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.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 | 78 | 79 | theorem Icc_eq_empty_iff : Icc a b = ∅ ↔ ¬a ≤ b := by |
rw [← coe_eq_empty, coe_Icc, Set.Icc_eq_empty_iff]
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Tactic.Abel
#align_import algebra.geom_sum fro... | Mathlib/Algebra/GeomSum.lean | 46 | 48 | theorem geom_sum_succ {x : α} {n : ℕ} :
∑ i ∈ range (n + 1), x ^ i = (x * ∑ i ∈ range n, x ^ i) + 1 := by |
simp only [mul_sum, ← pow_succ', sum_range_succ', pow_zero]
|
import Mathlib.Probability.ProbabilityMassFunction.Basic
#align_import probability.probability_mass_function.monad from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal
open MeasureTheory
namespac... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 74 | 80 | theorem toOuterMeasure_pure_apply : (pure a).toOuterMeasure s = if a ∈ s then 1 else 0 := by |
refine (toOuterMeasure_apply (pure a) s).trans ?_
split_ifs with ha
· refine (tsum_congr fun b => ?_).trans (tsum_ite_eq a 1)
exact ite_eq_left_iff.2 fun hb => symm (ite_eq_right_iff.2 fun h => (hb <| h.symm ▸ ha).elim)
· refine (tsum_congr fun b => ?_).trans tsum_zero
exact ite_eq_right_iff.2 fun hb =... |
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 90 | 92 | theorem empty_prod : (∅ : Set α) ×ˢ t = ∅ := by |
ext
exact false_and_iff _
|
import Mathlib.Algebra.MonoidAlgebra.Basic
#align_import algebra.monoid_algebra.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {k G : Type*} [Semiring k]
namespace AddMonoidAlgebra
section
variable [AddCancelCommMonoid G]
noncomputable def divOf (x : k[G]) (g... | Mathlib/Algebra/MonoidAlgebra/Division.lean | 133 | 135 | theorem modOf_apply_of_not_exists_add (x : k[G]) (g : G) (g' : G)
(h : ¬∃ d, g' = g + d) : (x %ᵒᶠ g) g' = x g' := by |
classical exact Finsupp.filter_apply_pos _ _ h
|
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.ring.divisibility from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
variable {α β : Type*}
section Semigroup
variable [Semigroup α] [Semigroup β]... | Mathlib/Algebra/Ring/Divisibility/Basic.lean | 31 | 38 | theorem MulEquiv.decompositionMonoid [DecompositionMonoid β] : DecompositionMonoid α where
primal a b c h := by |
rw [← map_dvd_iff f, map_mul] at h
obtain ⟨a₁, a₂, h⟩ := DecompositionMonoid.primal _ h
refine ⟨symm f a₁, symm f a₂, ?_⟩
simp_rw [← map_dvd_iff f, ← map_mul, eq_symm_apply]
iterate 2 erw [(f : α ≃* β).apply_symm_apply]
exact h
|
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
import Mathlib.Order.Antichain
#align_import combinatorics.simple_graph.coloring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open ... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 249 | 264 | theorem colorable_iff_exists_bdd_nat_coloring (n : ℕ) :
G.Colorable n ↔ ∃ C : G.Coloring ℕ, ∀ v, C v < n := by |
constructor
· rintro hc
have C : G.Coloring (Fin n) := hc.toColoring (by simp)
let f := Embedding.completeGraph (@Fin.valEmbedding n)
use f.toHom.comp C
intro v
cases' C with color valid
exact Fin.is_lt (color v)
· rintro ⟨C, Cf⟩
refine ⟨Coloring.mk ?_ ?_⟩
· exact fun v => ⟨C v, C... |
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ι : Type*} {R : Type*} {M₁ M₂ N₁ N₂ : Type*} {Mᵢ Nᵢ : ι → Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 163 | 175 | theorem posDef_prod_iff {R} [OrderedCommRing R] [Module R M₁] [Module R M₂]
{Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} :
(Q₁.prod Q₂).PosDef ↔ Q₁.PosDef ∧ Q₂.PosDef := by |
simp_rw [posDef_iff_nonneg, nonneg_prod_iff]
constructor
· rintro ⟨⟨hle₁, hle₂⟩, ha⟩
obtain ⟨ha₁, ha₂⟩ := anisotropic_of_prod ha
exact ⟨⟨hle₁, ha₁⟩, ⟨hle₂, ha₂⟩⟩
· rintro ⟨⟨hle₁, ha₁⟩, ⟨hle₂, ha₂⟩⟩
refine ⟨⟨hle₁, hle₂⟩, ?_⟩
rintro ⟨x₁, x₂⟩ (hx : Q₁ x₁ + Q₂ x₂ = 0)
rw [add_eq_zero_iff' (hle₁... |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section General
variable {α : Type*} {g : Gen... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 41 | 42 | theorem part_num_none_iff_s_none : g.partialNumerators.get? n = none ↔ g.s.get? n = none := by |
cases s_nth_eq : g.s.get? n <;> simp [partialNumerators, s_nth_eq]
|
import Mathlib.Algebra.Polynomial.UnitTrinomial
import Mathlib.RingTheory.Polynomial.GaussLemma
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.selmer from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Polynomial
open scoped Polynomial
variable ... | Mathlib/RingTheory/Polynomial/Selmer.lean | 71 | 82 | theorem X_pow_sub_X_sub_one_irreducible_rat (hn1 : n ≠ 1) : Irreducible (X ^ n - X - 1 : ℚ[X]) := by |
by_cases hn0 : n = 0
· rw [hn0, pow_zero, sub_sub, add_comm, ← sub_sub, sub_self, zero_sub]
exact Associated.irreducible ⟨-1, mul_neg_one X⟩ irreducible_X
have hp : (X ^ n - X - 1 : ℤ[X]) = trinomial 0 1 n (-1) (-1) 1 := by
simp only [trinomial, C_neg, C_1]; ring
have hn : 1 < n := Nat.one_lt_iff_ne_ze... |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.shapes.kernels from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
noncomputable section
universe v v₂ u u' u₂
open CategoryTheory
open CategoryTheory.Limits.WalkingParallelPair
namespace... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 369 | 372 | theorem kernelZeroIsoSource_inv :
kernelZeroIsoSource.inv = kernel.lift (0 : X ⟶ Y) (𝟙 X) (by simp) := by |
ext
simp [kernelZeroIsoSource]
|
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Quotient
import Mathlib.Combinatorics.Quiver.Path
#align_import category_theory.path_category from "leanprover-community/mathlib"@"c6dd521ebdce53bb372c527569dd7c25de53a08b"
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
section
def Paths (V : ... | Mathlib/CategoryTheory/PathCategory.lean | 124 | 135 | theorem ext_functor {C} [Category C] {F G : Paths V ⥤ C} (h_obj : F.obj = G.obj)
(h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath =
eqToHom (congr_fun h_obj a) ≫ G.map e.toPath ≫ eqToHom (congr_fun h_obj.symm b)) :
F = G := by |
fapply Functor.ext
· intro X
rw [h_obj]
· intro X Y f
induction' f with Y' Z' g e ih
· erw [F.map_id, G.map_id, Category.id_comp, eqToHom_trans, eqToHom_refl]
· erw [F.map_comp g (Quiver.Hom.toPath e), G.map_comp g (Quiver.Hom.toPath e), ih, h]
simp only [Category.id_comp, eqToHom_refl, eqT... |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 322 | 330 | theorem lift_rank_lt_rank_dual' {V : Type v} [AddCommGroup V] [Module K V]
(h : ℵ₀ ≤ Module.rank K V) :
Cardinal.lift.{u} (Module.rank K V) < Module.rank Kᵐᵒᵖ (V →ₗ[K] K) := by |
obtain ⟨⟨ι, b⟩⟩ := Module.Free.exists_basis (R := K) (M := V)
rw [← b.mk_eq_rank'', rank_dual_eq_card_dual_of_aleph0_le_rank' h,
← (b.constr ℕ (M' := K)).toEquiv.cardinal_eq, mk_arrow]
apply cantor'
erw [nat_lt_lift_iff, one_lt_iff_nontrivial]
infer_instance
|
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196f818146ff79bc0e7ccd9e2d"
universe u
open LawfulTraversable
open Function hiding comp
open Functor
attribute [functor_norm] LawfulTraversabl... | Mathlib/Control/Traversable/Lemmas.lean | 93 | 95 | theorem comp_sequence (x : t (F (G α))) :
sequence (Comp.mk <$> x) = Comp.mk (sequence <$> sequence x) := by |
simp only [sequence, traverse_map, id_comp]; rw [← comp_traverse]; simp [map_id]
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
open Nat
namespace List
@[simp]
theo... | Mathlib/Data/List/Sublists.lean | 372 | 375 | theorem pairwise_sublists {R} {l : List α} (H : Pairwise R l) :
Pairwise (fun l₁ l₂ => Lex R (reverse l₁) (reverse l₂)) (sublists l) := by |
have := (pairwise_reverse.2 H).sublists'
rwa [sublists'_reverse, pairwise_map] at this
|
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
#align_import category_theory.adhesive from "leanprover-community/mathlib"@"afff1f24a6b68d0077c9d63782a1d093e337758c"
namespace CategoryTheory
open Limits
universe v' u' v u
variable {J : Type v'} [Category.{u'} J] {... | Mathlib/CategoryTheory/Adhesive.lean | 289 | 304 | theorem adhesive_of_preserves_and_reflects (F : C ⥤ D) [Adhesive D]
[H₁ : ∀ {X Y S : C} (f : X ⟶ S) (g : Y ⟶ S) [Mono f], HasPullback f g]
[H₂ : ∀ {X Y S : C} (f : S ⟶ X) (g : S ⟶ Y) [Mono f], HasPushout f g]
[PreservesLimitsOfShape WalkingCospan F]
[ReflectsLimitsOfShape WalkingCospan F]
[Preserves... |
apply Adhesive.mk (hasPullback_of_mono_left := H₁) (hasPushout_of_mono_left := H₂)
intros W X Y Z f g h i hf H
rw [IsPushout.isVanKampen_iff]
refine IsVanKampenColimit.of_mapCocone F ?_
refine (IsVanKampenColimit.precompose_isIso_iff (diagramIsoSpan _).inv).mp ?_
refine IsVanKampenColimit.of_iso ?_ (Pushou... |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 280 | 283 | theorem union_zeroLocus (s s' : Set A) :
zeroLocus 𝒜 s ∪ zeroLocus 𝒜 s' = zeroLocus 𝒜 (Ideal.span s ⊓ Ideal.span s' : Ideal A) := by |
rw [zeroLocus_inf]
simp
|
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w
variable {α : Type u} {β : Type v}
open ... | Mathlib/RingTheory/Ideal/Basic.lean | 395 | 403 | theorem IsMaximal.exists_inv {I : Ideal α} (hI : I.IsMaximal) {x} (hx : x ∉ I) :
∃ y, ∃ i ∈ I, y * x + i = 1 := by |
cases' isMaximal_iff.1 hI with H₁ H₂
rcases mem_span_insert.1
(H₂ (span (insert x I)) x (Set.Subset.trans (subset_insert _ _) subset_span) hx
(subset_span (mem_insert _ _))) with
⟨y, z, hz, hy⟩
refine ⟨y, z, ?_, hy.symm⟩
rwa [← span_eq I]
|
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.convex.combination from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set Function
open scoped Classical
open Pointwise
... | Mathlib/Analysis/Convex/Combination.lean | 201 | 213 | theorem Convex.finsum_mem {ι : Sort*} {w : ι → R} {z : ι → E} {s : Set E} (hs : Convex R s)
(h₀ : ∀ i, 0 ≤ w i) (h₁ : ∑ᶠ i, w i = 1) (hz : ∀ i, w i ≠ 0 → z i ∈ s) :
(∑ᶠ i, w i • z i) ∈ s := by |
have hfin_w : (support (w ∘ PLift.down)).Finite := by
by_contra H
rw [finsum, dif_neg H] at h₁
exact zero_ne_one h₁
have hsub : support ((fun i => w i • z i) ∘ PLift.down) ⊆ hfin_w.toFinset :=
(support_smul_subset_left _ _).trans hfin_w.coe_toFinset.ge
rw [finsum_eq_sum_plift_of_support_subset hs... |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.PowerBasis
import Mathlib.RingTheory.PrincipalI... | Mathlib/RingTheory/AdjoinRoot.lean | 546 | 562 | theorem minpoly_root (hf : f ≠ 0) : minpoly K (root f) = f * C f.leadingCoeff⁻¹ := by |
have f'_monic : Monic _ := monic_mul_leadingCoeff_inv hf
refine (minpoly.unique K _ f'_monic ?_ ?_).symm
· rw [AlgHom.map_mul, aeval_eq, mk_self, zero_mul]
intro q q_monic q_aeval
have commutes : (lift (algebraMap K (AdjoinRoot f)) (root f) q_aeval).comp (mk q) = mk f := by
ext
· simp only [RingHom.c... |
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 | 200 | 213 | theorem ext {F G : C ⥤ D} (h_obj : ∀ X, F.obj X = G.obj X)
(h_map : ∀ X Y f,
F.map f = eqToHom (h_obj X) ≫ G.map f ≫ eqToHom (h_obj Y).symm := by | aesop_cat) :
F = G := by
match F, G with
| mk F_pre _ _ , mk G_pre _ _ =>
match F_pre, G_pre with -- Porting note: did not unfold the Prefunctor unlike Lean3
| Prefunctor.mk F_obj _ , Prefunctor.mk G_obj _ =>
obtain rfl : F_obj = G_obj := by
ext X
apply h_obj
congr
funext X Y f... |
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 | 441 | 443 | theorem image_const_sub_Ioc : (fun x => a - x) '' Ioc b c = Ico (a - c) (a - b) := by |
have := image_comp (fun x => a + x) fun x => -x; dsimp [Function.comp_def] at this
simp [sub_eq_add_neg, this, add_comm]
|
import Batteries.Data.List.Lemmas
import Batteries.Data.Array.Basic
import Batteries.Tactic.SeqFocus
import Batteries.Util.ProofWanted
namespace Array
theorem forIn_eq_data_forIn [Monad m]
(as : Array α) (b : β) (f : α → β → m (ForInStep β)) :
forIn as b f = forIn as.data b f := by
let rec loop : ∀ {i h b ... | .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | 106 | 113 | theorem mem_join : ∀ {L : Array (Array α)}, a ∈ L.join ↔ ∃ l, l ∈ L ∧ a ∈ l := by |
simp only [mem_def, join_data, List.mem_join, List.mem_map]
intro l
constructor
· rintro ⟨_, ⟨s, m, rfl⟩, h⟩
exact ⟨s, m, h⟩
· rintro ⟨s, h₁, h₂⟩
refine ⟨s.data, ⟨⟨s, h₁, rfl⟩, h₂⟩⟩
|
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Int
theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
#align int.... | Mathlib/Data/Int/Order/Units.lean | 63 | 67 | theorem units_pow_eq_pow_mod_two (u : ℤˣ) (n : ℕ) : u ^ n = u ^ (n % 2) := by |
conv =>
lhs
rw [← Nat.mod_add_div n 2];
rw [pow_add, pow_mul, units_sq, one_pow, mul_one]
|
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 102 | 131 | theorem derivative_succ_aux (n ν : ℕ) :
Polynomial.derivative (bernsteinPolynomial R (n + 1) (ν + 1)) =
(n + 1) * (bernsteinPolynomial R n ν - bernsteinPolynomial R n (ν + 1)) := by |
rw [bernsteinPolynomial]
suffices ((n + 1).choose (ν + 1) : R[X]) * ((↑(ν + 1 : ℕ) : R[X]) * X ^ ν) * (1 - X) ^ (n - ν) -
((n + 1).choose (ν + 1) : R[X]) * X ^ (ν + 1) * ((↑(n - ν) : R[X]) * (1 - X) ^ (n - ν - 1)) =
(↑(n + 1) : R[X]) * ((n.choose ν : R[X]) * X ^ ν * (1 - X) ^ (n - ν) -
(n.choos... |
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 142 | 164 | theorem sum_subgroup_units_eq_zero [Ring K] [NoZeroDivisors K]
{G : Subgroup Kˣ} [Fintype G] (hg : G ≠ ⊥) :
∑ x : G, (x.val : K) = 0 := by |
rw [Subgroup.ne_bot_iff_exists_ne_one] at hg
rcases hg with ⟨a, ha⟩
-- The action of a on G as an embedding
let a_mul_emb : G ↪ G := mulLeftEmbedding a
-- ... and leaves G unchanged
have h_unchanged : Finset.univ.map a_mul_emb = Finset.univ := by simp
-- Therefore the sum of x over a G is the sum of a x ... |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 110 | 113 | theorem withDensity_add_measure {m : MeasurableSpace α} (μ ν : Measure α) (f : α → ℝ≥0∞) :
(μ + ν).withDensity f = μ.withDensity f + ν.withDensity f := by |
ext1 s hs
simp only [withDensity_apply f hs, restrict_add, lintegral_add_measure, Measure.add_apply]
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 285 | 286 | theorem oangle_add_oangle_rev_neg_right (x y : V) : o.oangle x (-y) + o.oangle y (-x) = 0 := by |
rw [o.oangle_rev (-x), oangle_neg_left_eq_neg_right, add_neg_self]
|
import Mathlib.Algebra.Periodic
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.periodic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace Nat
open Nat Function
theorem periodic_gcd (a : ℕ) : P... | Mathlib/Data/Nat/Periodic.lean | 33 | 34 | theorem periodic_mod (a : ℕ) : Periodic (fun n => n % a) a := by |
simp only [forall_const, eq_self_iff_true, add_mod_right, Periodic]
|
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 74 | 76 | theorem image_restrict (f : α → β) (s t : Set α) :
s.restrict f '' (Subtype.val ⁻¹' t) = f '' (t ∩ s) := by |
rw [restrict_eq, image_comp, image_preimage_eq_inter_range, Subtype.range_coe]
|
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 162 | 168 | theorem sign_of_cycleType (f : Perm α) :
sign f = (-1 : ℤˣ) ^ (f.cycleType.sum + Multiset.card f.cycleType) := by |
rw [sign_of_cycleType']
induction' f.cycleType using Multiset.induction_on with a s ihs
· rfl
· rw [Multiset.map_cons, Multiset.prod_cons, Multiset.sum_cons, Multiset.card_cons, ihs]
simp only [pow_add, pow_one, mul_neg_one, neg_mul, mul_neg, mul_assoc, mul_one]
|
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Normed.Group.ZeroAtInfty
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Ana... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 145 | 153 | theorem isBigO_cocompact_zpow_neg_nat (k : ℕ) :
f =O[cocompact E] fun x => ‖x‖ ^ (-k : ℤ) := by |
obtain ⟨d, _, hd'⟩ := f.decay k 0
simp only [norm_iteratedFDeriv_zero] at hd'
simp_rw [Asymptotics.IsBigO, Asymptotics.IsBigOWith]
refine ⟨d, Filter.Eventually.filter_mono Filter.cocompact_le_cofinite ?_⟩
refine (Filter.eventually_cofinite_ne 0).mono fun x hx => ?_
rw [Real.norm_of_nonneg (zpow_nonneg (nor... |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 123 | 164 | theorem increasing_cantorFunction (h1 : 0 < c) (h2 : c < 1 / 2) {n : ℕ} {f g : ℕ → Bool}
(hn : ∀ k < n, f k = g k) (fn : f n = false) (gn : g n = true) :
cantorFunction c f < cantorFunction c g := by |
have h3 : c < 1 := by
apply h2.trans
norm_num
induction' n with n ih generalizing f g
· let f_max : ℕ → Bool := fun n => Nat.rec false (fun _ _ => true) n
have hf_max : ∀ n, f n → f_max n := by
intro n hn
cases n
· rw [fn] at hn
contradiction
apply rfl
let g_min : ... |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set hiding prod_eq
open Function MeasureTheory
open Filter hiding ma... | Mathlib/MeasureTheory/Group/Prod.lean | 279 | 291 | theorem ae_measure_preimage_mul_right_lt_top [IsMulLeftInvariant ν] (sm : MeasurableSet s)
(hμs : μ s ≠ ∞) : ∀ᵐ x ∂μ, ν ((fun y => y * x) ⁻¹' s) < ∞ := by |
refine ae_of_forall_measure_lt_top_ae_restrict' ν.inv _ ?_
intro A hA _ h3A
simp only [ν.inv_apply] at h3A
apply ae_lt_top (measurable_measure_mul_right ν sm)
have h1 := measure_mul_lintegral_eq μ ν sm (A⁻¹.indicator 1) (measurable_one.indicator hA.inv)
rw [lintegral_indicator _ hA.inv] at h1
simp_rw [Pi... |
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 | 115 | 118 | theorem chain_range_succ (r : ℕ → ℕ → Prop) (n a : ℕ) :
Chain r a (range n.succ) ↔ r a 0 ∧ ∀ m < n, r m m.succ := by |
rw [range_succ_eq_map, chain_cons, and_congr_right_iff, ← chain'_range_succ, range_succ_eq_map]
exact fun _ => Iff.rfl
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Prime
#align_import data.nat.choose.dvd from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
open Nat
namespace Prime
variable {p a b k : ℕ}
| Mathlib/Data/Nat/Choose/Dvd.lean | 24 | 29 | theorem dvd_choose_add (hp : Prime p) (hap : a < p) (hbp : b < p) (h : p ≤ a + b) :
p ∣ choose (a + b) a := by |
have h₁ : p ∣ (a + b)! := hp.dvd_factorial.2 h
rw [← add_choose_mul_factorial_mul_factorial, ← choose_symm_add, hp.dvd_mul, hp.dvd_mul,
hp.dvd_factorial, hp.dvd_factorial] at h₁
exact (h₁.resolve_right hbp.not_le).resolve_right hap.not_le
|
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 190 | 201 | theorem lowerSemicontinuous_aux {ι : Type*} {F : ι → α → E} {p : Filter ι} {f : α → E} {s : Set α}
(Ffs : ∀ x ∈ s, Tendsto (fun i => F i x) p (𝓝 (f x))) {v : ℝ≥0∞} (hv : v < eVariationOn f s) :
∀ᶠ n : ι in p, v < eVariationOn (F n) s := by |
obtain ⟨⟨n, ⟨u, um, us⟩⟩, hlt⟩ :
∃ p : ℕ × { u : ℕ → α // Monotone u ∧ ∀ i, u i ∈ s },
v < ∑ i ∈ Finset.range p.1, edist (f ((p.2 : ℕ → α) (i + 1))) (f ((p.2 : ℕ → α) i)) :=
lt_iSup_iff.mp hv
have : Tendsto (fun j => ∑ i ∈ Finset.range n, edist (F j (u (i + 1))) (F j (u i))) p
(𝓝 (∑ i ∈ Finset... |
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Init.Data.Ordering.Lemmas
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.NormNum
#align_import set_theory.ordinal.notation from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
set_option linter.uppercaseLean3 ... | Mathlib/SetTheory/Ordinal/Notation.lean | 1,207 | 1,210 | theorem fastGrowing_one : fastGrowing 1 = fun n => 2 * n := by |
rw [@fastGrowing_succ 1 0 rfl]; funext i; rw [two_mul, fastGrowing_zero]
suffices ∀ a b, Nat.succ^[a] b = b + a from this _ _
intro a b; induction a <;> simp [*, Function.iterate_succ', Nat.add_assoc, -Function.iterate_succ]
|
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 81 | 84 | theorem ContMDiffWithinAt.comp_of_eq {t : Set M'} {g : M' → M''} {x : M} {y : M'}
(hg : ContMDiffWithinAt I' I'' n g t y) (hf : ContMDiffWithinAt I I' n f s x)
(st : MapsTo f s t) (hx : f x = y) : ContMDiffWithinAt I I'' n (g ∘ f) s x := by |
subst hx; exact hg.comp x hf st
|
import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Analysis.PSeries
open Set Nat
open scoped Topology
-- This needs `Mathlib.Analysis.RCLike.Basic`, so we put it here
-- instead of in `Mathlib.NumberTheory.SmoothNumbers`.
lemma Nat.roughNumbersUpTo_card_le' (N k : ℕ) :
(roughNumbersUpTo N k).card ≤
... | Mathlib/NumberTheory/SumPrimeReciprocals.lean | 82 | 83 | theorem Nat.Primes.not_summable_one_div : ¬ Summable (fun p : Nat.Primes ↦ (1 / p : ℝ)) := by |
convert summable_subtype_iff_indicator.mp.mt not_summable_one_div_on_primes
|
import Mathlib.Algebra.Group.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
universe u
variable {α : Type u}
class Invertible [Mul α] [One α] (a : α) : Type u where
invOf... | Mathlib/Algebra/Group/Invertible/Defs.lean | 141 | 142 | theorem mul_mul_invOf_self_cancel' [Monoid α] (a b : α) {_ : Invertible b} : a * b * ⅟ b = a := by |
simp [mul_assoc]
|
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 319 | 330 | theorem map_div_right_divisors :
n.divisors.map ⟨fun d => (d, n / d), fun p₁ p₂ => congr_arg Prod.fst⟩ =
n.divisorsAntidiagonal := by |
ext ⟨d, nd⟩
simp only [mem_map, mem_divisorsAntidiagonal, Function.Embedding.coeFn_mk, mem_divisors,
Prod.ext_iff, exists_prop, and_left_comm, exists_eq_left]
constructor
· rintro ⟨⟨⟨k, rfl⟩, hn⟩, rfl⟩
rw [Nat.mul_div_cancel_left _ (left_ne_zero_of_mul hn).bot_lt]
exact ⟨rfl, hn⟩
· rintro ⟨rfl, h... |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 79 | 79 | theorem div_zero (h : a ≠ 0) : a / 0 = ∞ := by | simp [div_eq_mul_inv, h]
|
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
universe u v
namespace SimpleGraph
@[ext]
structure Subgraph {V : Type u} (G : SimpleGra... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 507 | 510 | theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) :
(sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by |
ext
simp
|
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Order.Interval.Finset.Nat
#align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bf... | Mathlib/Algebra/Polynomial/Inductions.lean | 45 | 46 | theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) := by |
rw [add_comm]; cases p; rfl
|
import Mathlib.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Function
def PFun (α β : Type*) :=
α → Part β
#align pfun PFun
infixr:25 " →. " => PFun
namespace PFun
variable {α β γ δ ε ι : Type*}
instance inhab... | Mathlib/Data/PFun.lean | 261 | 263 | theorem dom_of_mem_fix {f : α →. Sum β α} {a : α} {b : β} (h : b ∈ f.fix a) : (f a).Dom := by |
let ⟨h₁, h₂⟩ := Part.mem_assert_iff.1 h
rw [WellFounded.fixFEq] at h₂; exact h₂.fst.fst
|
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 | 92 | 94 | theorem aemeasurable_comp_iff {f : α → β} (hf : MeasurePreserving f μa μb)
(h₂ : MeasurableEmbedding f) {g : β → γ} : AEMeasurable (g ∘ f) μa ↔ AEMeasurable g μb := by |
rw [← hf.map_eq, h₂.aemeasurable_map_iff]
|
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
import Mathlib.Order.Antichain
#align_import combinatorics.simple_graph.coloring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open ... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 341 | 349 | theorem chromaticNumber_pos [Nonempty V] {n : ℕ} (hc : G.Colorable n) : 0 < G.chromaticNumber := by |
rw [hc.chromaticNumber_eq_sInf, Nat.cast_pos]
apply le_csInf (colorable_set_nonempty_of_colorable hc)
intro m hm
by_contra h'
simp only [not_le] at h'
obtain ⟨i, hi⟩ := hm.some (Classical.arbitrary V)
have h₁: i < 0 := lt_of_lt_of_le hi (Nat.le_of_lt_succ h')
exact Nat.not_lt_zero _ h₁
|
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
#align_import group_theory.perm.sign from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u v
o... | Mathlib/GroupTheory/Perm/Finite.lean | 176 | 184 | theorem Disjoint.extendDomain {p : β → Prop} [DecidablePred p] (f : α ≃ Subtype p)
{σ τ : Perm α} (h : Disjoint σ τ) : Disjoint (σ.extendDomain f) (τ.extendDomain f) := by |
intro b
by_cases pb : p b
· refine (h (f.symm ⟨b, pb⟩)).imp ?_ ?_ <;>
· intro h
rw [extendDomain_apply_subtype _ _ pb, h, apply_symm_apply, Subtype.coe_mk]
· left
rw [extendDomain_apply_not_subtype _ _ pb]
|
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
#align_import algebraic_geometry.stalks from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
universe v u v' u'
open Opposite Cate... | Mathlib/Geometry/RingedSpace/Stalks.lean | 171 | 177 | theorem congr {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y)
(h₁ : α = β) (x x' : X) (h₂ : x = x') :
stalkMap α x ≫ eqToHom (show X.stalk x = X.stalk x' by rw [h₂]) =
eqToHom (show Y.stalk (α.base x) = Y.stalk (β.base x') by rw [h₁, h₂]) ≫ stalkMap β x' := by |
ext
substs h₁ h₂
simp
|
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import Mathlib.Algebra.ContinuedFractions.TerminatedStable
import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#align_import algebra.continued_... | Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean | 269 | 276 | theorem of_correctness_atTop_of_terminates (terminates : (of v).Terminates) :
∀ᶠ n in atTop, v = (of v).convergents n := by |
rw [eventually_atTop]
obtain ⟨n, terminated_at_n⟩ : ∃ n, (of v).TerminatedAt n := terminates
use n
intro m m_geq_n
rw [convergents_stable_of_terminated m_geq_n terminated_at_n]
exact of_correctness_of_terminatedAt terminated_at_n
|
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Fil... | Mathlib/Analysis/ODE/Gronwall.lean | 86 | 89 | theorem gronwallBound_ε0 (δ K x : ℝ) : gronwallBound δ K 0 x = δ * exp (K * x) := by |
by_cases hK : K = 0
· simp only [gronwallBound_K0, hK, zero_mul, exp_zero, add_zero, mul_one]
· simp only [gronwallBound_of_K_ne_0 hK, zero_div, zero_mul, add_zero]
|
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
noncomputable secti... | Mathlib/CategoryTheory/Closed/Functor.lean | 156 | 159 | theorem frobeniusMorphism_iso_of_expComparison_iso (h : L ⊣ F) (A : C)
[i : IsIso (expComparison F A)] : IsIso (frobeniusMorphism F h A) := by |
rw [← frobeniusMorphism_mate F h] at i
exact @transferNatTransSelf_of_iso _ _ _ _ _ _ _ _ _ _ _ i
|
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import geometry.euclidean.angle.unoriented.basic from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
assert_not_exists HasFDerivAt
assert_not_exists ConformalAt
noncom... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 226 | 229 | theorem angle_add_angle_eq_pi_of_angle_eq_pi {x y : V} (z : V) (h : angle x y = π) :
angle x z + angle y z = π := by |
rcases angle_eq_pi_iff.1 h with ⟨_, ⟨r, ⟨hr, rfl⟩⟩⟩
rw [angle_smul_left_of_neg x z hr, angle_neg_left, add_sub_cancel]
|
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
noncomputable section
open scoped Manifold
open Bundle Set Topology
section SpecificFunctions
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 167 | 172 | theorem tangentMapWithin_id {p : TangentBundle I M} (hs : UniqueMDiffWithinAt I s p.proj) :
tangentMapWithin I I (id : M → M) s p = p := by |
simp only [tangentMapWithin, id]
rw [mfderivWithin_id]
· rcases p with ⟨⟩; rfl
· exact hs
|
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v'... | Mathlib/LinearAlgebra/Dimension/Free.lean | 63 | 66 | theorem FiniteDimensional.finrank_mul_finrank : finrank F K * finrank K A = finrank F A := by |
simp_rw [finrank]
rw [← toNat_lift.{w} (Module.rank F K), ← toNat_lift.{v} (Module.rank K A), ← toNat_mul,
lift_rank_mul_lift_rank, toNat_lift]
|
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 | 892 | 894 | theorem continuousWithinAt_iff_continuousAt {f : α → β} {s : Set α} {x : α} (h : s ∈ 𝓝 x) :
ContinuousWithinAt f s x ↔ ContinuousAt f x := by |
rw [← univ_inter s, continuousWithinAt_inter h, continuousWithinAt_univ]
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 121 | 123 | theorem mem_vadd_const_affineSegment {x y z : V} (p : P) :
z +ᵥ p ∈ affineSegment R (x +ᵥ p) (y +ᵥ p) ↔ z ∈ affineSegment R x y := by |
rw [← affineSegment_vadd_const_image, (vadd_right_injective p).mem_set_image]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.Order.Int... | Mathlib/Order/Filter/AtTopBot.lean | 270 | 273 | theorem Tendsto.eventually_forall_le_atBot {α β : Type*} [Preorder β] {l : Filter α}
{p : β → Prop} {f : α → β} (hf : Tendsto f l atBot) (h_evtl : ∀ᶠ x in atBot, p x) :
∀ᶠ x in l, ∀ y, y ≤ f x → p y := by |
rw [← Filter.eventually_forall_le_atBot] at h_evtl; exact (h_evtl.comap f).filter_mono hf.le_comap
|
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 178 | 181 | theorem angle_eq_zero_of_angle_eq_pi_right {p1 p2 p3 : P} (h : ∠ p1 p2 p3 = π) :
∠ p2 p3 p1 = 0 := by |
rw [angle_comm] at h
exact angle_eq_zero_of_angle_eq_pi_left h
|
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.lym from "leanprover-co... | Mathlib/Combinatorics/SetFamily/LYM.lean | 131 | 133 | theorem mem_falling : s ∈ falling k 𝒜 ↔ (∃ t ∈ 𝒜, s ⊆ t) ∧ s.card = k := by |
simp_rw [falling, mem_sup, mem_powersetCard]
aesop
|
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 217 | 218 | theorem equivRealProd_apply_le (z : ℂ) : ‖equivRealProd z‖ ≤ abs z := by |
simp [Prod.norm_def, abs_re_le_abs, abs_im_le_abs]
|
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 1,034 | 1,040 | theorem uniformity_hasBasis_open_symmetric :
HasBasis (𝓤 α) (fun V : Set (α × α) => V ∈ 𝓤 α ∧ IsOpen V ∧ SymmetricRel V) id := by |
simp only [← and_assoc]
refine uniformity_hasBasis_open.restrict fun s hs => ⟨symmetrizeRel s, ?_⟩
exact
⟨⟨symmetrize_mem_uniformity hs.1, IsOpen.inter hs.2 (hs.2.preimage continuous_swap)⟩,
symmetric_symmetrizeRel s, symmetrizeRel_subset_self s⟩
|
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
open Function
universe u v w
def WSeq (α) :=
Seq (Option α)
#align stream.wseq Stream'.WSeq
... | Mathlib/Data/Seq/WSeq.lean | 1,091 | 1,094 | theorem liftRel_nil (R : α → β → Prop) : LiftRel R nil nil := by |
rw [liftRel_destruct_iff]
-- Porting note: These 2 theorems should be excluded.
simp [-liftRel_pure_left, -liftRel_pure_right]
|
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 | 72 | 73 | theorem piIsoPi_inv_π {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) :
(piIsoPi α).inv ≫ Pi.π α i = piπ α i := by | simp [piIsoPi]
|
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
open Function
universe u v w
def WSeq (α) :=
Seq (Option α)
#align stream.wseq Stream'.WSeq
... | Mathlib/Data/Seq/WSeq.lean | 1,122 | 1,123 | theorem think_congr {s t : WSeq α} (h : s ~ʷ t) : think s ~ʷ think t := by |
unfold Equiv; simpa using h
|
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 | 1,315 | 1,319 | theorem clusterPt_iff_lift'_closure {F : Filter X} :
ClusterPt x F ↔ pure x ≤ (F.lift' closure) := by |
simp_rw [clusterPt_iff_forall_mem_closure,
(hasBasis_pure _).le_basis_iff F.basis_sets.lift'_closure, id, singleton_subset_iff, true_and,
exists_const]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 167 | 171 | theorem iterate_derivative_smul {S : Type*} [Monoid S] [DistribMulAction S R] [IsScalarTower S R R]
(s : S) (p : R[X]) (k : ℕ) : derivative^[k] (s • p) = s • derivative^[k] p := by |
induction' k with k ih generalizing p
· simp
· simp [ih]
|
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {α : Type*} [DecidableEq α]
def nextOr : ∀ (_ : List α) (_ _ : α), α
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 154 | 155 | theorem next_cons_cons_eq' (y z : α) (h : x ∈ y :: z :: l) (hx : x = y) :
next (y :: z :: l) x h = z := by | rw [next, nextOr, if_pos hx]
|
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Algebra.Group.Basic
open scoped Topology Pointwise
open MulAction Set Function
variable {G X : Type*} [TopologicalSpace G] [TopologicalSpace X]
[Group G] [TopologicalGroup G] [MulAction G X]
[SigmaCompactSpace G] [BaireSpace X] [T2Space X]
[Contin... | Mathlib/Topology/Algebra/Group/OpenMapping.lean | 112 | 121 | theorem MonoidHom.isOpenMap_of_sigmaCompact
{H : Type*} [Group H] [TopologicalSpace H] [BaireSpace H] [T2Space H] [ContinuousMul H]
(f : G →* H) (hf : Function.Surjective f) (h'f : Continuous f) :
IsOpenMap f := by |
let A : MulAction G H := MulAction.compHom _ f
have : ContinuousSMul G H := continuousSMul_compHom h'f
have : IsPretransitive G H := isPretransitive_compHom hf
have : f = (fun (g : G) ↦ g • (1 : H)) := by simp [MulAction.compHom_smul_def]
rw [this]
exact isOpenMap_smul_of_sigmaCompact _
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{𝕜 : Type*} [NontriviallyNormedField 𝕜]
{F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 102 | 104 | theorem iteratedDeriv_const_mul {n : ℕ} {f : 𝕜 → 𝕜} (h : ContDiff 𝕜 n f) (c : 𝕜) :
iteratedDeriv n (fun x => f (c * x)) = fun x => c ^ n * iteratedDeriv n f (c * x) := by |
simpa only [smul_eq_mul] using iteratedDeriv_const_smul h c
|
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
#align_import category_theory.limits.shapes.binary_products from "leanprover-community/mathlib"@"fec1d95fc61c750c1ddbb5b1f7f48b8e811a80d7"
... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 711 | 712 | theorem prod.comp_diag {X Y : C} [HasBinaryProduct Y Y] (f : X ⟶ Y) :
f ≫ diag Y = prod.lift f f := by | simp
|
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
open Real Set MeasureTheory MeasureTheory.Measure
section real
| Mathlib/MeasureTheory/Integral/Gamma.lean | 21 | 37 | theorem integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 0 < p) (hq : - 1 < q) :
∫ x in Ioi (0:ℝ), x ^ q * exp (- x ^ p) = (1 / p) * Gamma ((q + 1) / p) := by |
calc
_ = ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * exp (-x)) := by
rw [← integral_comp_rpow_Ioi _ (one_div_ne_zero (ne_of_gt hp)),
abs_eq_self.mpr (le_of_lt (one_div_pos.mpr hp))]
refine setIntegral_congr measurableSet_Ioi (fun _ hx => ?_)
rw [← rpow_mul (le_... |
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 2,163 | 2,168 | theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by |
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
apply subset_union_right
|
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.Order.Basic
#align_import topology.fiber_bundle.trivialization from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open TopologicalSpace Filter Set Bundle Function
... | Mathlib/Topology/FiberBundle/Trivialization.lean | 175 | 177 | theorem symm_apply_mk_proj {x : Z} (ex : x ∈ e.source) :
e.toPartialEquiv.symm (proj x, (e x).2) = x := by |
rw [← e.coe_fst ex, ← e.coe_coe, e.left_inv ex]
|
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open CategoryTheory.Limits CategoryTheory.Functor Opposite
section ArbitraryUniverses
variable {C : Type u₁} [Category.{v₁}... | Mathlib/CategoryTheory/Filtered/Final.lean | 108 | 117 | theorem IsFilteredOrEmpty.of_exists_of_isFiltered_of_fullyFaithful [IsFilteredOrEmpty D] [F.Full]
[F.Faithful] (h : ∀ d, ∃ c, Nonempty (d ⟶ F.obj c)) : IsFilteredOrEmpty C where
cocone_objs c c' := by |
obtain ⟨c₀, ⟨f⟩⟩ := h (IsFiltered.max (F.obj c) (F.obj c'))
exact ⟨c₀, F.preimage (IsFiltered.leftToMax _ _ ≫ f),
F.preimage (IsFiltered.rightToMax _ _ ≫ f), trivial⟩
cocone_maps {c c'} f g := by
obtain ⟨c₀, ⟨f₀⟩⟩ := h (IsFiltered.coeq (F.map f) (F.map g))
refine ⟨_, F.preimage (IsFiltered.coeq... |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 65 | 72 | theorem _root_.MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
evariance X μ < ∞ := by |
have := ENNReal.pow_lt_top (hX.sub <| memℒp_const <| μ[X]).2 2
rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, ← ENNReal.rpow_two] at this
simp only [coe_two, Pi.sub_apply, ENNReal.one_toReal, one_div] at this
rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_o... |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
#align_import category_theory.limits.constructions.zero_objects from "leanprover-community/mathlib"@"52a270e2ea4e342c2587c106f8be904524214a4... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 177 | 180 | theorem pullbackZeroZeroIso_inv_snd (X Y : C) [HasBinaryProduct X Y] :
(pullbackZeroZeroIso X Y).inv ≫ pullback.snd = prod.snd := by |
dsimp [pullbackZeroZeroIso]
simp
|
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Topology Manifold
open Set Bundle
section DerivativesProperties
variable
{𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 726 | 731 | theorem mfderivWithin_comp (hg : MDifferentiableWithinAt I' I'' g u (f x))
(hf : MDifferentiableWithinAt I I' f s x) (h : s ⊆ f ⁻¹' u) (hxs : UniqueMDiffWithinAt I s x) :
mfderivWithin I I'' (g ∘ f) s x =
(mfderivWithin I' I'' g u (f x)).comp (mfderivWithin I I' f s x) := by |
apply HasMFDerivWithinAt.mfderivWithin _ hxs
exact HasMFDerivWithinAt.comp x hg.hasMFDerivWithinAt hf.hasMFDerivWithinAt h
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Finite.Card
#align_import group_theory.subgroup.finite from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
variable {G : Type*} [Group G]
variable {A : Type*} [AddGroup A]
n... | Mathlib/Algebra/Group/Subgroup/Finite.lean | 259 | 270 | theorem mem_normalizer_fintype {S : Set G} [Finite S] {x : G} (h : ∀ n, n ∈ S → x * n * x⁻¹ ∈ S) :
x ∈ Subgroup.setNormalizer S := by |
haveI := Classical.propDecidable; cases nonempty_fintype S;
haveI := Set.fintypeImage S fun n => x * n * x⁻¹;
exact fun n =>
⟨h n, fun h₁ =>
have heq : (fun n => x * n * x⁻¹) '' S = S :=
Set.eq_of_subset_of_card_le (fun n ⟨y, hy⟩ => hy.2 ▸ h y hy.1)
(by rw [Set.card_imag... |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 230 | 232 | theorem HasDerivAt.norm_sq {f : ℝ → F} {f' : F} {x : ℝ} (hf : HasDerivAt f f' x) :
HasDerivAt (‖f ·‖ ^ 2) (2 * Inner.inner (f x) f') x := by |
simpa using hf.hasFDerivAt.norm_sq.hasDerivAt
|
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
varia... | Mathlib/CategoryTheory/Products/Bifunctor.lean | 31 | 34 | theorem map_id_comp (F : C × D ⥤ E) (W : C) {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((𝟙 W, f ≫ g) : (W, X) ⟶ (W, Z)) =
F.map ((𝟙 W, f) : (W, X) ⟶ (W, Y)) ≫ F.map ((𝟙 W, g) : (W, Y) ⟶ (W, Z)) := by |
rw [← Functor.map_comp, prod_comp, Category.comp_id]
|
import Mathlib.MeasureTheory.Decomposition.SignedLebesgue
import Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure
#align_import measure_theory.decomposition.radon_nikodym from "leanprover-community/mathlib"@"fc75855907eaa8ff39791039710f567f37d4556f"
noncomputable section
open scoped Classical MeasureTheory... | Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean | 479 | 487 | theorem integral_rnDeriv_smul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)
[SigmaFinite μ] {f : α → E} :
∫ x, (μ.rnDeriv ν x).toReal • f x ∂ν = ∫ x, f x ∂μ := by |
by_cases hf : Integrable f μ
· rw [← integral_univ, ← withDensityᵥ_apply ((integrable_rnDeriv_smul_iff hμν).mpr hf) .univ,
← integral_univ, ← withDensityᵥ_apply hf .univ, withDensityᵥ_rnDeriv_smul hμν hf]
· rw [integral_undef hf, integral_undef]
contrapose! hf
exact (integrable_rnDeriv_smul_iff hμν... |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 252 | 259 | theorem multiequalizer_ext {I : MulticospanIndex.{w} C} [HasMultiequalizer I]
[PreservesLimit I.multicospan (forget C)] (x y : ↑(multiequalizer I))
(h : ∀ t : I.L, Multiequalizer.ι I t x = Multiequalizer.ι I t y) : x = y := by |
apply Concrete.limit_ext
rintro (a | b)
· apply h
· rw [← limit.w I.multicospan (WalkingMulticospan.Hom.fst b), comp_apply, comp_apply]
simp [h]
|
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Bicategory.Functor.Pseudofunctor
#align_import category_theory.with_terminal from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
namespace CategoryTheory
universe v u
variable (C : Type u) [Category.{v} ... | Mathlib/CategoryTheory/WithTerminal.lean | 297 | 302 | theorem lift_map_liftStar {D : Type*} [Category D] {Z : D} (F : C ⥤ D) (M : ∀ x : C, F.obj x ⟶ Z)
(hM : ∀ (x y : C) (f : x ⟶ y), F.map f ≫ M y = M x) (x : C) :
(lift F M hM).map (starTerminal.from (incl.obj x)) ≫ (liftStar F M hM).hom =
(inclLift F M hM).hom.app x ≫ M x := by |
erw [Category.id_comp, Category.comp_id]
rfl
|
import Mathlib.Data.Matroid.IndepAxioms
open Set
namespace Matroid
variable {α : Type*} {M : Matroid α} {I B X : Set α}
section dual
@[simps] def dualIndepMatroid (M : Matroid α) : IndepMatroid α where
E := M.E
Indep I := I ⊆ M.E ∧ ∃ B, M.Base B ∧ Disjoint I B
indep_empty := ⟨empty_subset M.E, M.exists_b... | Mathlib/Data/Matroid/Dual.lean | 219 | 220 | theorem Indep.ssubset_ground [h : RkPos M✶] (hI : M.Indep I) : I ⊂ M.E := by |
obtain ⟨B, hB⟩ := hI.exists_base_superset; exact hB.2.trans_ssubset hB.1.ssubset_ground
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 86 | 87 | theorem left_ne_zero_of_oangle_ne_zero {x y : V} (h : o.oangle x y ≠ 0) : x ≠ 0 := by |
rintro rfl; simp at h
|
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 219 | 225 | theorem log_div_mul_self (b n : ℕ) : log b (n / b * b) = log b n := by |
rcases le_or_lt b 1 with hb | hb
· rw [log_of_left_le_one hb, log_of_left_le_one hb]
cases' lt_or_le n b with h h
· rw [div_eq_of_lt h, Nat.zero_mul, log_zero_right, log_of_lt h]
rw [log_mul_base hb (Nat.div_pos h (by omega)).ne', log_div_base,
Nat.sub_add_cancel (succ_le_iff.2 <| log_pos hb h)]
|
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 72 | 81 | theorem measurableSet_lineDifferentiableAt_uncurry (hf : Continuous f) :
MeasurableSet {p : E × E | LineDifferentiableAt 𝕜 f p.1 p.2} := by |
borelize 𝕜
let g : (E × E) → 𝕜 → F := fun p t ↦ f (p.1 + t • p.2)
have : Continuous g.uncurry :=
hf.comp <| (continuous_fst.comp continuous_fst).add
<| continuous_snd.smul (continuous_snd.comp continuous_fst)
have M_meas : MeasurableSet {q : (E × E) × 𝕜 | DifferentiableAt 𝕜 (g q.1) q.2} :=
meas... |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace Affine... | Mathlib/Analysis/Convex/Side.lean | 391 | 405 | theorem _root_.Wbtw.wOppSide₁₃ {s : AffineSubspace R P} {x y z : P} (h : Wbtw R x y z)
(hy : y ∈ s) : s.WOppSide x z := by |
rcases h with ⟨t, ⟨ht0, ht1⟩, rfl⟩
refine ⟨_, hy, _, hy, ?_⟩
rcases ht1.lt_or_eq with (ht1' | rfl); swap
· rw [lineMap_apply_one]; simp
rcases ht0.lt_or_eq with (ht0' | rfl); swap
· rw [lineMap_apply_zero]; simp
refine Or.inr (Or.inr ⟨1 - t, t, sub_pos.2 ht1', ht0', ?_⟩)
-- TODO: after lean4#2336 "simp... |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
#align_import analysis.complex.schwarz from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric Set Function Filter TopologicalSpace
open scoped Topology
namespace Complex
section Space... | Mathlib/Analysis/Complex/Schwarz.lean | 92 | 108 | theorem norm_dslope_le_div_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
(h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
‖dslope f c z‖ ≤ R₂ / R₁ := by |
have hR₁ : 0 < R₁ := nonempty_ball.1 ⟨z, hz⟩
have hR₂ : 0 < R₂ := nonempty_ball.1 ⟨f z, h_maps hz⟩
rcases eq_or_ne (dslope f c z) 0 with hc | hc
· rw [hc, norm_zero]; exact div_nonneg hR₂.le hR₁.le
rcases exists_dual_vector ℂ _ hc with ⟨g, hg, hgf⟩
have hg' : ‖g‖₊ = 1 := NNReal.eq hg
have hg₀ : ‖g‖₊ ≠ 0 ... |
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 | 138 | 141 | theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
(h : P.subtype hP = Q.subtype hQ) : P = Q := by |
rw [SetLike.ext_iff] at h ⊢
exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
|
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 347 | 372 | theorem measure_iUnion_congr_of_subset [Countable β] {s : β → Set α} {t : β → Set α}
(hsub : ∀ b, s b ⊆ t b) (h_le : ∀ b, μ (t b) ≤ μ (s b)) : μ (⋃ b, s b) = μ (⋃ b, t b) := by |
rcases Classical.em (∃ b, μ (t b) = ∞) with (⟨b, hb⟩ | htop)
· calc
μ (⋃ b, s b) = ∞ := top_unique (hb ▸ (h_le b).trans <| measure_mono <| subset_iUnion _ _)
_ = μ (⋃ b, t b) := Eq.symm <| top_unique <| hb ▸ measure_mono (subset_iUnion _ _)
push_neg at htop
refine le_antisymm (measure_mono (iUnion_... |
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.basis from "leanprover-community/mathlib"@"6c263e4bfc2e6714de30f22178b4d0ca4d149a76"
noncomputable section
open LinearMap Matrix Set Submodule
open Matrix
section BasisToMatrix
variable {ι... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 113 | 114 | theorem sum_toMatrix_smul_self [Fintype ι] : ∑ i : ι, e.toMatrix v i j • e i = v j := by |
simp_rw [e.toMatrix_apply, e.sum_repr]
|
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 102 | 102 | theorem or_inr {a b : Bool} (H : b) : a || b := by | cases a <;> simp [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 | 65 | 69 | theorem IicSnd_mono {r r' : ℝ} (h_le : r ≤ r') : ρ.IicSnd r ≤ ρ.IicSnd r' := by |
refine Measure.le_iff.2 fun s hs ↦ ?_
simp_rw [IicSnd_apply ρ _ hs]
refine measure_mono (prod_subset_prod_iff.mpr (Or.inl ⟨subset_rfl, Iic_subset_Iic.mpr ?_⟩))
exact mod_cast h_le
|
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {𝕜 E F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup E] [InnerProductSpace... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 82 | 83 | theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v ∈ (𝕜 ∙ u)ᗮ ↔ ⟪v, u⟫ = 0 := by |
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 150 | 154 | theorem mapRange_comp (f : ∀ i, β₁ i → β₂ i) (f₂ : ∀ i, β i → β₁ i) (hf : ∀ i, f i 0 = 0)
(hf₂ : ∀ i, f₂ i 0 = 0) (h : ∀ i, (f i ∘ f₂ i) 0 = 0) (g : Π₀ i : ι, β i) :
mapRange (fun i => f i ∘ f₂ i) h g = mapRange f hf (mapRange f₂ hf₂ g) := by |
ext
simp only [mapRange_apply]; rfl
|
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover-community/mathlib"@"b1abe23ae96fef89ad30d9f4362c307f72a55010"
open Function Set
open OrderDual (toDual ofDual)
universe u v w x
variable {α : Type u} {β : Type v}... | Mathlib/Order/Bounds/Basic.lean | 1,453 | 1,456 | theorem BddBelow.bddBelow_image2_of_bddAbove :
BddBelow s → BddAbove t → BddBelow (Set.image2 f s t) := by |
rintro ⟨a, ha⟩ ⟨b, hb⟩
exact ⟨f a b, mem_lowerBounds_image2_of_mem_lowerBounds_of_mem_upperBounds h₀ h₁ ha hb⟩
|
import Mathlib.Analysis.Normed.Field.Basic
#align_import topology.metric_space.cau_seq_filter from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
universe u v
open Set Filter
open scoped Classical
open Topology
variable {β : Type v}
theorem CauSeq.tendsto_limit [NormedRing β] [hn : ... | Mathlib/Topology/MetricSpace/CauSeqFilter.lean | 67 | 82 | theorem CauSeq.cauchySeq (f : CauSeq β norm) : CauchySeq f := by |
refine cauchy_iff.2 ⟨by infer_instance, fun s hs => ?_⟩
rcases mem_uniformity_dist.1 hs with ⟨ε, ⟨hε, hεs⟩⟩
cases' CauSeq.cauchy₂ f hε with N hN
exists { n | n ≥ N }.image f
simp only [exists_prop, mem_atTop_sets, mem_map, mem_image, ge_iff_le, mem_setOf_eq]
constructor
· exists N
intro b hb
exis... |
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