Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Analysis.SpecialFunctions.Pow.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 | 308 | 309 | theorem logb_le_logb_of_base_lt_one (h : 0 < x) (h₁ : 0 < y) : logb b x ≤ logb b y ↔ y ≤ x := by |
rw [logb, logb, div_le_div_right_of_neg (log_neg b_pos b_lt_one), log_le_log_iff h₁ h]
|
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.Probability.Independence.Basic
#align_import probability.integration from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
noncomputable section
open Set MeasureTheory
open scoped ENNReal MeasureTheory
variable {Ω : Type*... | Mathlib/Probability/Integration.lean | 45 | 73 | theorem lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator {Mf mΩ : MeasurableSpace Ω}
{μ : Measure Ω} (hMf : Mf ≤ mΩ) (c : ℝ≥0∞) {T : Set Ω} (h_meas_T : MeasurableSet T)
(h_ind : IndepSets {s | MeasurableSet[Mf] s} {T} μ) (h_meas_f : Measurable[Mf] f) :
(∫⁻ ω, f ω * T.indicator (fun _ => c) ω ∂μ)... |
revert f
have h_mul_indicator : ∀ g, Measurable g → Measurable fun a => g a * T.indicator (fun _ => c) a :=
fun g h_mg => h_mg.mul (measurable_const.indicator h_meas_T)
apply @Measurable.ennreal_induction _ Mf
· intro c' s' h_meas_s'
simp_rw [← inter_indicator_mul]
rw [lintegral_indicator _ (Measur... |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 87 | 92 | theorem inv_anti_mono (hI : I ≠ 0) (hJ : J ≠ 0) (hIJ : I ≤ J) : J⁻¹ ≤ I⁻¹ := by |
-- Porting note: in Lean3, introducing `x` would just give `x ∈ J⁻¹ → x ∈ I⁻¹`, but
-- in Lean4, it goes all the way down to the subtypes
intro x
simp only [val_eq_coe, mem_coe, mem_inv_iff hJ, mem_inv_iff hI]
exact fun h y hy => h y (hIJ hy)
|
import Mathlib.Combinatorics.SimpleGraph.Basic
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
structure Dart extends V × V where
adj : G.Adj fst snd
deriving DecidableEq
#align simple_graph.dart SimpleGraph.Dart
initialize_simps_projections Dart (+toProd, -fst, -snd)
attribute [simp] Dart.a... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 33 | 34 | theorem Dart.ext_iff (d₁ d₂ : G.Dart) : d₁ = d₂ ↔ d₁.toProd = d₂.toProd := by |
cases d₁; cases d₂; simp
|
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory
open CategoryTheory.Limits Opposite
universe w v u
var... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 477 | 479 | theorem sheafifyMap_id (P : Cᵒᵖ ⥤ D) : J.sheafifyMap (𝟙 P) = 𝟙 (J.sheafify P) := by |
dsimp [sheafifyMap, sheafify]
simp
|
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
namespace List
open Nat
@[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : ∀ l : List α, drop 1 l =... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,375 | 1,376 | theorem range_subset {m n : Nat} : range m ⊆ range n ↔ m ≤ n := by |
simp only [range_eq_range', range'_subset_right, lt_succ_self]
|
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
open scoped Pointwise
namespace Subgroup
variable {G : Type*} [Group G] (H K : Subgroup G) (S T : Set G)
@[to_additive "`S` and `T` are complements if ... | Mathlib/GroupTheory/Complement.lean | 144 | 153 | theorem isComplement_univ_left : IsComplement univ S ↔ ∃ g : G, S = {g} := by |
refine
⟨fun h => Set.exists_eq_singleton_iff_nonempty_subsingleton.mpr ⟨?_, fun a ha b hb => ?_⟩, ?_⟩
· obtain ⟨a, _⟩ := h.2 1
exact ⟨a.2.1, a.2.2⟩
· have : (⟨⟨_, mem_top a⁻¹⟩, ⟨a, ha⟩⟩ : (⊤ : Set G) × S) = ⟨⟨_, mem_top b⁻¹⟩, ⟨b, hb⟩⟩ :=
h.1 ((inv_mul_self a).trans (inv_mul_self b).symm)
exact ... |
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
#align_import logic.equiv.set from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
open Function Set
universe u v w z
variable {α : Sort u} {β : Sort v} {γ : Sort w}
namespace Equiv
@[simp]
th... | Mathlib/Logic/Equiv/Set.lean | 155 | 158 | theorem prod_assoc_symm_preimage {α β γ} {s : Set α} {t : Set β} {u : Set γ} :
(Equiv.prodAssoc α β γ).symm ⁻¹' (s ×ˢ t) ×ˢ u = s ×ˢ t ×ˢ u := by |
ext
simp [and_assoc]
|
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 73 | 75 | theorem moveLeft_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) := inferInstance
HEq (nim o).moveLeft fun i : o.out.α => nim (typein (· < ·) i) := by | rw [nim_def]; rfl
|
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Order.CauSeq.Completion
#align_import data.real.basic from "leanprover-community/mathlib"@"cb42593171ba005beaaf4549fcfe0dece9ada4c9"
assert_not_exists Finset
assert_not_exists Module
assert_not_exists Submonoid
assert_not_exists FloorRing
structure Real w... | Mathlib/Data/Real/Basic.lean | 332 | 334 | theorem mk_pos {f : CauSeq ℚ abs} : 0 < mk f ↔ Pos f := by |
rw [← mk_zero, mk_lt]
exact iff_of_eq (congr_arg Pos (sub_zero f))
|
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 | 118 | 119 | theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by |
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ]
|
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.RingTheory.Prime
import Mathlib.RingTheory.Polynomial.Content
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.eisenstein_criterion from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
open Polynomial Ideal.Quotient
v... | Mathlib/RingTheory/EisensteinCriterion.lean | 52 | 61 | theorem le_natDegree_of_map_eq_mul_X_pow {n : ℕ} {P : Ideal R} (hP : P.IsPrime) {q : R[X]}
{c : Polynomial (R ⧸ P)} (hq : map (mk P) q = c * X ^ n) (hc0 : c.degree = 0) :
n ≤ q.natDegree :=
Nat.cast_le.1
(calc
↑n = degree (q.map (mk P)) := by |
rw [hq, degree_mul, hc0, zero_add, degree_pow, degree_X, nsmul_one]
_ ≤ degree q := degree_map_le _ _
_ ≤ natDegree q := degree_le_natDegree
)
|
import Mathlib.Topology.Separation
import Mathlib.Topology.Bases
#align_import topology.dense_embedding from "leanprover-community/mathlib"@"148aefbd371a25f1cff33c85f20c661ce3155def"
noncomputable section
open Set Filter
open scoped Topology
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
structure D... | Mathlib/Topology/DenseEmbedding.lean | 117 | 124 | theorem tendsto_comap_nhds_nhds {d : δ} {a : α} (di : DenseInducing i)
(H : Tendsto h (𝓝 d) (𝓝 (i a))) (comm : h ∘ g = i ∘ f) : Tendsto f (comap g (𝓝 d)) (𝓝 a) := by |
have lim1 : map g (comap g (𝓝 d)) ≤ 𝓝 d := map_comap_le
replace lim1 : map h (map g (comap g (𝓝 d))) ≤ map h (𝓝 d) := map_mono lim1
rw [Filter.map_map, comm, ← Filter.map_map, map_le_iff_le_comap] at lim1
have lim2 : comap i (map h (𝓝 d)) ≤ comap i (𝓝 (i a)) := comap_mono H
rw [← di.nhds_eq_comap] at l... |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : ℕ) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 235 | 246 | theorem peval_polyOfInterest' (n : ℕ) (x y : 𝕎 k) :
peval (polyOfInterest p n) ![fun i => x.coeff i, fun i => y.coeff i] =
(x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) -
x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1) := by |
rw [peval_polyOfInterest]
have : (p : k) = 0 := CharP.cast_eq_zero k p
simp only [this, Nat.cast_pow, ne_eq, add_eq_zero, and_false, zero_pow, zero_mul, add_zero,
not_false_eq_true]
have sum_zero_pow_mul_pow_p (y : 𝕎 k) : ∑ x ∈ range (n + 1 + 1),
(0 : k) ^ x * y.coeff x ^ p ^ (n + 1 - x) = y.coeff 0... |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 63 | 64 | theorem unitors_equal : (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom := by |
coherence
|
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 430 | 441 | theorem exists_seq_norm_le_one_le_norm_sub' {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
(h : ¬FiniteDimensional 𝕜 E) :
∃ f : ℕ → E, (∀ n, ‖f n‖ ≤ R) ∧ Pairwise fun m n => 1 ≤ ‖f m - f n‖ := by |
have : IsSymm E fun x y : E => 1 ≤ ‖x - y‖ := by
constructor
intro x y hxy
rw [← norm_neg]
simpa
apply
exists_seq_of_forall_finset_exists' (fun x : E => ‖x‖ ≤ R) fun (x : E) (y : E) => 1 ≤ ‖x - y‖
rintro s -
exact exists_norm_le_le_norm_sub_of_finset hc hR h s
|
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.PEmpty
#align_import category_theory.monoidal.of_chosen_finite_products.basic from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
names... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean | 242 | 246 | theorem tensor_id (X₁ X₂ : C) : tensorHom ℬ (𝟙 X₁) (𝟙 X₂) = 𝟙 (tensorObj ℬ X₁ X₂) := by |
apply IsLimit.hom_ext (ℬ _ _).isLimit;
rintro ⟨⟨⟩⟩ <;>
· dsimp [tensorHom]
simp
|
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 | 153 | 159 | theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop}
(smul : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn))
(add : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by |
refine Exists.elim ?_ fun (h : x ∈ I • N) (H : p x h) => H
exact
smul_induction_on hx (fun a ha x hx => ⟨_, smul _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ =>
⟨_, add _ _ _ _ hx hy⟩
|
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 | 375 | 378 | theorem filter_lt_le_eq_Ioc [DecidablePred fun j => a < j ∧ j ≤ b] :
(univ.filter fun j => a < j ∧ j ≤ b) = Ioc a b := by |
ext
simp
|
import Mathlib.Topology.Sets.Closeds
#align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
variable (α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
namespace TopologicalSpace
@[mk_iff]
class NoetherianSpace : Prop where
wellFounded_open... | Mathlib/Topology/NoetherianSpace.lean | 223 | 264 | theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [NoetherianSpace α]
(Z : Set α) (H : Z ∈ irreducibleComponents α) :
∃ o : Set α, IsOpen o ∧ o ≠ ∅ ∧ o ≤ Z := by |
classical
let ι : Set (Set α) := irreducibleComponents α \ {Z}
have hι : ι.Finite := NoetherianSpace.finite_irreducibleComponents.subset Set.diff_subset
have hι' : Finite ι := by rwa [Set.finite_coe_iff]
let U := Z \ ⋃ (x : ι), x
have hU0 : U ≠ ∅ := fun r ↦ by
obtain ⟨Z', hZ'⟩ := isIrreducible_iff_sU... |
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import linear_algebra.affine_space.independent from "leanprover-c... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 72 | 81 | theorem affineIndependent_iff_of_fintype [Fintype ι] (p : ι → P) :
AffineIndependent k p ↔
∀ w : ι → k, ∑ i, w i = 0 → Finset.univ.weightedVSub p w = (0 : V) → ∀ i, w i = 0 := by |
constructor
· exact fun h w hw hs i => h Finset.univ w hw hs i (Finset.mem_univ _)
· intro h s w hw hs i hi
rw [Finset.weightedVSub_indicator_subset _ _ (Finset.subset_univ s)] at hs
rw [← Finset.sum_indicator_subset _ (Finset.subset_univ s)] at hw
replace h := h ((↑s : Set ι).indicator w) hw hs i
... |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 345 | 347 | theorem comp_add_left (hf : IntervalIntegrable f volume a b) (c : ℝ) :
IntervalIntegrable (fun x => f (c + x)) volume (a - c) (b - c) := by |
simpa only [add_comm] using IntervalIntegrable.comp_add_right hf c
|
import Mathlib.ModelTheory.Syntax
import Mathlib.ModelTheory.Semantics
import Mathlib.Algebra.Ring.Equiv
variable {α : Type*}
namespace FirstOrder
open FirstOrder
inductive ringFunc : ℕ → Type
| add : ringFunc 2
| mul : ringFunc 2
| neg : ringFunc 1
| zero : ringFunc 0
| one : ringFunc 0
deriving D... | Mathlib/ModelTheory/Algebra/Ring/Basic.lean | 138 | 140 | theorem card_ring : card Language.ring = 5 := by |
have : Fintype.card Language.ring.Symbols = 5 := rfl
simp [Language.card, this]
|
import Mathlib.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Noetherian
#align_import ring_theory.ideal.associated_prime from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [... | Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 83 | 103 | theorem exists_le_isAssociatedPrime_of_isNoetherianRing [H : IsNoetherianRing R] (x : M)
(hx : x ≠ 0) : ∃ P : Ideal R, IsAssociatedPrime P M ∧ (R ∙ x).annihilator ≤ P := by |
have : (R ∙ x).annihilator ≠ ⊤ := by
rwa [Ne, Ideal.eq_top_iff_one, Submodule.mem_annihilator_span_singleton, one_smul]
obtain ⟨P, ⟨l, h₁, y, rfl⟩, h₃⟩ :=
set_has_maximal_iff_noetherian.mpr H
{ P | (R ∙ x).annihilator ≤ P ∧ P ≠ ⊤ ∧ ∃ y : M, P = (R ∙ y).annihilator }
⟨(R ∙ x).annihilator, rfl.le... |
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
import Mathlib.Order.PartialSups
#align_import linear_algebra.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
universe u v w x y z u' v' w' y'
variable {R : Type u} {K : Ty... | Mathlib/LinearAlgebra/Prod.lean | 333 | 336 | theorem ker_prodMap (f : M →ₗ[R] M₂) (g : M₃ →ₗ[R] M₄) :
ker (LinearMap.prodMap f g) = Submodule.prod (ker f) (ker g) := by |
dsimp only [ker]
rw [← prodMap_comap_prod, Submodule.prod_bot]
|
import Mathlib.Algebra.Associated
import Mathlib.NumberTheory.Divisors
#align_import algebra.is_prime_pow from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
def IsPrimePow : Prop :=
∃ (p : R) (k : ℕ), Prime p ∧ 0 < k ∧ p... | Mathlib/Algebra/IsPrimePow.lean | 84 | 91 | theorem isPrimePow_nat_iff_bounded (n : ℕ) :
IsPrimePow n ↔ ∃ p : ℕ, p ≤ n ∧ ∃ k : ℕ, k ≤ n ∧ p.Prime ∧ 0 < k ∧ p ^ k = n := by |
rw [isPrimePow_nat_iff]
refine Iff.symm ⟨fun ⟨p, _, k, _, hp, hk, hn⟩ => ⟨p, k, hp, hk, hn⟩, ?_⟩
rintro ⟨p, k, hp, hk, rfl⟩
refine ⟨p, ?_, k, (Nat.lt_pow_self hp.one_lt _).le, hp, hk, rfl⟩
conv => { lhs; rw [← (pow_one p)] }
exact Nat.pow_le_pow_right hp.one_lt.le hk
|
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 353 | 354 | theorem braiding_tensorUnit_right (X : C) : (β_ X (𝟙_ C)).hom = (ρ_ X).hom ≫ (λ_ X).inv := by |
simp [← rightUnitor_inv_braiding]
|
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Topology.Algebra.UniformGroup
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Algebra.Algebra.Defs
import Mathlib.LinearAlgebra.Projection
import Mat... | Mathlib/Topology/Algebra/Module/Basic.lean | 222 | 234 | theorem LinearMap.continuous_on_pi {ι : Type*} {R : Type*} {M : Type*} [Finite ι] [Semiring R]
[TopologicalSpace R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [ContinuousAdd M]
[ContinuousSMul R M] (f : (ι → R) →ₗ[R] M) : Continuous f := by |
cases nonempty_fintype ι
classical
-- for the proof, write `f` in the standard basis, and use that each coordinate is a continuous
-- function.
have : (f : (ι → R) → M) = fun x => ∑ i : ι, x i • f fun j => if i = j then 1 else 0 := by
ext x
exact f.pi_apply_eq_sum_univ x
rw [this]
r... |
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 | 522 | 528 | theorem coeff_zero_eq_eval_zero (p : R[X]) : coeff p 0 = p.eval 0 :=
calc
coeff p 0 = coeff p 0 * 0 ^ 0 := by | simp
_ = p.eval 0 := by
symm
rw [eval_eq_sum]
exact Finset.sum_eq_single _ (fun b _ hb => by simp [zero_pow hb]) (by simp)
|
import Mathlib.Analysis.NormedSpace.AddTorsorBases
#align_import analysis.convex.intrinsic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open AffineSubspace Set
open scoped Pointwise
variable {𝕜 V W Q P : Type*}
section AddTorsor
variable (𝕜) [Ring 𝕜] [AddCommGroup V] [Modu... | Mathlib/Analysis/Convex/Intrinsic.lean | 147 | 149 | theorem intrinsicClosure_singleton (x : P) : intrinsicClosure 𝕜 ({x} : Set P) = {x} := by |
simpa only [intrinsicClosure, preimage_coe_affineSpan_singleton, closure_univ, image_univ,
Subtype.range_coe] using coe_affineSpan_singleton _ _ _
|
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 197 | 202 | theorem roots_C (x : R) : (C x).roots = 0 := by |
classical exact
if H : x = 0 then by rw [H, C_0, roots_zero]
else
Multiset.ext.mpr fun r => (by
rw [count_roots, count_zero, rootMultiplicity_eq_zero (not_isRoot_C _ _ H)])
|
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "leanprover-community/mathlib"@"3b88f4005dc2e28d42f974cc1ce838f0dafb39b8"
open MeasureTheory Set Filter TopologicalSpace
open scoped NNReal ENNReal Me... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 54 | 58 | theorem IicSnd_apply (r : ℝ) {s : Set α} (hs : MeasurableSet s) :
ρ.IicSnd r s = ρ (s ×ˢ Iic r) := by |
rw [IicSnd, fst_apply hs,
restrict_apply' (MeasurableSet.univ.prod (measurableSet_Iic : MeasurableSet (Iic r))), ←
prod_univ, prod_inter_prod, inter_univ, univ_inter]
|
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
namespace CategoryTheory
universe w v u
open Category Iso
-- intended to be used with explicit universe parameters
@[nolint checkUnivs]
class Bicate... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 485 | 485 | theorem unitors_inv_equal : (λ_ (𝟙 a)).inv = (ρ_ (𝟙 a)).inv := by | simp [Iso.inv_eq_inv]
|
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section
open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
variable {ι κ X X' Y Z α α' β β'... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 251 | 254 | theorem equicontinuousAt_finite [Finite ι] {F : ι → X → α} {x₀ : X} :
EquicontinuousAt F x₀ ↔ ∀ i, ContinuousAt (F i) x₀ := by |
simp [EquicontinuousAt, ContinuousAt, (nhds_basis_uniformity' (𝓤 α).basis_sets).tendsto_right_iff,
UniformSpace.ball, @forall_swap _ ι]
|
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 378 | 378 | theorem image_neg_Iic : Neg.neg '' Iic a = Ici (-a) := by | simp
|
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Data.Set.Pointwise.Support
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
import Mathlib.MeasureTheo... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 384 | 399 | theorem y_eq_zero_of_not_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (hx : x ∉ ball (0 : E) (1 + D)) :
y D x = 0 := by |
change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 0
have B : ∀ y, y ∈ ball x D → φ y = 0 := by
intro y hy
simp only [φ, indicator, mem_closedBall_zero_iff, ite_eq_right_iff, one_ne_zero]
intro h'y
have C : ball y D ⊆ ball 0 (1 + D) := by
apply ball_subset_ball'
rw [← dist_zero_right] at h'y
lina... |
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 176 | 178 | theorem sup_product_right (s : Finset β) (t : Finset γ) (f : β × γ → α) :
(s ×ˢ t).sup f = t.sup fun i' => s.sup fun i => f ⟨i, i'⟩ := by |
rw [sup_product_left, Finset.sup_comm]
|
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
namespace PNat
open Nat
def gcd (n m : ℕ+) : ℕ+ :=
⟨Nat.gcd (n : ℕ) (m : ℕ), Nat.gcd_pos_of_pos_left (m : ℕ) n.pos⟩
#align pnat.gcd PNat.gc... | Mathlib/Data/PNat/Prime.lean | 302 | 305 | theorem Coprime.gcd_mul (k : ℕ+) {m n : ℕ+} (h : m.Coprime n) :
k.gcd (m * n) = k.gcd m * k.gcd n := by |
rw [← coprime_coe] at h; apply eq
simp only [gcd_coe, mul_coe]; apply Nat.Coprime.gcd_mul k h
|
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.LinearAlgebra.Dual
#align_import analysis.calculus.lagrange_multipliers from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Set
open scoped Topology Fi... | Mathlib/Analysis/Calculus/LagrangeMultipliers.lean | 108 | 121 | theorem IsLocalExtrOn.exists_multipliers_of_hasStrictFDerivAt {ι : Type*} [Fintype ι]
{f : ι → E → ℝ} {f' : ι → E →L[ℝ] ℝ} (hextr : IsLocalExtrOn φ {x | ∀ i, f i x = f i x₀} x₀)
(hf' : ∀ i, HasStrictFDerivAt (f i) (f' i) x₀) (hφ' : HasStrictFDerivAt φ φ' x₀) :
∃ (Λ : ι → ℝ) (Λ₀ : ℝ), (Λ, Λ₀) ≠ 0 ∧ (∑ i, Λ i... |
letI := Classical.decEq ι
replace hextr : IsLocalExtrOn φ {x | (fun i => f i x) = fun i => f i x₀} x₀ := by
simpa only [Function.funext_iff] using hextr
rcases hextr.exists_linear_map_of_hasStrictFDerivAt (hasStrictFDerivAt_pi.2 fun i => hf' i)
hφ' with
⟨Λ, Λ₀, h0, hsum⟩
rcases (LinearEquiv.piRin... |
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 | 286 | 288 | theorem normAtPlace_apply_isReal {w : InfinitePlace K} (hw : IsReal w) (x : E K):
normAtPlace w x = ‖x.1 ⟨w, hw⟩‖ := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, dif_pos]
|
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 182 | 185 | theorem toComplex_comp_ofComplex : toComplex.comp ofComplex = AlgHom.id ℝ ℂ := by |
ext1
dsimp only [AlgHom.comp_apply, Subtype.coe_mk, AlgHom.id_apply]
rw [ofComplex_I, toComplex_ι, one_smul]
|
import Mathlib.MeasureTheory.Measure.AEMeasurable
#align_import dynamics.ergodic.measure_preserving from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
variable {α β γ δ : Type*} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ]
[MeasurableSpace δ]
namespace MeasureTheory
... | Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 182 | 186 | theorem exists_mem_iterate_mem [IsFiniteMeasure μ] (hf : MeasurePreserving f μ μ)
(hs : MeasurableSet s) (hs' : μ s ≠ 0) : ∃ x ∈ s, ∃ m ≠ 0, f^[m] x ∈ s := by |
rcases ENNReal.exists_nat_mul_gt hs' (measure_ne_top μ (Set.univ : Set α)) with ⟨N, hN⟩
rcases hf.exists_mem_iterate_mem_of_volume_lt_mul_volume hs hN with ⟨x, hx, m, hm, hmx⟩
exact ⟨x, hx, m, hm.1.ne', hmx⟩
|
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
#align_import topology.fiber_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
variable {ι B F X : Type*} [TopologicalSpace X]
open TopologicalSpace Filter Set Bundle Topology
... | Mathlib/Topology/FiberBundle/Basic.lean | 668 | 670 | theorem localTrivAt_apply (p : Z.TotalSpace) : (Z.localTrivAt p.1) p = ⟨p.1, p.2⟩ := by |
rw [localTrivAt, localTriv_apply, coordChange_self]
exact Z.mem_baseSet_at p.1
|
import Mathlib.Order.Monotone.Odd
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
#align_import analysis.special_functions.trigonometric.deriv from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open s... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | 35 | 41 | theorem hasStrictDerivAt_sin (x : ℂ) : HasStrictDerivAt sin (cos x) x := by |
simp only [cos, div_eq_mul_inv]
convert ((((hasStrictDerivAt_id x).neg.mul_const I).cexp.sub
((hasStrictDerivAt_id x).mul_const I).cexp).mul_const I).mul_const (2 : ℂ)⁻¹ using 1
simp only [Function.comp, id]
rw [sub_mul, mul_assoc, mul_assoc, I_mul_I, neg_one_mul, neg_neg, mul_one, one_mul, mul_assoc,
... |
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Functor
#align_import control.traversable.instances from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
universe u v
section Option
open Functor
variab... | Mathlib/Control/Traversable/Instances.lean | 31 | 32 | theorem Option.id_traverse {α} (x : Option α) : Option.traverse (pure : α → Id α) x = x := by |
cases x <;> rfl
|
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Control.Functor
import Mathlib.Data.SProd
import Mathlib.Util.CompileInductive
import Batteries.Tactic.Lint.Basic
#align_import data.list.defs from "leanprover-community/mathlib"@"d2d8742b0c21426362a9dacebc6005db895ca963"
-- Porting note
-- Many of the definitio... | Mathlib/Data/List/Defs.lean | 569 | 571 | theorem iterateTR_loop_eq (f : α → α) (a : α) (n : ℕ) (l : List α) :
iterateTR.loop f a n l = reverse l ++ iterate f a n := by |
induction n generalizing a l <;> simp [*]
|
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.biproducts from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w₁ w₂ v₁ v₂ u₁ u₂
noncomputable section
open ... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean | 349 | 351 | theorem biprodComparison'_comp_biprodComparison :
biprodComparison' F X Y ≫ biprodComparison F X Y = 𝟙 (F.obj X ⊞ F.obj Y) := by |
ext <;> simp [← Functor.map_comp]
|
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 87 | 88 | theorem vars_C : (C r : MvPolynomial σ R).vars = ∅ := by |
classical rw [vars_def, degrees_C, Multiset.toFinset_zero]
|
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 129 | 151 | theorem IsLindelof.elim_countable_subcover {ι : Type v} (hs : IsLindelof s) (U : ι → Set X)
(hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) :
∃ r : Set ι, r.Countable ∧ (s ⊆ ⋃ i ∈ r, U i) := by |
have hmono : ∀ ⦃s t : Set X⦄, s ⊆ t → (∃ r : Set ι, r.Countable ∧ t ⊆ ⋃ i ∈ r, U i)
→ (∃ r : Set ι, r.Countable ∧ s ⊆ ⋃ i ∈ r, U i) := by
intro _ _ hst ⟨r, ⟨hrcountable, hsub⟩⟩
exact ⟨r, hrcountable, Subset.trans hst hsub⟩
have hcountable_union : ∀ (S : Set (Set X)), S.Countable
→ (∀ s ∈ S, ∃ r... |
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.ApplyFun
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import category_theory.subobject.basic from "leanprover-community/mathlib"@"70fd9563a21e7b... | Mathlib/CategoryTheory/Subobject/Basic.lean | 440 | 445 | theorem ofMkLEMk_comp_ofMkLEMk {B A₁ A₂ A₃ : C} (f : A₁ ⟶ B) [Mono f] (g : A₂ ⟶ B) [Mono g]
(h : A₃ ⟶ B) [Mono h] (h₁ : mk f ≤ mk g) (h₂ : mk g ≤ mk h) :
ofMkLEMk f g h₁ ≫ ofMkLEMk g h h₂ = ofMkLEMk f h (h₁.trans h₂) := by |
simp only [ofMkLE, ofLEMk, ofLE, ofMkLEMk, ← Functor.map_comp_assoc underlying, assoc,
Iso.hom_inv_id_assoc]
congr 1
|
import Mathlib.Topology.VectorBundle.Basic
#align_import topology.vector_bundle.hom from "leanprover-community/mathlib"@"8905e5ed90859939681a725b00f6063e65096d95"
noncomputable section
open scoped Bundle
open Bundle Set ContinuousLinearMap
variable {𝕜₁ : Type*} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type*} [Non... | Mathlib/Topology/VectorBundle/Hom.lean | 177 | 183 | theorem continuousLinearMap_symm_apply' {b : B} (hb : b ∈ e₁.baseSet ∩ e₂.baseSet)
(L : F₁ →SL[σ] F₂) :
(continuousLinearMap σ e₁ e₂).symm b L =
(e₂.symmL 𝕜₂ b).comp (L.comp <| e₁.continuousLinearMapAt 𝕜₁ b) := by |
rw [symm_apply]
· rfl
· exact hb
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Nilpotent
import Mathlib.Order.Radical
def frattini (G : Type*) [Group G] : Subgroup G :=
Order.radical (Subgroup G)
variable {G H : Type*} [Group G] [Group H] {φ : G →* H} (hφ : Function.Surjective φ)
lemma... | Mathlib/GroupTheory/Frattini.lean | 59 | 74 | theorem frattini_nilpotent [Finite G] : Group.IsNilpotent (frattini G) := by |
-- We use the characterisation of nilpotency in terms of all Sylow subgroups being normal.
have q := (isNilpotent_of_finite_tfae (G := frattini G)).out 0 3
rw [q]; clear q
-- Consider each prime `p` and Sylow `p`-subgroup `P` of `frattini G`.
intro p p_prime P
-- The Frattini argument shows that the normal... |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 150 | 152 | theorem Even.zpow_pos_iff (hn : Even n) (h : n ≠ 0) : 0 < a ^ n ↔ a ≠ 0 := by |
obtain ⟨k, rfl⟩ := hn
rw [zpow_add' (by simp [em']), mul_self_pos, zpow_ne_zero_iff (by simpa using h)]
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import algebra.big_operators.finprod from "leanprover-community/mathlib"@"d6fad0e5bf... | Mathlib/Algebra/BigOperators/Finprod.lean | 198 | 201 | theorem finprod_of_isEmpty [IsEmpty α] (f : α → M) : ∏ᶠ i, f i = 1 := by |
rw [← finprod_one]
congr
simp [eq_iff_true_of_subsingleton]
|
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 61 | 73 | theorem closure_ball (x : E) {r : ℝ} (hr : r ≠ 0) : closure (ball x r) = closedBall x r := by |
refine Subset.antisymm closure_ball_subset_closedBall fun y hy => ?_
have : ContinuousWithinAt (fun c : ℝ => c • (y - x) + x) (Ico 0 1) 1 :=
((continuous_id.smul continuous_const).add continuous_const).continuousWithinAt
convert this.mem_closure _ _
· rw [one_smul, sub_add_cancel]
· simp [closure_Ico zer... |
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open scoped Classical
open Real Topology Filter ComplexConjugate Finset Set
namespace Complex
noncomputable def cpow (x y : ℂ) ... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 141 | 141 | theorem cpow_natCast (x : ℂ) (n : ℕ) : x ^ (n : ℂ) = x ^ n := by | simpa using cpow_nat_mul x n 1
|
import Mathlib.Order.RelClasses
import Mathlib.Data.List.Basic
#align_import data.list.lex from "leanprover-community/mathlib"@"d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d"
namespace List
open Nat
universe u
variable {α : Type u}
inductive Lex (r : α → α → Prop) : List α → List α → Prop
| nil {a l} : Lex r... | Mathlib/Data/List/Lex.lean | 177 | 179 | theorem ne_iff {l₁ l₂ : List α} (H : length l₁ ≤ length l₂) : Lex (· ≠ ·) l₁ l₂ ↔ l₁ ≠ l₂ := by |
classical
exact Decidable.List.Lex.ne_iff H
|
import Mathlib.SetTheory.Game.State
#align_import set_theory.game.domineering from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
namespace SetTheory
namespace PGame
namespace Domineering
open Function
@[simps!]
def shiftUp : ℤ × ℤ ≃ ℤ × ℤ :=
(Equiv.refl ℤ).prodCongr (Equiv.addRig... | Mathlib/SetTheory/Game/Domineering.lean | 79 | 83 | theorem fst_pred_mem_erase_of_mem_right {b : Board} {m : ℤ × ℤ} (h : m ∈ right b) :
(m.1 - 1, m.2) ∈ b.erase m := by |
rw [mem_right] at h
apply Finset.mem_erase_of_ne_of_mem _ h.2
exact ne_of_apply_ne Prod.fst (pred_ne_self m.1)
|
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Init.Data.Bool.Lemmas
#align_import analysis.box_integral.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open... | Mathlib/Analysis/BoxIntegral/Basic.lean | 527 | 532 | theorem to_subbox_aux (h : Integrable I l f vol) (hJ : J ≤ I) :
∃ y : F, HasIntegral J l f vol y ∧
Tendsto (integralSum f vol) (l.toFilteriUnion I (Prepartition.single I J hJ)) (𝓝 y) := by |
refine (cauchy_map_iff_exists_tendsto.1
(h.cauchy_map_integralSum_toFilteriUnion (.single I J hJ))).imp fun y hy ↦ ⟨?_, hy⟩
convert hy.comp (l.tendsto_embedBox_toFilteriUnion_top hJ) -- faster than `exact` here
|
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 309 | 315 | theorem ae_le_of_forall_setIntegral_le {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
(hf_le : ∀ s, MeasurableSet s → μ s < ∞ → (∫ x in s, f x ∂μ) ≤ ∫ x in s, g x ∂μ) :
f ≤ᵐ[μ] g := by |
rw [← eventually_sub_nonneg]
refine ae_nonneg_of_forall_setIntegral_nonneg (hg.sub hf) fun s hs => ?_
rw [integral_sub' hg.integrableOn hf.integrableOn, sub_nonneg]
exact hf_le s hs
|
import Mathlib.Data.Matrix.Basis
import Mathlib.RingTheory.TensorProduct.Basic
#align_import ring_theory.matrix_algebra from "leanprover-community/mathlib"@"6c351a8fb9b06e5a542fdf427bfb9f46724f9453"
suppress_compilation
universe u v w
open TensorProduct
open TensorProduct
open Algebra.TensorProduct
open Matri... | Mathlib/RingTheory/MatrixAlgebra.lean | 99 | 101 | theorem invFun_smul (a : A) (M : Matrix n n A) :
invFun R A n (a • M) = a ⊗ₜ 1 * invFun R A n M := by |
simp [invFun, Finset.mul_sum]
|
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 235 | 240 | theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by |
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
|
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 313 | 320 | theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
((↑) : ℍ → ℂ) '' closedBall (α := ℍ) z r = closedBall ↑(z.center r) (z.im * Real.sinh r) := by |
ext w; constructor
· rintro ⟨w, hw, rfl⟩
exact dist_le_iff_dist_coe_center_le.1 hw
· intro hw
lift w to ℍ using im_pos_of_dist_center_le hw
exact mem_image_of_mem _ (dist_le_iff_dist_coe_center_le.2 hw)
|
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 | 99 | 110 | theorem memberSubfamily_union_nonMemberSubfamily (a : α) (𝒜 : Finset (Finset α)) :
𝒜.memberSubfamily a ∪ 𝒜.nonMemberSubfamily a = 𝒜.image fun s => s.erase a := by |
ext s
simp only [mem_union, mem_memberSubfamily, mem_nonMemberSubfamily, mem_image, exists_prop]
constructor
· rintro (h | h)
· exact ⟨_, h.1, erase_insert h.2⟩
· exact ⟨_, h.1, erase_eq_of_not_mem h.2⟩
· rintro ⟨s, hs, rfl⟩
by_cases ha : a ∈ s
· exact Or.inl ⟨by rwa [insert_erase ha], not_me... |
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.manifold.cont_mdiff from "leanprover-community/mathlib"@"e5ab837fc252451f3eb9124ae6e7b6f57455e7b9"
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scope... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 116 | 154 | theorem contDiffWithinAt_localInvariantProp (n : ℕ∞) :
(contDiffGroupoid ∞ I).LocalInvariantProp (contDiffGroupoid ∞ I')
(ContDiffWithinAtProp I I' n) where
is_local {s x u f} u_open xu := by |
have : I.symm ⁻¹' (s ∩ u) ∩ range I = I.symm ⁻¹' s ∩ range I ∩ I.symm ⁻¹' u := by
simp only [inter_right_comm, preimage_inter]
rw [ContDiffWithinAtProp, ContDiffWithinAtProp, this]
symm
apply contDiffWithinAt_inter
have : u ∈ 𝓝 (I.symm (I x)) := by
rw [ModelWithCorners.left_inv]
... |
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
namespace MvPowerSeries
open Fi... | Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 107 | 137 | theorem mul_invOfUnit (φ : MvPowerSeries σ R) (u : Rˣ) (h : constantCoeff σ R φ = u) :
φ * invOfUnit φ u = 1 :=
ext fun n =>
letI := Classical.decEq (σ →₀ ℕ)
if H : n = 0 then by
rw [H]
simp [coeff_mul, support_single_ne_zero, h]
else by
classical
have : ((0 : σ →₀ ℕ), n) ∈ ant... | rw [mem_antidiagonal, zero_add]
rw [coeff_one, if_neg H, coeff_mul, ← Finset.insert_erase this,
Finset.sum_insert (Finset.not_mem_erase _ _), coeff_zero_eq_constantCoeff_apply, h,
coeff_invOfUnit, if_neg H, neg_mul, mul_neg, Units.mul_inv_cancel_left, ←
Finset.insert_erase this, Finset.su... |
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
namespace Int
theorem gcd_eq_one_iff_coprime {a b : ℤ} : Int.gcd a b ... | Mathlib/RingTheory/Int/Basic.lean | 49 | 50 | theorem coprime_iff_nat_coprime {a b : ℤ} : IsCoprime a b ↔ Nat.Coprime a.natAbs b.natAbs := by |
rw [← gcd_eq_one_iff_coprime, Nat.coprime_iff_gcd_eq_one, gcd_eq_natAbs]
|
import Mathlib.Tactic.Ring
#align_import algebra.group_power.identities from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {R : Type*} [CommRing R] {a b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R}
theorem sq_add_sq_mul_sq_add_sq :
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 +... | Mathlib/Algebra/Ring/Identities.lean | 55 | 60 | theorem sum_four_sq_mul_sum_four_sq :
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2) * (y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄) ^ 2 + (x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁) ... |
ring
|
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import ring_theory.graded_algebra.homogeneous_ideal from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
open SetLike Direc... | Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | 102 | 107 | theorem HomogeneousIdeal.ext' {I J : HomogeneousIdeal 𝒜} (h : ∀ i, ∀ x ∈ 𝒜 i, x ∈ I ↔ x ∈ J) :
I = J := by |
ext
rw [I.isHomogeneous.mem_iff, J.isHomogeneous.mem_iff]
apply forall_congr'
exact fun i ↦ h i _ (decompose 𝒜 _ i).2
|
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.IsomorphismClasses
import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
#align_import category_theory.limits.shapes.zero_morphisms from "leanpr... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 241 | 244 | theorem of_epi {X Y : C} (f : X ⟶ Y) [Epi f] (i : IsZero X) : IsZero Y := by |
have hf := i.eq_zero_of_src f
subst hf
exact IsZero.of_epi_zero X Y
|
import Mathlib.CategoryTheory.Abelian.Basic
#align_import category_theory.idempotents.basic from "leanprover-community/mathlib"@"3a061790136d13594ec10c7c90d202335ac5d854"
open CategoryTheory
open CategoryTheory.Category
open CategoryTheory.Limits
open CategoryTheory.Preadditive
open Opposite
namespace Catego... | Mathlib/CategoryTheory/Idempotents/Basic.lean | 63 | 92 | theorem isIdempotentComplete_iff_hasEqualizer_of_id_and_idempotent :
IsIdempotentComplete C ↔ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p := by |
constructor
· intro
intro X p hp
rcases IsIdempotentComplete.idempotents_split X p hp with ⟨Y, i, e, ⟨h₁, h₂⟩⟩
exact
⟨Nonempty.intro
{ cone := Fork.ofι i (show i ≫ 𝟙 X = i ≫ p by rw [comp_id, ← h₂, ← assoc, h₁, id_comp])
isLimit := by
apply Fork.IsLimit.mk'
... |
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 | 77 | 78 | theorem eval₂_monomial {n : ℕ} {r : R} : (monomial n r).eval₂ f x = f r * x ^ n := by |
simp [eval₂_eq_sum]
|
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 | 49 | 50 | theorem bernoulliFun_eval_zero (k : ℕ) : bernoulliFun k 0 = bernoulli k := by |
rw [bernoulliFun, Polynomial.eval_zero_map, Polynomial.bernoulli_eval_zero, eq_ratCast]
|
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
noncomputable section
open scoped Classical
open Topology Filter
open TopologicalSpace Set Metric Filter Function
attribute [local simp... | Mathlib/Topology/MetricSpace/PiNat.lean | 219 | 219 | theorem res_length (x : ℕ → α) (n : ℕ) : (res x n).length = n := by | induction n <;> simp [*]
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 132 | 161 | theorem even_odd_of_coprime (hc : Int.gcd x y = 1) :
x % 2 = 0 ∧ y % 2 = 1 ∨ x % 2 = 1 ∧ y % 2 = 0 := by |
cases' Int.emod_two_eq_zero_or_one x with hx hx <;>
cases' Int.emod_two_eq_zero_or_one y with hy hy
-- x even, y even
· exfalso
apply Nat.not_coprime_of_dvd_of_dvd (by decide : 1 < 2) _ _ hc
· apply Int.natCast_dvd.1
apply Int.dvd_of_emod_eq_zero hx
· apply Int.natCast_dvd.1
apply Int... |
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Topology.MetricSpace.Holder
import Mathlib.Topology.MetricSpace.MetricSeparated
#align_import measure_theory.measure.hausdorff from "leanprover-communit... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 159 | 226 | theorem borel_le_caratheodory (hm : IsMetric μ) : borel X ≤ μ.caratheodory := by |
rw [borel_eq_generateFrom_isClosed]
refine MeasurableSpace.generateFrom_le fun t ht => μ.isCaratheodory_iff_le.2 fun s => ?_
set S : ℕ → Set X := fun n => {x ∈ s | (↑n)⁻¹ ≤ infEdist x t}
have Ssep (n) : IsMetricSeparated (S n) t :=
⟨n⁻¹, ENNReal.inv_ne_zero.2 (ENNReal.natCast_ne_top _),
fun x hx y hy... |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [Decida... | Mathlib/Data/Matrix/Basis.lean | 51 | 54 | theorem stdBasisMatrix_add (i : m) (j : n) (a b : α) :
stdBasisMatrix i j (a + b) = stdBasisMatrix i j a + stdBasisMatrix i j b := by |
unfold stdBasisMatrix; ext
split_ifs with h <;> simp [h]
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 491 | 493 | theorem mem_nhds_subtype_iff_nhdsWithin {s : Set α} {a : s} {t : Set s} :
t ∈ 𝓝 a ↔ (↑) '' t ∈ 𝓝[s] (a : α) := by |
rw [← map_nhds_subtype_val, image_mem_map_iff Subtype.val_injective]
|
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α : Type*}
namespace Coheyting
variable [CoheytingAlgebra α] {a b : α}
def boundary (a : α) : α :=
a ⊓ ￢a
#align cohe... | Mathlib/Order/Heyting/Boundary.lean | 80 | 82 | theorem boundary_inf (a b : α) : ∂ (a ⊓ b) = ∂ a ⊓ b ⊔ a ⊓ ∂ b := by |
unfold boundary
rw [hnot_inf_distrib, inf_sup_left, inf_right_comm, ← inf_assoc]
|
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 167 | 169 | theorem repr_total (v) : b.repr (Finsupp.total _ _ _ b v) = v := by |
rw [← b.coe_repr_symm]
exact b.repr.apply_symm_apply v
|
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 161 | 164 | theorem roots_mul {p q : R[X]} (hpq : p * q ≠ 0) : (p * q).roots = p.roots + q.roots := by |
classical
exact Multiset.ext.mpr fun r => by
rw [count_add, count_roots, count_roots, count_roots, rootMultiplicity_mul hpq]
|
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multiset.basic from "leanprover-community/mathlib"@"6c5f73fd6f6cc83122788a80a27cdd54663609f4"
assert_not_exists MonoidWithZero
variable {F ι α β γ : Type*}
names... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 125 | 127 | theorem prod_filter_mul_prod_filter_not (p) [DecidablePred p] :
(s.filter p).prod * (s.filter (fun a ↦ ¬ p a)).prod = s.prod := by |
rw [← prod_add, filter_add_not]
|
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 | 858 | 859 | theorem sign_zero : (0 : Angle).sign = 0 := by |
rw [sign, sin_zero, _root_.sign_zero]
|
import Mathlib.Data.List.Sigma
#align_import data.list.alist from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v w
open List
variable {α : Type u} {β : α → Type v}
structure AList (β : α → Type v) : Type max u v where
entries : List (Sigma β)
nodupKeys : entri... | Mathlib/Data/List/AList.lean | 332 | 334 | theorem insert_insert {a} {b b' : β a} (s : AList β) :
(s.insert a b).insert a b' = s.insert a b' := by |
ext : 1; simp only [AList.insert_entries, List.kerase_cons_eq]
|
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
open MeasureTheory Filter Finset Real
noncomputable section
open scoped MeasureTheory ProbabilityTheory ENNReal NNReal
namespace ProbabilityTheory
variable {Ω ι ... | Mathlib/Probability/Moments.lean | 174 | 175 | theorem mgf_undef (hX : ¬Integrable (fun ω => exp (t * X ω)) μ) : mgf X μ t = 0 := by |
simp only [mgf, integral_undef hX]
|
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 204 | 206 | theorem get_finRange {n : ℕ} {i : ℕ} (h) :
(finRange n).get ⟨i, h⟩ = ⟨i, length_finRange n ▸ h⟩ := by |
simp only [finRange, get_range, get_pmap]
|
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, S... | Mathlib/Order/BooleanAlgebra.lean | 477 | 483 | theorem sup_eq_sdiff_sup_sdiff_sup_inf : x ⊔ y = x \ y ⊔ y \ x ⊔ x ⊓ y :=
Eq.symm <|
calc
x \ y ⊔ y \ x ⊔ x ⊓ y = (x \ y ⊔ y \ x ⊔ x) ⊓ (x \ y ⊔ y \ x ⊔ y) := by | rw [sup_inf_left]
_ = (x \ y ⊔ x ⊔ y \ x) ⊓ (x \ y ⊔ (y \ x ⊔ y)) := by ac_rfl
_ = (x ⊔ y \ x) ⊓ (x \ y ⊔ y) := by rw [sup_sdiff_right, sup_sdiff_right]
_ = x ⊔ y := by rw [sup_sdiff_self_right, sup_sdiff_self_left, inf_idem]
|
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 | 263 | 263 | theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) := by | aesop
|
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Topology.Category.TopCat.Limits.Products
universe w w' v u
open CategoryTheory Opposit... | Mathlib/Topology/Category/TopCat/Yoneda.lean | 48 | 58 | theorem piComparison_fac {α : Type} (X : α → TopCat) :
piComparison (yonedaPresheaf'.{w, w'} Y) (fun x ↦ op (X x)) =
(yonedaPresheaf' Y).map ((opCoproductIsoProduct X).inv ≫ (TopCat.sigmaIsoSigma X).inv.op) ≫
(equivEquivIso (sigmaEquiv Y (fun x ↦ (X x).1))).inv ≫ (Types.productIso _).inv := by |
rw [← Category.assoc, Iso.eq_comp_inv]
ext
simp only [yonedaPresheaf', unop_op, piComparison, types_comp_apply,
Types.productIso_hom_comp_eval_apply, Types.pi_lift_π_apply, comp_apply, TopCat.coe_of,
unop_comp, Quiver.Hom.unop_op, sigmaEquiv, equivEquivIso_hom, Equiv.toIso_inv,
Equiv.coe_fn_symm_mk, ... |
import Mathlib.Dynamics.Newton
import Mathlib.LinearAlgebra.Semisimple
open Algebra Polynomial
namespace Module.End
variable {K V : Type*} [Field K] [AddCommGroup V] [Module K V] [FiniteDimensional K V] {f : End K V}
theorem exists_isNilpotent_isSemisimple_of_separable_of_dvd_pow {P : K[X]} {k : ℕ}
(sep : P.... | Mathlib/LinearAlgebra/JordanChevalley.lean | 71 | 75 | theorem exists_isNilpotent_isSemisimple [PerfectField K] :
∃ᵉ (n ∈ adjoin K {f}) (s ∈ adjoin K {f}), IsNilpotent n ∧ IsSemisimple s ∧ f = n + s := by |
obtain ⟨g, k, sep, -, nil⟩ := exists_squarefree_dvd_pow_of_ne_zero (minpoly.ne_zero_of_finite K f)
rw [← PerfectField.separable_iff_squarefree] at sep
exact exists_isNilpotent_isSemisimple_of_separable_of_dvd_pow sep nil
|
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 | 109 | 110 | theorem isCoseparating_unop_iff (𝒢 : Set Cᵒᵖ) : IsCoseparating 𝒢.unop ↔ IsSeparating 𝒢 := by |
rw [← isSeparating_op_iff, Set.unop_op]
|
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
open Function Set
open scoped Classical
open Affine
variable {𝕜 E F ι : Type*} {π : ι → Type*}
section SMul
variable (𝕜) [OrderedSemiring 𝕜] [AddCommMonoi... | Mathlib/Analysis/Convex/Extreme.lean | 111 | 117 | theorem isExtreme_iInter {ι : Sort*} [Nonempty ι] {F : ι → Set E}
(hAF : ∀ i : ι, IsExtreme 𝕜 A (F i)) : IsExtreme 𝕜 A (⋂ i : ι, F i) := by |
obtain i := Classical.arbitrary ι
refine ⟨iInter_subset_of_subset i (hAF i).1, fun x₁ hx₁A x₂ hx₂A x hxF hx ↦ ?_⟩
simp_rw [mem_iInter] at hxF ⊢
have h := fun i ↦ (hAF i).2 hx₁A hx₂A (hxF i) hx
exact ⟨fun i ↦ (h i).1, fun i ↦ (h i).2⟩
|
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.CategoryTheory.Limits.Preserves.Finite
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open Limits
variable {C : Type u₁} [Category.{v₁} C] (J : Grothendiec... | Mathlib/CategoryTheory/Sites/Sheafification.lean | 131 | 138 | theorem isIso_toSheafify {P : Cᵒᵖ ⥤ D} (hP : Presheaf.IsSheaf J P) : IsIso (toSheafify J P) := by |
refine ⟨(sheafificationAdjunction J D |>.counit.app ⟨P, hP⟩).val, ?_, ?_⟩
· change _ = (𝟙 (sheafToPresheaf J D ⋙ 𝟭 (Cᵒᵖ ⥤ D)) : _).app ⟨P, hP⟩
rw [← sheafificationAdjunction J D |>.right_triangle]
rfl
· change (sheafToPresheaf _ _).map _ ≫ _ = _
change _ ≫ (sheafificationAdjunction J D).unit.app ((... |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 137 | 142 | theorem finsuppRight_apply (t : M ⊗[R] (ι →₀ N)) (i : ι) :
finsuppRight R M N ι t i = lTensor M (Finsupp.lapply i) t := by |
induction t using TensorProduct.induction_on with
| zero => simp
| tmul m f => simp [finsuppRight_apply_tmul_apply]
| add x y hx hy => simp [map_add, hx, hy]
|
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
open Monoid Coprod Multiplicative Subgroup Function
def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (φ : A ≃* B) :
Con (G ∗ Multiplicative ℤ) :=
conGen (fun x y => ∃ (a : A),
x = inr (ofAdd 1) * inl (a : G) ∧
... | Mathlib/GroupTheory/HNNExtension.lean | 344 | 346 | theorem unitsSMulGroup_snd (u : ℤˣ) (g : G) :
(unitsSMulGroup φ d u g).2 = ((d.compl u).equiv g).2 := by |
rcases Int.units_eq_one_or u with rfl | rfl <;> rfl
|
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.manifold.cont_mdiff from "leanprover-community/mathlib"@"e5ab837fc252451f3eb9124ae6e7b6f57455e7b9"
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scope... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 399 | 407 | theorem contMDiffWithinAt_iff_image {x : M} (he : e ∈ maximalAtlas I M)
(he' : e' ∈ maximalAtlas I' M') (hs : s ⊆ e.source) (hx : x ∈ e.source) (hy : f x ∈ e'.source) :
ContMDiffWithinAt I I' n f s x ↔
ContinuousWithinAt f s x ∧
ContDiffWithinAt 𝕜 n (e'.extend I' ∘ f ∘ (e.extend I).symm) (e.exten... |
rw [contMDiffWithinAt_iff_of_mem_maximalAtlas he he' hx hy, and_congr_right_iff]
refine fun _ => contDiffWithinAt_congr_nhds ?_
simp_rw [nhdsWithin_eq_iff_eventuallyEq, e.extend_symm_preimage_inter_range_eventuallyEq I hs hx]
|
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
noncomputable section
open scoped Classical
open Filter
open Function Set Filter
namespace BoxIntegral
variable {ι M : Type*} {... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 147 | 153 | theorem splitLower_ne_splitUpper (I : Box ι) (i : ι) (x : ℝ) :
I.splitLower i x ≠ I.splitUpper i x := by |
cases' le_or_lt x (I.lower i) with h
· rw [splitUpper_eq_self.2 h, splitLower_eq_bot.2 h]
exact WithBot.bot_ne_coe
· refine (disjoint_splitLower_splitUpper I i x).ne ?_
rwa [Ne, splitLower_eq_bot, not_le]
|
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 | 532 | 537 | theorem inv_def (z : K) : z⁻¹ = conj z * ((‖z‖ ^ 2)⁻¹ : ℝ) := by |
rcases eq_or_ne z 0 with (rfl | h₀)
· simp
· apply inv_eq_of_mul_eq_one_right
rw [← mul_assoc, mul_conj, ofReal_inv, ofReal_pow, mul_inv_cancel]
simpa
|
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 | 117 | 118 | theorem contDiffWithinAt_of_subsingleton [Subsingleton F] : ContDiffWithinAt 𝕜 n f s x := by |
rw [Subsingleton.elim f fun _ => 0]; exact contDiffWithinAt_const
|
import Mathlib.Topology.Algebra.UniformConvergence
#align_import topology.algebra.module.strong_topology from "leanprover-community/mathlib"@"8905e5ed90859939681a725b00f6063e65096d95"
open scoped Topology UniformConvergence
section General
variable {𝕜₁ 𝕜₂ : Type*} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜... | Mathlib/Topology/Algebra/Module/StrongTopology.lean | 168 | 177 | theorem continuousSMul [RingHomSurjective σ] [RingHomIsometric σ]
[TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜₂ F] (𝔖 : Set (Set E))
(h𝔖₃ : ∀ S ∈ 𝔖, Bornology.IsVonNBounded 𝕜₁ S) :
ContinuousSMul 𝕜₂ (UniformConvergenceCLM σ F 𝔖) := by |
letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
let φ : (UniformConvergenceCLM σ F 𝔖) →ₗ[𝕜₂] E → F :=
⟨⟨DFunLike.coe, fun _ _ => rfl⟩, fun _ _ => rfl⟩
exact UniformOnFun.continuousSMul_induced_of_image_bounded 𝕜₂ E F (Uniform... |
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