| 0, _ + 1 => 0
| n + 1, k + 1 => choose n k + choose n ... | Mathlib/Data/Nat/Choose/Basic.lean | 382 | 382 | theorem multichoose_zero_succ (k : ℕ) : multichoose 0 (k + 1) = 0 := by | simp [multichoose]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Sym.Card
open Finset Function
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V) {e : Sym2 V}
section EdgeFinset
variable {G₁ G₂ : SimpleGraph V} [Fintype G.edgeSet] [Fintype G₁.edgeSet] [Finty... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 94 | 95 | theorem edgeFinset_sup [Fintype (edgeSet (G₁ ⊔ G₂))] [DecidableEq V] :
(G₁ ⊔ G₂).edgeFinset = G₁.edgeFinset ∪ G₂.edgeFinset := by | simp [edgeFinset]
|
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 110 | 113 | theorem WithTop.coe_iInf [Nonempty ι] [InfSet α] {f : ι → α} (hf : BddBelow (range f)) :
↑(⨅ i, f i) = (⨅ i, f i : WithTop α) := by |
rw [iInf, iInf, WithTop.coe_sInf' (range_nonempty f) hf, ← range_comp]
rfl
|
import Mathlib.Data.Set.Lattice
#align_import order.concept from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
open Function OrderDual Set
variable {ι : Sort*} {α β γ : Type*} {κ : ι → Sort*} (r : α → β → Prop) {s s₁ s₂ : Set α}
{t t₁ t₂ : Set β}
def intentClosure (s : Set α) :... | Mathlib/Order/Concept.lean | 234 | 239 | theorem snd_subset_snd_iff : c.snd ⊆ d.snd ↔ d ≤ c := by |
refine ⟨fun h => ?_, fun h => ?_⟩
· rw [← fst_subset_fst_iff, ← c.closure_snd, ← d.closure_snd]
exact extentClosure_anti _ h
· rw [← c.closure_fst, ← d.closure_fst]
exact intentClosure_anti _ h
|
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Nat.Cast.NeZero
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {α β : T... | Mathlib/Data/Nat/Cast/Order.lean | 138 | 138 | theorem one_le_cast : 1 ≤ (n : α) ↔ 1 ≤ n := by | rw [← cast_one, cast_le]
|
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 | 545 | 556 | theorem essSup_measure_restrict (hs : IsFundamentalDomain G s μ) {f : α → ℝ≥0∞}
(hf : ∀ γ : G, ∀ x : α, f (γ • x) = f x) : essSup f (μ.restrict s) = essSup f μ := by |
refine le_antisymm (essSup_mono_measure' Measure.restrict_le_self) ?_
rw [essSup_eq_sInf (μ.restrict s) f, essSup_eq_sInf μ f]
refine sInf_le_sInf ?_
rintro a (ha : (μ.restrict s) {x : α | a < f x} = 0)
rw [Measure.restrict_apply₀' hs.nullMeasurableSet] at ha
refine measure_zero_of_invariant hs _ ?_ ha
i... |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 472 | 477 | theorem generateFrom_piiUnionInter_le {m : MeasurableSpace α} (π : ι → Set (Set α))
(h : ∀ n, generateFrom (π n) ≤ m) (S : Set ι) : generateFrom (piiUnionInter π S) ≤ m := by |
refine generateFrom_le ?_
rintro t ⟨ht_p, _, ft, hft_mem_pi, rfl⟩
refine Finset.measurableSet_biInter _ fun x hx_mem => (h x) _ ?_
exact measurableSet_generateFrom (hft_mem_pi x hx_mem)
|
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {α : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 505 | 508 | theorem isRotated_reverse_comm_iff : l.reverse ~r l' ↔ l ~r l'.reverse := by |
constructor <;>
· intro h
simpa using h.reverse
|
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
universe v v₁ v₂ u u₁ u₂
variable {U : Type*} [Quiver.{u + 1} U]
namespace Quiver
def Hom.cast {u v u' v... | Mathlib/Combinatorics/Quiver/Cast.lean | 148 | 152 | theorem eq_nil_of_length_zero {u v : U} (p : Path u v) (hzero : p.length = 0) :
p.cast (eq_of_length_zero p hzero) rfl = Path.nil := by |
cases p
· rfl
· simp only [Nat.succ_ne_zero, length_cons] at hzero
|
import Mathlib.NumberTheory.Liouville.Basic
#align_import number_theory.liouville.liouville_number from "leanprover-community/mathlib"@"04e80bb7e8510958cd9aacd32fe2dc147af0b9f1"
noncomputable section
open scoped Nat
open Real Finset
def liouvilleNumber (m : ℝ) : ℝ :=
∑' i : ℕ, 1 / m ^ i !
#align liouville_n... | Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean | 110 | 134 | theorem remainder_lt' (n : ℕ) {m : ℝ} (m1 : 1 < m) :
remainder m n < (1 - 1 / m)⁻¹ * (1 / m ^ (n + 1)!) :=
-- two useful inequalities
have m0 : 0 < m := zero_lt_one.trans m1
have mi : 1 / m < 1 := (div_lt_one m0).mpr m1
-- to show the strict inequality between these series, we prove that:
calc
(∑' i, ... |
simp only [pow_add, one_div, mul_inv, inv_pow]
-- factor the constant `(1 / m ^ (n + 1)!)` out of the series
_ = (∑' i, (1 / m) ^ i) * (1 / m ^ (n + 1)!) := tsum_mul_right
-- the series is the geometric series
_ = (1 - 1 / m)⁻¹ * (1 / m ^ (n + 1)!) := by rw [tsum_geometric_of_lt_one (by positivit... |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 353 | 355 | theorem ContinuousLinearEquiv.comp_contDiffOn_iff (e : F ≃L[𝕜] G) :
ContDiffOn 𝕜 n (e ∘ f) s ↔ ContDiffOn 𝕜 n f s := by |
simp [ContDiffOn, e.comp_contDiffWithinAt_iff]
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,214 | 1,215 | theorem tendsto_norm' {x : E} : Tendsto (fun a => ‖a‖) (𝓝 x) (𝓝 ‖x‖) := by |
simpa using tendsto_id.dist (tendsto_const_nhds : Tendsto (fun _a => (1 : E)) _ _)
|
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 | 333 | 346 | theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by |
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
Real.arcsin_neg]
rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;>
rcases lt_trichotomy x.im 0 with (hi | hi | hi)
· simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm]
· simp [hr, hr.not_le, hi]
· simp [hr, hr... |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 118 | 120 | theorem sum_tensor {P Q R S : C} {J : Type*} (s : Finset J) (f : P ⟶ Q) (g : J → (R ⟶ S)) :
(∑ j ∈ s, g j) ⊗ f = ∑ j ∈ s, g j ⊗ f := by |
simp only [tensorHom_def, sum_whiskerRight, Preadditive.sum_comp]
|
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#align_import linear_algebra.basis.bilinear from "leanprover-community/mathlib"@"87c54600fe3cdc7d32ff5b50873ac724d86aef8d"
namespace LinearMap
variable {ι₁ ι₂ : Type*}
variable {R R₂ S S₂ M N P Rₗ : Type*}
variable {Mₗ Nₗ Pₗ : Type*}
--... | Mathlib/LinearAlgebra/Basis/Bilinear.lean | 44 | 49 | theorem sum_repr_mul_repr_mulₛₗ {B : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (x y) :
((b₁.repr x).sum fun i xi => (b₂.repr y).sum fun j yj => ρ₁₂ xi • σ₁₂ yj • B (b₁ i) (b₂ j)) =
B x y := by |
conv_rhs => rw [← b₁.total_repr x, ← b₂.total_repr y]
simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum₂, map_sum, LinearMap.map_smulₛₗ₂,
LinearMap.map_smulₛₗ]
|
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
noncomputable section
open scoped Manifold
open Bundle Set Topology
section SpecificFunctions
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 503 | 509 | theorem mfderiv_neg (f : M → E') (x : M) :
(mfderiv I 𝓘(𝕜, E') (-f) x : TangentSpace I x →L[𝕜] E') =
(-mfderiv I 𝓘(𝕜, E') f x : TangentSpace I x →L[𝕜] E') := by |
simp_rw [mfderiv]
by_cases hf : MDifferentiableAt I 𝓘(𝕜, E') f x
· exact hf.hasMFDerivAt.neg.mfderiv
· rw [if_neg hf]; rw [← mdifferentiableAt_neg] at hf; rw [if_neg hf, neg_zero]
|
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.group.ext from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da12a06"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
@[to_additive (attr := ext)]
theorem Monoid.ext {M : Type u} ⦃m₁ m₂ : Mo... | Mathlib/Algebra/Group/Ext.lean | 103 | 106 | theorem CancelMonoid.toLeftCancelMonoid_injective {M : Type u} :
Function.Injective (@CancelMonoid.toLeftCancelMonoid M) := by |
rintro ⟨⟩ ⟨⟩ h
congr
|
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 | 388 | 400 | theorem ofDigits_add_ofDigits_eq_ofDigits_zipWith_of_length_eq {b : ℕ} {l1 l2 : List ℕ}
(h : l1.length = l2.length) :
ofDigits b l1 + ofDigits b l2 = ofDigits b (l1.zipWith (· + ·) l2) := by |
induction l1 generalizing l2 with
| nil => simp_all [eq_comm, List.length_eq_zero, ofDigits]
| cons hd₁ tl₁ ih₁ =>
induction l2 generalizing tl₁ with
| nil => simp_all
| cons hd₂ tl₂ ih₂ =>
simp_all only [List.length_cons, succ_eq_add_one, ofDigits_cons, add_left_inj,
eq_comm, List.zipW... |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.Analytic.Basic
#align_import measure_theory.integral.circle_integral from "leanprover-communit... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 130 | 131 | theorem circleMap_not_mem_ball (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ ∉ ball c R := by |
simp [dist_eq, le_abs_self]
|
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 124 | 128 | theorem log_conj_eq_ite (x : ℂ) : log (conj x) = if x.arg = π then log x else conj (log x) := by |
simp_rw [log, abs_conj, arg_conj, map_add, map_mul, conj_ofReal]
split_ifs with hx
· rw [hx]
simp_rw [ofReal_neg, conj_I, mul_neg, neg_mul]
|
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombination
#align_import number_theory.fermat4 from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
noncomputable section
open scope... | Mathlib/NumberTheory/FLT/Four.lean | 114 | 120 | theorem neg_of_minimal {a b c : ℤ} : Minimal a b c → Minimal a b (-c) := by |
rintro ⟨⟨ha, hb, heq⟩, h2⟩
constructor
· apply And.intro ha (And.intro hb _)
rw [heq]
exact (neg_sq c).symm
rwa [Int.natAbs_neg c]
|
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
open MeasureTheory Measure FiniteDimensional
variable {E F G W : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F]
[NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCo... | Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.lean | 101 | 151 | theorem integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable
{f f' : E → F} {g g' : E → G} {v : E} {B : F →L[ℝ] G →L[ℝ] W}
(hf'g : Integrable (fun x ↦ B (f' x) (g x)) μ) (hfg' : Integrable (fun x ↦ B (f x) (g' x)) μ)
(hfg : Integrable (fun x ↦ B (f x) (g x)) μ)
(hf : ∀ x, HasLineDerivAt ℝ f ... |
by_cases hW : CompleteSpace W; swap
· simp [integral, hW]
rcases eq_or_ne v 0 with rfl|hv
· have Hf' x : f' x = 0 := by
simpa [(hasLineDerivAt_zero (f := f) (x := x)).lineDeriv] using (hf x).lineDeriv.symm
have Hg' x : g' x = 0 := by
simpa [(hasLineDerivAt_zero (f := g) (x := x)).lineDeriv] usi... |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ι : Type*} {G : ι → Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [∀ i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 88 | 93 | theorem of_comp_eq_base (i : ι) : (of i).comp (φ i) = (base φ) := by |
ext x
apply (Con.eq _).2
refine ConGen.Rel.of _ _ ?_
simp only [MonoidHom.comp_apply, Set.mem_iUnion, Set.mem_range]
exact ⟨_, _, rfl, rfl⟩
|
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 203 | 214 | theorem map_bracket_eq : map f ⁅I, N⁆ = ⁅I, map f N⁆ := by |
rw [← coe_toSubmodule_eq_iff, coeSubmodule_map, lieIdeal_oper_eq_linear_span,
lieIdeal_oper_eq_linear_span, Submodule.map_span]
congr
ext m
constructor
· rintro ⟨-, ⟨⟨x, ⟨n, hn⟩, rfl⟩, hm⟩⟩
simp only [LieModuleHom.coe_toLinearMap, LieModuleHom.map_lie] at hm
exact ⟨x, ⟨f n, (mem_map (f n)).mpr ⟨n... |
import Mathlib.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
open Finset
section birkhoffAverage
variable (R : Type*) {α M : Type*} [DivisionSemiring R] [AddCommMonoid M] [Module R M]
def birkhoffAverage (f : α → α) (g : α → M) (n : ℕ) (x : α) : M := (n : R)⁻¹ • birkhoffSum f g n x
theorem bir... | Mathlib/Dynamics/BirkhoffSum/Average.lean | 68 | 70 | theorem birkhoffAverage_congr_ring' (S : Type*) [DivisionSemiring S] [Module S M] :
birkhoffAverage (α := α) (M := M) R = birkhoffAverage S := by |
ext; apply birkhoffAverage_congr_ring
|
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-community/mathlib"@"a347076985674932c0e91da09b9961ed0a79508c"
universe u
open Ordinal
namespace SetTheory
open scoped NaturalOps PGame
namespace PGame
noncomputable def b... | Mathlib/SetTheory/Game/Birthday.lean | 177 | 177 | theorem birthday_add_zero : (a + 0).birthday = a.birthday := by | simp
|
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 | 257 | 262 | theorem measure_mono_ae (H : s ≤ᵐ[μ] t) : μ s ≤ μ t :=
calc
μ s ≤ μ (s ∪ t) := measure_mono subset_union_left
_ = μ (t ∪ s \ t) := by | rw [union_diff_self, Set.union_comm]
_ ≤ μ t + μ (s \ t) := measure_union_le _ _
_ = μ t := by rw [ae_le_set.1 H, add_zero]
|
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 124 | 127 | theorem verschiebung_frobenius [CharP R p] (x : 𝕎 R) : verschiebung (frobenius x) = x * p := by |
ext ⟨i⟩
· rw [mul_charP_coeff_zero, verschiebung_coeff_zero]
· rw [mul_charP_coeff_succ, verschiebung_coeff_succ, coeff_frobenius_charP]
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 345 | 347 | theorem eval₂_at_one {S : Type*} [Semiring S] (f : R →+* S) : p.eval₂ f 1 = f (p.eval 1) := by |
convert eval₂_at_apply (p := p) f 1
simp
|
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.GroupTheory.Finiteness
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.finit... | Mathlib/RingTheory/Finiteness.lean | 247 | 256 | theorem fg_pi {ι : Type*} {M : ι → Type*} [Finite ι] [∀ i, AddCommMonoid (M i)]
[∀ i, Module R (M i)] {p : ∀ i, Submodule R (M i)} (hsb : ∀ i, (p i).FG) :
(Submodule.pi Set.univ p).FG := by |
classical
simp_rw [fg_def] at hsb ⊢
choose t htf hts using hsb
refine
⟨⋃ i, (LinearMap.single i : _ →ₗ[R] _) '' t i, Set.finite_iUnion fun i => (htf i).image _, ?_⟩
-- Note: #8386 changed `span_image` into `span_image _`
simp_rw [span_iUnion, span_image _, hts, Submodule.iSup_map_single]
|
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ring_theory.witt_vector.basic from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
noncomputable section
open MvPolynomial Function
variable... | Mathlib/RingTheory/WittVector/Basic.lean | 126 | 126 | theorem pow (n : ℕ) : mapFun f (x ^ n) = mapFun f x ^ n := by | map_fun_tac
|
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 114 | 123 | theorem truncation_eq_of_nonneg {f : α → ℝ} {A : ℝ} (h : ∀ x, 0 ≤ f x) :
truncation f A = indicator (Set.Ioc 0 A) id ∘ f := by |
ext x
rcases (h x).lt_or_eq with (hx | hx)
· simp only [truncation, indicator, hx, Set.mem_Ioc, id, Function.comp_apply, true_and_iff]
by_cases h'x : f x ≤ A
· have : -A < f x := by linarith [h x]
simp only [this, true_and_iff]
· simp only [h'x, and_false_iff]
· simp only [truncation, indicat... |
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 | 491 | 501 | theorem eventually_uniformity_iterate_comp_subset {s : Set (α × α)} (hs : s ∈ 𝓤 α) (n : ℕ) :
∀ᶠ t in (𝓤 α).smallSets, (t ○ ·)^[n] t ⊆ s := by |
suffices ∀ᶠ t in (𝓤 α).smallSets, t ⊆ s ∧ (t ○ ·)^[n] t ⊆ s from (eventually_and.1 this).2
induction' n with n ihn generalizing s
· simpa
rcases comp_mem_uniformity_sets hs with ⟨t, htU, hts⟩
refine (ihn htU).mono fun U hU => ?_
rw [Function.iterate_succ_apply']
exact
⟨hU.1.trans <| (subset_comp_sel... |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Finset.Pointwise
import Mathlib.Tactic.GCongr
#align_import combinatorics.additive.pluennecke_ruzsa from "leanprover-community/mathlib"@"4aab2abced69a9e579b1e6dc... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 176 | 179 | theorem card_div_mul_le_card_mul_mul_card_div (A B C : Finset α) :
(A / C).card * B.card ≤ (A * B).card * (B / C).card := by |
rw [div_eq_mul_inv, div_eq_mul_inv]
exact card_mul_mul_card_le_card_mul_mul_card_mul _ _ _
|
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
namespace Finset
def nonMemberSubfamily (a : α) (𝒜 : ... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 61 | 66 | theorem mem_memberSubfamily : s ∈ 𝒜.memberSubfamily a ↔ insert a s ∈ 𝒜 ∧ a ∉ s := by |
simp_rw [memberSubfamily, mem_image, mem_filter]
refine ⟨?_, fun h => ⟨insert a s, ⟨h.1, by simp⟩, erase_insert h.2⟩⟩
rintro ⟨s, ⟨hs1, hs2⟩, rfl⟩
rw [insert_erase hs2]
exact ⟨hs1, not_mem_erase _ _⟩
|
import Mathlib.Algebra.Module.Defs
import Mathlib.SetTheory.Cardinal.Basic
open Function
universe u v
namespace Cardinal
| Mathlib/Algebra/Module/Card.lean | 24 | 29 | theorem mk_le_of_module (R : Type u) (E : Type v)
[AddCommGroup E] [Ring R] [Module R E] [Nontrivial E] [NoZeroSMulDivisors R E] :
Cardinal.lift.{v} (#R) ≤ Cardinal.lift.{u} (#E) := by |
obtain ⟨x, hx⟩ : ∃ (x : E), x ≠ 0 := exists_ne 0
have : Injective (fun k ↦ k • x) := smul_left_injective R hx
exact lift_mk_le_lift_mk_of_injective this
|
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
open Function
universe u v w
def WSeq (α) :=
Seq (Option α)
#align stream.wseq Stream'.WSeq
... | Mathlib/Data/Seq/WSeq.lean | 1,004 | 1,020 | theorem exists_get?_of_mem {s : WSeq α} {a} (h : a ∈ s) : ∃ n, some a ∈ get? s n := by |
apply mem_rec_on h
· intro a' s' h
cases' h with h h
· exists 0
simp only [get?, drop, head_cons]
rw [h]
apply ret_mem
· cases' h with n h
exists n + 1
-- porting note (#10745): was `simp [get?]`.
simpa [get?]
· intro s' h
cases' h with n h
exists n
sim... |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 119 | 121 | theorem mul_charP_coeff_succ [CharP R p] (x : 𝕎 R) (i : ℕ) :
(x * p).coeff (i + 1) = x.coeff i ^ p := by |
rw [← frobenius_verschiebung, coeff_frobenius_charP, verschiebung_coeff_succ]
|
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 241 | 244 | theorem rightDistributor_inv_comp_biproduct_π {J : Type} [Fintype J] (f : J → C) (X : C) (j : J) :
(rightDistributor f X).inv ≫ (biproduct.π _ j ▷ X) = biproduct.π _ j := by |
simp [rightDistributor_inv, Preadditive.sum_comp, ← MonoidalCategory.comp_whiskerRight,
biproduct.ι_π, dite_whiskerRight, comp_dite]
|
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 | 977 | 992 | theorem integrable_withDensity_iff_integrable_coe_smul₀ {f : α → ℝ≥0} (hf : AEMeasurable f μ)
{g : α → E} :
Integrable g (μ.withDensity fun x => f x) ↔ Integrable (fun x => (f x : ℝ) • g x) μ :=
calc
Integrable g (μ.withDensity fun x => f x) ↔
Integrable g (μ.withDensity fun x => (hf.mk f x : ℝ≥0)... |
suffices (fun x => (f x : ℝ≥0∞)) =ᵐ[μ] (fun x => (hf.mk f x : ℝ≥0)) by
rw [withDensity_congr_ae this]
filter_upwards [hf.ae_eq_mk] with x hx
simp [hx]
_ ↔ Integrable (fun x => ((hf.mk f x : ℝ≥0) : ℝ) • g x) μ :=
integrable_withDensity_iff_integrable_coe_smul hf.measurable_mk
_ ↔... |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 56 | 58 | theorem disjoint_principal_nhdsSet : Disjoint (𝓟 s) (𝓝ˢ t) ↔ Disjoint (closure s) t := by |
rw [disjoint_principal_left, ← subset_interior_iff_mem_nhdsSet, interior_compl,
subset_compl_iff_disjoint_left]
|
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"b07688016d62f81d14508ff339ea3415558d6353"
universe u v
namespace SimpleGraph
open Walk
variable {V : Type u} (G : SimpleGraph V)
def IsAcy... | Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 68 | 80 | theorem isAcyclic_iff_forall_adj_isBridge :
G.IsAcyclic ↔ ∀ ⦃v w : V⦄, G.Adj v w → G.IsBridge s(v, w) := by |
simp_rw [isBridge_iff_adj_and_forall_cycle_not_mem]
constructor
· intro ha v w hvw
apply And.intro hvw
intro u p hp
cases ha p hp
· rintro hb v (_ | ⟨ha, p⟩) hp
· exact hp.not_of_nil
· apply (hb ha).2 _ hp
rw [Walk.edges_cons]
apply List.mem_cons_self
|
import Mathlib.Data.ENNReal.Real
#align_import data.real.conjugate_exponents from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open scoped ENNReal
namespace Real
@[mk_iff]
structure IsConjExponent (p q : ℝ) : Prop where
one_lt : 1 < p
inv_add_inv_conj : p⁻... | Mathlib/Data/Real/ConjExponents.lean | 101 | 102 | theorem mul_eq_add : p * q = p + q := by |
simpa only [sub_mul, sub_eq_iff_eq_add, one_mul] using h.sub_one_mul_conj
|
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 | 251 | 253 | theorem IsTheta.div {f₁ f₂ : α → 𝕜} {g₁ g₂ : α → 𝕜'} (h₁ : f₁ =Θ[l] g₁) (h₂ : f₂ =Θ[l] g₂) :
(fun x ↦ f₁ x / f₂ x) =Θ[l] fun x ↦ g₁ x / g₂ x := by |
simpa only [div_eq_mul_inv] using h₁.mul h₂.inv
|
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 | 239 | 240 | theorem two_zsmul_eq_pi_iff {θ : Angle} : (2 : ℤ) • θ = π ↔ θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ) := by |
rw [two_zsmul, ← two_nsmul, two_nsmul_eq_pi_iff]
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordinal.topology from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
universe u v
open Cardinal Order Topology
namespace Ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 76 | 82 | theorem isOpen_iff : IsOpen s ↔ ∀ o ∈ s, IsLimit o → ∃ a < o, Set.Ioo a o ⊆ s := by |
refine isOpen_iff_mem_nhds.trans <| forall₂_congr fun o ho => ?_
by_cases ho' : IsLimit o
· simp only [(nhdsBasis_Ioc ho'.1).mem_iff, ho', true_implies]
refine exists_congr fun a => and_congr_right fun ha => ?_
simp only [← Set.Ioo_insert_right ha, Set.insert_subset_iff, ho, true_and]
· simp [nhds_eq_p... |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 119 | 123 | theorem sign_inv (r : ℝ) : sign r⁻¹ = sign r := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn, sign_of_neg (inv_lt_zero.mpr hn)]
· rw [sign_zero, inv_zero, sign_zero]
· rw [sign_of_pos hp, sign_of_pos (inv_pos.mpr hp)]
|
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