Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 55 | 56 | theorem einfsep_zero : s.einfsep = 0 ↔ ∀ C > 0, ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ edist x y < C := by |
simp_rw [einfsep, ← _root_.bot_eq_zero, iInf_eq_bot, iInf_lt_iff, exists_prop]
|
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
#align_import category_theory.monoidal.Bimod from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba"
universe v₁ v₂ u₁ u₂
open Categor... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 680 | 683 | theorem inv_hom_id : inv P ≫ hom P = 𝟙 _ := by |
dsimp [hom, inv]
slice_lhs 3 4 => rw [coequalizer.π_desc]
rw [one_actLeft, Iso.inv_hom_id]
|
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 | 170 | 172 | theorem withDensity_const (c : ℝ≥0∞) : μ.withDensity (fun _ ↦ c) = c • μ := by |
ext1 s hs
simp [withDensity_apply _ hs]
|
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Order.Partition.Equipartition
import Mathlib.SetTheory.Ordinal.Basic
#align_import combinatorics.simple_graph.regularity.uniform from "leanprover-community/mathlib"@"bf7ef0e83e5b... | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 160 | 165 | theorem nonuniformWitnesses_spec (h : ¬G.IsUniform ε s t) :
ε ≤
|G.edgeDensity (G.nonuniformWitnesses ε s t).1 (G.nonuniformWitnesses ε s t).2 -
G.edgeDensity s t| := by |
rw [nonuniformWitnesses, dif_pos h]
exact (not_isUniform_iff.1 h).choose_spec.2.choose_spec.2.2.2
|
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 797 | 798 | theorem det_conj {M : Matrix m m α} (h : IsUnit M) (N : Matrix m m α) :
det (M * N * M⁻¹) = det N := by | rw [← h.unit_spec, ← coe_units_inv, det_units_conj]
|
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 | 102 | 104 | theorem dist_eq_re_im (z w : ℂ) : dist z w = √((z.re - w.re) ^ 2 + (z.im - w.im) ^ 2) := by |
rw [sq, sq]
rfl
|
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Bornology
universe u v w
variable {ι : Sort*} {α : Type u} {β :... | Mathlib/Topology/MetricSpace/Thickening.lean | 253 | 254 | theorem cthickening_max_zero (δ : ℝ) (E : Set α) : cthickening (max 0 δ) E = cthickening δ E := by |
cases le_total δ 0 <;> simp [cthickening_of_nonpos, *]
|
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists ... | Mathlib/RingTheory/Ideal/Operations.lean | 426 | 426 | theorem one_eq_top : (1 : Ideal R) = ⊤ := by | erw [Submodule.one_eq_range, LinearMap.range_id]
|
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 196 | 200 | theorem tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle (x + y) y) * ‖y‖ = ‖x‖ := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h
|
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,227 | 1,228 | theorem contDiff_pi : ContDiff 𝕜 n Φ ↔ ∀ i, ContDiff 𝕜 n fun x => Φ x i := by |
simp only [← contDiffOn_univ, contDiffOn_pi]
|
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 191 | 217 | theorem totient_prime_pow_succ {p : ℕ} (hp : p.Prime) (n : ℕ) : φ (p ^ (n + 1)) = p ^ n * (p - 1) :=
calc
φ (p ^ (n + 1)) = ((range (p ^ (n + 1))).filter (Coprime (p ^ (n + 1)))).card :=
totient_eq_card_coprime _
_ = (range (p ^ (n + 1)) \ (range (p ^ n)).image (· * p)).card :=
(congr_arg card
... |
have h1 : Function.Injective (· * p) := mul_left_injective₀ hp.ne_zero
have h2 : (range (p ^ n)).image (· * p) ⊆ range (p ^ (n + 1)) := fun a => by
simp only [mem_image, mem_range, exists_imp]
rintro b ⟨h, rfl⟩
rw [Nat.pow_succ]
exact (mul_lt_mul_right hp.pos).2 h
rw [... |
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,582 | 1,583 | theorem dist_div_div_le (a₁ a₂ b₁ b₂ : E) : dist (a₁ / a₂) (b₁ / b₂) ≤ dist a₁ b₁ + dist a₂ b₂ := by |
simpa only [div_eq_mul_inv, dist_inv_inv] using dist_mul_mul_le a₁ a₂⁻¹ b₁ b₂⁻¹
|
import Mathlib.Data.DFinsupp.Interval
import Mathlib.Data.DFinsupp.Multiset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.multiset.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset DFinsupp Function
open Pointwise
variable {α : Type*}
namespace Mu... | Mathlib/Data/Multiset/Interval.lean | 77 | 80 | theorem card_uIcc :
(uIcc s t).card = ∏ i ∈ s.toFinset ∪ t.toFinset, ((t.count i - s.count i : ℤ).natAbs + 1) := by |
simp_rw [uIcc_eq, Finset.card_map, DFinsupp.card_uIcc, Nat.card_uIcc, Multiset.toDFinsupp_apply,
toDFinsupp_support]
|
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNR... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,433 | 1,438 | theorem setToFun_indicator_const (hT : DominatedFinMeasAdditive μ T C) {s : Set α}
(hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : E) :
setToFun μ T hT (s.indicator fun _ => x) = T s x := by |
rw [setToFun_congr_ae hT (@indicatorConstLp_coeFn _ _ _ 1 _ _ _ hs hμs x).symm]
rw [L1.setToFun_eq_setToL1 hT]
exact L1.setToL1_indicatorConstLp hT hs hμs x
|
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryTheory.Triangulated.TriangleShift
#align_import category_theory.triangulated.pretriangulated from "leanprover-community/mathlib"@"6876fa15e3158ff3e4a4e2af1fb6e194... | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 156 | 159 | theorem comp_distTriang_mor_zero₃₁ (T : Triangle C) (H : T ∈ distTriang C) :
T.mor₃ ≫ T.mor₁⟦1⟧' = 0 := by |
have H₂ := rot_of_distTriang T.rotate (rot_of_distTriang T H)
simpa using comp_distTriang_mor_zero₁₂ T.rotate.rotate H₂
|
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 | 1,119 | 1,119 | theorem think_equiv (s : WSeq α) : think s ~ʷ s := by | unfold Equiv; simpa using Equiv.refl _
|
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.TFAE
#align_import ring_theory.valuation.basic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open scoped Classical
open Function Ideal
nonco... | Mathlib/RingTheory/Valuation/Basic.lean | 187 | 193 | theorem map_sum_le {ι : Type*} {s : Finset ι} {f : ι → R} {g : Γ₀} (hf : ∀ i ∈ s, v (f i) ≤ g) :
v (∑ i ∈ s, f i) ≤ g := by |
refine
Finset.induction_on s (fun _ => v.map_zero ▸ zero_le')
(fun a s has ih hf => ?_) hf
rw [Finset.forall_mem_insert] at hf; rw [Finset.sum_insert has]
exact v.map_add_le hf.1 (ih hf.2)
|
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Order.OrderClosed
#align_import topology.order.basic from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafe... | Mathlib/Topology/Order/Basic.lean | 322 | 323 | theorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] :
𝓝 (⊤ : α) = ⨅ (l) (h₂ : l < ⊤), 𝓟 (Ioi l) := by | simp [nhds_eq_order (⊤ : α)]
|
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.stream.init from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
set_option autoImplicit true
open Nat Function Option
namespace Stre... | Mathlib/Data/Stream/Init.lean | 674 | 679 | theorem get_tails : ∀ (n : Nat) (s : Stream' α), get (tails s) n = drop n (tail s) := by |
intro n; induction' n with n' ih
· intros
rfl
· intro s
rw [get_succ, drop_succ, tails_eq, tail_cons, ih]
|
import Mathlib.Algebra.FreeAlgebra
import Mathlib.Algebra.RingQuot
import Mathlib.Algebra.TrivSqZeroExt
import Mathlib.Algebra.Algebra.Operations
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import linear_algebra.tensor_algebra.basic from "leanprover-community/mathlib"@"b8d2eaa69d69ce8f03179a5cda774fc0cde984... | Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean | 153 | 155 | theorem ι_comp_lift {A : Type*} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) :
(lift R f).toLinearMap.comp (ι R) = f := by |
convert (lift R).symm_apply_apply f
|
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
variable {R K L : Type*} [CommRing R]
variable [Field K] [Field L] [DecidableEq L]
variable [Algebra R K] [Is... | Mathlib/RingTheory/ClassGroup.lean | 400 | 409 | theorem card_classGroup_eq_one_iff [IsDedekindDomain R] [Fintype (ClassGroup R)] :
Fintype.card (ClassGroup R) = 1 ↔ IsPrincipalIdealRing R := by |
constructor; swap; · intros; convert card_classGroup_eq_one (R := R)
rw [Fintype.card_eq_one_iff]
rintro ⟨I, hI⟩
have eq_one : ∀ J : ClassGroup R, J = 1 := fun J => (hI J).trans (hI 1).symm
refine ⟨fun I => ?_⟩
by_cases hI : I = ⊥
· rw [hI]; exact bot_isPrincipal
· exact (ClassGroup.mk0_eq_one_iff (mem... |
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.Factors
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Multiplicity
#align_import ring_theory.unique_factorization_domain from "leanprover-community/mathli... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,117 | 1,132 | theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
(hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
(hcp : ∀ {x y}, IsRelPrime x y → P x → P y → P (x * y)) : P a := by |
letI := Classical.decEq α
have P_of_associated : ∀ {x y}, Associated x y → P x → P y := by
rintro x y ⟨u, rfl⟩ hx
exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)
by_cases ha0 : a = 0
· rwa [ha0]
haveI : Nontrivial α := ⟨⟨_, _, ha0⟩⟩
letI : NormalizationMonoid α := Unique... |
import Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol
#align_import number_theory.legendre_symbol.norm_num from "leanprover-community/mathlib"@"e2621d935895abe70071ab828a4ee6e26a52afe4"
section Lemmas
namespace Mathlib.Meta.NormNum
def jacobiSymNat (a b : ℕ) : ℤ :=
jacobiSym a b
#align norm_num.jacobi_sym_... | Mathlib/Tactic/NormNum/LegendreSymbol.lean | 86 | 87 | theorem jacobiSymNat.one_left (b : ℕ) : jacobiSymNat 1 b = 1 := by |
rw [jacobiSymNat, Nat.cast_one, jacobiSym.one_left]
|
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
variable {s s₁ s₂ : Set α} {t ... | Mathlib/Data/Set/Function.lean | 906 | 912 | theorem SurjOn.inter_inter (h₁ : SurjOn f s₁ t₁) (h₂ : SurjOn f s₂ t₂) (h : InjOn f (s₁ ∪ s₂)) :
SurjOn f (s₁ ∩ s₂) (t₁ ∩ t₂) := by |
intro y hy
rcases h₁ hy.1 with ⟨x₁, hx₁, rfl⟩
rcases h₂ hy.2 with ⟨x₂, hx₂, heq⟩
obtain rfl : x₁ = x₂ := h (Or.inl hx₁) (Or.inr hx₂) heq.symm
exact mem_image_of_mem f ⟨hx₁, hx₂⟩
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Sym.Card
open Finset Function
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V) {e : Sym2 V}
theorem edgeFinset_deleteEdges [DecidableEq V] [Fintype G.edgeSet] (s : Finset (Sym2 V))
[Fintyp... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 383 | 387 | theorem le_minDegree_of_forall_le_degree [DecidableRel G.Adj] [Nonempty V] (k : ℕ)
(h : ∀ v, k ≤ G.degree v) : k ≤ G.minDegree := by |
rcases G.exists_minimal_degree_vertex with ⟨v, hv⟩
rw [hv]
apply h
|
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 468 | 483 | theorem generalized_eigenvec_disjoint_range_ker [FiniteDimensional K V] (f : End K V) (μ : K) :
Disjoint (f.genEigenrange μ (finrank K V))
(f.genEigenspace μ (finrank K V)) := by |
have h :=
calc
Submodule.comap ((f - algebraMap _ _ μ) ^ finrank K V)
(f.genEigenspace μ (finrank K V)) =
LinearMap.ker ((f - algebraMap _ _ μ) ^ finrank K V *
(f - algebraMap K (End K V) μ) ^ finrank K V) := by
rw [genEigenspace, OrderHom.coe_mk, ← LinearMap.ker... |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.affine_space.affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 528 | 529 | theorem lineMap_apply_module (p₀ p₁ : V1) (c : k) : lineMap p₀ p₁ c = (1 - c) • p₀ + c • p₁ := by |
simp [lineMap_apply_module', smul_sub, sub_smul]; abel
|
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 | 216 | 223 | theorem pow_card_sub_one_eq_one (a : K) (ha : a ≠ 0) : a ^ (q - 1) = 1 := by |
calc
a ^ (Fintype.card K - 1) = (Units.mk0 a ha ^ (Fintype.card K - 1) : Kˣ).1 := by
rw [Units.val_pow_eq_pow_val, Units.val_mk0]
_ = 1 := by
classical
rw [← Fintype.card_units, pow_card_eq_one]
rfl
|
import Mathlib.CategoryTheory.Preadditive.ProjectiveResolution
import Mathlib.Algebra.Homology.HomotopyCategory
import Mathlib.Tactic.SuppressCompilation
suppress_compilation
noncomputable section
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
open Category Limits Projective
set_... | Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean | 73 | 76 | theorem liftFOne_zero_comm {Y Z : C} (f : Y ⟶ Z) (P : ProjectiveResolution Y)
(Q : ProjectiveResolution Z) :
liftFOne f P Q ≫ Q.complex.d 1 0 = P.complex.d 1 0 ≫ liftFZero f P Q := by |
apply Q.exact₀.liftFromProjective_comp
|
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Set.Finite
#align_import order.conditionally_complete_lattice.finset from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
open Set
variable {ι α β γ : Type*}
section ConditionallyCompleteLinearOrder
variable [... | Mathlib/Order/ConditionallyCompleteLattice/Finset.lean | 42 | 44 | theorem Set.Nonempty.csSup_mem (h : s.Nonempty) (hs : s.Finite) : sSup s ∈ s := by |
lift s to Finset α using hs
exact Finset.Nonempty.csSup_mem h
|
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on from "leanprover-community/mathlib"@"8b8ba04e2f326f3f7cf24ad129beda58531ada61"
noncomputable section
open Set Filter TopologicalSpace MeasureTheory Function
ope... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 220 | 223 | theorem integrableOn_finite_iUnion [Finite β] {t : β → Set α} :
IntegrableOn f (⋃ i, t i) μ ↔ ∀ i, IntegrableOn f (t i) μ := by |
cases nonempty_fintype β
simpa using @integrableOn_finset_iUnion _ _ _ _ _ f μ Finset.univ t
|
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 | 282 | 283 | theorem coeff_X (n : ℕ) : coeff R n (X : R⟦X⟧) = if n = 1 then 1 else 0 := by |
rw [X_eq, coeff_monomial]
|
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
open Finset Int
namespace Int
variable (a b : ℤ) {r : ℤ} (hr : 0 < r)
lemma Ico_filter_dvd_eq : (Ico a b).filter (r ∣ ·) =
(Ico ⌈a / (r : ℚ)⌉ ⌈b... | Mathlib/Data/Int/CardIntervalMod.lean | 71 | 73 | theorem Ioc_filter_modEq_card (v : ℤ) : ((Ioc a b).filter (· ≡ v [ZMOD r])).card =
max (⌊(b - v) / (r : ℚ)⌋ - ⌊(a - v) / (r : ℚ)⌋) 0 := by |
simp [Ioc_filter_modEq_eq, Ioc_filter_dvd_eq, toNat_eq_max, hr]
|
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 124 | 128 | theorem Matrix.Represents.one : (1 : Matrix ι ι R).Represents b 1 := by |
delta Matrix.Represents PiToModule.fromMatrix
rw [LinearMap.comp_apply, AlgEquiv.toLinearMap_apply, _root_.map_one]
ext
rfl
|
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.LinearAlgebra.Trace
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosu... | Mathlib/RingTheory/Trace.lean | 586 | 599 | theorem det_traceMatrix_ne_zero' [IsSeparable K L] : det (traceMatrix K pb.basis) ≠ 0 := by |
suffices algebraMap K (AlgebraicClosure L) (det (traceMatrix K pb.basis)) ≠ 0 by
refine mt (fun ht => ?_) this
rw [ht, RingHom.map_zero]
haveI : FiniteDimensional K L := pb.finite
let e : Fin pb.dim ≃ (L →ₐ[K] AlgebraicClosure L) := (Fintype.equivFinOfCardEq ?_).symm
· rw [RingHom.map_det, RingHom.mapM... |
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 534 | 551 | theorem Sum.isConnected_iff [TopologicalSpace β] {s : Set (Sum α β)} :
IsConnected s ↔
(∃ t, IsConnected t ∧ s = Sum.inl '' t) ∨ ∃ t, IsConnected t ∧ s = Sum.inr '' t := by |
refine ⟨fun hs => ?_, ?_⟩
· obtain ⟨x | x, hx⟩ := hs.nonempty
· have h : s ⊆ range Sum.inl :=
hs.isPreconnected.subset_isClopen isClopen_range_inl ⟨.inl x, hx, x, rfl⟩
refine Or.inl ⟨Sum.inl ⁻¹' s, ?_, ?_⟩
· exact hs.preimage_of_isOpenMap Sum.inl_injective isOpenMap_inl h
· exact (ima... |
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
import Mathlib.Topology.Order.Monotone
#align_import topology.algebra.order.intermediate_value from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filt... | Mathlib/Topology/Order/IntermediateValue.lean | 546 | 548 | theorem intermediate_value_uIcc {a b : α} {f : α → δ} (hf : ContinuousOn f (uIcc a b)) :
uIcc (f a) (f b) ⊆ f '' uIcc a b := by |
cases le_total (f a) (f b) <;> simp [*, isPreconnected_uIcc.intermediate_value]
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 563 | 564 | theorem mem_nhds_prod_iff {x : X} {y : Y} {s : Set (X × Y)} :
s ∈ 𝓝 (x, y) ↔ ∃ u ∈ 𝓝 x, ∃ v ∈ 𝓝 y, u ×ˢ v ⊆ s := by | rw [nhds_prod_eq, mem_prod_iff]
|
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 564 | 564 | theorem one_div_one_div : 1 / (1 / a) = a := by | simp
|
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Tactic.Lift
import Mathlib.Tactic.Monotonicity.Attr
open Function
variable {β G M : Type*}
section Monoid
variable [Monoid M]
section Preorder
variable [Preorder M]
section DivInvMonoid
variable [DivInvMonoid G] [Preorder G] [CovariantClass G G (· ... | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 352 | 355 | theorem one_le_zpow {x : G} (H : 1 ≤ x) {n : ℤ} (hn : 0 ≤ n) : 1 ≤ x ^ n := by |
lift n to ℕ using hn
rw [zpow_natCast]
apply one_le_pow_of_one_le' H
|
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_zero from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862b9fb9a992fef2"
open Topology Filter Function
variable {α β G₀ : Type*}
class HasContin... | Mathlib/Topology/Algebra/GroupWithZero.lean | 236 | 245 | theorem ContinuousAt.comp_div_cases {f g : α → G₀} (h : α → G₀ → β) (hf : ContinuousAt f a)
(hg : ContinuousAt g a) (hh : g a ≠ 0 → ContinuousAt (↿h) (a, f a / g a))
(h2h : g a = 0 → Tendsto (↿h) (𝓝 a ×ˢ ⊤) (𝓝 (h a 0))) :
ContinuousAt (fun x => h x (f x / g x)) a := by |
show ContinuousAt (↿h ∘ fun x => (x, f x / g x)) a
by_cases hga : g a = 0
· rw [ContinuousAt]
simp_rw [comp_apply, hga, div_zero]
exact (h2h hga).comp (continuousAt_id.prod_mk tendsto_top)
· exact ContinuousAt.comp (hh hga) (continuousAt_id.prod (hf.div hg hga))
|
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section NatPowAssoc
variable (R : Type*) [Semiring R] {p : R[X]} (r : R) (p q : R[X]) {S : Type*}
[NonAssocSemiring S] [... | Mathlib/Algebra/Polynomial/Smeval.lean | 182 | 189 | theorem smeval_at_natCast (q : ℕ[X]): ∀(n : ℕ), q.smeval (n : S) = q.smeval n := by |
induction q using Polynomial.induction_on' with
| h_add p q ph qh =>
intro n
simp only [add_mul, smeval_add, ph, qh, Nat.cast_add]
| h_monomial n a =>
intro n
rw [smeval_monomial, smeval_monomial, nsmul_eq_mul, smul_eq_mul, Nat.cast_mul, Nat.cast_npow]
|
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Logic.Function.Basic
#align_import group_theory.semidirect_product from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable (N : Type*) (G : Type*) {H : Type*} [Group N] [Group G] [Group H]
... | Mathlib/GroupTheory/SemidirectProduct.lean | 300 | 300 | theorem map_inr (g : G) : map f₁ f₂ h (inr g) = inr (f₂ g) := by | simp [map]
|
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ι A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 116 | 124 | theorem gradedComm_of_tmul_of (i j : ι) (a : 𝒜 i) (b : ℬ j) :
gradedComm R 𝒜 ℬ (lof R _ 𝒜 i a ⊗ₜ lof R _ ℬ j b) =
(-1 : ℤˣ)^(j * i) • (lof R _ ℬ _ b ⊗ₜ lof R _ 𝒜 _ a) := by |
rw [gradedComm]
dsimp only [LinearEquiv.trans_apply, LinearEquiv.ofLinear_apply]
rw [TensorProduct.directSum_lof_tmul_lof, gradedCommAux_lof_tmul, Units.smul_def,
-- Note: #8386 specialized `map_smul` to `LinearEquiv.map_smul` to avoid timeouts.
zsmul_eq_smul_cast R, LinearEquiv.map_smul, TensorProduct.d... |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 322 | 324 | theorem eval_eq_sum : p.eval x = p.sum fun e a => a * x ^ e := by |
rw [eval, eval₂_eq_sum]
rfl
|
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 264 | 268 | theorem inner_indicatorConstLp_eq_inner_setIntegral [CompleteSpace E] [NormedSpace ℝ E]
(hs : MeasurableSet s) (hμs : μ s ≠ ∞) (c : E) (f : Lp E 2 μ) :
(⟪indicatorConstLp 2 hs hμs c, f⟫ : 𝕜) = ⟪c, ∫ x in s, f x ∂μ⟫ := by |
rw [← integral_inner (integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim hμs),
L2.inner_indicatorConstLp_eq_setIntegral_inner]
|
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 131 | 136 | theorem Asymptotics.isLittleO_sum_range_of_tendsto_zero {α : Type*} [NormedAddCommGroup α]
{f : ℕ → α} (h : Tendsto f atTop (𝓝 0)) :
(fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => (n : ℝ) := by |
have := ((isLittleO_one_iff ℝ).2 h).sum_range fun i => zero_le_one
simp only [sum_const, card_range, Nat.smul_one_eq_cast] at this
exact this tendsto_natCast_atTop_atTop
|
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Limits.Constructions.Fin... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 411 | 465 | theorem IsVanKampenColimit.map_reflective [HasColimitsOfShape J C]
{Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Gr.Full] [Gr.Faithful]
{F : J ⥤ D} {c : Cocone (F ⋙ Gr)} (H : IsVanKampenColimit c)
[∀ X (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)]
[∀ X (f : X ⟶ Gl.obj c.pt), PreservesL... |
have := adj.rightAdjointPreservesLimits
have : PreservesColimitsOfSize.{u', v'} Gl := adj.leftAdjointPreservesColimits
intro F' c' α f h hα
refine ⟨?_, H.isUniversal.map_reflective adj c' α f h hα⟩
intro ⟨hc'⟩ j
let α' := α ≫ (Functor.associator _ _ _).hom ≫ whiskerLeft F adj.counit ≫ F.rightUnitor.hom
h... |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Order.Hom.Basic
#align_import algebra.order.sub.basic from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
variable {α β : Type*}
section Add
variable [Preord... | Mathlib/Algebra/Order/Sub/Basic.lean | 25 | 28 | theorem AddHom.le_map_tsub [Preorder β] [Add β] [Sub β] [OrderedSub β] (f : AddHom α β)
(hf : Monotone f) (a b : α) : f a - f b ≤ f (a - b) := by |
rw [tsub_le_iff_right, ← f.map_add]
exact hf le_tsub_add
|
import Mathlib.Data.Nat.Choose.Dvd
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
#align_import ring_theory.polynomial.eisenstein.is_integral from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf1682e59106d6c942434a32"
universe u ... | Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean | 237 | 370 | theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
(hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
(hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
z ∈ adjoin R ({B.gen} : Set L) := by |
-- First define some abbreviations.
have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
hei.not_mem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
have := B.finite
set P := minpoly R B.gen with hP
obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
haveI : NoZeroSMulDi... |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 282 | 283 | theorem inner_neg_right (x y : F) : ⟪x, -y⟫ = -⟪x, y⟫ := by |
rw [← inner_conj_symm, inner_neg_left]; simp only [RingHom.map_neg, inner_conj_symm]
|
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 185 | 185 | theorem sub_half (a : R) : a - a / 2 = a / 2 := by | rw [sub_eq_iff_eq_add, add_halves']
|
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#align_import data.complex.exponential from "leanprover-community/mathlib"@"a8b2226cfb0a79f5986492053fc49b1a0c6aeffb"
open CauSeq Finset IsAbsoluteValue
open ... | Mathlib/Data/Complex/Exponential.lean | 779 | 780 | theorem exp_eq_exp_re_mul_sin_add_cos : exp x = exp x.re * (cos x.im + sin x.im * I) := by |
rw [← exp_add_mul_I, re_add_im]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
var... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 417 | 420 | theorem zero_rpow_of_pos {y : ℝ} (h : 0 < y) : (0 : ℝ≥0∞) ^ y = 0 := by |
rw [← ENNReal.coe_zero, ← ENNReal.some_eq_coe]
dsimp only [(· ^ ·), rpow, Pow.pow]
simp [h, asymm h, ne_of_gt h]
|
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-community/mathlib"@"9822b65bfc4ac74537d77ae318d27df1df662471"
universe u v w w₁ w₂
variable {R M}
theorem Submodule.exists_lieSubmodule_coe_eq_iff (p... | Mathlib/Algebra/Lie/Submodule.lean | 1,179 | 1,182 | theorem mem_idealRange_iff (h : IsIdealMorphism f) {y : L'} :
y ∈ idealRange f ↔ ∃ x : L, f x = y := by |
rw [f.isIdealMorphism_def] at h
rw [← LieSubmodule.mem_coe, ← LieIdeal.coe_toSubalgebra, h, f.range_coe, Set.mem_range]
|
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.special_functions.improper_inte... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 53 | 54 | theorem integral_exp_neg_Ioi (c : ℝ) : (∫ x : ℝ in Ioi c, exp (-x)) = exp (-c) := by |
simpa only [integral_comp_neg_Ioi] using integral_exp_Iic (-c)
|
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.Peel
#align_import group_... | Mathlib/GroupTheory/Exponent.lean | 193 | 200 | theorem exp_eq_one_iff : exponent G = 1 ↔ Subsingleton G := by |
refine ⟨fun eq_one => ⟨fun a b => ?a_eq_b⟩, fun h => le_antisymm ?le ?ge⟩
· rw [← pow_one a, ← pow_one b, ← eq_one, Monoid.pow_exponent_eq_one, Monoid.pow_exponent_eq_one]
· apply exponent_min' _ Nat.one_pos
simp [eq_iff_true_of_subsingleton]
· apply Nat.succ_le_of_lt
apply exponent_pos_of_exists 1 Nat... |
import Mathlib.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.Order.Filter.SmallSets
import Mathlib.Order.LiminfLimsup
import Mathlib.Data.Set.UnionLift
#align_import measure_theory.meas... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,354 | 1,356 | theorem measurableSet_preimage (hf : MeasurableEmbedding f) {s : Set β} :
MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (s ∩ range f) := by |
rw [← image_preimage_eq_inter_range, hf.measurableSet_image]
|
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Set Filter Topology
variable {X : Type*}
def OnePoint (X : Type*) :=
Option X
#ali... | Mathlib/Topology/Compactification/OnePoint.lean | 486 | 492 | theorem not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
¬Continuous (@CofiniteTopology.of (OnePoint X)).symm := by |
inhabit X
simp only [continuous_iff_continuousAt, ContinuousAt, not_forall]
use CofiniteTopology.of ↑(default : X)
simpa [nhds_coe_eq, nhds_discrete, CofiniteTopology.nhds_eq] using
(finite_singleton ((default : X) : OnePoint X)).infinite_compl
|
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 242 | 243 | theorem IsRelPrime.prod_right : (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i) := by |
simpa only [isRelPrime_comm] using IsRelPrime.prod_left (α := α)
|
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebra.finsupp from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
noncomputable section
open Set LinearMap Submodule
namespa... | Mathlib/LinearAlgebra/Finsupp.lean | 746 | 749 | theorem total_comp_lmapDomain (f : α → α') :
(Finsupp.total α' M' R v').comp (Finsupp.lmapDomain R R f) = Finsupp.total α M' R (v' ∘ f) := by |
ext
simp
|
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.Ideal.Maps
#align_import data.polynomial.div from "leanprover-community/mathlib"@"e1e7190efdcefc925cb36f257a8362ef22944204"
noncomputable section
open Polynomial
... | Mathlib/Algebra/Polynomial/Div.lean | 656 | 658 | theorem eval₂_modByMonic_eq_self_of_root [CommRing S] {f : R →+* S} {p q : R[X]} (hq : q.Monic)
{x : S} (hx : q.eval₂ f x = 0) : (p %ₘ q).eval₂ f x = p.eval₂ f x := by |
rw [modByMonic_eq_sub_mul_div p hq, eval₂_sub, eval₂_mul, hx, zero_mul, sub_zero]
|
import Mathlib.Algebra.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Data.Bracket
#align_import algebra.ring.commute from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w x
variable {α : Type u} {β : Type v} {γ : T... | Mathlib/Algebra/Ring/Commute.lean | 77 | 79 | theorem mul_self_sub_mul_self_eq' [NonUnitalNonAssocRing R] {a b : R} (h : Commute a b) :
a * a - b * b = (a - b) * (a + b) := by |
rw [mul_add, sub_mul, sub_mul, h.eq, sub_add_sub_cancel]
|
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 180 | 188 | theorem kahler_rotation_left (x y : V) (θ : Real.Angle) :
o.kahler (o.rotation θ x) y = conj (θ.expMapCircle : ℂ) * o.kahler x y := by |
-- Porting note: this needed the `Complex.conj_ofReal` instead of `RCLike.conj_ofReal`;
-- I believe this is because the respective coercions are no longer defeq, and
-- `Real.Angle.coe_expMapCircle` uses the `Complex` version.
simp only [o.rotation_apply, map_add, map_mul, LinearMap.map_smulₛₗ, RingHom.id_app... |
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Data.Finset.Basic
import Mathlib.Order.Interval.Finset.Defs
open Function
namespace Finset
class HasAntidiagonal (A : Type*) [AddMonoid A] where
antidiagonal : A → Finset (A × A)
mem_antidiagonal {n} {a} : a ∈ antidiagonal n ↔ a.fst + a.snd = n
exp... | Mathlib/Data/Finset/Antidiagonal.lean | 135 | 138 | theorem antidiagonal.fst_le {n : A} {kl : A × A} (hlk : kl ∈ antidiagonal n) : kl.1 ≤ n := by |
rw [le_iff_exists_add]
use kl.2
rwa [mem_antidiagonal, eq_comm] at hlk
|
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Mo... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 105 | 119 | theorem rank_eq_one_iff [Module.Free K V] :
Module.rank K V = 1 ↔ ∃ v₀ : V, v₀ ≠ 0 ∧ ∀ v, ∃ r : K, r • v₀ = v := by |
haveI := nontrivial_of_invariantBasisNumber K
refine ⟨fun h ↦ ?_, fun ⟨v₀, h, hv⟩ ↦ (rank_le_one_iff.2 ⟨v₀, hv⟩).antisymm ?_⟩
· obtain ⟨v₀, hv⟩ := rank_le_one_iff.1 h.le
refine ⟨v₀, fun hzero ↦ ?_, hv⟩
simp_rw [hzero, smul_zero, exists_const] at hv
haveI : Subsingleton V := .intro fun _ _ ↦ by simp_r... |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"b07688016d62f81d14508ff339ea3415558d6353"
universe u v
namespace SimpleGraph
open Walk
variable {V : Type u} (G : SimpleGraph V)
def IsAcy... | Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 88 | 115 | theorem IsAcyclic.path_unique {G : SimpleGraph V} (h : G.IsAcyclic) {v w : V} (p q : G.Path v w) :
p = q := by |
obtain ⟨p, hp⟩ := p
obtain ⟨q, hq⟩ := q
rw [Subtype.mk.injEq]
induction p with
| nil =>
cases (Walk.isPath_iff_eq_nil _).mp hq
rfl
| cons ph p ih =>
rw [isAcyclic_iff_forall_adj_isBridge] at h
specialize h ph
rw [isBridge_iff_adj_and_forall_walk_mem_edges] at h
replace h := h.2 (q.a... |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.cone.basic from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
assert_not_exists NormedSpace
assert_not_exists Real
open Set LinearMap
open scoped Classical
open Pointwise
variable {𝕜 E F G : Type*}
namespace Conve... | Mathlib/Analysis/Convex/Cone/Basic.lean | 381 | 384 | theorem Flat.pointed {S : ConvexCone 𝕜 E} (hS : S.Flat) : S.Pointed := by |
obtain ⟨x, hx, _, hxneg⟩ := hS
rw [Pointed, ← add_neg_self x]
exact add_mem S hx hxneg
|
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 481 | 485 | theorem LindelofSpace.elim_nhds_subcover [LindelofSpace X] (U : X → Set X) (hU : ∀ x, U x ∈ 𝓝 x) :
∃ t : Set X, t.Countable ∧ ⋃ x ∈ t, U x = univ := by |
obtain ⟨t, tc, -, s⟩ := IsLindelof.elim_nhds_subcover isLindelof_univ U fun x _ => hU x
use t, tc
apply top_unique s
|
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 | 309 | 310 | theorem _root_.Sbtw.angle₃₂₁_eq_pi {p₁ p₂ p₃ : P} (h : Sbtw ℝ p₁ p₂ p₃) : ∠ p₃ p₂ p₁ = π := by |
rw [← h.angle₁₂₃_eq_pi, angle_comm]
|
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot
import Mathlib.Order.Hom.Basic
#align_import data.multiset.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
universe v
... | Mathlib/Data/Multiset/Basic.lean | 2,669 | 2,675 | theorem replicate_inter (n : ℕ) (x : α) (s : Multiset α) :
replicate n x ∩ s = replicate (min n (s.count x)) x := by |
ext y
rw [count_inter, count_replicate, count_replicate]
by_cases h : y = x
· simp only [h, if_true]
· simp only [h, if_false, Nat.zero_min]
|
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.G... | Mathlib/RingTheory/Norm.lean | 151 | 166 | theorem norm_eq_zero_iff [IsDomain R] [IsDomain S] [Module.Free R S] [Module.Finite R S] {x : S} :
norm R x = 0 ↔ x = 0 := by |
constructor
on_goal 1 => let b := Module.Free.chooseBasis R S
swap
· rintro rfl; exact norm_zero
· letI := Classical.decEq (Module.Free.ChooseBasisIndex R S)
rw [norm_eq_matrix_det b, ← Matrix.exists_mulVec_eq_zero_iff]
rintro ⟨v, v_ne, hv⟩
rw [← b.equivFun.apply_symm_apply v, b.equivFun_symm_app... |
import Mathlib.CategoryTheory.Functor.Const
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.Yoneda
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.limits.cones from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
-- morphism le... | Mathlib/CategoryTheory/Limits/Cones.lean | 525 | 528 | theorem CoconeMorphism.ext {c c' : Cocone F} (f g : c ⟶ c') (w : f.hom = g.hom) : f = g := by |
cases f
cases g
congr
|
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 841 | 855 | theorem sup_mul_inf (I J : Ideal A) : (I ⊔ J) * (I ⊓ J) = I * J := by |
letI := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
have hgcd : gcd I J = I ⊔ J := by
rw [gcd_eq_normalize _ _, normalize_eq]
· rw [dvd_iff_le, sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
exact ⟨gcd_dvd_left _ _, gcd_dvd_right _ _⟩
· rw [dvd_gcd_iff, dvd_iff_le, dvd_iff_le]
simp
... |
import Mathlib.CategoryTheory.Comma.StructuredArrow
import Mathlib.CategoryTheory.IsConnected
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.CategoryTheory.Limits.Yoneda
import Mathlib.Category... | Mathlib/CategoryTheory/Limits/Final.lean | 411 | 427 | theorem cofinal_of_colimit_comp_coyoneda_iso_pUnit
(I : ∀ d, colimit (F ⋙ coyoneda.obj (op d)) ≅ PUnit) : Final F :=
⟨fun d => by
have : Nonempty (StructuredArrow d F) := by |
have := (I d).inv PUnit.unit
obtain ⟨j, y, rfl⟩ := Limits.Types.jointly_surjective'.{v, v} this
exact ⟨StructuredArrow.mk y⟩
apply zigzag_isConnected
rintro ⟨⟨⟨⟩⟩, X₁, f₁⟩ ⟨⟨⟨⟩⟩, X₂, f₂⟩
let y₁ := colimit.ι (F ⋙ coyoneda.obj (op d)) X₁ f₁
let y₂ := colimit.ι (F ⋙ coyoneda.obj (op d)) ... |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 79 | 80 | theorem sup_ndunion (s₁ s₂ : Multiset α) : (ndunion s₁ s₂).sup = s₁.sup ⊔ s₂.sup := by |
rw [← sup_dedup, dedup_ext.2, sup_dedup, sup_add]; simp
|
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
inductive Heap (α : Type u) where
| nil : Heap α
| node (a : α) (child sibling : Heap α) : Heap α
deriving Repr
def Heap.size : Heap α → Nat
| .nil => 0
| .node _ c s => c.size + 1 + s.size
def Heap.singleton (a : α) : Heap α := .... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 142 | 146 | theorem Heap.size_tail? {s : Heap α} (h : s.NoSibling) : s.tail? le = some s' →
s.size = s'.size + 1 := by |
simp only [Heap.tail?]; intro eq
match eq₂ : s.deleteMin le, eq with
| some (a, tl), rfl => exact size_deleteMin h eq₂
|
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Algebra.Module.Basic
import Mathlib.Topology.Separation
#align_import topology.support from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
open Function Set Filter Topology
variable {X α α' β γ δ M E R : Type*}
theorem tsup... | Mathlib/Topology/Support.lean | 370 | 372 | theorem HasCompactSupport.mul_left (hf : HasCompactSupport f') : HasCompactSupport (f * f') := by |
rw [hasCompactSupport_iff_eventuallyEq] at hf ⊢
exact hf.mono fun x hx => by simp_rw [Pi.mul_apply, hx, Pi.zero_apply, mul_zero]
|
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 | 155 | 156 | theorem card_pair (h : a ≠ b) : ({a, b} : Finset α).card = 2 := by |
rw [card_insert_of_not_mem (not_mem_singleton.2 h), card_singleton]
|
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finite.Basic
import Mathlib.Data.Set.Functor
import Mathlib.Data.Set.Lattice
#align_import data.set.finite from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists OrderedRing
assert_not_exists MonoidWithZero
open Set Fun... | Mathlib/Data/Set/Finite.lean | 1,505 | 1,509 | theorem Finite.exists_lt_map_eq_of_forall_mem [LinearOrder α] [Infinite α] {t : Set β} {f : α → β}
(hf : ∀ a, f a ∈ t) (ht : t.Finite) : ∃ a b, a < b ∧ f a = f b := by |
rw [← mapsTo_univ_iff] at hf
obtain ⟨a, -, b, -, h⟩ := infinite_univ.exists_lt_map_eq_of_mapsTo hf ht
exact ⟨a, b, h⟩
|
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 | 782 | 793 | theorem measure_inter_eq_of_measure_eq {s t u : Set α} (hs : MeasurableSet s) (h : μ t = μ u)
(htu : t ⊆ u) (ht_ne_top : μ t ≠ ∞) : μ (t ∩ s) = μ (u ∩ s) := by |
rw [h] at ht_ne_top
refine le_antisymm (by gcongr) ?_
have A : μ (u ∩ s) + μ (u \ s) ≤ μ (t ∩ s) + μ (u \ s) :=
calc
μ (u ∩ s) + μ (u \ s) = μ u := measure_inter_add_diff _ hs
_ = μ t := h.symm
_ = μ (t ∩ s) + μ (t \ s) := (measure_inter_add_diff _ hs).symm
_ ≤ μ (t ∩ s) + μ (u \ s) :... |
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
#align_import algebra.indicator_function from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
assert_not_exists MonoidWithZero
open Function
variable {α β ι M N : Type*}
namespace Set
section One
variable [On... | Mathlib/Algebra/Group/Indicator.lean | 304 | 308 | theorem mulIndicator_const_preimage (U : Set α) (s : Set M) (a : M) :
(U.mulIndicator fun _ => a) ⁻¹' s ∈ ({Set.univ, U, Uᶜ, ∅} : Set (Set α)) := by |
classical
rw [mulIndicator_const_preimage_eq_union]
split_ifs <;> simp
|
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Directed
import Mathlib.Order.Interval.Set.Basic
#align_import data.finset.basic from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
-- Assert that we define `Finset... | Mathlib/Data/Finset/Basic.lean | 2,266 | 2,269 | theorem sdiff_insert_of_not_mem {x : α} (h : x ∉ s) (t : Finset α) : s \ insert x t = s \ t := by |
refine Subset.antisymm (sdiff_subset_sdiff (Subset.refl _) (subset_insert _ _)) fun y hy => ?_
simp only [mem_sdiff, mem_insert, not_or] at hy ⊢
exact ⟨hy.1, fun hxy => h <| hxy ▸ hy.1, hy.2⟩
|
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 85 | 86 | theorem tsupport_normed_eq : tsupport (f.normed μ) = Metric.closedBall c f.rOut := by |
rw [tsupport, f.support_normed_eq, closure_ball _ f.rOut_pos.ne']
|
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 997 | 998 | theorem inner_self_eq_norm_sq (x : E) : re ⟪x, x⟫ = ‖x‖ ^ 2 := by |
rw [pow_two, inner_self_eq_norm_mul_norm]
|
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Tactic.GCongr
import Mathlib.Topology.Order.LeftRightNhds
#align_import algebra.continued_fractions.computation.approximation_corollar... | Mathlib/Algebra/ContinuedFractions/Computation/ApproximationCorollaries.lean | 144 | 146 | theorem of_convergence [TopologicalSpace K] [OrderTopology K] :
Filter.Tendsto (of v).convergents Filter.atTop <| 𝓝 v := by |
simpa [LinearOrderedAddCommGroup.tendsto_nhds, abs_sub_comm] using of_convergence_epsilon v
|
import Mathlib.Data.Matrix.Basic
#align_import data.matrix.block from "leanprover-community/mathlib"@"c060baa79af5ca092c54b8bf04f0f10592f59489"
variable {l m n o p q : Type*} {m' n' p' : o → Type*}
variable {R : Type*} {S : Type*} {α : Type*} {β : Type*}
open Matrix
namespace Matrix
theorem dotProduct_block [F... | Mathlib/Data/Matrix/Block.lean | 693 | 701 | theorem blockDiagonal'_transpose (M : ∀ i, Matrix (m' i) (n' i) α) :
(blockDiagonal' M)ᵀ = blockDiagonal' fun k => (M k)ᵀ := by |
ext ⟨ii, ix⟩ ⟨ji, jx⟩
simp only [transpose_apply, blockDiagonal'_apply]
split_ifs with h -- Porting note: was split_ifs <;> cc
· subst h; rfl
· simp_all only [not_true]
· simp_all only [not_true]
· rfl
|
import Mathlib.Probability.Process.Stopping
import Mathlib.Tactic.AdaptationNote
#align_import probability.process.hitting_time from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Order TopologicalSpace
open scoped Classical MeasureTheory NNReal ENNReal Topology
namespace ... | Mathlib/Probability/Process/HittingTime.lean | 215 | 225 | theorem hitting_mono {m₁ m₂ : ι} (hm : m₁ ≤ m₂) : hitting u s n m₁ ω ≤ hitting u s n m₂ ω := by |
by_cases h : ∃ j ∈ Set.Icc n m₁, u j ω ∈ s
· exact (hitting_eq_hitting_of_exists hm h).le
· simp_rw [hitting, if_neg h]
split_ifs with h'
· obtain ⟨j, hj₁, hj₂⟩ := h'
refine le_csInf ⟨j, hj₁, hj₂⟩ ?_
by_contra hneg; push_neg at hneg
obtain ⟨i, hi₁, hi₂⟩ := hneg
exact h ⟨i, ⟨hi₁.1.... |
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 | 474 | 491 | theorem wOppSide_iff_exists_left {s : AffineSubspace R P} {x y p₁ : P} (h : p₁ ∈ s) :
s.WOppSide x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y) := by |
constructor
· rintro ⟨p₁', hp₁', p₂', hp₂', h0 | h0 | ⟨r₁, r₂, hr₁, hr₂, hr⟩⟩
· rw [vsub_eq_zero_iff_eq] at h0
rw [h0]
exact Or.inl hp₁'
· refine Or.inr ⟨p₂', hp₂', ?_⟩
rw [h0]
exact SameRay.zero_right _
· refine Or.inr ⟨(-r₁ / r₂) • (p₁ -ᵥ p₁') +ᵥ p₂', s.smul_vsub_vadd_mem _ h ... |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 121 | 147 | theorem Memℒp.memℒp_of_exponent_le {p q : ℝ≥0∞} [IsFiniteMeasure μ] {f : α → E} (hfq : Memℒp f q μ)
(hpq : p ≤ q) : Memℒp f p μ := by |
cases' hfq with hfq_m hfq_lt_top
by_cases hp0 : p = 0
· rwa [hp0, memℒp_zero_iff_aestronglyMeasurable]
rw [← Ne] at hp0
refine ⟨hfq_m, ?_⟩
by_cases hp_top : p = ∞
· have hq_top : q = ∞ := by rwa [hp_top, top_le_iff] at hpq
rw [hp_top]
rwa [hq_top] at hfq_lt_top
have hp_pos : 0 < p.toReal := ENN... |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
open Function Set
open scoped ENNReal Classical
noncomputable section
variable {α β δ : Type*} [MeasurableSpace α] [MeasurableSpace β] {s : Set α} ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 103 | 110 | theorem tsum_indicator_apply_singleton [Countable α] [MeasurableSingletonClass α] (μ : Measure α)
(s : Set α) (hs : MeasurableSet s) : (∑' x : α, s.indicator (fun x => μ {x}) x) = μ s :=
calc
(∑' x : α, s.indicator (fun x => μ {x}) x) =
Measure.sum (fun a => μ {a} • Measure.dirac a) s := by |
simp only [Measure.sum_apply _ hs, Measure.smul_apply, smul_eq_mul, Measure.dirac_apply,
Set.indicator_apply, mul_ite, Pi.one_apply, mul_one, mul_zero]
_ = μ s := by rw [μ.sum_smul_dirac]
|
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[Topolo... | Mathlib/Topology/Inseparable.lean | 482 | 484 | theorem map_prod_map_mk_nhds (x : X) (y : Y) :
map (Prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) := by |
rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]
|
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 441 | 447 | theorem finPiFinEquiv_single {m : ℕ} {n : Fin m → ℕ} [∀ i, NeZero (n i)] (i : Fin m)
(j : Fin (n i)) :
(finPiFinEquiv (Pi.single i j : ∀ i : Fin m, Fin (n i)) : ℕ) =
j * ∏ j, n (Fin.castLE i.is_lt.le j) := by |
rw [finPiFinEquiv_apply, Fintype.sum_eq_single i, Pi.single_eq_same]
rintro x hx
rw [Pi.single_eq_of_ne hx, Fin.val_zero', zero_mul]
|
import Mathlib.Topology.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Function Topology
noncomputable section
namespace TopologicalSpace
universe u
variable {α : Type u} {β : Type*} ... | Mathlib/Topology/Bases.lean | 93 | 103 | theorem IsTopologicalBasis.diff_empty {s : Set (Set α)} (h : IsTopologicalBasis s) :
IsTopologicalBasis (s \ {∅}) := by |
refine ⟨?_, by rw [sUnion_diff_singleton_empty, h.sUnion_eq], ?_⟩
· rintro t₁ ⟨h₁, -⟩ t₂ ⟨h₂, -⟩ x hx
obtain ⟨t₃, h₃, hs⟩ := h.exists_subset_inter _ h₁ _ h₂ x hx
exact ⟨t₃, ⟨h₃, Nonempty.ne_empty ⟨x, hs.1⟩⟩, hs⟩
· rw [h.eq_generateFrom]
refine le_antisymm (generateFrom_anti diff_subset) (le_generateF... |
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.GDelta
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analysis.SpecificLimits.Basic
#align_import topology.urysohns_lemma from "lea... | Mathlib/Topology/UrysohnsLemma.lean | 415 | 458 | theorem exists_continuous_one_zero_of_isCompact_of_isGδ [RegularSpace X] [LocallyCompactSpace X]
{s t : Set X} (hs : IsCompact s) (h's : IsGδ s) (ht : IsClosed t) (hd : Disjoint s t) :
∃ f : C(X, ℝ), s = f ⁻¹' {1} ∧ EqOn f 0 t ∧ HasCompactSupport f
∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 := by |
rcases h's.eq_iInter_nat with ⟨U, U_open, hU⟩
obtain ⟨m, m_comp, -, sm, mt⟩ : ∃ m, IsCompact m ∧ IsClosed m ∧ s ⊆ interior m ∧ m ⊆ tᶜ :=
exists_compact_closed_between hs ht.isOpen_compl hd.symm.subset_compl_left
have A n : ∃ f : C(X, ℝ), EqOn f 1 s ∧ EqOn f 0 (U n ∩ interior m)ᶜ ∧ HasCompactSupport f
∧... |
import Mathlib.Data.List.Basic
#align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
open Nat
variable {α β : Type*}
namespace List
variable {l l₁ l₂ l₃ : List α} {a b : α} {m n : ℕ}
section InitsTails
@[simp]
theorem mem_inits : ∀ s t : List α, s ∈ in... | Mathlib/Data/List/Infix.lean | 435 | 437 | theorem tails_reverse (l : List α) : tails (reverse l) = reverse (map reverse l.inits) := by |
rw [inits_eq_tails l]
simp [reverse_involutive.comp_self, reverse_map]
|
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Limits.Cones
#align_import category_theory.limits.is_limit from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Functor Opposite
... | Mathlib/CategoryTheory/Limits/IsLimit.lean | 513 | 520 | theorem coneOfHom_fac {Y : C} (f : Y ⟶ X) : coneOfHom h f = (limitCone h).extend f := by |
dsimp [coneOfHom, limitCone, Cone.extend]
congr with j
have t := congrFun (h.hom.naturality f.op) ⟨𝟙 X⟩
dsimp at t
simp only [comp_id] at t
rw [congrFun (congrArg NatTrans.app t) j]
rfl
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 745 | 755 | theorem IsCycle.isConj_iff (hσ : IsCycle σ) (hτ : IsCycle τ) :
IsConj σ τ ↔ σ.support.card = τ.support.card where
mp h := by |
obtain ⟨π, rfl⟩ := (_root_.isConj_iff).1 h
refine Finset.card_bij (fun a _ => π a) (fun _ ha => ?_) (fun _ _ _ _ ab => π.injective ab)
fun b hb ↦ ⟨π⁻¹ b, ?_, π.apply_inv_self b⟩
· simp [mem_support.1 ha]
contrapose! hb
rw [mem_support, Classical.not_not] at hb
rw [mem_support, Classical... |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.convex.topology from "leanprover-community/mathlib"@"0e3aacdc98d25e0afe035c452d876... | Mathlib/Analysis/Convex/Topology.lean | 200 | 203 | theorem Convex.openSegment_closure_interior_subset_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E}
(hx : x ∈ closure s) (hy : y ∈ interior s) : openSegment 𝕜 x y ⊆ interior s := by |
rintro _ ⟨a, b, ha, hb, hab, rfl⟩
exact hs.combo_closure_interior_mem_interior hx hy ha.le hb hab
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
set_option autoImplicit true
open Function Set Order
open scoped Classical
universe u v w x y
structure Filter (α : Type*) where... | Mathlib/Order/Filter/Basic.lean | 3,360 | 3,367 | theorem Filter.map_surjOn_Iic_iff_le_map {m : α → β} :
SurjOn (map m) (Iic F) (Iic G) ↔ G ≤ map m F := by |
refine ⟨fun hm ↦ ?_, fun hm ↦ ?_⟩
· rcases hm right_mem_Iic with ⟨H, (hHF : H ≤ F), rfl⟩
exact map_mono hHF
· have : RightInvOn (F ⊓ comap m ·) (map m) (Iic G) :=
fun H (hHG : H ≤ G) ↦ by simpa [Filter.push_pull] using hHG.trans hm
exact this.surjOn fun H _ ↦ mem_Iic.mpr inf_le_left
|
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