Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k |
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import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 49 | 51 | theorem rank_one [StrongRankCondition R] [DecidableEq n] :
rank (1 : Matrix n n R) = Fintype.card n := by |
rw [rank, mulVecLin_one, LinearMap.range_id, finrank_top, finrank_pi]
|
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Group.Semiconj.Units
import Mathlib.Init.Classical
#align_import algebra.group_with_zero.semiconj from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
assert_not_exists DenselyOrdered
variable {α M₀ G₀ M₀' G₀' F F' :... | Mathlib/Algebra/GroupWithZero/Semiconj.lean | 29 | 30 | theorem zero_left [MulZeroClass G₀] (x y : G₀) : SemiconjBy 0 x y := by |
simp only [SemiconjBy, mul_zero, zero_mul]
|
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 | 111 | 113 | theorem isClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
IsClosed s ↔ ∀ i, IsClosed ((↑) ⁻¹' s : Set (U i)) := by |
simpa using isOpen_iff_coe_preimage_of_iSup_eq_top hU sᶜ
|
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 155 | 156 | theorem sub (hf : IsBoundedLinearMap 𝕜 f) (hg : IsBoundedLinearMap 𝕜 g) :
IsBoundedLinearMap 𝕜 fun e => f e - g e := by | simpa [sub_eq_add_neg] using add hf (neg hg)
|
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 | 99 | 102 | theorem factorization_prod_pow_eq_self {n : ℕ} (hn : n ≠ 0) : n.factorization.prod (· ^ ·) = n := by |
rw [factorization_eq_factors_multiset n]
simp only [← prod_toMultiset, factorization, Multiset.prod_coe, Multiset.toFinsupp_toMultiset]
exact prod_factors hn
|
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.normalized from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits
CategoryTheory.Subobject CategoryTheory.Idempotents DoldKan
non... | Mathlib/AlgebraicTopology/DoldKan/Normalized.lean | 91 | 92 | theorem PInfty_comp_PInftyToNormalizedMooreComplex (X : SimplicialObject A) :
PInfty ≫ PInftyToNormalizedMooreComplex X = PInftyToNormalizedMooreComplex X := by | aesop_cat
|
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.RingTheory.Ideal.Quotient
#align_import number_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open I... | Mathlib/NumberTheory/Multiplicity.lean | 46 | 48 | theorem dvd_geom_sum₂_iff_of_dvd_sub' {x y p : R} (h : p ∣ x - y) :
(p ∣ ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i)) ↔ p ∣ n * x ^ (n - 1) := by |
rw [geom_sum₂_comm, dvd_geom_sum₂_iff_of_dvd_sub]; simpa using h.neg_right
|
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.CauSeq.Basic
#align_import data.real.cau_seq from "leanprover-community/mathlib"@"9116dd6709f303dcf781632e15fdef382b0fc579"
open Finset IsAbsoluteValue
namespace IsCauSeq
variable {α β : Type*} [LinearOrderedField... | Mathlib/Algebra/Order/CauSeq/BigOperators.lean | 57 | 141 | theorem _root_.cauchy_product (ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n))
(hb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ, ∀ j ≥ i,
abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k -
∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε := by |
let ⟨P, hP⟩ := ha.bounded
let ⟨Q, hQ⟩ := hb.bounded
have hP0 : 0 < P := lt_of_le_of_lt (abs_nonneg _) (hP 0)
have hPε0 : 0 < ε / (2 * P) := div_pos ε0 (mul_pos (show (2 : α) > 0 by norm_num) hP0)
let ⟨N, hN⟩ := hb.cauchy₂ hPε0
have hQε0 : 0 < ε / (4 * Q) :=
div_pos ε0 (mul_pos (show (0 : α) < 4 by norm... |
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 53 | 54 | theorem mem_stepSet (s : σ) (S : Set σ) (a : α) : s ∈ M.stepSet S a ↔ ∃ t ∈ S, s ∈ M.step t a := by |
simp [stepSet]
|
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 158 | 163 | theorem Path.reverse_reverse [h : HasInvolutiveReverse V] {a b : V} (p : Path a b) :
p.reverse.reverse = p := by |
induction' p with _ _ _ _ h
· simp
· rw [Path.reverse, Path.reverse_comp, h, Path.reverse_toPath, Quiver.reverse_reverse]
rfl
|
import Mathlib.Topology.EMetricSpace.Basic
#align_import topology.metric_space.metric_separated from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open EMetric Set
noncomputable section
def IsMetricSeparated {X : Type*} [EMetricSpace X] (s t : Set X) :=
∃ r, r ≠ 0 ∧ ∀ x ∈ s, ∀ y ∈... | Mathlib/Topology/MetricSpace/MetricSeparated.lean | 106 | 109 | theorem finite_iUnion_left_iff {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set X}
{t : Set X} : IsMetricSeparated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, IsMetricSeparated (s i) t := by |
refine Finite.induction_on hI (by simp) @fun i I _ _ hI => ?_
rw [biUnion_insert, forall_mem_insert, union_left_iff, hI]
|
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 | 80 | 90 | theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by |
refine (betaIntegral_convergent_left hu v).trans ?_
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> · push_cast; ring
· norm_num
· norm_num
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Set.Lattice
#align_import group_theory.archimedean from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Set
variable {G : Type*} [LinearOrderedAddCommGroup G] [Archimedean G]
| Mathlib/GroupTheory/Archimedean.lean | 40 | 54 | theorem AddSubgroup.cyclic_of_min {H : AddSubgroup G} {a : G}
(ha : IsLeast { g : G | g ∈ H ∧ 0 < g } a) : H = AddSubgroup.closure {a} := by |
obtain ⟨⟨a_in, a_pos⟩, a_min⟩ := ha
refine le_antisymm ?_ (H.closure_le.mpr <| by simp [a_in])
intro g g_in
obtain ⟨k, ⟨nonneg, lt⟩, _⟩ := existsUnique_zsmul_near_of_pos' a_pos g
have h_zero : g - k • a = 0 := by
by_contra h
have h : a ≤ g - k • a := by
refine a_min ⟨?_, ?_⟩
· exact AddSu... |
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.RCLike.Basic
#align_import data.is_R_or_C.lemmas from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
variable {K E : Type*} [RCLike K]
namespace RCLike
@[simp, rclike_simps]
| Mathlib/Analysis/RCLike/Lemmas.lean | 71 | 74 | theorem reCLM_norm : ‖(reCLM : K →L[ℝ] ℝ)‖ = 1 := by |
apply le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _)
convert ContinuousLinearMap.ratio_le_opNorm (reCLM : K →L[ℝ] ℝ) (1 : K)
simp
|
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 89 | 90 | theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by |
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
|
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 111 | 111 | theorem card_insert_of_mem (h : a ∈ s) : card (insert a s) = s.card := by | rw [insert_eq_of_mem h]
|
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {α : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 27 | 28 | theorem eq_of_eq_of_eq {α} [OrderedSemiring α] {a b : α} (ha : a = 0) (hb : b = 0) : a + b = 0 := by |
simp [*]
|
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 102 | 106 | theorem IsSRGWith.card_neighborFinset_union_of_not_adj {v w : V} (h : G.IsSRGWith n k ℓ μ)
(hne : v ≠ w) (ha : ¬G.Adj v w) :
(G.neighborFinset v ∪ G.neighborFinset w).card = 2 * k - μ := by |
rw [← h.of_not_adj hne ha]
apply h.card_neighborFinset_union_eq
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 118 | 119 | theorem cosh_log {x : ℝ} (hx : 0 < x) : cosh (log x) = (x + x⁻¹) / 2 := by |
rw [cosh_eq, exp_neg, exp_log hx]
|
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (𝕜 E : Type*) {ι : Type*} [OrderedRing 𝕜] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 110 | 119 | theorem disjoint_or_exists_inter_eq_convexHull (hs : s ∈ K.faces) (ht : t ∈ K.faces) :
Disjoint (convexHull 𝕜 (s : Set E)) (convexHull 𝕜 ↑t) ∨
∃ u ∈ K.faces, convexHull 𝕜 (s : Set E) ∩ convexHull 𝕜 ↑t = convexHull 𝕜 ↑u := by |
classical
by_contra! h
refine h.2 (s ∩ t) (K.down_closed hs inter_subset_left fun hst => h.1 <|
disjoint_iff_inf_le.mpr <| (K.inter_subset_convexHull hs ht).trans ?_) ?_
· rw [← coe_inter, hst, coe_empty, convexHull_empty]
rfl
· rw [coe_inter, convexHull_inter_convexHull hs ht]
|
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Topology.MetricSpace.Holder
import Mathlib.Topology.MetricSpace.MetricSeparated
#align_import measure_theory.measure.hausdorff from "leanprover-communit... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 270 | 271 | theorem le_pre : μ ≤ pre m r ↔ ∀ s : Set X, diam s ≤ r → μ s ≤ m s := by |
simp only [pre, le_boundedBy, extend, le_iInf_iff]
|
import Mathlib.Algebra.Algebra.Subalgebra.Operations
import Mathlib.Algebra.Ring.Fin
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.ideal.quotient_operations from "leanprover-community/mathlib"@"b88d81c84530450a8989e918608e5960f015e6c8"
universe u v w
namespace Ideal
open Function RingHom
var... | Mathlib/RingTheory/Ideal/QuotientOperations.lean | 136 | 138 | theorem map_mk_eq_bot_of_le {I J : Ideal R} (h : I ≤ J) : I.map (Quotient.mk J) = ⊥ := by |
rw [map_eq_bot_iff_le_ker, mk_ker]
exact h
|
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Decomposition.Lebesgue
#align_import measure... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 125 | 149 | theorem measure_le_of_frequently_le [SecondCountableTopology α] [BorelSpace α] {ρ : Measure α}
(ν : Measure α) [IsLocallyFiniteMeasure ν] (hρ : ρ ≪ μ) (s : Set α)
(hs : ∀ x ∈ s, ∃ᶠ a in v.filterAt x, ρ a ≤ ν a) : ρ s ≤ ν s := by |
-- this follows from a covering argument using the sets satisfying `ρ a ≤ ν a`.
apply ENNReal.le_of_forall_pos_le_add fun ε εpos _ => ?_
obtain ⟨U, sU, U_open, νU⟩ : ∃ (U : Set α), s ⊆ U ∧ IsOpen U ∧ ν U ≤ ν s + ε :=
exists_isOpen_le_add s ν (ENNReal.coe_pos.2 εpos).ne'
let f : α → Set (Set α) := fun _ => ... |
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Order.Synonym
#align_import algebra.field.basic from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
open Function ... | Mathlib/Algebra/Field/Basic.lean | 135 | 135 | theorem inv_neg : (-a)⁻¹ = -a⁻¹ := by | rw [neg_inv]
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 101 | 104 | theorem isPiSystem_Iic_rat : IsPiSystem (⋃ a : ℚ, {Iic (a : ℝ)}) := by |
convert isPiSystem_image_Iic (((↑) : ℚ → ℝ) '' univ)
ext x
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and]
|
import Mathlib.Algebra.Order.CauSeq.Basic
#align_import data.real.cau_seq_completion from "leanprover-community/mathlib"@"cf4c49c445991489058260d75dae0ff2b1abca28"
variable {α : Type*} [LinearOrderedField α]
namespace CauSeq
section
variable (β : Type*) [Ring β] (abv : β → α) [IsAbsoluteValue abv]
class IsCo... | Mathlib/Algebra/Order/CauSeq/Completion.lean | 413 | 436 | theorem lim_inv {f : CauSeq β abv} (hf : ¬LimZero f) : lim (inv f hf) = (lim f)⁻¹ :=
have hl : lim f ≠ 0 := by | rwa [← lim_eq_zero_iff] at hf
lim_eq_of_equiv_const <|
show LimZero (inv f hf - const abv (lim f)⁻¹) from
have h₁ : ∀ (g f : CauSeq β abv) (hf : ¬LimZero f), LimZero (g - f * inv f hf * g) :=
fun g f hf => by
have h₂ : g - f * inv f hf * g = 1 * g - f * inv f hf * g := by rw [one_mul g]
... |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 546 | 551 | theorem smul_map_inv_div_map_inv_of_isMulTwoCocycle
{f : G × G → M} (hf : IsMulTwoCocycle f) (g : G) :
g • f (g⁻¹, g) / f (g, g⁻¹) = f (1, 1) / f (g, 1) := by |
have := hf g g⁻¹ g
simp only [mul_right_inv, mul_left_inv, map_one_fst_of_isMulTwoCocycle hf g] at this
exact div_eq_div_iff_mul_eq_mul.2 this.symm
|
import Mathlib.CategoryTheory.Functor.Flat
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.cover_preserving from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
universe w v₁ v₂ v₃ u₁ u₂ u₃
noncomputable section
open CategoryT... | Mathlib/CategoryTheory/Sites/CoverPreserving.lean | 126 | 158 | theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
(K : GrothendieckTopology D) (G : C ⥤ D) [RepresentablyFlat G] : CompatiblePreserving K G := by |
constructor
intro ℱ Z T x hx Y₁ Y₂ X f₁ f₂ g₁ g₂ hg₁ hg₂ e
-- First, `f₁` and `f₂` form a cone over `cospan g₁ g₂ ⋙ u`.
let c : Cone (cospan g₁ g₂ ⋙ G) :=
(Cones.postcompose (diagramIsoCospan (cospan g₁ g₂ ⋙ G)).inv).obj (PullbackCone.mk f₁ f₂ e)
/-
This can then be viewed as a cospan of structured a... |
import Mathlib.Algebra.Order.Ring.Int
#align_import data.int.least_greatest from "leanprover-community/mathlib"@"3342d1b2178381196f818146ff79bc0e7ccd9e2d"
namespace Int
def leastOfBdd {P : ℤ → Prop} [DecidablePred P] (b : ℤ) (Hb : ∀ z : ℤ, P z → b ≤ z)
(Hinh : ∃ z : ℤ, P z) : { lb : ℤ // P lb ∧ ∀ z : ℤ, P z... | Mathlib/Data/Int/LeastGreatest.lean | 61 | 68 | theorem exists_least_of_bdd
{P : ℤ → Prop}
(Hbdd : ∃ b : ℤ , ∀ z : ℤ , P z → b ≤ z)
(Hinh : ∃ z : ℤ , P z) : ∃ lb : ℤ , P lb ∧ ∀ z : ℤ , P z → lb ≤ z := by |
classical
let ⟨b , Hb⟩ := Hbdd
let ⟨lb , H⟩ := leastOfBdd b Hb Hinh
exact ⟨lb , H⟩
|
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.int.gcd from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
namespace Nat
... | Mathlib/Data/Int/GCD.lean | 146 | 154 | theorem exists_mul_emod_eq_gcd {k n : ℕ} (hk : gcd n k < k) : ∃ m, n * m % k = gcd n k := by |
have hk' := Int.ofNat_ne_zero.2 (ne_of_gt (lt_of_le_of_lt (zero_le (gcd n k)) hk))
have key := congr_arg (fun (m : ℤ) => (m % k).toNat) (gcd_eq_gcd_ab n k)
simp only at key
rw [Int.add_mul_emod_self_left, ← Int.natCast_mod, Int.toNat_natCast, mod_eq_of_lt hk] at key
refine ⟨(n.gcdA k % k).toNat, Eq.trans (In... |
import Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties
import Mathlib.RingTheory.RingHom.FiniteType
#align_import algebraic_geometry.morphisms.finite_type from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite ... | Mathlib/AlgebraicGeometry/Morphisms/FiniteType.lean | 65 | 71 | theorem locallyOfFiniteTypeOfComp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
[hf : LocallyOfFiniteType (f ≫ g)] : LocallyOfFiniteType f := by |
revert hf
rw [locallyOfFiniteType_eq]
apply RingHom.finiteType_is_local.affineLocally_of_comp
introv H
exact RingHom.FiniteType.of_comp_finiteType H
|
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 64 | 79 | theorem closure_mul_image_eq (hR : R ∈ rightTransversals (H : Set G)) (hR1 : (1 : G) ∈ R)
(hS : closure S = ⊤) : closure ((R * S).image fun g => g * (toFun hR g : G)⁻¹) = H := by |
have hU : closure ((R * S).image fun g => g * (toFun hR g : G)⁻¹) ≤ H := by
rw [closure_le]
rintro - ⟨g, -, rfl⟩
exact mul_inv_toFun_mem hR g
refine le_antisymm hU fun h hh => ?_
obtain ⟨g, hg, r, hr, rfl⟩ :=
show h ∈ _ from eq_top_iff.mp (closure_mul_image_mul_eq_top hR hR1 hS) (mem_top h)
suf... |
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 | 63 | 76 | theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by |
apply IntervalIntegrable.mul_continuousOn
· refine intervalIntegral.intervalIntegrable_cpow' ?_
rwa [sub_re, one_re, ← zero_sub, sub_lt_sub_iff_right]
· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : ℝ) ≤ 1 / 2)] at hx
apply ContinuousAt.cpow
· exact (continuo... |
import Mathlib.Topology.Homotopy.Basic
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Analysis.Convex.Basic
#align_import topology.homotopy.path from "leanprover-community/mathlib"@"bb9d1c5085e0b7ea619806a68c5021927cecb2a6"
universe u v
variable {X : Type u} {Y : Type v} [TopologicalSpace X] [Top... | Mathlib/Topology/Homotopy/Path.lean | 89 | 91 | theorem eval_one (F : Homotopy p₀ p₁) : F.eval 1 = p₁ := by |
ext t
simp [eval]
|
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
open Function Set
open scoped Classical
open Affine
variable {𝕜 E F ι : Type*} {π : ι → Type*}
section SMul
variable (𝕜) [OrderedSemiring 𝕜] [AddCommMonoi... | Mathlib/Analysis/Convex/Extreme.lean | 97 | 103 | theorem IsExtreme.inter (hAB : IsExtreme 𝕜 A B) (hAC : IsExtreme 𝕜 A C) :
IsExtreme 𝕜 A (B ∩ C) := by |
use Subset.trans inter_subset_left hAB.1
rintro x₁ hx₁A x₂ hx₂A x ⟨hxB, hxC⟩ hx
obtain ⟨hx₁B, hx₂B⟩ := hAB.2 hx₁A hx₂A hxB hx
obtain ⟨hx₁C, hx₂C⟩ := hAC.2 hx₁A hx₂A hxC hx
exact ⟨⟨hx₁B, hx₁C⟩, hx₂B, hx₂C⟩
|
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multiset.basic from "leanprover-community/mathlib"@"6c5f73fd6f6cc83122788a80a27cdd54663609f4"
assert_not_exists MonoidWithZero
variable {F ι α β γ : Type*}
names... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 99 | 100 | theorem prod_singleton (a : α) : prod {a} = a := by |
simp only [mul_one, prod_cons, ← cons_zero, eq_self_iff_true, prod_zero]
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.Topology.Algebra.Module.Basic
open Function
structure ContinuousAffineEquiv (k P₁ P₂ : Type*) {V₁ V₂ : Type*} [Ring k]
[AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
[AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P... | Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean | 65 | 67 | theorem toAffineEquiv_injective : Injective (toAffineEquiv : (P₁ ≃ᵃL[k] P₂) → P₁ ≃ᵃ[k] P₂) := by |
rintro ⟨e, econt, einv_cont⟩ ⟨e', e'cont, e'inv_cont⟩ H
congr
|
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.I... | Mathlib/LinearAlgebra/Dual.lean | 388 | 392 | theorem sum_dual_apply_smul_coord (f : Module.Dual R M) :
(∑ x, f (b x) • b.coord x) = f := by |
ext m
simp_rw [LinearMap.sum_apply, LinearMap.smul_apply, smul_eq_mul, mul_comm (f _), ← smul_eq_mul, ←
f.map_smul, ← _root_.map_sum, Basis.coord_apply, Basis.sum_repr]
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
... | Mathlib/RingTheory/PowerSeries/Basic.lean | 229 | 231 | theorem coeff_zero_eq_constantCoeff : ⇑(coeff R 0) = constantCoeff R := by |
rw [coeff, Finsupp.single_zero]
rfl
|
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import topology.algebra.module.multilinear from "leanprover-community/mathlib"@"f40476639bac089693a489c9e354ebd75dc0f886"
open Function Fin Set
universe u v w w₁ w₁' w₂ w₃ w₄
variable {R : Type u} {ι : Type v} {n ... | Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean | 113 | 114 | theorem ext_iff {f f' : ContinuousMultilinearMap R M₁ M₂} : f = f' ↔ ∀ x, f x = f' x := by |
rw [← toMultilinearMap_injective.eq_iff, MultilinearMap.ext_iff]; rfl
|
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 126 | 146 | theorem hurwitzZetaEven_one_sub_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
hurwitzZetaEven x (1 - 2 * k) =
-1 / (2 * k) * ((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
have h1 (n : ℕ) : (2 * k : ℂ) ≠ -n := by
rw [← Int.cast_ofNat, ← Int.cast_natCast, ← Int.cast_mul, ← Int.cast_natCast n, ← Int.cast_neg,
Ne, Int.cast_inj, ← Ne]
refine ne_of_gt ((neg_nonpos_of_nonneg n.cast_nonneg).trans_lt (mul_pos two_pos ?_))
exact Nat.cast_pos.mpr (Nat.pos_of_ne_zero hk)
have... |
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 | 137 | 143 | theorem pairwise_union :
(s ∪ t).Pairwise r ↔
s.Pairwise r ∧ t.Pairwise r ∧ ∀ a ∈ s, ∀ b ∈ t, a ≠ b → r a b ∧ r b a := by |
simp only [Set.Pairwise, mem_union, or_imp, forall_and]
exact
⟨fun H => ⟨H.1.1, H.2.2, H.1.2, fun x hx y hy hne => H.2.1 y hy x hx hne.symm⟩,
fun H => ⟨⟨H.1, H.2.2.1⟩, fun x hx y hy hne => H.2.2.2 y hy x hx hne.symm, H.2.1⟩⟩
|
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 71 | 74 | theorem rank_mul_le_left [StrongRankCondition R] (A : Matrix m n R) (B : Matrix n o R) :
(A * B).rank ≤ A.rank := by |
rw [rank, rank, mulVecLin_mul]
exact Cardinal.toNat_le_toNat (LinearMap.rank_comp_le_left _ _) (rank_lt_aleph0 _ _)
|
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Linarith.Frontend
#align_import algebra.quadratic_discriminant from "leanprover-community/mathlib"@"e085d1df33274f4b32f611f483aae678ba0b42df"
open Filter
section Ring
variable {R : ... | Mathlib/Algebra/QuadraticDiscriminant.lean | 63 | 70 | theorem quadratic_eq_zero_iff_discrim_eq_sq [NeZero (2 : R)] [NoZeroDivisors R]
(ha : a ≠ 0) (x : R) :
a * x * x + b * x + c = 0 ↔ discrim a b c = (2 * a * x + b) ^ 2 := by |
refine ⟨discrim_eq_sq_of_quadratic_eq_zero, fun h ↦ ?_⟩
rw [discrim] at h
have ha : 2 * 2 * a ≠ 0 := mul_ne_zero (mul_ne_zero (NeZero.ne _) (NeZero.ne _)) ha
apply mul_left_cancel₀ ha
linear_combination -h
|
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
#align_import analysis.calculus.deriv.pow from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
variable {... | Mathlib/Analysis/Calculus/Deriv/Pow.lean | 99 | 102 | theorem HasDerivAt.pow (hc : HasDerivAt c c' x) :
HasDerivAt (fun y => c y ^ n) ((n : 𝕜) * c x ^ (n - 1) * c') x := by |
rw [← hasDerivWithinAt_univ] at *
exact hc.pow n
|
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 | 158 | 161 | theorem mapRange_zero (f : ∀ i, β₁ i → β₂ i) (hf : ∀ i, f i 0 = 0) :
mapRange f hf (0 : Π₀ i, β₁ i) = 0 := by |
ext
simp only [mapRange_apply, coe_zero, Pi.zero_apply, hf]
|
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Instances.Real
#align_import topology.unit_interval from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
... | Mathlib/Topology/UnitInterval.lean | 323 | 324 | theorem affineHomeomorph_image_I (a b : 𝕜) (h : 0 < a) :
affineHomeomorph a b h.ne.symm '' Set.Icc 0 1 = Set.Icc b (a + b) := by | simp [h]
|
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 | 150 | 160 | theorem nonneg_prod_iff {R} [OrderedCommRing R] [Module R M₁] [Module R M₂]
{Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} :
(∀ x, 0 ≤ (Q₁.prod Q₂) x) ↔ (∀ x, 0 ≤ Q₁ x) ∧ ∀ x, 0 ≤ Q₂ x := by |
simp_rw [Prod.forall, prod_apply]
constructor
· intro h
constructor
· intro x; simpa only [add_zero, map_zero] using h x 0
· intro x; simpa only [zero_add, map_zero] using h 0 x
· rintro ⟨h₁, h₂⟩ x₁ x₂
exact add_nonneg (h₁ x₁) (h₂ x₂)
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 46 | 46 | theorem taylor_X : taylor r X = X + C r := by | simp only [taylor_apply, X_comp]
|
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : ℕ} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 82 | 101 | theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) := by |
rw [isUnit_iff]
apply le_antisymm (r.den : ℤ_[p]).2
rw [← not_lt, coe_natCast]
intro norm_denom_lt
have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
congr
rw_mod_cast [@Rat.mul_den_eq_num r]
rw [padicNormE.mul] at hr
have key : ‖(r.num : ℚ_[p])‖ < 1 := by
calc
_ = _ := hr.symm
... |
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.RingTheory.GradedAlgebra.Radical
#align_import algebraic_geometry.projective_spectrum.scheme from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomp... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 172 | 181 | theorem mk_mem_carrier (z : HomogeneousLocalization.NumDenSameDeg 𝒜 (.powers f)) :
HomogeneousLocalization.mk z ∈ carrier x ↔ z.num.1 ∈ x.1.asHomogeneousIdeal := by |
rw [carrier, Ideal.mem_comap, HomogeneousLocalization.algebraMap_apply,
HomogeneousLocalization.val_mk, Localization.mk_eq_mk', IsLocalization.mk'_eq_mul_mk'_one,
mul_comm, Ideal.unit_mul_mem_iff_mem, ← Ideal.mem_comap,
IsLocalization.comap_map_of_isPrime_disjoint (.powers f)]
· rfl
· infer_instance
... |
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.discrete_valuation_ring.basic from "leanprover-community/mathlib"@"c163ec99dfc664628ca15d215fce0a5b9c2... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 227 | 245 | theorem of_ufd_of_unique_irreducible [UniqueFactorizationMonoid R] (h₁ : ∃ p : R, Irreducible p)
(h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q) :
HasUnitMulPowIrreducibleFactorization R := by |
obtain ⟨p, hp⟩ := h₁
refine ⟨p, hp, ?_⟩
intro x hx
cases' WfDvdMonoid.exists_factors x hx with fx hfx
refine ⟨Multiset.card fx, ?_⟩
have H := hfx.2
rw [← Associates.mk_eq_mk_iff_associated] at H ⊢
rw [← H, ← Associates.prod_mk, Associates.mk_pow, ← Multiset.prod_replicate]
congr 1
symm
rw [Multis... |
import Mathlib.Topology.Separation
import Mathlib.Algebra.Group.Defs
#align_import topology.algebra.semigroup from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
@[to_additive
"Any nonempty compact Hausdorff additive semigroup where right-addition is continuous
contains an ... | Mathlib/Topology/Algebra/Semigroup.lean | 82 | 95 | theorem exists_idempotent_in_compact_subsemigroup {M} [Semigroup M] [TopologicalSpace M] [T2Space M]
(continuous_mul_left : ∀ r : M, Continuous (· * r)) (s : Set M) (snemp : s.Nonempty)
(s_compact : IsCompact s) (s_add : ∀ᵉ (x ∈ s) (y ∈ s), x * y ∈ s) :
∃ m ∈ s, m * m = m := by |
let M' := { m // m ∈ s }
letI : Semigroup M' :=
{ mul := fun p q => ⟨p.1 * q.1, s_add _ p.2 _ q.2⟩
mul_assoc := fun p q r => Subtype.eq (mul_assoc _ _ _) }
haveI : CompactSpace M' := isCompact_iff_compactSpace.mp s_compact
haveI : Nonempty M' := nonempty_subtype.mpr snemp
have : ∀ p : M', Continuou... |
import Mathlib.Probability.ProbabilityMassFunction.Basic
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.MeasureTheory.Integral.Bochner
namespace PMF
open MeasureTheory ENNReal TopologicalSpace
section General
variable {α : Type*} [MeasurableSpace α] [MeasurableSingletonClass α]
v... | Mathlib/Probability/ProbabilityMassFunction/Integrals.lean | 43 | 47 | theorem integral_eq_sum [Fintype α] (p : PMF α) (f : α → E) :
∫ a, f a ∂(p.toMeasure) = ∑ a, (p a).toReal • f a := by |
rw [integral_fintype _ (.of_finite _ f)]
congr with x; congr 2
exact PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)
|
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 | 161 | 162 | theorem Disjoint.symmDiff_eq_sup {a b : α} (h : Disjoint a b) : a ∆ b = a ⊔ b := by |
rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right]
|
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 145 | 148 | theorem _root_.MeasureTheory.Integrable.condDistrib_ae_map
(hY : AEMeasurable Y μ) (hf_int : Integrable f (μ.map fun a => (X a, Y a))) :
∀ᵐ b ∂μ.map X, Integrable (fun ω => f (b, ω)) (condDistrib Y X μ b) := by |
rw [condDistrib, ← Measure.fst_map_prod_mk₀ (X := X) hY]; exact hf_int.condKernel_ae
|
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section TopologicalGroup
variable [CommGroup α] [TopologicalSpace α] [TopologicalGroup α]
variable {f g : β → α} {a a₁... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 63 | 65 | theorem Multipliable.trans_div (hg : Multipliable g) (hfg : Multipliable fun b ↦ f b / g b) :
Multipliable f := by |
simpa only [div_mul_cancel] using hfg.mul hg
|
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
open NormedSpace -- For `NormedSpace.exp`.
section Star
variable {A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [Continu... | Mathlib/Analysis/NormedSpace/Star/Exponential.lean | 51 | 56 | theorem Commute.expUnitary {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
Commute (expUnitary a) (expUnitary b) :=
calc
selfAdjoint.expUnitary a * selfAdjoint.expUnitary b =
selfAdjoint.expUnitary b * selfAdjoint.expUnitary a := by |
rw [← h.expUnitary_add, ← h.symm.expUnitary_add, add_comm]
|
import Mathlib.Algebra.Polynomial.Degree.Lemmas
open Polynomial
namespace Mathlib.Tactic.ComputeDegree
section recursion_lemmas
variable {R : Type*}
section semiring
variable [Semiring R]
theorem natDegree_C_le (a : R) : natDegree (C a) ≤ 0 := (natDegree_C a).le
theorem natDegree_natCast_le (n : ℕ) : natDeg... | Mathlib/Tactic/ComputeDegree.lean | 150 | 155 | theorem natDegree_eq_of_le_of_coeff_ne_zero' {deg m o : ℕ} {c : R} {p : R[X]}
(h_natDeg_le : natDegree p ≤ m) (coeff_eq : coeff p o = c)
(coeff_ne_zero : c ≠ 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :
natDegree p = deg := by |
subst coeff_eq deg_eq_deg coeff_eq_deg
exact natDegree_eq_of_le_of_coeff_ne_zero ‹_› ‹_›
|
import Mathlib.Analysis.Complex.Basic
import Mathlib.FieldTheory.IntermediateField
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.instances.complex from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
section ComplexSubfield
open... | Mathlib/Topology/Instances/Complex.lean | 25 | 44 | theorem Complex.subfield_eq_of_closed {K : Subfield ℂ} (hc : IsClosed (K : Set ℂ)) :
K = ofReal.fieldRange ∨ K = ⊤ := by |
suffices range (ofReal' : ℝ → ℂ) ⊆ K by
rw [range_subset_iff, ← coe_algebraMap] at this
have :=
(Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top
(Subfield.toIntermediateField K this).toSubalgebra
simp_rw [← SetLike.coe_set_eq, IntermediateField.coe_toSubalgebra] a... |
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.nonarchimedean.adic_topology from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [CommRing R]
open S... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 92 | 103 | theorem hasBasis_nhds_zero_adic (I : Ideal R) :
HasBasis (@nhds R I.adicTopology (0 : R)) (fun _n : ℕ => True) fun n =>
((I ^ n : Ideal R) : Set R) :=
⟨by
intro U
rw [I.ringFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]
constructor
· rintro ⟨-, ⟨i, rfl⟩, h⟩
replace h : ↑... | simpa using h
exact ⟨i, trivial, h⟩
· rintro ⟨i, -, h⟩
exact ⟨(I ^ i : Ideal R), ⟨i, by simp⟩, h⟩⟩
|
import Batteries.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
namespace Batteries
namespace RBNode
open RBColor
attribute [simp] fold foldl foldr Any forM foldlM Ordered
@[simp] theorem min?_reverse (t : RBNode α) : t.reverse.min? = t.max? := by
unfold RBNode.max?; split <;> simp [RBNode.min?]
unfold RB... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 32 | 33 | theorem All_def {t : RBNode α} : t.All p ↔ ∀ x ∈ t, p x := by |
induction t <;> simp [or_imp, forall_and, *]
|
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Defs
import Mathlib.Order.WithBot
#align_import algebra.order.monoid.with_top ... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 128 | 128 | theorem add_top (a : WithTop α) : a + ⊤ = ⊤ := by | cases a <;> rfl
|
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Galois
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
import Mathlib.RingTheory.Norm
universe u
variable {K : Type u} [Field K]
open Polynomial IntermediateField AdjoinRoot
section Splits
lemma root_X_pow... | Mathlib/FieldTheory/KummerExtension.lean | 74 | 82 | theorem X_pow_sub_C_splits_of_isPrimitiveRoot
{n : ℕ} {ζ : K} (hζ : IsPrimitiveRoot ζ n) {α a : K} (e : α ^ n = a) :
(X ^ n - C a).Splits (RingHom.id _) := by |
cases n.eq_zero_or_pos with
| inl hn =>
rw [hn, pow_zero, ← C.map_one, ← map_sub]
exact splits_C _ _
| inr hn =>
rw [splits_iff_card_roots, ← nthRoots, hζ.card_nthRoots, natDegree_X_pow_sub_C, if_pos ⟨α, e⟩]
|
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal CategoryTh... | Mathlib/ModelTheory/Satisfiability.lean | 129 | 135 | theorem isSatisfiable_directed_union_iff {ι : Type*} [Nonempty ι] {T : ι → L.Theory}
(h : Directed (· ⊆ ·) T) : Theory.IsSatisfiable (⋃ i, T i) ↔ ∀ i, (T i).IsSatisfiable := by |
refine ⟨fun h' i => h'.mono (Set.subset_iUnion _ _), fun h' => ?_⟩
rw [isSatisfiable_iff_isFinitelySatisfiable, IsFinitelySatisfiable]
intro T0 hT0
obtain ⟨i, hi⟩ := h.exists_mem_subset_of_finset_subset_biUnion hT0
exact (h' i).mono hi
|
import Mathlib.Combinatorics.SetFamily.HarrisKleitman
import Mathlib.Combinatorics.SetFamily.Intersecting
#align_import combinatorics.set_family.kleitman from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Finset
open Fintype (card)
variable {ι α : Type*} [Fintype α] [DecidableEq... | Mathlib/Combinatorics/SetFamily/Kleitman.lean | 37 | 85 | theorem Finset.card_biUnion_le_of_intersecting (s : Finset ι) (f : ι → Finset (Finset α))
(hf : ∀ i ∈ s, (f i : Set (Finset α)).Intersecting) :
(s.biUnion f).card ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - s.card) := by |
have : DecidableEq ι := by
classical
infer_instance
obtain hs | hs := le_total (Fintype.card α) s.card
· rw [tsub_eq_zero_of_le hs, pow_zero]
refine (card_le_card <| biUnion_subset.2 fun i hi a ha ↦
mem_compl.2 <| not_mem_singleton.2 <| (hf _ hi).ne_bot ha).trans_eq ?_
rw [card_compl, Finty... |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder α] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 176 | 178 | theorem iUnion_Ioc_eq_Ioi_self_iff {f : ι → α} {a : α} :
⋃ i, Ioc a (f i) = Ioi a ↔ ∀ x, a < x → ∃ i, x ≤ f i := by |
simp [← Ioi_inter_Iic, ← inter_iUnion, subset_def]
|
import Mathlib.Data.Finset.Basic
variable {ι : Sort _} {π : ι → Sort _} {x : ∀ i, π i} [DecidableEq ι]
namespace Function
def updateFinset (x : ∀ i, π i) (s : Finset ι) (y : ∀ i : ↥s, π i) (i : ι) : π i :=
if hi : i ∈ s then y ⟨i, hi⟩ else x i
open Finset Equiv
theorem updateFinset_def {s : Finset ι} {y} :
... | Mathlib/Data/Finset/Update.lean | 43 | 50 | theorem update_eq_updateFinset {i y} :
Function.update x i y = updateFinset x {i} (uniqueElim y) := by |
congr with j
by_cases hj : j = i
· cases hj
simp only [dif_pos, Finset.mem_singleton, update_same, updateFinset]
exact uniqueElim_default (α := fun j : ({i} : Finset ι) => π j) y
· simp [hj, updateFinset]
|
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 136 | 141 | theorem ideal_oper_maxTrivSubmodule_eq_bot (I : LieIdeal R L) :
⁅I, maxTrivSubmodule R L M⁆ = ⊥ := by |
rw [← LieSubmodule.coe_toSubmodule_eq_iff, LieSubmodule.lieIdeal_oper_eq_linear_span,
LieSubmodule.bot_coeSubmodule, Submodule.span_eq_bot]
rintro m ⟨⟨x, hx⟩, ⟨⟨m, hm⟩, rfl⟩⟩
exact hm x
|
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 106 | 106 | theorem log_zero : log 0 = 0 := by | simp [log]
|
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.Analysis.Normed.MulAction
#align_import analysis.normed_space.continuous_linear_map from "leanprover-community/mathlib"@"fe18deda804e30c594e75a6e5fe0f7d14695289f"
open Metric ContinuousLinearMap
open Set Real
open NNReal
variable {𝕜 𝕜₂ E F G : Type*}... | Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean | 198 | 205 | theorem ContinuousLinearEquiv.homothety_inverse (a : ℝ) (ha : 0 < a) (f : E ≃ₛₗ[σ] F) :
(∀ x : E, ‖f x‖ = a * ‖x‖) → ∀ y : F, ‖f.symm y‖ = a⁻¹ * ‖y‖ := by |
intro hf y
calc
‖f.symm y‖ = a⁻¹ * (a * ‖f.symm y‖) := by
rw [← mul_assoc, inv_mul_cancel (ne_of_lt ha).symm, one_mul]
_ = a⁻¹ * ‖f (f.symm y)‖ := by rw [hf]
_ = a⁻¹ * ‖y‖ := by simp
|
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 201 | 201 | theorem zero_eval (v) : zero.eval v = pure [0] := by | simp [zero]
|
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 56 | 62 | theorem not_countably_generated_cocompact : ¬IsCountablyGenerated (cocompact ℚ) := by |
intro H
rcases exists_seq_tendsto (cocompact ℚ ⊓ 𝓝 0) with ⟨x, hx⟩
rw [tendsto_inf] at hx; rcases hx with ⟨hxc, hx0⟩
obtain ⟨n, hn⟩ : ∃ n : ℕ, x n ∉ insert (0 : ℚ) (range x) :=
(hxc.eventually hx0.isCompact_insert_range.compl_mem_cocompact).exists
exact hn (Or.inr ⟨n, rfl⟩)
|
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 152 | 155 | theorem condexp_ae_eq_condexpL1CLM (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1CLM F' hm μ (hf.toL1 f) := by |
refine (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => ?_)
rw [condexpL1_eq hf]
|
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 61 | 78 | theorem charpoly_sub_diagonal_degree_lt :
(M.charpoly - ∏ i : n, (X - C (M i i))).degree < ↑(Fintype.card n - 1) := by |
rw [charpoly, det_apply', ← insert_erase (mem_univ (Equiv.refl n)),
sum_insert (not_mem_erase (Equiv.refl n) univ), add_comm]
simp only [charmatrix_apply_eq, one_mul, Equiv.Perm.sign_refl, id, Int.cast_one,
Units.val_one, add_sub_cancel_right, Equiv.coe_refl]
rw [← mem_degreeLT]
apply Submodule.sum_mem... |
import Mathlib.FieldTheory.Galois
#align_import field_theory.polynomial_galois_group from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Polynomial
open FiniteDimensional
namespace Polynomial
variable {F : Type*} [Field F] (p q : F[X]) (E : Type*) [... | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 74 | 79 | theorem ext {σ τ : p.Gal} (h : ∀ x ∈ p.rootSet p.SplittingField, σ x = τ x) : σ = τ := by |
refine
AlgEquiv.ext fun x =>
(AlgHom.mem_equalizer σ.toAlgHom τ.toAlgHom x).mp
((SetLike.ext_iff.mp ?_ x).mpr Algebra.mem_top)
rwa [eq_top_iff, ← SplittingField.adjoin_rootSet, Algebra.adjoin_le_iff]
|
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 160 | 170 | theorem sum_cramer_apply {β} (s : Finset β) (f : n → β → α) (i : n) :
(∑ x ∈ s, cramer A (fun j => f j x) i) = cramer A (fun j : n => ∑ x ∈ s, f j x) i :=
calc
(∑ x ∈ s, cramer A (fun j => f j x) i) = (∑ x ∈ s, cramer A fun j => f j x) i :=
(Finset.sum_apply i s _).symm
_ = cramer A (fun j : n => ∑ ... |
rw [sum_cramer, cramer_apply, cramer_apply]
simp only [updateColumn]
congr with j
congr
apply Finset.sum_apply
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 404 | 408 | theorem fderivWithin.comp_derivWithin_of_eq {t : Set F} (hl : DifferentiableWithinAt 𝕜 l t y)
(hf : DifferentiableWithinAt 𝕜 f s x) (hs : MapsTo f s t) (hxs : UniqueDiffWithinAt 𝕜 s x)
(hy : y = f x) :
derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x) : F → E) (derivWithin f s x) := by |
rw [hy] at hl; exact fderivWithin.comp_derivWithin x hl hf hs hxs
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 506 | 508 | theorem measurableSet_graph (hf : Measurable f) :
MeasurableSet { p : α × ℝ | p.snd = f p.fst } := by |
simpa using measurableSet_region_between_cc hf hf MeasurableSet.univ
|
import Mathlib.Logic.Function.Iterate
import Mathlib.Init.Data.Int.Order
import Mathlib.Order.Compare
import Mathlib.Order.Max
import Mathlib.Order.RelClasses
import Mathlib.Tactic.Choose
#align_import order.monotone.basic from "leanprover-community/mathlib"@"554bb38de8ded0dafe93b7f18f0bfee6ef77dc5d"
open Functio... | Mathlib/Order/Monotone/Basic.lean | 1,014 | 1,018 | theorem Nat.rel_of_forall_rel_succ_of_le_of_lt (r : β → β → Prop) [IsTrans β r] {f : ℕ → β} {a : ℕ}
(h : ∀ n, a ≤ n → r (f n) (f (n + 1))) ⦃b c : ℕ⦄ (hab : a ≤ b) (hbc : b < c) :
r (f b) (f c) := by |
induction' hbc with k b_lt_k r_b_k
exacts [h _ hab, _root_.trans r_b_k (h _ (hab.trans_lt b_lt_k).le)]
|
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
... | Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 58 | 62 | theorem preserves_epi_of_preservesColimit {X Y : C} (f : X ⟶ Y) [PreservesColimit (span f f) F]
[Epi f] : Epi (F.map f) := by |
have := isColimitPushoutCoconeMapOfIsColimit F _ (PushoutCocone.isColimitMkIdId f)
simp_rw [F.map_id] at this
apply PushoutCocone.epi_of_isColimitMkIdId _ this
|
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Order.Filter.Archimedean
#align_import topology.instances.int from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/Int.lean | 84 | 85 | theorem cofinite_eq : (cofinite : Filter ℤ) = atBot ⊔ atTop := by |
rw [← cocompact_eq_cofinite, cocompact_eq_atBot_atTop]
|
import Mathlib.AlgebraicTopology.DoldKan.PInfty
#align_import algebraic_topology.dold_kan.decomposition from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Preadditive
Opposite Simplicial
noncomputable section
namespace Alge... | Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean | 120 | 124 | theorem id_φ : (id X n).φ = 𝟙 _ := by |
simp only [← P_add_Q_f (n + 1) (n + 1), φ]
congr 1
· simp only [id, PInfty_f, P_f_idem]
· exact Eq.trans (by congr; simp) (decomposition_Q n (n + 1)).symm
|
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular
import Mathlib.Topology.Category.CompHaus.EffectiveEpi
import Mathlib.Topology.Category.Stonean.Limits
import Mathlib.Topology.Category.CompHaus.EffectiveEpi
universe u
open CategoryTheory Limits
namespace Stonean
noncomputable
def struct {B X : St... | Mathlib/Topology/Category/Stonean/EffectiveEpi.lean | 103 | 121 | theorem effectiveEpiFamily_tfae
{α : Type} [Finite α] {B : Stonean.{u}}
(X : α → Stonean.{u}) (π : (a : α) → (X a ⟶ B)) :
TFAE
[ EffectiveEpiFamily X π
, Epi (Sigma.desc π)
, ∀ b : B, ∃ (a : α) (x : X a), π a x = b
] := by |
tfae_have 2 → 1
· intro
simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
tfae_have 1 → 2
· intro; infer_instance
tfae_have 3 ↔ 1
· erw [((CompHaus.effectiveEpiFamily_tfae
(fun a ↦ Stonean.toCompHaus.obj (X a)) (fun a ↦ Stonean.toCompHaus.map (π a))).o... |
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
open MeasureTheory Filter Finset Real
noncomputable section
open scoped MeasureTheory ProbabilityTheory ENNReal NNReal
namespace ProbabilityTheory
variable {Ω ι ... | Mathlib/Probability/Moments.lean | 73 | 77 | theorem centralMoment_one' [IsFiniteMeasure μ] (h_int : Integrable X μ) :
centralMoment X 1 μ = (1 - (μ Set.univ).toReal) * μ[X] := by |
simp only [centralMoment, Pi.sub_apply, pow_one]
rw [integral_sub h_int (integrable_const _)]
simp only [sub_mul, integral_const, smul_eq_mul, one_mul]
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Bits
import Mathlib.Data.Nat.Log
import Mathlib.Data.List.Indexes
import Mathlib.Data.List.Palindrome
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
impo... | Mathlib/Data/Nat/Digits.lean | 63 | 67 | theorem digitsAux_def (b : ℕ) (h : 2 ≤ b) (n : ℕ) (w : 0 < n) :
digitsAux b h n = (n % b) :: digitsAux b h (n / b) := by |
cases n
· cases w
· rw [digitsAux]
|
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 100 | 103 | theorem lcm_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀ a ∈ s₂, f a = g a) :
s₁.lcm f = s₂.lcm g := by |
subst hs
exact Finset.fold_congr hfg
|
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
open Function
open OrderDual (toDual ofDual)
variable {α β : Type*}
namespace Set
section Preorder
v... | Mathlib/Order/Interval/Set/Basic.lean | 181 | 181 | theorem left_mem_Ioo : a ∈ Ioo a b ↔ False := by | simp [lt_irrefl]
|
import Mathlib.CategoryTheory.Comma.StructuredArrow
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Functor.ReflectsIso
import Mathlib.CategoryTheory.Functor.EpiMono
#align_import category_theory.over from "leanprover-community/mathlib"@"8a318021995877a44630c898d0b2bc376fceef3b"
namespace Catego... | Mathlib/CategoryTheory/Comma/Over.lean | 59 | 63 | theorem OverMorphism.ext {X : T} {U V : Over X} {f g : U ⟶ V} (h : f.left = g.left) : f = g := by |
let ⟨_,b,_⟩ := f
let ⟨_,e,_⟩ := g
congr
simp only [eq_iff_true_of_subsingleton]
|
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
theorem riemannZeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : ℂ) ^ (2 * k - 1)
* (π : ℂ) ^ (2 * k) * bernoulli (2 * k) / (... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 220 | 224 | theorem riemannZeta_two : riemannZeta 2 = (π : ℂ) ^ 2 / 6 := by |
convert congr_arg ((↑) : ℝ → ℂ) hasSum_zeta_two.tsum_eq
· rw [← Nat.cast_two, zeta_nat_eq_tsum_of_gt_one one_lt_two]
simp only [push_cast]
· norm_cast
|
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.jacobson from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
set_option autoImplicit true
universe u
namespace Ideal
open Polynomial
... | Mathlib/RingTheory/Jacobson.lean | 108 | 117 | theorem isJacobson_of_surjective [H : IsJacobson R] :
(∃ f : R →+* S, Function.Surjective ↑f) → IsJacobson S := by |
rintro ⟨f, hf⟩
rw [isJacobson_iff_sInf_maximal]
intro p hp
use map f '' { J : Ideal R | comap f p ≤ J ∧ J.IsMaximal }
use fun j ⟨J, hJ, hmap⟩ => hmap ▸ (map_eq_top_or_isMaximal_of_surjective f hf hJ.right).symm
have : p = map f (comap f p).jacobson :=
(IsJacobson.out' _ <| hp.isRadical.comap f).symm ▸ ... |
import Mathlib.GroupTheory.Subgroup.Center
import Mathlib.GroupTheory.Submonoid.Centralizer
#align_import group_theory.subgroup.basic from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
open Function
open Int
variable {G : Type*} [Group G]
namespace Subgroup
variable {H K : Subgroup ... | Mathlib/GroupTheory/Subgroup/Centralizer.lean | 42 | 44 | theorem mem_centralizer_iff_commutator_eq_one {g : G} {s : Set G} :
g ∈ centralizer s ↔ ∀ h ∈ s, h * g * h⁻¹ * g⁻¹ = 1 := by |
simp only [mem_centralizer_iff, mul_inv_eq_iff_eq_mul, one_mul]
|
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComplex
#align_import algebraic_topology.dold_kan.gamma_comp_n from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
no... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 105 | 108 | theorem N₁Γ₀_hom_app_f_f (K : ChainComplex C ℕ) (n : ℕ) :
(N₁Γ₀.hom.app K).f.f n = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv.f.f n := by |
rw [N₁Γ₀_hom_app]
apply comp_id
|
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.pullbacks from "leanprover-community/mathlib"@"f11e306adb9f2a393539d2bb4293bf1b42caa7ac"
noncomputable section
universe v₁ v₂ u₁ u₂
-- Porting note: ne... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 132 | 134 | theorem PreservesPullback.iso_inv_fst :
(PreservesPullback.iso G f g).inv ≫ G.map pullback.fst = pullback.fst := by |
simp [PreservesPullback.iso, Iso.inv_comp_eq]
|
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 | 78 | 84 | theorem IsOpenMap.restrictPreimage (H : IsOpenMap f) (s : Set β) :
IsOpenMap (s.restrictPreimage f) := by |
intro t
suffices ∀ u, IsOpen u → Subtype.val ⁻¹' u = t →
∃ v, IsOpen v ∧ Subtype.val ⁻¹' v = s.restrictPreimage f '' t by
simpa [isOpen_induced_iff]
exact fun u hu e => ⟨f '' u, H u hu, by simp [← e, image_restrictPreimage]⟩
|
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Rat.Encodable
import Mathlib.Topology.GDelta
#align_import topology.instances.irrational from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Metric
open Filter Topology
protected theorem IsGδ.setOf_irrational : Is... | Mathlib/Topology/Instances/Irrational.lean | 78 | 89 | theorem eventually_forall_le_dist_cast_div (hx : Irrational x) (n : ℕ) :
∀ᶠ ε : ℝ in 𝓝 0, ∀ m : ℤ, ε ≤ dist x (m / n) := by |
have A : IsClosed (range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ)) :=
((isClosedMap_smul₀ (n⁻¹ : ℝ)).comp Int.closedEmbedding_coe_real.isClosedMap).isClosed_range
have B : x ∉ range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ) := by
rintro ⟨m, rfl⟩
simp at hx
rcases Metric.mem_nhds_iff.1 (A.isOpen_compl.mem_nhds B) with... |
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.NormedSpace.Completion
#align_import analysis.analytic.uniqueness from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type... | Mathlib/Analysis/Analytic/Uniqueness.lean | 32 | 70 | theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux [CompleteSpace F] {f : E → F} {U : Set E}
(hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0) :
EqOn f 0 U := by |
/- Let `u` be the set of points around which `f` vanishes. It is clearly open. We have to show
that its limit points in `U` still belong to it, from which the inclusion `U ⊆ u` will follow
by connectedness. -/
let u := {x | f =ᶠ[𝓝 x] 0}
suffices main : closure u ∩ U ⊆ u by
have Uu : U ⊆ u :=
h... |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α : Type*}
namespace Coheyting
variable [CoheytingAlgebra α] {a b : α}
def boundary (a : α) : α :=
a ⊓ ￢a
#align cohe... | Mathlib/Order/Heyting/Boundary.lean | 120 | 122 | theorem boundary_le_boundary_sup_sup_boundary_inf_right : ∂ b ≤ ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b) := by |
rw [sup_comm a, inf_comm]
exact boundary_le_boundary_sup_sup_boundary_inf_left
|
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.SumIntegralComparisons
import Mathlib.NumberTheory.Harmonic.Defs
| Mathlib/NumberTheory/Harmonic/Bounds.lean | 17 | 24 | theorem log_add_one_le_harmonic (n : ℕ) :
Real.log ↑(n+1) ≤ harmonic n := by |
calc _ = ∫ x in (1:ℕ)..↑(n+1), x⁻¹ := ?_
_ ≤ ∑ d ∈ Finset.Icc 1 n, (d:ℝ)⁻¹ := ?_
_ = harmonic n := ?_
· rw [Nat.cast_one, integral_inv (by simp [(show ¬ (1 : ℝ) ≤ 0 by norm_num)]), div_one]
· exact (inv_antitoneOn_Icc_right <| by norm_num).integral_le_sum_Ico (Nat.le_add_left 1 n)
· simp only [ha... |
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
noncomputable section
open scoped Classical
@[ext]
structure ComplexShape (ι : Type*) where
Rel : ι → ι → Prop
nex... | Mathlib/Algebra/Homology/ComplexShape.lean | 100 | 102 | theorem symm_symm (c : ComplexShape ι) : c.symm.symm = c := by |
ext
simp
|
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