Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.57k | proof stringlengths 5 7.36k | hint bool 2
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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 | 52 | 54 | theorem prod_univ_def [CommMonoid β] {n : ℕ} (f : Fin n → β) :
∏ i, f i = ((List.finRange n).map f).prod := by |
rw [← List.ofFn_eq_map, prod_ofFn]
| false |
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 147 | 148 | theorem TendstoUniformly.tendstoUniformlyOnFilter (h : TendstoUniformly F f p) :
TendstoUniformlyOnFilter F f p ⊤ := by | rwa [← tendstoUniformly_iff_tendstoUniformlyOnFilter]
| false |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 53 | 54 | theorem preimage_Ico (e : α ≃o β) (a b : β) : e ⁻¹' Ico a b = Ico (e.symm a) (e.symm b) := by |
simp [← Ici_inter_Iio]
| false |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 25 | 28 | theorem add_tsub_cancel_of_le (h : a ≤ b) : a + (b - a) = b := by |
refine le_antisymm ?_ le_add_tsub
obtain ⟨c, rfl⟩ := exists_add_of_le h
exact add_le_add_left add_tsub_le_left a
| false |
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [... | Mathlib/FieldTheory/SeparableClosure.lean | 94 | 96 | theorem map_mem_separableClosure_iff (i : E →ₐ[F] K) {x : E} :
i x ∈ separableClosure F K ↔ x ∈ separableClosure F E := by |
simp_rw [mem_separableClosure_iff, minpoly.algHom_eq i i.injective]
| false |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
#align_import topology.algebra.infinite_sum.ring from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Filter Finset Function
open... | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 34 | 35 | theorem HasSum.mul_left (a₂) (h : HasSum f a₁) : HasSum (fun i ↦ a₂ * f i) (a₂ * a₁) := by |
simpa only using h.map (AddMonoidHom.mulLeft a₂) (continuous_const.mul continuous_id)
| false |
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.NeZero
#align_import algebra.group_with_zero.defs from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
assert_not_exists DenselyOrdered
universe u
variable {M₀ M₀' : Type*} [MulZeroOneClass M₀] [Nontrivial M₀]
instance N... | Mathlib/Algebra/GroupWithZero/NeZero.lean | 55 | 59 | theorem inv_mul_cancel (h : a ≠ 0) : a⁻¹ * a = 1 :=
calc
a⁻¹ * a = a⁻¹ * a * a⁻¹ * a⁻¹⁻¹ := by | simp [inv_ne_zero h]
_ = a⁻¹ * a⁻¹⁻¹ := by simp [h]
_ = 1 := by simp [inv_ne_zero h]
| false |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.free_group from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Relation
universe u v w
variable {α : Type u... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 151 | 155 | theorem not_step_nil : ¬Step [] L := by |
generalize h' : [] = L'
intro h
cases' h with L₁ L₂
simp [List.nil_eq_append] at h'
| false |
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial In... | Mathlib/FieldTheory/PrimitiveElement.lean | 246 | 275 | theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by |
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
sim... | false |
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-community/mathlib"@"b90e72c7eebbe8de7c8293a80208ea2ba135c834"
universe u
open SetTheory PGame
open scoped NaturalOps PGame
namespace Ordinal
noncomputable def toPGame : Ordin... | Mathlib/SetTheory/Game/Ordinal.lean | 46 | 49 | theorem toPGame_def (o : Ordinal) :
have : IsWellOrder o.out.α (· < ·) := isWellOrder_out_lt o
o.toPGame = ⟨o.out.α, PEmpty, fun x => (typein (· < ·) x).toPGame, PEmpty.elim⟩ := by |
rw [toPGame]
| false |
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 126 | 128 | theorem union_prod : (s₁ ∪ s₂) ×ˢ t = s₁ ×ˢ t ∪ s₂ ×ˢ t := by |
ext ⟨x, y⟩
simp [or_and_right]
| false |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 82 | 83 | theorem encard_coe_eq_coe_finsetCard (s : Finset α) : encard (s : Set α) = s.card := by |
rw [Finite.encard_eq_coe_toFinset_card (Finset.finite_toSet s)]; simp
| false |
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 | 81 | 82 | theorem thickening_empty (δ : ℝ) : thickening δ (∅ : Set α) = ∅ := by |
simp only [thickening, setOf_false, infEdist_empty, not_top_lt]
| false |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 83 | 91 | theorem orthogonalFamily_eigenspaces :
OrthogonalFamily 𝕜 (fun μ => eigenspace T μ) fun μ => (eigenspace T μ).subtypeₗᵢ := by |
rintro μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩
by_cases hv' : v = 0
· simp [hv']
have H := hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector ⟨hv, hv'⟩)
rw [mem_eigenspace_iff] at hv hw
refine Or.resolve_left ?_ hμν.symm
simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
| false |
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.Basis
#align_import linear_algebra.affine_space.basis from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Affine
open Set
universe u₁ u₂ u₃ u₄
structure AffineBasis (ι : Type u₁) (k : Type u₂) {V ... | Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 174 | 179 | theorem coord_apply_ne (h : i ≠ j) : b.coord i (b j) = 0 := by |
-- Porting note:
-- in mathlib3 we didn't need to given the `fun j => j ≠ i` argument to `Subtype.coe_mk`,
-- but I don't think we can complain: this proof was over-golfed.
rw [coord, AffineMap.coe_mk, ← @Subtype.coe_mk _ (fun j => j ≠ i) j h.symm, ← b.basisOf_apply,
Basis.sumCoords_self_apply, sub_self]
| false |
import Mathlib.Algebra.PUnitInstances
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Ring
import Mathlib.Order.Hom.Lattice
#align_import algebra.ring.boolean_ring from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped symmDiff
variable {α β γ : Type*}
class BooleanRing (α) ... | Mathlib/Algebra/Ring/BooleanRing.lean | 66 | 72 | theorem add_self : a + a = 0 := by |
have : a + a = a + a + (a + a) :=
calc
a + a = (a + a) * (a + a) := by rw [mul_self]
_ = a * a + a * a + (a * a + a * a) := by rw [add_mul, mul_add]
_ = a + a + (a + a) := by rw [mul_self]
rwa [self_eq_add_left] at this
| false |
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 104 | 105 | theorem univ_nonempty_iff : (univ : Finset α).Nonempty ↔ Nonempty α := by |
rw [← coe_nonempty, coe_univ, Set.nonempty_iff_univ_nonempty]
| false |
import Mathlib.Algebra.MvPolynomial.Derivation
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.pderiv from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
universe u v
namespace MvPolynomial
open Set Function Finsupp
variable {R : ... | Mathlib/Algebra/MvPolynomial/PDeriv.lean | 120 | 122 | theorem pderiv_pow {i : σ} {f : MvPolynomial σ R} {n : ℕ} :
pderiv i (f ^ n) = n * f ^ (n - 1) * pderiv i f := by |
rw [(pderiv i).leibniz_pow f n, nsmul_eq_mul, smul_eq_mul, mul_assoc]
| false |
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.isometric_smul from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Set
open ENNReal Pointwise
universe u v w
vari... | Mathlib/Topology/MetricSpace/IsometricSMul.lean | 143 | 144 | theorem edist_inv [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul Gᵐᵒᵖ G]
(x y : G) : edist x⁻¹ y = edist x y⁻¹ := by | rw [← edist_inv_inv, inv_inv]
| false |
import Mathlib.Data.List.Range
import Mathlib.Algebra.Order.Ring.Nat
variable {α : Type*}
namespace List
@[simp]
theorem length_iterate (f : α → α) (a : α) (n : ℕ) : length (iterate f a n) = n := by
induction n generalizing a <;> simp [*]
@[simp]
theorem iterate_eq_nil {f : α → α} {a : α} {n : ℕ} : iterate f ... | Mathlib/Data/List/Iterate.lean | 39 | 41 | theorem mem_iterate {f : α → α} {a : α} {n : ℕ} {b : α} :
b ∈ iterate f a n ↔ ∃ m < n, b = f^[m] a := by |
simp [List.mem_iff_get, Fin.exists_iff, eq_comm (b := b)]
| false |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 70 | 72 | theorem Convex.thickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (thickening δ s) := by |
rw [← add_ball_zero]
exact hs.add (convex_ball 0 _)
| false |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
#align_import algebraic_geometry.morphisms.quasi_compact from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
noncomputable section
open CategoryTheory CategoryT... | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 114 | 120 | theorem quasiCompact_iff_affineProperty :
QuasiCompact f ↔ targetAffineLocally QuasiCompact.affineProperty f := by |
rw [quasiCompact_iff_forall_affine]
trans ∀ U : Y.affineOpens, IsCompact (f.1.base ⁻¹' (U : Set Y.carrier))
· exact ⟨fun h U => h U U.prop, fun h U hU => h ⟨U, hU⟩⟩
apply forall_congr'
exact fun _ => isCompact_iff_compactSpace
| false |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 161 | 163 | theorem degree_le_of_dvd {p q : R[X]} (h1 : p ∣ q) (h2 : q ≠ 0) : degree p ≤ degree q := by |
rcases h1 with ⟨q, rfl⟩; rw [mul_ne_zero_iff] at h2
exact degree_le_mul_left p h2.2
| false |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section Inter
@[simp]
theorem inter_nil (l : L... | Mathlib/Data/List/Lattice.lean | 139 | 140 | theorem inter_cons_of_not_mem (l₁ : List α) (h : a ∉ l₂) : (a :: l₁) ∩ l₂ = l₁ ∩ l₂ := by |
simp [Inter.inter, List.inter, h]
| false |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 36 | 41 | theorem zero_cpow_eq_nhds {b : ℂ} (hb : b ≠ 0) : (fun x : ℂ => (0 : ℂ) ^ x) =ᶠ[𝓝 b] 0 := by |
suffices ∀ᶠ x : ℂ in 𝓝 b, x ≠ 0 from
this.mono fun x hx ↦ by
dsimp only
rw [zero_cpow hx, Pi.zero_apply]
exact IsOpen.eventually_mem isOpen_ne hb
| false |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 49 | 50 | theorem intDegree_one : intDegree (1 : RatFunc K) = 0 := by |
rw [intDegree, num_one, denom_one, sub_self]
| false |
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set Filter ENNReal Topology NNReal TopologicalSpace
namespace MeasureTh... | Mathlib/MeasureTheory/Measure/Regular.lean | 349 | 353 | theorem _root_.Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular μ] :
μ A = ⨅ (U : Set α) (_ : A ⊆ U) (_ : IsOpen U), μ U := by |
refine le_antisymm (le_iInf₂ fun s hs => le_iInf fun _ => μ.mono hs) ?_
refine le_of_forall_lt' fun r hr => ?_
simpa only [iInf_lt_iff, exists_prop] using A.exists_isOpen_lt_of_lt r hr
| false |
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections
#align_import topology.sheaves.sheaf_condition.equalizer_products from "leanprover-community/mathlib"@"85d6221d32c37e68f05b2e42cde6cee658dae5... | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | 80 | 81 | theorem res_π (i : ι) : res F U ≫ limit.π _ ⟨i⟩ = F.map (Opens.leSupr U i).op := by |
rw [res, limit.lift_π, Fan.mk_π_app]
| false |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 117 | 151 | theorem generateMeasurable_eq_rec (s : Set (Set α)) :
{ t | GenerateMeasurable s t } =
⋃ (i : (Quotient.out (aleph 1).ord).α), generateMeasurableRec s i := by |
ext t; refine ⟨fun ht => ?_, fun ht => ?_⟩
· inhabit ω₁
induction' ht with u hu u _ IH f _ IH
· exact mem_iUnion.2 ⟨default, self_subset_generateMeasurableRec s _ hu⟩
· exact mem_iUnion.2 ⟨default, empty_mem_generateMeasurableRec s _⟩
· rcases mem_iUnion.1 IH with ⟨i, hi⟩
obtain ⟨j, hj⟩ := ex... | false |
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Localization.NumDen
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.rational_root from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
open scoped Polynomial
section ScaleRoots
var... | Mathlib/RingTheory/Polynomial/RationalRoot.lean | 39 | 44 | theorem scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero {p : A[X]} {r : A} {s : M}
(hr : aeval (mk' S r s) p = 0) : aeval (algebraMap A S r) (scaleRoots p s) = 0 := by |
convert scaleRoots_eval₂_eq_zero (algebraMap A S) hr
-- Porting note: added
funext
rw [aeval_def, mk'_spec' _ r s]
| false |
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 89 | 89 | theorem to_real_re (x : ℤ[i]) : ((x.re : ℤ) : ℝ) = (x : ℂ).re := by | simp [toComplex_def]
| false |
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 | 103 | 105 | theorem det_invOf [Invertible A] [Invertible A.det] : (⅟ A).det = ⅟ A.det := by |
letI := detInvertibleOfInvertible A
convert (rfl : _ = ⅟ A.det)
| false |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 115 | 116 | theorem addHaarMeasure_eq_volume : addHaarMeasure Icc01 = volume := by |
convert (addHaarMeasure_unique volume Icc01).symm; simp [Icc01]
| false |
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 | 126 | 144 | theorem ClassGroup.mk_eq_mk_of_coe_ideal {I J : (FractionalIdeal R⁰ <| FractionRing R)ˣ}
{I' J' : Ideal R} (hI : (I : FractionalIdeal R⁰ <| FractionRing R) = I')
(hJ : (J : FractionalIdeal R⁰ <| FractionRing R) = J') :
ClassGroup.mk I = ClassGroup.mk J ↔
∃ x y : R, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' ... |
rw [ClassGroup.mk_eq_mk]
constructor
· rintro ⟨x, rfl⟩
rw [Units.val_mul, hI, coe_toPrincipalIdeal, mul_comm,
spanSingleton_mul_coeIdeal_eq_coeIdeal] at hJ
exact ⟨_, _, sec_fst_ne_zero (R := R) le_rfl x.ne_zero,
sec_snd_ne_zero (R := R) le_rfl (x : FractionRing R), hJ⟩
· rintro ⟨x, y, hx, h... | false |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 136 | 141 | theorem Subgroup.eq_bot_or_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G}
(H : Subgroup G) [hp : Fact (Fintype.card G).Prime] : H = ⊥ ∨ H = ⊤ := by |
classical
have := card_subgroup_dvd_card H
rwa [Nat.card_eq_fintype_card (α := G), Nat.dvd_prime hp.1, ← Nat.card_eq_fintype_card,
← eq_bot_iff_card, card_eq_iff_eq_top] at this
| false |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Int
theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
#align int.... | Mathlib/Data/Int/Order/Units.lean | 45 | 46 | theorem units_coe_mul_self (u : ℤˣ) : (u * u : ℤ) = 1 := by |
rw [← Units.val_mul, units_mul_self, Units.val_one]
| false |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 114 | 119 | theorem isPerfectMatching_iff : M.IsPerfectMatching ↔ ∀ v, ∃! w, M.Adj v w := by |
refine ⟨?_, fun hm => ⟨fun v _ => hm v, fun v => ?_⟩⟩
· rintro ⟨hm, hs⟩ v
exact hm (hs v)
· obtain ⟨w, hw, -⟩ := hm v
exact M.edge_vert hw
| false |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 147 | 148 | theorem preimage_const_add_Icc : (fun x => a + x) ⁻¹' Icc b c = Icc (b - a) (c - a) := by |
simp [← Ici_inter_Iic]
| false |
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic.Abel
open Lean Elab Meta Tactic Qq
initialize registerTraceClass `abel
initialize registerTraceClass `abel.detail
structure Context where
α : Expr
univ :... | Mathlib/Tactic/Abel.lean | 128 | 130 | theorem const_add_term {α} [AddCommMonoid α] (k n x a a') (h : k + a = a') :
k + @term α _ n x a = term n x a' := by |
simp [h.symm, term, add_comm, add_assoc]
| false |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 105 | 114 | theorem count_toMultiset [DecidableEq α] (f : α →₀ ℕ) (a : α) : (toMultiset f).count a = f a :=
calc
(toMultiset f).count a = Finsupp.sum f (fun x n => (n • {x} : Multiset α).count a) := by |
rw [toMultiset_apply]; exact map_sum (Multiset.countAddMonoidHom a) _ f.support
_ = f.sum fun x n => n * ({x} : Multiset α).count a := by simp only [Multiset.count_nsmul]
_ = f a * ({a} : Multiset α).count a :=
sum_eq_single _
(fun a' _ H => by simp only [Multiset.count_singleton, if_false,... | false |
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set Filter ENNReal Topology NNReal TopologicalSpace
namespace MeasureTh... | Mathlib/MeasureTheory/Measure/Regular.lean | 215 | 219 | theorem measure_eq_iSup (H : InnerRegularWRT μ p q) (hU : q U) :
μ U = ⨆ (K) (_ : K ⊆ U) (_ : p K), μ K := by |
refine
le_antisymm (le_of_forall_lt fun r hr => ?_) (iSup₂_le fun K hK => iSup_le fun _ => μ.mono hK)
simpa only [lt_iSup_iff, exists_prop] using H hU r hr
| false |
import Mathlib.Topology.Connected.Basic
open Set Topology
universe u v
variable {α : Type u} {β : Type v} {ι : Type*} {π : ι → Type*} [TopologicalSpace α]
{s t u v : Set α}
section LocallyConnectedSpace
class LocallyConnectedSpace (α : Type*) [TopologicalSpace α] : Prop where
open_connected_basis : ∀ x,... | Mathlib/Topology/Connected/LocallyConnected.lean | 63 | 67 | theorem connectedComponentIn_mem_nhds [LocallyConnectedSpace α] {F : Set α} {x : α} (h : F ∈ 𝓝 x) :
connectedComponentIn F x ∈ 𝓝 x := by |
rw [(LocallyConnectedSpace.open_connected_basis x).mem_iff] at h
rcases h with ⟨s, ⟨h1s, hxs, h2s⟩, hsF⟩
exact mem_nhds_iff.mpr ⟨s, h2s.isPreconnected.subset_connectedComponentIn hxs hsF, h1s, hxs⟩
| false |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Rat
import Mathlib.Data.PNat.Defs
#align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
namespace Rat
o... | Mathlib/Data/Rat/Lemmas.lean | 71 | 76 | theorem den_mk (n d : ℤ) : (n /. d).den = if d = 0 then 1 else d.natAbs / n.gcd d := by |
have (m : ℕ) : Int.natAbs (m + 1) = m + 1 := by
rw [← Nat.cast_one, ← Nat.cast_add, Int.natAbs_cast]
rcases d with ((_ | _) | _) <;>
simp [divInt, mkRat, Rat.normalize, Nat.succPNat, Int.sign, Int.gcd,
if_neg (Nat.cast_add_one_ne_zero _), this]
| false |
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
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 80 | 83 | theorem LpAddConst_zero : LpAddConst 0 = 1 := by |
rw [LpAddConst, if_neg]
intro h
exact lt_irrefl _ h.1
| false |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section Sigma
variable {α : ι → Type*} {β : Type*} (s s₁ s₂ : Finset ι) (... | Mathlib/Data/Finset/Sigma.lean | 64 | 65 | theorem sigma_eq_empty : s.sigma t = ∅ ↔ ∀ i ∈ s, t i = ∅ := by |
simp only [← not_nonempty_iff_eq_empty, sigma_nonempty, not_exists, not_and]
| false |
import Mathlib.Algebra.Order.Pointwise
import Mathlib.Analysis.NormedSpace.SphereNormEquiv
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
open Set Function Metric MeasurableSpace intervalIntegral
open s... | Mathlib/MeasureTheory/Constructions/HaarToSphere.lean | 68 | 70 | theorem toSphere_apply_univ : μ.toSphere univ = dim E * μ (ball 0 1) := by |
nontriviality E
rw [toSphere_apply_univ', measure_diff_null (measure_singleton _)]
| false |
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Trace
#align_import ring_theory.complex from "leanprover-community/mathlib"@"9015c511549dc77a0f8d6eba021d8ac4bba20c82"
open Complex
theorem Algebra.leftMulMatrix_complex (z : ℂ) :
Algebra.leftMulMatrix Complex.basisOn... | Mathlib/RingTheory/Complex.lean | 31 | 34 | theorem Algebra.trace_complex_apply (z : ℂ) : Algebra.trace ℝ ℂ z = 2 * z.re := by |
rw [Algebra.trace_eq_matrix_trace Complex.basisOneI, Algebra.leftMulMatrix_complex,
Matrix.trace_fin_two]
exact (two_mul _).symm
| false |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#align_import combinatorics.simple_graph.degree_sum from "leanprover-community/mathlib"@"90659cbe25e59ec302e2fb92b00e9732160cc620"
open Finset
nam... | Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 98 | 106 | theorem dart_card_eq_twice_card_edges : Fintype.card G.Dart = 2 * G.edgeFinset.card := by |
classical
rw [← card_univ]
rw [@card_eq_sum_card_fiberwise _ _ _ Dart.edge _ G.edgeFinset fun d _h =>
by rw [mem_edgeFinset]; apply Dart.edge_mem]
rw [← mul_comm, sum_const_nat]
intro e h
apply G.dart_edge_fiber_card e
rwa [← mem_edgeFinset]
| false |
import Mathlib.Algebra.CharP.Basic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import algebra.char_p.local_ring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
| Mathlib/Algebra/CharP/LocalRing.lean | 25 | 67 | theorem charP_zero_or_prime_power (R : Type*) [CommRing R] [LocalRing R] (q : ℕ)
[char_R_q : CharP R q] : q = 0 ∨ IsPrimePow q := by |
-- Assume `q := char(R)` is not zero.
apply or_iff_not_imp_left.2
intro q_pos
let K := LocalRing.ResidueField R
haveI RM_char := ringChar.charP K
let r := ringChar K
let n := q.factorization r
-- `r := char(R/m)` is either prime or zero:
cases' CharP.char_is_prime_or_zero K r with r_prime r_zero
· ... | false |
import Mathlib.Topology.VectorBundle.Basic
#align_import topology.vector_bundle.hom from "leanprover-community/mathlib"@"8905e5ed90859939681a725b00f6063e65096d95"
noncomputable section
open scoped Bundle
open Bundle Set ContinuousLinearMap
variable {𝕜₁ : Type*} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type*} [Non... | Mathlib/Topology/VectorBundle/Hom.lean | 92 | 112 | theorem continuousOn_continuousLinearMapCoordChange [VectorBundle 𝕜₁ F₁ E₁] [VectorBundle 𝕜₂ F₂ E₂]
[MemTrivializationAtlas e₁] [MemTrivializationAtlas e₁'] [MemTrivializationAtlas e₂]
[MemTrivializationAtlas e₂'] :
ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂')
(e₁.baseSet ∩ e₂.baseS... |
have h₁ := (compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)).continuous
have h₂ := (ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)).continuous
have h₃ := continuousOn_coordChange 𝕜₁ e₁' e₁
have h₄ := continuousOn_coordChange 𝕜₂ e₂ e₂'
refine ((h₁.comp_continuousOn (h₄.mono ?_)).clm_comp (h₂.comp_continuo... | false |
import Mathlib.Algebra.Group.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
universe u
variable {α : Type u}
class Invertible [Mul α] [One α] (a : α) : Type u where
invOf... | Mathlib/Algebra/Group/Invertible/Defs.lean | 120 | 121 | theorem invOf_mul_self_assoc [Monoid α] (a b : α) [Invertible a] : ⅟ a * (a * b) = b := by |
rw [← mul_assoc, invOf_mul_self, one_mul]
| false |
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 | 62 | 65 | theorem lcm_dvd_iff {a : α} : s.lcm f ∣ a ↔ ∀ b ∈ s, f b ∣ a := by |
apply Iff.trans Multiset.lcm_dvd
simp only [Multiset.mem_map, and_imp, exists_imp]
exact ⟨fun k b hb ↦ k _ _ hb rfl, fun k a' b hb h ↦ h ▸ k _ hb⟩
| false |
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [Algebra F K]
section per... | Mathlib/FieldTheory/PurelyInseparable.lean | 281 | 283 | theorem mem_perfectClosure_iff_pow_mem (q : ℕ) [ExpChar F q] {x : E} :
x ∈ perfectClosure F E ↔ ∃ n : ℕ, x ^ q ^ n ∈ (algebraMap F E).range := by |
rw [mem_perfectClosure_iff, ringExpChar.eq F q]
| false |
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Tactic.Nontriviality
#align_import algebra.group_with_zero.commute from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
#align_import algebra.group_with_zero.power from "leanprover-communi... | Mathlib/Algebra/GroupWithZero/Commute.lean | 27 | 34 | theorem mul_inverse_rev' {a b : M₀} (h : Commute a b) :
inverse (a * b) = inverse b * inverse a := by |
by_cases hab : IsUnit (a * b)
· obtain ⟨⟨a, rfl⟩, b, rfl⟩ := h.isUnit_mul_iff.mp hab
rw [← Units.val_mul, inverse_unit, inverse_unit, inverse_unit, ← Units.val_mul, mul_inv_rev]
obtain ha | hb := not_and_or.mp (mt h.isUnit_mul_iff.mpr hab)
· rw [inverse_non_unit _ hab, inverse_non_unit _ ha, mul_zero]
· ... | false |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Order.Basic
import Mathlib.Order.Hom.Lattice
#align_import rin... | Mathlib/RingTheory/Filtration.lean | 67 | 71 | theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by |
induction' i with _ ih
· simp
· rw [pow_succ', mul_smul, add_assoc, add_comm 1, ← add_assoc]
exact (smul_mono_right _ ih).trans (F.smul_le _)
| false |
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Order.Archimedean
import Mathlib.GroupTheory.Coset
#align_import algebra.periodic from "leanprover-community/mathlib"@"3041... | Mathlib/Algebra/Periodic.lean | 143 | 144 | theorem Periodic.mul_const' [DivisionSemiring α] (h : Periodic f c) (a : α) :
Periodic (fun x => f (x * a)) (c / a) := by | simpa only [div_eq_mul_inv] using h.mul_const a
| false |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.localization.num_denom from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
variable {R : Type*} [CommRing R] (... | Mathlib/RingTheory/Localization/NumDen.lean | 70 | 72 | theorem mk'_num_den' (x : K) : algebraMap A K (num A x) / algebraMap A K (den A x) = x := by |
rw [← mk'_eq_div]
apply mk'_num_den
| false |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
namespace List
variable {α : Type*}
section Sym2
protected def sym2 : List α → List (Sym2 α)
| [] => []
| x :: xs => (x :: xs).map (fun y => s(x, y)) ++ xs.sym2
theorem mem_sym2_cons_iff {x : α} {xs : List α} {z : Sym2 α} :
z ∈ (x :: xs)... | Mathlib/Data/List/Sym.lean | 89 | 92 | theorem mem_sym2_iff {xs : List α} {z : Sym2 α} :
z ∈ xs.sym2 ↔ ∀ y ∈ z, y ∈ xs := by |
refine z.ind (fun a b => ?_)
simp [mk_mem_sym2_iff]
| false |
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#align_import linear_algebra.sesquilinear_form from "leanprover-community/mathlib"@"87c54600fe3cdc7d32ff5b50873ac724d86aef8d"
variable {R R₁ R₂ R₃ M M₁ M₂ M₃... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 246 | 252 | theorem isSymm_iff_eq_flip {B : LinearMap.BilinForm R M} : B.IsSymm ↔ B = B.flip := by |
constructor <;> intro h
· ext
rw [← h, flip_apply, RingHom.id_apply]
intro x y
conv_lhs => rw [h]
rfl
| false |
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {α : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 88 | 89 | theorem rotate'_length (l : List α) : rotate' l l.length = l := by |
rw [rotate'_eq_drop_append_take le_rfl]; simp
| false |
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : ℕ → Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 62 | 63 | theorem nth_of_card_le (hf : (setOf p).Finite) {n : ℕ} (hn : hf.toFinset.card ≤ n) :
nth p n = 0 := by | rw [nth, dif_pos hf, List.getD_eq_default]; rwa [Finset.length_sort]
| false |
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.order.group.min_max from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
section
variable {α : Type*} [Group α] [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)]
-- TODO... | Mathlib/Algebra/Order/Group/MinMax.lean | 63 | 64 | theorem max_div_div_right' (a b c : α) : max (a / c) (b / c) = max a b / c := by |
simpa only [div_eq_mul_inv] using max_mul_mul_right a b c⁻¹
| false |
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function
open Pointwise
variable {R : Type*} {R₁ : Type*} {R₂ : Type*} {R₃ : Type*}
variable {K : Type*}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*... | Mathlib/Algebra/Module/Submodule/Ker.lean | 96 | 100 | theorem ker_sup_ker_le_ker_comp_of_commute {f g : M →ₗ[R] M} (h : Commute f g) :
ker f ⊔ ker g ≤ ker (f ∘ₗ g) := by |
refine sup_le_iff.mpr ⟨?_, ker_le_ker_comp g f⟩
rw [← mul_eq_comp, h.eq, mul_eq_comp]
exact ker_le_ker_comp f g
| false |
import Mathlib.Analysis.NormedSpace.AddTorsorBases
#align_import analysis.convex.intrinsic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open AffineSubspace Set
open scoped Pointwise
variable {𝕜 V W Q P : Type*}
section AddTorsor
variable (𝕜) [Ring 𝕜] [AddCommGroup V] [Modu... | Mathlib/Analysis/Convex/Intrinsic.lean | 136 | 138 | theorem intrinsicInterior_singleton (x : P) : intrinsicInterior 𝕜 ({x} : Set P) = {x} := by |
simpa only [intrinsicInterior, preimage_coe_affineSpan_singleton, interior_univ, image_univ,
Subtype.range_coe] using coe_affineSpan_singleton _ _ _
| false |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w u₁ v₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 130 | 131 | theorem algebraMap_apply (x : R) : algebraMap R A x = algebraMap S A (algebraMap R S x) := by |
rw [algebraMap_eq R S A, RingHom.comp_apply]
| false |
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.terminated_stable from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n m : ℕ}
theorem te... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 45 | 58 | theorem convergents'Aux_stable_step_of_terminated {s : Stream'.Seq <| Pair K}
(terminated_at_n : s.TerminatedAt n) : convergents'Aux s (n + 1) = convergents'Aux s n := by |
change s.get? n = none at terminated_at_n
induction n generalizing s with
| zero => simp only [convergents'Aux, terminated_at_n, Stream'.Seq.head]
| succ n IH =>
cases s_head_eq : s.head with
| none => simp only [convergents'Aux, s_head_eq]
| some gp_head =>
have : s.tail.TerminatedAt n := by... | false |
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 128 | 129 | theorem toIocDiv_zsmul_sub_self (a b : α) : toIocDiv hp a b • p - b = -toIocMod hp a b := by |
rw [toIocMod, neg_sub]
| false |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.AlgebraicTopology.DoldKan.Notations
#align_import algebraic_topology.dold_kan.homotopies from "leanprover-community/mathlib"@"b12099d3b7febf4209824444dd836ef5ad96db55"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Preadditi... | Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean | 111 | 119 | theorem hσ'_eq {q n a m : ℕ} (ha : n = a + q) (hnm : c.Rel m n) :
(hσ' q n m hnm : X _[n] ⟶ X _[m]) =
((-1 : ℤ) ^ a • X.σ ⟨a, Nat.lt_succ_iff.mpr (Nat.le.intro (Eq.symm ha))⟩) ≫
eqToHom (by congr) := by |
simp only [hσ', hσ]
split_ifs
· omega
· have h' := tsub_eq_of_eq_add ha
congr
| false |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
open Part hiding some
def PartENat : Type :=
Part ℕ
#align part_enat ... | Mathlib/Data/Nat/PartENat.lean | 184 | 185 | theorem get_natCast' (x : ℕ) (h : (x : PartENat).Dom) : get (x : PartENat) h = x := by |
rw [← natCast_inj, natCast_get]
| false |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 84 | 89 | theorem OrthonormalBasis.measurePreserving_measurableEquiv (b : OrthonormalBasis ι ℝ F) :
MeasurePreserving b.measurableEquiv volume volume := by |
convert (b.measurableEquiv.symm.measurable.measurePreserving _).symm
rw [← (EuclideanSpace.basisFun ι ℝ).addHaar_eq_volume]
erw [MeasurableEquiv.coe_toEquiv_symm, Basis.map_addHaar _ b.repr.symm.toContinuousLinearEquiv]
exact b.addHaar_eq_volume.symm
| false |
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.symplectic_group from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Matrix
variable {l R : Type*}
namespace Matrix
variable (l) [DecidableEq l] (R) [CommRing R]
section JMatrixLemmas
def J : ... | Mathlib/LinearAlgebra/SymplecticGroup.lean | 43 | 46 | theorem J_transpose : (J l R)ᵀ = -J l R := by |
rw [J, fromBlocks_transpose, ← neg_one_smul R (fromBlocks _ _ _ _ : Matrix (l ⊕ l) (l ⊕ l) R),
fromBlocks_smul, Matrix.transpose_zero, Matrix.transpose_one, transpose_neg]
simp [fromBlocks]
| false |
import Mathlib.Algebra.Group.Support
import Mathlib.Data.Set.Pointwise.SMul
#align_import data.set.pointwise.support from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open Pointwise
open Function Set
section GroupWithZero
variable {α β γ : Type*} [GroupWithZero α] [MulAction α β]
... | Mathlib/Data/Set/Pointwise/Support.lean | 56 | 59 | theorem support_comp_inv_smul₀ [Zero γ] {c : α} (hc : c ≠ 0) (f : β → γ) :
(support fun x ↦ f (c⁻¹ • x)) = c • support f := by |
ext x
simp only [mem_smul_set_iff_inv_smul_mem₀ hc, mem_support]
| false |
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 131 | 157 | theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : f =O[atTop] fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
(fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atTop] fun x : ℝ => |x| ^ (-b) := by |
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved in-line rather than being formulated as a separate lemma.
have claim : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y →
... | false |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 94 | 97 | theorem toReal_lt_toReal (ha : a ≠ ∞) (hb : b ≠ ∞) : a.toReal < b.toReal ↔ a < b := by |
lift a to ℝ≥0 using ha
lift b to ℝ≥0 using hb
norm_cast
| false |
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Tactic.LiftLets
#align_import linear_algebra.clifford_algebra.even_equiv from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36d... | Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean | 82 | 86 | theorem neg_e0_mul_v (m : M) : -(e0 Q * v Q m) = v Q m * e0 Q := by |
refine neg_eq_of_add_eq_zero_right ((ι_mul_ι_add_swap _ _).trans ?_)
dsimp [QuadraticForm.polar]
simp only [add_zero, mul_zero, mul_one, zero_add, neg_zero, QuadraticForm.map_zero,
add_sub_cancel_right, sub_self, map_zero, zero_sub]
| false |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 585 | 587 | theorem toReal_sSup (s : Set ℝ≥0∞) (hf : ∀ r ∈ s, r ≠ ∞) :
(sSup s).toReal = sSup (ENNReal.toReal '' s) := by |
simp only [ENNReal.toReal, toNNReal_sSup s hf, NNReal.coe_sSup, Set.image_image]
| false |
import Mathlib.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
open Finset
section birkhoffAverage
variable (R : Type*) {α M : Type*} [DivisionSemiring R] [AddCommMonoid M] [Module R M]
def birkhoffAverage (f : α → α) (g : α → M) (n : ℕ) (x : α) : M := (n : R)⁻¹ • birkhoffSum f g n x
theorem bir... | Mathlib/Dynamics/BirkhoffSum/Average.lean | 57 | 61 | theorem map_birkhoffAverage (S : Type*) {F N : Type*}
[DivisionSemiring S] [AddCommMonoid N] [Module S N] [FunLike F M N]
[AddMonoidHomClass F M N] (g' : F) (f : α → α) (g : α → M) (n : ℕ) (x : α) :
g' (birkhoffAverage R f g n x) = birkhoffAverage S f (g' ∘ g) n x := by |
simp only [birkhoffAverage, map_inv_natCast_smul g' R S, map_birkhoffSum]
| false |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 83 | 84 | theorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by |
rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]
| false |
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 | 68 | 75 | theorem PiToModule.fromEnd_injective (hb : Submodule.span R (Set.range b) = ⊤) :
Function.Injective (PiToModule.fromEnd R b) := by |
intro x y e
ext m
obtain ⟨m, rfl⟩ : m ∈ LinearMap.range (Fintype.total R R b) := by
rw [(Fintype.range_total R b).trans hb]
exact Submodule.mem_top
exact (LinearMap.congr_fun e m : _)
| false |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 171 | 171 | theorem nonpos_iff {a : SignType} : a ≤ 0 ↔ a = -1 ∨ a = 0 := by | cases a <;> decide
| false |
import Mathlib.Topology.ContinuousFunction.Basic
#align_import topology.continuous_function.cocompact_map from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
universe u v w
open Filter Set
structure CocompactMap (α : Type u) (β : Type v) [TopologicalSpace α] [TopologicalSpace β] e... | Mathlib/Topology/ContinuousFunction/CocompactMap.lean | 185 | 195 | theorem isCompact_preimage [T2Space β] (f : CocompactMap α β) ⦃s : Set β⦄ (hs : IsCompact s) :
IsCompact (f ⁻¹' s) := by |
obtain ⟨t, ht, hts⟩ :=
mem_cocompact'.mp
(by
simpa only [preimage_image_preimage, preimage_compl] using
mem_map.mp
(cocompact_tendsto f <|
mem_cocompact.mpr ⟨s, hs, compl_subset_compl.mpr (image_preimage_subset f _)⟩))
exact
ht.of_isClosed_subset (hs.isClos... | false |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.lagrange from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Polynomial
section PolynomialDetermination
namespace Poly... | Mathlib/LinearAlgebra/Lagrange.lean | 63 | 67 | theorem eq_of_degrees_lt_of_eval_finset_eq (degree_f_lt : f.degree < s.card)
(degree_g_lt : g.degree < s.card) (eval_fg : ∀ x ∈ s, f.eval x = g.eval x) : f = g := by |
rw [← mem_degreeLT] at degree_f_lt degree_g_lt
refine eq_of_degree_sub_lt_of_eval_finset_eq _ ?_ eval_fg
rw [← mem_degreeLT]; exact Submodule.sub_mem _ degree_f_lt degree_g_lt
| false |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 110 | 115 | theorem indicator_le_thickenedIndicatorAux (δ : ℝ) (E : Set α) :
(E.indicator fun _ => (1 : ℝ≥0∞)) ≤ thickenedIndicatorAux δ E := by |
intro a
by_cases h : a ∈ E
· simp only [h, indicator_of_mem, thickenedIndicatorAux_one δ E h, le_refl]
· simp only [h, indicator_of_not_mem, not_false_iff, zero_le]
| false |
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import category_theory.sites.induced_topology from "leanprover-community/mathlib"@"ba43124c37cfe0009bbfc57505f9503ae0e8c1af"
namespace CategoryTheory
universe v u
open Limits Opposite Presieve
section
variable {C : Type*} [Category C] {D : Type*} [Catego... | Mathlib/CategoryTheory/Sites/InducedTopology.lean | 59 | 65 | theorem pushforward_cover_iff_cover_pullback {X : C} (S : Sieve X) :
K _ (S.functorPushforward G) ↔ ∃ T : K (G.obj X), T.val.functorPullback G = S := by |
constructor
· intro hS
exact ⟨⟨_, hS⟩, (Sieve.fullyFaithfulFunctorGaloisCoinsertion G X).u_l_eq S⟩
· rintro ⟨T, rfl⟩
exact Hld T
| false |
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 | 60 | 64 | theorem IsLindelof.compl_mem_sets_of_nhdsWithin (hs : IsLindelof s) {f : Filter X}
[CountableInterFilter f] (hf : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f) : sᶜ ∈ f := by |
refine hs.compl_mem_sets fun x hx ↦ ?_
rw [← disjoint_principal_right, disjoint_right_comm, (basis_sets _).disjoint_iff_left]
exact hf x hx
| false |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 110 | 111 | theorem disjoint_inv_right_iff : Disjoint f g⁻¹ ↔ Disjoint f g := by |
rw [disjoint_comm, disjoint_inv_left_iff, disjoint_comm]
| false |
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 87 | 89 | theorem coeff_coe_powerSeries (x : PowerSeries R) (n : ℕ) :
HahnSeries.coeff (x : LaurentSeries R) n = PowerSeries.coeff R n x := by |
rw [ofPowerSeries_apply_coeff]
| false |
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Algebra.Group.Basic
open scoped Topology Pointwise
open MulAction Set Function
variable {G X : Type*} [TopologicalSpace G] [TopologicalSpace X]
[Group G] [TopologicalGroup G] [MulAction G X]
[SigmaCompactSpace G] [BaireSpace X] [T2Space X]
[Contin... | Mathlib/Topology/Algebra/Group/OpenMapping.lean | 96 | 107 | theorem isOpenMap_smul_of_sigmaCompact (x : X) : IsOpenMap (fun (g : G) ↦ g • x) := by |
/- We have already proved the theorem around the basepoint of the orbit, in
`smul_singleton_mem_nhds_of_sigmaCompact`. The general statement follows around an arbitrary
point by changing basepoints. -/
simp_rw [isOpenMap_iff_nhds_le, Filter.le_map_iff]
intro g U hU
have : (· • x) = (· • (g • x)) ∘ (· * g⁻¹... | false |
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {α β : Type*} [LinearOrder α]
open Function
namespace Set
def projIci (a x : α) : Ici a := ⟨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 77 | 78 | theorem projIcc_of_right_le (hx : b ≤ x) : projIcc a b h x = ⟨b, right_mem_Icc.2 h⟩ := by |
simp [projIcc, hx, h]
| false |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 160 | 162 | theorem star_exp [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] (x : 𝔸) :
star (exp 𝕂 x) = exp 𝕂 (star x) := by |
simp_rw [exp_eq_tsum, ← star_pow, ← star_inv_natCast_smul, ← tsum_star]
| false |
import Mathlib.Order.Filter.Partial
import Mathlib.Topology.Basic
#align_import topology.partial from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter
open Topology
variable {X Y : Type*} [TopologicalSpace X]
theorem rtendsto_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
... | Mathlib/Topology/Partial.lean | 30 | 34 | theorem rtendsto'_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
RTendsto' r l (𝓝 x) ↔ ∀ s, IsOpen s → x ∈ s → r.preimage s ∈ l := by |
rw [rtendsto'_def]
apply all_mem_nhds_filter
apply Rel.preimage_mono
| false |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic
#align_import category_theory.monoidal.of_chosen_finite_products.symmetric from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
namespace CategoryTheory
... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean | 34 | 39 | theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
tensorHom ℬ f g ≫ (Limits.BinaryFan.braiding (ℬ Y Y').isLimit (ℬ Y' Y).isLimit).hom =
(Limits.BinaryFan.braiding (ℬ X X').isLimit (ℬ X' X).isLimit).hom ≫ tensorHom ℬ g f := by |
dsimp [tensorHom, Limits.BinaryFan.braiding]
apply (ℬ _ _).isLimit.hom_ext
rintro ⟨⟨⟩⟩ <;> · dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
| false |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.Topology.TietzeExtension
import Mathlib.Analysis.NormedSpace.HomeomorphBall
import Mathlib.Analysis.NormedSpace.RCLike
universe u u₁ v w
-- this is not an instance because Lean cannot determine `𝕜`.
theorem TietzeExtension.o... | Mathlib/Analysis/Complex/Tietze.lean | 105 | 118 | theorem exists_norm_eq_restrict_eq (f : s →ᵇ E) :
∃ g : X →ᵇ E, ‖g‖ = ‖f‖ ∧ g.restrict s = f := by |
by_cases hf : ‖f‖ = 0; · exact ⟨0, by aesop⟩
have := Metric.instTietzeExtensionClosedBall.{u, v} 𝕜 (0 : E) (by aesop : 0 < ‖f‖)
have hf' x : f x ∈ Metric.closedBall 0 ‖f‖ := by simpa using f.norm_coe_le_norm x
obtain ⟨g, hg_mem, hg⟩ := (f : C(s, E)).exists_forall_mem_restrict_eq hs hf'
simp only [Metric.mem... | false |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 60 | 63 | theorem tanh_half_dist (z w : ℍ) :
tanh (dist z w / 2) = dist (z : ℂ) w / dist (z : ℂ) (conj ↑w) := by |
rw [tanh_eq_sinh_div_cosh, sinh_half_dist, cosh_half_dist, div_div_div_comm, div_self, div_one]
positivity
| false |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section TietzeExten... | Mathlib/Topology/TietzeExtension.lean | 134 | 143 | theorem TietzeExtension.of_retract {Y : Type v} {Z : Type w} [TopologicalSpace Y]
[TopologicalSpace Z] [TietzeExtension.{u, w} Z] (ι : C(Y, Z)) (r : C(Z, Y))
(h : r.comp ι = .id Y) : TietzeExtension.{u, v} Y where
exists_restrict_eq' s hs f := by |
obtain ⟨g, hg⟩ := (ι.comp f).exists_restrict_eq hs
use r.comp g
ext1 x
have := congr(r.comp $(hg))
rw [← r.comp_assoc ι, h, f.id_comp] at this
congrm($this x)
| false |
import Mathlib.Algebra.Group.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
universe u
variable {α : Type u}
class Invertible [Mul α] [One α] (a : α) : Type u where
invOf... | Mathlib/Algebra/Group/Invertible/Defs.lean | 170 | 171 | theorem Invertible.congr [Monoid α] (a b : α) [Invertible a] [Invertible b] (h : a = b) :
⅟a = ⅟b := by | subst h; congr; apply Subsingleton.allEq
| false |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 177 | 178 | theorem mem_Lp_iff_memℒp {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ Memℒp f p μ := by |
simp [mem_Lp_iff_snorm_lt_top, Memℒp, f.stronglyMeasurable.aestronglyMeasurable]
| false |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 96 | 99 | theorem gcd_sub_self_left {m n : ℕ} (h : m ≤ n) : gcd (n - m) m = gcd n m := by |
calc
gcd (n - m) m = gcd (n - m + m) m := by rw [← gcd_add_self_left (n - m) m]
_ = gcd n m := by rw [Nat.sub_add_cancel h]
| false |
import Mathlib.Probability.Kernel.MeasurableIntegral
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.with_density from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
open MeasureTheory ProbabilityTheory
open scoped MeasureTheory ENNReal NNReal
namesp... | Mathlib/Probability/Kernel/WithDensity.lean | 135 | 144 | theorem withDensity_kernel_sum [Countable ι] (κ : ι → kernel α β) (hκ : ∀ i, IsSFiniteKernel (κ i))
(f : α → β → ℝ≥0∞) :
@withDensity _ _ _ _ (kernel.sum κ) (isSFiniteKernel_sum hκ) f =
kernel.sum fun i => withDensity (κ i) f := by |
by_cases hf : Measurable (Function.uncurry f)
· ext1 a
simp_rw [sum_apply, kernel.withDensity_apply _ hf, sum_apply,
withDensity_sum (fun n => κ n a) (f a)]
· simp_rw [withDensity_of_not_measurable _ hf]
exact sum_zero.symm
| false |
import Mathlib.Data.ZMod.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Tactic.IntervalCases
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.GroupTheory.SpecificGroups.Cyclic
#align_import group_theory.specific_groups.quaternion from "leanprover-community/mathlib"@"879155bff5af618b9062cbb2915... | Mathlib/GroupTheory/SpecificGroups/Quaternion.lean | 196 | 196 | theorem xa_sq (i : ZMod (2 * n)) : xa i ^ 2 = a n := by | simp [sq]
| false |
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