Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 |
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import Mathlib.Data.List.Duplicate
import Mathlib.Data.List.Sort
#align_import data.list.nodup_equiv_fin from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace List
variable {α : Type*}
section Sublist
theorem sublist_of_orderEmbedding_get?_eq {l l' : List α} (f : ℕ ↪o ℕ)
... | Mathlib/Data/List/NodupEquivFin.lean | 211 | 232 | theorem duplicate_iff_exists_distinct_get {l : List α} {x : α} :
l.Duplicate x ↔
∃ (n m : Fin l.length) (_ : n < m),
x = l.get n ∧ x = l.get m := by |
classical
rw [duplicate_iff_two_le_count, le_count_iff_replicate_sublist,
sublist_iff_exists_fin_orderEmbedding_get_eq]
constructor
· rintro ⟨f, hf⟩
refine ⟨f ⟨0, by simp⟩, f ⟨1, by simp⟩, f.lt_iff_lt.2 (Nat.zero_lt_one), ?_⟩
rw [← hf, ← hf]; simp
· rintro ⟨n, m, hnm, h, h'⟩
r... | 18 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Order.Hom.Bounded
import Mathlib.Algebra.GCDMonoid.Basic
#align_import ring_theory.chain_of_divisors from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {M : Type*} [CancelCommMonoidWithZero... | Mathlib/RingTheory/ChainOfDivisors.lean | 111 | 132 | theorem eq_second_of_chain_of_prime_dvd {p q r : Associates M} {n : ℕ} (hn : n ≠ 0)
{c : Fin (n + 1) → Associates M} (h₁ : StrictMono c)
(h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i) (hp : Prime p) (hr : r ∣ q) (hp' : p ∣ r) :
p = c 1 := by |
cases' n with n
· contradiction
obtain ⟨i, rfl⟩ := h₂.1 (dvd_trans hp' hr)
refine congr_arg c (eq_of_ge_of_not_gt ?_ fun hi => ?_)
· rw [Fin.le_iff_val_le_val, Fin.val_one, Nat.succ_le_iff, ← Fin.val_zero' (n.succ + 1), ←
Fin.lt_iff_val_lt_val, Fin.pos_iff_ne_zero]
rintro rfl
exact hp.not_unit ... | 18 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Prime
import Mathlib.Data.Nat.Digits
import Mathlib.RingTheory.Multiplicity
#align_import data.nat.multiplicity from "l... | Mathlib/Data/Nat/Multiplicity.lean | 138 | 158 | theorem multiplicity_factorial_mul_succ {n p : ℕ} (hp : p.Prime) :
multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1 := by |
have hp' := hp.prime
have h0 : 2 ≤ p := hp.two_le
have h1 : 1 ≤ p * n + 1 := Nat.le_add_left _ _
have h2 : p * n + 1 ≤ p * (n + 1) := by linarith
have h3 : p * n + 1 ≤ p * (n + 1) + 1 := by omega
have hm : multiplicity p (p * n)! ≠ ⊤ := by
rw [Ne, eq_top_iff_not_finite, Classical.not_not, finite_nat_if... | 19 |
import Mathlib.Analysis.Fourier.Inversion
open Real Complex Set MeasureTheory
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
open scoped FourierTransform
private theorem rexp_neg_deriv_aux :
∀ x ∈ univ, HasDerivWithinAt (rexp ∘ Neg.neg) (-rexp (-x)) univ x :=
fun x _ ↦ mul_neg_one (rexp (-x)... | Mathlib/Analysis/MellinInversion.lean | 44 | 67 | theorem mellin_eq_fourierIntegral (f : ℝ → E) {s : ℂ} :
mellin f s = 𝓕 (fun (u : ℝ) ↦ (Real.exp (-s.re * u) • f (Real.exp (-u)))) (s.im / (2 * π)) :=
calc
mellin f s
= ∫ (u : ℝ), Complex.exp (-s * u) • f (Real.exp (-u)) := by |
rw [mellin, ← rexp_neg_image_aux, integral_image_eq_integral_abs_deriv_smul
MeasurableSet.univ rexp_neg_deriv_aux rexp_neg_injOn_aux]
simp [rexp_cexp_aux]
_ = ∫ (u : ℝ), Complex.exp (↑(-2 * π * (u * (s.im / (2 * π)))) * I) •
(Real.exp (-s.re * u) • f (Real.exp (-u))) := by
congr
... | 19 |
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Tactic.ComputeDegree
#align_import linear_algebra.matrix.polynomial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
set_... | Mathlib/LinearAlgebra/Matrix/Polynomial.lean | 39 | 59 | theorem natDegree_det_X_add_C_le (A B : Matrix n n α) :
natDegree (det ((X : α[X]) • A.map C + B.map C : Matrix n n α[X])) ≤ Fintype.card n := by |
rw [det_apply]
refine (natDegree_sum_le _ _).trans ?_
refine Multiset.max_le_of_forall_le _ _ ?_
simp only [forall_apply_eq_imp_iff, true_and_iff, Function.comp_apply, Multiset.map_map,
Multiset.mem_map, exists_imp, Finset.mem_univ_val]
intro g
calc
natDegree (sign g • ∏ i : n, (X • A.map C + B.map... | 19 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Deprecated.Subring
#align_import deprecated.subfield from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
variable {F : Type*} [Field F] (S : Set F)
structure IsSubfield extends IsSubring S : Prop where
inv_mem : ∀ {x : F}, x ∈ S → x⁻... | Mathlib/Deprecated/Subfield.lean | 102 | 123 | theorem closure.isSubfield : IsSubfield (closure S) :=
{ closure.isSubmonoid with
add_mem := by |
intro a b ha hb
rcases id ha with ⟨p, hp, q, hq, rfl⟩
rcases id hb with ⟨r, hr, s, hs, rfl⟩
by_cases hq0 : q = 0
· rwa [hq0, div_zero, zero_add]
by_cases hs0 : s = 0
· rwa [hs0, div_zero, add_zero]
exact ⟨p * s + q * r,
IsAddSubmonoid.add_mem Ring.closure.isSubri... | 19 |
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Topology.Algebra.Polynomial
#align_import analysis.calculus.local_extr from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
namespace Polynomial
theorem card_roots_toFinset_... | Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean | 59 | 86 | theorem card_roots_le_derivative (p : ℝ[X]) :
Multiset.card p.roots ≤ Multiset.card (derivative p).roots + 1 :=
calc
Multiset.card p.roots = ∑ x ∈ p.roots.toFinset, p.roots.count x :=
(Multiset.toFinset_sum_count_eq _).symm
_ = ∑ x ∈ p.roots.toFinset, (p.roots.count x - 1 + 1) :=
(Eq.symm <| F... |
simp only [Finset.sum_add_distrib, Finset.card_eq_sum_ones, count_roots]
_ ≤ (∑ x ∈ p.roots.toFinset, p.derivative.rootMultiplicity x) +
((p.derivative.roots.toFinset \ p.roots.toFinset).card + 1) :=
(add_le_add
(Finset.sum_le_sum fun x _ => rootMultiplicity_sub_one_le_derivative_root... | 19 |
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Geometry.RingedSpace.SheafedSpace
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
#align_import algebraic_geometry.ringed_space from "leanprover-community/mathlib"@"5dc... | Mathlib/Geometry/RingedSpace/Basic.lean | 58 | 79 | theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
(h : IsUnit (X.presheaf.germ x f)) :
∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) := by |
obtain ⟨g', heq⟩ := h.exists_right_inv
obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x.1 g'
let W := U ⊓ V
have hxW : x.1 ∈ W := ⟨x.2, hxV⟩
-- Porting note: `erw` can't write into `HEq`, so this is replaced with another `HEq` in the
-- desired form
replace heq : (X.presheaf.germ ⟨x.val, hxW⟩) ((X.pres... | 19 |
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#align_import data.complex.exponential from "leanprover-community/mathlib"@"a8b2226cfb0a79f5986492053fc49b1a0c6aeffb"
open CauSeq Finset IsAbsoluteValue
open ... | Mathlib/Data/Complex/Exponential.lean | 1,285 | 1,309 | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m ∈ filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m ∈ filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m ∈ range (j - n), (1 / ((m + n).fac... |
refine sum_nbij' (· - n) (· + n) ?_ ?_ ?_ ?_ ?_ <;>
simp (config := { contextual := true }) [lt_tsub_iff_right, tsub_add_cancel_of_le]
_ ≤ ∑ m ∈ range (j - n), ((n.factorial : α) * (n.succ : α) ^ m)⁻¹ := by
simp_rw [one_div]
gcongr
rw [← Nat.cast_pow, ← Nat.cast_mul, Nat.cast_le... | 19 |
import Mathlib.Algebra.Algebra.Basic
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
import Mathlib.Data.Real.Archimedean
#align_import number_theory.class_number.admissible_abs from "leanprover-community/mathlib"@"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b"
namespace AbsoluteValue
open Int
| Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean | 31 | 52 | theorem exists_partition_int (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : ℤ} (hb : b ≠ 0) (A : Fin n → ℤ) :
∃ t : Fin n → Fin ⌈1 / ε⌉₊,
∀ i₀ i₁, t i₀ = t i₁ → ↑(abs (A i₁ % b - A i₀ % b)) < abs b • ε := by |
have hb' : (0 : ℝ) < ↑(abs b) := Int.cast_pos.mpr (abs_pos.mpr hb)
have hbε : 0 < abs b • ε := by
rw [Algebra.smul_def]
exact mul_pos hb' hε
have hfloor : ∀ i, 0 ≤ floor ((A i % b : ℤ) / abs b • ε : ℝ) :=
fun _ ↦ floor_nonneg.mpr (div_nonneg (cast_nonneg.mpr (emod_nonneg _ hb)) hbε.le)
refine ⟨fun ... | 19 |
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 | 98 | 121 | theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
{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) : ¬IntegrableOn g k := by |
let a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
let f' := a ∘ f
have h'd : ∀ᶠ x in l, DifferentiableAt ℝ f' x := by
filter_upwards [hd] with x hx using a.toContinuousLinearMap.differentiableAt.comp x hx
have h'f : Tendsto (fun x => ‖f' x‖) l atTop := hf.congr (fun x ↦ by sim... | 19 |
import Mathlib.Algebra.Module.Torsion
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import algebra.module.dedekind_domain from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
universe u v
variable {R : Type u} [CommRing R] [IsDomain R] {M : Type v} [AddCommGroup M] [Module R M]
... | Mathlib/Algebra/Module/DedekindDomain.lean | 37 | 59 | theorem isInternal_prime_power_torsion_of_is_torsion_by_ideal {I : Ideal R} (hI : I ≠ ⊥)
(hM : Module.IsTorsionBySet R M I) :
DirectSum.IsInternal fun p : (factors I).toFinset =>
torsionBySet R M (p ^ (factors I).count ↑p : Ideal R) := by |
let P := factors I
have prime_of_mem := fun p (hp : p ∈ P.toFinset) =>
prime_of_factor p (Multiset.mem_toFinset.mp hp)
apply torsionBySet_isInternal (p := fun p => p ^ P.count p) _
· convert hM
rw [← Finset.inf_eq_iInf, IsDedekindDomain.inf_prime_pow_eq_prod, ← Finset.prod_multiset_count,
← assoc... | 19 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"b07688016d62f81d14508ff339ea3415558d6353"
universe u v
namespace SimpleGraph
open Walk
variable {V : Type u} (G : SimpleGraph V)
def IsAcy... | Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 134 | 154 | theorem isTree_iff_existsUnique_path :
G.IsTree ↔ Nonempty V ∧ ∀ v w : V, ∃! p : G.Walk v w, p.IsPath := by |
classical
rw [isTree_iff, isAcyclic_iff_path_unique]
constructor
· rintro ⟨hc, hu⟩
refine ⟨hc.nonempty, ?_⟩
intro v w
let q := (hc v w).some.toPath
use q
simp only [true_and_iff, Path.isPath]
intro p hp
specialize hu ⟨p, hp⟩ q
exact Subtype.ext_iff.mp hu
· rintro ⟨hV, h⟩
r... | 19 |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.indexes from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
universe u v
open Function
namespace List
variable {α : Type u} {β : Type v}
section MapIdx
-- Porting n... | Mathlib/Data/List/Indexes.lean | 109 | 129 | theorem mapIdxGo_append : ∀ (f : ℕ → α → β) (l₁ l₂ : List α) (arr : Array β),
mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (List.toArray (mapIdx.go f l₁ arr)) := by |
intros f l₁ l₂ arr
generalize e : (l₁ ++ l₂).length = len
revert l₁ l₂ arr
induction' len with len ih <;> intros l₁ l₂ arr h
· have l₁_nil : l₁ = [] := by
cases l₁
· rfl
· contradiction
have l₂_nil : l₂ = [] := by
cases l₂
· rfl
· rw [List.length_append] at h; contradi... | 19 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 76 | 97 | theorem sinZeta_two_mul_nat_add_one (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
sinZeta x (2 * k + 1) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / (2 * k + 1)! *
((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [← (hasSum_nat_sinZeta x (?_ : 1 < re (2 * k + 1))).tsum_eq]
refine Eq.trans ?_ <| (congr_arg ofReal' (hasSum_one_div_nat_pow_mul_sin hk hx).tsum_eq).trans ?_
· rw [ofReal_tsum]
refine tsum_congr fun n ↦ ?_
rw [mul_comm (1 / _), mul_one_div, ofReal_div, mul_assoc (2 * π), mul_comm x n, ← mul_assoc]
... | 19 |
import Mathlib.Geometry.Euclidean.Inversion.Basic
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Tactic.AdaptationNote
open Metric Function AffineMap Set AffineSubspace
open scoped Topology RealInnerProductSpace
variable {E F : Type*} [NormedAddCommGrou... | Mathlib/Geometry/Euclidean/Inversion/Calculus.lean | 87 | 108 | theorem hasFDerivAt_inversion (hx : x ≠ c) :
HasFDerivAt (inversion c R)
((R / dist x c) ^ 2 • (reflection (ℝ ∙ (x - c))ᗮ : F →L[ℝ] F)) x := by |
rcases add_left_surjective c x with ⟨x, rfl⟩
have : HasFDerivAt (inversion c R) (?_ : F →L[ℝ] F) (c + x) := by
#adaptation_note /-- nightly-2024-03-16: simp was
simp (config := { unfoldPartialApp := true }) only [inversion] -/
simp only [inversion_def]
simp_rw [dist_eq_norm, div_pow, div_eq_mul_inv... | 19 |
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 355 | 377 | theorem Subalgebra.SeparatesPoints.rclike_to_real {A : StarSubalgebra 𝕜 C(X, 𝕜)}
(hA : A.SeparatesPoints) :
((A.restrictScalars ℝ).comap
(ofRealAm.compLeftContinuous ℝ continuous_ofReal)).SeparatesPoints := by |
intro x₁ x₂ hx
-- Let `f` in the subalgebra `A` separate the points `x₁`, `x₂`
obtain ⟨_, ⟨f, hfA, rfl⟩, hf⟩ := hA hx
let F : C(X, 𝕜) := f - const _ (f x₂)
-- Subtract the constant `f x₂` from `f`; this is still an element of the subalgebra
have hFA : F ∈ A := by
refine A.sub_mem hfA (@Eq.subst _ (· ∈... | 19 |
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic.LinearCombination
#align_import analysis.convex.specific_functions.basic from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
open Real Set NNReal
| Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | 39 | 58 | theorem strictConvexOn_exp : StrictConvexOn ℝ univ exp := by |
apply strictConvexOn_of_slope_strict_mono_adjacent convex_univ
rintro x y z - - hxy hyz
trans exp y
· have h1 : 0 < y - x := by linarith
have h2 : x - y < 0 := by linarith
rw [div_lt_iff h1]
calc
exp y - exp x = exp y - exp y * exp (x - y) := by rw [← exp_add]; ring_nf
_ = exp y * (1 - ... | 19 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
universe u... | Mathlib/Analysis/MeanInequalities.lean | 113 | 134 | theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
∏ i ∈ s, z i ^ w i ≤ ∑ i ∈ s, w i * z i := by |
-- If some number `z i` equals zero and has non-zero weight, then LHS is 0 and RHS is nonnegative.
by_cases A : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0
· rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi
-- I... | 19 |
import Mathlib.SetTheory.Ordinal.Arithmetic
namespace OrdinalApprox
universe u
variable {α : Type u}
variable [CompleteLattice α] (f : α →o α) (x : α)
open Function fixedPoints Cardinal Order OrderHom
set_option linter.unusedVariables false in
def lfpApprox (a : Ordinal.{u}) : α :=
sSup ({ f (lfpApprox b) | ... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 92 | 112 | theorem lfpApprox_add_one (h : x ≤ f x) (a : Ordinal) :
lfpApprox f x (a+1) = f (lfpApprox f x a) := by |
apply le_antisymm
· conv => left; unfold lfpApprox
apply sSup_le
simp only [Ordinal.add_one_eq_succ, lt_succ_iff, exists_prop, Set.union_singleton,
Set.mem_insert_iff, Set.mem_setOf_eq, forall_eq_or_imp, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂]
apply And.intro
· apply le_... | 19 |
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Order.Minimal
#align_import ring_theory.ideal.minimal_prime from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
section
variable {R S : Type*} [CommSemiring R] [CommSemiring S] (I J ... | Mathlib/RingTheory/Ideal/MinimalPrime.lean | 104 | 125 | theorem Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective {f : R →+* S}
(hf : Function.Injective f) (p) (H : p ∈ minimalPrimes R) :
∃ p' : Ideal S, p'.IsPrime ∧ p'.comap f = p := by |
have := H.1.1
have : Nontrivial (Localization (Submonoid.map f p.primeCompl)) := by
refine ⟨⟨1, 0, ?_⟩⟩
convert (IsLocalization.map_injective_of_injective p.primeCompl (Localization.AtPrime p)
(Localization <| p.primeCompl.map f) hf).ne one_ne_zero
· rw [map_one]
· rw [map_zero]
obtain ⟨M... | 19 |
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Invertible.Basic
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Data.Set.Basic
import Mathlib.Logic.Basic
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"1ac8d4304efba9d03fa720d06516fac845aa535... | Mathlib/Algebra/Group/Center.lean | 98 | 119 | theorem mul_mem_center [Mul M] {z₁ z₂ : M} (hz₁ : z₁ ∈ Set.center M) (hz₂ : z₂ ∈ Set.center M) :
z₁ * z₂ ∈ Set.center M where
comm a := calc
z₁ * z₂ * a = z₂ * z₁ * a := by | rw [hz₁.comm]
_ = z₂ * (z₁ * a) := by rw [hz₁.mid_assoc z₂]
_ = (a * z₁) * z₂ := by rw [hz₁.comm, hz₂.comm]
_ = a * (z₁ * z₂) := by rw [hz₂.right_assoc a z₁]
left_assoc (b c : M) := calc
z₁ * z₂ * (b * c) = z₁ * (z₂ * (b * c)) := by rw [hz₂.mid_assoc]
_ = z₁ * ((z₂ * b) * c) := by rw [hz₂.left_as... | 19 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {𝕜 : Type*} [RCLike 𝕜] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 55 | 75 | theorem LinearMap.bound_of_sphere_bound {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] 𝕜)
(h : ∀ z ∈ sphere (0 : E) r, ‖f z‖ ≤ c) (z : E) : ‖f z‖ ≤ c / r * ‖z‖ := by |
by_cases z_zero : z = 0
· rw [z_zero]
simp only [LinearMap.map_zero, norm_zero, mul_zero]
exact le_rfl
set z₁ := ((r : 𝕜) * (‖z‖ : 𝕜)⁻¹) • z with hz₁
have norm_f_z₁ : ‖f z₁‖ ≤ c := by
apply h
rw [mem_sphere_zero_iff_norm]
exact norm_smul_inv_norm' r_pos.le z_zero
have r_ne_zero : (r : �... | 19 |
import Mathlib.Data.SetLike.Fintype
import Mathlib.Algebra.Divisibility.Prod
import Mathlib.RingTheory.Nakayama
import Mathlib.RingTheory.SimpleModule
import Mathlib.Tactic.RSuffices
#align_import ring_theory.artinian from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
open Set Filter Po... | Mathlib/RingTheory/Artinian.lean | 175 | 195 | theorem IsArtinian.finite_of_linearIndependent [Nontrivial R] [IsArtinian R M] {s : Set M}
(hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by |
refine by_contradiction fun hf => (RelEmbedding.wellFounded_iff_no_descending_seq.1
(wellFounded_submodule_lt (R := R) (M := M))).elim' ?_
have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
have : ∀ n, (↑) ∘ f '' { m | n ≤ m } ⊆ s := by
rintro n x ⟨y, _, rfl⟩
exact (f y).2
have : ∀ a b : ℕ, a ≤ b ↔
... | 19 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 101 | 120 | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by |
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine Finset.sum_bij_ne_zero ?_ ?_ ?_ ?_ ?_
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans (Nat.le_add_right _ _))]
rw [tsub_le_iff_tsub_le, ... | 19 |
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 37 | 58 | theorem closure_mul_image_mul_eq_top
(hR : R ∈ rightTransversals (H : Set G)) (hR1 : (1 : G) ∈ R) (hS : closure S = ⊤) :
(closure ((R * S).image fun g => g * (toFun hR g : G)⁻¹)) * R = ⊤ := by |
let f : G → R := fun g => toFun hR g
let U : Set G := (R * S).image fun g => g * (f g : G)⁻¹
change (closure U : Set G) * R = ⊤
refine top_le_iff.mp fun g _ => ?_
refine closure_induction_right ?_ ?_ ?_ (eq_top_iff.mp hS (mem_top g))
· exact ⟨1, (closure U).one_mem, 1, hR1, one_mul 1⟩
· rintro - - s hs ⟨... | 19 |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Order.Filter.IndicatorFunction
open MeasureTheory
section DominatedConvergenceTheorem
open Set Filter TopologicalSpace ENNReal
open scoped Topology
namespace MeasureTheory
variable {α E G: Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E] [C... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 79 | 104 | theorem hasSum_integral_of_dominated_convergence {ι} [Countable ι] {F : ι → α → G} {f : α → G}
(bound : ι → α → ℝ) (hF_meas : ∀ n, AEStronglyMeasurable (F n) μ)
(h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound n a)
(bound_summable : ∀ᵐ a ∂μ, Summable fun n => bound n a)
(bound_integrable : Integrable (fun a => ... |
have hb_nonneg : ∀ᵐ a ∂μ, ∀ n, 0 ≤ bound n a :=
eventually_countable_forall.2 fun n => (h_bound n).mono fun a => (norm_nonneg _).trans
have hb_le_tsum : ∀ n, bound n ≤ᵐ[μ] fun a => ∑' n, bound n a := by
intro n
filter_upwards [hb_nonneg, bound_summable]
with _ ha0 ha_sum using le_tsum ha_sum _ fu... | 19 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 89 | 108 | theorem isPrimePow_iff_unique_prime_dvd {n : ℕ} : IsPrimePow n ↔ ∃! p : ℕ, p.Prime ∧ p ∣ n := by |
rw [isPrimePow_nat_iff]
constructor
· rintro ⟨p, k, hp, hk, rfl⟩
refine ⟨p, ⟨hp, dvd_pow_self _ hk.ne'⟩, ?_⟩
rintro q ⟨hq, hq'⟩
exact (Nat.prime_dvd_prime_iff_eq hq hp).1 (hq.dvd_of_dvd_pow hq')
rintro ⟨p, ⟨hp, hn⟩, hq⟩
rcases eq_or_ne n 0 with (rfl | hn₀)
· cases (hq 2 ⟨Nat.prime_two, dvd_zero... | 19 |
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Finset.Sum
import Mathlib.Logic.Embedding.Set
#align_import data.fintype.sum from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
universe u v
variable {α β : Type*}
open Finset
instance (α : Type u) (β : Type v) [Fintype α] [Fintyp... | Mathlib/Data/Fintype/Sum.lean | 79 | 100 | theorem Finset.exists_equiv_extend_of_card_eq [Fintype α] [DecidableEq β] {t : Finset β}
(hαt : Fintype.card α = t.card) {s : Finset α} {f : α → β} (hfst : Finset.image f s ⊆ t)
(hfs : Set.InjOn f s) : ∃ g : α ≃ t, ∀ i ∈ s, (g i : β) = f i := by |
classical
induction' s using Finset.induction with a s has H generalizing f
· obtain ⟨e⟩ : Nonempty (α ≃ ↥t) := by rwa [← Fintype.card_eq, Fintype.card_coe]
use e
simp
have hfst' : Finset.image f s ⊆ t := (Finset.image_mono _ (s.subset_insert a)).trans hfst
have hfs' : Set.InjOn f s := hf... | 19 |
import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
import Mathlib.RingTheory.Norm
#align_import linear_algebra.free_module.norm from "leanprover-community/mathlib"@"90b0d53ee6ffa910e5c2a977ce7e2fc704647974"
open Ideal Polynomial
open scoped Polynomial
variable {R S ι : Type*} [CommRing R] [IsDomain R] [IsPri... | Mathlib/LinearAlgebra/FreeModule/Norm.lean | 30 | 50 | theorem associated_norm_prod_smith [Fintype ι] (b : Basis ι R S) {f : S} (hf : f ≠ 0) :
Associated (Algebra.norm R f) (∏ i, smithCoeffs b _ (span_singleton_eq_bot.not.2 hf) i) := by |
have hI := span_singleton_eq_bot.not.2 hf
let b' := ringBasis b (span {f}) hI
classical
rw [← Matrix.det_diagonal, ← LinearMap.det_toLin b']
let e :=
(b'.equiv ((span {f}).selfBasis b hI) <| Equiv.refl _).trans
((LinearEquiv.coord S S f hf).restrictScalars R)
refine (LinearMap.associated_det_of_e... | 19 |
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.GaloisConnection
import Mathlib.Order.Hom.Basic
#align_import order.hom.order from "leanprover-community/mathlib"@"ba2245edf0c8bb155f1569fd9b9492a9b384cde6"
namespace OrderHom
variable {α β : Type*}
section Preorder
variable [Preorder α]
instance [Sem... | Mathlib/Order/Hom/Order.lean | 133 | 154 | theorem iterate_sup_le_sup_iff {α : Type*} [SemilatticeSup α] (f : α →o α) :
(∀ n₁ n₂ a₁ a₂, f^[n₁ + n₂] (a₁ ⊔ a₂) ≤ f^[n₁] a₁ ⊔ f^[n₂] a₂) ↔
∀ a₁ a₂, f (a₁ ⊔ a₂) ≤ f a₁ ⊔ a₂ := by |
constructor <;> intro h
· exact h 1 0
· intro n₁ n₂ a₁ a₂
have h' : ∀ n a₁ a₂, f^[n] (a₁ ⊔ a₂) ≤ f^[n] a₁ ⊔ a₂ := by
intro n
induction' n with n ih <;> intro a₁ a₂
· rfl
· calc
f^[n + 1] (a₁ ⊔ a₂) = f^[n] (f (a₁ ⊔ a₂)) := Function.iterate_succ_apply f n _
_ ≤ f^[n]... | 19 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 198 | 220 | theorem LipschitzOnWith.extend_finite_dimension {α : Type*} [PseudoMetricSpace α] {E' : Type*}
[NormedAddCommGroup E'] [NormedSpace ℝ E'] [FiniteDimensional ℝ E'] {s : Set α} {f : α → E'}
{K : ℝ≥0} (hf : LipschitzOnWith K f s) :
∃ g : α → E', LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s ... |
/- This result is already known for spaces `ι → ℝ`. We use a continuous linear equiv between
`E'` and such a space to transfer the result to `E'`. -/
let ι : Type _ := Basis.ofVectorSpaceIndex ℝ E'
let A := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
have LA : LipschitzWith ‖A.toContinuousL... | 19 |
import Mathlib.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Noetherian
#align_import ring_theory.ideal.associated_prime from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [... | Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 83 | 103 | theorem exists_le_isAssociatedPrime_of_isNoetherianRing [H : IsNoetherianRing R] (x : M)
(hx : x ≠ 0) : ∃ P : Ideal R, IsAssociatedPrime P M ∧ (R ∙ x).annihilator ≤ P := by |
have : (R ∙ x).annihilator ≠ ⊤ := by
rwa [Ne, Ideal.eq_top_iff_one, Submodule.mem_annihilator_span_singleton, one_smul]
obtain ⟨P, ⟨l, h₁, y, rfl⟩, h₃⟩ :=
set_has_maximal_iff_noetherian.mpr H
{ P | (R ∙ x).annihilator ≤ P ∧ P ≠ ⊤ ∧ ∃ y : M, P = (R ∙ y).annihilator }
⟨(R ∙ x).annihilator, rfl.le... | 19 |
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.CategoryTheory.Bicategory.LocallyDiscrete
#align_import category_theory.bicategory.coherence from "leanprover-community/mathlib"@"f187f1074fa1857c94589cc653c786ca... | Mathlib/CategoryTheory/Bicategory/Coherence.lean | 161 | 183 | theorem normalize_naturality {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
(preinclusion B).map ⟨p⟩ ◁ η ≫ (normalizeIso p g).hom =
(normalizeIso p f).hom ≫
(preinclusion B).map₂ (eqToHom (Discrete.ext _ _ (normalizeAux_congr p η))) := by |
rcases η with ⟨η'⟩; clear η;
induction η' with
| id => simp
| vcomp η θ ihf ihg =>
simp only [mk_vcomp, Bicategory.whiskerLeft_comp]
slice_lhs 2 3 => rw [ihg]
slice_lhs 1 2 => rw [ihf]
simp
-- p ≠ nil required! See the docstring of `normalizeAux`.
| whisker_left _ _ ih =>
dsimp
rw [... | 19 |
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 | 84 | 104 | theorem partialSections.directed :
Directed Superset fun G : FiniteDiagram J => partialSections F G.2 := by |
classical
intro A B
let ιA : FiniteDiagramArrow A.1 → FiniteDiagramArrow (A.1 ⊔ B.1) := fun f =>
⟨f.1, f.2.1, Finset.mem_union_left _ f.2.2.1, Finset.mem_union_left _ f.2.2.2.1, f.2.2.2.2⟩
let ιB : FiniteDiagramArrow B.1 → FiniteDiagramArrow (A.1 ⊔ B.1) := fun f =>
⟨f.1, f.2.1, Finset.mem_union_right _... | 19 |
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 | 67 | 87 | theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
(hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by |
rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
lift ε to ℝ≥0 using ε0.le
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
refine ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f ... | 19 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 85 | 105 | theorem reduce_to_p_prime {P : Prop} :
(∀ p > 0, MixedCharZero R p → P) ↔ ∀ p : ℕ, p.Prime → MixedCharZero R p → P := by |
constructor
· intro h q q_prime q_mixedChar
exact h q (Nat.Prime.pos q_prime) q_mixedChar
· intro h q q_pos q_mixedChar
rcases q_mixedChar.charP_quotient with ⟨I, hI_ne_top, _⟩
-- Krull's Thm: There exists a prime ideal `P` such that `I ≤ P`
rcases Ideal.exists_le_maximal I hI_ne_top with ⟨M, hM_... | 19 |
import Mathlib.Algebra.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
import Mathlib.RingTheory.Ideal.LocalRing
#align_import number_theory.class_number.admissible_card_pow_degree from "leanprover-community/mathlib"@"0b... | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 36 | 57 | theorem exists_eq_polynomial [Semiring Fq] {d : ℕ} {m : ℕ} (hm : Fintype.card Fq ^ d ≤ m)
(b : Fq[X]) (hb : natDegree b ≤ d) (A : Fin m.succ → Fq[X])
(hA : ∀ i, degree (A i) < degree b) : ∃ i₀ i₁, i₀ ≠ i₁ ∧ A i₁ = A i₀ := by |
-- Since there are > q^d elements of A, and only q^d choices for the highest `d` coefficients,
-- there must be two elements of A with the same coefficients at
-- `0`, ... `degree b - 1` ≤ `d - 1`.
-- In other words, the following map is not injective:
set f : Fin m.succ → Fin d → Fq := fun i j => (A i).coef... | 19 |
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.optional_stopping from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheory
vari... | Mathlib/Probability/Martingale/OptionalStopping.lean | 42 | 63 | theorem Submartingale.expected_stoppedValue_mono [SigmaFiniteFiltration μ 𝒢]
(hf : Submartingale f 𝒢 μ) (hτ : IsStoppingTime 𝒢 τ) (hπ : IsStoppingTime 𝒢 π) (hle : τ ≤ π)
{N : ℕ} (hbdd : ∀ ω, π ω ≤ N) : μ[stoppedValue f τ] ≤ μ[stoppedValue f π] := by |
rw [← sub_nonneg, ← integral_sub', stoppedValue_sub_eq_sum' hle hbdd]
· simp only [Finset.sum_apply]
have : ∀ i, MeasurableSet[𝒢 i] {ω : Ω | τ ω ≤ i ∧ i < π ω} := by
intro i
refine (hτ i).inter ?_
convert (hπ i).compl using 1
ext x
simp; rfl
rw [integral_finset_sum]
· ref... | 19 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.IndicatorConstPointwise
#align_import measure_theory.constructions.borel_space.metrizable from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Filter MeasureT... | Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean | 57 | 78 | theorem aemeasurable_of_tendsto_metrizable_ae {ι} {μ : Measure α} {f : ι → α → β} {g : α → β}
(u : Filter ι) [hu : NeBot u] [IsCountablyGenerated u] (hf : ∀ n, AEMeasurable (f n) μ)
(h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) : AEMeasurable g μ := by |
rcases u.exists_seq_tendsto with ⟨v, hv⟩
have h'f : ∀ n, AEMeasurable (f (v n)) μ := fun n => hf (v n)
set p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
have hp : ∀ᵐ x ∂μ, p x fun n => f (v n) x := by
filter_upwards [h_tendsto] with x hx using hx.comp hv
set aeSeqLim := f... | 19 |
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.Probability.Independence.Basic
#align_import probability.integration from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
noncomputable section
open Set MeasureTheory
open scoped ENNReal MeasureTheory
variable {Ω : Type*... | Mathlib/Probability/Integration.lean | 82 | 104 | theorem lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace
{Mf Mg mΩ : MeasurableSpace Ω} {μ : Measure Ω} (hMf : Mf ≤ mΩ) (hMg : Mg ≤ mΩ)
(h_ind : Indep Mf Mg μ) (h_meas_f : Measurable[Mf] f) (h_meas_g : Measurable[Mg] g) :
∫⁻ ω, f ω * g ω ∂μ = (∫⁻ ω, f ω ∂μ) * ∫⁻ ω, g ω ∂μ := by |
revert g
have h_measM_f : Measurable f := h_meas_f.mono hMf le_rfl
apply @Measurable.ennreal_induction _ Mg
· intro c s h_s
apply lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator hMf _ (hMg _ h_s) _ h_meas_f
apply indepSets_of_indepSets_of_le_right h_ind
rwa [singleton_subset_iff]
· i... | 19 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.Normed.Group.Completion
#align_import analysis.normed.group.hom_completion from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
noncomputable section
open Set NormedAddGroupHom UniformSpace
section Completion
variable {G... | Mathlib/Analysis/Normed/Group/HomCompletion.lean | 171 | 193 | theorem NormedAddGroupHom.ker_completion {f : NormedAddGroupHom G H} {C : ℝ}
(h : f.SurjectiveOnWith f.range C) :
(f.completion.ker : Set <| Completion G) = closure (toCompl.comp <| incl f.ker).range := by |
refine le_antisymm ?_ (closure_minimal f.ker_le_ker_completion f.completion.isClosed_ker)
rintro hatg (hatg_in : f.completion hatg = 0)
rw [SeminormedAddCommGroup.mem_closure_iff]
intro ε ε_pos
rcases h.exists_pos with ⟨C', C'_pos, hC'⟩
rcases exists_pos_mul_lt ε_pos (1 + C' * ‖f‖) with ⟨δ, δ_pos, hδ⟩
ob... | 20 |
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.quotient_nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
theorem Ideal.isRadical_iff_quotient_reduced {R : Type*} [CommRing R] (I : Ideal R) :
I.IsRad... | Mathlib/RingTheory/QuotientNilpotent.lean | 26 | 51 | theorem Ideal.IsNilpotent.induction_on (hI : IsNilpotent I)
{P : ∀ ⦃S : Type _⦄ [CommRing S], Ideal S → Prop}
(h₁ : ∀ ⦃S : Type _⦄ [CommRing S], ∀ I : Ideal S, I ^ 2 = ⊥ → P I)
(h₂ : ∀ ⦃S : Type _⦄ [CommRing S], ∀ I J : Ideal S, I ≤ J → P I →
P (J.map (Ideal.Quotient.mk I)) → P J) :
P I := by |
obtain ⟨n, hI : I ^ n = ⊥⟩ := hI
induction' n using Nat.strong_induction_on with n H generalizing S
by_cases hI' : I = ⊥
· subst hI'
apply h₁
rw [← Ideal.zero_eq_bot, zero_pow two_ne_zero]
cases' n with n
· rw [pow_zero, Ideal.one_eq_top] at hI
haveI := subsingleton_of_bot_eq_top hI.symm
ex... | 20 |
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 | 134 | 154 | theorem aeval_self_charpoly (M : Matrix n n R) : aeval M M.charpoly = 0 := by |
-- We begin with the fact $χ_M(t) I = adjugate (t I - M) * (t I - M)$,
-- as an identity in `Matrix n n R[X]`.
have h : M.charpoly • (1 : Matrix n n R[X]) = adjugate (charmatrix M) * charmatrix M :=
(adjugate_mul _).symm
-- Using the algebra isomorphism `Matrix n n R[X] ≃ₐ[R] Polynomial (Matrix n n R)`,
... | 20 |
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 | 172 | 195 | theorem ext [Fact p.Prime] {f g} (hf : IsPoly p f) (hg : IsPoly p g)
(h : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ),
ghostComponent n (f x) = ghostComponent n (g x)) :
∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R), f x = g x := by |
obtain ⟨φ, hf⟩ := hf
obtain ⟨ψ, hg⟩ := hg
intros
ext n
rw [hf, hg, poly_eq_of_wittPolynomial_bind_eq p φ ψ]
intro k
apply MvPolynomial.funext
intro x
simp only [hom_bind₁]
specialize h (ULift ℤ) (mk p fun i => ⟨x i⟩) k
simp only [ghostComponent_apply, aeval_eq_eval₂Hom] at h
apply (ULift.ringEq... | 20 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.PNat.Basic
import Mathlib.Data.PNat.Interval
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.IntervalCases
#align_import number_theory.ADE_inequality from "leanprover-community/math... | Mathlib/NumberTheory/ADEInequality.lean | 175 | 195 | theorem lt_three {p q r : ℕ+} (hpq : p ≤ q) (hqr : q ≤ r) (H : 1 < sumInv {p, q, r}) : p < 3 := by |
have h3 : (0 : ℚ) < 3 := by norm_num
contrapose! H
rw [sumInv_pqr]
have h3q := H.trans hpq
have h3r := h3q.trans hqr
have hp: (p : ℚ)⁻¹ ≤ 3⁻¹ := by
rw [inv_le_inv _ h3]
· assumption_mod_cast
· norm_num
have hq: (q : ℚ)⁻¹ ≤ 3⁻¹ := by
rw [inv_le_inv _ h3]
· assumption_mod_cast
· nor... | 20 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 130 | 153 | theorem thickenedIndicatorAux_tendsto_indicator_closure {δseq : ℕ → ℝ}
(δseq_lim : Tendsto δseq atTop (𝓝 0)) (E : Set α) :
Tendsto (fun n => thickenedIndicatorAux (δseq n) E) atTop
(𝓝 (indicator (closure E) fun _ => (1 : ℝ≥0∞))) := by |
rw [tendsto_pi_nhds]
intro x
by_cases x_mem_closure : x ∈ closure E
· simp_rw [thickenedIndicatorAux_one_of_mem_closure _ E x_mem_closure]
rw [show (indicator (closure E) fun _ => (1 : ℝ≥0∞)) x = 1 by
simp only [x_mem_closure, indicator_of_mem]]
exact tendsto_const_nhds
· rw [show (closure E)... | 20 |
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lattice
import Mathlib.Topology.Order.Lattice
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
#align_import measure_theory.constructions.borel_space.basic from "leanprover-c... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 54 | 74 | theorem borel_eq_generateFrom_Iio : borel α = .generateFrom (range Iio) := by |
refine le_antisymm ?_ (generateFrom_le ?_)
· rw [borel_eq_generateFrom_of_subbasis (@OrderTopology.topology_eq_generate_intervals α _ _ _)]
letI : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
have H : ∀ a : α, MeasurableSet (Iio a) := fun a => GenerateMeasurable.basic _ ⟨_, rfl⟩
refine... | 20 |
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7738ae4da1c5896191f72"
universe uR uS uT uA u₄
variable {R : Type uR} [Semiring R]
variable {S : Type uS} [CommSemiring S]
variable {T : Type uT}
vari... | Mathlib/Algebra/RingQuot.lean | 121 | 141 | theorem eqvGen_rel_eq (r : R → R → Prop) : EqvGen (Rel r) = RingConGen.Rel r := by |
ext x₁ x₂
constructor
· intro h
induction h with
| rel _ _ h => induction h with
| of => exact RingConGen.Rel.of _ _ ‹_›
| add_left _ h => exact h.add (RingConGen.Rel.refl _)
| mul_left _ h => exact h.mul (RingConGen.Rel.refl _)
| mul_right _ h => exact (RingConGen.Rel.refl _).mul... | 20 |
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 | 59 | 80 | theorem ringHom_ext {S} [Semiring S] {f g : R[X] →+* S} (h₁ : ∀ a, f (C a) = g (C a))
(h₂ : f X = g X) : f = g := by |
set f' := f.comp (toFinsuppIso R).symm.toRingHom with hf'
set g' := g.comp (toFinsuppIso R).symm.toRingHom with hg'
have A : f' = g' := by
-- Porting note: Was `ext; simp [..]; simpa [..] using h₂`.
ext : 1
· ext
simp [f', g', h₁, RingEquiv.toRingHom_eq_coe]
· refine MonoidHom.ext_mnat ?_
... | 20 |
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
noncomputable section
open scoped Classical
open scoped RealInnerProductSpace
namespace Affine
namespace Simplex
open Finset AffineSubspac... | Mathlib/Geometry/Euclidean/MongePoint.lean | 130 | 154 | theorem mongePoint_eq_affineCombination_of_pointsWithCircumcenter {n : ℕ}
(s : Simplex ℝ P (n + 2)) :
s.mongePoint =
(univ : Finset (PointsWithCircumcenterIndex (n + 2))).affineCombination ℝ
s.pointsWithCircumcenter (mongePointWeightsWithCircumcenter n) := by |
rw [mongePoint_eq_smul_vsub_vadd_circumcenter,
centroid_eq_affineCombination_of_pointsWithCircumcenter,
circumcenter_eq_affineCombination_of_pointsWithCircumcenter, affineCombination_vsub,
← LinearMap.map_smul, weightedVSub_vadd_affineCombination]
congr with i
rw [Pi.add_apply, Pi.smul_apply, smul_eq... | 20 |
import Batteries.Classes.SatisfiesM
namespace Array
theorem SatisfiesM_foldlM [Monad m] [LawfulMonad m]
{as : Array α} (motive : Nat → β → Prop) {init : β} (h0 : motive 0 init) {f : β → α → m β}
(hf : ∀ i : Fin as.size, ∀ b, motive i.1 b → SatisfiesM (motive (i.1 + 1)) (f b as[i])) :
SatisfiesM (motive... | .lake/packages/batteries/Batteries/Data/Array/Monadic.lean | 85 | 110 | theorem SatisfiesM_anyM_iff_exists [Monad m] [LawfulMonad m]
(p : α → m Bool) (as : Array α) (start stop) (q : Fin as.size → Prop)
(hp : ∀ i : Fin as.size, start ≤ i.1 → i.1 < stop → SatisfiesM (· = true ↔ q i) (p as[i])) :
SatisfiesM
(fun res => res = true ↔ ∃ i : Fin as.size, start ≤ i.1 ∧ i.1 < sto... |
cases Nat.le_total start (min stop as.size) with
| inl hstart =>
refine (SatisfiesM_anyM _ _ _ _ hstart
(fal := fun j => start ≤ j ∧ ¬ ∃ i : Fin as.size, start ≤ i.1 ∧ i.1 < j ∧ q i)
(tru := ∃ i : Fin as.size, start ≤ i.1 ∧ i.1 < stop ∧ q i) ?_ ?_).imp ?_
· exact ⟨Nat.le_refl _, fun ⟨i, h₁, h₂,... | 20 |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.Zify
import Mathlib.Data.Nat.Totient
#align_import number_theory.lucas_primality from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
| Mathlib/NumberTheory/LucasPrimality.lean | 42 | 63 | theorem lucas_primality (p : ℕ) (a : ZMod p) (ha : a ^ (p - 1) = 1)
(hd : ∀ q : ℕ, q.Prime → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1) : p.Prime := by |
have h0 : p ≠ 0 := by
rintro ⟨⟩
exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)
have h1 : p ≠ 1 := by
rintro ⟨⟩
exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)
have hp1 : 1 < p := lt_of_le_of_ne h0.bot_lt h1.symm
have order_of_a : orderOf a = p - 1 := by
apply orderOf_eq_of_pow_and_po... | 20 |
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Fintype.Card
#align_import data.finset.pi_induction from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Function
variable {ι : Type*} {α : ι → Type*} [Finite ι] [DecidableEq ι] [∀ i, DecidableEq (α i)]
namespace Finset
| Mathlib/Data/Finset/PiInduction.lean | 37 | 63 | theorem induction_on_pi_of_choice (r : ∀ i, α i → Finset (α i) → Prop)
(H_ex : ∀ (i) (s : Finset (α i)), s.Nonempty → ∃ x ∈ s, r i x (s.erase x))
{p : (∀ i, Finset (α i)) → Prop} (f : ∀ i, Finset (α i)) (h0 : p fun _ ↦ ∅)
(step :
∀ (g : ∀ i, Finset (α i)) (i : ι) (x : α i),
r i x (g i) → p g →... |
cases nonempty_fintype ι
induction' hs : univ.sigma f using Finset.strongInductionOn with s ihs generalizing f; subst s
rcases eq_empty_or_nonempty (univ.sigma f) with he | hne
· convert h0 using 1
simpa [funext_iff] using he
· rcases sigma_nonempty.1 hne with ⟨i, -, hi⟩
rcases H_ex i (f i) hi with ⟨... | 20 |
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 | 748 | 771 | theorem exists_upcrossings_of_not_bounded_under {ι : Type*} {l : Filter ι} {x : ι → ℝ}
(hf : liminf (fun i => (Real.nnabs (x i) : ℝ≥0∞)) l ≠ ∞)
(hbdd : ¬IsBoundedUnder (· ≤ ·) l fun i => |x i|) :
∃ a b : ℚ, a < b ∧ (∃ᶠ i in l, x i < a) ∧ ∃ᶠ i in l, ↑b < x i := by |
rw [isBoundedUnder_le_abs, not_and_or] at hbdd
obtain hbdd | hbdd := hbdd
· obtain ⟨R, hR⟩ := exists_frequently_lt_of_liminf_ne_top hf
obtain ⟨q, hq⟩ := exists_rat_gt R
refine ⟨q, q + 1, (lt_add_iff_pos_right _).2 zero_lt_one, ?_, ?_⟩
· refine fun hcon => hR ?_
filter_upwards [hcon] with x hx u... | 20 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
#align_import measure_theory.function.conditional_expectation.indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
open s... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean | 38 | 59 | theorem condexp_ae_eq_restrict_zero (hs : MeasurableSet[m] s) (hf : f =ᵐ[μ.restrict s] 0) :
μ[f|m] =ᵐ[μ.restrict s] 0 := by |
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
have : SigmaFinite ((μ.restrict s).trim hm) := by
rw [← restrict_trim hm _ hs]
exact Restrict.sigma... | 20 |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.init_tail from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
variable {p : ℕ} [hp : Fact p.Prime] (n : ℕ) {R : Type*} [CommRing R]
-- type as `\bbW`
local notat... | Mathlib/RingTheory/WittVector/InitTail.lean | 112 | 133 | theorem coeff_add_of_disjoint (x y : 𝕎 R) (h : ∀ n, x.coeff n = 0 ∨ y.coeff n = 0) :
(x + y).coeff n = x.coeff n + y.coeff n := by |
let P : ℕ → Prop := fun n => y.coeff n = 0
haveI : DecidablePred P := Classical.decPred P
set z := mk p fun n => if P n then x.coeff n else y.coeff n
have hx : select P z = x := by
ext1 n; rw [select, coeff_mk, coeff_mk]
split_ifs with hn
· rfl
· rw [(h n).resolve_right hn]
have hy : select (... | 20 |
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 142 | 164 | theorem sum_subgroup_units_eq_zero [Ring K] [NoZeroDivisors K]
{G : Subgroup Kˣ} [Fintype G] (hg : G ≠ ⊥) :
∑ x : G, (x.val : K) = 0 := by |
rw [Subgroup.ne_bot_iff_exists_ne_one] at hg
rcases hg with ⟨a, ha⟩
-- The action of a on G as an embedding
let a_mul_emb : G ↪ G := mulLeftEmbedding a
-- ... and leaves G unchanged
have h_unchanged : Finset.univ.map a_mul_emb = Finset.univ := by simp
-- Therefore the sum of x over a G is the sum of a x ... | 20 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 93 | 114 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by |
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [re_ofReal_mul, im_ofReal_mul, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left₀ _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_... | 20 |
import Mathlib.Init.Classical
import Mathlib.Order.FixedPoints
import Mathlib.Order.Zorn
#align_import set_theory.cardinal.schroeder_bernstein from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
open Set Function
open scoped Classical
universe u v
namespace Function
namespace Embedd... | Mathlib/SetTheory/Cardinal/SchroederBernstein.lean | 100 | 131 | theorem min_injective [I : Nonempty ι] : ∃ i, Nonempty (∀ j, β i ↪ β j) :=
let ⟨s, hs, ms⟩ :=
show ∃ s ∈ sets β, ∀ a ∈ sets β, s ⊆ a → a = s from
zorn_subset (sets β) fun c hc hcc =>
⟨⋃₀c, fun x ⟨p, hpc, hxp⟩ y ⟨q, hqc, hyq⟩ i hi =>
(hcc.total hpc hqc).elim (fun h => hc hqc x (h hxp) y hyq... |
simpa only [ne_eq, not_exists, not_forall, not_and] using h
let ⟨f, hf⟩ := Classical.axiom_of_choice h
have : f ∈ s :=
have : insert f s ∈ sets β := fun x hx y hy => by
cases' hx with hx hx <;> cases' hy with hy hy; · simp [hx, hy]
· subst x
exa... | 20 |
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Star.Pointwise
import Mathlib.Algebra.Group.Centralizer
variable {R : Type*} [Mul R] [StarMul R] {a : R} {s : Set R}
| Mathlib/Algebra/Star/Center.lean | 14 | 34 | theorem Set.star_mem_center (ha : a ∈ Set.center R) : star a ∈ Set.center R where
comm := by | simpa only [star_mul, star_star] using fun g =>
congr_arg star (((Set.mem_center_iff R).mp ha).comm <| star g).symm
left_assoc b c := calc
star a * (b * c) = star a * (star (star b) * star (star c)) := by rw [star_star, star_star]
_ = star a * star (star c * star b) := by rw [star_mul]
_ = star ((sta... | 20 |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Data.Set.Function
#align_import analysis.sum_integral_comparisons from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set MeasureTheory.MeasureSpace
variable {x₀ : ℝ} {a b : ℕ} {f : ℝ → ℝ}
theorem AntitoneOn.in... | Mathlib/Analysis/SumIntegralComparisons.lean | 126 | 147 | theorem AntitoneOn.sum_le_integral_Ico (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc a b)) :
(∑ i ∈ Finset.Ico a b, f (i + 1 : ℕ)) ≤ ∫ x in a..b, f x := by |
rw [(Nat.sub_add_cancel hab).symm, Nat.cast_add]
conv =>
congr
congr
congr
rw [← zero_add a]
· skip
· skip
· skip
rw [add_comm]
rw [← Finset.sum_Ico_add, Nat.Ico_zero_eq_range]
conv =>
lhs
congr
congr
· skip
ext
rw [add_assoc, Nat.cast_add]
apply Antito... | 20 |
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
import Mathlib.AlgebraicTopology.DoldKan.Decomposition
import Mathlib.CategoryTheory.Idempotents.HomologicalComplex
import Mathlib.CategoryTheory.Idempotents.KaroubiKaroubi
#align_import algebraic_topology.dold_kan.n_reflects_iso from "leanprover-community/mathlib"@"3... | Mathlib/AlgebraicTopology/DoldKan/NReflectsIso.lean | 68 | 92 | theorem compatibility_N₂_N₁_karoubi :
N₂ ⋙ (karoubiChainComplexEquivalence C ℕ).functor =
karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C ⋙
N₁ ⋙ (karoubiChainComplexEquivalence (Karoubi C) ℕ).functor ⋙
Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse _ := by |
refine CategoryTheory.Functor.ext (fun P => ?_) fun P Q f => ?_
· refine HomologicalComplex.ext ?_ ?_
· ext n
· rfl
· dsimp
simp only [karoubi_PInfty_f, comp_id, PInfty_f_naturality, id_comp, eqToHom_refl]
· rintro _ n (rfl : n + 1 = _)
ext
have h := (AlternatingFaceMapCompl... | 20 |
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 | 326 | 348 | theorem Memℒp.uniformIntegrable_of_identDistrib_aux {ι : Type*} {f : ι → α → E} {j : ι} {p : ℝ≥0∞}
(hp : 1 ≤ p) (hp' : p ≠ ∞) (hℒp : Memℒp (f j) p μ) (hfmeas : ∀ i, StronglyMeasurable (f i))
(hf : ∀ i, IdentDistrib (f i) (f j) μ μ) : UniformIntegrable f p μ := by |
refine uniformIntegrable_of' hp hp' hfmeas fun ε hε => ?_
by_cases hι : Nonempty ι
swap; · exact ⟨0, fun i => False.elim (hι <| Nonempty.intro i)⟩
obtain ⟨C, hC₁, hC₂⟩ := hℒp.snorm_indicator_norm_ge_pos_le (hfmeas _) hε
refine ⟨⟨C, hC₁.le⟩, fun i => le_trans (le_of_eq ?_) hC₂⟩
have : {x | (⟨C, hC₁.le⟩ : ℝ≥... | 20 |
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 109 | 129 | theorem content_X_mul {p : R[X]} : content (X * p) = content p := by |
rw [content, content, Finset.gcd_def, Finset.gcd_def]
refine congr rfl ?_
have h : (X * p).support = p.support.map ⟨Nat.succ, Nat.succ_injective⟩ := by
ext a
simp only [exists_prop, Finset.mem_map, Function.Embedding.coeFn_mk, Ne, mem_support_iff]
cases' a with a
· simp [coeff_X_mul_zero, Nat.suc... | 20 |
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 129 | 151 | theorem IsLindelof.elim_countable_subcover {ι : Type v} (hs : IsLindelof s) (U : ι → Set X)
(hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) :
∃ r : Set ι, r.Countable ∧ (s ⊆ ⋃ i ∈ r, U i) := by |
have hmono : ∀ ⦃s t : Set X⦄, s ⊆ t → (∃ r : Set ι, r.Countable ∧ t ⊆ ⋃ i ∈ r, U i)
→ (∃ r : Set ι, r.Countable ∧ s ⊆ ⋃ i ∈ r, U i) := by
intro _ _ hst ⟨r, ⟨hrcountable, hsub⟩⟩
exact ⟨r, hrcountable, Subset.trans hst hsub⟩
have hcountable_union : ∀ (S : Set (Set X)), S.Countable
→ (∀ s ∈ S, ∃ r... | 20 |
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.discrete_valuation_ring.basic from "leanprover-community/mathlib"@"c163ec99dfc664628ca15d215fce0a5b9c2... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 169 | 190 | theorem unique_irreducible ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
Associated p q := by |
rcases hR with ⟨ϖ, hϖ, hR⟩
suffices ∀ {p : R} (_ : Irreducible p), Associated p ϖ by
apply Associated.trans (this hp) (this hq).symm
clear hp hq p q
intro p hp
obtain ⟨n, hn⟩ := hR hp.ne_zero
have : Irreducible (ϖ ^ n) := hn.symm.irreducible hp
rcases lt_trichotomy n 1 with (H | rfl | H)
· obtain r... | 20 |
import Mathlib.Data.List.Duplicate
import Mathlib.Data.List.Sort
#align_import data.list.nodup_equiv_fin from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace List
variable {α : Type*}
section Sublist
| Mathlib/Data/List/NodupEquivFin.lean | 116 | 137 | theorem sublist_of_orderEmbedding_get?_eq {l l' : List α} (f : ℕ ↪o ℕ)
(hf : ∀ ix : ℕ, l.get? ix = l'.get? (f ix)) : l <+ l' := by |
induction' l with hd tl IH generalizing l' f
· simp
have : some hd = _ := hf 0
rw [eq_comm, List.get?_eq_some] at this
obtain ⟨w, h⟩ := this
let f' : ℕ ↪o ℕ :=
OrderEmbedding.ofMapLEIff (fun i => f (i + 1) - (f 0 + 1)) fun a b => by
dsimp only
rw [Nat.sub_le_sub_iff_right, OrderEmbedding.le... | 20 |
import Mathlib.Analysis.NormedSpace.Real
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed_space.riesz_lemma from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric
open Topology
variable {𝕜 : Type*} [Norm... | Mathlib/Analysis/NormedSpace/RieszLemma.lean | 83 | 105 | theorem riesz_lemma_of_norm_lt {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R) {F : Subspace 𝕜 E}
(hFc : IsClosed (F : Set E)) (hF : ∃ x : E, x ∉ F) :
∃ x₀ : E, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖ := by |
have Rpos : 0 < R := (norm_nonneg _).trans_lt hR
have : ‖c‖ / R < 1 := by
rw [div_lt_iff Rpos]
simpa using hR
rcases riesz_lemma hFc hF this with ⟨x, xF, hx⟩
have x0 : x ≠ 0 := fun H => by simp [H] at xF
obtain ⟨d, d0, dxlt, ledx, -⟩ :
∃ d : 𝕜, d ≠ 0 ∧ ‖d • x‖ < R ∧ R / ‖c‖ ≤ ‖d • x‖ ∧ ‖d‖⁻¹ ≤ R... | 20 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import analysis.special_functions.compare_exp from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Asympto... | Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 127 | 151 | theorem isLittleO_log_abs_re (hl : IsExpCmpFilter l) : (fun z => Real.log (abs z)) =o[l] re :=
calc
(fun z => Real.log (abs z)) =O[l] fun z => Real.log (√2) + Real.log (max z.re |z.im|) :=
IsBigO.of_bound 1 <|
(hl.tendsto_re.eventually_ge_atTop 1).mono fun z hz => by
have h2 : 0 < √2 := by | simp
have hz' : 1 ≤ abs z := hz.trans (re_le_abs z)
have hm₀ : 0 < max z.re |z.im| := lt_max_iff.2 (Or.inl <| one_pos.trans_le hz)
rw [one_mul, Real.norm_eq_abs, _root_.abs_of_nonneg (Real.log_nonneg hz')]
refine le_trans ?_ (le_abs_self _)
rw [← Real.log_mul, Real.log... | 20 |
import Mathlib.Algebra.Field.Subfield
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.uniform_field from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open uniformity Topology
... | Mathlib/Topology/Algebra/UniformField.lean | 72 | 93 | theorem continuous_hatInv [CompletableTopField K] {x : hat K} (h : x ≠ 0) :
ContinuousAt hatInv x := by |
refine denseInducing_coe.continuousAt_extend ?_
apply mem_of_superset (compl_singleton_mem_nhds h)
intro y y_ne
rw [mem_compl_singleton_iff] at y_ne
apply CompleteSpace.complete
have : (fun (x : K) => (↑x⁻¹: hat K)) =
((fun (y : K) => (↑y: hat K))∘(fun (x : K) => (x⁻¹ : K))) := by
unfold Function... | 20 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 141 | 162 | theorem monomial_mem_lifts_and_degree_eq {s : S} {n : ℕ} (hl : monomial n s ∈ lifts f) :
∃ q : R[X], map f q = monomial n s ∧ q.degree = (monomial n s).degree := by |
by_cases hzero : s = 0
· use 0
simp only [hzero, degree_zero, eq_self_iff_true, and_self_iff, monomial_zero_right,
Polynomial.map_zero]
rw [lifts_iff_set_range] at hl
obtain ⟨q, hq⟩ := hl
replace hq := (ext_iff.1 hq) n
have hcoeff : f (q.coeff n) = s := by
simp? [coeff_monomial] at hq says si... | 20 |
import Mathlib.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Order.Zorn
import Mathlib.Data.Finset.Order
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.Finite.Set
import Mathlib.Tactic.TFAE
#alig... | Mathlib/Order/CompactlyGenerated/Basic.lean | 83 | 105 | theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
CompleteLattice.IsCompactElement k ↔
∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s := by |
classical
constructor
· intro H ι s hs
obtain ⟨t, ht, ht'⟩ := H (Set.range s) hs
have : ∀ x : t, ∃ i, s i = x := fun x => ht x.prop
choose f hf using this
refine ⟨Finset.univ.image f, ht'.trans ?_⟩
rw [Finset.sup_le_iff]
intro b hb
rw [← show s (f ⟨b, hb⟩) = id b fro... | 20 |
import Mathlib.MeasureTheory.Integral.Lebesgue
#align_import measure_theory.measure.giry_monad from "leanprover-community/mathlib"@"56f4cd1ef396e9fd389b5d8371ee9ad91d163625"
noncomputable section
open scoped Classical
open ENNReal
open scoped Classical
open Set Filter
variable {α β : Type*}
namespace MeasureT... | Mathlib/MeasureTheory/Measure/GiryMonad.lean | 128 | 149 | theorem lintegral_join {m : Measure (Measure α)} {f : α → ℝ≥0∞} (hf : Measurable f) :
∫⁻ x, f x ∂join m = ∫⁻ μ, ∫⁻ x, f x ∂μ ∂m := by |
simp_rw [lintegral_eq_iSup_eapprox_lintegral hf, SimpleFunc.lintegral,
join_apply (SimpleFunc.measurableSet_preimage _ _)]
suffices
∀ (s : ℕ → Finset ℝ≥0∞) (f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞), (∀ n r, Measurable (f n r)) →
Monotone (fun n μ => ∑ r ∈ s n, r * f n r μ) →
⨆ n, ∑ r ∈ s n, r * ∫⁻ μ, f... | 20 |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.Layercake
#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
open Set
namespace MeasureTheory
variable {α : Type*} [MeasurableSpace α] {f : α → ℝ} (... | Mathlib/Analysis/SpecialFunctions/Pow/Integral.lean | 50 | 72 | theorem lintegral_rpow_eq_lintegral_meas_le_mul :
∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) := by |
have one_lt_p : -1 < p - 1 := by linarith
have obs : ∀ x : ℝ, ∫ t : ℝ in (0)..x, t ^ (p - 1) = x ^ p / p := by
intro x
rw [integral_rpow (Or.inl one_lt_p)]
simp [Real.zero_rpow p_pos.ne.symm]
set g := fun t : ℝ => t ^ (p - 1)
have g_nn : ∀ᵐ t ∂volume.restrict (Ioi (0 : ℝ)), 0 ≤ g t := by
filter... | 20 |
import Mathlib.Topology.Algebra.Valuation
import Mathlib.Topology.Algebra.WithZeroTopology
import Mathlib.Topology.Algebra.UniformField
#align_import topology.algebra.valued_field from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Filter Set
open Topology
section DivisionRing
v... | Mathlib/Topology/Algebra/ValuedField.lean | 51 | 72 | theorem Valuation.inversion_estimate {x y : K} {γ : Γ₀ˣ} (y_ne : y ≠ 0)
(h : v (x - y) < min (γ * (v y * v y)) (v y)) : v (x⁻¹ - y⁻¹) < γ := by |
have hyp1 : v (x - y) < γ * (v y * v y) := lt_of_lt_of_le h (min_le_left _ _)
have hyp1' : v (x - y) * (v y * v y)⁻¹ < γ := mul_inv_lt_of_lt_mul₀ hyp1
have hyp2 : v (x - y) < v y := lt_of_lt_of_le h (min_le_right _ _)
have key : v x = v y := Valuation.map_eq_of_sub_lt v hyp2
have x_ne : x ≠ 0 := by
intro... | 20 |
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.hall.finite from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Finset
universe u v
namespace HallMarriageTheorem
variable {ι : Type u} {α : Type v} [DecidableEq α] {t : ι → Finset α}
s... | Mathlib/Combinatorics/Hall/Finite.lean | 136 | 158 | theorem hall_cond_of_compl {ι : Type u} {t : ι → Finset α} {s : Finset ι}
(hus : s.card = (s.biUnion t).card) (ht : ∀ s : Finset ι, s.card ≤ (s.biUnion t).card)
(s' : Finset (sᶜ : Set ι)) : s'.card ≤ (s'.biUnion fun x' => t x' \ s.biUnion t).card := by |
haveI := Classical.decEq ι
have disj : Disjoint s (s'.image fun z => z.1) := by
simp only [disjoint_left, not_exists, mem_image, exists_prop, SetCoe.exists, exists_and_right,
exists_eq_right, Subtype.coe_mk]
intro x hx hc _
exact absurd hx hc
have : s'.card = (s ∪ s'.image fun z => z.1).card - ... | 20 |
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.ApplyFun
import Mathlib.Control.Fix
import Mathlib.Order.OmegaCompletePartialOrder
#align_import control.lawful_fix from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
universe u v
open scoped Classical
variable {α : Type*} {β : α →... | Mathlib/Control/LawfulFix.lean | 71 | 91 | theorem mem_iff (a : α) (b : β a) : b ∈ Part.fix f a ↔ ∃ i, b ∈ approx f i a := by |
by_cases h₀ : ∃ i : ℕ, (approx f i a).Dom
· simp only [Part.fix_def f h₀]
constructor <;> intro hh
· exact ⟨_, hh⟩
have h₁ := Nat.find_spec h₀
rw [dom_iff_mem] at h₁
cases' h₁ with y h₁
replace h₁ := approx_mono' f _ _ h₁
suffices y = b by
subst this
exact h₁
cases' hh w... | 20 |
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.pid from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
variable {R : Type*} [CommRing R]
open Ideal
open UniqueFactorizationMonoid
open scoped nonZer... | Mathlib/RingTheory/DedekindDomain/PID.lean | 78 | 102 | theorem FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_top {R A : Type*}
[CommRing R] [CommRing A] [Algebra R A] {S : Submonoid R} [IsLocalization S A]
(I : (FractionalIdeal S A)ˣ) {v : A} (hv : v ∈ (↑I⁻¹ : FractionalIdeal S A))
(h : Submodule.comap (Algebra.linearMap R A) ((I : Submodul... |
have hinv := I.mul_inv
set J := Submodule.comap (Algebra.linearMap R A) ((I : Submodule R A) * Submodule.span R {v})
have hJ : IsLocalization.coeSubmodule A J = ↑I * Submodule.span R {v} := by
-- Porting note: had to insert `val_eq_coe` into this rewrite.
-- Arguably this is because `Subtype.ext_iff` is ... | 20 |
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
namespace FractionalIdeal
open scoped Pointwise nonZeroDivisors
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free ℤ R] [Module.Finite ℤ R]
variable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]
th... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 106 | 128 | theorem abs_det_basis_change [NoZeroDivisors K] {ι : Type*} [Fintype ι]
[DecidableEq ι] (b : Basis ι ℤ R) (I : FractionalIdeal R⁰ K) (bI : Basis ι ℤ I) :
|(b.localizationLocalization ℚ ℤ⁰ K).det ((↑) ∘ bI)| = absNorm I := by |
have := IsFractionRing.nontrivial R K
let b₀ : Basis ι ℚ K := b.localizationLocalization ℚ ℤ⁰ K
let bI.num : Basis ι ℤ I.num := bI.map
((equivNum (nonZeroDivisors.coe_ne_zero _)).restrictScalars ℤ)
rw [absNorm_eq, ← Ideal.natAbs_det_basis_change b I.num bI.num, Int.cast_natAbs, Int.cast_abs,
Int.cast... | 20 |
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.SupIndep
#align_import group_theory.noncomm_pi_coprod from "leanprover-community/mathlib"@"6f9f36364eae3f42368b04858fd66d6d9ae730d8"
... | Mathlib/GroupTheory/NoncommPiCoprod.lean | 55 | 78 | theorem eq_one_of_noncommProd_eq_one_of_independent {ι : Type*} (s : Finset ι) (f : ι → G) (comm)
(K : ι → Subgroup G) (hind : CompleteLattice.Independent K) (hmem : ∀ x ∈ s, f x ∈ K x)
(heq1 : s.noncommProd f comm = 1) : ∀ i ∈ s, f i = 1 := by |
classical
revert heq1
induction' s using Finset.induction_on with i s hnmem ih
· simp
· have hcomm := comm.mono (Finset.coe_subset.2 <| Finset.subset_insert _ _)
simp only [Finset.forall_mem_insert] at hmem
have hmem_bsupr : s.noncommProd f hcomm ∈ ⨆ i ∈ (s : Set ι), K i := by
ref... | 21 |
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 105 | 128 | theorem KaehlerDifferential.submodule_span_range_eq_ideal :
Submodule.span S (Set.range fun s : S => (1 : S) ⊗ₜ[R] s - s ⊗ₜ[R] (1 : S)) =
(KaehlerDifferential.ideal R S).restrictScalars S := by |
apply le_antisymm
· rw [Submodule.span_le]
rintro _ ⟨s, rfl⟩
exact KaehlerDifferential.one_smul_sub_smul_one_mem_ideal _ _
· rintro x (hx : _ = _)
have : x - TensorProduct.lmul' (S := S) R x ⊗ₜ[R] (1 : S) = x := by
rw [hx, TensorProduct.zero_tmul, sub_zero]
rw [← this]
clear this hx
... | 21 |
import Mathlib.FieldTheory.Fixed
import Mathlib.FieldTheory.NormalClosure
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.GroupTheory.GroupAction.FixingSubgroup
#align_import field_theory.galois from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
open scoped Polynomial Interm... | Mathlib/FieldTheory/Galois.lean | 103 | 125 | theorem card_aut_eq_finrank [FiniteDimensional F E] [IsGalois F E] :
Fintype.card (E ≃ₐ[F] E) = finrank F E := by |
cases' Field.exists_primitive_element F E with α hα
let iso : F⟮α⟯ ≃ₐ[F] E :=
{ toFun := fun e => e.val
invFun := fun e => ⟨e, by rw [hα]; exact IntermediateField.mem_top⟩
left_inv := fun _ => by ext; rfl
right_inv := fun _ => rfl
map_mul' := fun _ _ => rfl
map_add' := fun _ _ => ... | 21 |
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
universe u v w
open Nat Function
open List
variable (F : PFunctor.{u})
-- Porting note: the ♯ tactic is never used
-- local prefix:0 "♯" => cast... | Mathlib/Data/PFunctor/Univariate/M.lean | 152 | 174 | theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
head' (x (succ n)) = head' (x (succ m)) := by |
suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
clear m n
intro n
cases' h₀ : x (succ n) with _ i₀ f₀
cases' h₁ : x 1 with _ i₁ f₁
dsimp only [head']
induction' n with n n_ih
· rw [h₁] at h₀
cases h₀
trivial
· have H := Hconsistent (succ n)
cases' h₂ : x (succ n) with _ i₂ f... | 21 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Set.Image
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Order.Monotone.Monovary
import Mathlib.Tactic.Abel
#align_impo... | Mathlib/Algebra/Order/Rearrangement.lean | 114 | 137 | theorem MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff (hfg : MonovaryOn f g s)
(hσ : { x | σ x ≠ x } ⊆ s) :
((∑ i ∈ s, f i • g (σ i)) = ∑ i ∈ s, f i • g i) ↔ MonovaryOn f (g ∘ σ) s := by |
classical
refine ⟨not_imp_not.1 fun h ↦ ?_, fun h ↦ (hfg.sum_smul_comp_perm_le_sum_smul hσ).antisymm ?_⟩
· rw [MonovaryOn] at h
push_neg at h
obtain ⟨x, hx, y, hy, hgxy, hfxy⟩ := h
set τ : Perm ι := (Equiv.swap x y).trans σ
have hτs : { x | τ x ≠ x } ⊆ s := by
refine (set_supp... | 21 |
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 | 116 | 138 | theorem setIntegral_abs_condexp_le {s : Set α} (hs : MeasurableSet[m] s) (f : α → ℝ) :
∫ x in s, |(μ[f|m]) x| ∂μ ≤ ∫ x in s, |f x| ∂μ := by |
by_cases hnm : m ≤ m0
swap
· simp_rw [condexp_of_not_le hnm, Pi.zero_apply, abs_zero, integral_zero]
positivity
by_cases hfint : Integrable f μ
swap
· simp only [condexp_undef hfint, Pi.zero_apply, abs_zero, integral_const, Algebra.id.smul_eq_mul,
mul_zero]
positivity
have : ∫ x in s, |(μ[f... | 21 |
import Mathlib.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.MvPowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.PowerSeries.Order
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
... | Mathlib/RingTheory/PowerSeries/Inverse.lean | 54 | 81 | theorem coeff_inv_aux (n : ℕ) (a : R) (φ : R⟦X⟧) :
coeff R n (inv.aux a φ) =
if n = 0 then a
else
-a *
∑ x ∈ antidiagonal n,
if x.2 < n then coeff R x.1 φ * coeff R x.2 (inv.aux a φ) else 0 := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [coeff, inv.aux, MvPowerSeries.coeff_inv_aux]
simp only [Finsupp.single_eq_zero]
split_ifs; · rfl
congr 1
symm
apply Finset.sum_nbij' (fun (a, b) ↦ (single () a, single () b))
fun (f, g) ↦ (f (), g ())
· aesop
· aesop
· ... | 21 |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
#align_import analysis.complex.schwarz from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric Set Function Filter TopologicalSpace
open scoped Topology
namespace Complex
section Space... | Mathlib/Analysis/Complex/Schwarz.lean | 65 | 88 | theorem schwarz_aux {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball c R₁))
(h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
‖dslope f c z‖ ≤ R₂ / R₁ := by |
have hR₁ : 0 < R₁ := nonempty_ball.1 ⟨z, hz⟩
suffices ∀ᶠ r in 𝓝[<] R₁, ‖dslope f c z‖ ≤ R₂ / r by
refine ge_of_tendsto ?_ this
exact (tendsto_const_nhds.div tendsto_id hR₁.ne').mono_left nhdsWithin_le_nhds
rw [mem_ball] at hz
filter_upwards [Ioo_mem_nhdsWithin_Iio ⟨hz, le_rfl⟩] with r hr
have hr₀ : ... | 21 |
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
#align_import category_theory.abelian.diagram_lemmas.four from "leanprover-community/mathlib"@"d34cbcf6c94953e965448c933cd9cc485115ebbd"
namespace CategoryTheory
open Category Limits Preadditive
namespace Abelian
va... | Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 95 | 120 | theorem epi_of_epi_of_epi_of_mono'
(hR₁ : (mk₂ (R₁.map' 1 2) (R₁.map' 2 3)).Exact)
(hR₂ : (mk₂ (R₂.map' 0 1) (R₂.map' 1 2)).Exact) (hR₂' : R₂.map' 1 3 = 0)
(h₀ : Epi (app' φ 0)) (h₂ : Epi (app' φ 2)) (h₃ : Mono (app' φ 3)) :
Epi (app' φ 1) := by |
rw [epi_iff_surjective_up_to_refinements]
intro A g₁
obtain ⟨A₁, π₁, _, f₂, h₁⟩ :=
surjective_up_to_refinements_of_epi (app' φ 2 _) (g₁ ≫ R₂.map' 1 2)
have h₂ : f₂ ≫ R₁.map' 2 3 = 0 := by
rw [← cancel_mono (app' φ 3 _), assoc, zero_comp, NatTrans.naturality, ← reassoc_of% h₁,
← R₂.map'_comp 1 2 3... | 21 |
import Mathlib.Topology.LocalAtTarget
import Mathlib.AlgebraicGeometry.Morphisms.Basic
#align_import algebraic_geometry.morphisms.open_immersion from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace... | Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean | 53 | 74 | theorem isOpenImmersion_is_local_at_target : PropertyIsLocalAtTarget @IsOpenImmersion := by |
constructor
· exact isOpenImmersion_respectsIso
· intros; infer_instance
· intro X Y f 𝒰 H
rw [isOpenImmersion_iff_stalk]
constructor
· apply (openEmbedding_iff_openEmbedding_of_iSup_eq_top 𝒰.iSup_opensRange f.1.base.2).mpr
intro i
have := ((isOpenImmersion_respectsIso.arrow_iso_iff
... | 21 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Tree.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.GCongr
import Mathlib... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 116 | 137 | theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by |
suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
have h := Nat.succ_dvd_centralBinom n
exact mod_cast this
induction' n using Nat.case_strong_induction_on with d hd
· simp
· simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
... | 21 |
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"1ac8d4304efba9d03fa720d06516fac845aa5353"
variable {M : Type*}
namespace Set
variable (M)
@[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) :... | Mathlib/Algebra/Ring/Center.lean | 46 | 67 | theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by | rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_natCast, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_one, neg_add_rev, add_mul, add_mul, add_mul,
neg_mul, one_mul, neg_mul 1, one_mul, ← neg_mul, ad... | 21 |
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
#align_import analysis.special_functions.integrals from "leanprover-community/mathlib"@"011... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 73 | 95 | theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) :
IntervalIntegrable (fun x => x ^ r) volume a b := by |
suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by
exact IntervalIntegrable.trans (this a).symm (this b)
have : ∀ c : ℝ, 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c := by
intro c hc
rw [intervalIntegrable_iff, uIoc_of_le hc]
have hderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x... | 21 |
import Mathlib.CategoryTheory.Sites.Sheaf
#align_import category_theory.sites.canonical from "leanprover-community/mathlib"@"9e7c80f638149bfb3504ba8ff48dfdbfc949fb1a"
universe v u
namespace CategoryTheory
open scoped Classical
open CategoryTheory Category Limits Sieve
variable {C : Type u} [Category.{v} C]
na... | Mathlib/CategoryTheory/Sites/Canonical.lean | 125 | 150 | theorem isSheafFor_trans (P : Cᵒᵖ ⥤ Type v) (R S : Sieve X)
(hR : Presieve.IsSheafFor P (R : Presieve X))
(hR' : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (_ : S f), Presieve.IsSeparatedFor P (R.pullback f : Presieve Y))
(hS : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (_ : R f), Presieve.IsSheafFor P (S.pullback f : Presieve Y)) :
Presieve.IsSheafFor... |
have : (bind R fun Y f _ => S.pullback f : Presieve X) ≤ S := by
rintro Z f ⟨W, f, g, hg, hf : S _, rfl⟩
apply hf
apply Presieve.isSheafFor_subsieve_aux P this
· apply isSheafFor_bind _ _ _ hR hS
intro Y f hf Z g
rw [← pullback_comp]
apply (hS (R.downward_closed hf _)).isSeparatedFor
· intr... | 21 |
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.SetTheory.Cardinal.Subfield
import Mathlib.LinearAlgebra.Dimension.RankNullity
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomput... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 171 | 193 | theorem linearIndependent_iff_card_eq_finrank_span {ι : Type*} [Fintype ι] {b : ι → V} :
LinearIndependent K b ↔ Fintype.card ι = (Set.range b).finrank K := by |
constructor
· intro h
exact (finrank_span_eq_card h).symm
· intro hc
let f := Submodule.subtype (span K (Set.range b))
let b' : ι → span K (Set.range b) := fun i =>
⟨b i, mem_span.2 fun p hp => hp (Set.mem_range_self _)⟩
have hs : ⊤ ≤ span K (Set.range b') := by
intro x
have h :... | 21 |
import Mathlib.Order.Filter.Partial
import Mathlib.Topology.Basic
#align_import topology.partial from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter
open Topology
variable {X Y : Type*} [TopologicalSpace X]
theorem rtendsto_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
... | Mathlib/Topology/Partial.lean | 61 | 83 | theorem pcontinuous_iff' {f : X →. Y} :
PContinuous f ↔ ∀ {x y} (h : y ∈ f x), PTendsto' f (𝓝 x) (𝓝 y) := by |
constructor
· intro h x y h'
simp only [ptendsto'_def, mem_nhds_iff]
rintro s ⟨t, tsubs, opent, yt⟩
exact ⟨f.preimage t, PFun.preimage_mono _ tsubs, h _ opent, ⟨y, yt, h'⟩⟩
intro hf s os
rw [isOpen_iff_nhds]
rintro x ⟨y, ys, fxy⟩ t
rw [mem_principal]
intro (h : f.preimage s ⊆ t)
change t ∈ ... | 21 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 91 | 113 | theorem cardinal_generateMeasurableRec_le (s : Set (Set α)) (i : ω₁) :
#(generateMeasurableRec s i) ≤ max #s 2 ^ aleph0.{u} := by |
apply (aleph 1).ord.out.wo.wf.induction i
intro i IH
have A := aleph0_le_aleph 1
have B : aleph 1 ≤ max #s 2 ^ aleph0.{u} :=
aleph_one_le_continuum.trans (power_le_power_right (le_max_right _ _))
have C : ℵ₀ ≤ max #s 2 ^ aleph0.{u} := A.trans B
have J : #(⋃ j : Iio i, generateMeasurableRec s j.1) ≤ max... | 21 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "leanprover-community/mathlib"@"3b88f4005dc2e28d42f974cc1ce838f0dafb39b8"
open MeasureTheory Set Filter TopologicalSpace
open scoped NNReal ENNReal Me... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 102 | 124 | theorem tendsto_IicSnd_atBot [IsFiniteMeasure ρ] {s : Set α} (hs : MeasurableSet s) :
Tendsto (fun r : ℚ ↦ ρ.IicSnd r s) atBot (𝓝 0) := by |
simp_rw [ρ.IicSnd_apply _ hs]
have h_empty : ρ (s ×ˢ ∅) = 0 := by simp only [prod_empty, measure_empty]
rw [← h_empty, ← Real.iInter_Iic_rat, prod_iInter]
suffices h_neg :
Tendsto (fun r : ℚ ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r : ℚ, s ×ˢ Iic ↑(-r)))) by
have h_inter_eq : ⋂ r : ℚ, s ×ˢ Iic ↑(-r) = ... | 21 |
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.BilinearForm.DualLattice
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Trace
#align_import ring_theory.dedekind_domain.... | Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean | 145 | 167 | theorem FiniteDimensional.exists_is_basis_integral :
∃ (s : Finset L) (b : Basis s K L), ∀ x, IsIntegral A (b x) := by |
letI := Classical.decEq L
letI : IsNoetherian K L := IsNoetherian.iff_fg.2 inferInstance
let s' := IsNoetherian.finsetBasisIndex K L
let bs' := IsNoetherian.finsetBasis K L
obtain ⟨y, hy, his'⟩ := exists_integral_multiples A K (Finset.univ.image bs')
have hy' : algebraMap A L y ≠ 0 := by
refine mt ((in... | 21 |
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.qpf.multivariate.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u
open MvFunctor
class MvQPF {n : ℕ} (F : TypeVec.{u} n → Type*) [MvFunctor F] where
P : MvPFunctor.{u} n
abs : ∀ {α}, P α → F α
... | Mathlib/Data/QPF/Multivariate/Basic.lean | 184 | 207 | theorem has_good_supp_iff {α : TypeVec n} (x : F α) :
(∀ p, LiftP p x ↔ ∀ (i), ∀ u ∈ supp x i, p i u) ↔
∃ a f, abs ⟨a, f⟩ = x ∧ ∀ i a' f', abs ⟨a', f'⟩ = x → f i '' univ ⊆ f' i '' univ := by |
constructor
· intro h
have : LiftP (supp x) x := by rw [h]; introv; exact id
rw [liftP_iff] at this
rcases this with ⟨a, f, xeq, h'⟩
refine ⟨a, f, xeq.symm, ?_⟩
intro a' f' h''
rintro hu u ⟨j, _h₂, hfi⟩
have hh : u ∈ supp x a' := by rw [← hfi]; apply h'
exact (mem_supp x _ u).mp hh ... | 21 |
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