Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.ApplyFun
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import category_theory.subobject.basic from "leanprover-community/mathlib"@"70fd9563a21e7b... | Mathlib/CategoryTheory/Subobject/Basic.lean | 264 | 266 | theorem le_of_comm {B : C} {X Y : Subobject B} (f : (X : C) ⟶ (Y : C)) (w : f ≫ Y.arrow = X.arrow) :
X ≤ Y := by |
convert mk_le_mk_of_comm _ w <;> simp
|
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 | 148 | 152 | theorem term_add_termg {α} [AddCommGroup α] (n₁ x a₁ n₂ a₂ n' a')
(h₁ : n₁ + n₂ = n') (h₂ : a₁ + a₂ = a') :
@termg α _ n₁ x a₁ + @termg α _ n₂ x a₂ = termg n' x a' := by |
simp only [termg, h₁.symm, add_zsmul, h₂.symm]
exact add_add_add_comm (n₁ • x) a₁ (n₂ • x) a₂
|
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 | 64 | 65 | theorem pderiv_def [DecidableEq σ] (i : σ) : pderiv i = mkDerivation R (Pi.single i 1) := by |
unfold pderiv; congr!
|
import Mathlib.FieldTheory.Minpoly.Field
#align_import ring_theory.power_basis from "leanprover-community/mathlib"@"d1d69e99ed34c95266668af4e288fc1c598b9a7f"
open Polynomial
open Polynomial
variable {R S T : Type*} [CommRing R] [Ring S] [Algebra R S]
variable {A B : Type*} [CommRing A] [CommRing B] [IsDomain B]... | Mathlib/RingTheory/PowerBasis.lean | 185 | 189 | theorem degree_minpolyGen [Nontrivial A] (pb : PowerBasis A S) :
degree (minpolyGen pb) = pb.dim := by |
unfold minpolyGen
rw [degree_sub_eq_left_of_degree_lt] <;> rw [degree_X_pow]
apply degree_sum_fin_lt
|
import Mathlib.Probability.Kernel.Composition
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.integral_comp_prod from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
noncomputable section
open scoped Topology ENNReal MeasureTheory ProbabilityTheory
op... | Mathlib/Probability/Kernel/IntegralCompProd.lean | 176 | 182 | theorem kernel.lintegral_fn_integral_sub ⦃f g : β × γ → E⦄ (F : E → ℝ≥0∞)
(hf : Integrable f ((κ ⊗ₖ η) a)) (hg : Integrable g ((κ ⊗ₖ η) a)) :
∫⁻ x, F (∫ y, f (x, y) - g (x, y) ∂η (a, x)) ∂κ a =
∫⁻ x, F (∫ y, f (x, y) ∂η (a, x) - ∫ y, g (x, y) ∂η (a, x)) ∂κ a := by |
refine lintegral_congr_ae ?_
filter_upwards [hf.compProd_mk_left_ae, hg.compProd_mk_left_ae] with _ h2f h2g
simp [integral_sub h2f h2g]
|
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 163 | 164 | theorem log_eq_one_iff' {b n : ℕ} : log b n = 1 ↔ b ≤ n ∧ n < b * b := by |
rw [log_eq_iff (Or.inl Nat.one_ne_zero), Nat.pow_add, Nat.pow_one]
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 439 | 443 | theorem isNormal_comap {S : Subgroupoid D} (Sn : IsNormal S) : IsNormal (comap φ S) where
wide c := by | rw [comap, mem_setOf, Functor.map_id]; apply Sn.wide
conj f γ hγ := by
simp_rw [inv_eq_inv f, comap, mem_setOf, Functor.map_comp, Functor.map_inv, ← inv_eq_inv]
exact Sn.conj _ hγ
|
import Mathlib.Order.Filter.Basic
import Mathlib.Data.PFun
#align_import order.filter.partial from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
universe u v w
namespace Filter
variable {α : Type u} {β : Type v} {γ : Type w}
open Filter
def rmap (r : Rel α β) (l : Filter α) : F... | Mathlib/Order/Filter/Partial.lean | 195 | 197 | theorem tendsto_iff_rtendsto (l₁ : Filter α) (l₂ : Filter β) (f : α → β) :
Tendsto f l₁ l₂ ↔ RTendsto (Function.graph f) l₁ l₂ := by |
simp [tendsto_def, Function.graph, rtendsto_def, Rel.core, Set.preimage]
|
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.calculus.fderiv.basic from "leanprover-community/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open To... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 246 | 276 | theorem HasFDerivWithinAt.lim (h : HasFDerivWithinAt f f' s x) {α : Type*} (l : Filter α)
{c : α → 𝕜} {d : α → E} {v : E} (dtop : ∀ᶠ n in l, x + d n ∈ s)
(clim : Tendsto (fun n => ‖c n‖) l atTop) (cdlim : Tendsto (fun n => c n • d n) l (𝓝 v)) :
Tendsto (fun n => c n • (f (x + d n) - f x)) l (𝓝 (f' v)) :=... |
have tendsto_arg : Tendsto (fun n => x + d n) l (𝓝[s] x) := by
conv in 𝓝[s] x => rw [← add_zero x]
rw [nhdsWithin, tendsto_inf]
constructor
· apply tendsto_const_nhds.add (tangentConeAt.lim_zero l clim cdlim)
· rwa [tendsto_principal]
have : (fun y => f y - f x - f' (y - x)) =o[𝓝[s] x] fun y... |
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 70 | 77 | theorem LocallyIntegrable.integrable_of_isBigO_atBot_atTop
[IsMeasurablyGenerated (atBot (α := α))] [IsMeasurablyGenerated (atTop (α := α))]
(hf : LocallyIntegrable f μ)
(ho : f =O[atBot] g) (hg : IntegrableAtFilter g atBot μ)
(ho' : f =O[atTop] g') (hg' : IntegrableAtFilter g' atTop μ) : Integrable f μ... |
refine integrable_iff_integrableAtFilter_atBot_atTop.mpr
⟨⟨ho.integrableAtFilter ?_ hg, ho'.integrableAtFilter ?_ hg'⟩, hf⟩
all_goals exact hf.aestronglyMeasurable.stronglyMeasurableAtFilter
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 125 | 125 | theorem mul_inv_lt_iff (h : 0 < b) : a * b⁻¹ < c ↔ a < b * c := by | rw [mul_comm, inv_mul_lt_iff h]
|
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 | 481 | 482 | theorem isSeparable_closure : IsSeparable (closure s) ↔ IsSeparable s := by |
simp only [IsSeparable, isClosed_closure.closure_subset_iff]
|
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.pfunctor.multivariate.M from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
set_option linter.uppercaseLean3 false
universe u
open MvFunctor
namespace MvPFunctor
open TypeVec... | Mathlib/Data/PFunctor/Multivariate/M.lean | 236 | 267 | theorem M.bisim {α : TypeVec n} (R : P.M α → P.M α → Prop)
(h :
∀ x y,
R x y →
∃ a f f₁ f₂,
M.dest P x = ⟨a, splitFun f f₁⟩ ∧
M.dest P y = ⟨a, splitFun f f₂⟩ ∧ ∀ i, R (f₁ i) (f₂ i))
(x y) (r : R x y) : x = y := by |
cases' x with a₁ f₁
cases' y with a₂ f₂
dsimp [mp] at *
have : a₁ = a₂ := by
refine
PFunctor.M.bisim (fun a₁ a₂ => ∃ x y, R x y ∧ x.1 = a₁ ∧ y.1 = a₂) ?_ _ _
⟨⟨a₁, f₁⟩, ⟨a₂, f₂⟩, r, rfl, rfl⟩
rintro _ _ ⟨⟨a₁, f₁⟩, ⟨a₂, f₂⟩, r, rfl, rfl⟩
rcases h _ _ r with ⟨a', f', f₁', f₂', e₁, e₂, h... |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
#align_import data.set.intervals.surj_on from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
variable {α : Type*} {β : Type*} [LinearOrder α] [PartialOrder β] {f : α → β}
open Set Function
open OrderDual (toDual)... | Mathlib/Order/Interval/Set/SurjOn.lean | 47 | 49 | theorem surjOn_Ioc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
(a b : α) : SurjOn f (Ioc a b) (Ioc (f a) (f b)) := by |
simpa using surjOn_Ico_of_monotone_surjective h_mono.dual h_surj (toDual b) (toDual a)
|
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
#align_import analysis.p_series from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
def SuccDiffBounded (C : ℕ) (u : ℕ → ℕ) : Prop :=... | Mathlib/Analysis/PSeries.lean | 127 | 130 | theorem sum_condensed_le (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
(∑ k ∈ range (n + 1), 2 ^ k • f (2 ^ k)) ≤ f 1 + 2 • ∑ k ∈ Ico 2 (2 ^ n + 1), f k := by |
convert add_le_add_left (nsmul_le_nsmul_right (sum_condensed_le' hf n) 2) (f 1)
simp [sum_range_succ', add_comm, pow_succ', mul_nsmul', sum_nsmul]
|
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory.GrothendieckTopology
open CategoryThe... | Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean | 70 | 76 | theorem sheafificationWhiskerLeftIso_hom_app (P : Cᵒᵖ ⥤ D) (F : D ⥤ E)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
PreservesLimit (W.index P).multicospan F] :
(sheafificationWhiskerLeftIso J P).hom.app F = (J.sheafifyCompIso... |
dsimp [sheafificationWhiskerLeftIso, sheafifyCompIso]
rw [Category.comp_id]
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{𝕜 : Type*} [NontriviallyNormedField 𝕜]
{F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 24 | 28 | theorem iteratedDerivWithin_add (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) :
iteratedDerivWithin n (f + g) s x =
iteratedDerivWithin n f s x + iteratedDerivWithin n g s x := by |
simp_rw [iteratedDerivWithin, iteratedFDerivWithin_add_apply hf hg h hx,
ContinuousMultilinearMap.add_apply]
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 237 | 239 | theorem weightedVSubOfPoint_const_smul (w : ι → k) (p : ι → P) (b : P) (c : k) :
s.weightedVSubOfPoint p b (c • w) = c • s.weightedVSubOfPoint p b w := by |
simp_rw [weightedVSubOfPoint_apply, smul_sum, Pi.smul_apply, smul_smul, smul_eq_mul]
|
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 168 | 169 | theorem Coprime.eq_one_of_dvd {k m : Nat} (H : Coprime k m) (d : k ∣ m) : k = 1 := by |
rw [← H.gcd_eq_one, gcd_eq_left d]
|
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 110 | 150 | theorem card_le_of_separated (s : Finset E) (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
(h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ 5 ^ finrank ℝ E := by |
/- We consider balls of radius `1/2` around the points in `s`. They are disjoint, and all
contained in the ball of radius `5/2`. A volume argument gives `s.card * (1/2)^dim ≤ (5/2)^dim`,
i.e., `s.card ≤ 5^dim`. -/
borelize E
let μ : Measure E := Measure.addHaar
let δ : ℝ := (1 : ℝ) / 2
let ρ : ℝ := (... |
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.QuasiSeparated
#align_import topology.sets.compacts from "leanprover-community/mathlib"@"8c1b484d6a214e059531e22f1be9898ed6c1fd47"
open Set
variable {α β γ : Type*} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
namespace TopologicalSp... | Mathlib/Topology/Sets/Compacts.lean | 125 | 129 | theorem coe_finset_sup {ι : Type*} {s : Finset ι} {f : ι → Compacts α} :
(↑(s.sup f) : Set α) = s.sup fun i => ↑(f i) := by |
refine Finset.cons_induction_on s rfl fun a s _ h => ?_
simp_rw [Finset.sup_cons, coe_sup, sup_eq_union]
congr
|
import Mathlib.Data.Matrix.Block
#align_import linear_algebra.matrix.symmetric from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β n m R : Type*}
namespace Matrix
open Matrix
def IsSymm (A : Matrix n n α) : Prop :=
Aᵀ = A
#align matrix.is_symm Matrix.IsSymm
instance... | Mathlib/LinearAlgebra/Matrix/Symmetric.lean | 86 | 89 | theorem IsSymm.pow [CommSemiring α] [Fintype n] [DecidableEq n] {A : Matrix n n α} (h : A.IsSymm)
(k : ℕ) :
(A ^ k).IsSymm := by |
rw [IsSymm, transpose_pow, h]
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 322 | 323 | theorem le_trailingDegree_X_pow (n : ℕ) : (n : ℕ∞) ≤ trailingDegree (X ^ n : R[X]) := by |
simpa only [C_1, one_mul] using le_trailingDegree_C_mul_X_pow n (1 : R)
|
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 | 234 | 235 | theorem dom_of_le_natCast {x : PartENat} {y : ℕ} (h : x ≤ y) : x.Dom := by |
exact dom_of_le_some h
|
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Join
#align_import data.list.of_fn from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
universe u
variable {α : Type u}
open Nat
namespace List
#noalign list.length_of_fn_aux
@[simp]
theorem length_ofFn_go {n} (f : Fin n ... | Mathlib/Data/List/OfFn.lean | 162 | 164 | theorem ofFn_fin_append {m n} (a : Fin m → α) (b : Fin n → α) :
List.ofFn (Fin.append a b) = List.ofFn a ++ List.ofFn b := by |
simp_rw [ofFn_add, Fin.append_left, Fin.append_right]
|
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
namespace PNat
open Nat
def gcd (n m : ℕ+) : ℕ+ :=
⟨Nat.gcd (n : ℕ) (m : ℕ), Nat.gcd_pos_of_pos_left (m : ℕ) n.pos⟩
#align pnat.gcd PNat.gc... | Mathlib/Data/PNat/Prime.lean | 232 | 235 | theorem Coprime.gcd_mul_left_cancel_right (m : ℕ+) {n k : ℕ+} :
k.Coprime m → m.gcd (k * n) = m.gcd n := by |
intro h; iterate 2 rw [gcd_comm]; symm;
apply Coprime.gcd_mul_left_cancel _ h
|
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
-- Porting note: added to make the syntax work below.
open scoped TensorProd... | Mathlib/RingTheory/Unramified/Basic.lean | 69 | 83 | theorem lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by |
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
... |
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_dimensional from "leanprover-community/mathlib"@"67e606eaea14c7854bdc556bd53d98aefdf76ec0"
noncomputable section
open Affine
section AffineSpace... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 563 | 566 | theorem ne₁₃_of_not_collinear {p₁ p₂ p₃ : P} (h : ¬Collinear k ({p₁, p₂, p₃} : Set P)) :
p₁ ≠ p₃ := by |
rintro rfl
simp [collinear_pair] at h
|
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.RingTheory.Algebraic
#align_import field_theory.ax_grothendieck from "leanprover-community/mathlib"@"4e529b03dd62b7b7d13806c3fb974d9d4848910e"
noncomputable section
open MvPolynomial Finset Function
| Mathlib/FieldTheory/AxGrothendieck.lean | 33 | 66 | theorem ax_grothendieck_of_locally_finite {ι K R : Type*} [Field K] [Finite K] [CommRing R]
[Finite ι] [Algebra K R] [Algebra.IsAlgebraic K R] (ps : ι → MvPolynomial ι R)
(hinj : Injective fun v i => MvPolynomial.eval v (ps i)) :
Surjective fun v i => MvPolynomial.eval v (ps i) := by |
classical
intro v
cases nonempty_fintype ι
/- `s` is the set of all coefficients of the polynomial, as well as all of
the coordinates of `v`, the point I am trying to find the preimage of. -/
let s : Finset R :=
(Finset.biUnion (univ : Finset ι) fun i => (ps i).support.image fun x => coef... |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 73 | 77 | theorem toDistrib_injective : Function.Injective (@toDistrib R) := by |
intro _ _ h
ext x y
· exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
|
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.Set.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.double_coset from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
-- Porting note: removed import
-- import Mathlib.Tac... | Mathlib/GroupTheory/DoubleCoset.lean | 69 | 73 | theorem eq_of_not_disjoint {H K : Subgroup G} {a b : G}
(h : ¬Disjoint (doset a H K) (doset b H K)) : doset a H K = doset b H K := by |
rw [disjoint_comm] at h
have ha : a ∈ doset b H K := mem_doset_of_not_disjoint h
apply doset_eq_of_mem ha
|
import Mathlib.Algebra.MvPolynomial.Funext
import Mathlib.Algebra.Ring.ULift
import Mathlib.RingTheory.WittVector.Basic
#align_import ring_theory.witt_vector.is_poly from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
namespace WittVector
universe u
variable {p : ℕ} {R S : Type u} {σ id... | Mathlib/RingTheory/WittVector/IsPoly.lean | 114 | 122 | theorem poly_eq_of_wittPolynomial_bind_eq' [Fact p.Prime] (f g : ℕ → MvPolynomial (idx × ℕ) ℤ)
(h : ∀ n, bind₁ f (wittPolynomial p _ n) = bind₁ g (wittPolynomial p _ n)) : f = g := by |
ext1 n
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
rw [← Function.funext_iff] at h
replace h :=
congr_arg (fun fam => bind₁ (MvPolynomial.map (Int.castRingHom ℚ) ∘ fam) (xInTermsOfW p ℚ n)) h
simpa only [Function.comp, map_bind₁, map_wittPolynomial, ← bind₁_bind₁,
bind₁_wi... |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 426 | 429 | theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x := by |
convert log_opow_mul_add hb zero_ne_one.symm hb (opow_pos x (zero_lt_one.trans hb))
using 1
rw [add_zero, mul_one]
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 541 | 557 | theorem norm_sub_le_of_mem_A {r x : ℝ} (hr : 0 < r) (ε : ℝ) {L₁ L₂ : F} (h₁ : x ∈ A f L₁ r ε)
(h₂ : x ∈ A f L₂ r ε) : ‖L₁ - L₂‖ ≤ 4 * ε := by |
suffices H : ‖(r / 2) • (L₁ - L₂)‖ ≤ r / 2 * (4 * ε) by
rwa [norm_smul, Real.norm_of_nonneg (half_pos hr).le, mul_le_mul_left (half_pos hr)] at H
calc
‖(r / 2) • (L₁ - L₂)‖ =
‖f (x + r / 2) - f x - (x + r / 2 - x) • L₂ -
(f (x + r / 2) - f x - (x + r / 2 - x) • L₁)‖ := by
simp [sm... |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
#align_import measure_theory.function.conditional_expectation.real from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 317 | 351 | theorem condexp_stronglyMeasurable_mul {f g : α → ℝ} (hf : StronglyMeasurable[m] f)
(hfg : Integrable (f * g) μ) (hg : Integrable g μ) : μ[f * g|m] =ᵐ[μ] f * μ[g|m] := by |
by_cases hm : m ≤ m0; swap; · simp_rw [condexp_of_not_le hm]; rw [mul_zero]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rw [mul_zero]
haveI : SigmaFinite (μ.trim hm) := hμm
obtain ⟨sets, sets_prop, h_univ⟩ := hf.exists_spanning_measurableSet_norm_le hm μ
simp... |
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
namespace PNat
open Nat
def gcd (n m : ℕ+) : ℕ+ :=
⟨Nat.gcd (n : ℕ) (m : ℕ), Nat.gcd_pos_of_pos_left (m : ℕ) n.pos⟩
#align pnat.gcd PNat.gc... | Mathlib/Data/PNat/Prime.lean | 198 | 201 | theorem Coprime.mul_right {k m n : ℕ+} : k.Coprime m → k.Coprime n → k.Coprime (m * n) := by |
repeat rw [← coprime_coe]
rw [mul_coe]
apply Nat.Coprime.mul_right
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,546 | 1,551 | theorem _root_.ENNReal.count_const_le_le_of_tsum_le [MeasurableSingletonClass α] {a : α → ℝ≥0∞}
(a_mble : Measurable a) {c : ℝ≥0∞} (tsum_le_c : ∑' i, a i ≤ c) {ε : ℝ≥0∞} (ε_ne_zero : ε ≠ 0)
(ε_ne_top : ε ≠ ∞) : Measure.count { i : α | ε ≤ a i } ≤ c / ε := by |
rw [← lintegral_count] at tsum_le_c
apply (MeasureTheory.meas_ge_le_lintegral_div a_mble.aemeasurable ε_ne_zero ε_ne_top).trans
exact ENNReal.div_le_div tsum_le_c rfl.le
|
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.Topology.MetricSpace.ThickenedIndicator
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDua... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 169 | 173 | theorem integral_add_compl₀ (hs : NullMeasurableSet s μ) (hfi : Integrable f μ) :
∫ x in s, f x ∂μ + ∫ x in sᶜ, f x ∂μ = ∫ x, f x ∂μ := by |
rw [
← integral_union_ae disjoint_compl_right.aedisjoint hs.compl hfi.integrableOn hfi.integrableOn,
union_compl_self, integral_univ]
|
import Mathlib.Topology.StoneCech
import Mathlib.Topology.Algebra.Semigroup
import Mathlib.Data.Stream.Init
#align_import combinatorics.hindman from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Filter
@[to_additive
"Addition of ultrafilters given by `∀ᶠ m in U+V, p m ↔ ∀ᶠ... | Mathlib/Combinatorics/Hindman.lean | 281 | 296 | theorem FP.finset_prod {M} [CommMonoid M] (a : Stream' M) (s : Finset ℕ) (hs : s.Nonempty) :
(s.prod fun i => a.get i) ∈ FP a := by |
refine FP_drop_subset_FP _ (s.min' hs) ?_
induction' s using Finset.strongInduction with s ih
rw [← Finset.mul_prod_erase _ _ (s.min'_mem hs), ← Stream'.head_drop]
rcases (s.erase (s.min' hs)).eq_empty_or_nonempty with h | h
· rw [h, Finset.prod_empty, mul_one]
exact FP.head _
· apply FP.cons
rw [S... |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 79 | 83 | theorem charmatrix_reindex (e : n ≃ m) :
charmatrix (reindex e e M) = reindex e e (charmatrix M) := by |
ext i j x
by_cases h : i = j
all_goals simp [h]
|
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.adjoin.fg from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
universe u v w
open Subsemiring Ring Submodule
open Pointwise
na... | Mathlib/RingTheory/Adjoin/FG.lean | 129 | 137 | theorem FG.prod {S : Subalgebra R A} {T : Subalgebra R B} (hS : S.FG) (hT : T.FG) :
(S.prod T).FG := by |
obtain ⟨s, hs⟩ := fg_def.1 hS
obtain ⟨t, ht⟩ := fg_def.1 hT
rw [← hs.2, ← ht.2]
exact fg_def.2 ⟨LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1}),
Set.Finite.union (Set.Finite.image _ (Set.Finite.union hs.1 (Set.finite_singleton _)))
(Set.Finite.image _ (Set.Finite.union ht.1 (Set.f... |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Nat.Lattice
#align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2"
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
noncomputable def dist (u v : V)... | Mathlib/Combinatorics/SimpleGraph/Metric.lean | 95 | 96 | theorem dist_eq_zero_of_not_reachable {u v : V} (h : ¬G.Reachable u v) : G.dist u v = 0 := by |
simp [h]
|
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import measure_theory.function.egorov from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open MeasureTheory NNReal ENNReal Topology
namespace MeasureTheory
open Set Filt... | Mathlib/MeasureTheory/Function/Egorov.lean | 146 | 157 | theorem measure_iUnionNotConvergentSeq (hε : 0 < ε) (hf : ∀ n, StronglyMeasurable (f n))
(hg : StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ∞)
(hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
μ (iUnionNotConvergentSeq hε hf hg hsm hs hfg) ≤ ENNReal.ofReal ε := by |
refine le_trans (measure_iUnion_le _) (le_trans
(ENNReal.tsum_le_tsum <| notConvergentSeqLTIndex_spec (half_pos hε) hf hg hsm hs hfg) ?_)
simp_rw [ENNReal.ofReal_mul (half_pos hε).le]
rw [ENNReal.tsum_mul_left, ← ENNReal.ofReal_tsum_of_nonneg, inv_eq_one_div, tsum_geometric_two,
← ENNReal.ofReal_mul (hal... |
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
open Function
open OrderDual (toDual ofDual)
variable {α β : Type*}
namespace Set
section Preorder
v... | Mathlib/Order/Interval/Set/Basic.lean | 199 | 199 | theorem left_mem_Ici : a ∈ Ici a := by | simp
|
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.TensorProduct.Basic
#align_import data.matrix.kronecker from "leanpr... | Mathlib/Data/Matrix/Kronecker.lean | 209 | 219 | theorem kroneckerMapBilinear_mul_mul [CommSemiring R] [Fintype m] [Fintype m']
[NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β] [NonUnitalNonAssocSemiring γ]
[Module R α] [Module R β] [Module R γ] (f : α →ₗ[R] β →ₗ[R] γ)
(h_comm : ∀ a b a' b', f (a * b) (a' * b') = f a a' * f b b') (A : Matrix l ... |
ext ⟨i, i'⟩ ⟨j, j'⟩
simp only [kroneckerMapBilinear_apply_apply, mul_apply, ← Finset.univ_product_univ,
Finset.sum_product, kroneckerMap_apply]
simp_rw [map_sum f, LinearMap.sum_apply, map_sum, h_comm]
|
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Order.Group.Action
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.Basic
#align_import algebra.module.submodule.pointwise from "leanprover-community/... | Mathlib/Algebra/Module/Submodule/Pointwise.lean | 269 | 272 | theorem smul_le_self_of_tower {α : Type*} [Semiring α] [Module α R] [Module α M]
[SMulCommClass α R M] [IsScalarTower α R M] (a : α) (S : Submodule R M) : a • S ≤ S := by |
rintro y ⟨x, hx, rfl⟩
exact smul_of_tower_mem _ a hx
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 207 | 210 | theorem natDegree_derivative_le (p : R[X]) : p.derivative.natDegree ≤ p.natDegree - 1 := by |
by_cases p0 : p.natDegree = 0
· simp [p0, derivative_of_natDegree_zero]
· exact Nat.le_sub_one_of_lt (natDegree_derivative_lt p0)
|
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
@[pp_with_univ]
def TypeVec (n : ℕ) :=
Fin2 n → Type*
#align typevec TypeVec
instance {n} : Inh... | Mathlib/Data/TypeVec.lean | 576 | 579 | theorem snd_diag {α : TypeVec n} : TypeVec.prod.snd ⊚ (prod.diag : α ⟹ _) = id := by |
funext i; induction i with
| fz => rfl
| fs _ i_ih => apply i_ih
|
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 | 411 | 421 | theorem prod_eq_prod_iff :
s ×ˢ t = s₁ ×ˢ t₁ ↔ s = s₁ ∧ t = t₁ ∨ (s = ∅ ∨ t = ∅) ∧ (s₁ = ∅ ∨ t₁ = ∅) := by |
symm
rcases eq_empty_or_nonempty (s ×ˢ t) with h | h
· simp_rw [h, @eq_comm _ ∅, prod_eq_empty_iff, prod_eq_empty_iff.mp h, true_and_iff,
or_iff_right_iff_imp]
rintro ⟨rfl, rfl⟩
exact prod_eq_empty_iff.mp h
rw [prod_eq_prod_iff_of_nonempty h]
rw [nonempty_iff_ne_empty, Ne, prod_eq_empty_iff] at... |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 167 | 170 | theorem hasFiniteIntegral_const_iff {c : β} :
HasFiniteIntegral (fun _ : α => c) μ ↔ c = 0 ∨ μ univ < ∞ := by |
simp [HasFiniteIntegral, lintegral_const, lt_top_iff_ne_top, ENNReal.mul_eq_top,
or_iff_not_imp_left]
|
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 238 | 247 | theorem trace_tensorProduct : compr₂ (mapBilinear R M N M N) (trace R (M ⊗ N)) =
compl₁₂ (lsmul R R : R →ₗ[R] R →ₗ[R] R) (trace R M) (trace R N) := by |
apply
(compl₁₂_inj (show Surjective (dualTensorHom R M M) from (dualTensorHomEquiv R M M).surjective)
(show Surjective (dualTensorHom R N N) from (dualTensorHomEquiv R N N).surjective)).1
ext f m g n
simp only [AlgebraTensorModule.curry_apply, toFun_eq_coe, TensorProduct.curry_apply,
coe_restrict... |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {α : Type v} {β : Type w}
namespace Matrix
def col (w : m → α) : Matrix m Unit α :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 254 | 258 | theorem updateRow_conjTranspose [DecidableEq n] [Star α] :
updateRow Mᴴ j (star c) = (updateColumn M j c)ᴴ := by |
rw [conjTranspose, conjTranspose, transpose_map, transpose_map, updateRow_transpose,
map_updateColumn]
rfl
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 229 | 231 | theorem volume_pi_Ioo_toReal {a b : ι → ℝ} (h : a ≤ b) :
(volume (pi univ fun i => Ioo (a i) (b i))).toReal = ∏ i, (b i - a i) := by |
simp only [volume_pi_Ioo, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]
|
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
#align_import algebraic_geometry.stalks from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
universe v u v' u'
open Opposite Cate... | Mathlib/Geometry/RingedSpace/Stalks.lean | 150 | 162 | theorem comp {X Y Z : PresheafedSpace.{_, _, v} C} (α : X ⟶ Y) (β : Y ⟶ Z) (x : X) :
stalkMap (α ≫ β) x =
(stalkMap β (α.base x) : Z.stalk (β.base (α.base x)) ⟶ Y.stalk (α.base x)) ≫
(stalkMap α x : Y.stalk (α.base x) ⟶ X.stalk x) := by |
dsimp [stalkMap, stalkFunctor, stalkPushforward]
-- We can't use `ext` here due to https://github.com/leanprover/std4/pull/159
refine colimit.hom_ext fun U => ?_
induction U with | h U => ?_
cases U
simp only [whiskeringLeft_obj_obj, comp_obj, op_obj, unop_op, OpenNhds.inclusion_obj,
ι_colimMap_assoc, ... |
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace Relation
open Multiset Prod
variable {α : Type*}
def CutExpand (r : α → α → Prop) (s' s : Multise... | Mathlib/Logic/Hydra.lean | 126 | 133 | theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s := by |
induction s using Multiset.induction with
| empty => exact Acc.intro 0 fun s h ↦ (not_cutExpand_zero s h).elim
| cons a s ihs =>
rw [← s.singleton_add a]
rw [forall_mem_cons] at hs
exact (hs.1.prod_gameAdd <| ihs fun a ha ↦ hs.2 a ha).of_fibration _ (cutExpand_fibration r)
|
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace List
variable {α β : Type*}
section FormPerm
variable [DecidableEq α] (l :... | Mathlib/GroupTheory/Perm/List.lean | 385 | 389 | theorem mem_of_formPerm_ne_self (l : List α) (x : α) (h : formPerm l x ≠ x) : x ∈ l := by |
suffices x ∈ { y | formPerm l y ≠ y } by
rw [← mem_toFinset]
exact support_formPerm_le' _ this
simpa using h
|
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "l... | Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 89 | 91 | theorem exp_blockDiagonal' (v : ∀ i, Matrix (n' i) (n' i) 𝔸) :
exp 𝕂 (blockDiagonal' v) = blockDiagonal' (exp 𝕂 v) := by |
simp_rw [exp_eq_tsum, ← blockDiagonal'_pow, ← blockDiagonal'_smul, ← blockDiagonal'_tsum]
|
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 | 305 | 309 | theorem filter_range_nth_subset_insert (k : ℕ) :
(range (nth p (k + 1))).filter p ⊆ insert (nth p k) ((range (nth p k)).filter p) := by |
intro a ha
simp only [mem_insert, mem_filter, mem_range] at ha ⊢
exact (le_nth_of_lt_nth_succ ha.1 ha.2).eq_or_lt.imp_right fun h => ⟨h, ha.2⟩
|
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 | 778 | 781 | theorem tendstoLocallyUniformly_iff_filter :
TendstoLocallyUniformly F f p ↔ ∀ x, TendstoUniformlyOnFilter F f p (𝓝 x) := by |
simpa [← tendstoLocallyUniformlyOn_univ, ← nhdsWithin_univ] using
@tendstoLocallyUniformlyOn_iff_filter _ _ _ _ F f univ p _
|
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Finset.Antidiagonal
import Mathlib.Data.Finset.Card
import Mathlib.Data.Multiset.NatAntidiagonal
#align_import data.finset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function
namespace Finset
name... | Mathlib/Data/Finset/NatAntidiagonal.lean | 78 | 86 | theorem antidiagonal_succ' (n : ℕ) :
antidiagonal (n + 1) =
cons (n + 1, 0)
((antidiagonal n).map
(Embedding.prodMap (Embedding.refl _) ⟨Nat.succ, Nat.succ_injective⟩))
(by simp) := by |
apply eq_of_veq
rw [cons_val, map_val]
exact Multiset.Nat.antidiagonal_succ'
|
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd"
namespace Polynomial
open Polynomial Finsupp Finset
open... | Mathlib/Algebra/Polynomial/Reverse.lean | 92 | 92 | theorem revAt_zero (N : ℕ) : revAt N 0 = N := by | simp
|
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 | 1,111 | 1,112 | theorem liftRel_think_right (R : α → β → Prop) (s t) : LiftRel R s (think t) ↔ LiftRel R s t := by |
rw [liftRel_destruct_iff, liftRel_destruct_iff]; simp
|
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.basic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical Topology Filter ENNReal ... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 235 | 236 | theorem derivWithin_zero_of_isolated (h : 𝓝[s \ {x}] x = ⊥) : derivWithin f s x = 0 := by |
rw [derivWithin, fderivWithin_zero_of_isolated h, ContinuousLinearMap.zero_apply]
|
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Data.Set.Pointwise.Support
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
import Mathlib.MeasureTheo... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 357 | 362 | theorem y_neg (D : ℝ) (x : E) : y D (-x) = y D x := by |
apply convolution_neg_of_neg_eq
· filter_upwards with x
simp only [w_def, Real.rpow_natCast, mul_inv_rev, smul_neg, u_neg, smul_eq_mul, forall_const]
· filter_upwards with x
simp only [φ, indicator, mem_closedBall, dist_zero_right, norm_neg, forall_const]
|
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "leanprover-community/mathlib"@"9b2660e1b25419042c8da10bf411aa3c67f14383"
open Nat
def ack : ℕ → ℕ → ℕ
| 0, n => n + 1
| m + 1, 0 ... | Mathlib/Computability/Ackermann.lean | 97 | 105 | theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 := by |
induction' n with n IH
· rfl
· rw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2,
Nat.mul_sub_left_distrib, ← Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right]
have H : 2 * 3 ≤ 2 * 2 ^ 3 := by norm_num
apply H.trans
rw [_root_.mul_le_mul_left two_pos]
ex... |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 153 | 154 | theorem U_add_one (n : ℤ) : U R (n + 1) = 2 * X * U R n - U R (n - 1) := by |
linear_combination (norm := ring_nf) U_add_two R (n - 1)
|
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 | 500 | 509 | theorem Measure.FiniteAtFilter.integrableAtFilter {l : Filter α} [IsMeasurablyGenerated l]
(hfm : StronglyMeasurableAtFilter f l μ) (hμ : μ.FiniteAtFilter l)
(hf : l.IsBoundedUnder (· ≤ ·) (norm ∘ f)) : IntegrableAtFilter f l μ := by |
obtain ⟨C, hC⟩ : ∃ C, ∀ᶠ s in l.smallSets, ∀ x ∈ s, ‖f x‖ ≤ C :=
hf.imp fun C hC => eventually_smallSets.2 ⟨_, hC, fun t => id⟩
rcases (hfm.eventually.and (hμ.eventually.and hC)).exists_measurable_mem_of_smallSets with
⟨s, hsl, hsm, hfm, hμ, hC⟩
refine ⟨s, hsl, ⟨hfm, hasFiniteIntegral_restrict_of_bounded... |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 447 | 450 | theorem attach_affineCombination_coe (s : Finset P) (w : P → k) :
s.attach.affineCombination k ((↑) : s → P) (w ∘ (↑)) = s.affineCombination k id w := by |
classical rw [attach_affineCombination_of_injective s w ((↑) : s → P) Subtype.coe_injective,
univ_eq_attach, attach_image_val]
|
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, S... | Mathlib/Order/BooleanAlgebra.lean | 461 | 462 | theorem inf_sdiff_right_comm : x \ z ⊓ y = (x ⊓ y) \ z := by |
rw [inf_comm x, inf_comm, inf_sdiff_assoc]
|
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.UpperLower.Basic
#align_import algebra.order.upper_lower from "leanprover-community/mathlib"@"c0c52abb75074ed8b73a948341f50521fbf43b4c"
open Function Set
open Pointw... | Mathlib/Algebra/Order/UpperLower.lean | 97 | 99 | theorem IsUpperSet.div_left (ht : IsUpperSet t) : IsLowerSet (s / t) := by |
rw [div_eq_mul_inv]
exact ht.inv.mul_left
|
import Mathlib.RingTheory.FiniteType
#align_import ring_theory.rees_algebra from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open Polynomial
def reesAlgebra : Subalgebra... | Mathlib/RingTheory/ReesAlgebra.lean | 98 | 108 | theorem adjoin_monomial_eq_reesAlgebra :
Algebra.adjoin R (Submodule.map (monomial 1 : R →ₗ[R] R[X]) I : Set R[X]) = reesAlgebra I := by |
apply le_antisymm
· apply Algebra.adjoin_le _
rintro _ ⟨r, hr, rfl⟩
exact reesAlgebra.monomial_mem.mpr (by rwa [pow_one])
· intro p hp
rw [p.as_sum_support]
apply Subalgebra.sum_mem _ _
rintro i -
exact monomial_mem_adjoin_monomial (hp i)
|
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import analysis.special_functions.gaussian from "leanprover-community/mathlib"@"7982767093ae38cba236487f9c9dd9cd99f63c16"
noncomputable section
open Real Set Measu... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 57 | 60 | theorem rpow_mul_exp_neg_mul_sq_isLittleO_exp_neg {b : ℝ} (hb : 0 < b) (s : ℝ) :
(fun x : ℝ => x ^ s * exp (-b * x ^ 2)) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) := by |
simp_rw [← rpow_two]
exact rpow_mul_exp_neg_mul_rpow_isLittleO_exp_neg s one_lt_two hb
|
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.foral... | Mathlib/Data/Sum/Basic.lean | 57 | 58 | theorem eq_right_iff_getRight_eq {b : β} : x = inr b ↔ ∃ h, x.getRight h = b := by |
cases x <;> simp
|
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
variable {R : Type*} [CommRing R]
namespace Ideal
open Submodule
variable (R) in
def isPrincipalSubmonoid : Submonoid (Ideal R) where
carrier := ... | Mathlib/RingTheory/Ideal/IsPrincipal.lean | 81 | 84 | theorem associatesEquivIsPrincipal_map_one :
(associatesEquivIsPrincipal R 1 : Ideal R) = 1 := by |
rw [Associates.one_eq_mk_one, ← Associates.quotient_mk_eq_mk, associatesEquivIsPrincipal_apply,
span_singleton_one, one_eq_top]
|
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 140 | 151 | theorem revzip_powersetAux_lemma {α : Type*} [DecidableEq α] (l : List α) {l' : List (Multiset α)}
(H : ∀ ⦃x : _ × _⦄, x ∈ revzip l' → x.1 + x.2 = ↑l) :
revzip l' = l'.map fun x => (x, (l : Multiset α) - x) := by |
have :
Forall₂ (fun (p : Multiset α × Multiset α) (s : Multiset α) => p = (s, ↑l - s)) (revzip l')
((revzip l').map Prod.fst) := by
rw [forall₂_map_right_iff, forall₂_same]
rintro ⟨s, t⟩ h
dsimp
rw [← H h, add_tsub_cancel_left]
rw [← forall₂_eq_eq_eq, forall₂_map_right_iff]
simpa using ... |
import Mathlib.Data.Int.AbsoluteValue
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
#align_import linear_algebra.matrix.absolute_value from "leanprover-community/mathlib"@"ab0a2959c83b06280ef576bc830d4aa5fe8c8e61"
open Matrix
namespace Matrix
open Equiv Finset
variable {R S : Type*} [CommRing R] [Nontr... | Mathlib/LinearAlgebra/Matrix/AbsoluteValue.lean | 64 | 73 | theorem det_sum_smul_le {ι : Type*} (s : Finset ι) {c : ι → R} {A : ι → Matrix n n R}
{abv : AbsoluteValue R S} {x : S} (hx : ∀ k i j, abv (A k i j) ≤ x) {y : S}
(hy : ∀ k, abv (c k) ≤ y) :
abv (det (∑ k ∈ s, c k • A k)) ≤
Nat.factorial (Fintype.card n) • (Finset.card s • y * x) ^ Fintype.card n := by |
simpa only [smul_mul_assoc] using
det_sum_le s fun k i j =>
calc
abv (c k * A k i j) = abv (c k) * abv (A k i j) := abv.map_mul _ _
_ ≤ y * x := mul_le_mul (hy k) (hx k i j) (abv.nonneg _) ((abv.nonneg _).trans (hy k))
|
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Defs
import Mathlib.Order.WithBot
#align_import algebra.order.monoid.with_top ... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 161 | 161 | theorem coe_add_eq_top_iff {y : WithTop α} : ↑x + y = ⊤ ↔ y = ⊤ := by | simp
|
import Mathlib.Init.Align
import Mathlib.Topology.PartialHomeomorph
#align_import geometry.manifold.charted_space from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db"
noncomputable section
open TopologicalSpace Topology
universe u
variable {H : Type u} {H' : Type*} {M : Type*} {M' : Ty... | Mathlib/Geometry/Manifold/ChartedSpace.lean | 529 | 551 | theorem closedUnderRestriction_iff_id_le (G : StructureGroupoid H) :
ClosedUnderRestriction G ↔ idRestrGroupoid ≤ G := by |
constructor
· intro _i
rw [StructureGroupoid.le_iff]
rintro e ⟨s, hs, hes⟩
refine G.mem_of_eqOnSource ?_ hes
convert closedUnderRestriction' G.id_mem hs
-- Porting note: was
-- change s = _ ∩ _
-- rw [hs.interior_eq]
-- simp only [mfld_simps]
ext
· rw [PartialHomeomorph.rest... |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 522 | 525 | theorem prod_divisorsAntidiagonal {M : Type*} [CommMonoid M] (f : ℕ → ℕ → M) {n : ℕ} :
∏ i ∈ n.divisorsAntidiagonal, f i.1 i.2 = ∏ i ∈ n.divisors, f i (n / i) := by |
rw [← map_div_right_divisors, Finset.prod_map]
rfl
|
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 | 49 | 53 | theorem pow_div_pow_eventuallyEq_atBot {p q : ℕ} :
(fun x : 𝕜 => x ^ p / x ^ q) =ᶠ[atBot] fun x => x ^ ((p : ℤ) - q) := by |
apply (eventually_lt_atBot (0 : 𝕜)).mono fun x hx => _
intro x hx
simp [zpow_sub₀ hx.ne]
|
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 52 | 57 | theorem Definable.map_expansion {L' : FirstOrder.Language} [L'.Structure M] (h : A.Definable L s)
(φ : L →ᴸ L') [φ.IsExpansionOn M] : A.Definable L' s := by |
obtain ⟨ψ, rfl⟩ := h
refine ⟨(φ.addConstants A).onFormula ψ, ?_⟩
ext x
simp only [mem_setOf_eq, LHom.realize_onFormula]
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,794 | 1,798 | theorem blsub_congr {o₁ o₂ : Ordinal.{u}} (f : ∀ a < o₁, Ordinal.{max u v}) (ho : o₁ = o₂) :
blsub.{_, v} o₁ f = blsub.{_, v} o₂ fun a h => f a (h.trans_eq ho.symm) := by |
subst ho
-- Porting note: `rfl` is required.
rfl
|
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.RingTheory.Noetherian
#align_import ring_theory.localization.submodule from "leanprover-community/mathlib"@"1ebb20602a8caef435ce47f6373e1aa40851a177"
variable {R : Type*} [CommRing R] (M : Submonoid R) ... | Mathlib/RingTheory/Localization/Submodule.lean | 82 | 84 | theorem coeSubmodule_span_singleton (x : R) :
coeSubmodule S (Ideal.span {x}) = Submodule.span R {(algebraMap R S) x} := by |
rw [coeSubmodule_span, Set.image_singleton]
|
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section
open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
variable {ι κ X X' Y Z α α' β β'... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 955 | 959 | theorem Filter.Tendsto.continuousAt_of_equicontinuousAt {l : Filter ι} [l.NeBot] {F : ι → X → α}
{f : X → α} {x₀ : X} (h₁ : Tendsto F l (𝓝 f)) (h₂ : EquicontinuousAt F x₀) :
ContinuousAt f x₀ := by |
rw [← continuousWithinAt_univ, ← equicontinuousWithinAt_univ, tendsto_pi_nhds] at *
exact continuousWithinAt_of_equicontinuousWithinAt (fun x _ ↦ h₁ x) (h₁ x₀) h₂
|
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.projection from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350df46599bdd7213"
noncomputable section Ring
variable {R : Type*} [Ring R] {E : Type*} [AddCommGroup E] [Module R E]
variable {F : Type*} [Ad... | Mathlib/LinearAlgebra/Projection.lean | 430 | 433 | theorem codRestrict_apply_cod {f : M →ₗ[S] M} (h : IsProj m f) (x : m) : h.codRestrict x = x := by |
ext
rw [codRestrict_apply]
exact h.map_id x x.2
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 104 | 109 | theorem summable_of_summable_hasFDerivAt (hu : Summable u)
(hf : ∀ n x, HasFDerivAt (f n) (f' n x) x) (hf' : ∀ n x, ‖f' n x‖ ≤ u n)
(hf0 : Summable fun n => f n x₀) (x : E) : Summable fun n => f n x := by |
let _ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
exact summable_of_summable_hasFDerivAt_of_isPreconnected hu isOpen_univ isPreconnected_univ
(fun n x _ => hf n x) (fun n x _ => hf' n x) (mem_univ _) hf0 (mem_univ _)
|
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.Set.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.double_coset from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
-- Porting note: removed import
-- import Mathlib.Tac... | Mathlib/GroupTheory/DoubleCoset.lean | 162 | 169 | theorem union_quotToDoset (H K : Subgroup G) : ⋃ q, quotToDoset H K q = Set.univ := by |
ext x
simp only [Set.mem_iUnion, quotToDoset, mem_doset, SetLike.mem_coe, exists_prop, Set.mem_univ,
iff_true_iff]
use mk H K x
obtain ⟨h, k, h3, h4, h5⟩ := mk_out'_eq_mul H K x
refine ⟨h⁻¹, H.inv_mem h3, k⁻¹, K.inv_mem h4, ?_⟩
simp only [h5, Subgroup.coe_mk, ← mul_assoc, one_mul, mul_left_inv, mul_inv... |
import Mathlib.Order.Filter.EventuallyConst
import Mathlib.Order.PartialSups
import Mathlib.Algebra.Module.Submodule.IterateMapComap
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Nilpotent.Lemmas
#align_import ring_theory.noetherian from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3... | Mathlib/RingTheory/Noetherian.lean | 626 | 629 | theorem isNoetherianRing_of_surjective (R) [Ring R] (S) [Ring S] (f : R →+* S)
(hf : Function.Surjective f) [H : IsNoetherianRing R] : IsNoetherianRing S := by |
rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at H ⊢
exact OrderEmbedding.wellFounded (Ideal.orderEmbeddingOfSurjective f hf).dual H
|
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 33 | 33 | theorem log_re (x : ℂ) : x.log.re = x.abs.log := by | simp [log]
|
import Mathlib.Logic.Equiv.Defs
#align_import data.erased from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
universe u
def Erased (α : Sort u) : Sort max 1 u :=
Σ's : α → Prop, ∃ a, (fun b => a = b) = s
#align erased Erased
namespace Erased
@[inline]
def mk {α} (a : α) : Erased... | Mathlib/Data/Erased.lean | 68 | 68 | theorem out_inj {α} (a b : Erased α) (h : a.out = b.out) : a = b := by | simpa using congr_arg mk h
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import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.CategoryTheory.Whiskering
import Mathlib.CategoryTheory.EssentialImage
import Mathlib.Tactic.CategoryTheory.Slice
#align_import category_theory.equivalence from "leanprover-community/mathlib"@"9aba7801eeec... | Mathlib/CategoryTheory/Equivalence.lean | 385 | 386 | theorem cancel_unit_right {X Y : C} (f f' : X ⟶ Y) :
f ≫ e.unit.app Y = f' ≫ e.unit.app Y ↔ f = f' := by | simp only [cancel_mono]
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 220 | 222 | theorem midpoint_eq_smul_add (x y : V) : midpoint R x y = (⅟ 2 : R) • (x + y) := by |
rw [midpoint_eq_iff, pointReflection_apply, vsub_eq_sub, vadd_eq_add, sub_add_eq_add_sub, ←
two_smul R, smul_smul, mul_invOf_self, one_smul, add_sub_cancel_left]
|
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 | 713 | 713 | theorem tail_think (s : WSeq α) : tail (think s) = (tail s).think := by | simp [tail]
|
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
@[simp]
theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
#align list.reduce_option_cons_of_some... | Mathlib/Data/List/ReduceOption.lean | 35 | 41 | theorem reduceOption_map {l : List (Option α)} {f : α → β} :
reduceOption (map (Option.map f) l) = map f (reduceOption l) := by |
induction' l with hd tl hl
· simp only [reduceOption_nil, map_nil]
· cases hd <;>
simpa [true_and_iff, Option.map_some', map, eq_self_iff_true,
reduceOption_cons_of_some] using hl
|
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 | 113 | 114 | theorem convexHull_diam (s : Set E) : Metric.diam (convexHull ℝ s) = Metric.diam s := by |
simp only [Metric.diam, convexHull_ediam]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Sort
#align_import data.polynomial.basic from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
set_option linter.uppercaseLean3 false
noncomputable section
structure ... | Mathlib/Algebra/Polynomial/Basic.lean | 220 | 222 | theorem toFinsupp_neg {R : Type u} [Ring R] (a : R[X]) : (-a).toFinsupp = -a.toFinsupp := by |
cases a
rw [← ofFinsupp_neg]
|
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.convex.body from "leanprover-community/mathlib"@"858a10cf68fd6c06872950fc58c4dcc68d465591"
open scoped Pointwise Topology NNReal
variable {V : Type*}
struc... | Mathlib/Analysis/Convex/Body.lean | 93 | 97 | theorem zero_mem_of_symmetric (K : ConvexBody V) (h_symm : ∀ x ∈ K, - x ∈ K) : 0 ∈ K := by |
obtain ⟨x, hx⟩ := K.nonempty
rw [show 0 = (1/2 : ℝ) • x + (1/2 : ℝ) • (- x) by field_simp]
apply convex_iff_forall_pos.mp K.convex hx (h_symm x hx)
all_goals linarith
|
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 158 | 160 | theorem real_smul_posPart_neg (r : ℝ) (hr : r < 0) :
(r • j).posPart = (-r).toNNReal • j.negPart := by |
rw [real_smul_def, ← smul_negPart, if_neg (not_le.2 hr), neg_posPart]
|
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 | 481 | 488 | theorem ContinuousLinearEquiv.contDiffWithinAt_comp_iff (e : G ≃L[𝕜] E) :
ContDiffWithinAt 𝕜 n (f ∘ e) (e ⁻¹' s) (e.symm x) ↔ ContDiffWithinAt 𝕜 n f s x := by |
constructor
· intro H
simpa [← preimage_comp, (· ∘ ·)] using H.comp_continuousLinearMap (e.symm : E →L[𝕜] G)
· intro H
rw [← e.apply_symm_apply x, ← e.coe_coe] at H
exact H.comp_continuousLinearMap _
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 335 | 338 | theorem cos_eq_real_cos_iff_eq_or_eq_neg {θ : Angle} {ψ : ℝ} :
cos θ = Real.cos ψ ↔ θ = ψ ∨ θ = -ψ := by |
induction θ using Real.Angle.induction_on
exact cos_eq_iff_coe_eq_or_eq_neg
|
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