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.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 82 | 85 | theorem eval₂_X_pow {n : ℕ} : (X ^ n).eval₂ f x = x ^ n := by |
rw [X_pow_eq_monomial]
convert eval₂_monomial f x (n := n) (r := 1)
simp
|
import Mathlib.Data.Set.Image
import Mathlib.Data.List.GetD
#align_import data.set.list from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4"
open List
variable {α β : Type*} (l : List α)
namespace Set
theorem range_list_map (f : α → β) : range (map f) = { l | ∀ x ∈ l, x ∈ range f } :=... | Mathlib/Data/Set/List.lean | 52 | 57 | theorem range_list_getD (d : α) : (range fun n => l.getD n d) = insert d { x | x ∈ l } :=
calc
(range fun n => l.getD n d) = (fun o : Option α => o.getD d) '' range l.get? := by |
simp only [← range_comp, (· ∘ ·), getD_eq_getD_get?]
_ = insert d { x | x ∈ l } := by
simp only [range_list_get?, image_insert_eq, Option.getD, image_image, image_id']
|
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 | 118 | 134 | theorem exists_Lp_half (p : ℝ≥0∞) {δ : ℝ≥0∞} (hδ : δ ≠ 0) :
∃ η : ℝ≥0∞,
0 < η ∧
∀ (f g : α → E), AEStronglyMeasurable f μ → AEStronglyMeasurable g μ →
snorm f p μ ≤ η → snorm g p μ ≤ η → snorm (f + g) p μ < δ := by |
have :
Tendsto (fun η : ℝ≥0∞ => LpAddConst p * (η + η)) (𝓝[>] 0) (𝓝 (LpAddConst p * (0 + 0))) :=
(ENNReal.Tendsto.const_mul (tendsto_id.add tendsto_id)
(Or.inr (LpAddConst_lt_top p).ne)).mono_left
nhdsWithin_le_nhds
simp only [add_zero, mul_zero] at this
rcases (((tendsto_order.1 this).... |
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 95 | 110 | theorem _root_.TendstoUniformlyOn.cderiv (hF : TendstoUniformlyOn F f φ (cthickening δ K))
(hδ : 0 < δ) (hFn : ∀ᶠ n in φ, ContinuousOn (F n) (cthickening δ K)) :
TendstoUniformlyOn (cderiv δ ∘ F) (cderiv δ f) φ K := by |
rcases φ.eq_or_neBot with rfl | hne
· simp only [TendstoUniformlyOn, eventually_bot, imp_true_iff]
have e1 : ContinuousOn f (cthickening δ K) := TendstoUniformlyOn.continuousOn hF hFn
rw [tendstoUniformlyOn_iff] at hF ⊢
rintro ε hε
filter_upwards [hF (ε * δ) (mul_pos hε hδ), hFn] with n h h' z hz
simp_rw... |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
def symmDiff [Sup α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 347 | 347 | theorem top_symmDiff' : ⊤ ∆ a = ￢a := by | simp [symmDiff]
|
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Topology Manifold
open Set Bundle
section DerivativesProperties
variable
{𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 635 | 640 | theorem Filter.EventuallyEq.mfderiv_eq (hL : f₁ =ᶠ[𝓝 x] f) :
mfderiv I I' f₁ x = (mfderiv I I' f x : _) := by |
have A : f₁ x = f x := (mem_of_mem_nhds hL : _)
rw [← mfderivWithin_univ, ← mfderivWithin_univ]
rw [← nhdsWithin_univ] at hL
exact hL.mfderivWithin_eq (uniqueMDiffWithinAt_univ I) A
|
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 132 | 135 | theorem liftOn₂_mk {α : Type*} (f : M × S → M × S → α)
(wd : ∀ (p q p' q' : M × S), p ≈ p' → q ≈ q' → f p q = f p' q') (m m' : M)
(s s' : S) : liftOn₂ (mk m s) (mk m' s') f wd = f ⟨m, s⟩ ⟨m', s'⟩ := by |
convert Quotient.liftOn₂_mk f wd _ _
|
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 256 | 259 | theorem finsuppTensorFinsupp_single (i : ι) (m : M) (k : κ) (n : N) :
finsuppTensorFinsupp R S M N ι κ (Finsupp.single i m ⊗ₜ Finsupp.single k n) =
Finsupp.single (i, k) (m ⊗ₜ n) := by |
simp [finsuppTensorFinsupp]
|
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
namespace List
open Nat
@[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : ∀ l : List α, drop 1 l =... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 746 | 750 | theorem pairwise_map {l : List α} :
(l.map f).Pairwise R ↔ l.Pairwise fun a b => R (f a) (f b) := by |
induction l
· simp
· simp only [map, pairwise_cons, forall_mem_map_iff, *]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
set_option autoImplicit true
open Function Set Order
open scoped Classical
universe u v w x y
structure Filter (α : Type*) where... | Mathlib/Order/Filter/Basic.lean | 216 | 217 | theorem sInter_mem {s : Set (Set α)} (hfin : s.Finite) : ⋂₀ s ∈ f ↔ ∀ U ∈ s, U ∈ f := by |
rw [sInter_eq_biInter, biInter_mem hfin]
|
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 | 635 | 635 | theorem disconnect_le : S.disconnect ≤ S := by | rw [le_iff]; rintro _ _ _ ⟨⟩; assumption
|
import Mathlib.Data.Matroid.IndepAxioms
open Set
namespace Matroid
variable {α : Type*} {M : Matroid α} {I B X : Set α}
section dual
@[simps] def dualIndepMatroid (M : Matroid α) : IndepMatroid α where
E := M.E
Indep I := I ⊆ M.E ∧ ∃ B, M.Base B ∧ Disjoint I B
indep_empty := ⟨empty_subset M.E, M.exists_b... | Mathlib/Data/Matroid/Dual.lean | 203 | 207 | theorem Base.inter_basis_iff_compl_inter_basis_dual (hB : M.Base B) (hX : X ⊆ M.E := by | aesop_mat):
M.Basis (B ∩ X) X ↔ M✶.Basis ((M.E \ B) ∩ (M.E \ X)) (M.E \ X) := by
refine ⟨hB.compl_inter_basis_of_inter_basis, fun h ↦ ?_⟩
simpa [inter_eq_self_of_subset_right hX, inter_eq_self_of_subset_right hB.subset_ground] using
hB.compl_base_dual.compl_inter_basis_of_inter_basis h
|
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
#align_import analysis.special_functions.gamma.bohr_mollerup from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
set_option linter.uppercaseLean3 false
noncomputable section
open Filter Set MeasureTheory
open scoped Na... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 164 | 173 | theorem convexOn_log_Gamma : ConvexOn ℝ (Ioi 0) (log ∘ Gamma) := by |
refine convexOn_iff_forall_pos.mpr ⟨convex_Ioi _, fun x hx y hy a b ha hb hab => ?_⟩
have : b = 1 - a := by linarith
subst this
simp_rw [Function.comp_apply, smul_eq_mul]
simp only [mem_Ioi] at hx hy
rw [← log_rpow, ← log_rpow, ← log_mul]
· gcongr
exact Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma ... |
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
noncomputable section
open scoped Classical
open Filter
open Function Set Filter
namespace BoxIntegral
variable {ι M : Type*} {... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 88 | 94 | theorem splitLower_def [DecidableEq ι] {i x} (h : x ∈ Ioo (I.lower i) (I.upper i))
(h' : ∀ j, I.lower j < update I.upper i x j :=
(forall_update_iff I.upper fun j y => I.lower j < y).2
⟨h.1, fun j _ => I.lower_lt_upper _⟩) :
I.splitLower i x = (⟨I.lower, update I.upper i x, h'⟩ : Box ι) := by |
simp (config := { unfoldPartialApp := true }) only [splitLower, mk'_eq_coe, min_eq_left h.2.le,
update, and_self]
|
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 296 | 309 | theorem taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x)
(hf : ContDiffOn ℝ n f (Icc x₀ x))
(hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) :
∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x =
iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * ... |
have gcont : ContinuousOn id (Icc x₀ x) := Continuous.continuousOn (by continuity)
have gdiff : ∀ x_1 : ℝ, x_1 ∈ Ioo x₀ x → HasDerivAt id ((fun _ : ℝ => (1 : ℝ)) x_1) x_1 :=
fun _ _ => hasDerivAt_id _
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff fun _ _ => ... |
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.Order.T5
#align_import topology.instances.ennreal from "leanprove... | Mathlib/Topology/Instances/ENNReal.lean | 243 | 256 | theorem hasBasis_nhds_of_ne_top' (xt : x ≠ ∞) :
(𝓝 x).HasBasis (· ≠ 0) (fun ε => Icc (x - ε) (x + ε)) := by |
rcases (zero_le x).eq_or_gt with rfl | x0
· simp_rw [zero_tsub, zero_add, ← bot_eq_zero, Icc_bot, ← bot_lt_iff_ne_bot]
exact nhds_bot_basis_Iic
· refine (nhds_basis_Ioo' ⟨_, x0⟩ ⟨_, xt.lt_top⟩).to_hasBasis ?_ fun ε ε0 => ?_
· rintro ⟨a, b⟩ ⟨ha, hb⟩
rcases exists_between (tsub_pos_of_lt ha) with ⟨ε,... |
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 | 103 | 107 | theorem aeval_modByMonic_eq_self_of_root [Algebra R S] {p q : R[X]} (hq : q.Monic) {x : S}
(hx : aeval x q = 0) : aeval x (p %ₘ q) = aeval x p := by |
--`eval₂_modByMonic_eq_self_of_root` doesn't work here as it needs commutativity
rw [modByMonic_eq_sub_mul_div p hq, _root_.map_sub, _root_.map_mul, hx, zero_mul,
sub_zero]
|
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.Order.Filter.AtTopBot
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mat... | Mathlib/Algebra/Lie/Nilpotent.lean | 504 | 508 | theorem ucs_le_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) :
(⊥ : LieSubmodule R L M).ucs k ≤ N₁ := by |
rw [← ucs_eq_self_of_normalizer_eq_self h k]
mono
simp
|
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Data.Set.Constructions
#align_import topology.category.Compactum from "leanprov... | Mathlib/Topology/Category/Compactum.lean | 282 | 350 | theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by |
-- Notation to be used in this proof.
let fsu := Finset (Set (Ultrafilter X))
let ssu := Set (Set (Ultrafilter X))
let ι : fsu → ssu := fun x ↦ ↑x
let T0 : ssu := { S | ∃ A ∈ F, S = basic A }
let AA := X.str ⁻¹' {x}
let T1 := insert AA T0
let T2 := finiteInterClosure T1
intro cond
-- If F contains ... |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Fil... | Mathlib/Analysis/ODE/Gronwall.lean | 263 | 295 | theorem ODE_solution_unique_of_mem_Icc_left
(hf : ContinuousOn f (Icc a b))
(hf' : ∀ t ∈ Ioc a b, HasDerivWithinAt f (v t (f t)) (Iic t) t)
(hfs : ∀ t ∈ Ioc a b, f t ∈ s t)
(hg : ContinuousOn g (Icc a b))
(hg' : ∀ t ∈ Ioc a b, HasDerivWithinAt g (v t (g t)) (Iic t) t)
(hgs : ∀ t ∈ Ioc a b, g t ∈... |
have hv' t : LipschitzOnWith K (Neg.neg ∘ (v (-t))) (s (-t)) := by
rw [← one_mul K]
exact LipschitzWith.id.neg.comp_lipschitzOnWith (hv _)
have hmt1 : MapsTo Neg.neg (Icc (-b) (-a)) (Icc a b) :=
fun _ ht ↦ ⟨le_neg.mp ht.2, neg_le.mp ht.1⟩
have hmt2 : MapsTo Neg.neg (Ico (-b) (-a)) (Ioc a b) :=
fu... |
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 | 92 | 92 | theorem volume_Ioc {a b : ℝ} : volume (Ioc a b) = ofReal (b - a) := by | simp [volume_val]
|
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 286 | 291 | theorem range_image_pred_top_sub (n : ℕ) :
((Finset.range n).image fun j => n - 1 - j) = Finset.range n := by |
cases n
· rw [range_zero, image_empty]
· rw [Finset.range_eq_Ico, Nat.Ico_image_const_sub_eq_Ico (Nat.zero_le _)]
simp_rw [succ_sub_succ, Nat.sub_zero, Nat.sub_self]
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Multinomial
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped C... | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | 40 | 122 | theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_aux {Du Eu Fu Gu : Type u}
[NormedAddCommGroup Du] [NormedSpace 𝕜 Du] [NormedAddCommGroup Eu] [NormedSpace 𝕜 Eu]
[NormedAddCommGroup Fu] [NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu]
(B : Eu →L[𝕜] Fu →L[𝕜] Gu) {f : Du... |
/- We argue by induction on `n`. The bound is trivial for `n = 0`. For `n + 1`, we write
the `(n+1)`-th derivative as the `n`-th derivative of the derivative `B f g' + B f' g`,
and apply the inductive assumption to each of those two terms. For this induction to make sense,
the spaces of linear maps that ... |
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Order.CauSeq.Completion
#align_import data.real.basic from "leanprover-community/mathlib"@"cb42593171ba005beaaf4549fcfe0dece9ada4c9"
assert_not_exists Finset
assert_not_exists Module
assert_not_exists Submonoid
assert_not_exists FloorRing
structure Real w... | Mathlib/Data/Real/Basic.lean | 328 | 328 | theorem mk_neg {f : CauSeq ℚ abs} : mk (-f) = -mk f := by | simp [mk, ← ofCauchy_neg]
|
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 | 134 | 137 | theorem neg_coe_pi : -(π : Angle) = π := by |
rw [← coe_neg, angle_eq_iff_two_pi_dvd_sub]
use -1
simp [two_mul, sub_eq_add_neg]
|
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Basic
import Mathlib.Order.CompleteLatticeIntervals
#align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
variable {α β ι ι' :... | Mathlib/Order/SupIndep.lean | 92 | 96 | theorem SupIndep.le_sup_iff (hs : s.SupIndep f) (hts : t ⊆ s) (hi : i ∈ s) (hf : ∀ i, f i ≠ ⊥) :
f i ≤ t.sup f ↔ i ∈ t := by |
refine ⟨fun h => ?_, le_sup⟩
by_contra hit
exact hf i (disjoint_self.1 <| (hs hts hi hit).mono_right h)
|
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Tactic.Lift
import Mathlib.Tactic.Monotonicity.Attr
open Function
variable {β G M : Type*}
section Monoid
variable [Monoid M]
section Preorder
variable [Preorder M]
section Left
variable [CovariantClass M M (· * ·) (· ≤ ·)] {x : M}
@[to_additive (... | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 71 | 77 | theorem one_lt_pow' {a : M} (ha : 1 < a) {k : ℕ} (hk : k ≠ 0) : 1 < a ^ k := by |
rcases Nat.exists_eq_succ_of_ne_zero hk with ⟨l, rfl⟩
clear hk
induction' l with l IH
· rw [pow_succ]; simpa using ha
· rw [pow_succ]
exact one_lt_mul'' IH ha
|
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Init.Data.Fin.Basic
#align_import data.list.nodup from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
universe u v
open Nat Function
variable {α : Type u} {β : Type v} {l l₁ l₂ : List α} {r : α → α ... | Mathlib/Data/List/Nodup.lean | 149 | 152 | theorem not_nodup_of_get_eq_of_ne (xs : List α) (n m : Fin xs.length)
(h : xs.get n = xs.get m) (hne : n ≠ m) : ¬Nodup xs := by |
rw [nodup_iff_injective_get]
exact fun hinj => hne (hinj h)
|
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 | 835 | 837 | theorem setIntegral_mono_on_ae (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) :
∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ := by |
refine setIntegral_mono_ae_restrict hf hg ?_; rwa [EventuallyLE, ae_restrict_iff' hs]
|
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "�... | Mathlib/Analysis/RCLike/Basic.lean | 782 | 783 | theorem norm_sq_re_conj_add (x : K) : ‖conj x + x‖ ^ 2 = re (conj x + x) ^ 2 := by |
rw [add_comm, norm_sq_re_add_conj]
|
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 | 945 | 960 | theorem integrable_withDensity_iff_integrable_coe_smul {f : α → ℝ≥0} (hf : Measurable f)
{g : α → E} :
Integrable g (μ.withDensity fun x => f x) ↔ Integrable (fun x => (f x : ℝ) • g x) μ := by |
by_cases H : AEStronglyMeasurable (fun x : α => (f x : ℝ) • g x) μ
· simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, HasFiniteIntegral, H,
true_and_iff]
rw [lintegral_withDensity_eq_lintegral_mul₀' hf.coe_nnreal_ennreal.aemeasurable]
· rw [iff_iff_eq]
congr
ext1 x
si... |
import Mathlib.Algebra.Polynomial.Basic
#align_import data.polynomial.monomial from "leanprover-community/mathlib"@"220f71ba506c8958c9b41bd82226b3d06b0991e8"
noncomputable section
namespace Polynomial
open Polynomial
universe u
variable {R : Type u} {a b : R} {m n : ℕ}
variable [Semiring R] {p q r : R[X]}
th... | Mathlib/Algebra/Polynomial/Monomial.lean | 39 | 56 | theorem card_support_le_one_iff_monomial {f : R[X]} :
Finset.card f.support ≤ 1 ↔ ∃ n a, f = monomial n a := by |
constructor
· intro H
rw [Finset.card_le_one_iff_subset_singleton] at H
rcases H with ⟨n, hn⟩
refine ⟨n, f.coeff n, ?_⟩
ext i
by_cases hi : i = n
· simp [hi, coeff_monomial]
· have : f.coeff i = 0 := by
rw [← not_mem_support_iff]
exact fun hi' => hi (Finset.mem_singleton... |
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Set.Image
import Mathlib.Order.Atoms
import Mathlib.Tactic.ApplyFun
#align_import g... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 144 | 145 | theorem div_mem {x y : M} (hx : x ∈ H) (hy : y ∈ H) : x / y ∈ H := by |
rw [div_eq_mul_inv]; exact mul_mem hx (inv_mem hy)
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
set_option autoImplicit true
universe u v
open Function Order
noncomputable section
def NatOrdinal : Type _ :=
... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 255 | 258 | theorem nadd_le_nadd_left (h : b ≤ c) (a) : a ♯ b ≤ a ♯ c := by |
rcases lt_or_eq_of_le h with (h | rfl)
· exact (nadd_lt_nadd_left h a).le
· exact le_rfl
|
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
open Nat Qq Lean Meta
namespace Mathlib.Meta.NormNum
theorem not_prime_mul_of_ble (a b n : ℕ) (h : a * b = n) (h₁ : a.ble 1 = fals... | Mathlib/Tactic/NormNum/Prime.lean | 90 | 95 | theorem minFacHelper_3 {n k k' : ℕ} (e : k + 2 = k') (nk : (n % k).beq 0 = false)
(h : MinFacHelper n k) : MinFacHelper n k' := by |
refine minFacHelper_1 e h λ h2 ↦ ?_
have nk := Nat.ne_of_beq_eq_false nk
rw [← Nat.dvd_iff_mod_eq_zero, ← h2] at nk
exact nk <| minFac_dvd n
|
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 99 | 107 | theorem opNorm_zero_iff [RingHomIsometric σ₁₂] : ‖f‖ = 0 ↔ f = 0 :=
Iff.intro
(fun hn => ContinuousLinearMap.ext fun x => norm_le_zero_iff.1
(calc
_ ≤ ‖f‖ * ‖x‖ := le_opNorm _ _
_ = _ := by | rw [hn, zero_mul]))
(by
rintro rfl
exact opNorm_zero)
|
import Mathlib.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Integral.DominatedConve... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,244 | 1,249 | theorem integral_deriv_eq_sub' (f) (hderiv : deriv f = f')
(hdiff : ∀ x ∈ uIcc a b, DifferentiableAt ℝ f x) (hcont : ContinuousOn f' (uIcc a b)) :
∫ y in a..b, f' y = f b - f a := by |
rw [← hderiv, integral_deriv_eq_sub hdiff]
rw [hderiv]
exact hcont.intervalIntegrable
|
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Topology.Category.TopCat.Limits.Basic
#align_import topology.category.Top.limits.konig from "leanprover-community/mathlib"@"dbdf71cee7bb20367cb7e37279c08b0c218cf967"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 ... | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | 107 | 124 | theorem partialSections.closed [∀ j : J, T2Space (F.obj j)] {G : Finset J}
(H : Finset (FiniteDiagramArrow G)) : IsClosed (partialSections F H) := by |
have :
partialSections F H =
⋂ (f : FiniteDiagramArrow G) (_ : f ∈ H), {u | F.map f.2.2.2.2 (u f.1) = u f.2.1} := by
ext1
simp only [Set.mem_iInter, Set.mem_setOf_eq]
rfl
rw [this]
apply isClosed_biInter
intro f _
-- Porting note: can't see through forget
have : T2Space ((forget TopCa... |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semirin... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 100 | 102 | theorem trinomial_trailingCoeff (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).trailingCoeff = u := by |
rw [trailingCoeff, trinomial_natTrailingDegree hkm hmn hu, trinomial_trailing_coeff' hkm hmn]
|
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ι : Type*} {R : Type*} {M₁ M₂ N₁ N₂ : Type*} {Mᵢ Nᵢ : ι → Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 328 | 339 | theorem nonneg_pi_iff [Fintype ι] {R} [OrderedCommRing R] [∀ i, Module R (Mᵢ i)]
{Q : ∀ i, QuadraticForm R (Mᵢ i)} : (∀ x, 0 ≤ pi Q x) ↔ ∀ i x, 0 ≤ Q i x := by |
simp_rw [pi, sum_apply, comp_apply, LinearMap.proj_apply]
constructor
-- TODO: does this generalize to a useful lemma independent of `QuadraticForm`?
· intro h i x
classical
convert h (Pi.single i x) using 1
rw [Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hji => ?_, Pi.single_eq_same]... |
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Init.Data.Ordering.Lemmas
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.NormNum
#align_import set_theory.ordinal.notation from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
set_option linter.uppercaseLean3 ... | Mathlib/SetTheory/Ordinal/Notation.lean | 970 | 1,014 | theorem repr_opow (o₁ o₂) [NF o₁] [NF o₂] : repr (o₁ ^ o₂) = repr o₁ ^ repr o₂ := by |
cases' e₁ : split o₁ with a m
cases' nf_repr_split e₁ with N₁ r₁
cases' a with a0 n a'
· cases' m with m
· by_cases h : o₂ = 0 <;> simp [opow_def, opowAux2, opow, e₁, h, r₁]
have := mt repr_inj.1 h
rw [zero_opow this]
· cases' e₂ : split' o₂ with b' k
cases' nf_repr_split' e₂ with _ r... |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.add from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal F... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 353 | 357 | theorem HasFDerivAtFilter.sum (h : ∀ i ∈ u, HasFDerivAtFilter (A i) (A' i) x L) :
HasFDerivAtFilter (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x L := by |
simp only [hasFDerivAtFilter_iff_isLittleO] at *
convert IsLittleO.sum h
simp [ContinuousLinearMap.sum_apply]
|
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
#align_import category_theory.limits.shapes.binary_products from "leanprover-community/mathlib"@"fec1d95fc61c750c1ddbb5b1f7f48b8e811a80d7"
... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 810 | 811 | theorem prod.diag_map {X Y : C} (f : X ⟶ Y) [HasBinaryProduct X X] [HasBinaryProduct Y Y] :
diag X ≫ prod.map f f = f ≫ diag Y := by | simp
|
import Mathlib.Geometry.Manifold.Sheaf.Smooth
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
noncomputable section
universe u
variable {𝕜 : Type u} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
{EM : Type*} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM]
{HM : Type*} [TopologicalSpace HM] (IM : ModelWit... | Mathlib/Geometry/Manifold/Sheaf/LocallyRingedSpace.lean | 43 | 98 | theorem smoothSheafCommRing.isUnit_stalk_iff {x : M}
(f : (smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf.stalk x) :
IsUnit f ↔ f ∉ RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) := by |
constructor
· rintro ⟨⟨f, g, hf, hg⟩, rfl⟩ (h' : smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x f = 0)
simpa [h'] using congr_arg (smoothSheafCommRing.eval IM 𝓘(𝕜) M 𝕜 x) hf
· let S := (smoothSheafCommRing IM 𝓘(𝕜) M 𝕜).presheaf
-- Suppose that `f`, in the stalk at `x`, is nonzero at `x`
rintro (hf :... |
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Instances.Real
#align_import topology.unit_interval from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
... | Mathlib/Topology/UnitInterval.lean | 193 | 198 | theorem mul_pos_mem_iff {a t : ℝ} (ha : 0 < a) : a * t ∈ I ↔ t ∈ Set.Icc (0 : ℝ) (1 / a) := by |
constructor <;> rintro ⟨h₁, h₂⟩ <;> constructor
· exact nonneg_of_mul_nonneg_right h₁ ha
· rwa [le_div_iff ha, mul_comm]
· exact mul_nonneg ha.le h₁
· rwa [le_div_iff ha, mul_comm] at h₂
|
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 64 | 79 | theorem continuous_right_toIcoMod : ContinuousWithinAt (toIcoMod hp a) (Ici x) x := by |
intro s h
rw [Filter.mem_map, mem_nhdsWithin_iff_exists_mem_nhds_inter]
haveI : Nontrivial 𝕜 := ⟨⟨0, p, hp.ne⟩⟩
simp_rw [mem_nhds_iff_exists_Ioo_subset] at h ⊢
obtain ⟨l, u, hxI, hIs⟩ := h
let d := toIcoDiv hp a x • p
have hd := toIcoMod_mem_Ico hp a x
simp_rw [subset_def, mem_inter_iff]
refine ⟨_, ... |
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 | 165 | 165 | theorem nonneg_iff_ne_neg_one {a : SignType} : 0 ≤ a ↔ a ≠ -1 := by | cases a <;> decide
|
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 | 566 | 569 | theorem Integrable.smul_measure {f : α → β} (h : Integrable f μ) {c : ℝ≥0∞} (hc : c ≠ ∞) :
Integrable f (c • μ) := by |
rw [← memℒp_one_iff_integrable] at h ⊢
exact h.smul_measure hc
|
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 | 1,473 | 1,476 | theorem MeasurePreserving.set_lintegral_comp_preimage {mb : MeasurableSpace β} {ν : Measure β}
{g : α → β} (hg : MeasurePreserving g μ ν) {s : Set β} (hs : MeasurableSet s) {f : β → ℝ≥0∞}
(hf : Measurable f) : ∫⁻ a in g ⁻¹' s, f (g a) ∂μ = ∫⁻ b in s, f b ∂ν := by |
rw [← hg.map_eq, set_lintegral_map hs hf hg.measurable]
|
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure α :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 139 | 141 | theorem count_ne_zero' (hs' : s.Nonempty) (s_mble : MeasurableSet s) : count s ≠ 0 := by |
rw [Ne, count_eq_zero_iff' s_mble]
exact hs'.ne_empty
|
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.lhopital from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Filter Set
open scoped Filter Topology Pointwise
variable {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f f... | Mathlib/Analysis/Calculus/LHopital.lean | 144 | 174 | theorem lhopital_zero_atTop_on_Ioi (hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x)
(hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioi a, g' x ≠ 0)
(hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l :=... |
obtain ⟨a', haa', ha'⟩ : ∃ a', a < a' ∧ 0 < a' := ⟨1 + max a 0,
⟨lt_of_le_of_lt (le_max_left a 0) (lt_one_add _),
lt_of_le_of_lt (le_max_right a 0) (lt_one_add _)⟩⟩
have fact1 : ∀ x : ℝ, x ∈ Ioo 0 a'⁻¹ → x ≠ 0 := fun _ hx => (ne_of_lt hx.1).symm
have fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹ := fun _ hx => lt_... |
import Mathlib.Order.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
open Function OrderDual
universe u
variable {ι α β : Type*}
section
variable (α β)
instance Prod.instHImp [HImp α] [HImp β] : HImp (α × β) :=
⟨fun a b => (a.1 ... | Mathlib/Order/Heyting/Basic.lean | 521 | 521 | theorem sdiff_sup_cancel (h : b ≤ a) : a \ b ⊔ b = a := by | rw [sup_comm, sup_sdiff_cancel_right h]
|
import Batteries.Data.UnionFind.Basic
namespace Batteries.UnionFind
@[simp] theorem arr_empty : empty.arr = #[] := rfl
@[simp] theorem parent_empty : empty.parent a = a := rfl
@[simp] theorem rank_empty : empty.rank a = 0 := rfl
@[simp] theorem rootD_empty : empty.rootD a = a := rfl
@[simp] theorem arr_push {m : Un... | .lake/packages/batteries/Batteries/Data/UnionFind/Lemmas.lean | 136 | 139 | theorem equiv_union {self : UnionFind} {x y : Fin self.size} :
Equiv (union self x y) a b ↔
Equiv self a b ∨ Equiv self a x ∧ Equiv self y b ∨ Equiv self a y ∧ Equiv self x b := by |
simp [union]; rw [equiv_link (by simp [← rootD_eq_self, rootD_rootD])]; simp [equiv_find]
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 110 | 114 | theorem natDegree_opRingEquiv (p : R[X]ᵐᵒᵖ) : (opRingEquiv R p).natDegree = (unop p).natDegree := by |
by_cases p0 : p = 0
· simp only [p0, _root_.map_zero, natDegree_zero, unop_zero]
· simp only [p0, natDegree_eq_support_max', Ne, AddEquivClass.map_eq_zero_iff, not_false_iff,
support_opRingEquiv, unop_eq_zero_iff]
|
import Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
#align_import geometry.manifold.bump_function from "leanprover-community/mathlib"@"b018406ad2f2a73223a3a9e198ccae61e6f05318"
universe uE uF uH uM
va... | Mathlib/Geometry/Manifold/BumpFunction.lean | 124 | 128 | theorem support_eq_symm_image :
support f = (extChartAt I c).symm '' (ball (extChartAt I c c) f.rOut ∩ range I) := by |
rw [f.support_eq_inter_preimage, ← extChartAt_source I,
← (extChartAt I c).symm_image_target_inter_eq', inter_comm,
ball_inter_range_eq_ball_inter_target]
|
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 105 | 108 | theorem char_zero_of_expChar_one (p : ℕ) [hp : CharP R p] [hq : ExpChar R 1] : p = 0 := by |
cases hq
· exact CharP.eq R hp inferInstance
· exact False.elim (CharP.char_ne_one R 1 rfl)
|
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalculus
#align_import analysis.special_functions.non_integrable from "leanprover-community/mathlib"@"55ec6e9af7d3e0043f57e394cb06a72f6275273e"
open scoped MeasureTheory Topology Interval NNReal ENNReal
open MeasureTh... | Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean | 52 | 96 | theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
[CompleteSpace E] {f : ℝ → E} {g : ℝ → F}
{k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
(hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
(hfg : deriv f =O[l] g) : ¬Inte... |
intro hgi
obtain ⟨C, hC₀, s, hsl, hsub, hfd, hg⟩ :
∃ (C : ℝ) (_ : 0 ≤ C), ∃ s ∈ l, (∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k) ∧
(∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z) ∧
∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖ := by
rcases hfg.exists_nonneg with ⟨C, C₀, hC⟩
have... |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Coset
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Hom
#align_import group_theory.group_action.... | Mathlib/GroupTheory/GroupAction/Quotient.lean | 120 | 123 | theorem _root_.QuotientGroup.out'_conj_pow_minimalPeriod_mem (a : α) (q : α ⧸ H) :
q.out'⁻¹ * a ^ Function.minimalPeriod (a • ·) q * q.out' ∈ H := by |
rw [mul_assoc, ← QuotientGroup.eq', QuotientGroup.out_eq', ← smul_eq_mul, Quotient.mk_smul_out',
eq_comm, pow_smul_eq_iff_minimalPeriod_dvd]
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 200 | 203 | theorem nhdsWithin_le_nhds {a : α} {s : Set α} : 𝓝[s] a ≤ 𝓝 a := by |
rw [← nhdsWithin_univ]
apply nhdsWithin_le_of_mem
exact univ_mem
|
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 404 | 405 | theorem Icc_subset_Ici_self : Icc a b ⊆ Ici a := by |
simpa [← coe_subset] using Set.Icc_subset_Ici_self
|
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.ContinuousFunction.Ordered
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.homotopy.basic from "leanprover-community/mathlib"@"11c53f174270aa43140c0b26dabce5fc4a253e80"
noncomputable section
universe u v ... | Mathlib/Topology/Homotopy/Basic.lean | 222 | 224 | theorem symm_symm {f₀ f₁ : C(X, Y)} (F : Homotopy f₀ f₁) : F.symm.symm = F := by |
ext
simp
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Function Set Filter
open scoped Topology Filter
variable... | Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 172 | 176 | theorem support_eq : Function.support f = Metric.ball c f.rOut := by |
simp only [toFun, support_comp_eq_preimage, ContDiffBumpBase.support _ _ f.one_lt_rOut_div_rIn]
ext x
simp only [mem_ball_iff_norm, sub_zero, norm_smul, mem_preimage, Real.norm_eq_abs, abs_inv,
abs_of_pos f.rIn_pos, ← div_eq_inv_mul, div_lt_div_right f.rIn_pos]
|
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 86 | 92 | theorem sub [AddGroup β] [TopologicalAddGroup β] {f g : α → β} (hfm : AEStronglyMeasurable' m f μ)
(hgm : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f - g) μ := by |
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
rcases hgm with ⟨g', hg'_meas, hg_ae⟩
refine ⟨f' - g', hf'_meas.sub hg'_meas, hf_ae.mp (hg_ae.mono fun x hx1 hx2 => ?_)⟩
simp_rw [Pi.sub_apply]
rw [hx1, hx2]
|
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Function Set Filter Topology TopologicalSpace
open scoped... | Mathlib/Topology/Separation.lean | 965 | 967 | theorem nhds_inter_eq_singleton_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
(hx : x ∈ s) : ∃ U ∈ 𝓝 x, U ∩ s = {x} := by |
simpa using (𝓝 x).basis_sets.exists_inter_eq_singleton_of_mem_discrete hx
|
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Init.Data.List.Instances
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.list.basic... | Mathlib/Data/List/Basic.lean | 885 | 889 | theorem nthLe_cons_aux {l : List α} {a : α} {n} (hn : n ≠ 0) (h : n < (a :: l).length) :
n - 1 < l.length := by |
contrapose! h
rw [length_cons]
omega
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 68 | 74 | theorem borel_eq_generateFrom_Iic_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Iic (a : ℝ)}) := by |
rw [borel_eq_generateFrom_Ioi_rat, iUnion_singleton_eq_range, iUnion_singleton_eq_range]
refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;>
rintro _ ⟨q, rfl⟩ <;>
dsimp only <;>
[rw [← compl_Iic]; rw [← compl_Ioi]] <;>
exact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q))
|
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 139 | 142 | theorem not_isDiag_mk_of_mem_offDiag {a : α × α} (h : a ∈ s.offDiag) :
¬ (Sym2.mk a).IsDiag := by |
rw [Sym2.isDiag_iff_proj_eq]
exact (mem_offDiag.1 h).2.2
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 556 | 558 | theorem support_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :
(p.concat h).support = p.support.concat w := by |
induction p <;> simp [*, concat_nil]
|
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.fin_range from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
universe u
open List
| Mathlib/Data/List/FinRange.lean | 79 | 82 | theorem Equiv.Perm.map_finRange_perm {n : ℕ} (σ : Equiv.Perm (Fin n)) :
map σ (finRange n) ~ finRange n := by |
rw [perm_ext_iff_of_nodup ((nodup_finRange n).map σ.injective) <| nodup_finRange n]
simpa [mem_map, mem_finRange, true_and_iff, iff_true_iff] using σ.surjective
|
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.ZornAtoms
#align_import order.filter.ultrafilter from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
universe u v
variable {α : Type u} {β : Type v} {γ : Type*}
open Set Filter Function
open scoped Classical
open Filter
inst... | Mathlib/Order/Filter/Ultrafilter.lean | 135 | 135 | theorem compl_mem_iff_not_mem : sᶜ ∈ f ↔ s ∉ f := by | rw [← compl_not_mem_iff, compl_compl]
|
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {α : Type u} {β : Type v} (l :... | Mathlib/Data/List/GetD.lean | 102 | 105 | theorem getD_eq_getD_get? (n : ℕ) : l.getD n d = (l.get? n).getD d := by |
cases Nat.lt_or_ge n l.length with
| inl h => rw [getD_eq_get _ _ h, get?_eq_get h, Option.getD_some]
| inr h => rw [getD_eq_default _ _ h, get?_eq_none.mpr h, Option.getD_none]
|
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.Prime
import Mathlib.Data.List.Prime
import Mathlib.Data.List.Sort
import Mathlib.Data.List.Chain
#align_import data.nat.factors from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
open Bool Subtype
open Nat
namespac... | Mathlib/Data/Nat/Factors.lean | 174 | 185 | theorem factors_unique {n : ℕ} {l : List ℕ} (h₁ : prod l = n) (h₂ : ∀ p ∈ l, Prime p) :
l ~ factors n := by |
refine perm_of_prod_eq_prod ?_ ?_ ?_
· rw [h₁]
refine (prod_factors ?_).symm
rintro rfl
rw [prod_eq_zero_iff] at h₁
exact Prime.ne_zero (h₂ 0 h₁) rfl
· simp_rw [← prime_iff]
exact h₂
· simp_rw [← prime_iff]
exact fun p => prime_of_mem_factors
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike �... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 83 | 108 | theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0 := by |
suffices ∀ a b : ι, a < b → ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0 by
cases' h₀.lt_or_lt with ha hb
· exact this _ _ ha
· rw [inner_eq_zero_symm]
exact this _ _ hb
clear h₀ a b
intro a b h₀
revert a
apply wellFounded_lt.induction b
intro b ih a h₀
simp only [gramSchmidt_def 𝕜 f b... |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingleton
#align_import combinatorics.composition from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open List
variable {n : ℕ}
... | Mathlib/Combinatorics/Enumerative/Composition.lean | 571 | 574 | theorem single_embedding {n : ℕ} (h : 0 < n) (i : Fin n) :
((single n h).embedding (0 : Fin 1)) i = i := by |
ext
simp
|
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 379 | 435 | theorem radius_right_inv_pos_of_radius_pos_aux1 (n : ℕ) (p : ℕ → ℝ) (hp : ∀ k, 0 ≤ p k) {r a : ℝ}
(hr : 0 ≤ r) (ha : 0 ≤ a) :
∑ k ∈ Ico 2 (n + 1),
a ^ k *
∑ c ∈ ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
r ^ c.length * ∏ j, p (c.blocksFun j) ≤
∑ j ∈ I... |
simp_rw [mul_sum]
congr! with k _ c
rw [prod_mul_distrib, prod_mul_distrib, prod_pow_eq_pow_sum, Composition.sum_blocksFun,
prod_const, card_fin]
ring
_ ≤
∑ d ∈ compPartialSumTarget 2 (n + 1) n,
∏ j : Fin d.2.length, r * (a ^ d.2.blocksFun j * p (d.2.blocksFun j)) ... |
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
import Mathlib.Order.PartialSups
#align_import linear_algebra.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
universe u v w x y z u' v' w' y'
variable {R : Type u} {K : Ty... | Mathlib/LinearAlgebra/Prod.lean | 256 | 256 | theorem fst_eq_coprod : fst R M M₂ = coprod LinearMap.id 0 := by | ext; simp
|
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 | 167 | 169 | theorem inter_eq_nil_iff_disjoint : l₁ ∩ l₂ = [] ↔ Disjoint l₁ l₂ := by |
simp only [eq_nil_iff_forall_not_mem, mem_inter_iff, not_and]
rfl
|
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 488 | 489 | theorem _root_.Sbtw.oangle₃₂₁_eq_pi {p₁ p₂ p₃ : P} (h : Sbtw ℝ p₁ p₂ p₃) : ∡ p₃ p₂ p₁ = π := by |
rw [oangle_eq_pi_iff_oangle_rev_eq_pi, ← h.oangle₁₂₃_eq_pi]
|
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 192 | 193 | theorem nhds_atTop [Preorder α] : 𝓝 atTop = ⨅ x : α, 𝓟 (Iic (𝓟 (Ici x))) := by |
simp only [atTop, nhds_iInf, nhds_principal]
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Pointwise Bornology
u... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 80 | 81 | theorem le_infEdist {d} : d ≤ infEdist x s ↔ ∀ y ∈ s, d ≤ edist x y := by |
simp only [infEdist, le_iInf_iff]
|
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.RingTheory.Localization.InvSubmonoid
#align_import algebraic_geometry.AffineScheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 218 | 234 | theorem Scheme.map_PrimeSpectrum_basicOpen_of_affine
(X : Scheme) [IsAffine X] (f : Scheme.Γ.obj (op X)) :
X.isoSpec.hom ⁻¹ᵁ PrimeSpectrum.basicOpen f = X.basicOpen f := by |
rw [← basicOpen_eq_of_affine]
trans
X.isoSpec.hom ⁻¹ᵁ (Scheme.Spec.obj (op (Scheme.Γ.obj (op X)))).basicOpen
((inv (X.isoSpec.hom.1.c.app (op ((Opens.map (inv X.isoSpec.hom).val.base).obj ⊤)))) f)
· congr
rw [← IsIso.inv_eq_inv, IsIso.inv_inv, IsIso.Iso.inv_inv, NatIso.app_hom]
-- Porting not... |
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Filter Finset
noncomputable section
open scoped Topology MeasureTheory ENNReal NNR... | Mathlib/Probability/IdentDistrib.lean | 141 | 145 | theorem ae_snd (h : IdentDistrib f g μ ν) {p : γ → Prop} (pmeas : MeasurableSet {x | p x})
(hp : ∀ᵐ x ∂μ, p (f x)) : ∀ᵐ x ∂ν, p (g x) := by |
apply (ae_map_iff h.aemeasurable_snd pmeas).1
rw [← h.map_eq]
exact (ae_map_iff h.aemeasurable_fst pmeas).2 hp
|
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
open scoped Pointwise
namespace Subgroup
variable {G : Type*} [Group G] (H K : Subgroup G) (S T : Set G)
@[to_additive "`S` and `T` are complements if ... | Mathlib/GroupTheory/Complement.lean | 453 | 456 | theorem equiv_snd_eq_one_of_mem_of_one_mem {g : G} (h1 : 1 ∈ T) (hg : g ∈ S) :
(hST.equiv g).snd = ⟨1, h1⟩ := by |
ext
rw [equiv_snd_eq_inv_mul, equiv_fst_eq_self_of_mem_of_one_mem _ h1 hg, inv_mul_self]
|
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 796 | 798 | theorem UniformSpace.mem_closure_iff_symm_ball {s : Set α} {x} :
x ∈ closure s ↔ ∀ {V}, V ∈ 𝓤 α → SymmetricRel V → (s ∩ ball x V).Nonempty := by |
simp [mem_closure_iff_nhds_basis (hasBasis_nhds x), Set.Nonempty]
|
import Mathlib.FieldTheory.Finite.Basic
#align_import field_theory.chevalley_warning from "leanprover-community/mathlib"@"e001509c11c4d0f549d91d89da95b4a0b43c714f"
universe u v
section FiniteField
open MvPolynomial
open Function hiding eval
open Finset FiniteField
variable {K σ ι : Type*} [Fintype K] [Field ... | Mathlib/FieldTheory/ChevalleyWarning.lean | 53 | 97 | theorem MvPolynomial.sum_eval_eq_zero (f : MvPolynomial σ K)
(h : f.totalDegree < (q - 1) * Fintype.card σ) : ∑ x, eval x f = 0 := by |
haveI : DecidableEq K := Classical.decEq K
calc
∑ x, eval x f = ∑ x : σ → K, ∑ d ∈ f.support, f.coeff d * ∏ i, x i ^ d i := by
simp only [eval_eq']
_ = ∑ d ∈ f.support, ∑ x : σ → K, f.coeff d * ∏ i, x i ^ d i := sum_comm
_ = 0 := sum_eq_zero ?_
intro d hd
obtain ⟨i, hi⟩ : ∃ i, d i < q - 1 := ... |
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.quaternion from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
@[inherit_doc] scoped[Quaternion... | Mathlib/Analysis/Quaternion.lean | 224 | 225 | theorem continuous_im : Continuous fun q : ℍ => q.im := by |
simpa only [← sub_self_re] using continuous_id.sub (continuous_coe.comp continuous_re)
|
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
set_option autoImplicit true
open Filter Set Function
variable {α β γ ι ι' : Type*}
namespace Filter
section Relation
... | Mathlib/Order/LiminfLimsup.lean | 282 | 284 | theorem isCobounded_top : IsCobounded r ⊤ ↔ Nonempty α := by |
simp (config := { contextual := true }) [IsCobounded, eq_univ_iff_forall,
exists_true_iff_nonempty]
|
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 68 | 74 | theorem nnnorm_approxOn_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
{y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
‖approxOn f hf s y₀ h₀ n x - f x‖₊ ≤ ‖f x - y₀‖₊ := by |
have := edist_approxOn_le hf h₀ x n
rw [edist_comm y₀] at this
simp only [edist_nndist, nndist_eq_nnnorm] at this
exact mod_cast this
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingleton
#align_import combinatorics.composition from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open List
variable {n : ℕ}
... | Mathlib/Combinatorics/Enumerative/Composition.lean | 949 | 952 | theorem blocks_sum : c.blocks.sum = n := by |
have : c.blocks.take c.length = c.blocks := take_all_of_le (by simp [blocks])
rw [← this, c.blocks_partial_sum c.length_lt_card_boundaries, c.boundary_length]
rfl
|
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 336 | 337 | theorem ord_compl_of_not_prime (n p : ℕ) (hp : ¬p.Prime) : ord_compl[p] n = n := by |
simp [factorization_eq_zero_of_non_prime n hp]
|
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 146 | 149 | theorem δ_comp_σ_of_le {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : i ≤ Fin.castSucc j) :
X.σ j.succ ≫ X.δ (Fin.castSucc i) = X.δ i ≫ X.σ j := by |
dsimp [δ, σ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_σ_of_le H]
|
import Mathlib.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section LinearOrder
variable [TopologicalSpace α] [LinearOrder α]
section OrderTopology
variable [OrderTopology α]
open List ... | Mathlib/Topology/Order/LeftRightNhds.lean | 173 | 176 | theorem nhdsWithin_Iio_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by |
convert (config := {preTransparency := .default})
nhdsWithin_Ioi_eq_bot_iff (a := OrderDual.toDual a) using 4
exact ofDual_covBy_ofDual_iff
|
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 156 | 162 | theorem exists_finite_submodule_left_of_finite' (s : Set (M₁ ⊗[R] N₁)) (hs : s.Finite) :
∃ (M' : Submodule R M) (hM : M' ≤ M₁), Module.Finite R M' ∧
s ⊆ LinearMap.range ((inclusion hM).rTensor N₁) := by |
obtain ⟨M', _, hM, _, hfin, _, h⟩ := exists_finite_submodule_of_finite' s hs
refine ⟨M', hM, hfin, ?_⟩
rw [← LinearMap.rTensor_comp_lTensor] at h
exact h.trans (LinearMap.range_comp_le_range _ _)
|
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u
open Function
namespace Set
variable {α β : Type u} {s : Set α} ... | Mathlib/Data/Set/Functor.lean | 155 | 156 | theorem image_image_val_eq_restrict_image {δ : Type*} {f : α → δ} : f '' γ = β.restrict f '' γ := by |
ext; simp
|
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.euclidean.circumcenter from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
noncomputable section
open scoped Classical
o... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 48 | 56 | theorem dist_eq_iff_dist_orthogonalProjection_eq {s : AffineSubspace ℝ P} [Nonempty s]
[HasOrthogonalProjection s.direction] {p1 p2 : P} (p3 : P) (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) :
dist p1 p3 = dist p2 p3 ↔
dist p1 (orthogonalProjection s p3) = dist p2 (orthogonalProjection s p3) := by |
rw [← mul_self_inj_of_nonneg dist_nonneg dist_nonneg, ←
mul_self_inj_of_nonneg dist_nonneg dist_nonneg,
dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq p3 hp1,
dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq p3 hp2]
simp
|
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 64 | 83 | theorem integral_comp_smul (f : E → F) (R : ℝ) :
∫ x, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x, f x ∂μ := by |
by_cases hF : CompleteSpace F; swap
· simp [integral, hF]
rcases eq_or_ne R 0 with (rfl | hR)
· simp only [zero_smul, integral_const]
rcases Nat.eq_zero_or_pos (finrank ℝ E) with (hE | hE)
· have : Subsingleton E := finrank_zero_iff.1 hE
have : f = fun _ => f 0 := by ext x; rw [Subsingleton.elim ... |
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 1,347 | 1,355 | theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₁} (h : q₁ ∈ stmts₁ q₂)
(hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by |
induction q₂ with
simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
at h hs
| branch p q₁ q₂ IH₁ IH₂ => rcases h with (rfl | h | h); exacts [hs, IH₁ h hs.1, IH₂ h hs.2]
| goto l => subst h; exact hs
| halt => subst h; trivial
| _ _ q IH => rcases h with (rf... |
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 | 530 | 535 | theorem addHaar_sphere_of_ne_zero (x : E) {r : ℝ} (hr : r ≠ 0) : μ (sphere x r) = 0 := by |
rcases hr.lt_or_lt with (h | h)
· simp only [empty_diff, measure_empty, ← closedBall_diff_ball, closedBall_eq_empty.2 h]
· rw [← closedBall_diff_ball,
measure_diff ball_subset_closedBall measurableSet_ball measure_ball_lt_top.ne,
addHaar_ball_of_pos μ _ h, addHaar_closedBall μ _ h.le, tsub_self]
|
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.Order.T5
#align_import topology.instances.ennreal from "leanprove... | Mathlib/Topology/Instances/ENNReal.lean | 393 | 404 | theorem tendsto_finset_prod_of_ne_top {ι : Type*} {f : ι → α → ℝ≥0∞} {x : Filter α} {a : ι → ℝ≥0∞}
(s : Finset ι) (h : ∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) (h' : ∀ i ∈ s, a i ≠ ∞) :
Tendsto (fun b => ∏ c ∈ s, f c b) x (𝓝 (∏ c ∈ s, a c)) := by |
induction' s using Finset.induction with a s has IH
· simp [tendsto_const_nhds]
simp only [Finset.prod_insert has]
apply Tendsto.mul (h _ (Finset.mem_insert_self _ _))
· right
exact (prod_lt_top fun i hi => h' _ (Finset.mem_insert_of_mem hi)).ne
· exact IH (fun i hi => h _ (Finset.mem_insert_of_mem hi)... |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 230 | 232 | theorem log_neg_natCast_nonneg (n : ℕ) : 0 ≤ log (-n) := by |
rw [← log_neg_eq_log, neg_neg]
exact log_natCast_nonneg _
|
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