Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 119 | 121 | theorem add (hf : Martingale f ℱ μ) (hg : Martingale g ℱ μ) : Martingale (f + g) ℱ μ := by |
refine ⟨hf.adapted.add hg.adapted, fun i j hij => ?_⟩
exact (condexp_add (hf.integrable j) (hg.integrable j)).trans ((hf.2 i j hij).add (hg.2 i j hij))
|
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot
import Mathlib.Order.Hom.Basic
#align_import data.multiset.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
universe v
... | Mathlib/Data/Multiset/Basic.lean | 2,563 | 2,565 | theorem count_inter (a : α) (s t : Multiset α) : count a (s ∩ t) = min (count a s) (count a t) := by |
apply @Nat.add_left_cancel (count a (s - t))
rw [← count_add, sub_add_inter, count_sub, Nat.sub_add_min_cancel]
|
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Algebra.Module.StrongTopology
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Tactic.SuppressCompilation
#align_import analysis.nor... | Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean | 307 | 313 | theorem opNorm_le_of_unit_norm [NormedSpace ℝ E] [NormedSpace ℝ F] {f : E →L[ℝ] F} {C : ℝ}
(hC : 0 ≤ C) (hf : ∀ x, ‖x‖ = 1 → ‖f x‖ ≤ C) : ‖f‖ ≤ C := by |
refine opNorm_le_bound' f hC fun x hx => ?_
have H₁ : ‖‖x‖⁻¹ • x‖ = 1 := by rw [norm_smul, norm_inv, norm_norm, inv_mul_cancel hx]
have H₂ := hf _ H₁
rwa [map_smul, norm_smul, norm_inv, norm_norm, ← div_eq_inv_mul, _root_.div_le_iff] at H₂
exact (norm_nonneg x).lt_of_ne' hx
|
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Localization.AsSubring
import Mathlib.Algebra.Ring.Subring.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
#align_import ring_theory.valuation.valuation_subring from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0d... | Mathlib/RingTheory/Valuation/ValuationSubring.lean | 218 | 221 | theorem valuation_lt_one_iff (a : A) : a ∈ LocalRing.maximalIdeal A ↔ A.valuation a < 1 := by |
rw [LocalRing.mem_maximalIdeal]
dsimp [nonunits]; rw [valuation_eq_one_iff]
exact (A.valuation_le_one a).lt_iff_ne.symm
|
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
open Nat Function
namespace List
theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : ∀ {a'}, a' ∈ l → R a a' :=
(pairwise_cons.1 p).1 _
theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
(pairwise_cons.1 p).2
theorem... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 168 | 171 | theorem pairwise_bind {R : β → β → Prop} {l : List α} {f : α → List β} :
List.Pairwise R (l.bind f) ↔
(∀ a ∈ l, Pairwise R (f a)) ∧ Pairwise (fun a₁ a₂ => ∀ x ∈ f a₁, ∀ y ∈ f a₂, R x y) l := by |
simp [List.bind, pairwise_join, pairwise_map]
|
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 1,380 | 1,386 | theorem snorm_exponent_top_lim_le_liminf_snorm_exponent_top {ι} [Nonempty ι] [Countable ι]
[LinearOrder ι] {f : ι → α → F} {f_lim : α → F}
(h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
snorm f_lim ∞ μ ≤ atTop.liminf fun n => snorm (f n) ∞ μ := by |
rw [snorm_exponent_top_lim_eq_essSup_liminf h_lim]
simp_rw [snorm_exponent_top, snormEssSup]
exact ENNReal.essSup_liminf_le fun n => fun x => (‖f n x‖₊ : ℝ≥0∞)
|
import Mathlib.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function Set
namespace Cardinal
variable {α : Type u} {c d : Cardinal.{u}}
noncomputable def toNat : Cardinal →*₀ ℕ :=
ENat.toNat.com... | Mathlib/SetTheory/Cardinal/ToNat.lean | 134 | 134 | theorem mk_toNat_eq_card [Fintype α] : toNat #α = Fintype.card α := by | simp
|
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Complex
theorem ... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 329 | 330 | theorem abs_cpow_real (x : ℂ) (y : ℝ) : abs (x ^ (y : ℂ)) = Complex.abs x ^ y := by |
rw [abs_cpow_of_imp] <;> simp
|
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 | 602 | 603 | theorem iSup_neBot {f : ι → Filter α} : (⨆ i, f i).NeBot ↔ ∃ i, (f i).NeBot := by |
simp [neBot_iff]
|
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 | 196 | 201 | theorem char_dvd_card_solutions_of_add_lt {f₁ f₂ : MvPolynomial σ K}
(h : f₁.totalDegree + f₂.totalDegree < Fintype.card σ) :
p ∣ Fintype.card { x : σ → K // eval x f₁ = 0 ∧ eval x f₂ = 0 } := by |
let F : Bool → MvPolynomial σ K := fun b => cond b f₂ f₁
have : (∑ b : Bool, (F b).totalDegree) < Fintype.card σ := (add_comm _ _).trans_lt h
simpa only [Bool.forall_bool] using char_dvd_card_solutions_of_fintype_sum_lt p this
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 611 | 613 | theorem lt_iterate_pos_iff {x : ℝ} {m : ℤ} {n : ℕ} (hn : 0 < n) :
x + n * m < f^[n] x ↔ x + m < f x := by |
simpa only [not_le] using not_congr (f.iterate_pos_le_iff hn)
|
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.MeasureTheory.Group.FundamentalDomain
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.RingTheory.Localization.Module
#align_import algebra.module.zlattice from "leanprover-community/mathlib"@"a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3"
n... | Mathlib/Algebra/Module/Zlattice/Basic.lean | 192 | 197 | theorem fract_eq_fract (m n : E) : fract b m = fract b n ↔ -m + n ∈ span ℤ (Set.range b) := by |
classical
rw [eq_comm, Basis.ext_elem_iff b]
simp_rw [repr_fract_apply, Int.fract_eq_fract, eq_comm, Basis.mem_span_iff_repr_mem,
sub_eq_neg_add, map_add, map_neg, Finsupp.coe_add, Finsupp.coe_neg, Pi.add_apply,
Pi.neg_apply, ← eq_intCast (algebraMap ℤ K) _, Set.mem_range]
|
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Common
#align_import data.fin.basic from "leanprover-community/mathlib"@"3a2b5524a138b5d0b818b858b516d4ac8a484b03"
assert_not_exists Monoid
universe u v
open Fin Na... | Mathlib/Data/Fin/Basic.lean | 1,865 | 1,865 | theorem coe_fin_one (a : Fin 1) : (a : ℕ) = 0 := by | simp [Subsingleton.elim a 0]
|
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set.Countable
import Mathlib.Logic.Small.Set
import Mathlib.Order.SuccPred.CompleteLinearOrder
import Mathlib.SetTheory.Cardinal.SchroederBernstein
#align_import set_theory.cardinal.basic f... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,291 | 1,292 | theorem aleph0_le_lift {c : Cardinal.{u}} : ℵ₀ ≤ lift.{v} c ↔ ℵ₀ ≤ c := by |
rw [← lift_aleph0.{u,v}, lift_le]
|
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}... | Mathlib/RingTheory/JacobsonIdeal.lean | 134 | 143 | theorem eq_jacobson_iff_sInf_maximal :
I.jacobson = I ↔ ∃ M : Set (Ideal R), (∀ J ∈ M, IsMaximal J ∨ J = ⊤) ∧ I = sInf M := by |
use fun hI => ⟨{ J : Ideal R | I ≤ J ∧ J.IsMaximal }, ⟨fun _ hJ => Or.inl hJ.right, hI.symm⟩⟩
rintro ⟨M, hM, hInf⟩
refine le_antisymm (fun x hx => ?_) le_jacobson
rw [hInf, mem_sInf]
intro I hI
cases' hM I hI with is_max is_top
· exact (mem_sInf.1 hx) ⟨le_sInf_iff.1 (le_of_eq hInf) I hI, is_max⟩
· exac... |
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 | 122 | 133 | theorem sym2_eq_image : s.sym2 = (s ×ˢ s).image Sym2.mk := by |
ext z
refine z.ind fun x y ↦ ?_
rw [mk_mem_sym2_iff, mem_image]
constructor
· intro h
use (x, y)
simp only [mem_product, h, and_self, true_and]
· rintro ⟨⟨a, b⟩, h⟩
simp only [mem_product, Sym2.eq_iff] at h
obtain ⟨h, (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)⟩ := h
<;> simp [h]
|
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheory.Torsion
import Mathlib.RingTheory.Coprime.Ideal
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Set.Lattice
#align_import algebra.module.torsion from "leanpr... | Mathlib/Algebra/Module/Torsion.lean | 281 | 289 | theorem torsionBySet_eq_torsionBySet_span :
torsionBySet R M s = torsionBySet R M (Ideal.span s) := by |
refine le_antisymm (fun x hx => ?_) (torsionBySet_le_torsionBySet_of_subset subset_span)
rw [mem_torsionBySet_iff] at hx ⊢
suffices Ideal.span s ≤ Ideal.torsionOf R M x by
rintro ⟨a, ha⟩
exact this ha
rw [Ideal.span_le]
exact fun a ha => hx ⟨a, ha⟩
|
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 28 | 30 | theorem coprime_multiset_prod_left_iff {m : Multiset ℕ} {k : ℕ} :
Coprime m.prod k ↔ ∀ n ∈ m, Coprime n k := by |
induction m using Quotient.inductionOn; simpa using coprime_list_prod_left_iff
|
import Mathlib.Order.Filter.Basic
import Mathlib.Algebra.Module.Pi
#align_import order.filter.germ from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
namespace Filter
variable {α β γ δ : Type*} {l : Filter α} {f g h : α → β}
theorem const_eventuallyEq' [NeBot l] {a b : β} : (∀ᶠ _ in ... | Mathlib/Order/Filter/Germ.lean | 209 | 211 | theorem map_id : map id = (id : Germ l β → Germ l β) := by |
ext ⟨f⟩
rfl
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Interval.Finset
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tactic.Linarith
#align_import algebra.big_operators.intervals from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open Nat
variable {α M ... | Mathlib/Algebra/BigOperators/Intervals.lean | 222 | 237 | theorem prod_Ico_reflect (f : ℕ → M) (k : ℕ) {m n : ℕ} (h : m ≤ n + 1) :
(∏ j ∈ Ico k m, f (n - j)) = ∏ j ∈ Ico (n + 1 - m) (n + 1 - k), f j := by |
have : ∀ i < m, i ≤ n := by
intro i hi
exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)
cases' lt_or_le k m with hkm hkm
· rw [← Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]
refine (prod_image ?_).symm
simp only [mem_Ico]
rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij
rw [← tsub... |
import Mathlib.Data.Fintype.Parity
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.FieldTheory.Finite.Basic
#align_import number_theory.legendre_symbol.quadratic_char.basic from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section quadraticChar
open MulChar
... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 176 | 180 | theorem quadraticChar_eq_neg_one_iff_not_one {a : F} (ha : a ≠ 0) :
quadraticChar F a = -1 ↔ ¬quadraticChar F a = 1 := by |
refine ⟨fun h => ?_, fun h₂ => (or_iff_right h₂).mp (quadraticChar_dichotomy ha)⟩
rw [h]
norm_num
|
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4972407cd4f1222"
noncomputable section
@[ext]
structure Cubic (R : Type*) where
(a b c d : R)
#align cubic Cubic
namespace Cubic
open Cubic Polynomial
open Polynom... | Mathlib/Algebra/CubicDiscriminant.lean | 567 | 575 | theorem disc_eq_prod_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
φ P.disc = (φ P.a * φ P.a * (x - y) * (x - z) * (y - z)) ^ 2 := by |
simp only [disc, RingHom.map_add, RingHom.map_sub, RingHom.map_mul, map_pow]
-- Porting note: Replaced `simp only [RingHom.map_one, map_bit0, map_bit1]` with f4, f18, f27
have f4 : φ 4 = 4 := map_natCast φ 4
have f18 : φ 18 = 18 := map_natCast φ 18
have f27 : φ 27 = 27 := map_natCast φ 27
rw [f4, f18, f27,... |
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
theorem isCobounde... | Mathlib/Order/LiminfLimsup.lean | 1,191 | 1,193 | theorem sdiff_liminf (a : α) : a \ liminf u f = limsup (fun b => a \ u b) f := by |
rw [← compl_inj_iff]
simp only [sdiff_eq, limsup_compl, (· ∘ ·), compl_inf, compl_compl, sup_liminf]
|
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.Dual
#align_import linear_algebra.clifford_algebra.contraction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 138 | 141 | theorem contractRight_mul_ι (a : M) (b : CliffordAlgebra Q) :
b * ι Q a⌊d = d a • b - b⌊d * ι Q a := by |
rw [contractRight_eq, reverse.map_mul, reverse_ι, contractLeft_ι_mul, map_sub, map_smul,
reverse_reverse, reverse.map_mul, reverse_ι, contractRight_eq]
|
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 281 | 282 | theorem inv_inv_of_invertible [Invertible A] : A⁻¹⁻¹ = A := by |
simp only [← invOf_eq_nonsing_inv, invOf_invOf]
|
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... |
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 | 450 | 453 | theorem le_symmDiff_iff_left : a ≤ a ∆ b ↔ Disjoint a b := by |
refine ⟨fun h => ?_, fun h => h.symmDiff_eq_sup.symm ▸ le_sup_left⟩
rw [symmDiff_eq_sup_sdiff_inf] at h
exact disjoint_iff_inf_le.mpr (le_sdiff_iff.1 <| inf_le_of_left_le h).le
|
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 | 1,264 | 1,270 | theorem integrable_of_integrable_trim (hm : m ≤ m0) (hf_int : Integrable f (μ'.trim hm)) :
Integrable f μ' := by |
obtain ⟨hf_meas_ae, hf⟩ := hf_int
refine ⟨aestronglyMeasurable_of_aestronglyMeasurable_trim hm hf_meas_ae, ?_⟩
rw [HasFiniteIntegral] at hf ⊢
rwa [lintegral_trim_ae hm _] at hf
exact AEStronglyMeasurable.ennnorm hf_meas_ae
|
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 205 | 218 | theorem invertible_iff_generator_nonzero (I : FractionalIdeal R₁⁰ K)
[Submodule.IsPrincipal (I : Submodule R₁ K)] :
I * I⁻¹ = 1 ↔ generator (I : Submodule R₁ K) ≠ 0 := by |
constructor
· intro hI hg
apply ne_zero_of_mul_eq_one _ _ hI
rw [eq_spanSingleton_of_principal I, hg, spanSingleton_zero]
· intro hg
apply invertible_of_principal
rw [eq_spanSingleton_of_principal I]
intro hI
have := mem_spanSingleton_self R₁⁰ (generator (I : Submodule R₁ K))
rw [hI, ... |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Function Nat
namespace Int
variable {a b : ℤ} {n : ℕ}
theorem natAbs_eq_iff_mul_self_eq {a b : ℤ} : a.natAbs = b.natAbs ↔ a * a = b * b := by
rw [← a... | Mathlib/Data/Int/Order/Lemmas.lean | 45 | 50 | theorem dvd_div_of_mul_dvd {a b c : ℤ} (h : a * b ∣ c) : b ∣ c / a := by |
rcases eq_or_ne a 0 with (rfl | ha)
· simp only [Int.ediv_zero, Int.dvd_zero]
rcases h with ⟨d, rfl⟩
refine ⟨d, ?_⟩
rw [mul_assoc, Int.mul_ediv_cancel_left _ ha]
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,556 | 1,557 | theorem dist_self_div_right (a b : E) : dist a (a / b) = ‖b‖ := by |
rw [div_eq_mul_inv, dist_self_mul_right, norm_inv']
|
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 411 | 413 | theorem IsCompact.prod_nhdsSet_eq_biSup {K : Set Y} (hK : IsCompact K) (l : Filter X) :
l ×ˢ (𝓝ˢ K) = ⨆ y ∈ K, l ×ˢ 𝓝 y := by |
simp only [prod_comm (f := l), hK.nhdsSet_prod_eq_biSup, map_iSup]
|
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Data.Finsupp.Antidiagonal
import Mathlib.Order.SymmDiff
import Mathlib.RingTheory.Adjoin.Basic
#align_import data.mv_polynomial.basic fr... | Mathlib/Algebra/MvPolynomial/Basic.lean | 976 | 978 | theorem constantCoeff_monomial [DecidableEq σ] (d : σ →₀ ℕ) (r : R) :
constantCoeff (monomial d r) = if d = 0 then r else 0 := by |
rw [constantCoeff_eq, coeff_monomial]
|
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 1,094 | 1,100 | theorem norm_compLp_sub_le (hg : LipschitzWith c g) (g0 : g 0 = 0) (f f' : Lp E p μ) :
‖hg.compLp g0 f - hg.compLp g0 f'‖ ≤ c * ‖f - f'‖ := by |
apply Lp.norm_le_mul_norm_of_ae_le_mul
filter_upwards [hg.coeFn_compLp g0 f, hg.coeFn_compLp g0 f',
Lp.coeFn_sub (hg.compLp g0 f) (hg.compLp g0 f'), Lp.coeFn_sub f f'] with a ha1 ha2 ha3 ha4
simp only [ha1, ha2, ha3, ha4, ← dist_eq_norm, Pi.sub_apply, Function.comp_apply]
exact hg.dist_le_mul (f a) (f' a)
|
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 275 | 277 | theorem polar_smul_left (a : R) (x y : M) : polar Q (a • x) y = a * polar Q x y := by |
obtain ⟨B, h⟩ := Q.exists_companion
simp_rw [polar, h, Q.map_smul, LinearMap.map_smul₂, sub_sub, add_sub_cancel_left, smul_eq_mul]
|
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.Disjoint
#align_import data.real.basic from "leanprover-community/mathlib"@"cb42593171ba005beaaf4549fcfe0dece9ada4c9"
open scoped Classical
open Pointwise CauSeq
namespace Real
... | Mathlib/Data/Real/Archimedean.lean | 281 | 283 | theorem sInf_nonneg (S : Set ℝ) (hS : ∀ x ∈ S, (0 : ℝ) ≤ x) : 0 ≤ sInf S := by |
rcases S.eq_empty_or_nonempty with (rfl | hS₂)
exacts [sInf_empty.ge, le_csInf hS₂ hS]
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Topology.Constructions
#align_import measure_theory.constructions.pi from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Function Set MeasureTheory... | Mathlib/MeasureTheory/Constructions/Pi.lean | 182 | 184 | theorem piPremeasure_pi_eval {s : Set (∀ i, α i)} :
piPremeasure m (pi univ fun i => eval i '' s) = piPremeasure m s := by |
simp only [eval, piPremeasure_pi']; rfl
|
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α β : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 168 | 176 | theorem eq_iff_join_eq (L L' : List (List α)) :
L = L' ↔ L.join = L'.join ∧ map length L = map length L' := by |
refine ⟨fun H => by simp [H], ?_⟩
rintro ⟨join_eq, length_eq⟩
apply ext_get
· have : length (map length L) = length (map length L') := by rw [length_eq]
simpa using this
· intro n h₁ h₂
rw [← drop_take_succ_join_eq_get', ← drop_take_succ_join_eq_get', join_eq, length_eq]
|
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.Sylow
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.TFAE
#align_import group_theory.nilpotent from "leanprover-community/mathlib"@"2bbc7e3884ba234309d2a43b19144... | Mathlib/GroupTheory/Nilpotent.lean | 805 | 810 | theorem nilpotencyClass_pi [Fintype η] [∀ i, IsNilpotent (Gs i)] :
Group.nilpotencyClass (∀ i, Gs i) = Finset.univ.sup fun i => Group.nilpotencyClass (Gs i) := by |
apply eq_of_forall_ge_iff
intro k
simp only [Finset.sup_le_iff, ← lowerCentralSeries_eq_bot_iff_nilpotencyClass_le,
lowerCentralSeries_pi_of_finite, pi_eq_bot_iff, Finset.mem_univ, true_imp_iff]
|
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 | 61 | 71 | theorem list_reverse_induction (p : List α → Prop) (base : p [])
(ind : ∀ (l : List α) (e : α), p l → p (l ++ [e])) : (∀ (l : List α), p l) := by |
let q := fun l ↦ p (reverse l)
have pq : ∀ l, p (reverse l) → q l := by simp only [q, reverse_reverse]; intro; exact id
have qp : ∀ l, q (reverse l) → p l := by simp only [q, reverse_reverse]; intro; exact id
intro l
apply qp
generalize (reverse l) = l
induction' l with head tail ih
· apply pq; simp on... |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 159 | 175 | theorem le_degrees_add {p q : MvPolynomial σ R} (h : p.degrees.Disjoint q.degrees) :
p.degrees ≤ (p + q).degrees := by |
classical
apply Finset.sup_le
intro d hd
rw [Multiset.disjoint_iff_ne] at h
obtain rfl | h0 := eq_or_ne d 0
· rw [toMultiset_zero]; apply Multiset.zero_le
· refine Finset.le_sup_of_le (b := d) ?_ le_rfl
rw [mem_support_iff, coeff_add]
suffices q.coeff d = 0 by rwa [this, add_zero, coeff, ← Finsup... |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {α β R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 169 | 179 | theorem isDiag_fromBlocks_iff [Zero α] {A : Matrix m m α} {B : Matrix m n α} {C : Matrix n m α}
{D : Matrix n n α} : (A.fromBlocks B C D).IsDiag ↔ A.IsDiag ∧ B = 0 ∧ C = 0 ∧ D.IsDiag := by |
constructor
· intro h
refine ⟨fun i j hij => ?_, ext fun i j => ?_, ext fun i j => ?_, fun i j hij => ?_⟩
· exact h (Sum.inl_injective.ne hij)
· exact h Sum.inl_ne_inr
· exact h Sum.inr_ne_inl
· exact h (Sum.inr_injective.ne hij)
· rintro ⟨ha, hb, hc, hd⟩
convert IsDiag.fromBlocks ha hd
|
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.Set.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.double_coset from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
-- Porting note: removed import
-- import Mathlib.Tac... | Mathlib/GroupTheory/DoubleCoset.lean | 105 | 114 | theorem rel_bot_eq_right_group_rel (H : Subgroup G) :
(setoid ↑H ↑(⊥ : Subgroup G)).Rel = (QuotientGroup.rightRel H).Rel := by |
ext a b
rw [rel_iff, Setoid.Rel, QuotientGroup.rightRel_apply]
constructor
· rintro ⟨b, hb, a, rfl : a = 1, rfl⟩
change b * a * 1 * a⁻¹ ∈ H
rwa [mul_one, mul_inv_cancel_right]
· rintro (h : b * a⁻¹ ∈ H)
exact ⟨b * a⁻¹, h, 1, rfl, by rw [mul_one, inv_mul_cancel_right]⟩
|
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Order.Nonneg.Field
import Mathlib.Algebra.Order.Nonneg.Floor
import Mathlib.Data.Real.Pointwise
import Mathlib.Order.ConditionallyCompleteLattice.Group
imp... | Mathlib/Data/Real/NNReal.lean | 933 | 934 | theorem lt_inv_iff_mul_lt {r p : ℝ≥0} (h : p ≠ 0) : r < p⁻¹ ↔ r * p < 1 := by |
rw [← mul_lt_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h, mul_comm]
|
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 125 | 134 | theorem sdiff_compl_neighborFinset_inter_eq {v w : V} (h : G.Adj v w) :
((G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ) \ ({w} ∪ {v}) =
(G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ := by |
ext
simp only [and_imp, mem_union, mem_sdiff, mem_compl, and_iff_left_iff_imp, mem_neighborFinset,
mem_inter, mem_singleton]
rintro hnv hnw (rfl | rfl)
· exact hnv h
· apply hnw
rwa [adj_comm]
|
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Topological... | Mathlib/Probability/Martingale/Upcrossing.lean | 219 | 223 | theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by |
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ
|
import Mathlib.Data.List.Range
import Mathlib.Data.Multiset.Range
#align_import data.multiset.nodup from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
open Function List
variable {α β γ : Type*} {r : α → α → Prop} {s t : Multiset α} {a : α}
-- nodup
def Nodup (s ... | Mathlib/Data/Multiset/Nodup.lean | 190 | 192 | theorem Nodup.mem_erase_iff [DecidableEq α] {a b : α} {l} (d : Nodup l) :
a ∈ l.erase b ↔ a ≠ b ∧ a ∈ l := by |
rw [d.erase_eq_filter b, mem_filter, and_comm]
|
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.locally_convex.with_seminorms from "leanprover-community/mathlib"@"... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 350 | 357 | theorem WithSeminorms.separating_of_T1 [T1Space E] (hp : WithSeminorms p) (x : E) (hx : x ≠ 0) :
∃ i, p i x ≠ 0 := by |
have := ((t1Space_TFAE E).out 0 9).mp (inferInstanceAs <| T1Space E)
by_contra! h
refine hx (this ?_)
rw [hp.hasBasis_zero_ball.specializes_iff]
rintro ⟨s, r⟩ (hr : 0 < r)
simp only [ball_finset_sup_eq_iInter _ _ _ hr, mem_iInter₂, mem_ball_zero, h, hr, forall_true_iff]
|
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Adjunction.Over
#align_import category_theory.limits.shapes.diagonal from "leanprover-community/mathlib"@"f6bab67886fb92c3e2f539cc90a83815f69a189d"
open CategoryTheory
noncomput... | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 272 | 275 | theorem pullback.diagonal_comp (f : X ⟶ Y) (g : Y ⟶ Z) [HasPullback f f] [HasPullback g g]
[HasPullback (f ≫ g) (f ≫ g)] :
diagonal (f ≫ g) = diagonal f ≫ (pullbackDiagonalMapIdIso f f g).inv ≫ pullback.snd := by |
ext <;> simp
|
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.interval from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
open Function OrderDual Set
variable {α β γ δ : Type*} {ι : Sort*} {κ : ι → Sort*}
@[ext (flat :=... | Mathlib/Order/Interval/Basic.lean | 420 | 423 | theorem dual_map (f : α →o β) (s : Interval α) : dual (s.map f) = s.dual.map (OrderHom.dual f) := by |
cases s
· rfl
· exact WithBot.map_comm rfl _
|
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.idempotents.karoubi from "leanprover-community/mathlib"@"200eda15d8ff5669854ff6bcc10aaf37cb70498f"
noncomputable section
open CategoryT... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 291 | 292 | theorem decompId_i_toKaroubi (X : C) : decompId_i ((toKaroubi C).obj X) = 𝟙 _ := by |
rfl
|
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 1,094 | 1,100 | theorem IsBigOWith.add (h₁ : IsBigOWith c₁ l f₁ g) (h₂ : IsBigOWith c₂ l f₂ g) :
IsBigOWith (c₁ + c₂) l (fun x => f₁ x + f₂ x) g := by |
rw [IsBigOWith_def] at *
filter_upwards [h₁, h₂] with x hx₁ hx₂ using
calc
‖f₁ x + f₂ x‖ ≤ c₁ * ‖g x‖ + c₂ * ‖g x‖ := norm_add_le_of_le hx₁ hx₂
_ = (c₁ + c₂) * ‖g x‖ := (add_mul _ _ _).symm
|
import Mathlib.Algebra.Algebra.Tower
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Algebra.Field.Equiv
#align_import ring_theory.localization.fraction_ring from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
variable (R : Type*) [CommRing R] {M : Submonoid R} (S : Type*) [... | Mathlib/RingTheory/Localization/FractionRing.lean | 180 | 184 | theorem mk'_eq_zero_iff_eq_zero [Algebra R K] [IsFractionRing R K] {x : R} {y : nonZeroDivisors R} :
mk' K x y = 0 ↔ x = 0 := by |
refine ⟨fun hxy => ?_, fun h => by rw [h, mk'_zero]⟩
simp_rw [mk'_eq_zero_iff, mul_left_coe_nonZeroDivisors_eq_zero_iff] at hxy
exact (exists_const _).mp hxy
|
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈... | Mathlib/Topology/Basic.lean | 870 | 873 | theorem nhds_basis_opens' (x : X) :
(𝓝 x).HasBasis (fun s : Set X => s ∈ 𝓝 x ∧ IsOpen s) fun x => x := by |
convert nhds_basis_opens x using 2
exact and_congr_left_iff.2 IsOpen.mem_nhds_iff
|
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 | 688 | 690 | theorem toReal_coe_eq_self_sub_two_pi_iff {θ : ℝ} :
(θ : Angle).toReal = θ - 2 * π ↔ θ ∈ Set.Ioc π (3 * π) := by |
convert @toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff θ 1 <;> norm_num
|
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
open Nat Function
namespace List
theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : ∀ {a'}, a' ∈ l → R a a' :=
(pairwise_cons.1 p).1 _
theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
(pairwise_cons.1 p).2
theorem... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 205 | 219 | theorem map_get_sublist {l : List α} {is : List (Fin l.length)} (h : is.Pairwise (·.val < ·.val)) :
is.map (get l) <+ l := by |
suffices ∀ n l', l' = l.drop n → (∀ i ∈ is, n ≤ i) → map (get l) is <+ l'
from this 0 l (by simp) (by simp)
intro n l' hl' his
induction is generalizing n l' with
| nil => simp
| cons hd tl IH =>
simp; cases hl'
have := IH h.of_cons (hd+1) _ rfl (pairwise_cons.mp h).1
specialize his hd (.head... |
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 | 471 | 477 | theorem continuous_sub_right (a : ℝ≥0∞) : Continuous fun x : ℝ≥0∞ => x - a := by |
by_cases a_infty : a = ∞
· simp [a_infty, continuous_const]
· rw [show (fun x => x - a) = (fun p : ℝ≥0∞ × ℝ≥0∞ => p.fst - p.snd) ∘ fun x => ⟨x, a⟩ by rfl]
apply ContinuousOn.comp_continuous continuousOn_sub (continuous_id'.prod_mk continuous_const)
intro x
simp only [a_infty, Ne, mem_setOf_eq, Prod.m... |
import Mathlib.Algebra.FreeAlgebra
import Mathlib.Algebra.RingQuot
import Mathlib.Algebra.TrivSqZeroExt
import Mathlib.Algebra.Algebra.Operations
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import linear_algebra.tensor_algebra.basic from "leanprover-community/mathlib"@"b8d2eaa69d69ce8f03179a5cda774fc0cde984... | Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean | 295 | 304 | theorem ι_eq_algebraMap_iff (x : M) (r : R) : ι R x = algebraMap R _ r ↔ x = 0 ∧ r = 0 := by |
refine ⟨fun h => ?_, ?_⟩
· letI : Module Rᵐᵒᵖ M := Module.compHom _ ((RingHom.id R).fromOpposite mul_comm)
haveI : IsCentralScalar R M := ⟨fun r m => rfl⟩
have hf0 : toTrivSqZeroExt (ι R x) = (0, x) := lift_ι_apply _ _
rw [h, AlgHom.commutes] at hf0
have : r = 0 ∧ 0 = x := Prod.ext_iff.1 hf0
ex... |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.discrete_valuation_ring.tfae from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variab... | Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean | 92 | 150 | theorem maximalIdeal_isPrincipal_of_isDedekindDomain [LocalRing R] [IsDomain R]
[IsDedekindDomain R] : (maximalIdeal R).IsPrincipal := by |
classical
by_cases ne_bot : maximalIdeal R = ⊥
· rw [ne_bot]; infer_instance
obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ maximalIdeal R, a ≠ (0 : R) := by
by_contra! h'; apply ne_bot; rwa [eq_bot_iff]
have hle : Ideal.span {a} ≤ maximalIdeal R := by rwa [Ideal.span_le, Set.singleton_subset_iff]
have : (Ideal.span {a}... |
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
#align_import category_theory.limits.shapes.biproducts from "leanprover-community/mathlib"@"ac3ae212f394f508df43e37aa093722fa9b65d31"
noncomputab... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 1,023 | 1,026 | theorem biproduct.ι_matrix (m : ∀ j k, f j ⟶ g k) (j : J) :
biproduct.ι f j ≫ biproduct.matrix m = biproduct.lift fun k => m j k := by |
ext
simp [biproduct.matrix]
|
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dfinsupp from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
variable {ι : Type*} {R : Type*} {S : Type*} {M : ι → Type*} {N : Type*}
n... | Mathlib/LinearAlgebra/DFinsupp.lean | 463 | 471 | theorem lsum_comp_mapRange_toSpanSingleton [∀ m : R, Decidable (m ≠ 0)] (p : ι → Submodule R N)
{v : ι → N} (hv : ∀ i : ι, v i ∈ p i) :
(lsum ℕ (M := fun i ↦ ↥(p i)) fun i => (p i).subtype : _ →ₗ[R] _).comp
((mapRange.linearMap fun i => LinearMap.toSpanSingleton R (↥(p i)) ⟨v i, hv i⟩ :
_ ... |
ext
simp
|
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 102 | 102 | theorem snorm_exponent_top {f : α → F} : snorm f ∞ μ = snormEssSup f μ := by | simp [snorm]
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped Classical Topology
open Filter Asymptotics Set
variable {𝕜... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 228 | 231 | theorem iteratedFDeriv_eq_equiv_comp : iteratedFDeriv 𝕜 n f =
ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F ∘ iteratedDeriv n f := by |
rw [iteratedDeriv_eq_equiv_comp, ← Function.comp.assoc, LinearIsometryEquiv.self_comp_symm,
Function.id_comp]
|
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.ideal from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
namespace IsLocalization
section CommRing
variable {R : Type*} [CommRing R] (M : Submonoid R... | Mathlib/RingTheory/Localization/Ideal.lean | 209 | 214 | theorem bot_lt_comap_prime [IsDomain R] (hM : M ≤ R⁰) (p : Ideal S) [hpp : p.IsPrime]
(hp0 : p ≠ ⊥) : ⊥ < Ideal.comap (algebraMap R S) p := by |
haveI : IsDomain S := isDomain_of_le_nonZeroDivisors _ hM
rw [← Ideal.comap_bot_of_injective (algebraMap R S) (IsLocalization.injective _ hM)]
convert (orderIsoOfPrime M S).lt_iff_lt.mpr (show (⟨⊥, Ideal.bot_prime⟩ :
{ p : Ideal S // p.IsPrime }) < ⟨p, hpp⟩ from hp0.bot_lt)
|
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.monoidal.functor from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
open CategoryTheory
universe v₁ v₂ v₃ u... | Mathlib/CategoryTheory/Monoidal/Functor.lean | 178 | 182 | theorem LaxMonoidalFunctor.associativity_inv (F : LaxMonoidalFunctor C D) (X Y Z : C) :
F.obj X ◁ F.μ Y Z ≫ F.μ X (Y ⊗ Z) ≫ F.map (α_ X Y Z).inv =
(α_ (F.obj X) (F.obj Y) (F.obj Z)).inv ≫ F.μ X Y ▷ F.obj Z ≫ F.μ (X ⊗ Y) Z := by |
rw [Iso.eq_inv_comp, ← F.associativity_assoc, ← F.toFunctor.map_comp, Iso.hom_inv_id,
F.toFunctor.map_id, comp_id]
|
import Mathlib.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
#align_import probability.independence.zero_one from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
open MeasureTheory MeasurableSpace
open scoped MeasureTheory ENNReal
namespace Probabili... | Mathlib/Probability/Independence/ZeroOne.lean | 64 | 74 | theorem condexp_eq_zero_or_one_of_condIndepSet_self
[StandardBorelSpace Ω] [Nonempty Ω]
(hm : m ≤ m0) [hμ : IsFiniteMeasure μ] {t : Set Ω} (ht : MeasurableSet t)
(h_indep : CondIndepSet m hm t t μ) :
∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 := by |
have h := ae_of_ae_trim hm (kernel.measure_eq_zero_or_one_of_indepSet_self h_indep)
filter_upwards [condexpKernel_ae_eq_condexp hm ht, h] with ω hω_eq hω
rw [← hω_eq, ENNReal.toReal_eq_zero_iff, ENNReal.toReal_eq_one_iff]
cases hω with
| inl h => exact Or.inl (Or.inl h)
| inr h => exact Or.inr h
|
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set
open Convex Pointwise
variable {𝕜 𝕝 E F β : Type*}
open Function Se... | Mathlib/Analysis/Convex/Strict.lean | 222 | 227 | theorem StrictConvex.preimage_add_right (hs : StrictConvex 𝕜 s) (z : E) :
StrictConvex 𝕜 ((fun x => z + x) ⁻¹' s) := by |
intro x hx y hy hxy a b ha hb hab
refine preimage_interior_subset_interior_preimage (continuous_add_left _) ?_
have h := hs hx hy ((add_right_injective _).ne hxy) ha hb hab
rwa [smul_add, smul_add, add_add_add_comm, ← _root_.add_smul, hab, one_smul] at h
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.PUnitInstances
import Mathlib.GroupTheory.Congruence.Basic
open FreeMonoid Function List Set
namespace Monoid
@[to_additive "The minimal additive congruence relation `c` on `FreeAddMonoid (M ⊕ N)`... | Mathlib/GroupTheory/Coprod/Basic.lean | 520 | 521 | theorem snd_comp_toProd : (MonoidHom.snd M N).comp toProd = snd := by |
rw [← fst_prod_snd, MonoidHom.snd_comp_prod]
|
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 | 171 | 173 | theorem log_pos_iff (hx : 0 < x) : 0 < log x ↔ 1 < x := by |
rw [← log_one]
exact log_lt_log_iff zero_lt_one hx
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,595 | 1,598 | theorem abs_dist_sub_le_dist_mul_mul (a₁ a₂ b₁ b₂ : E) :
|dist a₁ b₁ - dist a₂ b₂| ≤ dist (a₁ * a₂) (b₁ * b₂) := by |
simpa only [dist_mul_left, dist_mul_right, dist_comm b₂] using
abs_dist_sub_le (a₁ * a₂) (b₁ * b₂) (b₁ * a₂)
|
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... |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 1,871 | 1,878 | theorem range_toLp [NormedField 𝕜] [NormedSpace 𝕜 E] :
(LinearMap.range (toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ)).toAddSubgroup =
MeasureTheory.Lp.boundedContinuousFunction E p μ := by |
refine SetLike.ext' ?_
have := (linearIsometryBoundedOfCompact α E 𝕜).surjective
convert Function.Surjective.range_comp this (BoundedContinuousFunction.toLp (E := E) p μ 𝕜)
rw [← BoundedContinuousFunction.range_toLp p μ (𝕜 := 𝕜), Submodule.coe_toAddSubgroup,
LinearMap.range_coe]
|
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.LinearAlgebra.Matrix.PosDef
open Finset Matrix
namespace SimpleGraph
variable {V : Type*} (R : Type*)
variable [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj]
def degMatrix [AddMonoidWithOne R] : Matrix V V R := Matrix.diago... | Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean | 103 | 106 | theorem lapMatrix_toLin'_apply_eq_zero_iff_forall_adj (x : V → ℝ) :
Matrix.toLin' (G.lapMatrix ℝ) x = 0 ↔ ∀ i j : V, G.Adj i j → x i = x j := by |
rw [← (posSemidef_lapMatrix ℝ G).toLinearMap₂'_zero_iff, star_trivial,
lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_adj]
|
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
#align_import data.pnat.xgcd from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Nat
namespace PNat
structure XgcdType where
wp : ℕ
x : ℕ
y : ℕ
zp : ℕ
ap : ℕ
bp : ℕ
deriving Inhabited
#alig... | Mathlib/Data/PNat/Xgcd.lean | 353 | 355 | theorem reduce_b {u : XgcdType} (h : u.r ≠ 0) : u.reduce = u.step.reduce.flip := by |
rw [reduce]
exact if_neg h
|
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 | 140 | 146 | theorem homothety_norm [RingHomIsometric σ₁₂] [Nontrivial E] (f : E →SL[σ₁₂] F) {a : ℝ}
(hf : ∀ x, ‖f x‖ = a * ‖x‖) : ‖f‖ = a := by |
obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0
rw [← norm_pos_iff] at hx
have ha : 0 ≤ a := by simpa only [hf, hx, mul_nonneg_iff_of_pos_right] using norm_nonneg (f x)
apply le_antisymm (f.opNorm_le_bound ha fun y => le_of_eq (hf y))
simpa only [hf, hx, mul_le_mul_right] using f.le_opNorm x
|
import Mathlib.MeasureTheory.Measure.Trim
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
#align_import measure_theory.measure.ae_measurable from "leanprover-community/mathlib"@"3310acfa9787aa171db6d4cba3945f6f275fe9f2"
open scoped Classical
open MeasureTheory MeasureTheory.Measure Filter Set Funct... | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | 311 | 325 | theorem aemeasurable_Ioi_of_forall_Ioc {β} {mβ : MeasurableSpace β} [LinearOrder α]
[(atTop : Filter α).IsCountablyGenerated] {x : α} {g : α → β}
(g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))) :
AEMeasurable g (μ.restrict (Ioi x)) := by |
haveI : Nonempty α := ⟨x⟩
obtain ⟨u, hu_tendsto⟩ := exists_seq_tendsto (atTop : Filter α)
have Ioi_eq_iUnion : Ioi x = ⋃ n : ℕ, Ioc x (u n) := by
rw [iUnion_Ioc_eq_Ioi_self_iff.mpr _]
exact fun y _ => (hu_tendsto.eventually (eventually_ge_atTop y)).exists
rw [Ioi_eq_iUnion, aemeasurable_iUnion_iff]
i... |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
namespace Ideal
universe u v
variable {R : Type u} [CommRing R]
variable {S : Type v} [CommRing S] (f : R →+* S)
variable (p : Ideal R) (... | Mathlib/NumberTheory/RamificationInertia.lean | 153 | 156 | theorem ramificationIdx_eq_factors_count (hp0 : map f p ≠ ⊥) (hP : P.IsPrime) (hP0 : P ≠ ⊥) :
ramificationIdx f p P = (factors (map f p)).count P := by |
rw [IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hp0 hP hP0,
factors_eq_normalizedFactors]
|
import Mathlib.Topology.MetricSpace.Isometry
#align_import topology.metric_space.gluing from "leanprover-community/mathlib"@"e1a7bdeb4fd826b7e71d130d34988f0a2d26a177"
noncomputable section
universe u v w
open Function Set Uniformity Topology
namespace Metric
section Sum
variable {X : Type u} {Y : Type v} {Z ... | Mathlib/Topology/MetricSpace/Gluing.lean | 221 | 225 | theorem Sum.dist_eq_glueDist {p q : X ⊕ Y} (x : X) (y : Y) :
Sum.dist p q =
glueDist (fun _ : Unit => Nonempty.some ⟨x⟩) (fun _ : Unit => Nonempty.some ⟨y⟩) 1 p q := by |
cases p <;> cases q <;> first |rfl|simp [Sum.dist, glueDist, dist_comm, add_comm,
add_left_comm, add_assoc]
|
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 968 | 968 | theorem mul_inv_eq_one : a * b⁻¹ = 1 ↔ a = b := by | rw [mul_eq_one_iff_eq_inv, inv_inv]
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 310 | 313 | theorem differentiableWithinAt_euclidean :
DifferentiableWithinAt 𝕜 f t y ↔ ∀ i, DifferentiableWithinAt 𝕜 (fun x => f x i) t y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableWithinAt_iff, differentiableWithinAt_pi]
rfl
|
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 | 1,668 | 1,670 | theorem map_eq_replicate_iff {l : List α} {f : α → β} {b : β} :
l.map f = replicate l.length b ↔ ∀ x ∈ l, f x = b := by |
simp [eq_replicate]
|
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {α : Type u} {β : Type v} {ι : Sort w} (r r' s : α → α → Prop)
local infixl:50 " ≼ " => r
def Directed (f : ι → α) :=
∀ x y, ∃ z, ... | Mathlib/Order/Directed.lean | 254 | 257 | theorem directedOn_pair' (h : Reflexive r) {a b : α} (hab : a ≼ b) :
DirectedOn r ({b, a} : Set α) := by |
rw [Set.pair_comm]
apply directedOn_pair h hab
|
import Mathlib.CategoryTheory.Functor.Const
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.Yoneda
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.limits.cones from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
-- morphism le... | Mathlib/CategoryTheory/Limits/Cones.lean | 304 | 307 | theorem ConeMorphism.ext {c c' : Cone F} (f g : c ⟶ c') (w : f.hom = g.hom) : f = g := by |
cases f
cases g
congr
|
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Category
structure Iso {... | Mathlib/CategoryTheory/Iso.lean | 232 | 233 | theorem hom_comp_eq_id (α : X ≅ Y) {f : Y ⟶ X} : α.hom ≫ f = 𝟙 X ↔ f = α.inv := by |
rw [← eq_inv_comp, comp_id]
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open Set hiding restrict restrict_apply
open Filter ENNReal
open Function (support)
open scoped Cla... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 66 | 67 | theorem coe_injective ⦃f g : α →ₛ β⦄ (H : (f : α → β) = g) : f = g := by |
cases f; cases g; congr
|
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 175 | 194 | theorem eq_zero_of_polynomial_mem_map_range (I : Ideal R[X]) (x : ((Quotient.mk I).comp C).range)
(hx : C x ∈ I.map (Polynomial.mapRingHom ((Quotient.mk I).comp C).rangeRestrict)) : x = 0 := by |
let i := ((Quotient.mk I).comp C).rangeRestrict
have hi' : RingHom.ker (Polynomial.mapRingHom i) ≤ I := by
refine fun f hf => polynomial_mem_ideal_of_coeff_mem_ideal I f fun n => ?_
rw [mem_comap, ← Quotient.eq_zero_iff_mem, ← RingHom.comp_apply]
rw [RingHom.mem_ker, coe_mapRingHom] at hf
replace h... |
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.monoidal.category from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
open CategoryTheory.Category
open CategoryTheory.Iso
namespace CategoryTheory
... | Mathlib/CategoryTheory/Monoidal/Category.lean | 346 | 348 | theorem inv_whiskerLeft (X : C) {Y Z : C} (f : Y ⟶ Z) [IsIso f] :
inv (X ◁ f) = X ◁ inv f := by |
aesop_cat
|
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 | 1,300 | 1,311 | theorem mem_iSup_of_directed {ι} [hι : Nonempty ι] {K : ι → Subgroup G} (hK : Directed (· ≤ ·) K)
{x : G} : x ∈ (iSup K : Subgroup G) ↔ ∃ i, x ∈ K i := by |
refine ⟨?_, fun ⟨i, hi⟩ ↦ le_iSup K i hi⟩
suffices x ∈ closure (⋃ i, (K i : Set G)) → ∃ i, x ∈ K i by
simpa only [closure_iUnion, closure_eq (K _)] using this
refine fun hx ↦ closure_induction hx (fun _ ↦ mem_iUnion.1) ?_ ?_ ?_
· exact hι.elim fun i ↦ ⟨i, (K i).one_mem⟩
· rintro x y ⟨i, hi⟩ ⟨j, hj⟩
r... |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 56 | 70 | theorem exists_coeff_ne_zero_mem_comap_of_non_zero_divisor_root_mem {r : S}
(r_non_zero_divisor : ∀ {x}, x * r = 0 → x = 0) (hr : r ∈ I) {p : R[X]} :
p ≠ 0 → p.eval₂ f r = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ I.comap f := by |
refine p.recOnHorner ?_ ?_ ?_
· intro h
contradiction
· intro p a coeff_eq_zero a_ne_zero _ _ hp
refine ⟨0, ?_, coeff_zero_mem_comap_of_root_mem hr hp⟩
simp [coeff_eq_zero, a_ne_zero]
· intro p p_nonzero ih _ hp
rw [eval₂_mul, eval₂_X] at hp
obtain ⟨i, hi, mem⟩ := ih p_nonzero (r_non_zero_d... |
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u w
open CategoryTheory CategoryTheory.Limits
variable {ι : Type*}
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
o... | Mathlib/Algebra/Homology/ImageToKernel.lean | 214 | 216 | theorem homology'.π_desc {D : V} (k : (kernelSubobject g : V) ⟶ D)
(p : imageToKernel f g w ≫ k = 0) : homology'.π f g w ≫ homology'.desc f g w k p = k := by |
simp [homology'.π, homology'.desc]
|
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheory.Torsion
import Mathlib.RingTheory.Coprime.Ideal
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Set.Lattice
#align_import algebra.module.torsion from "leanpr... | Mathlib/Algebra/Module/Torsion.lean | 452 | 462 | theorem iSup_torsionBy_eq_torsionBy_prod :
⨆ i ∈ S, torsionBy R M (q i) = torsionBy R M (∏ i ∈ S, q i) := by |
rw [← torsionBySet_span_singleton_eq, Ideal.submodule_span_eq, ←
Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf, ←
iSup_torsionBySet_ideal_eq_torsionBySet_iInf]
· congr
ext : 1
congr
ext : 1
exact (torsionBySet_span_singleton_eq _).symm
exact fun i hi j hj ij => (Ideal.sup_eq_... |
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.extend_deriv from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Filter Set Metric Contin... | Mathlib/Analysis/Calculus/FDeriv/Extend.lean | 37 | 106 | theorem has_fderiv_at_boundary_of_tendsto_fderiv {f : E → F} {s : Set E} {x : E} {f' : E →L[ℝ] F}
(f_diff : DifferentiableOn ℝ f s) (s_conv : Convex ℝ s) (s_open : IsOpen s)
(f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y)
(h : Tendsto (fun y => fderiv ℝ f y) (𝓝[s] x) (𝓝 f')) :
HasFDerivWithinAt f... |
classical
-- one can assume without loss of generality that `x` belongs to the closure of `s`, as the
-- statement is empty otherwise
by_cases hx : x ∉ closure s
· rw [← closure_closure] at hx; exact hasFDerivWithinAt_of_nmem_closure hx
push_neg at hx
rw [HasFDerivWithinAt, hasFDerivAtFilter_... |
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Algebraic
#align_import field_theory.minpoly.field from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
open scoped Classical
open Polynomial Set Function minpoly
namespace... | Mathlib/FieldTheory/Minpoly/Field.lean | 220 | 221 | theorem zero : minpoly A (0 : B) = X := by |
simpa only [add_zero, C_0, sub_eq_add_neg, neg_zero, RingHom.map_zero] using eq_X_sub_C B (0 : A)
|
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 124 | 126 | theorem vars_mul [DecidableEq σ] (φ ψ : MvPolynomial σ R) : (φ * ψ).vars ⊆ φ.vars ∪ ψ.vars := by |
simp_rw [vars_def, ← Multiset.toFinset_add, Multiset.toFinset_subset]
exact Multiset.subset_of_le (degrees_mul φ ψ)
|
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 164 | 164 | theorem kernelSubobjectMap_id : kernelSubobjectMap (𝟙 (Arrow.mk f)) = 𝟙 _ := by | aesop_cat
|
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 2,077 | 2,079 | theorem pair_eq_pair_iff {x y z w : α} :
({x, y} : Set α) = {z, w} ↔ x = z ∧ y = w ∨ x = w ∧ y = z := by |
simp [subset_antisymm_iff, insert_subset_iff]; aesop
|
import Mathlib.MeasureTheory.Integral.IntegrableOn
#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace Bornology
open scoped Topology Interval ENNReal
variabl... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 205 | 212 | theorem locallyIntegrableOn_of_locallyIntegrable_restrict [OpensMeasurableSpace X]
(hf : LocallyIntegrable f (μ.restrict s)) : LocallyIntegrableOn f s μ := by |
intro x _
obtain ⟨t, ht_mem, ht_int⟩ := hf x
obtain ⟨u, hu_sub, hu_o, hu_mem⟩ := mem_nhds_iff.mp ht_mem
refine ⟨_, inter_mem_nhdsWithin s (hu_o.mem_nhds hu_mem), ?_⟩
simpa only [IntegrableOn, Measure.restrict_restrict hu_o.measurableSet, inter_comm] using
ht_int.mono_set hu_sub
|
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Index
import Mathlib.Order.Interval.Finset.Nat
import Mat... | Mathlib/GroupTheory/OrderOfElement.lean | 197 | 198 | theorem orderOf_eq_zero_iff' : orderOf x = 0 ↔ ∀ n : ℕ, 0 < n → x ^ n ≠ 1 := by |
simp_rw [orderOf_eq_zero_iff, isOfFinOrder_iff_pow_eq_one, not_exists, not_and]
|
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
open LinearMap (BilinForm)
universe u v w
variable {R : Type*} {M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M]
variable {R₁ : Type*} {M₁ : Type*} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁]
va... | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 100 | 105 | theorem isOrtho_smul_left {x y : M₄} {a : R₄} (ha : a ≠ 0) :
IsOrtho G (a • x) y ↔ IsOrtho G x y := by |
dsimp only [IsOrtho]
rw [map_smul]
simp only [LinearMap.smul_apply, smul_eq_mul, mul_eq_zero, or_iff_right_iff_imp]
exact fun a ↦ (ha a).elim
|
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Fintype.Basic
#align_import data.fintype.pi from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*}
open Finset
namespace Fintype
variable [DecidableEq α] [Fintype α] {γ δ : α → Type*} {s : ∀ a, Finset (γ a)}
def pi... | Mathlib/Data/Fintype/Pi.lean | 34 | 42 | theorem mem_piFinset {t : ∀ a, Finset (δ a)} {f : ∀ a, δ a} : f ∈ piFinset t ↔ ∀ a, f a ∈ t a := by |
constructor
· simp only [piFinset, mem_map, and_imp, forall_prop_of_true, exists_prop, mem_univ, exists_imp,
mem_pi]
rintro g hg hgf a
rw [← hgf]
exact hg a
· simp only [piFinset, mem_map, forall_prop_of_true, exists_prop, mem_univ, mem_pi]
exact fun hf => ⟨fun a _ => f a, hf, rfl⟩
|
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