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.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 123 | 125 | theorem isAlgebraic_int [Nontrivial R] (n : ℤ) : IsAlgebraic R (n : A) := by |
rw [← _root_.map_intCast (algebraMap R A)]
exact isAlgebraic_algebraMap (Int.cast n)
|
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 265 | 268 | theorem map_comp_heq' (hobj : ∀ X : C, F.obj X = G.obj X)
(hmap : ∀ {X Y} (f : X ⟶ Y), HEq (F.map f) (G.map f)) :
HEq (F.map (f ≫ g)) (G.map (f ≫ g)) := by |
rw [Functor.hext hobj fun _ _ => hmap]
|
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 | 255 | 257 | theorem doubleton_eq [DecidableEq α] {x y : α} (h : x ≠ y) : ({x, y} : List α) = [x, y] := by |
rw [insert_neg, singleton_eq]
rwa [singleton_eq, mem_singleton]
|
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 629 | 632 | theorem tendstoLocallyUniformly_iff_forall_tendsto :
TendstoLocallyUniformly F f p ↔
∀ x, Tendsto (fun y : ι × α => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤 β) := by |
simp [← tendstoLocallyUniformlyOn_univ, isOpen_univ.tendstoLocallyUniformlyOn_iff_forall_tendsto]
|
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 | 456 | 463 | theorem norm_div_sin_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.sin (o.oangle y (y - x)) = ‖y - x‖ := by |
have hs : (o.oangle y (y - x)).sign = 1 := by
rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
InnerProductGeometry.norm_div_sin_angle_sub_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inr (o.l... |
import Mathlib.MeasureTheory.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set Function Filter
open scoped Classical NNReal Topology ENNReal
... | Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 82 | 84 | theorem measure_inter_union (h : s₁ ∩ s₂ ⊆ ∅) (h₁ : IsCaratheodory m s₁) {t : Set α} :
m (t ∩ (s₁ ∪ s₂)) = m (t ∩ s₁) + m (t ∩ s₂) := by |
rw [h₁, Set.inter_assoc, Set.union_inter_cancel_left, inter_diff_assoc, union_diff_cancel_left h]
|
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 | 972 | 979 | theorem isOpen_inter_eq_singleton_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
(hx : x ∈ s) : ∃ U : Set X, IsOpen U ∧ U ∩ s = {x} := by |
obtain ⟨U, hU_nhds, hU_inter⟩ := nhds_inter_eq_singleton_of_mem_discrete hx
obtain ⟨t, ht_sub, ht_open, ht_x⟩ := mem_nhds_iff.mp hU_nhds
refine ⟨t, ht_open, Set.Subset.antisymm ?_ ?_⟩
· exact hU_inter ▸ Set.inter_subset_inter_left s ht_sub
· rw [Set.subset_inter_iff, Set.singleton_subset_iff, Set.singleton_s... |
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 | 169 | 170 | theorem lowerSemicontinuousOn_univ_iff : LowerSemicontinuousOn f univ ↔ LowerSemicontinuous f := by |
simp [LowerSemicontinuousOn, LowerSemicontinuous, lowerSemicontinuousWithinAt_univ_iff]
|
import Mathlib.Geometry.Manifold.MFDeriv.Basic
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Geometry.Manifold.Algebra.LieGroup
#align_import geometry.manifold.vector_bundle.smooth_section from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Bundle Filter Function
... | Mathlib/Geometry/Manifold/VectorBundle/SmoothSection.lean | 212 | 217 | theorem coe_zsmul (s : Cₛ^n⟮I; F, V⟯) (z : ℤ) : ⇑(z • s : Cₛ^n⟮I; F, V⟯) = z • ⇑s := by |
cases' z with n n
· refine (coe_nsmul s n).trans ?_
simp only [Int.ofNat_eq_coe, natCast_zsmul]
· refine (congr_arg Neg.neg (coe_nsmul s (n + 1))).trans ?_
simp only [negSucc_zsmul, neg_inj]
|
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.localization.localization_localization from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Function
namespace ... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 198 | 231 | theorem isLocalization_of_submonoid_le (M N : Submonoid R) (h : M ≤ N) [IsLocalization M S]
[IsLocalization N T] [Algebra S T] [IsScalarTower R S T] :
IsLocalization (N.map (algebraMap R S)) T :=
{ map_units' := by |
rintro ⟨_, ⟨y, hy, rfl⟩⟩
convert IsLocalization.map_units T ⟨y, hy⟩
exact (IsScalarTower.algebraMap_apply _ _ _ _).symm
surj' := fun y => by
obtain ⟨⟨x, s⟩, e⟩ := IsLocalization.surj N y
refine ⟨⟨algebraMap R S x, _, _, s.prop, rfl⟩, ?_⟩
simpa [← IsScalarTower.algebraMap_apply] ... |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 218 | 218 | theorem ofReal_pos {p : ℝ} : 0 < ENNReal.ofReal p ↔ 0 < p := by | simp [ENNReal.ofReal]
|
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
#align_import algebraic_geometry.pullbacks from "leanprover-community/mathlib"@"7316286ff2942aa14e540add9058c6b0aa1c8070"
set_opt... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 152 | 156 | theorem t'_snd_snd (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ pullback.snd ≫ pullback.snd = pullback.fst ≫ pullback.fst ≫ pullback.fst := by |
simp only [t', Category.assoc, pullbackSymmetry_hom_comp_snd_assoc,
pullbackRightPullbackFstIso_inv_fst_assoc, pullback.lift_fst_assoc, t_snd,
pullbackRightPullbackFstIso_hom_fst_assoc]
|
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 | 772 | 772 | theorem head!_eq_head? [Inhabited α] (l : List α) : head! l = (head? l).iget := by | cases l <;> rfl
|
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTheory.Polynomial.Content
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.polynomial.basic from "leanprover-commun... | Mathlib/RingTheory/Polynomial/Basic.lean | 67 | 68 | theorem mem_degreeLE {n : WithBot ℕ} {f : R[X]} : f ∈ degreeLE R n ↔ degree f ≤ n := by |
simp only [degreeLE, Submodule.mem_iInf, degree_le_iff_coeff_zero, LinearMap.mem_ker]; rfl
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.Real.Sqrt
import Mathlib.Tactic.Polyrith
#align_import algebra.star.chsh from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
universe u
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure Is... | Mathlib/Algebra/Star/CHSH.lean | 104 | 112 | theorem CHSH_id [CommRing R] {A₀ A₁ B₀ B₁ : R} (A₀_inv : A₀ ^ 2 = 1) (A₁_inv : A₁ ^ 2 = 1)
(B₀_inv : B₀ ^ 2 = 1) (B₁_inv : B₁ ^ 2 = 1) :
(2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) * (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) =
4 * (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) := by |
-- polyrith suggests:
linear_combination
(2 * B₀ * B₁ + 2) * A₀_inv + (B₀ ^ 2 - 2 * B₀ * B₁ + B₁ ^ 2) * A₁_inv +
(A₀ ^ 2 + 2 * A₀ * A₁ + 1) * B₀_inv +
(A₀ ^ 2 - 2 * A₀ * A₁ + 1) * B₁_inv
|
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "leanprover-community/mathlib"@"9b2660e1b25419042c8da10bf411aa3c67f14383"
open Nat
def ack : ℕ → ℕ → ℕ
| 0, n => n + 1
| m + 1, 0 ... | Mathlib/Computability/Ackermann.lean | 175 | 185 | theorem add_lt_ack : ∀ m n, m + n < ack m n
| 0, n => by simp
| m + 1, 0 => by simpa using add_lt_ack m 1
| m + 1, n + 1 =>
calc
m + 1 + n + 1 ≤ m + (m + n + 2) := by | omega
_ < ack m (m + n + 2) := add_lt_ack _ _
_ ≤ ack m (ack (m + 1) n) :=
ack_mono_right m <| le_of_eq_of_le (by rw [succ_eq_add_one]; ring_nf)
<| succ_le_of_lt <| add_lt_ack (m + 1) n
_ = ack (m + 1) (n + 1) := (ack_succ_succ m n).symm
|
import Mathlib.Data.Multiset.Bind
import Mathlib.Control.Traversable.Lemmas
import Mathlib.Control.Traversable.Instances
#align_import data.multiset.functor from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
universe u
namespace Multiset
open List
instance functor : Functor Multiset... | Mathlib/Data/Multiset/Functor.lean | 108 | 116 | theorem comp_traverse {G H : Type _ → Type _} [Applicative G] [Applicative H] [CommApplicative G]
[CommApplicative H] {α β γ : Type _} (g : α → G β) (h : β → H γ) (x : Multiset α) :
traverse (Comp.mk ∘ Functor.map h ∘ g) x =
Comp.mk (Functor.map (traverse h) (traverse g x)) := by |
refine Quotient.inductionOn x ?_
intro
simp only [traverse, quot_mk_to_coe, lift_coe, Coe.coe, Function.comp_apply, Functor.map_map,
functor_norm]
simp only [Function.comp, lift_coe]
|
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 94 | 95 | theorem monic_scaleRoots_iff {p : R[X]} (s : R) : Monic (scaleRoots p s) ↔ Monic p := by |
simp only [Monic, leadingCoeff, natDegree_scaleRoots, coeff_scaleRoots_natDegree]
|
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 1,053 | 1,071 | theorem isConnected_iff_sUnion_disjoint_open {s : Set α} :
IsConnected s ↔
∀ U : Finset (Set α), (∀ u v : Set α, u ∈ U → v ∈ U → (s ∩ (u ∩ v)).Nonempty → u = v) →
(∀ u ∈ U, IsOpen u) → (s ⊆ ⋃₀ ↑U) → ∃ u ∈ U, s ⊆ u := by |
rw [IsConnected, isPreconnected_iff_subset_of_disjoint]
refine ⟨fun ⟨hne, h⟩ U hU hUo hsU => ?_, fun h => ⟨?_, fun u v hu hv hs hsuv => ?_⟩⟩
· induction U using Finset.induction_on with
| empty => exact absurd (by simpa using hsU) hne.not_subset_empty
| @insert u U uU IH =>
simp only [← forall_cond... |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 223 | 227 | theorem edist_efixedPoint_le' {s : Set α} (hsc : IsComplete s) (hsf : MapsTo f s s)
(hf : ContractingWith K <| hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
edist x (efixedPoint' f hsc hsf hf x hxs hx) ≤ edist x (f x) / (1 - K) := by |
convert hf.apriori_edist_iterate_efixedPoint_le' hsc hsf hxs hx 0
rw [pow_zero, mul_one]
|
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 | 88 | 101 | theorem liouville_theorem_aux {f : ℂ → F} (hf : Differentiable ℂ f) (hb : IsBounded (range f))
(z w : ℂ) : f z = f w := by |
suffices ∀ c, deriv f c = 0 from is_const_of_deriv_eq_zero hf this z w
clear z w; intro c
obtain ⟨C, C₀, hC⟩ : ∃ C > (0 : ℝ), ∀ z, ‖f z‖ ≤ C := by
rcases isBounded_iff_forall_norm_le.1 hb with ⟨C, hC⟩
exact
⟨max C 1, lt_max_iff.2 (Or.inr zero_lt_one), fun z =>
(hC (f z) (mem_range_self _)).... |
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.metric_space.emetric_space from "leanprover-community/mathlib"@"c8f3055... | Mathlib/Topology/EMetricSpace/Basic.lean | 903 | 904 | theorem diam_le_iff {d : ℝ≥0∞} : diam s ≤ d ↔ ∀ x ∈ s, ∀ y ∈ s, edist x y ≤ d := by |
simp only [diam, iSup_le_iff]
|
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-community/mathlib"@"fefd8a38be7811574cd2ec2f77d3a393a407f112"
open scoped Classical
open Affine
open Finset Function
variable {𝕜 E ι : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Independent.lean | 158 | 166 | theorem convexIndependent_set_iff_not_mem_convexHull_diff {s : Set E} :
ConvexIndependent 𝕜 ((↑) : s → E) ↔ ∀ x ∈ s, x ∉ convexHull 𝕜 (s \ {x}) := by |
rw [convexIndependent_set_iff_inter_convexHull_subset]
constructor
· rintro hs x hxs hx
exact (hs _ Set.diff_subset ⟨hxs, hx⟩).2 (Set.mem_singleton _)
· rintro hs t ht x ⟨hxs, hxt⟩
by_contra h
exact hs _ hxs (convexHull_mono (Set.subset_diff_singleton ht h) hxt)
|
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#align_import data.rat.floor from "leanprover-community/mathlib"@"e1bccd6e40ae78370f01659715d3c948716e3b7e"
open Int
namespace Rat
variable {α : Type*} [LinearOrderedField α] [FloorRi... | Mathlib/Data/Rat/Floor.lean | 75 | 76 | theorem ceil_cast (x : ℚ) : ⌈(x : α)⌉ = ⌈x⌉ := by |
rw [← neg_inj, ← floor_neg, ← floor_neg, ← Rat.cast_neg, Rat.floor_cast]
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 828 | 830 | theorem frontier_prod_univ_eq (s : Set X) :
frontier (s ×ˢ (univ : Set Y)) = frontier s ×ˢ univ := by |
simp [frontier_prod_eq]
|
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 141 | 144 | theorem expChar_is_prime_or_one (q : ℕ) [hq : ExpChar R q] : Nat.Prime q ∨ q = 1 := by |
cases hq with
| zero => exact .inr rfl
| prime hp => exact .inl hp
|
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
import Mathlib.Data.Set.Lattice
#align_import group_theory.sylow from "leanprove... | Mathlib/GroupTheory/Sylow.lean | 538 | 543 | theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
{H : Subgroup G} (hH : Fintype.card H = p ^ n) :
Fintype.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
card (G ⧸ H) [MOD p] := by |
rw [← Fintype.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
|
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped NNReal ENNReal
open Set MeasureTheory Classical
variable {α : Type*} [LinearOrder α] {E : Type*} [PseudoEMetr... | Mathlib/Analysis/ConstantSpeed.lean | 85 | 99 | theorem hasConstantSpeedOnWith_iff_variationOnFromTo_eq :
HasConstantSpeedOnWith f s l ↔ LocallyBoundedVariationOn f s ∧
∀ ⦃x⦄ (_ : x ∈ s) ⦃y⦄ (_ : y ∈ s), variationOnFromTo f s x y = l * (y - x) := by |
constructor
· rintro h; refine ⟨h.hasLocallyBoundedVariationOn, fun x xs y ys => ?_⟩
rw [hasConstantSpeedOnWith_iff_ordered] at h
rcases le_total x y with (xy | yx)
· rw [variationOnFromTo.eq_of_le f s xy, h xs ys xy]
exact ENNReal.toReal_ofReal (mul_nonneg l.prop (sub_nonneg.mpr xy))
· rw [v... |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 129 | 131 | theorem condCount_eq_zero_iff (hs : s.Finite) : condCount s t = 0 ↔ s ∩ t = ∅ := by |
simp [condCount, cond_apply _ hs.measurableSet, Measure.count_apply_eq_top, Set.not_infinite.2 hs,
Measure.count_apply_finite _ (hs.inter_of_left _)]
|
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_dimensional from "leanprover-community/mathlib"@"67e606eaea14c7854bdc556bd53d98aefdf76ec0"
noncomputable section
open Affine
section AffineSpace... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 578 | 589 | theorem Collinear.mem_affineSpan_of_mem_of_ne {s : Set P} (h : Collinear k s) {p₁ p₂ p₃ : P}
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₁p₂ : p₁ ≠ p₂) : p₃ ∈ line[k, p₁, p₂] := by |
rw [collinear_iff_of_mem hp₁] at h
rcases h with ⟨v, h⟩
rcases h p₂ hp₂ with ⟨r₂, rfl⟩
rcases h p₃ hp₃ with ⟨r₃, rfl⟩
rw [vadd_left_mem_affineSpan_pair]
refine ⟨r₃ / r₂, ?_⟩
have h₂ : r₂ ≠ 0 := by
rintro rfl
simp at hp₁p₂
simp [smul_smul, h₂]
|
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 129 | 145 | theorem inducing_iff_inducing_of_iSup_eq_top (h : Continuous f) :
Inducing f ↔ ∀ i, Inducing ((U i).1.restrictPreimage f) := by |
simp_rw [← inducing_subtype_val.of_comp_iff, inducing_iff_nhds, restrictPreimage,
MapsTo.coe_restrict, restrict_eq, ← @Filter.comap_comap _ _ _ _ _ f]
constructor
· intro H i x
rw [Function.comp_apply, ← H, ← inducing_subtype_val.nhds_eq_comap]
· intro H x
obtain ⟨i, hi⟩ :=
Opens.mem_iSup.mp
... |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 456 | 458 | theorem Prime.pow_dvd_iff_dvd_ord_proj {p k n : ℕ} (pp : Prime p) (hn : n ≠ 0) :
p ^ k ∣ n ↔ p ^ k ∣ ord_proj[p] n := by |
rw [pow_dvd_pow_iff_le_right pp.one_lt, pp.pow_dvd_iff_le_factorization hn]
|
import Mathlib.Order.Disjoint
#align_import order.prop_instances from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
instance Prop.instDistribLattice : DistribLattice Prop where
sup := Or
le_sup_left := @Or.inl
le_sup_right := @Or.inr
sup_le := fun _ _ _ => Or.rec
inf := And
... | Mathlib/Order/PropInstances.lean | 88 | 90 | theorem isCompl_iff [∀ i, BoundedOrder (α' i)] {f g : ∀ i, α' i} :
IsCompl f g ↔ ∀ i, IsCompl (f i) (g i) := by |
simp_rw [_root_.isCompl_iff, disjoint_iff, codisjoint_iff, forall_and]
|
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 | 145 | 147 | theorem inv_id : inv (@Eq α) = @Eq α := by |
ext x y
constructor <;> apply Eq.symm
|
import Mathlib.MeasureTheory.SetSemiring
open MeasurableSpace Set
namespace MeasureTheory
variable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}
structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where
empty_mem : ∅ ∈ 𝒜
compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜
union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
... | Mathlib/MeasureTheory/SetAlgebra.lean | 138 | 145 | theorem generateSetAlgebra_mono {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ) :
generateSetAlgebra 𝒜 ⊆ generateSetAlgebra ℬ := by |
intro s hs
induction hs with
| base t t_mem => exact self_subset_generateSetAlgebra (h t_mem)
| empty => exact isSetAlgebra_generateSetAlgebra.empty_mem
| compl t _ t_mem => exact isSetAlgebra_generateSetAlgebra.compl_mem t_mem
| union t u _ _ t_mem u_mem => exact isSetAlgebra_generateSetAlgebra.union_mem ... |
import Mathlib.Data.Set.Function
import Mathlib.Logic.Relation
import Mathlib.Logic.Pairwise
#align_import data.set.pairwise.basic from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Function Order Set
variable {α β γ ι ι' : Type*} {r p q : α → α → Prop}
section Pairwise
variabl... | Mathlib/Data/Set/Pairwise/Basic.lean | 173 | 175 | theorem pairwise_insert_of_symmetric (hr : Symmetric r) :
(insert a s).Pairwise r ↔ s.Pairwise r ∧ ∀ b ∈ s, a ≠ b → r a b := by |
simp only [pairwise_insert, hr.iff a, and_self_iff]
|
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 | 59 | 69 | theorem sq_of_gcd_eq_one {a b c : ℤ} (h : Int.gcd a b = 1) (heq : a * b = c ^ 2) :
∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 := by |
have h' : IsUnit (GCDMonoid.gcd a b) := by
rw [← coe_gcd, h, Int.ofNat_one]
exact isUnit_one
obtain ⟨d, ⟨u, hu⟩⟩ := exists_associated_pow_of_mul_eq_pow h' heq
use d
rw [← hu]
cases' Int.units_eq_one_or u with hu' hu' <;>
· rw [hu']
simp
|
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 | 581 | 590 | theorem addHaar_parallelepiped (b : Basis ι ℝ G) (v : ι → G) :
b.addHaar (parallelepiped v) = ENNReal.ofReal |b.det v| := by |
have : FiniteDimensional ℝ G := FiniteDimensional.of_fintype_basis b
have A : parallelepiped v = b.constr ℕ v '' parallelepiped b := by
rw [image_parallelepiped]
-- Porting note: was `congr 1 with i` but Lean 4 `congr` applies `ext` first
refine congr_arg _ <| funext fun i ↦ ?_
exact (b.constr_basi... |
import Batteries.Data.DList
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Cases
#align_import data.dlist from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
universe u
#align dlist Batteries.DList
namespace Batteries.DList
open Function
variable {α : Type u}
#align dlist.of_list... | Mathlib/Data/DList/Defs.lean | 69 | 69 | theorem toList_empty : toList (@empty α) = [] := by | simp
|
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 123 | 136 | theorem pdf_eq {X : Ω → E} {s : Set E} (hms : MeasurableSet s)
(hu : IsUniform X s ℙ μ) : pdf X ℙ μ =ᵐ[μ] s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) := by |
by_cases hnt : μ s = ∞
· simp [pdf_eq_zero_of_measure_eq_zero_or_top hu (Or.inr hnt), hnt]
by_cases hns : μ s = 0
· filter_upwards [measure_zero_iff_ae_nmem.mp hns,
pdf_eq_zero_of_measure_eq_zero_or_top hu (Or.inl hns)] with x hx h'x
simp [hx, h'x, hns]
have : HasPDF X ℙ μ := hasPDF hns hnt hu
ha... |
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Order.DenselyOrdered
open Set Filter TopologicalSpace Bornology
open scoped ENNReal NNReal Uniformity Topology
universe u v w
variable {α : Type u} {β : Typ... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 196 | 197 | theorem dist_triangle_right (x y z : α) : dist x y ≤ dist x z + dist y z := by |
rw [dist_comm y]; apply dist_triangle
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 135 | 139 | theorem hasDerivAt_tsum (hu : Summable u) (hg : ∀ n y, HasDerivAt (g n) (g' n y) y)
(hg' : ∀ n y, ‖g' n y‖ ≤ u n) (hg0 : Summable fun n => g n y₀) (y : 𝕜) :
HasDerivAt (fun z => ∑' n, g n z) (∑' n, g' n y) y := by |
exact hasDerivAt_tsum_of_isPreconnected hu isOpen_univ isPreconnected_univ
(fun n y _ => hg n y) (fun n y _ => hg' n y) (mem_univ _) hg0 (mem_univ _)
|
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
universe u
namespace PMF
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENN... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 136 | 136 | theorem mem_support_seq_iff : b ∈ (seq q p).support ↔ ∃ f ∈ q.support, b ∈ f '' p.support := by | simp
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Data.Complex.Cardinality
import Mathlib.Data.Fin.VecNotation
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import data.complex.module from "leanprover-community/mathlib"@"c7bce2818663f456335892ddbdd1809f111a... | Mathlib/Data/Complex/Module.lean | 179 | 180 | theorem rank_real_complex'.{u} : Cardinal.lift.{u} (Module.rank ℝ ℂ) = 2 := by |
rw [← finrank_eq_rank, finrank_real_complex, Cardinal.lift_natCast, Nat.cast_ofNat]
|
import Mathlib.Algebra.Polynomial.Module.AEval
#align_import data.polynomial.module from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
universe u v
open Polynomial BigOperators
@[nolint unusedArguments]
def PolynomialModule (R M : Type*) [CommRing R] [AddCommGroup M] [Module R M] := ℕ ... | Mathlib/Algebra/Polynomial/Module/Basic.lean | 324 | 333 | theorem comp_eval (p : R[X]) (q : PolynomialModule R M) (r : R) :
eval r (comp p q) = eval (p.eval r) q := by |
rw [← LinearMap.comp_apply]
apply induction_linear q
· simp_rw [map_zero]
· intro _ _ e₁ e₂
simp_rw [map_add, e₁, e₂]
· intro i m
rw [LinearMap.comp_apply, comp_single, eval_single, eval_smul, eval_single, pow_zero, one_smul,
Polynomial.eval_pow]
|
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 68 | 70 | theorem pow_two : t ^ 2 = 1 := by |
rcases ht with ⟨w, i, rfl⟩
simp
|
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 | 158 | 162 | theorem coe_ideal_span_singleton_div_self {x : R₁} (hx : x ≠ 0) :
(Ideal.span ({x} : Set R₁) : FractionalIdeal R₁⁰ K) / Ideal.span ({x} : Set R₁) = 1 := by |
rw [coeIdeal_span_singleton,
spanSingleton_div_self K <|
(map_ne_zero_iff _ <| NoZeroSMulDivisors.algebraMap_injective R₁ K).mpr hx]
|
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 | 115 | 123 | theorem integralSum_sub_partitions (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F)
{π₁ π₂ : TaggedPrepartition I} (h₁ : π₁.IsPartition) (h₂ : π₂.IsPartition) :
integralSum f vol π₁ - integralSum f vol π₂ =
∑ J ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,
(vol J (f <| (π₁.infPrepartition π₂.toPrepartition... |
rw [← integralSum_inf_partition f vol π₁ h₂, ← integralSum_inf_partition f vol π₂ h₁,
integralSum, integralSum, Finset.sum_sub_distrib]
simp only [infPrepartition_toPrepartition, inf_comm]
|
import Mathlib.Order.Filter.EventuallyConst
import Mathlib.Order.PartialSups
import Mathlib.Algebra.Module.Submodule.IterateMapComap
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Nilpotent.Lemmas
#align_import ring_theory.noetherian from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3... | Mathlib/RingTheory/Noetherian.lean | 359 | 361 | theorem set_has_maximal_iff_noetherian :
(∀ a : Set <| Submodule R M, a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬M' < I) ↔ IsNoetherian R M := by |
rw [isNoetherian_iff_wellFounded, WellFounded.wellFounded_iff_has_min]
|
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped NNReal ENNReal
open Set MeasureTheory Classical
variable {α : Type*} [LinearOrder α] {E : Type*} [PseudoEMetr... | Mathlib/Analysis/ConstantSpeed.lean | 259 | 277 | theorem has_unit_speed_naturalParameterization (f : α → E) {s : Set α}
(hf : LocallyBoundedVariationOn f s) {a : α} (as : a ∈ s) :
HasUnitSpeedOn (naturalParameterization f s a) (variationOnFromTo f s a '' s) := by |
dsimp only [HasUnitSpeedOn]
rw [hasConstantSpeedOnWith_iff_ordered]
rintro _ ⟨b, bs, rfl⟩ _ ⟨c, cs, rfl⟩ h
rcases le_total c b with (cb | bc)
· rw [NNReal.coe_one, one_mul, le_antisymm h (variationOnFromTo.monotoneOn hf as cs bs cb),
sub_self, ENNReal.ofReal_zero, Icc_self, eVariationOn.subsingleton]
... |
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.special_functions.improper_inte... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 76 | 90 | theorem integrableOn_Ioi_rpow_iff {s t : ℝ} (ht : 0 < t) :
IntegrableOn (fun x ↦ x ^ s) (Ioi t) ↔ s < -1 := by |
refine ⟨fun h ↦ ?_, fun h ↦ integrableOn_Ioi_rpow_of_lt h ht⟩
contrapose! h
intro H
have H' : IntegrableOn (fun x ↦ x ^ s) (Ioi (max 1 t)) :=
H.mono (Set.Ioi_subset_Ioi (le_max_right _ _)) le_rfl
have : IntegrableOn (fun x ↦ x⁻¹) (Ioi (max 1 t)) := by
apply H'.mono' measurable_inv.aestronglyMeasurabl... |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
#align_import combinatorics.simple_graph.clique from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
open Finset Fintype Function SimpleGraph.Walk... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 55 | 66 | theorem isClique_iff_induce_eq : G.IsClique s ↔ G.induce s = ⊤ := by |
rw [isClique_iff]
constructor
· intro h
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [comap_adj, Subtype.coe_mk, top_adj, Ne, Subtype.mk_eq_mk]
exact ⟨Adj.ne, h hv hw⟩
· intro h v hv w hw hne
have h2 : (G.induce s).Adj ⟨v, hv⟩ ⟨w, hw⟩ = _ := rfl
conv_lhs at h2 => rw [h]
simp only [top_adj, ne_eq, Subt... |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈... | Mathlib/Topology/Basic.lean | 1,784 | 1,787 | theorem Continuous.range_subset_closure_image_dense {f : X → Y} (hf : Continuous f)
(hs : Dense s) : range f ⊆ closure (f '' s) := by |
rw [← image_univ, ← hs.closure_eq]
exact image_closure_subset_closure_image hf
|
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#align_import analysis.normed.group.add_torsor from "leanprover-community/mathlib"@"837f72de63ad6cd96519cde5f1ffd5ed8d280ad0"
noncomputable section
open NNReal Topo... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 142 | 143 | theorem dist_vsub_cancel_left (x y z : P) : dist (x -ᵥ y) (x -ᵥ z) = dist y z := by |
rw [dist_eq_norm, vsub_sub_vsub_cancel_left, dist_comm, dist_eq_norm_vsub V]
|
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Multiset.Antidiagonal
#align_import data.finsupp.antidiagonal from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
namespace Finsupp
open Finset
universe u
variable {α : Type u} [Decidab... | Mathlib/Data/Finsupp/Antidiagonal.lean | 61 | 79 | theorem antidiagonal_single (a : α) (n : ℕ) :
antidiagonal (single a n) = (antidiagonal n).map
(Function.Embedding.prodMap ⟨_, single_injective a⟩ ⟨_, single_injective a⟩) := by |
ext ⟨x, y⟩
simp only [mem_antidiagonal, mem_map, mem_antidiagonal, Function.Embedding.coe_prodMap,
Function.Embedding.coeFn_mk, Prod.map_apply, Prod.mk.injEq, Prod.exists]
constructor
· intro h
refine ⟨x a, y a, DFunLike.congr_fun h a |>.trans single_eq_same, ?_⟩
simp_rw [DFunLike.ext_iff, ← forall... |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 48 | 50 | theorem zero_scaleRoots (s : R) : scaleRoots 0 s = 0 := by |
ext
simp
|
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform_space.cauchy from "leanprover-community/mathlib"@"22131150f88a2d125713ffa0f4693e3355b1eb49"
universe u v
open scoped Classical
open Filter TopologicalSpace Set Uni... | Mathlib/Topology/UniformSpace/Cauchy.lean | 516 | 530 | theorem TotallyBounded.exists_subset {s : Set α} (hs : TotallyBounded s) {U : Set (α × α)}
(hU : U ∈ 𝓤 α) : ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, { x | (x, y) ∈ U } := by |
rcases comp_symm_of_uniformity hU with ⟨r, hr, rs, rU⟩
rcases hs r hr with ⟨k, fk, ks⟩
let u := k ∩ { y | ∃ x ∈ s, (x, y) ∈ r }
choose f hfs hfr using fun x : u => x.coe_prop.2
refine ⟨range f, ?_, ?_, ?_⟩
· exact range_subset_iff.2 hfs
· haveI : Fintype u := (fk.inter_of_left _).fintype
exact finite... |
import Mathlib.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
#align_import order.bounded from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {α : Type*} {r : α → α → Prop} {s t : Set α}
theorem Bounded.mono (hst : s ⊆ t) (hs : Bounded r t) : Bounde... | Mathlib/Order/Bounded.lean | 116 | 118 | theorem unbounded_lt_iff_unbounded_le [Preorder α] [NoMaxOrder α] :
Unbounded (· < ·) s ↔ Unbounded (· ≤ ·) s := by |
simp_rw [← not_bounded_iff, bounded_le_iff_bounded_lt]
|
import Mathlib.AlgebraicTopology.DoldKan.Homotopies
import Mathlib.Tactic.Ring
#align_import algebraic_topology.dold_kan.faces from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Limits CategoryTheory.Category
CategoryTheory.Preadditive CategoryTheor... | Mathlib/AlgebraicTopology/DoldKan/Faces.lean | 69 | 139 | theorem comp_Hσ_eq {Y : C} {n a q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ)
(hnaq : n = a + q) :
φ ≫ (Hσ q).f (n + 1) =
-φ ≫ X.δ ⟨a + 1, Nat.succ_lt_succ (Nat.lt_succ_iff.mpr (Nat.le.intro hnaq.symm))⟩ ≫
X.σ ⟨a, Nat.lt_succ_iff.mpr (Nat.le.intro hnaq.symm)⟩ := by |
have hnaq_shift : ∀ d : ℕ, n + d = a + d + q := by
intro d
rw [add_assoc, add_comm d, ← add_assoc, hnaq]
rw [Hσ, Homotopy.nullHomotopicMap'_f (c_mk (n + 2) (n + 1) rfl) (c_mk (n + 1) n rfl),
hσ'_eq hnaq (c_mk (n + 1) n rfl), hσ'_eq (hnaq_shift 1) (c_mk (n + 2) (n + 1) rfl)]
simp only [AlternatingFace... |
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
#align_import algebra.lie.skew_adjoint from "leanprover-community/mathlib"@"075b3f7d19b9da85a0b54b3e33055a74fc388dec"
universe u v w w₁
section SkewAdjointEndomorphisms
open LinearMap (BilinF... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 77 | 80 | theorem skewAdjointLieSubalgebraEquiv_apply
(f : skewAdjointLieSubalgebra (B.compl₁₂ (Qₗ := N) (Qₗ' := N) ↑e ↑e)) :
↑(skewAdjointLieSubalgebraEquiv B e f) = e.lieConj f := by |
simp [skewAdjointLieSubalgebraEquiv]
|
import Mathlib.NumberTheory.ModularForms.SlashInvariantForms
import Mathlib.NumberTheory.ModularForms.CongruenceSubgroups
noncomputable section
open ModularForm UpperHalfPlane Matrix
namespace SlashInvariantForm
| Mathlib/NumberTheory/ModularForms/Identities.lean | 22 | 32 | theorem vAdd_width_periodic (N : ℕ) (k n : ℤ) (f : SlashInvariantForm (Gamma N) k) (z : ℍ) :
f (((N * n) : ℝ) +ᵥ z) = f z := by |
norm_cast
rw [← modular_T_zpow_smul z (N * n)]
have Hn := (ModularGroup_T_pow_mem_Gamma N (N * n) (by simp))
simp only [zpow_natCast, Int.natAbs_ofNat] at Hn
convert (SlashInvariantForm.slash_action_eqn' k (Gamma N) f ⟨((ModularGroup.T ^ (N * n))), Hn⟩ z)
unfold SpecialLinearGroup.coeToGL
simp only [Fin.... |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.Data.ULift
#align_import category_theory.discrete_category from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
-- morphism levels before object levels. See note [Category... | Mathlib/CategoryTheory/DiscreteCategory.lean | 56 | 57 | theorem Discrete.mk_as {α : Type u₁} (X : Discrete α) : Discrete.mk X.as = X := by |
rfl
|
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Order.UpperLower.Basic
open Set
namespace Nat
@[simp]
theorem range_cast_int : range ((↑) : ℕ → ℤ) = Ici 0 :=
Subset.antisymm (range_subset_iff.2 Int.ofNat_nonneg) CanLift.prf
theorem image_cast_int_Icc (a b : ℕ) : (↑) '' I... | Mathlib/Data/Nat/Cast/SetInterval.lean | 37 | 38 | theorem image_cast_int_Iic (a : ℕ) : (↑) '' Iic a = Icc (0 : ℤ) a := by |
rw [← Icc_bot, image_cast_int_Icc]; rfl
|
import Mathlib.Algebra.Star.Order
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.StdBasis
#align_import linear_algebra.matrix.dot_product from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
variable {m n p R : Type*}
namespace Matrix
section Semiring
variable [Semiring... | Mathlib/LinearAlgebra/Matrix/DotProduct.lean | 41 | 45 | theorem dotProduct_stdBasis_eq_mul [DecidableEq n] (v : n → R) (c : R) (i : n) :
dotProduct v (LinearMap.stdBasis R (fun _ => R) i c) = v i * c := by |
rw [dotProduct, Finset.sum_eq_single i, LinearMap.stdBasis_same]
· exact fun _ _ hb => by rw [LinearMap.stdBasis_ne _ _ _ _ hb, mul_zero]
· exact fun hi => False.elim (hi <| Finset.mem_univ _)
|
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.Linarith
#align_import set_theory.cardinal.ordinal from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f"
noncomputa... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 298 | 298 | theorem aleph'_pos {o : Ordinal} (ho : 0 < o) : 0 < aleph' o := by | rwa [← aleph'_zero, aleph'_lt]
|
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 | 54 | 60 | theorem rdrop_eq_reverse_drop_reverse : l.rdrop n = reverse (l.reverse.drop n) := by |
rw [rdrop]
induction' l using List.reverseRecOn with xs x IH generalizing n
· simp
· cases n
· simp [take_append]
· simp [take_append_eq_append_take, IH]
|
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 | 200 | 205 | theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by |
simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs,
div_mul_cancel₀ _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff]
rw [← ofReal_div, arg_real_mul]
exact div_pos (abs.pos hy) (abs.pos hx)
|
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import topology.c... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | 143 | 149 | theorem induced_of_isLimit {F : J ⥤ TopCat.{max v u}} (C : Cone F) (hC : IsLimit C) :
C.pt.str = ⨅ j, (F.obj j).str.induced (C.π.app j) := by |
let homeo := homeoOfIso (hC.conePointUniqueUpToIso (limitConeInfiIsLimit F))
refine homeo.inducing.induced.trans ?_
change induced homeo (⨅ j : J, _) = _
simp [induced_iInf, induced_compose]
rfl
|
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [... | Mathlib/FieldTheory/SeparableClosure.lean | 186 | 192 | theorem IsSepClosed.separableClosure_eq_bot_iff [IsSepClosed E] :
separableClosure F E = ⊥ ↔ IsSepClosed F := by |
refine ⟨fun h ↦ IsSepClosed.of_exists_root _ fun p _ hirr hsep ↦ ?_,
fun _ ↦ IntermediateField.eq_bot_of_isSepClosed_of_isSeparable _⟩
obtain ⟨x, hx⟩ := IsSepClosed.exists_aeval_eq_zero E p (degree_pos_of_irreducible hirr).ne' hsep
obtain ⟨x, rfl⟩ := h ▸ mem_separableClosure_iff.2 (hsep.of_dvd <| minpoly.dvd... |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 266 | 270 | theorem natDegree_C (a : R) : natDegree (C a) = 0 := by |
by_cases ha : a = 0
· have : C a = 0 := by rw [ha, C_0]
rw [natDegree, degree_eq_bot.2 this, WithBot.unbot'_bot]
· rw [natDegree, degree_C ha, WithBot.unbot_zero']
|
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.Star.Unitary
#align_import linear_algebra.unitary_group from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
universe u ... | Mathlib/LinearAlgebra/UnitaryGroup.lean | 76 | 80 | theorem det_of_mem_unitary {A : Matrix n n α} (hA : A ∈ Matrix.unitaryGroup n α) :
A.det ∈ unitary α := by |
constructor
· simpa [star, det_transpose] using congr_arg det hA.1
· simpa [star, det_transpose] using congr_arg det hA.2
|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 119 | 123 | theorem Finite.encard_lt_top (h : s.Finite) : s.encard < ⊤ := by |
refine h.induction_on (by simp) ?_
rintro a t hat _ ht'
rw [encard_insert_of_not_mem hat]
exact lt_tsub_iff_right.1 ht'
|
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 102 | 105 | theorem conj (w : W) : cs.IsReflection (w * t * w⁻¹) := by |
obtain ⟨u, i, rfl⟩ := ht
use w * u, i
group
|
import Mathlib.Topology.ContinuousOn
import Mathlib.Order.Minimal
open Set Classical
variable {X : Type*} {Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Preirreducible
def IsPreirreducible (s : Set X) : Prop :=
∀ u v : Set X, IsOpen u → IsOpen v → (s ∩ u).Nonempty → (s ∩ v).Nonempt... | Mathlib/Topology/Irreducible.lean | 332 | 339 | theorem IsPreirreducible.preimage (ht : IsPreirreducible t) {f : Y → X}
(hf : OpenEmbedding f) : IsPreirreducible (f ⁻¹' t) := by |
rintro U V hU hV ⟨x, hx, hx'⟩ ⟨y, hy, hy'⟩
obtain ⟨_, h₁, ⟨y, h₂, rfl⟩, ⟨y', h₃, h₄⟩⟩ :=
ht _ _ (hf.isOpenMap _ hU) (hf.isOpenMap _ hV) ⟨f x, hx, Set.mem_image_of_mem f hx'⟩
⟨f y, hy, Set.mem_image_of_mem f hy'⟩
cases hf.inj h₄
exact ⟨y, h₁, h₂, h₃⟩
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Sym.Card
open Finset Function
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V) {e : Sym2 V}
section EdgeFinset
variable {G₁ G₂ : SimpleGraph V} [Fintype G.edgeSet] [Fintype G₁.edgeSet] [Finty... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 78 | 78 | theorem edgeFinset_ssubset_edgeFinset : G₁.edgeFinset ⊂ G₂.edgeFinset ↔ G₁ < G₂ := by | simp
|
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 | 867 | 874 | theorem ContinuousOn.congr_mono {f g : α → β} {s s₁ : Set α} (h : ContinuousOn f s)
(h' : EqOn g f s₁) (h₁ : s₁ ⊆ s) : ContinuousOn g s₁ := by |
intro x hx
unfold ContinuousWithinAt
have A := (h x (h₁ hx)).mono h₁
unfold ContinuousWithinAt at A
rw [← h' hx] at A
exact A.congr' h'.eventuallyEq_nhdsWithin.symm
|
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
def symmDiff [Sup α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 184 | 185 | theorem sdiff_symmDiff_eq_sup : (a \ b) ∆ b = a ⊔ b := by |
rw [symmDiff_comm, symmDiff_sdiff_eq_sup, sup_comm]
|
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter unitInterval Set Fun... | Mathlib/Topology/Connected/PathConnected.lean | 342 | 354 | theorem trans_symm (γ : Path x y) (γ' : Path y z) : (γ.trans γ').symm = γ'.symm.trans γ.symm := by |
ext t
simp only [trans_apply, ← one_div, symm_apply, not_le, Function.comp_apply]
split_ifs with h h₁ h₂ <;> rw [coe_symm_eq] at h
· have ht : (t : ℝ) = 1 / 2 := by linarith
norm_num [ht]
· refine congr_arg _ (Subtype.ext ?_)
norm_num [sub_sub_eq_add_sub, mul_sub]
· refine congr_arg _ (Subtype.ext ... |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 32 | 32 | theorem lift_top (g : Set α → Filter β) : (⊤ : Filter α).lift g = g univ := by | simp [Filter.lift]
|
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 349 | 355 | theorem linearIsometryEquiv_comp_rightAngleRotation (φ : E ≃ₗᵢ[ℝ] E)
(hφ : 0 < LinearMap.det (φ.toLinearEquiv : E →ₗ[ℝ] E)) (x : E) : φ (J x) = J (φ x) := by |
convert (o.rightAngleRotation_map φ (φ x)).symm
· simp
· symm
rwa [← o.map_eq_iff_det_pos φ.toLinearEquiv] at hφ
rw [@Fact.out (finrank ℝ E = 2), Fintype.card_fin]
|
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 | 898 | 899 | theorem sign_ne_zero_iff {θ : Angle} : θ.sign ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ π := by |
rw [← not_or, ← sign_eq_zero_iff]
|
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 277 | 298 | theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
(h' : Pairwise fun i j => 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E := by |
classical
have finj : Function.Injective f := by
intro i j hij
by_contra h
have : 1 - goodδ E ≤ ‖f i - f j‖ := h' h
simp only [hij, norm_zero, sub_self] at this
linarith [goodδ_lt_one E]
let s := Finset.image f Finset.univ
have s_card : s.card = n := by rw [Finset.card_image_of_injective _ ... |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-community/mathlib"@"fefd8a38be7811574cd2ec2f77d3a393a407f112"
open scoped Classical
open Affine
open Finset Function
variable {𝕜 E ι : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Independent.lean | 133 | 141 | theorem convexIndependent_iff_not_mem_convexHull_diff {p : ι → E} :
ConvexIndependent 𝕜 p ↔ ∀ i s, p i ∉ convexHull 𝕜 (p '' (s \ {i})) := by |
refine ⟨fun hc i s h => ?_, fun h s i hi => ?_⟩
· rw [hc.mem_convexHull_iff] at h
exact h.2 (Set.mem_singleton _)
· by_contra H
refine h i s ?_
rw [Set.diff_singleton_eq_self H]
exact hi
|
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 | 429 | 430 | theorem Antiperiodic.sub_eq [AddGroup α] [InvolutiveNeg β] (h : Antiperiodic f c) (x : α) :
f (x - c) = -f x := by | simp only [← neg_eq_iff_eq_neg, ← h (x - c), sub_add_cancel]
|
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 124 | 124 | theorem lie_le_left : ⁅I, J⁆ ≤ I := by | rw [lie_comm]; exact lie_le_right I J
|
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
#align_import model_theory.basic from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
set_option autoImplicit true
universe u v u' v' w w'
open Cardinal
open Cardinal
namespace FirstOrder
-- intended to b... | Mathlib/ModelTheory/Basic.lean | 253 | 256 | theorem card_functions_sum (i : ℕ) :
#((L.sum L').Functions i)
= (Cardinal.lift.{u'} #(L.Functions i) + Cardinal.lift.{u} #(L'.Functions i) : Cardinal) := by |
simp [Language.sum]
|
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 190 | 206 | theorem mem_sym_iff {m : Sym α n} : m ∈ s.sym n ↔ ∀ a ∈ m, a ∈ s := by |
induction' n with n ih
· refine mem_singleton.trans ⟨?_, fun _ ↦ Sym.eq_nil_of_card_zero _⟩
rintro rfl
exact fun a ha ↦ (Finset.not_mem_empty _ ha).elim
refine mem_sup.trans ⟨?_, fun h ↦ ?_⟩
· rintro ⟨a, ha, he⟩ b hb
rw [mem_image] at he
obtain ⟨m, he, rfl⟩ := he
rw [Sym.mem_cons] at hb
... |
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
#align_import analysis.special_functions.stirling from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open scoped Topology Real Nat Asymptotics
open Finset Filter Nat Real
namespace... | Mathlib/Analysis/SpecialFunctions/Stirling.lean | 61 | 62 | theorem stirlingSeq_one : stirlingSeq 1 = exp 1 / √2 := by |
rw [stirlingSeq, pow_one, factorial_one, cast_one, mul_one, mul_one_div, one_div_div]
|
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 | 863 | 863 | theorem sign_coe_pi : (π : Angle).sign = 0 := by | rw [sign, sin_coe_pi, _root_.sign_zero]
|
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 | 364 | 366 | theorem t0Space_iff_or_not_mem_closure (X : Type u) [TopologicalSpace X] :
T0Space X ↔ Pairwise fun a b : X => a ∉ closure ({b} : Set X) ∨ b ∉ closure ({a} : Set X) := by |
simp only [t0Space_iff_not_inseparable, inseparable_iff_mem_closure, not_and_or]
|
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 114 | 132 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by |
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, ← intervalIntegral.int... |
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 | 488 | 491 | theorem norm_div_tan_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.tan (o.oangle (x - y) x) = ‖x‖ := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
exact (-o).norm_div_tan_oangle_sub_right_of_oangle_eq_pi_div_two h
|
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 42 | 44 | theorem preimage_Ioi (e : α ≃o β) (b : β) : e ⁻¹' Ioi b = Ioi (e.symm b) := by |
ext x
simp [← e.lt_iff_lt]
|
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 169 | 170 | theorem eqToIso_trans {X Y Z : C} (p : X = Y) (q : Y = Z) :
eqToIso p ≪≫ eqToIso q = eqToIso (p.trans q) := by | ext; simp
|
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 126 | 133 | theorem mul_lt_mul_iff_of_le_of_le [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂)
(hb : b₁ ≤ b₂) : a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂ := by |
refine ⟨lt_or_lt_of_mul_lt_mul, fun h => ?_⟩
cases' h with ha' hb'
· exact mul_lt_mul_of_lt_of_le ha' hb
· exact mul_lt_mul_of_le_of_lt ha hb'
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped Classical Topology
open Filter Asymptotics Set
variable {𝕜... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 285 | 288 | theorem iteratedDeriv_succ : iteratedDeriv (n + 1) f = deriv (iteratedDeriv n f) := by |
ext x
rw [← iteratedDerivWithin_univ, ← iteratedDerivWithin_univ, ← derivWithin_univ]
exact iteratedDerivWithin_succ uniqueDiffWithinAt_univ
|
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Tactic.Common
#align_import data.vector from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
assert_not_exists Monoid
universe u v w
def Vector (α : Type u) (n : ℕ) :=
{ l : List α // l.length = n }
#align vector Vector
namespace Vecto... | Mathlib/Data/Vector/Defs.lean | 253 | 254 | theorem toList_cons (a : α) (v : Vector α n) : toList (cons a v) = a :: toList v := by |
cases v; rfl
|
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Bornology
universe u v w
variable {ι : Sort*} {α : Type u} {β :... | Mathlib/Topology/MetricSpace/Thickening.lean | 242 | 244 | theorem cthickening_of_nonpos {δ : ℝ} (hδ : δ ≤ 0) (E : Set α) : cthickening δ E = closure E := by |
ext x
simp [mem_closure_iff_infEdist_zero, cthickening, ENNReal.ofReal_eq_zero.2 hδ]
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,056 | 2,069 | theorem mex_lt_ord_succ_mk {ι : Type u} (f : ι → Ordinal.{u}) :
mex.{_, u} f < (succ #ι).ord := by |
by_contra! h
apply (lt_succ #ι).not_le
have H := fun a => exists_of_lt_mex ((typein_lt_self a).trans_le h)
let g : (succ #ι).ord.out.α → ι := fun a => Classical.choose (H a)
have hg : Injective g := fun a b h' => by
have Hf : ∀ x, f (g x) =
typein ((· < ·) : (succ #ι).ord.out.α → (succ #ι).ord.ou... |
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 125 | 129 | theorem revzip_powersetAux {l : List α} ⦃x⦄ (h : x ∈ revzip (powersetAux l)) : x.1 + x.2 = ↑l := by |
rw [revzip, powersetAux_eq_map_coe, ← map_reverse, zip_map, ← revzip, List.mem_map] at h
simp only [Prod.map_apply, Prod.exists] at h
rcases h with ⟨l₁, l₂, h, rfl, rfl⟩
exact Quot.sound (revzip_sublists _ _ _ h)
|
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