Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#align_import data.complex.exponential from "leanprover-community/mathlib"@"a8b2226cfb0a79f5986492053fc49b1a0c6aeffb"
open CauSeq Finset IsAbsoluteValue
open ... | Mathlib/Data/Complex/Exponential.lean | 1,259 | 1,259 | theorem exp_lt_one_iff {x : ℝ} : exp x < 1 ↔ x < 0 := by | rw [← exp_zero, exp_lt_exp]
|
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists ... | Mathlib/RingTheory/Ideal/Operations.lean | 1,008 | 1,009 | theorem IsRadical.inf (hI : IsRadical I) (hJ : IsRadical J) : IsRadical (I ⊓ J) := by |
rw [IsRadical, radical_inf]; exact inf_le_inf hI hJ
|
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 | 496 | 498 | theorem addHaar_closedBall' (x : E) {r : ℝ} (hr : 0 ≤ r) :
μ (closedBall x r) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 1) := by |
rw [← addHaar_closedBall_mul μ x hr zero_le_one, mul_one]
|
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 553 | 555 | theorem toLp_sub (f g : α →ₛ E) (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
toLp (f - g) (hf.sub hg) = toLp f hf - toLp g hg := by |
simp only [sub_eq_add_neg, ← toLp_neg, ← toLp_add]
|
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
noncomputable section
open scoped Classical
open ... | Mathlib/Geometry/Euclidean/Basic.lean | 467 | 477 | theorem dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq
{s : AffineSubspace ℝ P} [Nonempty s] [HasOrthogonalProjection s.direction] {p1 : P} (p2 : P)
(hp1 : p1 ∈ s) :
dist p1 p2 * dist p1 p2 =
dist p1 (orthogonalProjection s p2) * dist p1 (orthogonalProjection s p2) +
... |
rw [dist_comm p2 _, dist_eq_norm_vsub V p1 _, dist_eq_norm_vsub V p1 _, dist_eq_norm_vsub V _ p2,
← vsub_add_vsub_cancel p1 (orthogonalProjection s p2) p2,
norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero]
exact Submodule.inner_right_of_mem_orthogonal (vsub_orthogonalProjection_mem_direction p2 hp... |
import Mathlib.Data.Finset.Card
#align_import data.finset.option from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
variable {α β : Type*}
open Function
namespace Finset
def insertNone : Finset α ↪o Finset (Option α) :=
(OrderEmbedding.ofMapLEIff fun s => cons none (s.map Embeddi... | Mathlib/Data/Finset/Option.lean | 98 | 99 | theorem mem_eraseNone {s : Finset (Option α)} {x : α} : x ∈ eraseNone s ↔ some x ∈ s := by |
simp [eraseNone]
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Function Set Filter
open scoped Topology Filter
variable... | Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 118 | 120 | theorem one_lt_rOut_div_rIn {c : E} (f : ContDiffBump c) : 1 < f.rOut / f.rIn := by |
rw [one_lt_div f.rIn_pos]
exact f.rIn_lt_rOut
|
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Image
#align_import data.complex.basic from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
open Set Function
structure Complex : Type where... | Mathlib/Data/Complex/Basic.lean | 792 | 795 | theorem normSq_sub (z w : ℂ) : normSq (z - w) = normSq z + normSq w - 2 * (z * conj w).re := by |
rw [sub_eq_add_neg, normSq_add]
simp only [RingHom.map_neg, mul_neg, neg_re, normSq_neg]
ring
|
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Norm
import Mathlib.Topology.Instances.Complex
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import number_theory.number_field.embeddings from "leanprov... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 385 | 392 | theorem isComplex_iff {w : InfinitePlace K} :
IsComplex w ↔ ¬ComplexEmbedding.IsReal (embedding w) := by |
refine ⟨?_, fun h => ⟨embedding w, h, mk_embedding w⟩⟩
rintro ⟨φ, ⟨hφ, rfl⟩⟩
contrapose! hφ
cases mk_eq_iff.mp (mk_embedding (mk φ)) with
| inl h => rwa [h] at hφ
| inr h => rwa [← ComplexEmbedding.isReal_conjugate_iff, h] at hφ
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.Order.Int... | Mathlib/Order/Filter/AtTopBot.lean | 1,755 | 1,757 | theorem tendsto_comp_val_Ioi_atTop [SemilatticeSup α] [NoMaxOrder α] {a : α} {f : α → β}
{l : Filter β} : Tendsto (fun x : Ioi a => f x) atTop l ↔ Tendsto f atTop l := by |
rw [← map_val_Ioi_atTop a, tendsto_map'_iff, Function.comp_def]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 190 | 193 | theorem HasDerivAt.cpow (hf : HasDerivAt f f' x) (hg : HasDerivAt g g' x)
(h0 : f x ∈ slitPlane) : HasDerivAt (fun x => f x ^ g x)
(g x * f x ^ (g x - 1) * f' + f x ^ g x * Complex.log (f x) * g') x := by |
simpa only [aux] using (hf.hasFDerivAt.cpow hg h0).hasDerivAt
|
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.Ideal.Maps
#align_import data.polynomial.div from "leanprover-community/mathlib"@"e1e7190efdcefc925cb36f257a8362ef22944204"
noncomputable section
open Polynomial
... | Mathlib/Algebra/Polynomial/Div.lean | 638 | 646 | theorem mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero {b : R[X]} {P : R[X][X]} :
P ∈ Ideal.span {C (X - C a), X - C b} ↔ (P.eval b).eval a = 0 := by |
rw [Ideal.mem_span_pair]
constructor <;> intro h
· rcases h with ⟨_, _, rfl⟩
simp only [eval_C, eval_X, eval_add, eval_sub, eval_mul, add_zero, mul_zero, sub_self]
· rcases dvd_iff_isRoot.mpr h with ⟨p, hp⟩
rcases @X_sub_C_dvd_sub_C_eval _ b _ P with ⟨q, hq⟩
exact ⟨C p, q, by rw [mul_comm, mul_comm... |
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.Factors
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Multiplicity
#align_import ring_theory.unique_factorization_domain from "leanprover-community/mathli... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,825 | 1,828 | theorem factors_one [Nontrivial α] : factors (1 : Associates α) = 0 := by |
apply eq_of_prod_eq_prod
rw [Associates.factors_prod]
exact Multiset.prod_zero
|
import Mathlib.Logic.Equiv.Nat
import Mathlib.Data.PNat.Basic
import Mathlib.Order.Directed
import Mathlib.Data.Countable.Defs
import Mathlib.Order.RelIso.Basic
import Mathlib.Data.Fin.Basic
#align_import logic.encodable.basic from "leanprover-community/mathlib"@"7c523cb78f4153682c2929e3006c863bfef463d0"
open Opt... | Mathlib/Logic/Encodable/Basic.lean | 192 | 195 | theorem mem_decode₂' [Encodable α] {n : ℕ} {a : α} :
a ∈ decode₂ α n ↔ a ∈ decode n ∧ encode a = n := by |
simpa [decode₂, bind_eq_some] using
⟨fun ⟨_, h₁, rfl, h₂⟩ => ⟨h₁, h₂⟩, fun ⟨h₁, h₂⟩ => ⟨_, h₁, rfl, h₂⟩⟩
|
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 | 247 | 248 | theorem tprod_eq_one_iff (hf : Multipliable f) : ∏' i, f i = 1 ↔ ∀ x, f x = 1 := by |
rw [← hasProd_one_iff, hf.hasProd_iff]
|
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Init.Data.List.Instances
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.list.basic... | Mathlib/Data/List/Basic.lean | 858 | 861 | theorem tail_append_of_ne_nil (l l' : List α) (h : l ≠ []) : (l ++ l').tail = l.tail ++ l' := by |
cases l
· contradiction
· simp
|
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 | 491 | 493 | theorem Antiperiodic.sub_nsmul_eq [AddGroup α] [AddGroup β] (h : Antiperiodic f c) (n : ℕ) :
f (x - n • c) = (-1) ^ n • f x := by |
simpa only [Int.reduceNeg, natCast_zsmul] using h.sub_zsmul_eq n
|
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Basic
#align_import data.polynomial.induction from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
noncomputable section
open Finsupp Finset
namespace Polynomial
open Polynomial
universe u v w x y z
variable {R ... | Mathlib/Algebra/Polynomial/Induction.lean | 82 | 94 | theorem mem_span_C_coeff : f ∈ Ideal.span { g : R[X] | ∃ i : ℕ, g = C (coeff f i) } := by |
let p := Ideal.span { g : R[X] | ∃ i : ℕ, g = C (coeff f i) }
nth_rw 1 [(sum_C_mul_X_pow_eq f).symm]
refine Submodule.sum_mem _ fun n _hn => ?_
dsimp
have : C (coeff f n) ∈ p := by
apply subset_span
rw [mem_setOf_eq]
use n
have : monomial n (1 : R) • C (coeff f n) ∈ p := p.smul_mem _ this
con... |
import Mathlib.Topology.ContinuousOn
import Mathlib.Data.Set.BoolIndicator
open Set Filter Topology TopologicalSpace Classical
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Clopen
protected theorem IsClopen.isOpen (hs : IsClo... | Mathlib/Topology/Clopen.lean | 151 | 154 | theorem continuousOn_boolIndicator_iff_isClopen (s U : Set X) :
ContinuousOn U.boolIndicator s ↔ IsClopen (((↑) : s → X) ⁻¹' U) := by |
rw [continuousOn_iff_continuous_restrict, ← continuous_boolIndicator_iff_isClopen]
rfl
|
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 | 910 | 912 | theorem exists_nhds_ne_neBot (X : Type*) [TopologicalSpace X] [CompactSpace X] [Infinite X] :
∃ z : X, (𝓝[≠] z).NeBot := by |
simpa [AccPt] using (@infinite_univ X _).exists_accPt_principal
|
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on from "leanprover-community/mathlib"@"8b8ba04e2f326f3f7cf24ad129beda58531ada61"
noncomputable section
open Set Filter TopologicalSpace MeasureTheory Function
ope... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 99 | 99 | theorem integrableOn_empty : IntegrableOn f ∅ μ := by | simp [IntegrableOn, integrable_zero_measure]
|
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 | 810 | 822 | theorem Integrable.smul_essSup {𝕜 : Type*} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β]
{f : α → 𝕜} (hf : Integrable f μ) {g : α → β}
(g_aestronglyMeasurable : AEStronglyMeasurable g μ)
(ess_sup_g : essSup (fun x => (‖g x‖₊ : ℝ≥0∞)) μ ≠ ∞) :
Integrable (fun x : α => f x • g x) μ := by |
rw [← memℒp_one_iff_integrable] at *
refine ⟨hf.1.smul g_aestronglyMeasurable, ?_⟩
have h : (1 : ℝ≥0∞) / 1 = 1 / 1 + 1 / ∞ := by norm_num
have hg' : snorm g ∞ μ ≠ ∞ := by rwa [snorm_exponent_top]
calc
snorm (fun x : α => f x • g x) 1 μ ≤ _ := by
simpa using MeasureTheory.snorm_smul_le_mul_snorm g_a... |
import Mathlib.Data.Sum.Order
import Mathlib.Order.InitialSeg
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
#align_import set_theory.ordinal.basic from "leanprover-community/mathlib"@"8ea5598db6caeddde6cb734aa179cc2408dbd345"
assert_not_exists Module
assert_not_exists Field
noncomputabl... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,132 | 1,136 | theorem typein_le_typein' (o : Ordinal) {x x' : o.out.α} :
@typein _ (· < ·) (isWellOrder_out_lt _) x ≤ @typein _ (· < ·) (isWellOrder_out_lt _) x'
↔ x ≤ x' := by |
rw [typein_le_typein]
exact not_lt
|
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space
#align_import measure_theory.function.uniform_integrable from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
open scoped Classical MeasureTheory NNReal ENNReal... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 424 | 447 | theorem unifIntegrable_fin (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) {n : ℕ} {f : Fin n → α → β}
(hf : ∀ i, Memℒp (f i) p μ) : UnifIntegrable f p μ := by |
revert f
induction' n with n h
· intro f hf
-- Porting note (#10754): added this instance
have : Subsingleton (Fin Nat.zero) := subsingleton_fin_zero
exact unifIntegrable_subsingleton hp_one hp_top hf
intro f hfLp ε hε
let g : Fin n → α → β := fun k => f k
have hgLp : ∀ i, Memℒp (g i) p μ := fun ... |
import Mathlib.Topology.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Function Topology
noncomputable section
namespace TopologicalSpace
universe u
variable {α : Type u} {β : Type*} ... | Mathlib/Topology/Bases.lean | 421 | 429 | theorem _root_.Pairwise.countable_of_isOpen_disjoint [SeparableSpace α] {ι : Type*}
{s : ι → Set α} (hd : Pairwise (Disjoint on s)) (ho : ∀ i, IsOpen (s i))
(hne : ∀ i, (s i).Nonempty) : Countable ι := by |
rcases exists_countable_dense α with ⟨u, u_countable, u_dense⟩
choose f hfu hfs using fun i ↦ u_dense.exists_mem_open (ho i) (hne i)
have f_inj : Injective f := fun i j hij ↦
hd.eq <| not_disjoint_iff.2 ⟨f i, hfs i, hij.symm ▸ hfs j⟩
have := u_countable.to_subtype
exact (f_inj.codRestrict hfu).countable
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 1,590 | 1,593 | theorem map_direction (s : AffineSubspace k P₁) :
(s.map f).direction = s.direction.map f.linear := by |
rw [direction_eq_vectorSpan, direction_eq_vectorSpan, coe_map,
AffineMap.vectorSpan_image_eq_submodule_map]
|
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.RingTheory.Valuation.RankOne
import Mathlib.Topology.Algebra.Valuation
noncomputable section
open Filter Set Valuation
open scoped NNReal
variable {K : Type*} [hK : NormedField K] (h : IsNonarchimedean (norm : K → ℝ))
namespace Valued
variable {L : Typ... | Mathlib/Topology/Algebra/NormedValued.lean | 70 | 72 | theorem norm_add_le (x y : L) : norm (x + y) ≤ max (norm x) (norm y) := by |
simp only [norm, NNReal.coe_le_coe, le_max_iff, StrictMono.le_iff_le hv.strictMono]
exact le_max_iff.mp (Valuation.map_add_le_max' val.v _ _)
|
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 | 816 | 819 | theorem CompactSpace.elim_nhds_subcover [CompactSpace X] (U : X → Set X) (hU : ∀ x, U x ∈ 𝓝 x) :
∃ t : Finset X, ⋃ x ∈ t, U x = ⊤ := by |
obtain ⟨t, -, s⟩ := IsCompact.elim_nhds_subcover isCompact_univ U fun x _ => hU x
exact ⟨t, top_unique s⟩
|
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F ... | Mathlib/MeasureTheory/Group/Integral.lean | 70 | 74 | theorem integral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x * g) ∂μ) = ∫ x, f x ∂μ := by |
have h_mul : MeasurableEmbedding fun x => x * g :=
(MeasurableEquiv.mulRight g).measurableEmbedding
rw [← h_mul.integral_map, map_mul_right_eq_self]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 101 | 102 | theorem logb_div_base {a b : ℝ} (h₁ : a ≠ 0) (h₂ : b ≠ 0) (c : ℝ) :
logb (a / b) c = ((logb a c)⁻¹ - (logb b c)⁻¹)⁻¹ := by | rw [← inv_logb_div_base h₁ h₂ c, inv_inv]
|
import Mathlib.Topology.Instances.ENNReal
import Mathlib.MeasureTheory.Measure.Dirac
#align_import probability.probability_mass_function.basic from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal M... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 115 | 133 | theorem apply_eq_one_iff (p : PMF α) (a : α) : p a = 1 ↔ p.support = {a} := by |
refine ⟨fun h => Set.Subset.antisymm (fun a' ha' => by_contra fun ha => ?_)
fun a' ha' => ha'.symm ▸ (p.mem_support_iff a).2 fun ha => zero_ne_one <| ha.symm.trans h,
fun h => _root_.trans (symm <| tsum_eq_single a
fun a' ha' => (p.apply_eq_zero_iff a').2 (h.symm ▸ ha')) p.tsum_coe⟩
suffices 1 < ∑' a... |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {α : Type*} [DecidableEq α]
def nextOr : ∀ (_ : List α) (_ _ : α), α
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 62 | 73 | theorem nextOr_eq_nextOr_of_mem_of_ne (xs : List α) (x d d' : α) (x_mem : x ∈ xs)
(x_ne : x ≠ xs.getLast (ne_nil_of_mem x_mem)) : nextOr xs x d = nextOr xs x d' := by |
induction' xs with y ys IH
· cases x_mem
cases' ys with z zs
· simp at x_mem x_ne
contradiction
by_cases h : x = y
· rw [h, nextOr_self_cons_cons, nextOr_self_cons_cons]
· rw [nextOr, nextOr, IH]
· simpa [h] using x_mem
· simpa using x_ne
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Rat
import Mathlib.Data.PNat.Defs
#align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
namespace Rat
o... | Mathlib/Data/Rat/Lemmas.lean | 114 | 119 | theorem add_num_den (q r : ℚ) :
q + r = (q.num * r.den + q.den * r.num : ℤ) /. (↑q.den * ↑r.den : ℤ) := by |
have hqd : (q.den : ℤ) ≠ 0 := Int.natCast_ne_zero_iff_pos.2 q.den_pos
have hrd : (r.den : ℤ) ≠ 0 := Int.natCast_ne_zero_iff_pos.2 r.den_pos
conv_lhs => rw [← num_divInt_den q, ← num_divInt_den r, divInt_add_divInt _ _ hqd hrd]
rw [mul_comm r.num q.den]
|
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Units
#align_import algebra.divisibility.units from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da12a06"
variable {α : Type*}
namespace Units
end IsUnit
section CommMonoid
variable [CommMonoid α]
theorem isUnit_iff_dvd... | Mathlib/Algebra/Divisibility/Units.lean | 211 | 212 | theorem isRelPrime_mul_unit_right_right : IsRelPrime y (z * x) ↔ IsRelPrime y z := by |
rw [mul_comm, isRelPrime_mul_unit_left_right hu]
|
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 169 | 179 | theorem closureHasCore (f : E →ₗ.[R] F) : f.closure.HasCore f.domain := by |
refine ⟨f.le_closure.1, ?_⟩
congr
ext x y hxy
· simp only [domRestrict_domain, Submodule.mem_inf, and_iff_left_iff_imp]
intro hx
exact f.le_closure.1 hx
let z : f.closure.domain := ⟨y.1, f.le_closure.1 y.2⟩
have hyz : (y : E) = z := by simp
rw [f.le_closure.2 hyz]
exact domRestrict_apply (hxy.t... |
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#align_import data.complex.exponential from "leanprover-community/mathlib"@"a8b2226cfb0a79f5986492053fc49b1a0c6aeffb"
open CauSeq Finset IsAbsoluteValue
open ... | Mathlib/Data/Complex/Exponential.lean | 1,127 | 1,127 | theorem cosh_sq_sub_sinh_sq (x : ℝ) : cosh x ^ 2 - sinh x ^ 2 = 1 := by | rw [← ofReal_inj]; simp
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
... | Mathlib/RingTheory/PowerSeries/Basic.lean | 628 | 628 | theorem rescale_zero_apply : rescale 0 X = C R (constantCoeff R X) := by | simp
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.add from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
variable... | Mathlib/Analysis/Calculus/Deriv/Add.lean | 153 | 155 | theorem HasDerivAtFilter.sum (h : ∀ i ∈ u, HasDerivAtFilter (A i) (A' i) x L) :
HasDerivAtFilter (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x L := by |
simpa [ContinuousLinearMap.sum_apply] using (HasFDerivAtFilter.sum h).hasDerivAtFilter
|
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set.Countable
import Mathlib.Logic.Small.Set
import Mathlib.Order.SuccPred.CompleteLinearOrder
import Mathlib.SetTheory.Cardinal.SchroederBernstein
#align_import set_theory.cardinal.basic f... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,074 | 2,081 | theorem mk_set_eq_nat_iff_finset {α} {s : Set α} {n : ℕ} :
#s = n ↔ ∃ t : Finset α, (t : Set α) = s ∧ t.card = n := by |
constructor
· intro h
lift s to Finset α using lt_aleph0_iff_set_finite.1 (h.symm ▸ nat_lt_aleph0 n)
simpa using h
· rintro ⟨t, rfl, rfl⟩
exact mk_coe_finset
|
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 | 591 | 593 | theorem add_le_right_iff_mul_omega_le {a b : Ordinal} : a + b ≤ b ↔ a * omega ≤ b := by |
rw [← add_eq_right_iff_mul_omega_le]
exact (add_isNormal a).le_iff_eq
|
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathlib.Order.RelSeries
#align_import order.jordan_holder from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u
... | Mathlib/Order/JordanHolder.lean | 432 | 437 | theorem length_eq_zero_of_head_eq_head_of_last_eq_last_of_length_eq_zero
{s₁ s₂ : CompositionSeries X} (hb : s₁.head = s₂.head)
(ht : s₁.last = s₂.last) (hs₁ : s₁.length = 0) : s₂.length = 0 := by |
have : Fin.last s₂.length = (0 : Fin s₂.length.succ) :=
s₂.injective (hb.symm.trans ((congr_arg s₁ (Fin.ext (by simp [hs₁]))).trans ht)).symm
simpa [Fin.ext_iff]
|
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 | 216 | 218 | theorem eventually_nhdsSet_iUnion₂ {ι : Sort*} {p : ι → Prop} {s : ι → Set X} {P : X → Prop} :
(∀ᶠ x in 𝓝ˢ (⋃ (i) (_ : p i), s i), P x) ↔ ∀ i, p i → ∀ᶠ x in 𝓝ˢ (s i), P x := by |
simp only [nhdsSet_iUnion, eventually_iSup]
|
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 175 | 178 | theorem coeff_C_mul (p : R[X]) : coeff (C a * p) n = a * coeff p n := by |
rcases p with ⟨p⟩
simp_rw [← monomial_zero_left, ← ofFinsupp_single, ← ofFinsupp_mul, coeff]
exact AddMonoidAlgebra.single_zero_mul_apply p a n
|
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn
#align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Function Set Filter Metric
open scoped Topology Classical NNReal
n... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean | 101 | 108 | theorem approximates_deriv_on_open_nhds (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
∃ s : Set E, a ∈ s ∧ IsOpen s ∧
ApproximatesLinearOn f (f' : E →L[𝕜] F) s (‖(f'.symm : F →L[𝕜] E)‖₊⁻¹ / 2) := by |
simp only [← and_assoc]
refine ((nhds_basis_opens a).exists_iff fun s t => ApproximatesLinearOn.mono_set).1 ?_
exact
hf.approximates_deriv_on_nhds <|
f'.subsingleton_or_nnnorm_symm_pos.imp id fun hf' => half_pos <| inv_pos.2 hf'
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.Order.Int... | Mathlib/Order/Filter/AtTopBot.lean | 2,025 | 2,036 | theorem tendsto_of_subseq_tendsto {α ι : Type*} {x : ι → α} {f : Filter α} {l : Filter ι}
[l.IsCountablyGenerated]
(hxy : ∀ ns : ℕ → ι, Tendsto ns atTop l →
∃ ms : ℕ → ℕ, Tendsto (fun n => x (ns <| ms n)) atTop f) :
Tendsto x l f := by |
contrapose! hxy
obtain ⟨s, hs, hfreq⟩ : ∃ s ∈ f, ∃ᶠ n in l, x n ∉ s := by
rwa [not_tendsto_iff_exists_frequently_nmem] at hxy
obtain ⟨y, hy_tendsto, hy_freq⟩ := exists_seq_forall_of_frequently hfreq
refine ⟨y, hy_tendsto, fun ms hms_tendsto ↦ ?_⟩
rcases (hms_tendsto.eventually_mem hs).exists with ⟨n, hn⟩... |
import Batteries.Classes.SatisfiesM
namespace Array
theorem SatisfiesM_foldlM [Monad m] [LawfulMonad m]
{as : Array α} (motive : Nat → β → Prop) {init : β} (h0 : motive 0 init) {f : β → α → m β}
(hf : ∀ i : Fin as.size, ∀ b, motive i.1 b → SatisfiesM (motive (i.1 + 1)) (f b as[i])) :
SatisfiesM (motive... | .lake/packages/batteries/Batteries/Data/Array/Monadic.lean | 112 | 126 | theorem SatisfiesM_foldrM [Monad m] [LawfulMonad m]
{as : Array α} (motive : Nat → β → Prop)
{init : β} (h0 : motive as.size init) {f : α → β → m β}
(hf : ∀ i : Fin as.size, ∀ b, motive (i.1 + 1) b → SatisfiesM (motive i.1) (f as[i] b)) :
SatisfiesM (motive 0) (as.foldrM f init) := by |
let rec go {i b} (hi : i ≤ as.size) (H : motive i b) :
SatisfiesM (motive 0) (foldrM.fold f as 0 i hi b) := by
unfold foldrM.fold; simp; split
· next hi => exact .pure (hi ▸ H)
· next hi =>
split; {simp at hi}
· next i hi' =>
exact (hf ⟨i, hi'⟩ b H).bind fun _ => go _
simp [fold... |
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 592 | 599 | theorem zpow_eq_one_iff_dvd (h : IsPrimitiveRoot ζ k) (l : ℤ) : ζ ^ l = 1 ↔ (k : ℤ) ∣ l := by |
by_cases h0 : 0 ≤ l
· lift l to ℕ using h0; rw [zpow_natCast]; norm_cast; exact h.pow_eq_one_iff_dvd l
· have : 0 ≤ -l := by simp only [not_le, neg_nonneg] at h0 ⊢; exact le_of_lt h0
lift -l to ℕ using this with l' hl'
rw [← dvd_neg, ← hl']
norm_cast
rw [← h.pow_eq_one_iff_dvd, ← inv_inj, ← zpow_... |
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 | 190 | 194 | theorem card_parts_equitabilise (hm : m ≠ 0) : (P.equitabilise h).parts.card = a + b := by |
rw [← filter_true_of_mem fun x => card_eq_of_mem_parts_equitabilise, filter_or,
card_union_of_disjoint, P.card_filter_equitabilise_small _ hm, P.card_filter_equitabilise_big]
-- Porting note (#11187): was `infer_instance`
exact disjoint_filter.2 fun x _ h₀ h₁ => Nat.succ_ne_self m <| h₁.symm.trans h₀
|
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 798 | 802 | theorem norm_indicatorConstLp_top (hμs_ne_zero : μ s ≠ 0) :
‖indicatorConstLp ∞ hs hμs c‖ = ‖c‖ := by |
rw [Lp.norm_def, snorm_congr_ae indicatorConstLp_coeFn,
snorm_indicator_const' hs hμs_ne_zero ENNReal.top_ne_zero, ENNReal.top_toReal, _root_.div_zero,
ENNReal.rpow_zero, mul_one, ENNReal.coe_toReal, coe_nnnorm]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.invertible from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
variable {α : Type*} [LinearOrderedSemiring α] {a : α}
@[simp]
theorem invOf_pos [I... | Mathlib/Algebra/Order/Invertible.lean | 29 | 31 | theorem invOf_nonneg [Invertible a] : 0 ≤ ⅟ a ↔ 0 ≤ a :=
haveI : 0 < a * ⅟ a := by | simp only [mul_invOf_self, zero_lt_one]
⟨fun h => (pos_of_mul_pos_left this h).le, fun h => (pos_of_mul_pos_right this h).le⟩
|
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
suppress_compilation
open Set
variable {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L]
(H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R H]
{M : Type*} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L ... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 61 | 69 | theorem lie_mem_weightSpace_of_mem_weightSpace {χ₁ χ₂ : H → R} {x : L} {m : M}
(hx : x ∈ rootSpace H χ₁) (hm : m ∈ weightSpace M χ₂) : ⁅x, m⁆ ∈ weightSpace M (χ₁ + χ₂) := by |
rw [weightSpace, LieSubmodule.mem_iInf]
intro y
replace hx : x ∈ weightSpaceOf L (χ₁ y) y := by
rw [rootSpace, weightSpace, LieSubmodule.mem_iInf] at hx; exact hx y
replace hm : m ∈ weightSpaceOf M (χ₂ y) y := by
rw [weightSpace, LieSubmodule.mem_iInf] at hm; exact hm y
exact lie_mem_maxGenEigenspace... |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {α : Type*} [DecidableEq α]
def nextOr : ∀ (_ : List α) (_ _ : α), α
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 180 | 204 | theorem next_getLast_cons (h : x ∈ l) (y : α) (h : x ∈ y :: l) (hy : x ≠ y)
(hx : x = getLast (y :: l) (cons_ne_nil _ _)) (hl : Nodup l) : next (y :: l) x h = y := by |
rw [next, get, ← dropLast_append_getLast (cons_ne_nil y l), hx, nextOr_concat]
subst hx
intro H
obtain ⟨⟨_ | k, hk⟩, hk'⟩ := get_of_mem H
· rw [← Option.some_inj] at hk'
rw [← get?_eq_get, dropLast_eq_take, get?_take, get?_zero, head?_cons,
Option.some_inj] at hk'
· exact hy (Eq.symm hk')
r... |
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 146 | 149 | theorem content_C_mul (r : R) (p : R[X]) : (C r * p).content = normalize r * p.content := by |
by_cases h0 : r = 0; · simp [h0]
rw [content]; rw [content]; rw [← Finset.gcd_mul_left]
refine congr (congr rfl ?_) ?_ <;> ext <;> simp [h0, mem_support_iff]
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import data.nat.choose.central from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
namespace Nat
def centralBinom (n : ℕ) :=
(2 * n).choose n
#alig... | Mathlib/Data/Nat/Choose/Central.lean | 88 | 98 | theorem four_pow_lt_mul_centralBinom (n : ℕ) (n_big : 4 ≤ n) : 4 ^ n < n * centralBinom n := by |
induction' n using Nat.strong_induction_on with n IH
rcases lt_trichotomy n 4 with (hn | rfl | hn)
· clear IH; exact False.elim ((not_lt.2 n_big) hn)
· norm_num [centralBinom, choose]
obtain ⟨n, rfl⟩ : ∃ m, n = m + 1 := Nat.exists_eq_succ_of_ne_zero (Nat.not_eq_zero_of_lt hn)
calc
4 ^ (n + 1) < 4 * (n ... |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 1,124 | 1,127 | theorem vectorSpan_range_eq_span_range_vsub_right (p : ι → P) (i0 : ι) :
vectorSpan k (Set.range p) = Submodule.span k (Set.range fun i : ι => p i -ᵥ p i0) := by |
rw [vectorSpan_eq_span_vsub_set_right k (Set.mem_range_self i0), ← Set.range_comp]
congr
|
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.NormedSpace.Completion
#align_import analysis.complex.liouville from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Metric Set Filter Asymptotics ... | Mathlib/Analysis/Complex/Liouville.lean | 134 | 145 | theorem eq_const_of_tendsto_cocompact [Nontrivial E] {f : E → F} (hf : Differentiable ℂ f) {c : F}
(hb : Tendsto f (cocompact E) (𝓝 c)) : f = Function.const E c := by |
have h_bdd : Bornology.IsBounded (Set.range f) := by
obtain ⟨s, hs, hs_bdd⟩ := Metric.exists_isBounded_image_of_tendsto hb
obtain ⟨t, ht, hts⟩ := mem_cocompact.mp hs
apply ht.image hf.continuous |>.isBounded.union hs_bdd |>.subset
simpa [Set.image_union, Set.image_univ] using Set.image_subset _ <| ca... |
import Mathlib.Algebra.Module.Hom
import Mathlib.Algebra.Module.LinearMap.End
#align_import algebra.module.equiv from "leanprover-community/mathlib"@"ea94d7cd54ad9ca6b7710032868abb7c6a104c9c"
open Function
universe u u' v w x y z
variable {R : Type*} {R₁ : Type*} {R₂ : Type*} {R₃ : Type*}
variable {k : Type*} {K... | Mathlib/Algebra/Module/Equiv.lean | 1,169 | 1,174 | theorem arrowCongr_comp {N N₂ N₃ : Sort _} [AddCommMonoid N] [AddCommMonoid N₂] [AddCommMonoid N₃]
[Module R N] [Module R N₂] [Module R N₃] (e₁ : M ≃ₗ[R] N) (e₂ : M₂ ≃ₗ[R] N₂) (e₃ : M₃ ≃ₗ[R] N₃)
(f : M →ₗ[R] M₂) (g : M₂ →ₗ[R] M₃) :
arrowCongr e₁ e₃ (g.comp f) = (arrowCongr e₂ e₃ g).comp (arrowCongr e₁ e₂ f)... |
ext
simp only [symm_apply_apply, arrowCongr_apply, LinearMap.comp_apply]
|
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Order.Filter.AtTopBot
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
import Mathlib.MeasureTheory.... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,054 | 1,065 | theorem integral_eq_zero_of_hasDerivAt_of_integrable
(hderiv : ∀ x, HasDerivAt f (f' x) x) (hf' : Integrable f') (hf : Integrable f) :
∫ x, f' x = 0 := by |
by_cases hE : CompleteSpace E; swap
· simp [integral, hE]
have A : Tendsto f atBot (𝓝 0) :=
tendsto_zero_of_hasDerivAt_of_integrableOn_Iic (a := 0) (fun x _hx ↦ hderiv x)
hf'.integrableOn hf.integrableOn
have B : Tendsto f atTop (𝓝 0) :=
tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi (a := 0) (... |
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.ContinuousOn
import Mathlib.Topology.Instances.ENNReal
#align_import topology.semicontinuous from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal
open Set Function Filter
variable {α : Type*} [... | Mathlib/Topology/Semicontinuous.lean | 1,254 | 1,258 | theorem continuousOn_iff_lower_upperSemicontinuousOn {f : α → γ} :
ContinuousOn f s ↔ LowerSemicontinuousOn f s ∧ UpperSemicontinuousOn f s := by |
simp only [ContinuousOn, continuousWithinAt_iff_lower_upperSemicontinuousWithinAt]
exact
⟨fun H => ⟨fun x hx => (H x hx).1, fun x hx => (H x hx).2⟩, fun H x hx => ⟨H.1 x hx, H.2 x hx⟩⟩
|
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
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 572 | 574 | theorem zpow_eq_one_iff {a : Solution₁ d} (h : IsFundamental a) (n : ℤ) : a ^ n = 1 ↔ n = 0 := by |
rw [← zpow_zero a]
exact ⟨fun H => h.y_strictMono.injective (congr_arg Solution₁.y H), fun H => H ▸ rfl⟩
|
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 215 | 218 | theorem image_bot (s : Set α) : (⊥ : Rel α β).image s = ∅ := by |
rw [Set.eq_empty_iff_forall_not_mem]
intro x h
simp [mem_image, Bot.bot] at h
|
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 | 159 | 166 | theorem sublists_append (l₁ l₂ : List α) :
sublists (l₁ ++ l₂) = (sublists l₂) >>= (fun x => (sublists l₁).map (· ++ x)) := by |
simp only [sublists, foldr_append]
induction l₁ with
| nil => simp
| cons a l₁ ih =>
rw [foldr_cons, ih]
simp [List.bind, join_join, Function.comp]
|
import Mathlib.Algebra.Polynomial.Taylor
import Mathlib.FieldTheory.RatFunc.AsPolynomial
#align_import field_theory.laurent from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
namespace RatFunc
noncomputable section
open Polynomial
open scoped Classical nonZeroDivisors Po... | Mathlib/FieldTheory/Laurent.lean | 108 | 108 | theorem laurent_at_zero : laurent 0 f = f := by | induction f using RatFunc.induction_on; simp
|
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Function Set Filter Topology TopologicalSpace
open scoped... | Mathlib/Topology/Separation.lean | 2,517 | 2,523 | theorem totallySeparatedSpace_of_t1_of_basis_clopen [T1Space X]
(h : IsTopologicalBasis { s : Set X | IsClopen s }) : TotallySeparatedSpace X := by |
constructor
rintro x - y - hxy
rcases h.mem_nhds_iff.mp (isOpen_ne.mem_nhds hxy) with ⟨U, hU, hxU, hyU⟩
exact ⟨U, Uᶜ, hU.isOpen, hU.compl.isOpen, hxU, fun h => hyU h rfl, (union_compl_self U).superset,
disjoint_compl_right⟩
|
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.order.archimedean from "leanprover-community/mathlib"@"6f413f3f7330b94c92a5a27488fdc74e6d483a78"
open Int Set
variable {α : Type*}
class Archimedean (... | Mathlib/Algebra/Order/Archimedean.lean | 287 | 303 | theorem exists_rat_btwn {x y : α} (h : x < y) : ∃ q : ℚ, x < q ∧ (q : α) < y := by |
cases' exists_nat_gt (y - x)⁻¹ with n nh
cases' exists_floor (x * n) with z zh
refine ⟨(z + 1 : ℤ) / n, ?_⟩
have n0' := (inv_pos.2 (sub_pos.2 h)).trans nh
have n0 := Nat.cast_pos.1 n0'
rw [Rat.cast_div_of_ne_zero, Rat.cast_natCast, Rat.cast_intCast, div_lt_iff n0']
· refine ⟨(lt_div_iff n0').2 <| (lt_iff... |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.Order.Filter.ENNReal
#align_import measure_theory.function.ess_sup from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open MeasureTheory Filter Set TopologicalSpace
open ENNReal MeasureTheory NNReal
variable {... | Mathlib/MeasureTheory/Function/EssSup.lean | 200 | 202 | theorem essInf_antitone_measure {f : α → β} (hμν : μ ≪ ν) : essInf f ν ≤ essInf f μ := by |
refine liminf_le_liminf_of_le (Measure.ae_le_iff_absolutelyContinuous.mpr hμν) ?_ ?_
all_goals isBoundedDefault
|
import Mathlib.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
import Mathlib.Order.SetNotation
#align_import order.complete_lattice from "leanprover-community/mathlib"@"5709b0d8725255e76f47debca6400c07b5c2... | Mathlib/Order/CompleteLattice.lean | 1,208 | 1,209 | theorem iSup_sup [Nonempty ι] {f : ι → α} {a : α} : (⨆ x, f x) ⊔ a = ⨆ x, f x ⊔ a := by |
rw [iSup_sup_eq, iSup_const]
|
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.TensorProduct.Basic
suppress_compilation
universe u v w
open scoped TensorProduct
class CoalgebraStruct (R : Type u) (A : Type v)
[CommSemiring R] [AddCommMonoid A] [Module R A] where
comul : A →ₗ[R] A ⊗... | Mathlib/RingTheory/Coalgebra/Basic.lean | 205 | 207 | theorem comul_comp_lsingle (i : ι) :
comul ∘ₗ (lsingle i : A →ₗ[R] _) = TensorProduct.map (lsingle i) (lsingle i) ∘ₗ comul := by |
ext; simp
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 745 | 747 | theorem sbtw_iff_right_ne_and_left_mem_image_Ioi {x y z : P} :
Sbtw R x y z ↔ z ≠ y ∧ x ∈ lineMap z y '' Set.Ioi (1 : R) := by |
rw [sbtw_comm, sbtw_iff_left_ne_and_right_mem_image_Ioi]
|
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722"
open Finset
open Multiset
open Polynomial
universe u w
variable {R : Type u} {ι : Type w}
namespace Polynomial
variable (s : Finset ι)
sectio... | Mathlib/Algebra/Polynomial/BigOperators.lean | 321 | 326 | theorem natDegree_prod (h : ∀ i ∈ s, f i ≠ 0) :
(∏ i ∈ s, f i).natDegree = ∑ i ∈ s, (f i).natDegree := by |
nontriviality R
apply natDegree_prod'
rw [prod_ne_zero_iff]
intro x hx; simp [h x hx]
|
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheory.Torsion
import Mathlib.RingTheory.Coprime.Ideal
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Set.Lattice
#align_import algebra.module.torsion from "leanpr... | Mathlib/Algebra/Module/Torsion.lean | 341 | 344 | theorem isTorsionBySet_iff_is_torsion_by_span :
IsTorsionBySet R M s ↔ IsTorsionBySet R M (Ideal.span s) := by |
rw [isTorsionBySet_iff_torsionBySet_eq_top, isTorsionBySet_iff_torsionBySet_eq_top,
torsionBySet_eq_torsionBySet_span]
|
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 | 700 | 705 | theorem toIocDiv_wcovBy_toIcoDiv (a b : α) : toIocDiv hp a b ⩿ toIcoDiv hp a b := by |
suffices toIocDiv hp a b = toIcoDiv hp a b ∨ toIocDiv hp a b + 1 = toIcoDiv hp a b by
rwa [wcovBy_iff_eq_or_covBy, ← Order.succ_eq_iff_covBy]
rw [eq_comm, ← not_modEq_iff_toIcoDiv_eq_toIocDiv, eq_comm, ←
modEq_iff_toIcoDiv_eq_toIocDiv_add_one]
exact em' _
|
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 | 84 | 84 | theorem gold_sub_goldConj : φ - ψ = √5 := by | ring
|
import Mathlib.CategoryTheory.GlueData
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.Tactic.Generalize
import Mathlib.CategoryTheory.Elementwise
#align_import topology.gluing from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d... | Mathlib/Topology/Gluing.lean | 104 | 115 | theorem isOpen_iff (U : Set 𝖣.glued) : IsOpen U ↔ ∀ i, IsOpen (𝖣.ι i ⁻¹' U) := by |
delta CategoryTheory.GlueData.ι
simp_rw [← Multicoequalizer.ι_sigmaπ 𝖣.diagram]
rw [← (homeoOfIso (Multicoequalizer.isoCoequalizer 𝖣.diagram).symm).isOpen_preimage]
rw [coequalizer_isOpen_iff]
dsimp only [GlueData.diagram_l, GlueData.diagram_left, GlueData.diagram_r, GlueData.diagram_right,
parallelPai... |
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Monotone.Basic
#align_import order.iterate from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
open Function
open Function (Commute)
namespace Monotone
variable {α : Type*} [Preorder α] {f : α → α} {x y : ℕ → α}
| Mathlib/Order/Iterate.lean | 42 | 48 | theorem seq_le_seq (hf : Monotone f) (n : ℕ) (h₀ : x 0 ≤ y 0) (hx : ∀ k < n, x (k + 1) ≤ f (x k))
(hy : ∀ k < n, f (y k) ≤ y (k + 1)) : x n ≤ y n := by |
induction' n with n ihn
· exact h₀
· refine (hx _ n.lt_succ_self).trans ((hf <| ihn ?_ ?_).trans (hy _ n.lt_succ_self))
· exact fun k hk => hx _ (hk.trans n.lt_succ_self)
· exact fun k hk => hy _ (hk.trans n.lt_succ_self)
|
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 76 | 85 | theorem norm_le_iff [NumberField K] (x : K) (r : ℝ) :
‖canonicalEmbedding K x‖ ≤ r ↔ ∀ φ : K →+* ℂ, ‖φ x‖ ≤ r := by |
obtain hr | hr := lt_or_le r 0
· obtain ⟨φ⟩ := (inferInstance : Nonempty (K →+* ℂ))
refine iff_of_false ?_ ?_
· exact (hr.trans_le (norm_nonneg _)).not_le
· exact fun h => hr.not_le (le_trans (norm_nonneg _) (h φ))
· lift r to NNReal using hr
simp_rw [← coe_nnnorm, nnnorm_eq, NNReal.coe_le_coe, F... |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 228 | 236 | theorem rtakeWhile_eq_nil_iff : rtakeWhile p l = [] ↔ ∀ hl : l ≠ [], ¬p (l.getLast hl) := by |
induction' l using List.reverseRecOn with l a
· simp only [rtakeWhile, takeWhile, reverse_nil, true_iff]
intro f; contradiction
· simp only [rtakeWhile, reverse_append, takeWhile, reverse_eq_nil_iff, getLast_append, ne_eq,
append_eq_nil, and_false, not_false_eq_true, forall_true_left]
refine ⟨fun h... |
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 615 | 626 | theorem zpow_of_gcd_eq_one (h : IsPrimitiveRoot ζ k) (i : ℤ) (hi : i.gcd k = 1) :
IsPrimitiveRoot (ζ ^ i) k := by |
by_cases h0 : 0 ≤ i
· lift i to ℕ using h0
rw [zpow_natCast]
exact h.pow_of_coprime i hi
have : 0 ≤ -i := by simp only [not_le, neg_nonneg] at h0 ⊢; exact le_of_lt h0
lift -i to ℕ using this with i' hi'
rw [← inv_iff, ← zpow_neg, ← hi', zpow_natCast]
apply h.pow_of_coprime
rw [Int.gcd, ← Int.natA... |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Symmetric
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.Algebra.DirectSum.Decomposition
#align_import analysis.inner_product_space.proje... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 70 | 177 | theorem exists_norm_eq_iInf_of_complete_convex {K : Set F} (ne : K.Nonempty) (h₁ : IsComplete K)
(h₂ : Convex ℝ K) : ∀ u : F, ∃ v ∈ K, ‖u - v‖ = ⨅ w : K, ‖u - w‖ := fun u => by
let δ := ⨅ w : K, ‖u - w‖
letI : Nonempty K := ne.to_subtype
have zero_le_δ : 0 ≤ δ := le_ciInf fun _ => norm_nonneg _
have δ_le : ... |
have hδ : ∀ n : ℕ, δ < δ + 1 / (n + 1) := fun n =>
lt_add_of_le_of_pos le_rfl Nat.one_div_pos_of_nat
have h := fun n => exists_lt_of_ciInf_lt (hδ n)
let w : ℕ → K := fun n => Classical.choose (h n)
exact ⟨w, fun n => Classical.choose_spec (h n)⟩
rcases exists_seq with ⟨w, hw⟩
have norm_tendst... |
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Integral.Average
#align_import analysis.convex.integral from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Mea... | Mathlib/Analysis/Convex/Integral.lean | 56 | 81 | theorem Convex.integral_mem [IsProbabilityMeasure μ] (hs : Convex ℝ s) (hsc : IsClosed s)
(hf : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ) : (∫ x, f x ∂μ) ∈ s := by |
borelize E
rcases hfi.aestronglyMeasurable with ⟨g, hgm, hfg⟩
haveI : SeparableSpace (range g ∩ s : Set E) :=
(hgm.isSeparable_range.mono inter_subset_left).separableSpace
obtain ⟨y₀, h₀⟩ : (range g ∩ s).Nonempty := by
rcases (hf.and hfg).exists with ⟨x₀, h₀⟩
exact ⟨f x₀, by simp only [h₀.2, mem_ra... |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 118 | 120 | theorem norm_self_mul_star {x : E} : ‖x * x⋆‖ = ‖x‖ * ‖x‖ := by |
nth_rw 1 [← star_star x]
simp only [norm_star_mul_self, norm_star]
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.MeasureTheory.Constructio... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,241 | 1,248 | theorem integral_target_eq_integral_abs_det_fderiv_smul {f : PartialHomeomorph E E}
(hf' : ∀ x ∈ f.source, HasFDerivAt f (f' x) x) (g : E → F) :
∫ x in f.target, g x ∂μ = ∫ x in f.source, |(f' x).det| • g (f x) ∂μ := by |
have : f '' f.source = f.target := PartialEquiv.image_source_eq_target f.toPartialEquiv
rw [← this]
apply integral_image_eq_integral_abs_det_fderiv_smul μ f.open_source.measurableSet _ f.injOn
intro x hx
exact (hf' x hx).hasFDerivWithinAt
|
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 542 | 545 | theorem tan_oangle_add_left_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
Real.Angle.tan (o.oangle (x + r • o.rotation (π / 2 : ℝ) x) (r • o.rotation (π / 2 : ℝ) x)) =
r⁻¹ := by |
rw [o.oangle_add_left_smul_rotation_pi_div_two h, Real.Angle.tan_coe, Real.tan_arctan]
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 132 | 133 | theorem sameCycle_apply_right : SameCycle f x (f y) ↔ SameCycle f x y := by |
rw [sameCycle_comm, sameCycle_apply_left, sameCycle_comm]
|
import Mathlib.FieldTheory.Galois
#align_import field_theory.polynomial_galois_group from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Polynomial
open FiniteDimensional
namespace Polynomial
variable {F : Type*} [Field F] (p q : F[X]) (E : Type*) [... | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 392 | 399 | theorem restrictComp_surjective (hq : q.natDegree ≠ 0) :
Function.Surjective (restrictComp p q hq) := by |
-- Porting note: was
-- simp only [restrictComp, restrict_surjective]
haveI : Fact (Splits (algebraMap F (SplittingField (comp p q))) p) :=
⟨splits_in_splittingField_of_comp p q hq⟩
rw [restrictComp]
exact restrict_surjective _ _
|
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 98 | 102 | theorem commProb_def' : commProb G = Nat.card (ConjClasses G) / Nat.card G := by |
rw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]
by_cases h : (Nat.card G : ℚ) = 0
· rw [h, zero_mul, div_zero, div_zero]
· exact mul_div_mul_right _ _ h
|
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 | 118 | 120 | theorem hasFiniteIntegral_iff_edist (f : α → β) :
HasFiniteIntegral f μ ↔ (∫⁻ a, edist (f a) 0 ∂μ) < ∞ := by |
simp only [hasFiniteIntegral_iff_norm, edist_dist, dist_zero_right]
|
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 175 | 211 | theorem of_exponent_ge {p q : ℝ≥0∞} {f : ∀ i, E i} (hfq : Memℓp f q) (hpq : q ≤ p) : Memℓp f p := by |
rcases ENNReal.trichotomy₂ hpq with
(⟨rfl, rfl⟩ | ⟨rfl, rfl⟩ | ⟨rfl, hp⟩ | ⟨rfl, rfl⟩ | ⟨hq, rfl⟩ | ⟨hq, _, hpq'⟩)
· exact hfq
· apply memℓp_infty
obtain ⟨C, hC⟩ := (hfq.finite_dsupport.image fun i => ‖f i‖).bddAbove
use max 0 C
rintro x ⟨i, rfl⟩
by_cases hi : f i = 0
· simp [hi]
· ex... |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
set_option autoImplicit true
open Filter Set Function
variable {α β γ ι ι' : Type*}
namespace Filter
section Relation
... | Mathlib/Order/LiminfLimsup.lean | 83 | 84 | theorem isBounded_principal (s : Set α) : IsBounded r (𝓟 s) ↔ ∃ t, ∀ x ∈ s, r x t := by |
simp [IsBounded, subset_def]
|
import Mathlib.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import Mathlib.Topology.ShrinkingLemma
#align_import geometry.manifold.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 157 | 162 | theorem exists_pos_of_mem {x} (hx : x ∈ s) : ∃ i, 0 < f i x := by |
by_contra! h
have H : ∀ i, f i x = 0 := fun i ↦ le_antisymm (h i) (f.nonneg i x)
have := f.sum_eq_one hx
simp_rw [H] at this
simpa
|
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 | 572 | 572 | theorem Icc_diff_both (a b : α) : Icc a b \ {a, b} = Ioo a b := by | simp [← coe_inj]
|
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Tactic.Conv
import Mathlib.Util.Qq
set_option autoImplicit true
-- In this file we would like to be able to use multi-character auto-implicits.
set_option relaxedAutoImplicit true
namespace Mathlib.Tactic
open Lean hiding Rat
open Qq Me... | Mathlib/Tactic/Ring/RingNF.lean | 121 | 121 | theorem nat_rawCast_1 : (Nat.rawCast 1 : R) = 1 := by | simp
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.Instances.NNReal
#align_import analysis.normed.group.infinite_sum from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Topology NNReal
open Finset Filter Metric
variabl... | Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 40 | 46 | theorem cauchySeq_finset_iff_vanishing_norm {f : ι → E} :
(CauchySeq fun s : Finset ι => ∑ i ∈ s, f i) ↔
∀ ε > (0 : ℝ), ∃ s : Finset ι, ∀ t, Disjoint t s → ‖∑ i ∈ t, f i‖ < ε := by |
rw [cauchySeq_finset_iff_sum_vanishing, nhds_basis_ball.forall_iff]
· simp only [ball_zero_eq, Set.mem_setOf_eq]
· rintro s t hst ⟨s', hs'⟩
exact ⟨s', fun t' ht' => hst <| hs' _ ht'⟩
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.Ring.NegOnePow
#align_import analysis.special_functions.trigonometric.basic from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open scoped Classical
open Top... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 859 | 863 | theorem sin_pi_div_sixteen : sin (π / 16) = √(2 - √(2 + √2)) / 2 := by |
trans sin (π / 2 ^ 4)
· congr
norm_num
· simp
|
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Tactic.NthRewrite
#align_import algebra.regular.basic from "leanprover-community/mathlib"@"5cd3c25312f210fec96ba1edb2aebfb2ccf2010f"... | Mathlib/Algebra/Regular/Basic.lean | 312 | 314 | theorem isRegular_mul_iff : IsRegular (a * b) ↔ IsRegular a ∧ IsRegular b := by |
refine Iff.trans ?_ isRegular_mul_and_mul_iff
exact ⟨fun ab => ⟨ab, by rwa [mul_comm]⟩, fun rab => rab.1⟩
|
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
namespace Ideal
universe u v
variable {R : Type u} [CommRing R]
variable {S : Type v} [CommRing S] (f : R →+* S)
variable (p : Ideal R) (... | Mathlib/NumberTheory/RamificationInertia.lean | 661 | 679 | theorem finrank_prime_pow_ramificationIdx [IsDedekindDomain S] (hP0 : P ≠ ⊥)
[p.IsMaximal] [P.IsPrime] (he : e ≠ 0) :
finrank (R ⧸ p) (S ⧸ P ^ e) =
e *
@finrank (R ⧸ p) (S ⧸ P) _ _
(@Algebra.toModule _ _ _ _ <|
@Quotient.algebraQuotientOfRamificationIdxNeZero _ _ _ _ _ _ _ ⟨h... |
letI : NeZero e := ⟨he⟩
letI : Algebra (R ⧸ p) (S ⧸ P) := Quotient.algebraQuotientOfRamificationIdxNeZero f p P
letI := Ideal.Quotient.field p
have hdim := rank_prime_pow_ramificationIdx _ _ _ hP0 he
by_cases hP : FiniteDimensional (R ⧸ p) (S ⧸ P)
· haveI := hP
haveI := (finiteDimensional_iff_of_rank_e... |
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Finite.Set
#align_import combinatorics.simple_graph.ends.defs from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
universe u
variable {V : Type u} (G : SimpleGraph V... | Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 229 | 242 | theorem infinite_iff_in_all_ranges {K : Finset V} (C : G.ComponentCompl K) :
C.supp.Infinite ↔ ∀ (L) (h : K ⊆ L), ∃ D : G.ComponentCompl L, D.hom h = C := by |
classical
constructor
· rintro Cinf L h
obtain ⟨v, ⟨vK, rfl⟩, vL⟩ := Set.Infinite.nonempty (Set.Infinite.diff Cinf L.finite_toSet)
exact ⟨componentComplMk _ vL, rfl⟩
· rintro h Cfin
obtain ⟨D, e⟩ := h (K ∪ Cfin.toFinset) Finset.subset_union_left
obtain ⟨v, vD⟩ := D.nonempty
... |
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Rel
import Mathlib.Data.Set.Finite
import Mathlib.Data.Sym.Sym2
#align_import combinatorics.simple_graph.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
-- Porting note: using `aesop` for automation
-- Porting n... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 752 | 756 | theorem incidenceSet_inter_incidenceSet_of_not_adj (h : ¬G.Adj a b) (hn : a ≠ b) :
G.incidenceSet a ∩ G.incidenceSet b = ∅ := by |
simp_rw [Set.eq_empty_iff_forall_not_mem, Set.mem_inter_iff, not_and]
intro u ha hb
exact h (G.adj_of_mem_incidenceSet hn ha hb)
|
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
#align_import analysis.special_functions.gamma.bohr_mollerup from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
set_option linter.uppercaseLean3 false
noncomputable section
open Filter Set MeasureTheory
open scoped Na... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 201 | 210 | theorem f_nat_eq (hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hn : n ≠ 0) :
f n = f 1 + log (n - 1)! := by |
refine Nat.le_induction (by simp) (fun m hm IH => ?_) n (Nat.one_le_iff_ne_zero.2 hn)
have A : 0 < (m : ℝ) := Nat.cast_pos.2 hm
simp only [hf_feq A, Nat.cast_add, Nat.cast_one, Nat.add_succ_sub_one, add_zero]
rw [IH, add_assoc, ← log_mul (Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero _)) A.ne', ←
Nat.cast_mu... |
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation
open scoped Topology
variable {X Y A} [TopologicalSpace X] [TopologicalSpace A]
theorem embedding_toPullbackDiag (f : X → Y) : Embedding (toPullbackDiag f) :=
Embedding.mk' _ (injective_toPullbackDiag f) fun x ↦ by
rw [toPullbackDiag,... | Mathlib/Topology/SeparatedMap.lean | 144 | 147 | theorem IsLocallyInjective_iff_openEmbedding {f : X → Y} :
IsLocallyInjective f ↔ OpenEmbedding (toPullbackDiag f) := by |
rw [isLocallyInjective_iff_isOpen_diagonal, ← range_toPullbackDiag]
exact ⟨fun h ↦ ⟨embedding_toPullbackDiag f, h⟩, fun h ↦ h.isOpen_range⟩
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.