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import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measure (π i)} [∀ i, SigmaFinite (μ i)] [DecidableEq δ]
variable {s t : Finset δ} {f g : (∀ i, π i) → ℝ≥0∞} {x y : ∀ i, π i} {i : δ}
def lmarginal (μ : ∀ i, Measure (π i)) (s : Finset δ) (f : (∀ i, π i) → ℝ≥0∞)
(x : ∀ i, π i) : ℝ≥0∞ :=
∫⁻ y : ∀ i : s, π i, f (updateFinset x s y) ∂Measure.pi fun i : s => μ i
-- Note: this notation is not a binder. This is more convenient since it returns a function.
@[inherit_doc]
notation "∫⋯∫⁻_" s ", " f " ∂" μ:70 => lmarginal μ s f
@[inherit_doc]
notation "∫⋯∫⁻_" s ", " f => lmarginal (fun _ ↦ volume) s f
variable (μ)
theorem _root_.Measurable.lmarginal (hf : Measurable f) : Measurable (∫⋯∫⁻_s, f ∂μ) := by
refine Measurable.lintegral_prod_right ?_
refine hf.comp ?_
rw [measurable_pi_iff]; intro i
by_cases hi : i ∈ s
· simp [hi, updateFinset]
exact measurable_pi_iff.1 measurable_snd _
· simp [hi, updateFinset]
exact measurable_pi_iff.1 measurable_fst _
@[simp] theorem lmarginal_empty (f : (∀ i, π i) → ℝ≥0∞) : ∫⋯∫⁻_∅, f ∂μ = f := by
ext1 x
simp_rw [lmarginal, Measure.pi_of_empty fun i : (∅ : Finset δ) => μ i]
apply lintegral_dirac'
exact Subsingleton.measurable
| Mathlib/MeasureTheory/Integral/Marginal.lean | 105 | 108 | theorem lmarginal_congr {x y : ∀ i, π i} (f : (∀ i, π i) → ℝ≥0∞)
(h : ∀ i ∉ s, x i = y i) :
(∫⋯∫⁻_s, f ∂μ) x = (∫⋯∫⁻_s, f ∂μ) y := by |
dsimp [lmarginal, updateFinset_def]; rcongr; exact h _ ‹_›
|
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
open MeasureTheory Filter Finset Real
noncomputable section
open scoped MeasureTheory ProbabilityTheory ENNReal NNReal
namespace ProbabilityTheory
variable {Ω ι : Type*} {m : MeasurableSpace Ω} {X : Ω → ℝ} {p : ℕ} {μ : Measure Ω}
def moment (X : Ω → ℝ) (p : ℕ) (μ : Measure Ω) : ℝ :=
μ[X ^ p]
#align probability_theory.moment ProbabilityTheory.moment
def centralMoment (X : Ω → ℝ) (p : ℕ) (μ : Measure Ω) : ℝ := by
have m := fun (x : Ω) => μ[X] -- Porting note: Lean deems `μ[(X - fun x => μ[X]) ^ p]` ambiguous
exact μ[(X - m) ^ p]
#align probability_theory.central_moment ProbabilityTheory.centralMoment
@[simp]
theorem moment_zero (hp : p ≠ 0) : moment 0 p μ = 0 := by
simp only [moment, hp, zero_pow, Ne, not_false_iff, Pi.zero_apply, integral_const,
smul_eq_mul, mul_zero, integral_zero]
#align probability_theory.moment_zero ProbabilityTheory.moment_zero
@[simp]
theorem centralMoment_zero (hp : p ≠ 0) : centralMoment 0 p μ = 0 := by
simp only [centralMoment, hp, Pi.zero_apply, integral_const, smul_eq_mul,
mul_zero, zero_sub, Pi.pow_apply, Pi.neg_apply, neg_zero, zero_pow, Ne, not_false_iff]
#align probability_theory.central_moment_zero ProbabilityTheory.centralMoment_zero
theorem centralMoment_one' [IsFiniteMeasure μ] (h_int : Integrable X μ) :
centralMoment X 1 μ = (1 - (μ Set.univ).toReal) * μ[X] := by
simp only [centralMoment, Pi.sub_apply, pow_one]
rw [integral_sub h_int (integrable_const _)]
simp only [sub_mul, integral_const, smul_eq_mul, one_mul]
#align probability_theory.central_moment_one' ProbabilityTheory.centralMoment_one'
@[simp]
theorem centralMoment_one [IsProbabilityMeasure μ] : centralMoment X 1 μ = 0 := by
by_cases h_int : Integrable X μ
· rw [centralMoment_one' h_int]
simp only [measure_univ, ENNReal.one_toReal, sub_self, zero_mul]
· simp only [centralMoment, Pi.sub_apply, pow_one]
have : ¬Integrable (fun x => X x - integral μ X) μ := by
refine fun h_sub => h_int ?_
have h_add : X = (fun x => X x - integral μ X) + fun _ => integral μ X := by ext1 x; simp
rw [h_add]
exact h_sub.add (integrable_const _)
rw [integral_undef this]
#align probability_theory.central_moment_one ProbabilityTheory.centralMoment_one
theorem centralMoment_two_eq_variance [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
centralMoment X 2 μ = variance X μ := by rw [hX.variance_eq]; rfl
#align probability_theory.central_moment_two_eq_variance ProbabilityTheory.centralMoment_two_eq_variance
section MomentGeneratingFunction
variable {t : ℝ}
def mgf (X : Ω → ℝ) (μ : Measure Ω) (t : ℝ) : ℝ :=
μ[fun ω => exp (t * X ω)]
#align probability_theory.mgf ProbabilityTheory.mgf
def cgf (X : Ω → ℝ) (μ : Measure Ω) (t : ℝ) : ℝ :=
log (mgf X μ t)
#align probability_theory.cgf ProbabilityTheory.cgf
@[simp]
theorem mgf_zero_fun : mgf 0 μ t = (μ Set.univ).toReal := by
simp only [mgf, Pi.zero_apply, mul_zero, exp_zero, integral_const, smul_eq_mul, mul_one]
#align probability_theory.mgf_zero_fun ProbabilityTheory.mgf_zero_fun
@[simp]
theorem cgf_zero_fun : cgf 0 μ t = log (μ Set.univ).toReal := by simp only [cgf, mgf_zero_fun]
#align probability_theory.cgf_zero_fun ProbabilityTheory.cgf_zero_fun
@[simp]
theorem mgf_zero_measure : mgf X (0 : Measure Ω) t = 0 := by simp only [mgf, integral_zero_measure]
#align probability_theory.mgf_zero_measure ProbabilityTheory.mgf_zero_measure
@[simp]
theorem cgf_zero_measure : cgf X (0 : Measure Ω) t = 0 := by
simp only [cgf, log_zero, mgf_zero_measure]
#align probability_theory.cgf_zero_measure ProbabilityTheory.cgf_zero_measure
@[simp]
theorem mgf_const' (c : ℝ) : mgf (fun _ => c) μ t = (μ Set.univ).toReal * exp (t * c) := by
simp only [mgf, integral_const, smul_eq_mul]
#align probability_theory.mgf_const' ProbabilityTheory.mgf_const'
-- @[simp] -- Porting note: `simp only` already proves this
theorem mgf_const (c : ℝ) [IsProbabilityMeasure μ] : mgf (fun _ => c) μ t = exp (t * c) := by
simp only [mgf_const', measure_univ, ENNReal.one_toReal, one_mul]
#align probability_theory.mgf_const ProbabilityTheory.mgf_const
@[simp]
theorem cgf_const' [IsFiniteMeasure μ] (hμ : μ ≠ 0) (c : ℝ) :
cgf (fun _ => c) μ t = log (μ Set.univ).toReal + t * c := by
simp only [cgf, mgf_const']
rw [log_mul _ (exp_pos _).ne']
· rw [log_exp _]
· rw [Ne, ENNReal.toReal_eq_zero_iff, Measure.measure_univ_eq_zero]
simp only [hμ, measure_ne_top μ Set.univ, or_self_iff, not_false_iff]
#align probability_theory.cgf_const' ProbabilityTheory.cgf_const'
@[simp]
theorem cgf_const [IsProbabilityMeasure μ] (c : ℝ) : cgf (fun _ => c) μ t = t * c := by
simp only [cgf, mgf_const, log_exp]
#align probability_theory.cgf_const ProbabilityTheory.cgf_const
@[simp]
theorem mgf_zero' : mgf X μ 0 = (μ Set.univ).toReal := by
simp only [mgf, zero_mul, exp_zero, integral_const, smul_eq_mul, mul_one]
#align probability_theory.mgf_zero' ProbabilityTheory.mgf_zero'
-- @[simp] -- Porting note: `simp only` already proves this
theorem mgf_zero [IsProbabilityMeasure μ] : mgf X μ 0 = 1 := by
simp only [mgf_zero', measure_univ, ENNReal.one_toReal]
#align probability_theory.mgf_zero ProbabilityTheory.mgf_zero
@[simp]
| Mathlib/Probability/Moments.lean | 166 | 166 | theorem cgf_zero' : cgf X μ 0 = log (μ Set.univ).toReal := by | simp only [cgf, mgf_zero']
|
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Set
open Filter
namespace Filter
variable {α β γ δ : Type*} {ι : Sort*}
section Prod
variable {s : Set α} {t : Set β} {f : Filter α} {g : Filter β}
protected def prod (f : Filter α) (g : Filter β) : Filter (α × β) :=
f.comap Prod.fst ⊓ g.comap Prod.snd
#align filter.prod Filter.prod
instance instSProd : SProd (Filter α) (Filter β) (Filter (α × β)) where
sprod := Filter.prod
theorem prod_mem_prod (hs : s ∈ f) (ht : t ∈ g) : s ×ˢ t ∈ f ×ˢ g :=
inter_mem_inf (preimage_mem_comap hs) (preimage_mem_comap ht)
#align filter.prod_mem_prod Filter.prod_mem_prod
theorem mem_prod_iff {s : Set (α × β)} {f : Filter α} {g : Filter β} :
s ∈ f ×ˢ g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ s := by
simp only [SProd.sprod, Filter.prod]
constructor
· rintro ⟨t₁, ⟨s₁, hs₁, hts₁⟩, t₂, ⟨s₂, hs₂, hts₂⟩, rfl⟩
exact ⟨s₁, hs₁, s₂, hs₂, fun p ⟨h, h'⟩ => ⟨hts₁ h, hts₂ h'⟩⟩
· rintro ⟨t₁, ht₁, t₂, ht₂, h⟩
exact mem_inf_of_inter (preimage_mem_comap ht₁) (preimage_mem_comap ht₂) h
#align filter.mem_prod_iff Filter.mem_prod_iff
@[simp]
theorem prod_mem_prod_iff [f.NeBot] [g.NeBot] : s ×ˢ t ∈ f ×ˢ g ↔ s ∈ f ∧ t ∈ g :=
⟨fun h =>
let ⟨_s', hs', _t', ht', H⟩ := mem_prod_iff.1 h
(prod_subset_prod_iff.1 H).elim
(fun ⟨hs's, ht't⟩ => ⟨mem_of_superset hs' hs's, mem_of_superset ht' ht't⟩) fun h =>
h.elim (fun hs'e => absurd hs'e (nonempty_of_mem hs').ne_empty) fun ht'e =>
absurd ht'e (nonempty_of_mem ht').ne_empty,
fun h => prod_mem_prod h.1 h.2⟩
#align filter.prod_mem_prod_iff Filter.prod_mem_prod_iff
theorem mem_prod_principal {s : Set (α × β)} :
s ∈ f ×ˢ 𝓟 t ↔ { a | ∀ b ∈ t, (a, b) ∈ s } ∈ f := by
rw [← @exists_mem_subset_iff _ f, mem_prod_iff]
refine exists_congr fun u => Iff.rfl.and ⟨?_, fun h => ⟨t, mem_principal_self t, ?_⟩⟩
· rintro ⟨v, v_in, hv⟩ a a_in b b_in
exact hv (mk_mem_prod a_in <| v_in b_in)
· rintro ⟨x, y⟩ ⟨hx, hy⟩
exact h hx y hy
#align filter.mem_prod_principal Filter.mem_prod_principal
theorem mem_prod_top {s : Set (α × β)} :
s ∈ f ×ˢ (⊤ : Filter β) ↔ { a | ∀ b, (a, b) ∈ s } ∈ f := by
rw [← principal_univ, mem_prod_principal]
simp only [mem_univ, forall_true_left]
#align filter.mem_prod_top Filter.mem_prod_top
theorem eventually_prod_principal_iff {p : α × β → Prop} {s : Set β} :
(∀ᶠ x : α × β in f ×ˢ 𝓟 s, p x) ↔ ∀ᶠ x : α in f, ∀ y : β, y ∈ s → p (x, y) := by
rw [eventually_iff, eventually_iff, mem_prod_principal]
simp only [mem_setOf_eq]
#align filter.eventually_prod_principal_iff Filter.eventually_prod_principal_iff
theorem comap_prod (f : α → β × γ) (b : Filter β) (c : Filter γ) :
comap f (b ×ˢ c) = comap (Prod.fst ∘ f) b ⊓ comap (Prod.snd ∘ f) c := by
erw [comap_inf, Filter.comap_comap, Filter.comap_comap]
#align filter.comap_prod Filter.comap_prod
theorem prod_top : f ×ˢ (⊤ : Filter β) = f.comap Prod.fst := by
dsimp only [SProd.sprod]
rw [Filter.prod, comap_top, inf_top_eq]
#align filter.prod_top Filter.prod_top
theorem top_prod : (⊤ : Filter α) ×ˢ g = g.comap Prod.snd := by
dsimp only [SProd.sprod]
rw [Filter.prod, comap_top, top_inf_eq]
theorem sup_prod (f₁ f₂ : Filter α) (g : Filter β) : (f₁ ⊔ f₂) ×ˢ g = (f₁ ×ˢ g) ⊔ (f₂ ×ˢ g) := by
dsimp only [SProd.sprod]
rw [Filter.prod, comap_sup, inf_sup_right, ← Filter.prod, ← Filter.prod]
#align filter.sup_prod Filter.sup_prod
theorem prod_sup (f : Filter α) (g₁ g₂ : Filter β) : f ×ˢ (g₁ ⊔ g₂) = (f ×ˢ g₁) ⊔ (f ×ˢ g₂) := by
dsimp only [SProd.sprod]
rw [Filter.prod, comap_sup, inf_sup_left, ← Filter.prod, ← Filter.prod]
#align filter.prod_sup Filter.prod_sup
theorem eventually_prod_iff {p : α × β → Prop} :
(∀ᶠ x in f ×ˢ g, p x) ↔
∃ pa : α → Prop, (∀ᶠ x in f, pa x) ∧ ∃ pb : β → Prop, (∀ᶠ y in g, pb y) ∧
∀ {x}, pa x → ∀ {y}, pb y → p (x, y) := by
simpa only [Set.prod_subset_iff] using @mem_prod_iff α β p f g
#align filter.eventually_prod_iff Filter.eventually_prod_iff
theorem tendsto_fst : Tendsto Prod.fst (f ×ˢ g) f :=
tendsto_inf_left tendsto_comap
#align filter.tendsto_fst Filter.tendsto_fst
theorem tendsto_snd : Tendsto Prod.snd (f ×ˢ g) g :=
tendsto_inf_right tendsto_comap
#align filter.tendsto_snd Filter.tendsto_snd
theorem Tendsto.fst {h : Filter γ} {m : α → β × γ} (H : Tendsto m f (g ×ˢ h)) :
Tendsto (fun a ↦ (m a).1) f g :=
tendsto_fst.comp H
theorem Tendsto.snd {h : Filter γ} {m : α → β × γ} (H : Tendsto m f (g ×ˢ h)) :
Tendsto (fun a ↦ (m a).2) f h :=
tendsto_snd.comp H
theorem Tendsto.prod_mk {h : Filter γ} {m₁ : α → β} {m₂ : α → γ}
(h₁ : Tendsto m₁ f g) (h₂ : Tendsto m₂ f h) : Tendsto (fun x => (m₁ x, m₂ x)) f (g ×ˢ h) :=
tendsto_inf.2 ⟨tendsto_comap_iff.2 h₁, tendsto_comap_iff.2 h₂⟩
#align filter.tendsto.prod_mk Filter.Tendsto.prod_mk
theorem tendsto_prod_swap : Tendsto (Prod.swap : α × β → β × α) (f ×ˢ g) (g ×ˢ f) :=
tendsto_snd.prod_mk tendsto_fst
#align filter.tendsto_prod_swap Filter.tendsto_prod_swap
theorem Eventually.prod_inl {la : Filter α} {p : α → Prop} (h : ∀ᶠ x in la, p x) (lb : Filter β) :
∀ᶠ x in la ×ˢ lb, p (x : α × β).1 :=
tendsto_fst.eventually h
#align filter.eventually.prod_inl Filter.Eventually.prod_inl
theorem Eventually.prod_inr {lb : Filter β} {p : β → Prop} (h : ∀ᶠ x in lb, p x) (la : Filter α) :
∀ᶠ x in la ×ˢ lb, p (x : α × β).2 :=
tendsto_snd.eventually h
#align filter.eventually.prod_inr Filter.Eventually.prod_inr
theorem Eventually.prod_mk {la : Filter α} {pa : α → Prop} (ha : ∀ᶠ x in la, pa x) {lb : Filter β}
{pb : β → Prop} (hb : ∀ᶠ y in lb, pb y) : ∀ᶠ p in la ×ˢ lb, pa (p : α × β).1 ∧ pb p.2 :=
(ha.prod_inl lb).and (hb.prod_inr la)
#align filter.eventually.prod_mk Filter.Eventually.prod_mk
theorem EventuallyEq.prod_map {δ} {la : Filter α} {fa ga : α → γ} (ha : fa =ᶠ[la] ga)
{lb : Filter β} {fb gb : β → δ} (hb : fb =ᶠ[lb] gb) :
Prod.map fa fb =ᶠ[la ×ˢ lb] Prod.map ga gb :=
(Eventually.prod_mk ha hb).mono fun _ h => Prod.ext h.1 h.2
#align filter.eventually_eq.prod_map Filter.EventuallyEq.prod_map
theorem EventuallyLE.prod_map {δ} [LE γ] [LE δ] {la : Filter α} {fa ga : α → γ} (ha : fa ≤ᶠ[la] ga)
{lb : Filter β} {fb gb : β → δ} (hb : fb ≤ᶠ[lb] gb) :
Prod.map fa fb ≤ᶠ[la ×ˢ lb] Prod.map ga gb :=
Eventually.prod_mk ha hb
#align filter.eventually_le.prod_map Filter.EventuallyLE.prod_map
theorem Eventually.curry {la : Filter α} {lb : Filter β} {p : α × β → Prop}
(h : ∀ᶠ x in la ×ˢ lb, p x) : ∀ᶠ x in la, ∀ᶠ y in lb, p (x, y) := by
rcases eventually_prod_iff.1 h with ⟨pa, ha, pb, hb, h⟩
exact ha.mono fun a ha => hb.mono fun b hb => h ha hb
#align filter.eventually.curry Filter.Eventually.curry
protected lemma Frequently.uncurry {la : Filter α} {lb : Filter β} {p : α → β → Prop}
(h : ∃ᶠ x in la, ∃ᶠ y in lb, p x y) : ∃ᶠ xy in la ×ˢ lb, p xy.1 xy.2 :=
mt (fun h ↦ by simpa only [not_frequently] using h.curry) h
theorem Eventually.diag_of_prod {p : α × α → Prop} (h : ∀ᶠ i in f ×ˢ f, p i) :
∀ᶠ i in f, p (i, i) := by
obtain ⟨t, ht, s, hs, hst⟩ := eventually_prod_iff.1 h
apply (ht.and hs).mono fun x hx => hst hx.1 hx.2
#align filter.eventually.diag_of_prod Filter.Eventually.diag_of_prod
theorem Eventually.diag_of_prod_left {f : Filter α} {g : Filter γ} {p : (α × α) × γ → Prop} :
(∀ᶠ x in (f ×ˢ f) ×ˢ g, p x) → ∀ᶠ x : α × γ in f ×ˢ g, p ((x.1, x.1), x.2) := by
intro h
obtain ⟨t, ht, s, hs, hst⟩ := eventually_prod_iff.1 h
exact (ht.diag_of_prod.prod_mk hs).mono fun x hx => by simp only [hst hx.1 hx.2]
#align filter.eventually.diag_of_prod_left Filter.Eventually.diag_of_prod_left
theorem Eventually.diag_of_prod_right {f : Filter α} {g : Filter γ} {p : α × γ × γ → Prop} :
(∀ᶠ x in f ×ˢ (g ×ˢ g), p x) → ∀ᶠ x : α × γ in f ×ˢ g, p (x.1, x.2, x.2) := by
intro h
obtain ⟨t, ht, s, hs, hst⟩ := eventually_prod_iff.1 h
exact (ht.prod_mk hs.diag_of_prod).mono fun x hx => by simp only [hst hx.1 hx.2]
#align filter.eventually.diag_of_prod_right Filter.Eventually.diag_of_prod_right
theorem tendsto_diag : Tendsto (fun i => (i, i)) f (f ×ˢ f) :=
tendsto_iff_eventually.mpr fun _ hpr => hpr.diag_of_prod
#align filter.tendsto_diag Filter.tendsto_diag
theorem prod_iInf_left [Nonempty ι] {f : ι → Filter α} {g : Filter β} :
(⨅ i, f i) ×ˢ g = ⨅ i, f i ×ˢ g := by
dsimp only [SProd.sprod]
rw [Filter.prod, comap_iInf, iInf_inf]
simp only [Filter.prod, eq_self_iff_true]
#align filter.prod_infi_left Filter.prod_iInf_left
theorem prod_iInf_right [Nonempty ι] {f : Filter α} {g : ι → Filter β} :
(f ×ˢ ⨅ i, g i) = ⨅ i, f ×ˢ g i := by
dsimp only [SProd.sprod]
rw [Filter.prod, comap_iInf, inf_iInf]
simp only [Filter.prod, eq_self_iff_true]
#align filter.prod_infi_right Filter.prod_iInf_right
@[mono, gcongr]
theorem prod_mono {f₁ f₂ : Filter α} {g₁ g₂ : Filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) :
f₁ ×ˢ g₁ ≤ f₂ ×ˢ g₂ :=
inf_le_inf (comap_mono hf) (comap_mono hg)
#align filter.prod_mono Filter.prod_mono
@[gcongr]
theorem prod_mono_left (g : Filter β) {f₁ f₂ : Filter α} (hf : f₁ ≤ f₂) : f₁ ×ˢ g ≤ f₂ ×ˢ g :=
Filter.prod_mono hf rfl.le
#align filter.prod_mono_left Filter.prod_mono_left
@[gcongr]
theorem prod_mono_right (f : Filter α) {g₁ g₂ : Filter β} (hf : g₁ ≤ g₂) : f ×ˢ g₁ ≤ f ×ˢ g₂ :=
Filter.prod_mono rfl.le hf
#align filter.prod_mono_right Filter.prod_mono_right
theorem prod_comap_comap_eq.{u, v, w, x} {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x}
{f₁ : Filter α₁} {f₂ : Filter α₂} {m₁ : β₁ → α₁} {m₂ : β₂ → α₂} :
comap m₁ f₁ ×ˢ comap m₂ f₂ = comap (fun p : β₁ × β₂ => (m₁ p.1, m₂ p.2)) (f₁ ×ˢ f₂) := by
simp only [SProd.sprod, Filter.prod, comap_comap, comap_inf, (· ∘ ·)]
#align filter.prod_comap_comap_eq Filter.prod_comap_comap_eq
theorem prod_comm' : f ×ˢ g = comap Prod.swap (g ×ˢ f) := by
simp only [SProd.sprod, Filter.prod, comap_comap, (· ∘ ·), inf_comm, Prod.swap, comap_inf]
#align filter.prod_comm' Filter.prod_comm'
theorem prod_comm : f ×ˢ g = map (fun p : β × α => (p.2, p.1)) (g ×ˢ f) := by
rw [prod_comm', ← map_swap_eq_comap_swap]
rfl
#align filter.prod_comm Filter.prod_comm
theorem mem_prod_iff_left {s : Set (α × β)} :
s ∈ f ×ˢ g ↔ ∃ t ∈ f, ∀ᶠ y in g, ∀ x ∈ t, (x, y) ∈ s := by
simp only [mem_prod_iff, prod_subset_iff]
refine exists_congr fun _ => Iff.rfl.and <| Iff.trans ?_ exists_mem_subset_iff
exact exists_congr fun _ => Iff.rfl.and forall₂_swap
theorem mem_prod_iff_right {s : Set (α × β)} :
s ∈ f ×ˢ g ↔ ∃ t ∈ g, ∀ᶠ x in f, ∀ y ∈ t, (x, y) ∈ s := by
rw [prod_comm, mem_map, mem_prod_iff_left]; rfl
@[simp]
| Mathlib/Order/Filter/Prod.lean | 288 | 291 | theorem map_fst_prod (f : Filter α) (g : Filter β) [NeBot g] : map Prod.fst (f ×ˢ g) = f := by |
ext s
simp only [mem_map, mem_prod_iff_left, mem_preimage, eventually_const, ← subset_def,
exists_mem_subset_iff]
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import measure_theory.integral.lebesgue from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
assert_not_exists NormedSpace
set_option autoImplicit true
noncomputable section
open Set hiding restrict restrict_apply
open Filter ENNReal
open Function (support)
open scoped Classical
open Topology NNReal ENNReal MeasureTheory
namespace MeasureTheory
local infixr:25 " →ₛ " => SimpleFunc
variable {α β γ δ : Type*}
section Lintegral
open SimpleFunc
variable {m : MeasurableSpace α} {μ ν : Measure α}
irreducible_def lintegral {_ : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) : ℝ≥0∞ :=
⨆ (g : α →ₛ ℝ≥0∞) (_ : ⇑g ≤ f), g.lintegral μ
#align measure_theory.lintegral MeasureTheory.lintegral
@[inherit_doc MeasureTheory.lintegral]
notation3 "∫⁻ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => lintegral μ r
@[inherit_doc MeasureTheory.lintegral]
notation3 "∫⁻ "(...)", "r:60:(scoped f => lintegral volume f) => r
@[inherit_doc MeasureTheory.lintegral]
notation3"∫⁻ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => lintegral (Measure.restrict μ s) r
@[inherit_doc MeasureTheory.lintegral]
notation3"∫⁻ "(...)" in "s", "r:60:(scoped f => lintegral (Measure.restrict volume s) f) => r
theorem SimpleFunc.lintegral_eq_lintegral {m : MeasurableSpace α} (f : α →ₛ ℝ≥0∞) (μ : Measure α) :
∫⁻ a, f a ∂μ = f.lintegral μ := by
rw [MeasureTheory.lintegral]
exact le_antisymm (iSup₂_le fun g hg => lintegral_mono hg <| le_rfl)
(le_iSup₂_of_le f le_rfl le_rfl)
#align measure_theory.simple_func.lintegral_eq_lintegral MeasureTheory.SimpleFunc.lintegral_eq_lintegral
@[mono]
theorem lintegral_mono' {m : MeasurableSpace α} ⦃μ ν : Measure α⦄ (hμν : μ ≤ ν) ⦃f g : α → ℝ≥0∞⦄
(hfg : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂ν := by
rw [lintegral, lintegral]
exact iSup_mono fun φ => iSup_mono' fun hφ => ⟨le_trans hφ hfg, lintegral_mono (le_refl φ) hμν⟩
#align measure_theory.lintegral_mono' MeasureTheory.lintegral_mono'
-- workaround for the known eta-reduction issue with `@[gcongr]`
@[gcongr] theorem lintegral_mono_fn' ⦃f g : α → ℝ≥0∞⦄ (hfg : ∀ x, f x ≤ g x) (h2 : μ ≤ ν) :
lintegral μ f ≤ lintegral ν g :=
lintegral_mono' h2 hfg
theorem lintegral_mono ⦃f g : α → ℝ≥0∞⦄ (hfg : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ :=
lintegral_mono' (le_refl μ) hfg
#align measure_theory.lintegral_mono MeasureTheory.lintegral_mono
-- workaround for the known eta-reduction issue with `@[gcongr]`
@[gcongr] theorem lintegral_mono_fn ⦃f g : α → ℝ≥0∞⦄ (hfg : ∀ x, f x ≤ g x) :
lintegral μ f ≤ lintegral μ g :=
lintegral_mono hfg
theorem lintegral_mono_nnreal {f g : α → ℝ≥0} (h : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ :=
lintegral_mono fun a => ENNReal.coe_le_coe.2 (h a)
#align measure_theory.lintegral_mono_nnreal MeasureTheory.lintegral_mono_nnreal
theorem iSup_lintegral_measurable_le_eq_lintegral (f : α → ℝ≥0∞) :
⨆ (g : α → ℝ≥0∞) (_ : Measurable g) (_ : g ≤ f), ∫⁻ a, g a ∂μ = ∫⁻ a, f a ∂μ := by
apply le_antisymm
· exact iSup_le fun i => iSup_le fun _ => iSup_le fun h'i => lintegral_mono h'i
· rw [lintegral]
refine iSup₂_le fun i hi => le_iSup₂_of_le i i.measurable <| le_iSup_of_le hi ?_
exact le_of_eq (i.lintegral_eq_lintegral _).symm
#align measure_theory.supr_lintegral_measurable_le_eq_lintegral MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral
theorem lintegral_mono_set {_ : MeasurableSpace α} ⦃μ : Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞}
(hst : s ⊆ t) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in t, f x ∂μ :=
lintegral_mono' (Measure.restrict_mono hst (le_refl μ)) (le_refl f)
#align measure_theory.lintegral_mono_set MeasureTheory.lintegral_mono_set
theorem lintegral_mono_set' {_ : MeasurableSpace α} ⦃μ : Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞}
(hst : s ≤ᵐ[μ] t) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in t, f x ∂μ :=
lintegral_mono' (Measure.restrict_mono' hst (le_refl μ)) (le_refl f)
#align measure_theory.lintegral_mono_set' MeasureTheory.lintegral_mono_set'
theorem monotone_lintegral {_ : MeasurableSpace α} (μ : Measure α) : Monotone (lintegral μ) :=
lintegral_mono
#align measure_theory.monotone_lintegral MeasureTheory.monotone_lintegral
@[simp]
theorem lintegral_const (c : ℝ≥0∞) : ∫⁻ _, c ∂μ = c * μ univ := by
rw [← SimpleFunc.const_lintegral, ← SimpleFunc.lintegral_eq_lintegral, SimpleFunc.coe_const]
rfl
#align measure_theory.lintegral_const MeasureTheory.lintegral_const
theorem lintegral_zero : ∫⁻ _ : α, 0 ∂μ = 0 := by simp
#align measure_theory.lintegral_zero MeasureTheory.lintegral_zero
theorem lintegral_zero_fun : lintegral μ (0 : α → ℝ≥0∞) = 0 :=
lintegral_zero
#align measure_theory.lintegral_zero_fun MeasureTheory.lintegral_zero_fun
-- @[simp] -- Porting note (#10618): simp can prove this
theorem lintegral_one : ∫⁻ _, (1 : ℝ≥0∞) ∂μ = μ univ := by rw [lintegral_const, one_mul]
#align measure_theory.lintegral_one MeasureTheory.lintegral_one
theorem set_lintegral_const (s : Set α) (c : ℝ≥0∞) : ∫⁻ _ in s, c ∂μ = c * μ s := by
rw [lintegral_const, Measure.restrict_apply_univ]
#align measure_theory.set_lintegral_const MeasureTheory.set_lintegral_const
theorem set_lintegral_one (s) : ∫⁻ _ in s, 1 ∂μ = μ s := by rw [set_lintegral_const, one_mul]
#align measure_theory.set_lintegral_one MeasureTheory.set_lintegral_one
theorem set_lintegral_const_lt_top [IsFiniteMeasure μ] (s : Set α) {c : ℝ≥0∞} (hc : c ≠ ∞) :
∫⁻ _ in s, c ∂μ < ∞ := by
rw [lintegral_const]
exact ENNReal.mul_lt_top hc (measure_ne_top (μ.restrict s) univ)
#align measure_theory.set_lintegral_const_lt_top MeasureTheory.set_lintegral_const_lt_top
theorem lintegral_const_lt_top [IsFiniteMeasure μ] {c : ℝ≥0∞} (hc : c ≠ ∞) : ∫⁻ _, c ∂μ < ∞ := by
simpa only [Measure.restrict_univ] using set_lintegral_const_lt_top (univ : Set α) hc
#align measure_theory.lintegral_const_lt_top MeasureTheory.lintegral_const_lt_top
section
variable (μ)
theorem exists_measurable_le_lintegral_eq (f : α → ℝ≥0∞) :
∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ := by
rcases eq_or_ne (∫⁻ a, f a ∂μ) 0 with h₀ | h₀
· exact ⟨0, measurable_zero, zero_le f, h₀.trans lintegral_zero.symm⟩
rcases exists_seq_strictMono_tendsto' h₀.bot_lt with ⟨L, _, hLf, hL_tendsto⟩
have : ∀ n, ∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ L n < ∫⁻ a, g a ∂μ := by
intro n
simpa only [← iSup_lintegral_measurable_le_eq_lintegral f, lt_iSup_iff, exists_prop] using
(hLf n).2
choose g hgm hgf hLg using this
refine
⟨fun x => ⨆ n, g n x, measurable_iSup hgm, fun x => iSup_le fun n => hgf n x, le_antisymm ?_ ?_⟩
· refine le_of_tendsto' hL_tendsto fun n => (hLg n).le.trans <| lintegral_mono fun x => ?_
exact le_iSup (fun n => g n x) n
· exact lintegral_mono fun x => iSup_le fun n => hgf n x
#align measure_theory.exists_measurable_le_lintegral_eq MeasureTheory.exists_measurable_le_lintegral_eq
end
theorem lintegral_eq_nnreal {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : Measure α) :
∫⁻ a, f a ∂μ =
⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ x, ↑(φ x) ≤ f x), (φ.map ((↑) : ℝ≥0 → ℝ≥0∞)).lintegral μ := by
rw [lintegral]
refine
le_antisymm (iSup₂_le fun φ hφ => ?_) (iSup_mono' fun φ => ⟨φ.map ((↑) : ℝ≥0 → ℝ≥0∞), le_rfl⟩)
by_cases h : ∀ᵐ a ∂μ, φ a ≠ ∞
· let ψ := φ.map ENNReal.toNNReal
replace h : ψ.map ((↑) : ℝ≥0 → ℝ≥0∞) =ᵐ[μ] φ := h.mono fun a => ENNReal.coe_toNNReal
have : ∀ x, ↑(ψ x) ≤ f x := fun x => le_trans ENNReal.coe_toNNReal_le_self (hφ x)
exact
le_iSup_of_le (φ.map ENNReal.toNNReal) (le_iSup_of_le this (ge_of_eq <| lintegral_congr h))
· have h_meas : μ (φ ⁻¹' {∞}) ≠ 0 := mt measure_zero_iff_ae_nmem.1 h
refine le_trans le_top (ge_of_eq <| (iSup_eq_top _).2 fun b hb => ?_)
obtain ⟨n, hn⟩ : ∃ n : ℕ, b < n * μ (φ ⁻¹' {∞}) := exists_nat_mul_gt h_meas (ne_of_lt hb)
use (const α (n : ℝ≥0)).restrict (φ ⁻¹' {∞})
simp only [lt_iSup_iff, exists_prop, coe_restrict, φ.measurableSet_preimage, coe_const,
ENNReal.coe_indicator, map_coe_ennreal_restrict, SimpleFunc.map_const, ENNReal.coe_natCast,
restrict_const_lintegral]
refine ⟨indicator_le fun x hx => le_trans ?_ (hφ _), hn⟩
simp only [mem_preimage, mem_singleton_iff] at hx
simp only [hx, le_top]
#align measure_theory.lintegral_eq_nnreal MeasureTheory.lintegral_eq_nnreal
theorem exists_simpleFunc_forall_lintegral_sub_lt_of_pos {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞)
{ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ φ : α →ₛ ℝ≥0,
(∀ x, ↑(φ x) ≤ f x) ∧
∀ ψ : α →ₛ ℝ≥0, (∀ x, ↑(ψ x) ≤ f x) → (map (↑) (ψ - φ)).lintegral μ < ε := by
rw [lintegral_eq_nnreal] at h
have := ENNReal.lt_add_right h hε
erw [ENNReal.biSup_add] at this <;> [skip; exact ⟨0, fun x => zero_le _⟩]
simp_rw [lt_iSup_iff, iSup_lt_iff, iSup_le_iff] at this
rcases this with ⟨φ, hle : ∀ x, ↑(φ x) ≤ f x, b, hbφ, hb⟩
refine ⟨φ, hle, fun ψ hψ => ?_⟩
have : (map (↑) φ).lintegral μ ≠ ∞ := ne_top_of_le_ne_top h (by exact le_iSup₂ (α := ℝ≥0∞) φ hle)
rw [← ENNReal.add_lt_add_iff_left this, ← add_lintegral, ← SimpleFunc.map_add @ENNReal.coe_add]
refine (hb _ fun x => le_trans ?_ (max_le (hle x) (hψ x))).trans_lt hbφ
norm_cast
simp only [add_apply, sub_apply, add_tsub_eq_max]
rfl
#align measure_theory.exists_simple_func_forall_lintegral_sub_lt_of_pos MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos
theorem iSup_lintegral_le {ι : Sort*} (f : ι → α → ℝ≥0∞) :
⨆ i, ∫⁻ a, f i a ∂μ ≤ ∫⁻ a, ⨆ i, f i a ∂μ := by
simp only [← iSup_apply]
exact (monotone_lintegral μ).le_map_iSup
#align measure_theory.supr_lintegral_le MeasureTheory.iSup_lintegral_le
theorem iSup₂_lintegral_le {ι : Sort*} {ι' : ι → Sort*} (f : ∀ i, ι' i → α → ℝ≥0∞) :
⨆ (i) (j), ∫⁻ a, f i j a ∂μ ≤ ∫⁻ a, ⨆ (i) (j), f i j a ∂μ := by
convert (monotone_lintegral μ).le_map_iSup₂ f with a
simp only [iSup_apply]
#align measure_theory.supr₂_lintegral_le MeasureTheory.iSup₂_lintegral_le
theorem le_iInf_lintegral {ι : Sort*} (f : ι → α → ℝ≥0∞) :
∫⁻ a, ⨅ i, f i a ∂μ ≤ ⨅ i, ∫⁻ a, f i a ∂μ := by
simp only [← iInf_apply]
exact (monotone_lintegral μ).map_iInf_le
#align measure_theory.le_infi_lintegral MeasureTheory.le_iInf_lintegral
theorem le_iInf₂_lintegral {ι : Sort*} {ι' : ι → Sort*} (f : ∀ i, ι' i → α → ℝ≥0∞) :
∫⁻ a, ⨅ (i) (h : ι' i), f i h a ∂μ ≤ ⨅ (i) (h : ι' i), ∫⁻ a, f i h a ∂μ := by
convert (monotone_lintegral μ).map_iInf₂_le f with a
simp only [iInf_apply]
#align measure_theory.le_infi₂_lintegral MeasureTheory.le_iInf₂_lintegral
theorem lintegral_mono_ae {f g : α → ℝ≥0∞} (h : ∀ᵐ a ∂μ, f a ≤ g a) :
∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ := by
rcases exists_measurable_superset_of_null h with ⟨t, hts, ht, ht0⟩
have : ∀ᵐ x ∂μ, x ∉ t := measure_zero_iff_ae_nmem.1 ht0
rw [lintegral, lintegral]
refine iSup_le fun s => iSup_le fun hfs => le_iSup_of_le (s.restrict tᶜ) <| le_iSup_of_le ?_ ?_
· intro a
by_cases h : a ∈ t <;>
simp only [restrict_apply s ht.compl, mem_compl_iff, h, not_true, not_false_eq_true,
indicator_of_not_mem, zero_le, not_false_eq_true, indicator_of_mem]
exact le_trans (hfs a) (_root_.by_contradiction fun hnfg => h (hts hnfg))
· refine le_of_eq (SimpleFunc.lintegral_congr <| this.mono fun a hnt => ?_)
by_cases hat : a ∈ t <;> simp only [restrict_apply s ht.compl, mem_compl_iff, hat, not_true,
not_false_eq_true, indicator_of_not_mem, not_false_eq_true, indicator_of_mem]
exact (hnt hat).elim
#align measure_theory.lintegral_mono_ae MeasureTheory.lintegral_mono_ae
theorem set_lintegral_mono_ae {s : Set α} {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g)
(hfg : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ :=
lintegral_mono_ae <| (ae_restrict_iff <| measurableSet_le hf hg).2 hfg
#align measure_theory.set_lintegral_mono_ae MeasureTheory.set_lintegral_mono_ae
theorem set_lintegral_mono {s : Set α} {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g)
(hfg : ∀ x ∈ s, f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ :=
set_lintegral_mono_ae hf hg (ae_of_all _ hfg)
#align measure_theory.set_lintegral_mono MeasureTheory.set_lintegral_mono
theorem set_lintegral_mono_ae' {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s)
(hfg : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ :=
lintegral_mono_ae <| (ae_restrict_iff' hs).2 hfg
theorem set_lintegral_mono' {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s)
(hfg : ∀ x ∈ s, f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ :=
set_lintegral_mono_ae' hs (ae_of_all _ hfg)
theorem set_lintegral_le_lintegral (s : Set α) (f : α → ℝ≥0∞) :
∫⁻ x in s, f x ∂μ ≤ ∫⁻ x, f x ∂μ :=
lintegral_mono' Measure.restrict_le_self le_rfl
theorem lintegral_congr_ae {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ :=
le_antisymm (lintegral_mono_ae <| h.le) (lintegral_mono_ae <| h.symm.le)
#align measure_theory.lintegral_congr_ae MeasureTheory.lintegral_congr_ae
theorem lintegral_congr {f g : α → ℝ≥0∞} (h : ∀ a, f a = g a) : ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ := by
simp only [h]
#align measure_theory.lintegral_congr MeasureTheory.lintegral_congr
theorem set_lintegral_congr {f : α → ℝ≥0∞} {s t : Set α} (h : s =ᵐ[μ] t) :
∫⁻ x in s, f x ∂μ = ∫⁻ x in t, f x ∂μ := by rw [Measure.restrict_congr_set h]
#align measure_theory.set_lintegral_congr MeasureTheory.set_lintegral_congr
theorem set_lintegral_congr_fun {f g : α → ℝ≥0∞} {s : Set α} (hs : MeasurableSet s)
(hfg : ∀ᵐ x ∂μ, x ∈ s → f x = g x) : ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ := by
rw [lintegral_congr_ae]
rw [EventuallyEq]
rwa [ae_restrict_iff' hs]
#align measure_theory.set_lintegral_congr_fun MeasureTheory.set_lintegral_congr_fun
theorem lintegral_ofReal_le_lintegral_nnnorm (f : α → ℝ) :
∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ x, ‖f x‖₊ ∂μ := by
simp_rw [← ofReal_norm_eq_coe_nnnorm]
refine lintegral_mono fun x => ENNReal.ofReal_le_ofReal ?_
rw [Real.norm_eq_abs]
exact le_abs_self (f x)
#align measure_theory.lintegral_of_real_le_lintegral_nnnorm MeasureTheory.lintegral_ofReal_le_lintegral_nnnorm
theorem lintegral_nnnorm_eq_of_ae_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ᵐ[μ] f) :
∫⁻ x, ‖f x‖₊ ∂μ = ∫⁻ x, ENNReal.ofReal (f x) ∂μ := by
apply lintegral_congr_ae
filter_upwards [h_nonneg] with x hx
rw [Real.nnnorm_of_nonneg hx, ENNReal.ofReal_eq_coe_nnreal hx]
#align measure_theory.lintegral_nnnorm_eq_of_ae_nonneg MeasureTheory.lintegral_nnnorm_eq_of_ae_nonneg
theorem lintegral_nnnorm_eq_of_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ f) :
∫⁻ x, ‖f x‖₊ ∂μ = ∫⁻ x, ENNReal.ofReal (f x) ∂μ :=
lintegral_nnnorm_eq_of_ae_nonneg (Filter.eventually_of_forall h_nonneg)
#align measure_theory.lintegral_nnnorm_eq_of_nonneg MeasureTheory.lintegral_nnnorm_eq_of_nonneg
theorem lintegral_iSup {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) :
∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ := by
set c : ℝ≥0 → ℝ≥0∞ := (↑)
set F := fun a : α => ⨆ n, f n a
refine le_antisymm ?_ (iSup_lintegral_le _)
rw [lintegral_eq_nnreal]
refine iSup_le fun s => iSup_le fun hsf => ?_
refine ENNReal.le_of_forall_lt_one_mul_le fun a ha => ?_
rcases ENNReal.lt_iff_exists_coe.1 ha with ⟨r, rfl, _⟩
have ha : r < 1 := ENNReal.coe_lt_coe.1 ha
let rs := s.map fun a => r * a
have eq_rs : rs.map c = (const α r : α →ₛ ℝ≥0∞) * map c s := rfl
have eq : ∀ p, rs.map c ⁻¹' {p} = ⋃ n, rs.map c ⁻¹' {p} ∩ { a | p ≤ f n a } := by
intro p
rw [← inter_iUnion]; nth_rw 1 [← inter_univ (map c rs ⁻¹' {p})]
refine Set.ext fun x => and_congr_right fun hx => true_iff_iff.2 ?_
by_cases p_eq : p = 0
· simp [p_eq]
simp only [coe_map, mem_preimage, Function.comp_apply, mem_singleton_iff] at hx
subst hx
have : r * s x ≠ 0 := by rwa [Ne, ← ENNReal.coe_eq_zero]
have : s x ≠ 0 := right_ne_zero_of_mul this
have : (rs.map c) x < ⨆ n : ℕ, f n x := by
refine lt_of_lt_of_le (ENNReal.coe_lt_coe.2 ?_) (hsf x)
suffices r * s x < 1 * s x by simpa
exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this)
rcases lt_iSup_iff.1 this with ⟨i, hi⟩
exact mem_iUnion.2 ⟨i, le_of_lt hi⟩
have mono : ∀ r : ℝ≥0∞, Monotone fun n => rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a } := by
intro r i j h
refine inter_subset_inter_right _ ?_
simp_rw [subset_def, mem_setOf]
intro x hx
exact le_trans hx (h_mono h x)
have h_meas : ∀ n, MeasurableSet {a : α | map c rs a ≤ f n a} := fun n =>
measurableSet_le (SimpleFunc.measurable _) (hf n)
calc
(r : ℝ≥0∞) * (s.map c).lintegral μ = ∑ r ∈ (rs.map c).range, r * μ (rs.map c ⁻¹' {r}) := by
rw [← const_mul_lintegral, eq_rs, SimpleFunc.lintegral]
_ = ∑ r ∈ (rs.map c).range, r * μ (⋃ n, rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) := by
simp only [(eq _).symm]
_ = ∑ r ∈ (rs.map c).range, ⨆ n, r * μ (rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) :=
(Finset.sum_congr rfl fun x _ => by
rw [measure_iUnion_eq_iSup (mono x).directed_le, ENNReal.mul_iSup])
_ = ⨆ n, ∑ r ∈ (rs.map c).range, r * μ (rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) := by
refine ENNReal.finset_sum_iSup_nat fun p i j h ↦ ?_
gcongr _ * μ ?_
exact mono p h
_ ≤ ⨆ n : ℕ, ((rs.map c).restrict { a | (rs.map c) a ≤ f n a }).lintegral μ := by
gcongr with n
rw [restrict_lintegral _ (h_meas n)]
refine le_of_eq (Finset.sum_congr rfl fun r _ => ?_)
congr 2 with a
refine and_congr_right ?_
simp (config := { contextual := true })
_ ≤ ⨆ n, ∫⁻ a, f n a ∂μ := by
simp only [← SimpleFunc.lintegral_eq_lintegral]
gcongr with n a
simp only [map_apply] at h_meas
simp only [coe_map, restrict_apply _ (h_meas _), (· ∘ ·)]
exact indicator_apply_le id
#align measure_theory.lintegral_supr MeasureTheory.lintegral_iSup
theorem lintegral_iSup' {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, AEMeasurable (f n) μ)
(h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ := by
simp_rw [← iSup_apply]
let p : α → (ℕ → ℝ≥0∞) → Prop := fun _ f' => Monotone f'
have hp : ∀ᵐ x ∂μ, p x fun i => f i x := h_mono
have h_ae_seq_mono : Monotone (aeSeq hf p) := by
intro n m hnm x
by_cases hx : x ∈ aeSeqSet hf p
· exact aeSeq.prop_of_mem_aeSeqSet hf hx hnm
· simp only [aeSeq, hx, if_false, le_rfl]
rw [lintegral_congr_ae (aeSeq.iSup hf hp).symm]
simp_rw [iSup_apply]
rw [lintegral_iSup (aeSeq.measurable hf p) h_ae_seq_mono]
congr with n
exact lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae hf hp n)
#align measure_theory.lintegral_supr' MeasureTheory.lintegral_iSup'
theorem lintegral_tendsto_of_tendsto_of_monotone {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞}
(hf : ∀ n, AEMeasurable (f n) μ) (h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x)
(h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 <| F x)) :
Tendsto (fun n => ∫⁻ x, f n x ∂μ) atTop (𝓝 <| ∫⁻ x, F x ∂μ) := by
have : Monotone fun n => ∫⁻ x, f n x ∂μ := fun i j hij =>
lintegral_mono_ae (h_mono.mono fun x hx => hx hij)
suffices key : ∫⁻ x, F x ∂μ = ⨆ n, ∫⁻ x, f n x ∂μ by
rw [key]
exact tendsto_atTop_iSup this
rw [← lintegral_iSup' hf h_mono]
refine lintegral_congr_ae ?_
filter_upwards [h_mono, h_tendsto] with _ hx_mono hx_tendsto using
tendsto_nhds_unique hx_tendsto (tendsto_atTop_iSup hx_mono)
#align measure_theory.lintegral_tendsto_of_tendsto_of_monotone MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone
theorem lintegral_eq_iSup_eapprox_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) :
∫⁻ a, f a ∂μ = ⨆ n, (eapprox f n).lintegral μ :=
calc
∫⁻ a, f a ∂μ = ∫⁻ a, ⨆ n, (eapprox f n : α → ℝ≥0∞) a ∂μ := by
congr; ext a; rw [iSup_eapprox_apply f hf]
_ = ⨆ n, ∫⁻ a, (eapprox f n : α → ℝ≥0∞) a ∂μ := by
apply lintegral_iSup
· measurability
· intro i j h
exact monotone_eapprox f h
_ = ⨆ n, (eapprox f n).lintegral μ := by
congr; ext n; rw [(eapprox f n).lintegral_eq_lintegral]
#align measure_theory.lintegral_eq_supr_eapprox_lintegral MeasureTheory.lintegral_eq_iSup_eapprox_lintegral
theorem exists_pos_set_lintegral_lt_of_measure_lt {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {ε : ℝ≥0∞}
(hε : ε ≠ 0) : ∃ δ > 0, ∀ s, μ s < δ → ∫⁻ x in s, f x ∂μ < ε := by
rcases exists_between (pos_iff_ne_zero.mpr hε) with ⟨ε₂, hε₂0, hε₂ε⟩
rcases exists_between hε₂0 with ⟨ε₁, hε₁0, hε₁₂⟩
rcases exists_simpleFunc_forall_lintegral_sub_lt_of_pos h hε₁0.ne' with ⟨φ, _, hφ⟩
rcases φ.exists_forall_le with ⟨C, hC⟩
use (ε₂ - ε₁) / C, ENNReal.div_pos_iff.2 ⟨(tsub_pos_iff_lt.2 hε₁₂).ne', ENNReal.coe_ne_top⟩
refine fun s hs => lt_of_le_of_lt ?_ hε₂ε
simp only [lintegral_eq_nnreal, iSup_le_iff]
intro ψ hψ
calc
(map (↑) ψ).lintegral (μ.restrict s) ≤
(map (↑) φ).lintegral (μ.restrict s) + (map (↑) (ψ - φ)).lintegral (μ.restrict s) := by
rw [← SimpleFunc.add_lintegral, ← SimpleFunc.map_add @ENNReal.coe_add]
refine SimpleFunc.lintegral_mono (fun x => ?_) le_rfl
simp only [add_tsub_eq_max, le_max_right, coe_map, Function.comp_apply, SimpleFunc.coe_add,
SimpleFunc.coe_sub, Pi.add_apply, Pi.sub_apply, ENNReal.coe_max (φ x) (ψ x)]
_ ≤ (map (↑) φ).lintegral (μ.restrict s) + ε₁ := by
gcongr
refine le_trans ?_ (hφ _ hψ).le
exact SimpleFunc.lintegral_mono le_rfl Measure.restrict_le_self
_ ≤ (SimpleFunc.const α (C : ℝ≥0∞)).lintegral (μ.restrict s) + ε₁ := by
gcongr
exact SimpleFunc.lintegral_mono (fun x ↦ ENNReal.coe_le_coe.2 (hC x)) le_rfl
_ = C * μ s + ε₁ := by
simp only [← SimpleFunc.lintegral_eq_lintegral, coe_const, lintegral_const,
Measure.restrict_apply, MeasurableSet.univ, univ_inter, Function.const]
_ ≤ C * ((ε₂ - ε₁) / C) + ε₁ := by gcongr
_ ≤ ε₂ - ε₁ + ε₁ := by gcongr; apply mul_div_le
_ = ε₂ := tsub_add_cancel_of_le hε₁₂.le
#align measure_theory.exists_pos_set_lintegral_lt_of_measure_lt MeasureTheory.exists_pos_set_lintegral_lt_of_measure_lt
theorem tendsto_set_lintegral_zero {ι} {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {l : Filter ι}
{s : ι → Set α} (hl : Tendsto (μ ∘ s) l (𝓝 0)) :
Tendsto (fun i => ∫⁻ x in s i, f x ∂μ) l (𝓝 0) := by
simp only [ENNReal.nhds_zero, tendsto_iInf, tendsto_principal, mem_Iio,
← pos_iff_ne_zero] at hl ⊢
intro ε ε0
rcases exists_pos_set_lintegral_lt_of_measure_lt h ε0.ne' with ⟨δ, δ0, hδ⟩
exact (hl δ δ0).mono fun i => hδ _
#align measure_theory.tendsto_set_lintegral_zero MeasureTheory.tendsto_set_lintegral_zero
theorem le_lintegral_add (f g : α → ℝ≥0∞) :
∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ ≤ ∫⁻ a, f a + g a ∂μ := by
simp only [lintegral]
refine ENNReal.biSup_add_biSup_le' (p := fun h : α →ₛ ℝ≥0∞ => h ≤ f)
(q := fun h : α →ₛ ℝ≥0∞ => h ≤ g) ⟨0, zero_le f⟩ ⟨0, zero_le g⟩ fun f' hf' g' hg' => ?_
exact le_iSup₂_of_le (f' + g') (add_le_add hf' hg') (add_lintegral _ _).ge
#align measure_theory.le_lintegral_add MeasureTheory.le_lintegral_add
-- Use stronger lemmas `lintegral_add_left`/`lintegral_add_right` instead
theorem lintegral_add_aux {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) :
∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ :=
calc
∫⁻ a, f a + g a ∂μ =
∫⁻ a, (⨆ n, (eapprox f n : α → ℝ≥0∞) a) + ⨆ n, (eapprox g n : α → ℝ≥0∞) a ∂μ := by
simp only [iSup_eapprox_apply, hf, hg]
_ = ∫⁻ a, ⨆ n, (eapprox f n + eapprox g n : α → ℝ≥0∞) a ∂μ := by
congr; funext a
rw [ENNReal.iSup_add_iSup_of_monotone]
· simp only [Pi.add_apply]
· intro i j h
exact monotone_eapprox _ h a
· intro i j h
exact monotone_eapprox _ h a
_ = ⨆ n, (eapprox f n).lintegral μ + (eapprox g n).lintegral μ := by
rw [lintegral_iSup]
· congr
funext n
rw [← SimpleFunc.add_lintegral, ← SimpleFunc.lintegral_eq_lintegral]
simp only [Pi.add_apply, SimpleFunc.coe_add]
· measurability
· intro i j h a
dsimp
gcongr <;> exact monotone_eapprox _ h _
_ = (⨆ n, (eapprox f n).lintegral μ) + ⨆ n, (eapprox g n).lintegral μ := by
refine (ENNReal.iSup_add_iSup_of_monotone ?_ ?_).symm <;>
· intro i j h
exact SimpleFunc.lintegral_mono (monotone_eapprox _ h) le_rfl
_ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := by
rw [lintegral_eq_iSup_eapprox_lintegral hf, lintegral_eq_iSup_eapprox_lintegral hg]
#align measure_theory.lintegral_add_aux MeasureTheory.lintegral_add_aux
@[simp]
theorem lintegral_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞) :
∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := by
refine le_antisymm ?_ (le_lintegral_add _ _)
rcases exists_measurable_le_lintegral_eq μ fun a => f a + g a with ⟨φ, hφm, hφ_le, hφ_eq⟩
calc
∫⁻ a, f a + g a ∂μ = ∫⁻ a, φ a ∂μ := hφ_eq
_ ≤ ∫⁻ a, f a + (φ a - f a) ∂μ := lintegral_mono fun a => le_add_tsub
_ = ∫⁻ a, f a ∂μ + ∫⁻ a, φ a - f a ∂μ := lintegral_add_aux hf (hφm.sub hf)
_ ≤ ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ :=
add_le_add_left (lintegral_mono fun a => tsub_le_iff_left.2 <| hφ_le a) _
#align measure_theory.lintegral_add_left MeasureTheory.lintegral_add_left
theorem lintegral_add_left' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (g : α → ℝ≥0∞) :
∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := by
rw [lintegral_congr_ae hf.ae_eq_mk, ← lintegral_add_left hf.measurable_mk,
lintegral_congr_ae (hf.ae_eq_mk.add (ae_eq_refl g))]
#align measure_theory.lintegral_add_left' MeasureTheory.lintegral_add_left'
theorem lintegral_add_right' (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : AEMeasurable g μ) :
∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := by
simpa only [add_comm] using lintegral_add_left' hg f
#align measure_theory.lintegral_add_right' MeasureTheory.lintegral_add_right'
@[simp]
theorem lintegral_add_right (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g) :
∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ :=
lintegral_add_right' f hg.aemeasurable
#align measure_theory.lintegral_add_right MeasureTheory.lintegral_add_right
@[simp]
theorem lintegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) : ∫⁻ a, f a ∂c • μ = c * ∫⁻ a, f a ∂μ := by
simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.mul_iSup, smul_eq_mul]
#align measure_theory.lintegral_smul_measure MeasureTheory.lintegral_smul_measure
lemma set_lintegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) (s : Set α) :
∫⁻ a in s, f a ∂(c • μ) = c * ∫⁻ a in s, f a ∂μ := by
rw [Measure.restrict_smul, lintegral_smul_measure]
@[simp]
theorem lintegral_sum_measure {m : MeasurableSpace α} {ι} (f : α → ℝ≥0∞) (μ : ι → Measure α) :
∫⁻ a, f a ∂Measure.sum μ = ∑' i, ∫⁻ a, f a ∂μ i := by
simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_sum, ENNReal.tsum_eq_iSup_sum]
rw [iSup_comm]
congr; funext s
induction' s using Finset.induction_on with i s hi hs
· simp
simp only [Finset.sum_insert hi, ← hs]
refine (ENNReal.iSup_add_iSup ?_).symm
intro φ ψ
exact
⟨⟨φ ⊔ ψ, fun x => sup_le (φ.2 x) (ψ.2 x)⟩,
add_le_add (SimpleFunc.lintegral_mono le_sup_left le_rfl)
(Finset.sum_le_sum fun j _ => SimpleFunc.lintegral_mono le_sup_right le_rfl)⟩
#align measure_theory.lintegral_sum_measure MeasureTheory.lintegral_sum_measure
theorem hasSum_lintegral_measure {ι} {_ : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : ι → Measure α) :
HasSum (fun i => ∫⁻ a, f a ∂μ i) (∫⁻ a, f a ∂Measure.sum μ) :=
(lintegral_sum_measure f μ).symm ▸ ENNReal.summable.hasSum
#align measure_theory.has_sum_lintegral_measure MeasureTheory.hasSum_lintegral_measure
@[simp]
theorem lintegral_add_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ ν : Measure α) :
∫⁻ a, f a ∂(μ + ν) = ∫⁻ a, f a ∂μ + ∫⁻ a, f a ∂ν := by
simpa [tsum_fintype] using lintegral_sum_measure f fun b => cond b μ ν
#align measure_theory.lintegral_add_measure MeasureTheory.lintegral_add_measure
@[simp]
theorem lintegral_finset_sum_measure {ι} {m : MeasurableSpace α} (s : Finset ι) (f : α → ℝ≥0∞)
(μ : ι → Measure α) : ∫⁻ a, f a ∂(∑ i ∈ s, μ i) = ∑ i ∈ s, ∫⁻ a, f a ∂μ i := by
rw [← Measure.sum_coe_finset, lintegral_sum_measure, ← Finset.tsum_subtype']
simp only [Finset.coe_sort_coe]
#align measure_theory.lintegral_finset_sum_measure MeasureTheory.lintegral_finset_sum_measure
@[simp]
theorem lintegral_zero_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) :
∫⁻ a, f a ∂(0 : Measure α) = 0 := by
simp [lintegral]
#align measure_theory.lintegral_zero_measure MeasureTheory.lintegral_zero_measure
@[simp]
theorem lintegral_of_isEmpty {α} [MeasurableSpace α] [IsEmpty α] (μ : Measure α) (f : α → ℝ≥0∞) :
∫⁻ x, f x ∂μ = 0 := by
have : Subsingleton (Measure α) := inferInstance
convert lintegral_zero_measure f
theorem set_lintegral_empty (f : α → ℝ≥0∞) : ∫⁻ x in ∅, f x ∂μ = 0 := by
rw [Measure.restrict_empty, lintegral_zero_measure]
#align measure_theory.set_lintegral_empty MeasureTheory.set_lintegral_empty
theorem set_lintegral_univ (f : α → ℝ≥0∞) : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by
rw [Measure.restrict_univ]
#align measure_theory.set_lintegral_univ MeasureTheory.set_lintegral_univ
theorem set_lintegral_measure_zero (s : Set α) (f : α → ℝ≥0∞) (hs' : μ s = 0) :
∫⁻ x in s, f x ∂μ = 0 := by
convert lintegral_zero_measure _
exact Measure.restrict_eq_zero.2 hs'
#align measure_theory.set_lintegral_measure_zero MeasureTheory.set_lintegral_measure_zero
theorem lintegral_finset_sum' (s : Finset β) {f : β → α → ℝ≥0∞}
(hf : ∀ b ∈ s, AEMeasurable (f b) μ) :
∫⁻ a, ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ a, f b a ∂μ := by
induction' s using Finset.induction_on with a s has ih
· simp
· simp only [Finset.sum_insert has]
rw [Finset.forall_mem_insert] at hf
rw [lintegral_add_left' hf.1, ih hf.2]
#align measure_theory.lintegral_finset_sum' MeasureTheory.lintegral_finset_sum'
theorem lintegral_finset_sum (s : Finset β) {f : β → α → ℝ≥0∞} (hf : ∀ b ∈ s, Measurable (f b)) :
∫⁻ a, ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ a, f b a ∂μ :=
lintegral_finset_sum' s fun b hb => (hf b hb).aemeasurable
#align measure_theory.lintegral_finset_sum MeasureTheory.lintegral_finset_sum
@[simp]
theorem lintegral_const_mul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ :=
calc
∫⁻ a, r * f a ∂μ = ∫⁻ a, ⨆ n, (const α r * eapprox f n) a ∂μ := by
congr
funext a
rw [← iSup_eapprox_apply f hf, ENNReal.mul_iSup]
simp
_ = ⨆ n, r * (eapprox f n).lintegral μ := by
rw [lintegral_iSup]
· congr
funext n
rw [← SimpleFunc.const_mul_lintegral, ← SimpleFunc.lintegral_eq_lintegral]
· intro n
exact SimpleFunc.measurable _
· intro i j h a
exact mul_le_mul_left' (monotone_eapprox _ h _) _
_ = r * ∫⁻ a, f a ∂μ := by rw [← ENNReal.mul_iSup, lintegral_eq_iSup_eapprox_lintegral hf]
#align measure_theory.lintegral_const_mul MeasureTheory.lintegral_const_mul
theorem lintegral_const_mul'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ := by
have A : ∫⁻ a, f a ∂μ = ∫⁻ a, hf.mk f a ∂μ := lintegral_congr_ae hf.ae_eq_mk
have B : ∫⁻ a, r * f a ∂μ = ∫⁻ a, r * hf.mk f a ∂μ :=
lintegral_congr_ae (EventuallyEq.fun_comp hf.ae_eq_mk _)
rw [A, B, lintegral_const_mul _ hf.measurable_mk]
#align measure_theory.lintegral_const_mul'' MeasureTheory.lintegral_const_mul''
theorem lintegral_const_mul_le (r : ℝ≥0∞) (f : α → ℝ≥0∞) :
r * ∫⁻ a, f a ∂μ ≤ ∫⁻ a, r * f a ∂μ := by
rw [lintegral, ENNReal.mul_iSup]
refine iSup_le fun s => ?_
rw [ENNReal.mul_iSup, iSup_le_iff]
intro hs
rw [← SimpleFunc.const_mul_lintegral, lintegral]
refine le_iSup_of_le (const α r * s) (le_iSup_of_le (fun x => ?_) le_rfl)
exact mul_le_mul_left' (hs x) _
#align measure_theory.lintegral_const_mul_le MeasureTheory.lintegral_const_mul_le
theorem lintegral_const_mul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ := by
by_cases h : r = 0
· simp [h]
apply le_antisymm _ (lintegral_const_mul_le r f)
have rinv : r * r⁻¹ = 1 := ENNReal.mul_inv_cancel h hr
have rinv' : r⁻¹ * r = 1 := by
rw [mul_comm]
exact rinv
have := lintegral_const_mul_le (μ := μ) r⁻¹ fun x => r * f x
simp? [(mul_assoc _ _ _).symm, rinv'] at this says
simp only [(mul_assoc _ _ _).symm, rinv', one_mul] at this
simpa [(mul_assoc _ _ _).symm, rinv] using mul_le_mul_left' this r
#align measure_theory.lintegral_const_mul' MeasureTheory.lintegral_const_mul'
theorem lintegral_mul_const (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r := by simp_rw [mul_comm, lintegral_const_mul r hf]
#align measure_theory.lintegral_mul_const MeasureTheory.lintegral_mul_const
| Mathlib/MeasureTheory/Integral/Lebesgue.lean | 727 | 728 | theorem lintegral_mul_const'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r := by | simp_rw [mul_comm, lintegral_const_mul'' r hf]
|
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
theorem inducing_induced (f : X → Y) : @Inducing X Y (TopologicalSpace.induced f ‹_›) _ f :=
@Inducing.mk _ _ (TopologicalSpace.induced f ‹_›) _ _ rfl
theorem inducing_id : Inducing (@id X) :=
⟨induced_id.symm⟩
#align inducing_id inducing_id
protected theorem Inducing.comp (hg : Inducing g) (hf : Inducing f) :
Inducing (g ∘ f) :=
⟨by rw [hf.induced, hg.induced, induced_compose]⟩
#align inducing.comp Inducing.comp
theorem Inducing.of_comp_iff (hg : Inducing g) :
Inducing (g ∘ f) ↔ Inducing f := by
refine ⟨fun h ↦ ?_, hg.comp⟩
rw [inducing_iff, hg.induced, induced_compose, h.induced]
#align inducing.inducing_iff Inducing.of_comp_iff
theorem inducing_of_inducing_compose
(hf : Continuous f) (hg : Continuous g) (hgf : Inducing (g ∘ f)) : Inducing f :=
⟨le_antisymm (by rwa [← continuous_iff_le_induced])
(by
rw [hgf.induced, ← induced_compose]
exact induced_mono hg.le_induced)⟩
#align inducing_of_inducing_compose inducing_of_inducing_compose
theorem inducing_iff_nhds : Inducing f ↔ ∀ x, 𝓝 x = comap f (𝓝 (f x)) :=
(inducing_iff _).trans (induced_iff_nhds_eq f)
#align inducing_iff_nhds inducing_iff_nhds
namespace Inducing
theorem nhds_eq_comap (hf : Inducing f) : ∀ x : X, 𝓝 x = comap f (𝓝 <| f x) :=
inducing_iff_nhds.1 hf
#align inducing.nhds_eq_comap Inducing.nhds_eq_comap
theorem basis_nhds {p : ι → Prop} {s : ι → Set Y} (hf : Inducing f) {x : X}
(h_basis : (𝓝 (f x)).HasBasis p s) : (𝓝 x).HasBasis p (preimage f ∘ s) :=
hf.nhds_eq_comap x ▸ h_basis.comap f
theorem nhdsSet_eq_comap (hf : Inducing f) (s : Set X) :
𝓝ˢ s = comap f (𝓝ˢ (f '' s)) := by
simp only [nhdsSet, sSup_image, comap_iSup, hf.nhds_eq_comap, iSup_image]
#align inducing.nhds_set_eq_comap Inducing.nhdsSet_eq_comap
theorem map_nhds_eq (hf : Inducing f) (x : X) : (𝓝 x).map f = 𝓝[range f] f x :=
hf.induced.symm ▸ map_nhds_induced_eq x
#align inducing.map_nhds_eq Inducing.map_nhds_eq
theorem map_nhds_of_mem (hf : Inducing f) (x : X) (h : range f ∈ 𝓝 (f x)) :
(𝓝 x).map f = 𝓝 (f x) :=
hf.induced.symm ▸ map_nhds_induced_of_mem h
#align inducing.map_nhds_of_mem Inducing.map_nhds_of_mem
-- Porting note (#10756): new lemma
theorem mapClusterPt_iff (hf : Inducing f) {x : X} {l : Filter X} :
MapClusterPt (f x) l f ↔ ClusterPt x l := by
delta MapClusterPt ClusterPt
rw [← Filter.push_pull', ← hf.nhds_eq_comap, map_neBot_iff]
theorem image_mem_nhdsWithin (hf : Inducing f) {x : X} {s : Set X} (hs : s ∈ 𝓝 x) :
f '' s ∈ 𝓝[range f] f x :=
hf.map_nhds_eq x ▸ image_mem_map hs
#align inducing.image_mem_nhds_within Inducing.image_mem_nhdsWithin
theorem tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : Inducing g) :
Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) := by
rw [hg.nhds_eq_comap, tendsto_comap_iff]
#align inducing.tendsto_nhds_iff Inducing.tendsto_nhds_iff
theorem continuousAt_iff (hg : Inducing g) {x : X} :
ContinuousAt f x ↔ ContinuousAt (g ∘ f) x :=
hg.tendsto_nhds_iff
#align inducing.continuous_at_iff Inducing.continuousAt_iff
| Mathlib/Topology/Maps.lean | 132 | 134 | theorem continuous_iff (hg : Inducing g) :
Continuous f ↔ Continuous (g ∘ f) := by |
simp_rw [continuous_iff_continuousAt, hg.continuousAt_iff]
|
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Tactic.IntervalCases
namespace Polynomial
section IsDomain
variable {R : Type*} [CommRing R] [IsDomain R]
| Mathlib/Algebra/Polynomial/SpecificDegree.lean | 22 | 34 | theorem Monic.irreducible_iff_roots_eq_zero_of_degree_le_three {p : R[X]} (hp : p.Monic)
(hp2 : 2 ≤ p.natDegree) (hp3 : p.natDegree ≤ 3) : Irreducible p ↔ p.roots = 0 := by |
have hp0 : p ≠ 0 := hp.ne_zero
have hp1 : p ≠ 1 := by rintro rfl; rw [natDegree_one] at hp2; cases hp2
rw [hp.irreducible_iff_lt_natDegree_lt hp1]
simp_rw [show p.natDegree / 2 = 1 from
(Nat.div_le_div_right hp3).antisymm
(by apply Nat.div_le_div_right (c := 2) hp2),
show Finset.Ioc 0 1 = {1} from rfl,
Finset.mem_singleton, Multiset.eq_zero_iff_forall_not_mem, mem_roots hp0, ← dvd_iff_isRoot]
refine ⟨fun h r ↦ h _ (monic_X_sub_C r) (natDegree_X_sub_C r), fun h q hq hq1 ↦ ?_⟩
rw [hq.eq_X_add_C hq1, ← sub_neg_eq_add, ← C_neg]
apply h
|
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Algebra.Homology.QuasiIso
#align_import category_theory.preadditive.injective_resolution from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
noncomputable section
universe v u
namespace CategoryTheory
open Limits HomologicalComplex CochainComplex
variable {C : Type u} [Category.{v} C] [HasZeroObject C] [HasZeroMorphisms C]
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_nonempty_instance]
structure InjectiveResolution (Z : C) where
cocomplex : CochainComplex C ℕ
injective : ∀ n, Injective (cocomplex.X n) := by infer_instance
[hasHomology : ∀ i, cocomplex.HasHomology i]
ι : (single₀ C).obj Z ⟶ cocomplex
quasiIso : QuasiIso ι := by infer_instance
set_option linter.uppercaseLean3 false in
#align category_theory.InjectiveResolution CategoryTheory.InjectiveResolution
open InjectiveResolution in
attribute [instance] injective hasHomology InjectiveResolution.quasiIso
class HasInjectiveResolution (Z : C) : Prop where
out : Nonempty (InjectiveResolution Z)
#align category_theory.has_injective_resolution CategoryTheory.HasInjectiveResolution
attribute [inherit_doc HasInjectiveResolution] HasInjectiveResolution.out
section
variable (C)
class HasInjectiveResolutions : Prop where
out : ∀ Z : C, HasInjectiveResolution Z
#align category_theory.has_injective_resolutions CategoryTheory.HasInjectiveResolutions
attribute [instance 100] HasInjectiveResolutions.out
end
namespace InjectiveResolution
variable {Z : C} (I : InjectiveResolution Z)
lemma cocomplex_exactAt_succ (n : ℕ) :
I.cocomplex.ExactAt (n + 1) := by
rw [← quasiIsoAt_iff_exactAt I.ι (n + 1) (exactAt_succ_single_obj _ _)]
infer_instance
lemma exact_succ (n : ℕ):
(ShortComplex.mk _ _ (I.cocomplex.d_comp_d n (n + 1) (n + 2))).Exact :=
(HomologicalComplex.exactAt_iff' _ n (n + 1) (n + 2) (by simp)
(by simp only [CochainComplex.next]; rfl)).1 (I.cocomplex_exactAt_succ n)
@[simp]
theorem ι_f_succ (n : ℕ) : I.ι.f (n + 1) = 0 :=
(isZero_single_obj_X _ _ _ _ (by simp)).eq_of_src _ _
set_option linter.uppercaseLean3 false in
#align category_theory.InjectiveResolution.ι_f_succ CategoryTheory.InjectiveResolution.ι_f_succ
-- Porting note (#10618): removed @[simp] simp can prove this
@[reassoc]
theorem ι_f_zero_comp_complex_d :
I.ι.f 0 ≫ I.cocomplex.d 0 1 = 0 := by
simp
set_option linter.uppercaseLean3 false in
#align category_theory.InjectiveResolution.ι_f_zero_comp_complex_d CategoryTheory.InjectiveResolution.ι_f_zero_comp_complex_d
-- Porting note (#10618): removed @[simp] simp can prove this
| Mathlib/CategoryTheory/Preadditive/InjectiveResolution.lean | 111 | 113 | theorem complex_d_comp (n : ℕ) :
I.cocomplex.d n (n + 1) ≫ I.cocomplex.d (n + 1) (n + 2) = 0 := by |
simp
|
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
#align_import linear_algebra.matrix.to_linear_equiv from "leanprover-community/mathlib"@"e42cfdb03b7902f8787a1eb552cb8f77766b45b9"
variable {n : Type*} [Fintype n]
namespace Matrix
section LinearEquiv
open LinearMap
variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
section Nondegenerate
open Matrix
| Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean | 114 | 132 | theorem exists_mulVec_eq_zero_iff_aux {K : Type*} [DecidableEq n] [Field K] {M : Matrix n n K} :
(∃ v ≠ 0, M *ᵥ v = 0) ↔ M.det = 0 := by |
constructor
· rintro ⟨v, hv, mul_eq⟩
contrapose! hv
exact eq_zero_of_mulVec_eq_zero hv mul_eq
· contrapose!
intro h
have : Function.Injective (Matrix.toLin' M) := by
simpa only [← LinearMap.ker_eq_bot, ker_toLin'_eq_bot_iff, not_imp_not] using h
have :
M *
LinearMap.toMatrix'
((LinearEquiv.ofInjectiveEndo (Matrix.toLin' M) this).symm : (n → K) →ₗ[K] n → K) =
1 := by
refine Matrix.toLin'.injective (LinearMap.ext fun v => ?_)
rw [Matrix.toLin'_mul, Matrix.toLin'_one, Matrix.toLin'_toMatrix', LinearMap.comp_apply]
exact (LinearEquiv.ofInjectiveEndo (Matrix.toLin' M) this).apply_symm_apply v
exact Matrix.det_ne_zero_of_right_inverse this
|
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open scoped Classical
open Real Topology Filter ComplexConjugate Finset Set
namespace Complex
noncomputable def cpow (x y : ℂ) : ℂ :=
if x = 0 then if y = 0 then 1 else 0 else exp (log x * y)
#align complex.cpow Complex.cpow
noncomputable instance : Pow ℂ ℂ :=
⟨cpow⟩
@[simp]
theorem cpow_eq_pow (x y : ℂ) : cpow x y = x ^ y :=
rfl
#align complex.cpow_eq_pow Complex.cpow_eq_pow
theorem cpow_def (x y : ℂ) : x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) :=
rfl
#align complex.cpow_def Complex.cpow_def
theorem cpow_def_of_ne_zero {x : ℂ} (hx : x ≠ 0) (y : ℂ) : x ^ y = exp (log x * y) :=
if_neg hx
#align complex.cpow_def_of_ne_zero Complex.cpow_def_of_ne_zero
@[simp]
theorem cpow_zero (x : ℂ) : x ^ (0 : ℂ) = 1 := by simp [cpow_def]
#align complex.cpow_zero Complex.cpow_zero
@[simp]
theorem cpow_eq_zero_iff (x y : ℂ) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by
simp only [cpow_def]
split_ifs <;> simp [*, exp_ne_zero]
#align complex.cpow_eq_zero_iff Complex.cpow_eq_zero_iff
@[simp]
theorem zero_cpow {x : ℂ} (h : x ≠ 0) : (0 : ℂ) ^ x = 0 := by simp [cpow_def, *]
#align complex.zero_cpow Complex.zero_cpow
| Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 58 | 72 | theorem zero_cpow_eq_iff {x : ℂ} {a : ℂ} : (0 : ℂ) ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 := by |
constructor
· intro hyp
simp only [cpow_def, eq_self_iff_true, if_true] at hyp
by_cases h : x = 0
· subst h
simp only [if_true, eq_self_iff_true] at hyp
right
exact ⟨rfl, hyp.symm⟩
· rw [if_neg h] at hyp
left
exact ⟨h, hyp.symm⟩
· rintro (⟨h, rfl⟩ | ⟨rfl, rfl⟩)
· exact zero_cpow h
· exact cpow_zero _
|
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 Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.Matrix
#align_import analysis.normed_space.finite_dimension from "leanprover-community/mathlib"@"9425b6f8220e53b059f5a4904786c3c4b50fc057"
universe u v w x
noncomputable section
open Set FiniteDimensional TopologicalSpace Filter Asymptotics Classical Topology
NNReal Metric
section CompleteField
variable {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {F : Type w} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {F' : Type x}
[AddCommGroup F'] [Module 𝕜 F'] [TopologicalSpace F'] [TopologicalAddGroup F']
[ContinuousSMul 𝕜 F'] [CompleteSpace 𝕜]
theorem ContinuousLinearMap.continuous_det : Continuous fun f : E →L[𝕜] E => f.det := by
change Continuous fun f : E →L[𝕜] E => LinearMap.det (f : E →ₗ[𝕜] E)
-- Porting note: this could be easier with `det_cases`
by_cases h : ∃ s : Finset E, Nonempty (Basis (↥s) 𝕜 E)
· rcases h with ⟨s, ⟨b⟩⟩
haveI : FiniteDimensional 𝕜 E := FiniteDimensional.of_fintype_basis b
simp_rw [LinearMap.det_eq_det_toMatrix_of_finset b]
refine Continuous.matrix_det ?_
exact
((LinearMap.toMatrix b b).toLinearMap.comp
(ContinuousLinearMap.coeLM 𝕜)).continuous_of_finiteDimensional
· -- Porting note: was `unfold LinearMap.det`
rw [LinearMap.det_def]
simpa only [h, MonoidHom.one_apply, dif_neg, not_false_iff] using continuous_const
#align continuous_linear_map.continuous_det ContinuousLinearMap.continuous_det
irreducible_def lipschitzExtensionConstant (E' : Type*) [NormedAddCommGroup E'] [NormedSpace ℝ E']
[FiniteDimensional ℝ E'] : ℝ≥0 :=
let A := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
max (‖A.symm.toContinuousLinearMap‖₊ * ‖A.toContinuousLinearMap‖₊) 1
#align lipschitz_extension_constant lipschitzExtensionConstant
theorem lipschitzExtensionConstant_pos (E' : Type*) [NormedAddCommGroup E'] [NormedSpace ℝ E']
[FiniteDimensional ℝ E'] : 0 < lipschitzExtensionConstant E' := by
rw [lipschitzExtensionConstant]
exact zero_lt_one.trans_le (le_max_right _ _)
#align lipschitz_extension_constant_pos lipschitzExtensionConstant_pos
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 := by
let ι : Type _ := Basis.ofVectorSpaceIndex ℝ E'
let A := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
have LA : LipschitzWith ‖A.toContinuousLinearMap‖₊ A := by apply A.lipschitz
have L : LipschitzOnWith (‖A.toContinuousLinearMap‖₊ * K) (A ∘ f) s :=
LA.comp_lipschitzOnWith hf
obtain ⟨g, hg, gs⟩ :
∃ g : α → ι → ℝ, LipschitzWith (‖A.toContinuousLinearMap‖₊ * K) g ∧ EqOn (A ∘ f) g s :=
L.extend_pi
refine ⟨A.symm ∘ g, ?_, ?_⟩
· have LAsymm : LipschitzWith ‖A.symm.toContinuousLinearMap‖₊ A.symm := by
apply A.symm.lipschitz
apply (LAsymm.comp hg).weaken
rw [lipschitzExtensionConstant, ← mul_assoc]
exact mul_le_mul' (le_max_left _ _) le_rfl
· intro x hx
have : A (f x) = g x := gs hx
simp only [(· ∘ ·), ← this, A.symm_apply_apply]
#align lipschitz_on_with.extend_finite_dimension LipschitzOnWith.extend_finite_dimension
theorem LinearMap.exists_antilipschitzWith [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F)
(hf : LinearMap.ker f = ⊥) : ∃ K > 0, AntilipschitzWith K f := by
cases subsingleton_or_nontrivial E
· exact ⟨1, zero_lt_one, AntilipschitzWith.of_subsingleton⟩
· rw [LinearMap.ker_eq_bot] at hf
let e : E ≃L[𝕜] LinearMap.range f := (LinearEquiv.ofInjective f hf).toContinuousLinearEquiv
exact ⟨_, e.nnnorm_symm_pos, e.antilipschitz⟩
#align linear_map.exists_antilipschitz_with LinearMap.exists_antilipschitzWith
open Function in
theorem LinearMap.injective_iff_antilipschitz [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F) :
Injective f ↔ ∃ K > 0, AntilipschitzWith K f := by
constructor
· rw [← LinearMap.ker_eq_bot]
exact f.exists_antilipschitzWith
· rintro ⟨K, -, H⟩
exact H.injective
open Function in
theorem ContinuousLinearMap.isOpen_injective [FiniteDimensional 𝕜 E] :
IsOpen { L : E →L[𝕜] F | Injective L } := by
rw [isOpen_iff_eventually]
rintro φ₀ hφ₀
rcases φ₀.injective_iff_antilipschitz.mp hφ₀ with ⟨K, K_pos, H⟩
have : ∀ᶠ φ in 𝓝 φ₀, ‖φ - φ₀‖₊ < K⁻¹ := eventually_nnnorm_sub_lt _ <| inv_pos_of_pos K_pos
filter_upwards [this] with φ hφ
apply φ.injective_iff_antilipschitz.mpr
exact ⟨(K⁻¹ - ‖φ - φ₀‖₊)⁻¹, inv_pos_of_pos (tsub_pos_of_lt hφ),
H.add_sub_lipschitzWith (φ - φ₀).lipschitz hφ⟩
protected theorem LinearIndependent.eventually {ι} [Finite ι] {f : ι → E}
(hf : LinearIndependent 𝕜 f) : ∀ᶠ g in 𝓝 f, LinearIndependent 𝕜 g := by
cases nonempty_fintype ι
simp only [Fintype.linearIndependent_iff'] at hf ⊢
rcases LinearMap.exists_antilipschitzWith _ hf with ⟨K, K0, hK⟩
have : Tendsto (fun g : ι → E => ∑ i, ‖g i - f i‖) (𝓝 f) (𝓝 <| ∑ i, ‖f i - f i‖) :=
tendsto_finset_sum _ fun i _ =>
Tendsto.norm <| ((continuous_apply i).tendsto _).sub tendsto_const_nhds
simp only [sub_self, norm_zero, Finset.sum_const_zero] at this
refine (this.eventually (gt_mem_nhds <| inv_pos.2 K0)).mono fun g hg => ?_
replace hg : ∑ i, ‖g i - f i‖₊ < K⁻¹ := by
rw [← NNReal.coe_lt_coe]
push_cast
exact hg
rw [LinearMap.ker_eq_bot]
refine (hK.add_sub_lipschitzWith (LipschitzWith.of_dist_le_mul fun v u => ?_) hg).injective
simp only [dist_eq_norm, LinearMap.lsum_apply, Pi.sub_apply, LinearMap.sum_apply,
LinearMap.comp_apply, LinearMap.proj_apply, LinearMap.smulRight_apply, LinearMap.id_apply, ←
Finset.sum_sub_distrib, ← smul_sub, ← sub_smul, NNReal.coe_sum, coe_nnnorm, Finset.sum_mul]
refine norm_sum_le_of_le _ fun i _ => ?_
rw [norm_smul, mul_comm]
gcongr
exact norm_le_pi_norm (v - u) i
#align linear_independent.eventually LinearIndependent.eventually
theorem isOpen_setOf_linearIndependent {ι : Type*} [Finite ι] :
IsOpen { f : ι → E | LinearIndependent 𝕜 f } :=
isOpen_iff_mem_nhds.2 fun _ => LinearIndependent.eventually
#align is_open_set_of_linear_independent isOpen_setOf_linearIndependent
theorem isOpen_setOf_nat_le_rank (n : ℕ) :
IsOpen { f : E →L[𝕜] F | ↑n ≤ (f : E →ₗ[𝕜] F).rank } := by
simp only [LinearMap.le_rank_iff_exists_linearIndependent_finset, setOf_exists, ← exists_prop]
refine isOpen_biUnion fun t _ => ?_
have : Continuous fun f : E →L[𝕜] F => fun x : (t : Set E) => f x :=
continuous_pi fun x => (ContinuousLinearMap.apply 𝕜 F (x : E)).continuous
exact isOpen_setOf_linearIndependent.preimage this
#align is_open_set_of_nat_le_rank isOpen_setOf_nat_le_rank
theorem Basis.opNNNorm_le {ι : Type*} [Fintype ι] (v : Basis ι 𝕜 E) {u : E →L[𝕜] F} (M : ℝ≥0)
(hu : ∀ i, ‖u (v i)‖₊ ≤ M) : ‖u‖₊ ≤ Fintype.card ι • ‖v.equivFunL.toContinuousLinearMap‖₊ * M :=
u.opNNNorm_le_bound _ fun e => by
set φ := v.equivFunL.toContinuousLinearMap
calc
‖u e‖₊ = ‖u (∑ i, v.equivFun e i • v i)‖₊ := by rw [v.sum_equivFun]
_ = ‖∑ i, v.equivFun e i • (u <| v i)‖₊ := by simp [map_sum, LinearMap.map_smul]
_ ≤ ∑ i, ‖v.equivFun e i • (u <| v i)‖₊ := nnnorm_sum_le _ _
_ = ∑ i, ‖v.equivFun e i‖₊ * ‖u (v i)‖₊ := by simp only [nnnorm_smul]
_ ≤ ∑ i, ‖v.equivFun e i‖₊ * M := by gcongr; apply hu
_ = (∑ i, ‖v.equivFun e i‖₊) * M := by rw [Finset.sum_mul]
_ ≤ Fintype.card ι • (‖φ‖₊ * ‖e‖₊) * M := by
gcongr
calc
∑ i, ‖v.equivFun e i‖₊ ≤ Fintype.card ι • ‖φ e‖₊ := Pi.sum_nnnorm_apply_le_nnnorm _
_ ≤ Fintype.card ι • (‖φ‖₊ * ‖e‖₊) := nsmul_le_nsmul_right (φ.le_opNNNorm e) _
_ = Fintype.card ι • ‖φ‖₊ * M * ‖e‖₊ := by simp only [smul_mul_assoc, mul_right_comm]
#align basis.op_nnnorm_le Basis.opNNNorm_le
@[deprecated (since := "2024-02-02")] alias Basis.op_nnnorm_le := Basis.opNNNorm_le
theorem Basis.opNorm_le {ι : Type*} [Fintype ι] (v : Basis ι 𝕜 E) {u : E →L[𝕜] F} {M : ℝ}
(hM : 0 ≤ M) (hu : ∀ i, ‖u (v i)‖ ≤ M) :
‖u‖ ≤ Fintype.card ι • ‖v.equivFunL.toContinuousLinearMap‖ * M := by
simpa using NNReal.coe_le_coe.mpr (v.opNNNorm_le ⟨M, hM⟩ hu)
#align basis.op_norm_le Basis.opNorm_le
@[deprecated (since := "2024-02-02")] alias Basis.op_norm_le := Basis.opNorm_le
theorem Basis.exists_opNNNorm_le {ι : Type*} [Finite ι] (v : Basis ι 𝕜 E) :
∃ C > (0 : ℝ≥0), ∀ {u : E →L[𝕜] F} (M : ℝ≥0), (∀ i, ‖u (v i)‖₊ ≤ M) → ‖u‖₊ ≤ C * M := by
cases nonempty_fintype ι
exact
⟨max (Fintype.card ι • ‖v.equivFunL.toContinuousLinearMap‖₊) 1,
zero_lt_one.trans_le (le_max_right _ _), fun {u} M hu =>
(v.opNNNorm_le M hu).trans <| mul_le_mul_of_nonneg_right (le_max_left _ _) (zero_le M)⟩
#align basis.exists_op_nnnorm_le Basis.exists_opNNNorm_le
@[deprecated (since := "2024-02-02")] alias Basis.exists_op_nnnorm_le := Basis.exists_opNNNorm_le
theorem Basis.exists_opNorm_le {ι : Type*} [Finite ι] (v : Basis ι 𝕜 E) :
∃ C > (0 : ℝ), ∀ {u : E →L[𝕜] F} {M : ℝ}, 0 ≤ M → (∀ i, ‖u (v i)‖ ≤ M) → ‖u‖ ≤ C * M := by
obtain ⟨C, hC, h⟩ := v.exists_opNNNorm_le (F := F)
-- Porting note: used `Subtype.forall'` below
refine ⟨C, hC, ?_⟩
intro u M hM H
simpa using h ⟨M, hM⟩ H
#align basis.exists_op_norm_le Basis.exists_opNorm_le
@[deprecated (since := "2024-02-02")] alias Basis.exists_op_norm_le := Basis.exists_opNorm_le
instance [FiniteDimensional 𝕜 E] [SecondCountableTopology F] :
SecondCountableTopology (E →L[𝕜] F) := by
set d := FiniteDimensional.finrank 𝕜 E
suffices
∀ ε > (0 : ℝ), ∃ n : (E →L[𝕜] F) → Fin d → ℕ, ∀ f g : E →L[𝕜] F, n f = n g → dist f g ≤ ε from
Metric.secondCountable_of_countable_discretization fun ε ε_pos =>
⟨Fin d → ℕ, by infer_instance, this ε ε_pos⟩
intro ε ε_pos
obtain ⟨u : ℕ → F, hu : DenseRange u⟩ := exists_dense_seq F
let v := FiniteDimensional.finBasis 𝕜 E
obtain
⟨C : ℝ, C_pos : 0 < C, hC :
∀ {φ : E →L[𝕜] F} {M : ℝ}, 0 ≤ M → (∀ i, ‖φ (v i)‖ ≤ M) → ‖φ‖ ≤ C * M⟩ :=
v.exists_opNorm_le (E := E) (F := F)
have h_2C : 0 < 2 * C := mul_pos zero_lt_two C_pos
have hε2C : 0 < ε / (2 * C) := div_pos ε_pos h_2C
have : ∀ φ : E →L[𝕜] F, ∃ n : Fin d → ℕ, ‖φ - (v.constrL <| u ∘ n)‖ ≤ ε / 2 := by
intro φ
have : ∀ i, ∃ n, ‖φ (v i) - u n‖ ≤ ε / (2 * C) := by
simp only [norm_sub_rev]
intro i
have : φ (v i) ∈ closure (range u) := hu _
obtain ⟨n, hn⟩ : ∃ n, ‖u n - φ (v i)‖ < ε / (2 * C) := by
rw [mem_closure_iff_nhds_basis Metric.nhds_basis_ball] at this
specialize this (ε / (2 * C)) hε2C
simpa [dist_eq_norm]
exact ⟨n, le_of_lt hn⟩
choose n hn using this
use n
replace hn : ∀ i : Fin d, ‖(φ - (v.constrL <| u ∘ n)) (v i)‖ ≤ ε / (2 * C) := by simp [hn]
have : C * (ε / (2 * C)) = ε / 2 := by
rw [eq_div_iff (two_ne_zero : (2 : ℝ) ≠ 0), mul_comm, ← mul_assoc,
mul_div_cancel₀ _ (ne_of_gt h_2C)]
specialize hC (le_of_lt hε2C) hn
rwa [this] at hC
choose n hn using this
set Φ := fun φ : E →L[𝕜] F => v.constrL <| u ∘ n φ
change ∀ z, dist z (Φ z) ≤ ε / 2 at hn
use n
intro x y hxy
calc
dist x y ≤ dist x (Φ x) + dist (Φ x) y := dist_triangle _ _ _
_ = dist x (Φ x) + dist y (Φ y) := by simp [Φ, hxy, dist_comm]
_ ≤ ε := by linarith [hn x, hn y]
theorem AffineSubspace.closed_of_finiteDimensional {P : Type*} [MetricSpace P]
[NormedAddTorsor E P] (s : AffineSubspace 𝕜 P) [FiniteDimensional 𝕜 s.direction] :
IsClosed (s : Set P) :=
s.isClosed_direction_iff.mp s.direction.closed_of_finiteDimensional
#align affine_subspace.closed_of_finite_dimensional AffineSubspace.closed_of_finiteDimensional
open ContinuousLinearMap
def ContinuousLinearEquiv.piRing (ι : Type*) [Fintype ι] [DecidableEq ι] :
((ι → 𝕜) →L[𝕜] E) ≃L[𝕜] ι → E :=
{ LinearMap.toContinuousLinearMap.symm.trans (LinearEquiv.piRing 𝕜 E ι 𝕜) with
continuous_toFun := by
refine continuous_pi fun i => ?_
exact (ContinuousLinearMap.apply 𝕜 E (Pi.single i 1)).continuous
continuous_invFun := by
simp_rw [LinearEquiv.invFun_eq_symm, LinearEquiv.trans_symm, LinearEquiv.symm_symm]
-- Note: added explicit type and removed `change` that tried to achieve the same
refine AddMonoidHomClass.continuous_of_bound
(LinearMap.toContinuousLinearMap.toLinearMap.comp
(LinearEquiv.piRing 𝕜 E ι 𝕜).symm.toLinearMap)
(Fintype.card ι : ℝ) fun g => ?_
rw [← nsmul_eq_mul]
refine opNorm_le_bound _ (nsmul_nonneg (norm_nonneg g) (Fintype.card ι)) fun t => ?_
simp_rw [LinearMap.coe_comp, LinearEquiv.coe_toLinearMap, Function.comp_apply,
LinearMap.coe_toContinuousLinearMap', LinearEquiv.piRing_symm_apply]
apply le_trans (norm_sum_le _ _)
rw [smul_mul_assoc]
refine Finset.sum_le_card_nsmul _ _ _ fun i _ => ?_
rw [norm_smul, mul_comm]
gcongr <;> apply norm_le_pi_norm }
#align continuous_linear_equiv.pi_ring ContinuousLinearEquiv.piRing
theorem continuousOn_clm_apply {X : Type*} [TopologicalSpace X] [FiniteDimensional 𝕜 E]
{f : X → E →L[𝕜] F} {s : Set X} : ContinuousOn f s ↔ ∀ y, ContinuousOn (fun x => f x y) s := by
refine ⟨fun h y => (ContinuousLinearMap.apply 𝕜 F y).continuous.comp_continuousOn h, fun h => ?_⟩
let d := finrank 𝕜 E
have hd : d = finrank 𝕜 (Fin d → 𝕜) := (finrank_fin_fun 𝕜).symm
let e₁ : E ≃L[𝕜] Fin d → 𝕜 := ContinuousLinearEquiv.ofFinrankEq hd
let e₂ : (E →L[𝕜] F) ≃L[𝕜] Fin d → F :=
(e₁.arrowCongr (1 : F ≃L[𝕜] F)).trans (ContinuousLinearEquiv.piRing (Fin d))
rw [← f.id_comp, ← e₂.symm_comp_self]
exact e₂.symm.continuous.comp_continuousOn (continuousOn_pi.mpr fun i => h _)
#align continuous_on_clm_apply continuousOn_clm_apply
| Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 606 | 608 | theorem continuous_clm_apply {X : Type*} [TopologicalSpace X] [FiniteDimensional 𝕜 E]
{f : X → E →L[𝕜] F} : Continuous f ↔ ∀ y, Continuous (f · y) := by |
simp_rw [continuous_iff_continuousOn_univ, continuousOn_clm_apply]
|
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSeparatingOn
open Set MeasureTheory
namespace MeasurableSpace
variable {α β : Type*}
class CountablyGenerated (α : Type*) [m : MeasurableSpace α] : Prop where
isCountablyGenerated : ∃ b : Set (Set α), b.Countable ∧ m = generateFrom b
#align measurable_space.countably_generated MeasurableSpace.CountablyGenerated
def countableGeneratingSet (α : Type*) [MeasurableSpace α] [h : CountablyGenerated α] :
Set (Set α) :=
insert ∅ h.isCountablyGenerated.choose
lemma countable_countableGeneratingSet [MeasurableSpace α] [h : CountablyGenerated α] :
Set.Countable (countableGeneratingSet α) :=
Countable.insert _ h.isCountablyGenerated.choose_spec.1
lemma generateFrom_countableGeneratingSet [m : MeasurableSpace α] [h : CountablyGenerated α] :
generateFrom (countableGeneratingSet α) = m :=
(generateFrom_insert_empty _).trans <| h.isCountablyGenerated.choose_spec.2.symm
lemma empty_mem_countableGeneratingSet [MeasurableSpace α] [CountablyGenerated α] :
∅ ∈ countableGeneratingSet α := mem_insert _ _
lemma nonempty_countableGeneratingSet [MeasurableSpace α] [CountablyGenerated α] :
Set.Nonempty (countableGeneratingSet α) :=
⟨∅, mem_insert _ _⟩
lemma measurableSet_countableGeneratingSet [MeasurableSpace α] [CountablyGenerated α]
{s : Set α} (hs : s ∈ countableGeneratingSet α) :
MeasurableSet s := by
rw [← generateFrom_countableGeneratingSet (α := α)]
exact measurableSet_generateFrom hs
def natGeneratingSequence (α : Type*) [MeasurableSpace α] [CountablyGenerated α] : ℕ → (Set α) :=
enumerateCountable (countable_countableGeneratingSet (α := α)) ∅
lemma generateFrom_natGeneratingSequence (α : Type*) [m : MeasurableSpace α]
[CountablyGenerated α] : generateFrom (range (natGeneratingSequence _)) = m := by
rw [natGeneratingSequence, range_enumerateCountable_of_mem _ empty_mem_countableGeneratingSet,
generateFrom_countableGeneratingSet]
lemma measurableSet_natGeneratingSequence [MeasurableSpace α] [CountablyGenerated α] (n : ℕ) :
MeasurableSet (natGeneratingSequence α n) :=
measurableSet_countableGeneratingSet $ Set.enumerateCountable_mem _
empty_mem_countableGeneratingSet n
theorem CountablyGenerated.comap [m : MeasurableSpace β] [h : CountablyGenerated β] (f : α → β) :
@CountablyGenerated α (.comap f m) := by
rcases h with ⟨⟨b, hbc, rfl⟩⟩
rw [comap_generateFrom]
letI := generateFrom (preimage f '' b)
exact ⟨_, hbc.image _, rfl⟩
theorem CountablyGenerated.sup {m₁ m₂ : MeasurableSpace β} (h₁ : @CountablyGenerated β m₁)
(h₂ : @CountablyGenerated β m₂) : @CountablyGenerated β (m₁ ⊔ m₂) := by
rcases h₁ with ⟨⟨b₁, hb₁c, rfl⟩⟩
rcases h₂ with ⟨⟨b₂, hb₂c, rfl⟩⟩
exact @mk _ (_ ⊔ _) ⟨_, hb₁c.union hb₂c, generateFrom_sup_generateFrom⟩
instance (priority := 100) [MeasurableSpace α] [Countable α] : CountablyGenerated α where
isCountablyGenerated := by
refine ⟨⋃ y, {measurableAtom y}, countable_iUnion (fun i ↦ countable_singleton _), ?_⟩
refine le_antisymm ?_ (generateFrom_le (by simp [MeasurableSet.measurableAtom_of_countable]))
intro s hs
have : s = ⋃ y ∈ s, measurableAtom y := by
apply Subset.antisymm
· intro x hx
simpa using ⟨x, hx, by simp⟩
· simp only [iUnion_subset_iff]
intro x hx
exact measurableAtom_subset hs hx
rw [this]
apply MeasurableSet.biUnion (to_countable s) (fun x _hx ↦ ?_)
apply measurableSet_generateFrom
simp
instance [MeasurableSpace α] [CountablyGenerated α] {p : α → Prop} :
CountablyGenerated { x // p x } := .comap _
instance [MeasurableSpace α] [CountablyGenerated α] [MeasurableSpace β] [CountablyGenerated β] :
CountablyGenerated (α × β) :=
.sup (.comap Prod.fst) (.comap Prod.snd)
section SeparatesPoints
class SeparatesPoints (α : Type*) [m : MeasurableSpace α] : Prop where
separates : ∀ x y : α, (∀ s, MeasurableSet s → (x ∈ s → y ∈ s)) → x = y
theorem separatesPoints_def [MeasurableSpace α] [hs : SeparatesPoints α] {x y : α}
(h : ∀ s, MeasurableSet s → (x ∈ s → y ∈ s)) : x = y := hs.separates _ _ h
| Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 144 | 147 | theorem exists_measurableSet_of_ne [MeasurableSpace α] [SeparatesPoints α] {x y : α}
(h : x ≠ y) : ∃ s, MeasurableSet s ∧ x ∈ s ∧ y ∉ s := by |
contrapose! h
exact separatesPoints_def h
|
import Mathlib.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
open Function
namespace Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*} {f f' : α → β → γ} {g g' : α → β → γ → δ}
variable {s s' : Set α} {t t' : Set β} {u u' : Set γ} {v : Set δ} {a a' : α} {b b' : β} {c c' : γ}
{d d' : δ}
theorem mem_image2_iff (hf : Injective2 f) : f a b ∈ image2 f s t ↔ a ∈ s ∧ b ∈ t :=
⟨by
rintro ⟨a', ha', b', hb', h⟩
rcases hf h with ⟨rfl, rfl⟩
exact ⟨ha', hb'⟩, fun ⟨ha, hb⟩ => mem_image2_of_mem ha hb⟩
#align set.mem_image2_iff Set.mem_image2_iff
theorem image2_subset (hs : s ⊆ s') (ht : t ⊆ t') : image2 f s t ⊆ image2 f s' t' := by
rintro _ ⟨a, ha, b, hb, rfl⟩
exact mem_image2_of_mem (hs ha) (ht hb)
#align set.image2_subset Set.image2_subset
theorem image2_subset_left (ht : t ⊆ t') : image2 f s t ⊆ image2 f s t' :=
image2_subset Subset.rfl ht
#align set.image2_subset_left Set.image2_subset_left
theorem image2_subset_right (hs : s ⊆ s') : image2 f s t ⊆ image2 f s' t :=
image2_subset hs Subset.rfl
#align set.image2_subset_right Set.image2_subset_right
theorem image_subset_image2_left (hb : b ∈ t) : (fun a => f a b) '' s ⊆ image2 f s t :=
forall_mem_image.2 fun _ ha => mem_image2_of_mem ha hb
#align set.image_subset_image2_left Set.image_subset_image2_left
theorem image_subset_image2_right (ha : a ∈ s) : f a '' t ⊆ image2 f s t :=
forall_mem_image.2 fun _ => mem_image2_of_mem ha
#align set.image_subset_image2_right Set.image_subset_image2_right
theorem forall_image2_iff {p : γ → Prop} :
(∀ z ∈ image2 f s t, p z) ↔ ∀ x ∈ s, ∀ y ∈ t, p (f x y) :=
⟨fun h x hx y hy => h _ ⟨x, hx, y, hy, rfl⟩, fun h _ ⟨x, hx, y, hy, hz⟩ => hz ▸ h x hx y hy⟩
#align set.forall_image2_iff Set.forall_image2_iff
@[simp]
theorem image2_subset_iff {u : Set γ} : image2 f s t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, f x y ∈ u :=
forall_image2_iff
#align set.image2_subset_iff Set.image2_subset_iff
theorem image2_subset_iff_left : image2 f s t ⊆ u ↔ ∀ a ∈ s, (fun b => f a b) '' t ⊆ u := by
simp_rw [image2_subset_iff, image_subset_iff, subset_def, mem_preimage]
#align set.image2_subset_iff_left Set.image2_subset_iff_left
theorem image2_subset_iff_right : image2 f s t ⊆ u ↔ ∀ b ∈ t, (fun a => f a b) '' s ⊆ u := by
simp_rw [image2_subset_iff, image_subset_iff, subset_def, mem_preimage, @forall₂_swap α]
#align set.image2_subset_iff_right Set.image2_subset_iff_right
variable (f)
-- Porting note: Removing `simp` - LHS does not simplify
lemma image_prod : (fun x : α × β ↦ f x.1 x.2) '' s ×ˢ t = image2 f s t :=
ext fun _ ↦ by simp [and_assoc]
#align set.image_prod Set.image_prod
@[simp] lemma image_uncurry_prod (s : Set α) (t : Set β) : uncurry f '' s ×ˢ t = image2 f s t :=
image_prod _
#align set.image_uncurry_prod Set.image_uncurry_prod
@[simp] lemma image2_mk_eq_prod : image2 Prod.mk s t = s ×ˢ t := ext <| by simp
#align set.image2_mk_eq_prod Set.image2_mk_eq_prod
-- Porting note: Removing `simp` - LHS does not simplify
lemma image2_curry (f : α × β → γ) (s : Set α) (t : Set β) :
image2 (fun a b ↦ f (a, b)) s t = f '' s ×ˢ t := by
simp [← image_uncurry_prod, uncurry]
#align set.image2_curry Set.image2_curry
theorem image2_swap (s : Set α) (t : Set β) : image2 f s t = image2 (fun a b => f b a) t s := by
ext
constructor <;> rintro ⟨a, ha, b, hb, rfl⟩ <;> exact ⟨b, hb, a, ha, rfl⟩
#align set.image2_swap Set.image2_swap
variable {f}
theorem image2_union_left : image2 f (s ∪ s') t = image2 f s t ∪ image2 f s' t := by
simp_rw [← image_prod, union_prod, image_union]
#align set.image2_union_left Set.image2_union_left
theorem image2_union_right : image2 f s (t ∪ t') = image2 f s t ∪ image2 f s t' := by
rw [← image2_swap, image2_union_left, image2_swap f, image2_swap f]
#align set.image2_union_right Set.image2_union_right
lemma image2_inter_left (hf : Injective2 f) :
image2 f (s ∩ s') t = image2 f s t ∩ image2 f s' t := by
simp_rw [← image_uncurry_prod, inter_prod, image_inter hf.uncurry]
#align set.image2_inter_left Set.image2_inter_left
lemma image2_inter_right (hf : Injective2 f) :
image2 f s (t ∩ t') = image2 f s t ∩ image2 f s t' := by
simp_rw [← image_uncurry_prod, prod_inter, image_inter hf.uncurry]
#align set.image2_inter_right Set.image2_inter_right
@[simp]
theorem image2_empty_left : image2 f ∅ t = ∅ :=
ext <| by simp
#align set.image2_empty_left Set.image2_empty_left
@[simp]
theorem image2_empty_right : image2 f s ∅ = ∅ :=
ext <| by simp
#align set.image2_empty_right Set.image2_empty_right
theorem Nonempty.image2 : s.Nonempty → t.Nonempty → (image2 f s t).Nonempty :=
fun ⟨_, ha⟩ ⟨_, hb⟩ => ⟨_, mem_image2_of_mem ha hb⟩
#align set.nonempty.image2 Set.Nonempty.image2
@[simp]
theorem image2_nonempty_iff : (image2 f s t).Nonempty ↔ s.Nonempty ∧ t.Nonempty :=
⟨fun ⟨_, a, ha, b, hb, _⟩ => ⟨⟨a, ha⟩, b, hb⟩, fun h => h.1.image2 h.2⟩
#align set.image2_nonempty_iff Set.image2_nonempty_iff
theorem Nonempty.of_image2_left (h : (Set.image2 f s t).Nonempty) : s.Nonempty :=
(image2_nonempty_iff.1 h).1
#align set.nonempty.of_image2_left Set.Nonempty.of_image2_left
theorem Nonempty.of_image2_right (h : (Set.image2 f s t).Nonempty) : t.Nonempty :=
(image2_nonempty_iff.1 h).2
#align set.nonempty.of_image2_right Set.Nonempty.of_image2_right
@[simp]
theorem image2_eq_empty_iff : image2 f s t = ∅ ↔ s = ∅ ∨ t = ∅ := by
rw [← not_nonempty_iff_eq_empty, image2_nonempty_iff, not_and_or]
simp [not_nonempty_iff_eq_empty]
#align set.image2_eq_empty_iff Set.image2_eq_empty_iff
theorem Subsingleton.image2 (hs : s.Subsingleton) (ht : t.Subsingleton) (f : α → β → γ) :
(image2 f s t).Subsingleton := by
rw [← image_prod]
apply (hs.prod ht).image
theorem image2_inter_subset_left : image2 f (s ∩ s') t ⊆ image2 f s t ∩ image2 f s' t :=
Monotone.map_inf_le (fun _ _ ↦ image2_subset_right) s s'
#align set.image2_inter_subset_left Set.image2_inter_subset_left
theorem image2_inter_subset_right : image2 f s (t ∩ t') ⊆ image2 f s t ∩ image2 f s t' :=
Monotone.map_inf_le (fun _ _ ↦ image2_subset_left) t t'
#align set.image2_inter_subset_right Set.image2_inter_subset_right
@[simp]
theorem image2_singleton_left : image2 f {a} t = f a '' t :=
ext fun x => by simp
#align set.image2_singleton_left Set.image2_singleton_left
@[simp]
theorem image2_singleton_right : image2 f s {b} = (fun a => f a b) '' s :=
ext fun x => by simp
#align set.image2_singleton_right Set.image2_singleton_right
theorem image2_singleton : image2 f {a} {b} = {f a b} := by simp
#align set.image2_singleton Set.image2_singleton
@[simp]
theorem image2_insert_left : image2 f (insert a s) t = (fun b => f a b) '' t ∪ image2 f s t := by
rw [insert_eq, image2_union_left, image2_singleton_left]
#align set.image2_insert_left Set.image2_insert_left
@[simp]
theorem image2_insert_right : image2 f s (insert b t) = (fun a => f a b) '' s ∪ image2 f s t := by
rw [insert_eq, image2_union_right, image2_singleton_right]
#align set.image2_insert_right Set.image2_insert_right
@[congr]
theorem image2_congr (h : ∀ a ∈ s, ∀ b ∈ t, f a b = f' a b) : image2 f s t = image2 f' s t := by
ext
constructor <;> rintro ⟨a, ha, b, hb, rfl⟩ <;> exact ⟨a, ha, b, hb, by rw [h a ha b hb]⟩
#align set.image2_congr Set.image2_congr
theorem image2_congr' (h : ∀ a b, f a b = f' a b) : image2 f s t = image2 f' s t :=
image2_congr fun a _ b _ => h a b
#align set.image2_congr' Set.image2_congr'
#noalign set.image3
#noalign set.mem_image3
#noalign set.image3_mono
#noalign set.image3_congr
#noalign set.image3_congr'
#noalign set.image2_image2_left
#noalign set.image2_image2_right
theorem image_image2 (f : α → β → γ) (g : γ → δ) :
g '' image2 f s t = image2 (fun a b => g (f a b)) s t := by
simp only [← image_prod, image_image]
#align set.image_image2 Set.image_image2
theorem image2_image_left (f : γ → β → δ) (g : α → γ) :
image2 f (g '' s) t = image2 (fun a b => f (g a) b) s t := by
ext; simp
#align set.image2_image_left Set.image2_image_left
theorem image2_image_right (f : α → γ → δ) (g : β → γ) :
image2 f s (g '' t) = image2 (fun a b => f a (g b)) s t := by
ext; simp
#align set.image2_image_right Set.image2_image_right
@[simp]
theorem image2_left (h : t.Nonempty) : image2 (fun x _ => x) s t = s := by
simp [nonempty_def.mp h, ext_iff]
#align set.image2_left Set.image2_left
@[simp]
theorem image2_right (h : s.Nonempty) : image2 (fun _ y => y) s t = t := by
simp [nonempty_def.mp h, ext_iff]
#align set.image2_right Set.image2_right
lemma image2_range (f : α' → β' → γ) (g : α → α') (h : β → β') :
image2 f (range g) (range h) = range fun x : α × β ↦ f (g x.1) (h x.2) := by
simp_rw [← image_univ, image2_image_left, image2_image_right, ← image_prod, univ_prod_univ]
theorem image2_assoc {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε} {g' : β → γ → ε'}
(h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
image2 f (image2 g s t) u = image2 f' s (image2 g' t u) :=
eq_of_forall_subset_iff fun _ ↦ by simp only [image2_subset_iff, forall_image2_iff, h_assoc]
#align set.image2_assoc Set.image2_assoc
theorem image2_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : image2 f s t = image2 g t s :=
(image2_swap _ _ _).trans <| by simp_rw [h_comm]
#align set.image2_comm Set.image2_comm
theorem image2_left_comm {f : α → δ → ε} {g : β → γ → δ} {f' : α → γ → δ'} {g' : β → δ' → ε}
(h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
image2 f s (image2 g t u) = image2 g' t (image2 f' s u) := by
rw [image2_swap f', image2_swap f]
exact image2_assoc fun _ _ _ => h_left_comm _ _ _
#align set.image2_left_comm Set.image2_left_comm
theorem image2_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
image2 f (image2 g s t) u = image2 g' (image2 f' s u) t := by
rw [image2_swap g, image2_swap g']
exact image2_assoc fun _ _ _ => h_right_comm _ _ _
#align set.image2_right_comm Set.image2_right_comm
theorem image2_image2_image2_comm {f : ε → ζ → ν} {g : α → β → ε} {h : γ → δ → ζ} {f' : ε' → ζ' → ν}
{g' : α → γ → ε'} {h' : β → δ → ζ'}
(h_comm : ∀ a b c d, f (g a b) (h c d) = f' (g' a c) (h' b d)) :
image2 f (image2 g s t) (image2 h u v) = image2 f' (image2 g' s u) (image2 h' t v) := by
ext; constructor
· rintro ⟨_, ⟨a, ha, b, hb, rfl⟩, _, ⟨c, hc, d, hd, rfl⟩, rfl⟩
exact ⟨_, ⟨a, ha, c, hc, rfl⟩, _, ⟨b, hb, d, hd, rfl⟩, (h_comm _ _ _ _).symm⟩
· rintro ⟨_, ⟨a, ha, c, hc, rfl⟩, _, ⟨b, hb, d, hd, rfl⟩, rfl⟩
exact ⟨_, ⟨a, ha, b, hb, rfl⟩, _, ⟨c, hc, d, hd, rfl⟩, h_comm _ _ _ _⟩
#align set.image2_image2_image2_comm Set.image2_image2_image2_comm
theorem image_image2_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(image2 f s t).image g = image2 f' (s.image g₁) (t.image g₂) := by
simp_rw [image_image2, image2_image_left, image2_image_right, h_distrib]
#align set.image_image2_distrib Set.image_image2_distrib
theorem image_image2_distrib_left {g : γ → δ} {f' : α' → β → δ} {g' : α → α'}
(h_distrib : ∀ a b, g (f a b) = f' (g' a) b) :
(image2 f s t).image g = image2 f' (s.image g') t :=
(image_image2_distrib h_distrib).trans <| by rw [image_id']
#align set.image_image2_distrib_left Set.image_image2_distrib_left
theorem image_image2_distrib_right {g : γ → δ} {f' : α → β' → δ} {g' : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' a (g' b)) :
(image2 f s t).image g = image2 f' s (t.image g') :=
(image_image2_distrib h_distrib).trans <| by rw [image_id']
#align set.image_image2_distrib_right Set.image_image2_distrib_right
theorem image2_image_left_comm {f : α' → β → γ} {g : α → α'} {f' : α → β → δ} {g' : δ → γ}
(h_left_comm : ∀ a b, f (g a) b = g' (f' a b)) :
image2 f (s.image g) t = (image2 f' s t).image g' :=
(image_image2_distrib_left fun a b => (h_left_comm a b).symm).symm
#align set.image2_image_left_comm Set.image2_image_left_comm
theorem image_image2_right_comm {f : α → β' → γ} {g : β → β'} {f' : α → β → δ} {g' : δ → γ}
(h_right_comm : ∀ a b, f a (g b) = g' (f' a b)) :
image2 f s (t.image g) = (image2 f' s t).image g' :=
(image_image2_distrib_right fun a b => (h_right_comm a b).symm).symm
#align set.image_image2_right_comm Set.image_image2_right_comm
theorem image2_distrib_subset_left {f : α → δ → ε} {g : β → γ → δ} {f₁ : α → β → β'}
{f₂ : α → γ → γ'} {g' : β' → γ' → ε} (h_distrib : ∀ a b c, f a (g b c) = g' (f₁ a b) (f₂ a c)) :
image2 f s (image2 g t u) ⊆ image2 g' (image2 f₁ s t) (image2 f₂ s u) := by
rintro _ ⟨a, ha, _, ⟨b, hb, c, hc, rfl⟩, rfl⟩
rw [h_distrib]
exact mem_image2_of_mem (mem_image2_of_mem ha hb) (mem_image2_of_mem ha hc)
#align set.image2_distrib_subset_left Set.image2_distrib_subset_left
theorem image2_distrib_subset_right {f : δ → γ → ε} {g : α → β → δ} {f₁ : α → γ → α'}
{f₂ : β → γ → β'} {g' : α' → β' → ε} (h_distrib : ∀ a b c, f (g a b) c = g' (f₁ a c) (f₂ b c)) :
image2 f (image2 g s t) u ⊆ image2 g' (image2 f₁ s u) (image2 f₂ t u) := by
rintro _ ⟨_, ⟨a, ha, b, hb, rfl⟩, c, hc, rfl⟩
rw [h_distrib]
exact mem_image2_of_mem (mem_image2_of_mem ha hc) (mem_image2_of_mem hb hc)
#align set.image2_distrib_subset_right Set.image2_distrib_subset_right
| Mathlib/Data/Set/NAry.lean | 325 | 329 | theorem image_image2_antidistrib {g : γ → δ} {f' : β' → α' → δ} {g₁ : β → β'} {g₂ : α → α'}
(h_antidistrib : ∀ a b, g (f a b) = f' (g₁ b) (g₂ a)) :
(image2 f s t).image g = image2 f' (t.image g₁) (s.image g₂) := by |
rw [image2_swap f]
exact image_image2_distrib fun _ _ => h_antidistrib _ _
|
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
section Mul
-- Porting note (#11215): TODO: generalize to `WithTop`
@[mono, gcongr]
| Mathlib/Data/ENNReal/Operations.lean | 33 | 41 | theorem mul_lt_mul (ac : a < c) (bd : b < d) : a * b < c * d := by |
rcases lt_iff_exists_nnreal_btwn.1 ac with ⟨a', aa', a'c⟩
lift a to ℝ≥0 using ne_top_of_lt aa'
rcases lt_iff_exists_nnreal_btwn.1 bd with ⟨b', bb', b'd⟩
lift b to ℝ≥0 using ne_top_of_lt bb'
norm_cast at *
calc
↑(a * b) < ↑(a' * b') := coe_lt_coe.2 (mul_lt_mul₀ aa' bb')
_ ≤ c * d := mul_le_mul' a'c.le b'd.le
|
import Mathlib.ModelTheory.Syntax
import Mathlib.ModelTheory.Semantics
import Mathlib.Algebra.Ring.Equiv
variable {α : Type*}
namespace FirstOrder
open FirstOrder
inductive ringFunc : ℕ → Type
| add : ringFunc 2
| mul : ringFunc 2
| neg : ringFunc 1
| zero : ringFunc 0
| one : ringFunc 0
deriving DecidableEq
def Language.ring : Language :=
{ Functions := ringFunc
Relations := fun _ => Empty }
namespace Ring
open ringFunc Language
instance (n : ℕ) : DecidableEq (Language.ring.Functions n) := by
dsimp [Language.ring]; infer_instance
instance (n : ℕ) : DecidableEq (Language.ring.Relations n) := by
dsimp [Language.ring]; infer_instance
abbrev addFunc : Language.ring.Functions 2 := add
abbrev mulFunc : Language.ring.Functions 2 := mul
abbrev negFunc : Language.ring.Functions 1 := neg
abbrev zeroFunc : Language.ring.Functions 0 := zero
abbrev oneFunc : Language.ring.Functions 0 := one
instance (α : Type*) : Zero (Language.ring.Term α) :=
{ zero := Constants.term zeroFunc }
theorem zero_def (α : Type*) : (0 : Language.ring.Term α) = Constants.term zeroFunc := rfl
instance (α : Type*) : One (Language.ring.Term α) :=
{ one := Constants.term oneFunc }
theorem one_def (α : Type*) : (1 : Language.ring.Term α) = Constants.term oneFunc := rfl
instance (α : Type*) : Add (Language.ring.Term α) :=
{ add := addFunc.apply₂ }
theorem add_def (α : Type*) (t₁ t₂ : Language.ring.Term α) :
t₁ + t₂ = addFunc.apply₂ t₁ t₂ := rfl
instance (α : Type*) : Mul (Language.ring.Term α) :=
{ mul := mulFunc.apply₂ }
theorem mul_def (α : Type*) (t₁ t₂ : Language.ring.Term α) :
t₁ * t₂ = mulFunc.apply₂ t₁ t₂ := rfl
instance (α : Type*) : Neg (Language.ring.Term α) :=
{ neg := negFunc.apply₁ }
theorem neg_def (α : Type*) (t : Language.ring.Term α) :
-t = negFunc.apply₁ t := rfl
instance : Fintype Language.ring.Symbols :=
⟨⟨Multiset.ofList
[Sum.inl ⟨2, .add⟩,
Sum.inl ⟨2, .mul⟩,
Sum.inl ⟨1, .neg⟩,
Sum.inl ⟨0, .zero⟩,
Sum.inl ⟨0, .one⟩], by
dsimp [Language.Symbols]; decide⟩, by
intro x
dsimp [Language.Symbols]
rcases x with ⟨_, f⟩ | ⟨_, f⟩
· cases f <;> decide
· cases f ⟩
@[simp]
theorem card_ring : card Language.ring = 5 := by
have : Fintype.card Language.ring.Symbols = 5 := rfl
simp [Language.card, this]
open Language ring Structure
class CompatibleRing (R : Type*) [Add R] [Mul R] [Neg R] [One R] [Zero R]
extends Language.ring.Structure R where
funMap_add : ∀ x, funMap addFunc x = x 0 + x 1
funMap_mul : ∀ x, funMap mulFunc x = x 0 * x 1
funMap_neg : ∀ x, funMap negFunc x = -x 0
funMap_zero : ∀ x, funMap (zeroFunc : Language.ring.Constants) x = 0
funMap_one : ∀ x, funMap (oneFunc : Language.ring.Constants) x = 1
open CompatibleRing
attribute [simp] funMap_add funMap_mul funMap_neg funMap_zero funMap_one
section
variable {R : Type*} [Add R] [Mul R] [Neg R] [One R] [Zero R] [CompatibleRing R]
@[simp]
theorem realize_add (x y : ring.Term α) (v : α → R) :
Term.realize v (x + y) = Term.realize v x + Term.realize v y := by
simp [add_def, funMap_add]
@[simp]
| Mathlib/ModelTheory/Algebra/Ring/Basic.lean | 185 | 187 | theorem realize_mul (x y : ring.Term α) (v : α → R) :
Term.realize v (x * y) = Term.realize v x * Term.realize v y := by |
simp [mul_def, funMap_mul]
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import measure_theory.integral.lebesgue from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
assert_not_exists NormedSpace
set_option autoImplicit true
noncomputable section
open Set hiding restrict restrict_apply
open Filter ENNReal
open Function (support)
open scoped Classical
open Topology NNReal ENNReal MeasureTheory
namespace MeasureTheory
local infixr:25 " →ₛ " => SimpleFunc
variable {α β γ δ : Type*}
section Lintegral
open SimpleFunc
variable {m : MeasurableSpace α} {μ ν : Measure α}
irreducible_def lintegral {_ : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) : ℝ≥0∞ :=
⨆ (g : α →ₛ ℝ≥0∞) (_ : ⇑g ≤ f), g.lintegral μ
#align measure_theory.lintegral MeasureTheory.lintegral
@[inherit_doc MeasureTheory.lintegral]
notation3 "∫⁻ "(...)", "r:60:(scoped f => f)" ∂"μ:70 => lintegral μ r
@[inherit_doc MeasureTheory.lintegral]
notation3 "∫⁻ "(...)", "r:60:(scoped f => lintegral volume f) => r
@[inherit_doc MeasureTheory.lintegral]
notation3"∫⁻ "(...)" in "s", "r:60:(scoped f => f)" ∂"μ:70 => lintegral (Measure.restrict μ s) r
@[inherit_doc MeasureTheory.lintegral]
notation3"∫⁻ "(...)" in "s", "r:60:(scoped f => lintegral (Measure.restrict volume s) f) => r
theorem SimpleFunc.lintegral_eq_lintegral {m : MeasurableSpace α} (f : α →ₛ ℝ≥0∞) (μ : Measure α) :
∫⁻ a, f a ∂μ = f.lintegral μ := by
rw [MeasureTheory.lintegral]
exact le_antisymm (iSup₂_le fun g hg => lintegral_mono hg <| le_rfl)
(le_iSup₂_of_le f le_rfl le_rfl)
#align measure_theory.simple_func.lintegral_eq_lintegral MeasureTheory.SimpleFunc.lintegral_eq_lintegral
@[mono]
theorem lintegral_mono' {m : MeasurableSpace α} ⦃μ ν : Measure α⦄ (hμν : μ ≤ ν) ⦃f g : α → ℝ≥0∞⦄
(hfg : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂ν := by
rw [lintegral, lintegral]
exact iSup_mono fun φ => iSup_mono' fun hφ => ⟨le_trans hφ hfg, lintegral_mono (le_refl φ) hμν⟩
#align measure_theory.lintegral_mono' MeasureTheory.lintegral_mono'
-- workaround for the known eta-reduction issue with `@[gcongr]`
@[gcongr] theorem lintegral_mono_fn' ⦃f g : α → ℝ≥0∞⦄ (hfg : ∀ x, f x ≤ g x) (h2 : μ ≤ ν) :
lintegral μ f ≤ lintegral ν g :=
lintegral_mono' h2 hfg
theorem lintegral_mono ⦃f g : α → ℝ≥0∞⦄ (hfg : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ :=
lintegral_mono' (le_refl μ) hfg
#align measure_theory.lintegral_mono MeasureTheory.lintegral_mono
-- workaround for the known eta-reduction issue with `@[gcongr]`
@[gcongr] theorem lintegral_mono_fn ⦃f g : α → ℝ≥0∞⦄ (hfg : ∀ x, f x ≤ g x) :
lintegral μ f ≤ lintegral μ g :=
lintegral_mono hfg
theorem lintegral_mono_nnreal {f g : α → ℝ≥0} (h : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ :=
lintegral_mono fun a => ENNReal.coe_le_coe.2 (h a)
#align measure_theory.lintegral_mono_nnreal MeasureTheory.lintegral_mono_nnreal
theorem iSup_lintegral_measurable_le_eq_lintegral (f : α → ℝ≥0∞) :
⨆ (g : α → ℝ≥0∞) (_ : Measurable g) (_ : g ≤ f), ∫⁻ a, g a ∂μ = ∫⁻ a, f a ∂μ := by
apply le_antisymm
· exact iSup_le fun i => iSup_le fun _ => iSup_le fun h'i => lintegral_mono h'i
· rw [lintegral]
refine iSup₂_le fun i hi => le_iSup₂_of_le i i.measurable <| le_iSup_of_le hi ?_
exact le_of_eq (i.lintegral_eq_lintegral _).symm
#align measure_theory.supr_lintegral_measurable_le_eq_lintegral MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral
theorem lintegral_mono_set {_ : MeasurableSpace α} ⦃μ : Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞}
(hst : s ⊆ t) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in t, f x ∂μ :=
lintegral_mono' (Measure.restrict_mono hst (le_refl μ)) (le_refl f)
#align measure_theory.lintegral_mono_set MeasureTheory.lintegral_mono_set
theorem lintegral_mono_set' {_ : MeasurableSpace α} ⦃μ : Measure α⦄ {s t : Set α} {f : α → ℝ≥0∞}
(hst : s ≤ᵐ[μ] t) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in t, f x ∂μ :=
lintegral_mono' (Measure.restrict_mono' hst (le_refl μ)) (le_refl f)
#align measure_theory.lintegral_mono_set' MeasureTheory.lintegral_mono_set'
theorem monotone_lintegral {_ : MeasurableSpace α} (μ : Measure α) : Monotone (lintegral μ) :=
lintegral_mono
#align measure_theory.monotone_lintegral MeasureTheory.monotone_lintegral
@[simp]
theorem lintegral_const (c : ℝ≥0∞) : ∫⁻ _, c ∂μ = c * μ univ := by
rw [← SimpleFunc.const_lintegral, ← SimpleFunc.lintegral_eq_lintegral, SimpleFunc.coe_const]
rfl
#align measure_theory.lintegral_const MeasureTheory.lintegral_const
theorem lintegral_zero : ∫⁻ _ : α, 0 ∂μ = 0 := by simp
#align measure_theory.lintegral_zero MeasureTheory.lintegral_zero
theorem lintegral_zero_fun : lintegral μ (0 : α → ℝ≥0∞) = 0 :=
lintegral_zero
#align measure_theory.lintegral_zero_fun MeasureTheory.lintegral_zero_fun
-- @[simp] -- Porting note (#10618): simp can prove this
theorem lintegral_one : ∫⁻ _, (1 : ℝ≥0∞) ∂μ = μ univ := by rw [lintegral_const, one_mul]
#align measure_theory.lintegral_one MeasureTheory.lintegral_one
theorem set_lintegral_const (s : Set α) (c : ℝ≥0∞) : ∫⁻ _ in s, c ∂μ = c * μ s := by
rw [lintegral_const, Measure.restrict_apply_univ]
#align measure_theory.set_lintegral_const MeasureTheory.set_lintegral_const
theorem set_lintegral_one (s) : ∫⁻ _ in s, 1 ∂μ = μ s := by rw [set_lintegral_const, one_mul]
#align measure_theory.set_lintegral_one MeasureTheory.set_lintegral_one
theorem set_lintegral_const_lt_top [IsFiniteMeasure μ] (s : Set α) {c : ℝ≥0∞} (hc : c ≠ ∞) :
∫⁻ _ in s, c ∂μ < ∞ := by
rw [lintegral_const]
exact ENNReal.mul_lt_top hc (measure_ne_top (μ.restrict s) univ)
#align measure_theory.set_lintegral_const_lt_top MeasureTheory.set_lintegral_const_lt_top
theorem lintegral_const_lt_top [IsFiniteMeasure μ] {c : ℝ≥0∞} (hc : c ≠ ∞) : ∫⁻ _, c ∂μ < ∞ := by
simpa only [Measure.restrict_univ] using set_lintegral_const_lt_top (univ : Set α) hc
#align measure_theory.lintegral_const_lt_top MeasureTheory.lintegral_const_lt_top
section
variable (μ)
theorem exists_measurable_le_lintegral_eq (f : α → ℝ≥0∞) :
∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ := by
rcases eq_or_ne (∫⁻ a, f a ∂μ) 0 with h₀ | h₀
· exact ⟨0, measurable_zero, zero_le f, h₀.trans lintegral_zero.symm⟩
rcases exists_seq_strictMono_tendsto' h₀.bot_lt with ⟨L, _, hLf, hL_tendsto⟩
have : ∀ n, ∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ L n < ∫⁻ a, g a ∂μ := by
intro n
simpa only [← iSup_lintegral_measurable_le_eq_lintegral f, lt_iSup_iff, exists_prop] using
(hLf n).2
choose g hgm hgf hLg using this
refine
⟨fun x => ⨆ n, g n x, measurable_iSup hgm, fun x => iSup_le fun n => hgf n x, le_antisymm ?_ ?_⟩
· refine le_of_tendsto' hL_tendsto fun n => (hLg n).le.trans <| lintegral_mono fun x => ?_
exact le_iSup (fun n => g n x) n
· exact lintegral_mono fun x => iSup_le fun n => hgf n x
#align measure_theory.exists_measurable_le_lintegral_eq MeasureTheory.exists_measurable_le_lintegral_eq
end
theorem lintegral_eq_nnreal {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : Measure α) :
∫⁻ a, f a ∂μ =
⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ x, ↑(φ x) ≤ f x), (φ.map ((↑) : ℝ≥0 → ℝ≥0∞)).lintegral μ := by
rw [lintegral]
refine
le_antisymm (iSup₂_le fun φ hφ => ?_) (iSup_mono' fun φ => ⟨φ.map ((↑) : ℝ≥0 → ℝ≥0∞), le_rfl⟩)
by_cases h : ∀ᵐ a ∂μ, φ a ≠ ∞
· let ψ := φ.map ENNReal.toNNReal
replace h : ψ.map ((↑) : ℝ≥0 → ℝ≥0∞) =ᵐ[μ] φ := h.mono fun a => ENNReal.coe_toNNReal
have : ∀ x, ↑(ψ x) ≤ f x := fun x => le_trans ENNReal.coe_toNNReal_le_self (hφ x)
exact
le_iSup_of_le (φ.map ENNReal.toNNReal) (le_iSup_of_le this (ge_of_eq <| lintegral_congr h))
· have h_meas : μ (φ ⁻¹' {∞}) ≠ 0 := mt measure_zero_iff_ae_nmem.1 h
refine le_trans le_top (ge_of_eq <| (iSup_eq_top _).2 fun b hb => ?_)
obtain ⟨n, hn⟩ : ∃ n : ℕ, b < n * μ (φ ⁻¹' {∞}) := exists_nat_mul_gt h_meas (ne_of_lt hb)
use (const α (n : ℝ≥0)).restrict (φ ⁻¹' {∞})
simp only [lt_iSup_iff, exists_prop, coe_restrict, φ.measurableSet_preimage, coe_const,
ENNReal.coe_indicator, map_coe_ennreal_restrict, SimpleFunc.map_const, ENNReal.coe_natCast,
restrict_const_lintegral]
refine ⟨indicator_le fun x hx => le_trans ?_ (hφ _), hn⟩
simp only [mem_preimage, mem_singleton_iff] at hx
simp only [hx, le_top]
#align measure_theory.lintegral_eq_nnreal MeasureTheory.lintegral_eq_nnreal
theorem exists_simpleFunc_forall_lintegral_sub_lt_of_pos {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞)
{ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ φ : α →ₛ ℝ≥0,
(∀ x, ↑(φ x) ≤ f x) ∧
∀ ψ : α →ₛ ℝ≥0, (∀ x, ↑(ψ x) ≤ f x) → (map (↑) (ψ - φ)).lintegral μ < ε := by
rw [lintegral_eq_nnreal] at h
have := ENNReal.lt_add_right h hε
erw [ENNReal.biSup_add] at this <;> [skip; exact ⟨0, fun x => zero_le _⟩]
simp_rw [lt_iSup_iff, iSup_lt_iff, iSup_le_iff] at this
rcases this with ⟨φ, hle : ∀ x, ↑(φ x) ≤ f x, b, hbφ, hb⟩
refine ⟨φ, hle, fun ψ hψ => ?_⟩
have : (map (↑) φ).lintegral μ ≠ ∞ := ne_top_of_le_ne_top h (by exact le_iSup₂ (α := ℝ≥0∞) φ hle)
rw [← ENNReal.add_lt_add_iff_left this, ← add_lintegral, ← SimpleFunc.map_add @ENNReal.coe_add]
refine (hb _ fun x => le_trans ?_ (max_le (hle x) (hψ x))).trans_lt hbφ
norm_cast
simp only [add_apply, sub_apply, add_tsub_eq_max]
rfl
#align measure_theory.exists_simple_func_forall_lintegral_sub_lt_of_pos MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos
theorem iSup_lintegral_le {ι : Sort*} (f : ι → α → ℝ≥0∞) :
⨆ i, ∫⁻ a, f i a ∂μ ≤ ∫⁻ a, ⨆ i, f i a ∂μ := by
simp only [← iSup_apply]
exact (monotone_lintegral μ).le_map_iSup
#align measure_theory.supr_lintegral_le MeasureTheory.iSup_lintegral_le
theorem iSup₂_lintegral_le {ι : Sort*} {ι' : ι → Sort*} (f : ∀ i, ι' i → α → ℝ≥0∞) :
⨆ (i) (j), ∫⁻ a, f i j a ∂μ ≤ ∫⁻ a, ⨆ (i) (j), f i j a ∂μ := by
convert (monotone_lintegral μ).le_map_iSup₂ f with a
simp only [iSup_apply]
#align measure_theory.supr₂_lintegral_le MeasureTheory.iSup₂_lintegral_le
theorem le_iInf_lintegral {ι : Sort*} (f : ι → α → ℝ≥0∞) :
∫⁻ a, ⨅ i, f i a ∂μ ≤ ⨅ i, ∫⁻ a, f i a ∂μ := by
simp only [← iInf_apply]
exact (monotone_lintegral μ).map_iInf_le
#align measure_theory.le_infi_lintegral MeasureTheory.le_iInf_lintegral
theorem le_iInf₂_lintegral {ι : Sort*} {ι' : ι → Sort*} (f : ∀ i, ι' i → α → ℝ≥0∞) :
∫⁻ a, ⨅ (i) (h : ι' i), f i h a ∂μ ≤ ⨅ (i) (h : ι' i), ∫⁻ a, f i h a ∂μ := by
convert (monotone_lintegral μ).map_iInf₂_le f with a
simp only [iInf_apply]
#align measure_theory.le_infi₂_lintegral MeasureTheory.le_iInf₂_lintegral
theorem lintegral_mono_ae {f g : α → ℝ≥0∞} (h : ∀ᵐ a ∂μ, f a ≤ g a) :
∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ := by
rcases exists_measurable_superset_of_null h with ⟨t, hts, ht, ht0⟩
have : ∀ᵐ x ∂μ, x ∉ t := measure_zero_iff_ae_nmem.1 ht0
rw [lintegral, lintegral]
refine iSup_le fun s => iSup_le fun hfs => le_iSup_of_le (s.restrict tᶜ) <| le_iSup_of_le ?_ ?_
· intro a
by_cases h : a ∈ t <;>
simp only [restrict_apply s ht.compl, mem_compl_iff, h, not_true, not_false_eq_true,
indicator_of_not_mem, zero_le, not_false_eq_true, indicator_of_mem]
exact le_trans (hfs a) (_root_.by_contradiction fun hnfg => h (hts hnfg))
· refine le_of_eq (SimpleFunc.lintegral_congr <| this.mono fun a hnt => ?_)
by_cases hat : a ∈ t <;> simp only [restrict_apply s ht.compl, mem_compl_iff, hat, not_true,
not_false_eq_true, indicator_of_not_mem, not_false_eq_true, indicator_of_mem]
exact (hnt hat).elim
#align measure_theory.lintegral_mono_ae MeasureTheory.lintegral_mono_ae
theorem set_lintegral_mono_ae {s : Set α} {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g)
(hfg : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ :=
lintegral_mono_ae <| (ae_restrict_iff <| measurableSet_le hf hg).2 hfg
#align measure_theory.set_lintegral_mono_ae MeasureTheory.set_lintegral_mono_ae
theorem set_lintegral_mono {s : Set α} {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g)
(hfg : ∀ x ∈ s, f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ :=
set_lintegral_mono_ae hf hg (ae_of_all _ hfg)
#align measure_theory.set_lintegral_mono MeasureTheory.set_lintegral_mono
theorem set_lintegral_mono_ae' {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s)
(hfg : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ :=
lintegral_mono_ae <| (ae_restrict_iff' hs).2 hfg
theorem set_lintegral_mono' {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s)
(hfg : ∀ x ∈ s, f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ :=
set_lintegral_mono_ae' hs (ae_of_all _ hfg)
theorem set_lintegral_le_lintegral (s : Set α) (f : α → ℝ≥0∞) :
∫⁻ x in s, f x ∂μ ≤ ∫⁻ x, f x ∂μ :=
lintegral_mono' Measure.restrict_le_self le_rfl
theorem lintegral_congr_ae {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ :=
le_antisymm (lintegral_mono_ae <| h.le) (lintegral_mono_ae <| h.symm.le)
#align measure_theory.lintegral_congr_ae MeasureTheory.lintegral_congr_ae
theorem lintegral_congr {f g : α → ℝ≥0∞} (h : ∀ a, f a = g a) : ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ := by
simp only [h]
#align measure_theory.lintegral_congr MeasureTheory.lintegral_congr
theorem set_lintegral_congr {f : α → ℝ≥0∞} {s t : Set α} (h : s =ᵐ[μ] t) :
∫⁻ x in s, f x ∂μ = ∫⁻ x in t, f x ∂μ := by rw [Measure.restrict_congr_set h]
#align measure_theory.set_lintegral_congr MeasureTheory.set_lintegral_congr
theorem set_lintegral_congr_fun {f g : α → ℝ≥0∞} {s : Set α} (hs : MeasurableSet s)
(hfg : ∀ᵐ x ∂μ, x ∈ s → f x = g x) : ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ := by
rw [lintegral_congr_ae]
rw [EventuallyEq]
rwa [ae_restrict_iff' hs]
#align measure_theory.set_lintegral_congr_fun MeasureTheory.set_lintegral_congr_fun
theorem lintegral_ofReal_le_lintegral_nnnorm (f : α → ℝ) :
∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ x, ‖f x‖₊ ∂μ := by
simp_rw [← ofReal_norm_eq_coe_nnnorm]
refine lintegral_mono fun x => ENNReal.ofReal_le_ofReal ?_
rw [Real.norm_eq_abs]
exact le_abs_self (f x)
#align measure_theory.lintegral_of_real_le_lintegral_nnnorm MeasureTheory.lintegral_ofReal_le_lintegral_nnnorm
theorem lintegral_nnnorm_eq_of_ae_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ᵐ[μ] f) :
∫⁻ x, ‖f x‖₊ ∂μ = ∫⁻ x, ENNReal.ofReal (f x) ∂μ := by
apply lintegral_congr_ae
filter_upwards [h_nonneg] with x hx
rw [Real.nnnorm_of_nonneg hx, ENNReal.ofReal_eq_coe_nnreal hx]
#align measure_theory.lintegral_nnnorm_eq_of_ae_nonneg MeasureTheory.lintegral_nnnorm_eq_of_ae_nonneg
theorem lintegral_nnnorm_eq_of_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ f) :
∫⁻ x, ‖f x‖₊ ∂μ = ∫⁻ x, ENNReal.ofReal (f x) ∂μ :=
lintegral_nnnorm_eq_of_ae_nonneg (Filter.eventually_of_forall h_nonneg)
#align measure_theory.lintegral_nnnorm_eq_of_nonneg MeasureTheory.lintegral_nnnorm_eq_of_nonneg
theorem lintegral_iSup {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) :
∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ := by
set c : ℝ≥0 → ℝ≥0∞ := (↑)
set F := fun a : α => ⨆ n, f n a
refine le_antisymm ?_ (iSup_lintegral_le _)
rw [lintegral_eq_nnreal]
refine iSup_le fun s => iSup_le fun hsf => ?_
refine ENNReal.le_of_forall_lt_one_mul_le fun a ha => ?_
rcases ENNReal.lt_iff_exists_coe.1 ha with ⟨r, rfl, _⟩
have ha : r < 1 := ENNReal.coe_lt_coe.1 ha
let rs := s.map fun a => r * a
have eq_rs : rs.map c = (const α r : α →ₛ ℝ≥0∞) * map c s := rfl
have eq : ∀ p, rs.map c ⁻¹' {p} = ⋃ n, rs.map c ⁻¹' {p} ∩ { a | p ≤ f n a } := by
intro p
rw [← inter_iUnion]; nth_rw 1 [← inter_univ (map c rs ⁻¹' {p})]
refine Set.ext fun x => and_congr_right fun hx => true_iff_iff.2 ?_
by_cases p_eq : p = 0
· simp [p_eq]
simp only [coe_map, mem_preimage, Function.comp_apply, mem_singleton_iff] at hx
subst hx
have : r * s x ≠ 0 := by rwa [Ne, ← ENNReal.coe_eq_zero]
have : s x ≠ 0 := right_ne_zero_of_mul this
have : (rs.map c) x < ⨆ n : ℕ, f n x := by
refine lt_of_lt_of_le (ENNReal.coe_lt_coe.2 ?_) (hsf x)
suffices r * s x < 1 * s x by simpa
exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this)
rcases lt_iSup_iff.1 this with ⟨i, hi⟩
exact mem_iUnion.2 ⟨i, le_of_lt hi⟩
have mono : ∀ r : ℝ≥0∞, Monotone fun n => rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a } := by
intro r i j h
refine inter_subset_inter_right _ ?_
simp_rw [subset_def, mem_setOf]
intro x hx
exact le_trans hx (h_mono h x)
have h_meas : ∀ n, MeasurableSet {a : α | map c rs a ≤ f n a} := fun n =>
measurableSet_le (SimpleFunc.measurable _) (hf n)
calc
(r : ℝ≥0∞) * (s.map c).lintegral μ = ∑ r ∈ (rs.map c).range, r * μ (rs.map c ⁻¹' {r}) := by
rw [← const_mul_lintegral, eq_rs, SimpleFunc.lintegral]
_ = ∑ r ∈ (rs.map c).range, r * μ (⋃ n, rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) := by
simp only [(eq _).symm]
_ = ∑ r ∈ (rs.map c).range, ⨆ n, r * μ (rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) :=
(Finset.sum_congr rfl fun x _ => by
rw [measure_iUnion_eq_iSup (mono x).directed_le, ENNReal.mul_iSup])
_ = ⨆ n, ∑ r ∈ (rs.map c).range, r * μ (rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) := by
refine ENNReal.finset_sum_iSup_nat fun p i j h ↦ ?_
gcongr _ * μ ?_
exact mono p h
_ ≤ ⨆ n : ℕ, ((rs.map c).restrict { a | (rs.map c) a ≤ f n a }).lintegral μ := by
gcongr with n
rw [restrict_lintegral _ (h_meas n)]
refine le_of_eq (Finset.sum_congr rfl fun r _ => ?_)
congr 2 with a
refine and_congr_right ?_
simp (config := { contextual := true })
_ ≤ ⨆ n, ∫⁻ a, f n a ∂μ := by
simp only [← SimpleFunc.lintegral_eq_lintegral]
gcongr with n a
simp only [map_apply] at h_meas
simp only [coe_map, restrict_apply _ (h_meas _), (· ∘ ·)]
exact indicator_apply_le id
#align measure_theory.lintegral_supr MeasureTheory.lintegral_iSup
theorem lintegral_iSup' {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, AEMeasurable (f n) μ)
(h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ := by
simp_rw [← iSup_apply]
let p : α → (ℕ → ℝ≥0∞) → Prop := fun _ f' => Monotone f'
have hp : ∀ᵐ x ∂μ, p x fun i => f i x := h_mono
have h_ae_seq_mono : Monotone (aeSeq hf p) := by
intro n m hnm x
by_cases hx : x ∈ aeSeqSet hf p
· exact aeSeq.prop_of_mem_aeSeqSet hf hx hnm
· simp only [aeSeq, hx, if_false, le_rfl]
rw [lintegral_congr_ae (aeSeq.iSup hf hp).symm]
simp_rw [iSup_apply]
rw [lintegral_iSup (aeSeq.measurable hf p) h_ae_seq_mono]
congr with n
exact lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae hf hp n)
#align measure_theory.lintegral_supr' MeasureTheory.lintegral_iSup'
theorem lintegral_tendsto_of_tendsto_of_monotone {f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞}
(hf : ∀ n, AEMeasurable (f n) μ) (h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x)
(h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 <| F x)) :
Tendsto (fun n => ∫⁻ x, f n x ∂μ) atTop (𝓝 <| ∫⁻ x, F x ∂μ) := by
have : Monotone fun n => ∫⁻ x, f n x ∂μ := fun i j hij =>
lintegral_mono_ae (h_mono.mono fun x hx => hx hij)
suffices key : ∫⁻ x, F x ∂μ = ⨆ n, ∫⁻ x, f n x ∂μ by
rw [key]
exact tendsto_atTop_iSup this
rw [← lintegral_iSup' hf h_mono]
refine lintegral_congr_ae ?_
filter_upwards [h_mono, h_tendsto] with _ hx_mono hx_tendsto using
tendsto_nhds_unique hx_tendsto (tendsto_atTop_iSup hx_mono)
#align measure_theory.lintegral_tendsto_of_tendsto_of_monotone MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone
theorem lintegral_eq_iSup_eapprox_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) :
∫⁻ a, f a ∂μ = ⨆ n, (eapprox f n).lintegral μ :=
calc
∫⁻ a, f a ∂μ = ∫⁻ a, ⨆ n, (eapprox f n : α → ℝ≥0∞) a ∂μ := by
congr; ext a; rw [iSup_eapprox_apply f hf]
_ = ⨆ n, ∫⁻ a, (eapprox f n : α → ℝ≥0∞) a ∂μ := by
apply lintegral_iSup
· measurability
· intro i j h
exact monotone_eapprox f h
_ = ⨆ n, (eapprox f n).lintegral μ := by
congr; ext n; rw [(eapprox f n).lintegral_eq_lintegral]
#align measure_theory.lintegral_eq_supr_eapprox_lintegral MeasureTheory.lintegral_eq_iSup_eapprox_lintegral
theorem exists_pos_set_lintegral_lt_of_measure_lt {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {ε : ℝ≥0∞}
(hε : ε ≠ 0) : ∃ δ > 0, ∀ s, μ s < δ → ∫⁻ x in s, f x ∂μ < ε := by
rcases exists_between (pos_iff_ne_zero.mpr hε) with ⟨ε₂, hε₂0, hε₂ε⟩
rcases exists_between hε₂0 with ⟨ε₁, hε₁0, hε₁₂⟩
rcases exists_simpleFunc_forall_lintegral_sub_lt_of_pos h hε₁0.ne' with ⟨φ, _, hφ⟩
rcases φ.exists_forall_le with ⟨C, hC⟩
use (ε₂ - ε₁) / C, ENNReal.div_pos_iff.2 ⟨(tsub_pos_iff_lt.2 hε₁₂).ne', ENNReal.coe_ne_top⟩
refine fun s hs => lt_of_le_of_lt ?_ hε₂ε
simp only [lintegral_eq_nnreal, iSup_le_iff]
intro ψ hψ
calc
(map (↑) ψ).lintegral (μ.restrict s) ≤
(map (↑) φ).lintegral (μ.restrict s) + (map (↑) (ψ - φ)).lintegral (μ.restrict s) := by
rw [← SimpleFunc.add_lintegral, ← SimpleFunc.map_add @ENNReal.coe_add]
refine SimpleFunc.lintegral_mono (fun x => ?_) le_rfl
simp only [add_tsub_eq_max, le_max_right, coe_map, Function.comp_apply, SimpleFunc.coe_add,
SimpleFunc.coe_sub, Pi.add_apply, Pi.sub_apply, ENNReal.coe_max (φ x) (ψ x)]
_ ≤ (map (↑) φ).lintegral (μ.restrict s) + ε₁ := by
gcongr
refine le_trans ?_ (hφ _ hψ).le
exact SimpleFunc.lintegral_mono le_rfl Measure.restrict_le_self
_ ≤ (SimpleFunc.const α (C : ℝ≥0∞)).lintegral (μ.restrict s) + ε₁ := by
gcongr
exact SimpleFunc.lintegral_mono (fun x ↦ ENNReal.coe_le_coe.2 (hC x)) le_rfl
_ = C * μ s + ε₁ := by
simp only [← SimpleFunc.lintegral_eq_lintegral, coe_const, lintegral_const,
Measure.restrict_apply, MeasurableSet.univ, univ_inter, Function.const]
_ ≤ C * ((ε₂ - ε₁) / C) + ε₁ := by gcongr
_ ≤ ε₂ - ε₁ + ε₁ := by gcongr; apply mul_div_le
_ = ε₂ := tsub_add_cancel_of_le hε₁₂.le
#align measure_theory.exists_pos_set_lintegral_lt_of_measure_lt MeasureTheory.exists_pos_set_lintegral_lt_of_measure_lt
theorem tendsto_set_lintegral_zero {ι} {f : α → ℝ≥0∞} (h : ∫⁻ x, f x ∂μ ≠ ∞) {l : Filter ι}
{s : ι → Set α} (hl : Tendsto (μ ∘ s) l (𝓝 0)) :
Tendsto (fun i => ∫⁻ x in s i, f x ∂μ) l (𝓝 0) := by
simp only [ENNReal.nhds_zero, tendsto_iInf, tendsto_principal, mem_Iio,
← pos_iff_ne_zero] at hl ⊢
intro ε ε0
rcases exists_pos_set_lintegral_lt_of_measure_lt h ε0.ne' with ⟨δ, δ0, hδ⟩
exact (hl δ δ0).mono fun i => hδ _
#align measure_theory.tendsto_set_lintegral_zero MeasureTheory.tendsto_set_lintegral_zero
theorem le_lintegral_add (f g : α → ℝ≥0∞) :
∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ ≤ ∫⁻ a, f a + g a ∂μ := by
simp only [lintegral]
refine ENNReal.biSup_add_biSup_le' (p := fun h : α →ₛ ℝ≥0∞ => h ≤ f)
(q := fun h : α →ₛ ℝ≥0∞ => h ≤ g) ⟨0, zero_le f⟩ ⟨0, zero_le g⟩ fun f' hf' g' hg' => ?_
exact le_iSup₂_of_le (f' + g') (add_le_add hf' hg') (add_lintegral _ _).ge
#align measure_theory.le_lintegral_add MeasureTheory.le_lintegral_add
-- Use stronger lemmas `lintegral_add_left`/`lintegral_add_right` instead
theorem lintegral_add_aux {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) :
∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ :=
calc
∫⁻ a, f a + g a ∂μ =
∫⁻ a, (⨆ n, (eapprox f n : α → ℝ≥0∞) a) + ⨆ n, (eapprox g n : α → ℝ≥0∞) a ∂μ := by
simp only [iSup_eapprox_apply, hf, hg]
_ = ∫⁻ a, ⨆ n, (eapprox f n + eapprox g n : α → ℝ≥0∞) a ∂μ := by
congr; funext a
rw [ENNReal.iSup_add_iSup_of_monotone]
· simp only [Pi.add_apply]
· intro i j h
exact monotone_eapprox _ h a
· intro i j h
exact monotone_eapprox _ h a
_ = ⨆ n, (eapprox f n).lintegral μ + (eapprox g n).lintegral μ := by
rw [lintegral_iSup]
· congr
funext n
rw [← SimpleFunc.add_lintegral, ← SimpleFunc.lintegral_eq_lintegral]
simp only [Pi.add_apply, SimpleFunc.coe_add]
· measurability
· intro i j h a
dsimp
gcongr <;> exact monotone_eapprox _ h _
_ = (⨆ n, (eapprox f n).lintegral μ) + ⨆ n, (eapprox g n).lintegral μ := by
refine (ENNReal.iSup_add_iSup_of_monotone ?_ ?_).symm <;>
· intro i j h
exact SimpleFunc.lintegral_mono (monotone_eapprox _ h) le_rfl
_ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := by
rw [lintegral_eq_iSup_eapprox_lintegral hf, lintegral_eq_iSup_eapprox_lintegral hg]
#align measure_theory.lintegral_add_aux MeasureTheory.lintegral_add_aux
@[simp]
theorem lintegral_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞) :
∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := by
refine le_antisymm ?_ (le_lintegral_add _ _)
rcases exists_measurable_le_lintegral_eq μ fun a => f a + g a with ⟨φ, hφm, hφ_le, hφ_eq⟩
calc
∫⁻ a, f a + g a ∂μ = ∫⁻ a, φ a ∂μ := hφ_eq
_ ≤ ∫⁻ a, f a + (φ a - f a) ∂μ := lintegral_mono fun a => le_add_tsub
_ = ∫⁻ a, f a ∂μ + ∫⁻ a, φ a - f a ∂μ := lintegral_add_aux hf (hφm.sub hf)
_ ≤ ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ :=
add_le_add_left (lintegral_mono fun a => tsub_le_iff_left.2 <| hφ_le a) _
#align measure_theory.lintegral_add_left MeasureTheory.lintegral_add_left
theorem lintegral_add_left' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (g : α → ℝ≥0∞) :
∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := by
rw [lintegral_congr_ae hf.ae_eq_mk, ← lintegral_add_left hf.measurable_mk,
lintegral_congr_ae (hf.ae_eq_mk.add (ae_eq_refl g))]
#align measure_theory.lintegral_add_left' MeasureTheory.lintegral_add_left'
theorem lintegral_add_right' (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : AEMeasurable g μ) :
∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ := by
simpa only [add_comm] using lintegral_add_left' hg f
#align measure_theory.lintegral_add_right' MeasureTheory.lintegral_add_right'
@[simp]
theorem lintegral_add_right (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g) :
∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ :=
lintegral_add_right' f hg.aemeasurable
#align measure_theory.lintegral_add_right MeasureTheory.lintegral_add_right
@[simp]
theorem lintegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) : ∫⁻ a, f a ∂c • μ = c * ∫⁻ a, f a ∂μ := by
simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.mul_iSup, smul_eq_mul]
#align measure_theory.lintegral_smul_measure MeasureTheory.lintegral_smul_measure
lemma set_lintegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) (s : Set α) :
∫⁻ a in s, f a ∂(c • μ) = c * ∫⁻ a in s, f a ∂μ := by
rw [Measure.restrict_smul, lintegral_smul_measure]
@[simp]
theorem lintegral_sum_measure {m : MeasurableSpace α} {ι} (f : α → ℝ≥0∞) (μ : ι → Measure α) :
∫⁻ a, f a ∂Measure.sum μ = ∑' i, ∫⁻ a, f a ∂μ i := by
simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_sum, ENNReal.tsum_eq_iSup_sum]
rw [iSup_comm]
congr; funext s
induction' s using Finset.induction_on with i s hi hs
· simp
simp only [Finset.sum_insert hi, ← hs]
refine (ENNReal.iSup_add_iSup ?_).symm
intro φ ψ
exact
⟨⟨φ ⊔ ψ, fun x => sup_le (φ.2 x) (ψ.2 x)⟩,
add_le_add (SimpleFunc.lintegral_mono le_sup_left le_rfl)
(Finset.sum_le_sum fun j _ => SimpleFunc.lintegral_mono le_sup_right le_rfl)⟩
#align measure_theory.lintegral_sum_measure MeasureTheory.lintegral_sum_measure
theorem hasSum_lintegral_measure {ι} {_ : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : ι → Measure α) :
HasSum (fun i => ∫⁻ a, f a ∂μ i) (∫⁻ a, f a ∂Measure.sum μ) :=
(lintegral_sum_measure f μ).symm ▸ ENNReal.summable.hasSum
#align measure_theory.has_sum_lintegral_measure MeasureTheory.hasSum_lintegral_measure
@[simp]
theorem lintegral_add_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) (μ ν : Measure α) :
∫⁻ a, f a ∂(μ + ν) = ∫⁻ a, f a ∂μ + ∫⁻ a, f a ∂ν := by
simpa [tsum_fintype] using lintegral_sum_measure f fun b => cond b μ ν
#align measure_theory.lintegral_add_measure MeasureTheory.lintegral_add_measure
@[simp]
theorem lintegral_finset_sum_measure {ι} {m : MeasurableSpace α} (s : Finset ι) (f : α → ℝ≥0∞)
(μ : ι → Measure α) : ∫⁻ a, f a ∂(∑ i ∈ s, μ i) = ∑ i ∈ s, ∫⁻ a, f a ∂μ i := by
rw [← Measure.sum_coe_finset, lintegral_sum_measure, ← Finset.tsum_subtype']
simp only [Finset.coe_sort_coe]
#align measure_theory.lintegral_finset_sum_measure MeasureTheory.lintegral_finset_sum_measure
@[simp]
theorem lintegral_zero_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) :
∫⁻ a, f a ∂(0 : Measure α) = 0 := by
simp [lintegral]
#align measure_theory.lintegral_zero_measure MeasureTheory.lintegral_zero_measure
@[simp]
theorem lintegral_of_isEmpty {α} [MeasurableSpace α] [IsEmpty α] (μ : Measure α) (f : α → ℝ≥0∞) :
∫⁻ x, f x ∂μ = 0 := by
have : Subsingleton (Measure α) := inferInstance
convert lintegral_zero_measure f
theorem set_lintegral_empty (f : α → ℝ≥0∞) : ∫⁻ x in ∅, f x ∂μ = 0 := by
rw [Measure.restrict_empty, lintegral_zero_measure]
#align measure_theory.set_lintegral_empty MeasureTheory.set_lintegral_empty
theorem set_lintegral_univ (f : α → ℝ≥0∞) : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by
rw [Measure.restrict_univ]
#align measure_theory.set_lintegral_univ MeasureTheory.set_lintegral_univ
theorem set_lintegral_measure_zero (s : Set α) (f : α → ℝ≥0∞) (hs' : μ s = 0) :
∫⁻ x in s, f x ∂μ = 0 := by
convert lintegral_zero_measure _
exact Measure.restrict_eq_zero.2 hs'
#align measure_theory.set_lintegral_measure_zero MeasureTheory.set_lintegral_measure_zero
theorem lintegral_finset_sum' (s : Finset β) {f : β → α → ℝ≥0∞}
(hf : ∀ b ∈ s, AEMeasurable (f b) μ) :
∫⁻ a, ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ a, f b a ∂μ := by
induction' s using Finset.induction_on with a s has ih
· simp
· simp only [Finset.sum_insert has]
rw [Finset.forall_mem_insert] at hf
rw [lintegral_add_left' hf.1, ih hf.2]
#align measure_theory.lintegral_finset_sum' MeasureTheory.lintegral_finset_sum'
theorem lintegral_finset_sum (s : Finset β) {f : β → α → ℝ≥0∞} (hf : ∀ b ∈ s, Measurable (f b)) :
∫⁻ a, ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ a, f b a ∂μ :=
lintegral_finset_sum' s fun b hb => (hf b hb).aemeasurable
#align measure_theory.lintegral_finset_sum MeasureTheory.lintegral_finset_sum
@[simp]
theorem lintegral_const_mul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ :=
calc
∫⁻ a, r * f a ∂μ = ∫⁻ a, ⨆ n, (const α r * eapprox f n) a ∂μ := by
congr
funext a
rw [← iSup_eapprox_apply f hf, ENNReal.mul_iSup]
simp
_ = ⨆ n, r * (eapprox f n).lintegral μ := by
rw [lintegral_iSup]
· congr
funext n
rw [← SimpleFunc.const_mul_lintegral, ← SimpleFunc.lintegral_eq_lintegral]
· intro n
exact SimpleFunc.measurable _
· intro i j h a
exact mul_le_mul_left' (monotone_eapprox _ h _) _
_ = r * ∫⁻ a, f a ∂μ := by rw [← ENNReal.mul_iSup, lintegral_eq_iSup_eapprox_lintegral hf]
#align measure_theory.lintegral_const_mul MeasureTheory.lintegral_const_mul
theorem lintegral_const_mul'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ := by
have A : ∫⁻ a, f a ∂μ = ∫⁻ a, hf.mk f a ∂μ := lintegral_congr_ae hf.ae_eq_mk
have B : ∫⁻ a, r * f a ∂μ = ∫⁻ a, r * hf.mk f a ∂μ :=
lintegral_congr_ae (EventuallyEq.fun_comp hf.ae_eq_mk _)
rw [A, B, lintegral_const_mul _ hf.measurable_mk]
#align measure_theory.lintegral_const_mul'' MeasureTheory.lintegral_const_mul''
theorem lintegral_const_mul_le (r : ℝ≥0∞) (f : α → ℝ≥0∞) :
r * ∫⁻ a, f a ∂μ ≤ ∫⁻ a, r * f a ∂μ := by
rw [lintegral, ENNReal.mul_iSup]
refine iSup_le fun s => ?_
rw [ENNReal.mul_iSup, iSup_le_iff]
intro hs
rw [← SimpleFunc.const_mul_lintegral, lintegral]
refine le_iSup_of_le (const α r * s) (le_iSup_of_le (fun x => ?_) le_rfl)
exact mul_le_mul_left' (hs x) _
#align measure_theory.lintegral_const_mul_le MeasureTheory.lintegral_const_mul_le
theorem lintegral_const_mul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ := by
by_cases h : r = 0
· simp [h]
apply le_antisymm _ (lintegral_const_mul_le r f)
have rinv : r * r⁻¹ = 1 := ENNReal.mul_inv_cancel h hr
have rinv' : r⁻¹ * r = 1 := by
rw [mul_comm]
exact rinv
have := lintegral_const_mul_le (μ := μ) r⁻¹ fun x => r * f x
simp? [(mul_assoc _ _ _).symm, rinv'] at this says
simp only [(mul_assoc _ _ _).symm, rinv', one_mul] at this
simpa [(mul_assoc _ _ _).symm, rinv] using mul_le_mul_left' this r
#align measure_theory.lintegral_const_mul' MeasureTheory.lintegral_const_mul'
theorem lintegral_mul_const (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r := by simp_rw [mul_comm, lintegral_const_mul r hf]
#align measure_theory.lintegral_mul_const MeasureTheory.lintegral_mul_const
theorem lintegral_mul_const'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r := by simp_rw [mul_comm, lintegral_const_mul'' r hf]
#align measure_theory.lintegral_mul_const'' MeasureTheory.lintegral_mul_const''
theorem lintegral_mul_const_le (r : ℝ≥0∞) (f : α → ℝ≥0∞) :
(∫⁻ a, f a ∂μ) * r ≤ ∫⁻ a, f a * r ∂μ := by
simp_rw [mul_comm, lintegral_const_mul_le r f]
#align measure_theory.lintegral_mul_const_le MeasureTheory.lintegral_mul_const_le
theorem lintegral_mul_const' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
∫⁻ a, f a * r ∂μ = (∫⁻ a, f a ∂μ) * r := by simp_rw [mul_comm, lintegral_const_mul' r f hr]
#align measure_theory.lintegral_mul_const' MeasureTheory.lintegral_mul_const'
theorem lintegral_lintegral_mul {β} [MeasurableSpace β] {ν : Measure β} {f : α → ℝ≥0∞}
{g : β → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g ν) :
∫⁻ x, ∫⁻ y, f x * g y ∂ν ∂μ = (∫⁻ x, f x ∂μ) * ∫⁻ y, g y ∂ν := by
simp [lintegral_const_mul'' _ hg, lintegral_mul_const'' _ hf]
#align measure_theory.lintegral_lintegral_mul MeasureTheory.lintegral_lintegral_mul
-- TODO: Need a better way of rewriting inside of an integral
theorem lintegral_rw₁ {f f' : α → β} (h : f =ᵐ[μ] f') (g : β → ℝ≥0∞) :
∫⁻ a, g (f a) ∂μ = ∫⁻ a, g (f' a) ∂μ :=
lintegral_congr_ae <| h.mono fun a h => by dsimp only; rw [h]
#align measure_theory.lintegral_rw₁ MeasureTheory.lintegral_rw₁
-- TODO: Need a better way of rewriting inside of an integral
theorem lintegral_rw₂ {f₁ f₁' : α → β} {f₂ f₂' : α → γ} (h₁ : f₁ =ᵐ[μ] f₁') (h₂ : f₂ =ᵐ[μ] f₂')
(g : β → γ → ℝ≥0∞) : ∫⁻ a, g (f₁ a) (f₂ a) ∂μ = ∫⁻ a, g (f₁' a) (f₂' a) ∂μ :=
lintegral_congr_ae <| h₁.mp <| h₂.mono fun _ h₂ h₁ => by dsimp only; rw [h₁, h₂]
#align measure_theory.lintegral_rw₂ MeasureTheory.lintegral_rw₂
theorem lintegral_indicator_le (f : α → ℝ≥0∞) (s : Set α) :
∫⁻ a, s.indicator f a ∂μ ≤ ∫⁻ a in s, f a ∂μ := by
simp only [lintegral]
apply iSup_le (fun g ↦ (iSup_le (fun hg ↦ ?_)))
have : g ≤ f := hg.trans (indicator_le_self s f)
refine le_iSup_of_le g (le_iSup_of_le this (le_of_eq ?_))
rw [lintegral_restrict, SimpleFunc.lintegral]
congr with t
by_cases H : t = 0
· simp [H]
congr with x
simp only [mem_preimage, mem_singleton_iff, mem_inter_iff, iff_self_and]
rintro rfl
contrapose! H
simpa [H] using hg x
@[simp]
theorem lintegral_indicator (f : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) :
∫⁻ a, s.indicator f a ∂μ = ∫⁻ a in s, f a ∂μ := by
apply le_antisymm (lintegral_indicator_le f s)
simp only [lintegral, ← restrict_lintegral_eq_lintegral_restrict _ hs, iSup_subtype']
refine iSup_mono' (Subtype.forall.2 fun φ hφ => ?_)
refine ⟨⟨φ.restrict s, fun x => ?_⟩, le_rfl⟩
simp [hφ x, hs, indicator_le_indicator]
#align measure_theory.lintegral_indicator MeasureTheory.lintegral_indicator
theorem lintegral_indicator₀ (f : α → ℝ≥0∞) {s : Set α} (hs : NullMeasurableSet s μ) :
∫⁻ a, s.indicator f a ∂μ = ∫⁻ a in s, f a ∂μ := by
rw [← lintegral_congr_ae (indicator_ae_eq_of_ae_eq_set hs.toMeasurable_ae_eq),
lintegral_indicator _ (measurableSet_toMeasurable _ _),
Measure.restrict_congr_set hs.toMeasurable_ae_eq]
#align measure_theory.lintegral_indicator₀ MeasureTheory.lintegral_indicator₀
theorem lintegral_indicator_const_le (s : Set α) (c : ℝ≥0∞) :
∫⁻ a, s.indicator (fun _ => c) a ∂μ ≤ c * μ s :=
(lintegral_indicator_le _ _).trans (set_lintegral_const s c).le
theorem lintegral_indicator_const₀ {s : Set α} (hs : NullMeasurableSet s μ) (c : ℝ≥0∞) :
∫⁻ a, s.indicator (fun _ => c) a ∂μ = c * μ s := by
rw [lintegral_indicator₀ _ hs, set_lintegral_const]
theorem lintegral_indicator_const {s : Set α} (hs : MeasurableSet s) (c : ℝ≥0∞) :
∫⁻ a, s.indicator (fun _ => c) a ∂μ = c * μ s :=
lintegral_indicator_const₀ hs.nullMeasurableSet c
#align measure_theory.lintegral_indicator_const MeasureTheory.lintegral_indicator_const
theorem set_lintegral_eq_const {f : α → ℝ≥0∞} (hf : Measurable f) (r : ℝ≥0∞) :
∫⁻ x in { x | f x = r }, f x ∂μ = r * μ { x | f x = r } := by
have : ∀ᵐ x ∂μ, x ∈ { x | f x = r } → f x = r := ae_of_all μ fun _ hx => hx
rw [set_lintegral_congr_fun _ this]
· rw [lintegral_const, Measure.restrict_apply MeasurableSet.univ, Set.univ_inter]
· exact hf (measurableSet_singleton r)
#align measure_theory.set_lintegral_eq_const MeasureTheory.set_lintegral_eq_const
theorem lintegral_indicator_one_le (s : Set α) : ∫⁻ a, s.indicator 1 a ∂μ ≤ μ s :=
(lintegral_indicator_const_le _ _).trans <| (one_mul _).le
@[simp]
theorem lintegral_indicator_one₀ (hs : NullMeasurableSet s μ) : ∫⁻ a, s.indicator 1 a ∂μ = μ s :=
(lintegral_indicator_const₀ hs _).trans <| one_mul _
@[simp]
theorem lintegral_indicator_one (hs : MeasurableSet s) : ∫⁻ a, s.indicator 1 a ∂μ = μ s :=
(lintegral_indicator_const hs _).trans <| one_mul _
#align measure_theory.lintegral_indicator_one MeasureTheory.lintegral_indicator_one
theorem lintegral_add_mul_meas_add_le_le_lintegral {f g : α → ℝ≥0∞} (hle : f ≤ᵐ[μ] g)
(hg : AEMeasurable g μ) (ε : ℝ≥0∞) :
∫⁻ a, f a ∂μ + ε * μ { x | f x + ε ≤ g x } ≤ ∫⁻ a, g a ∂μ := by
rcases exists_measurable_le_lintegral_eq μ f with ⟨φ, hφm, hφ_le, hφ_eq⟩
calc
∫⁻ x, f x ∂μ + ε * μ { x | f x + ε ≤ g x } = ∫⁻ x, φ x ∂μ + ε * μ { x | f x + ε ≤ g x } := by
rw [hφ_eq]
_ ≤ ∫⁻ x, φ x ∂μ + ε * μ { x | φ x + ε ≤ g x } := by
gcongr
exact fun x => (add_le_add_right (hφ_le _) _).trans
_ = ∫⁻ x, φ x + indicator { x | φ x + ε ≤ g x } (fun _ => ε) x ∂μ := by
rw [lintegral_add_left hφm, lintegral_indicator₀, set_lintegral_const]
exact measurableSet_le (hφm.nullMeasurable.measurable'.add_const _) hg.nullMeasurable
_ ≤ ∫⁻ x, g x ∂μ := lintegral_mono_ae (hle.mono fun x hx₁ => ?_)
simp only [indicator_apply]; split_ifs with hx₂
exacts [hx₂, (add_zero _).trans_le <| (hφ_le x).trans hx₁]
#align measure_theory.lintegral_add_mul_meas_add_le_le_lintegral MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegral
theorem mul_meas_ge_le_lintegral₀ {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (ε : ℝ≥0∞) :
ε * μ { x | ε ≤ f x } ≤ ∫⁻ a, f a ∂μ := by
simpa only [lintegral_zero, zero_add] using
lintegral_add_mul_meas_add_le_le_lintegral (ae_of_all _ fun x => zero_le (f x)) hf ε
#align measure_theory.mul_meas_ge_le_lintegral₀ MeasureTheory.mul_meas_ge_le_lintegral₀
theorem mul_meas_ge_le_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) (ε : ℝ≥0∞) :
ε * μ { x | ε ≤ f x } ≤ ∫⁻ a, f a ∂μ :=
mul_meas_ge_le_lintegral₀ hf.aemeasurable ε
#align measure_theory.mul_meas_ge_le_lintegral MeasureTheory.mul_meas_ge_le_lintegral
lemma meas_le_lintegral₀ {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
{s : Set α} (hs : ∀ x ∈ s, 1 ≤ f x) : μ s ≤ ∫⁻ a, f a ∂μ := by
apply le_trans _ (mul_meas_ge_le_lintegral₀ hf 1)
rw [one_mul]
exact measure_mono hs
lemma lintegral_le_meas {s : Set α} {f : α → ℝ≥0∞} (hf : ∀ a, f a ≤ 1) (h'f : ∀ a ∈ sᶜ, f a = 0) :
∫⁻ a, f a ∂μ ≤ μ s := by
apply (lintegral_mono (fun x ↦ ?_)).trans (lintegral_indicator_one_le s)
by_cases hx : x ∈ s
· simpa [hx] using hf x
· simpa [hx] using h'f x hx
theorem lintegral_eq_top_of_measure_eq_top_ne_zero {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
(hμf : μ {x | f x = ∞} ≠ 0) : ∫⁻ x, f x ∂μ = ∞ :=
eq_top_iff.mpr <|
calc
∞ = ∞ * μ { x | ∞ ≤ f x } := by simp [mul_eq_top, hμf]
_ ≤ ∫⁻ x, f x ∂μ := mul_meas_ge_le_lintegral₀ hf ∞
#align measure_theory.lintegral_eq_top_of_measure_eq_top_ne_zero MeasureTheory.lintegral_eq_top_of_measure_eq_top_ne_zero
theorem setLintegral_eq_top_of_measure_eq_top_ne_zero (hf : AEMeasurable f (μ.restrict s))
(hμf : μ ({x ∈ s | f x = ∞}) ≠ 0) : ∫⁻ x in s, f x ∂μ = ∞ :=
lintegral_eq_top_of_measure_eq_top_ne_zero hf <|
mt (eq_bot_mono <| by rw [← setOf_inter_eq_sep]; exact Measure.le_restrict_apply _ _) hμf
#align measure_theory.set_lintegral_eq_top_of_measure_eq_top_ne_zero MeasureTheory.setLintegral_eq_top_of_measure_eq_top_ne_zero
theorem measure_eq_top_of_lintegral_ne_top (hf : AEMeasurable f μ) (hμf : ∫⁻ x, f x ∂μ ≠ ∞) :
μ {x | f x = ∞} = 0 :=
of_not_not fun h => hμf <| lintegral_eq_top_of_measure_eq_top_ne_zero hf h
#align measure_theory.measure_eq_top_of_lintegral_ne_top MeasureTheory.measure_eq_top_of_lintegral_ne_top
theorem measure_eq_top_of_setLintegral_ne_top (hf : AEMeasurable f (μ.restrict s))
(hμf : ∫⁻ x in s, f x ∂μ ≠ ∞) : μ ({x ∈ s | f x = ∞}) = 0 :=
of_not_not fun h => hμf <| setLintegral_eq_top_of_measure_eq_top_ne_zero hf h
#align measure_theory.measure_eq_top_of_set_lintegral_ne_top MeasureTheory.measure_eq_top_of_setLintegral_ne_top
theorem meas_ge_le_lintegral_div {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0)
(hε' : ε ≠ ∞) : μ { x | ε ≤ f x } ≤ (∫⁻ a, f a ∂μ) / ε :=
(ENNReal.le_div_iff_mul_le (Or.inl hε) (Or.inl hε')).2 <| by
rw [mul_comm]
exact mul_meas_ge_le_lintegral₀ hf ε
#align measure_theory.meas_ge_le_lintegral_div MeasureTheory.meas_ge_le_lintegral_div
theorem ae_eq_of_ae_le_of_lintegral_le {f g : α → ℝ≥0∞} (hfg : f ≤ᵐ[μ] g) (hf : ∫⁻ x, f x ∂μ ≠ ∞)
(hg : AEMeasurable g μ) (hgf : ∫⁻ x, g x ∂μ ≤ ∫⁻ x, f x ∂μ) : f =ᵐ[μ] g := by
have : ∀ n : ℕ, ∀ᵐ x ∂μ, g x < f x + (n : ℝ≥0∞)⁻¹ := by
intro n
simp only [ae_iff, not_lt]
have : ∫⁻ x, f x ∂μ + (↑n)⁻¹ * μ { x : α | f x + (n : ℝ≥0∞)⁻¹ ≤ g x } ≤ ∫⁻ x, f x ∂μ :=
(lintegral_add_mul_meas_add_le_le_lintegral hfg hg n⁻¹).trans hgf
rw [(ENNReal.cancel_of_ne hf).add_le_iff_nonpos_right, nonpos_iff_eq_zero, mul_eq_zero] at this
exact this.resolve_left (ENNReal.inv_ne_zero.2 (ENNReal.natCast_ne_top _))
refine hfg.mp ((ae_all_iff.2 this).mono fun x hlt hle => hle.antisymm ?_)
suffices Tendsto (fun n : ℕ => f x + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (f x)) from
ge_of_tendsto' this fun i => (hlt i).le
simpa only [inv_top, add_zero] using
tendsto_const_nhds.add (ENNReal.tendsto_inv_iff.2 ENNReal.tendsto_nat_nhds_top)
#align measure_theory.ae_eq_of_ae_le_of_lintegral_le MeasureTheory.ae_eq_of_ae_le_of_lintegral_le
@[simp]
theorem lintegral_eq_zero_iff' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
∫⁻ a, f a ∂μ = 0 ↔ f =ᵐ[μ] 0 :=
have : ∫⁻ _ : α, 0 ∂μ ≠ ∞ := by simp [lintegral_zero, zero_ne_top]
⟨fun h =>
(ae_eq_of_ae_le_of_lintegral_le (ae_of_all _ <| zero_le f) this hf
(h.trans lintegral_zero.symm).le).symm,
fun h => (lintegral_congr_ae h).trans lintegral_zero⟩
#align measure_theory.lintegral_eq_zero_iff' MeasureTheory.lintegral_eq_zero_iff'
@[simp]
theorem lintegral_eq_zero_iff {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂μ = 0 ↔ f =ᵐ[μ] 0 :=
lintegral_eq_zero_iff' hf.aemeasurable
#align measure_theory.lintegral_eq_zero_iff MeasureTheory.lintegral_eq_zero_iff
theorem lintegral_pos_iff_support {f : α → ℝ≥0∞} (hf : Measurable f) :
(0 < ∫⁻ a, f a ∂μ) ↔ 0 < μ (Function.support f) := by
simp [pos_iff_ne_zero, hf, Filter.EventuallyEq, ae_iff, Function.support]
#align measure_theory.lintegral_pos_iff_support MeasureTheory.lintegral_pos_iff_support
theorem setLintegral_pos_iff {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α} :
0 < ∫⁻ a in s, f a ∂μ ↔ 0 < μ (Function.support f ∩ s) := by
rw [lintegral_pos_iff_support hf, Measure.restrict_apply (measurableSet_support hf)]
theorem lintegral_iSup_ae {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n))
(h_mono : ∀ n, ∀ᵐ a ∂μ, f n a ≤ f n.succ a) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ := by
let ⟨s, hs⟩ := exists_measurable_superset_of_null (ae_iff.1 (ae_all_iff.2 h_mono))
let g n a := if a ∈ s then 0 else f n a
have g_eq_f : ∀ᵐ a ∂μ, ∀ n, g n a = f n a :=
(measure_zero_iff_ae_nmem.1 hs.2.2).mono fun a ha n => if_neg ha
calc
∫⁻ a, ⨆ n, f n a ∂μ = ∫⁻ a, ⨆ n, g n a ∂μ :=
lintegral_congr_ae <| g_eq_f.mono fun a ha => by simp only [ha]
_ = ⨆ n, ∫⁻ a, g n a ∂μ :=
(lintegral_iSup (fun n => measurable_const.piecewise hs.2.1 (hf n))
(monotone_nat_of_le_succ fun n a => ?_))
_ = ⨆ n, ∫⁻ a, f n a ∂μ := by simp only [lintegral_congr_ae (g_eq_f.mono fun _a ha => ha _)]
simp only [g]
split_ifs with h
· rfl
· have := Set.not_mem_subset hs.1 h
simp only [not_forall, not_le, mem_setOf_eq, not_exists, not_lt] at this
exact this n
#align measure_theory.lintegral_supr_ae MeasureTheory.lintegral_iSup_ae
theorem lintegral_sub' {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hg_fin : ∫⁻ a, g a ∂μ ≠ ∞)
(h_le : g ≤ᵐ[μ] f) : ∫⁻ a, f a - g a ∂μ = ∫⁻ a, f a ∂μ - ∫⁻ a, g a ∂μ := by
refine ENNReal.eq_sub_of_add_eq hg_fin ?_
rw [← lintegral_add_right' _ hg]
exact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx)
#align measure_theory.lintegral_sub' MeasureTheory.lintegral_sub'
theorem lintegral_sub {f g : α → ℝ≥0∞} (hg : Measurable g) (hg_fin : ∫⁻ a, g a ∂μ ≠ ∞)
(h_le : g ≤ᵐ[μ] f) : ∫⁻ a, f a - g a ∂μ = ∫⁻ a, f a ∂μ - ∫⁻ a, g a ∂μ :=
lintegral_sub' hg.aemeasurable hg_fin h_le
#align measure_theory.lintegral_sub MeasureTheory.lintegral_sub
theorem lintegral_sub_le' (f g : α → ℝ≥0∞) (hf : AEMeasurable f μ) :
∫⁻ x, g x ∂μ - ∫⁻ x, f x ∂μ ≤ ∫⁻ x, g x - f x ∂μ := by
rw [tsub_le_iff_right]
by_cases hfi : ∫⁻ x, f x ∂μ = ∞
· rw [hfi, add_top]
exact le_top
· rw [← lintegral_add_right' _ hf]
gcongr
exact le_tsub_add
#align measure_theory.lintegral_sub_le' MeasureTheory.lintegral_sub_le'
theorem lintegral_sub_le (f g : α → ℝ≥0∞) (hf : Measurable f) :
∫⁻ x, g x ∂μ - ∫⁻ x, f x ∂μ ≤ ∫⁻ x, g x - f x ∂μ :=
lintegral_sub_le' f g hf.aemeasurable
#align measure_theory.lintegral_sub_le MeasureTheory.lintegral_sub_le
theorem lintegral_strict_mono_of_ae_le_of_frequently_ae_lt {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ)
(hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h_le : f ≤ᵐ[μ] g) (h : ∃ᵐ x ∂μ, f x ≠ g x) :
∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ := by
contrapose! h
simp only [not_frequently, Ne, Classical.not_not]
exact ae_eq_of_ae_le_of_lintegral_le h_le hfi hg h
#align measure_theory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt MeasureTheory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt
theorem lintegral_strict_mono_of_ae_le_of_ae_lt_on {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ)
(hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h_le : f ≤ᵐ[μ] g) {s : Set α} (hμs : μ s ≠ 0)
(h : ∀ᵐ x ∂μ, x ∈ s → f x < g x) : ∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ :=
lintegral_strict_mono_of_ae_le_of_frequently_ae_lt hg hfi h_le <|
((frequently_ae_mem_iff.2 hμs).and_eventually h).mono fun _x hx => (hx.2 hx.1).ne
#align measure_theory.lintegral_strict_mono_of_ae_le_of_ae_lt_on MeasureTheory.lintegral_strict_mono_of_ae_le_of_ae_lt_on
theorem lintegral_strict_mono {f g : α → ℝ≥0∞} (hμ : μ ≠ 0) (hg : AEMeasurable g μ)
(hfi : ∫⁻ x, f x ∂μ ≠ ∞) (h : ∀ᵐ x ∂μ, f x < g x) : ∫⁻ x, f x ∂μ < ∫⁻ x, g x ∂μ := by
rw [Ne, ← Measure.measure_univ_eq_zero] at hμ
refine lintegral_strict_mono_of_ae_le_of_ae_lt_on hg hfi (ae_le_of_ae_lt h) hμ ?_
simpa using h
#align measure_theory.lintegral_strict_mono MeasureTheory.lintegral_strict_mono
theorem set_lintegral_strict_mono {f g : α → ℝ≥0∞} {s : Set α} (hsm : MeasurableSet s)
(hs : μ s ≠ 0) (hg : Measurable g) (hfi : ∫⁻ x in s, f x ∂μ ≠ ∞)
(h : ∀ᵐ x ∂μ, x ∈ s → f x < g x) : ∫⁻ x in s, f x ∂μ < ∫⁻ x in s, g x ∂μ :=
lintegral_strict_mono (by simp [hs]) hg.aemeasurable hfi ((ae_restrict_iff' hsm).mpr h)
#align measure_theory.set_lintegral_strict_mono MeasureTheory.set_lintegral_strict_mono
theorem lintegral_iInf_ae {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n))
(h_mono : ∀ n : ℕ, f n.succ ≤ᵐ[μ] f n) (h_fin : ∫⁻ a, f 0 a ∂μ ≠ ∞) :
∫⁻ a, ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ a, f n a ∂μ :=
have fn_le_f0 : ∫⁻ a, ⨅ n, f n a ∂μ ≤ ∫⁻ a, f 0 a ∂μ :=
lintegral_mono fun a => iInf_le_of_le 0 le_rfl
have fn_le_f0' : ⨅ n, ∫⁻ a, f n a ∂μ ≤ ∫⁻ a, f 0 a ∂μ := iInf_le_of_le 0 le_rfl
(ENNReal.sub_right_inj h_fin fn_le_f0 fn_le_f0').1 <|
show ∫⁻ a, f 0 a ∂μ - ∫⁻ a, ⨅ n, f n a ∂μ = ∫⁻ a, f 0 a ∂μ - ⨅ n, ∫⁻ a, f n a ∂μ from
calc
∫⁻ a, f 0 a ∂μ - ∫⁻ a, ⨅ n, f n a ∂μ = ∫⁻ a, f 0 a - ⨅ n, f n a ∂μ :=
(lintegral_sub (measurable_iInf h_meas)
(ne_top_of_le_ne_top h_fin <| lintegral_mono fun a => iInf_le _ _)
(ae_of_all _ fun a => iInf_le _ _)).symm
_ = ∫⁻ a, ⨆ n, f 0 a - f n a ∂μ := congr rfl (funext fun a => ENNReal.sub_iInf)
_ = ⨆ n, ∫⁻ a, f 0 a - f n a ∂μ :=
(lintegral_iSup_ae (fun n => (h_meas 0).sub (h_meas n)) fun n =>
(h_mono n).mono fun a ha => tsub_le_tsub le_rfl ha)
_ = ⨆ n, ∫⁻ a, f 0 a ∂μ - ∫⁻ a, f n a ∂μ :=
(have h_mono : ∀ᵐ a ∂μ, ∀ n : ℕ, f n.succ a ≤ f n a := ae_all_iff.2 h_mono
have h_mono : ∀ n, ∀ᵐ a ∂μ, f n a ≤ f 0 a := fun n =>
h_mono.mono fun a h => by
induction' n with n ih
· exact le_rfl
· exact le_trans (h n) ih
congr_arg iSup <|
funext fun n =>
lintegral_sub (h_meas _) (ne_top_of_le_ne_top h_fin <| lintegral_mono_ae <| h_mono n)
(h_mono n))
_ = ∫⁻ a, f 0 a ∂μ - ⨅ n, ∫⁻ a, f n a ∂μ := ENNReal.sub_iInf.symm
#align measure_theory.lintegral_infi_ae MeasureTheory.lintegral_iInf_ae
theorem lintegral_iInf {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, Measurable (f n)) (h_anti : Antitone f)
(h_fin : ∫⁻ a, f 0 a ∂μ ≠ ∞) : ∫⁻ a, ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ a, f n a ∂μ :=
lintegral_iInf_ae h_meas (fun n => ae_of_all _ <| h_anti n.le_succ) h_fin
#align measure_theory.lintegral_infi MeasureTheory.lintegral_iInf
theorem lintegral_iInf' {f : ℕ → α → ℝ≥0∞} (h_meas : ∀ n, AEMeasurable (f n) μ)
(h_anti : ∀ᵐ a ∂μ, Antitone (fun i ↦ f i a)) (h_fin : ∫⁻ a, f 0 a ∂μ ≠ ∞) :
∫⁻ a, ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ a, f n a ∂μ := by
simp_rw [← iInf_apply]
let p : α → (ℕ → ℝ≥0∞) → Prop := fun _ f' => Antitone f'
have hp : ∀ᵐ x ∂μ, p x fun i => f i x := h_anti
have h_ae_seq_mono : Antitone (aeSeq h_meas p) := by
intro n m hnm x
by_cases hx : x ∈ aeSeqSet h_meas p
· exact aeSeq.prop_of_mem_aeSeqSet h_meas hx hnm
· simp only [aeSeq, hx, if_false]
exact le_rfl
rw [lintegral_congr_ae (aeSeq.iInf h_meas hp).symm]
simp_rw [iInf_apply]
rw [lintegral_iInf (aeSeq.measurable h_meas p) h_ae_seq_mono]
· congr
exact funext fun n ↦ lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae h_meas hp n)
· rwa [lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae h_meas hp 0)]
| Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,081 | 1,106 | theorem lintegral_iInf_directed_of_measurable {mα : MeasurableSpace α} [Countable β]
{f : β → α → ℝ≥0∞} {μ : Measure α} (hμ : μ ≠ 0) (hf : ∀ b, Measurable (f b))
(hf_int : ∀ b, ∫⁻ a, f b a ∂μ ≠ ∞) (h_directed : Directed (· ≥ ·) f) :
∫⁻ a, ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ a, f b a ∂μ := by |
cases nonempty_encodable β
cases isEmpty_or_nonempty β
· simp only [iInf_of_empty, lintegral_const,
ENNReal.top_mul (Measure.measure_univ_ne_zero.mpr hμ)]
inhabit β
have : ∀ a, ⨅ b, f b a = ⨅ n, f (h_directed.sequence f n) a := by
refine fun a =>
le_antisymm (le_iInf fun n => iInf_le _ _)
(le_iInf fun b => iInf_le_of_le (Encodable.encode b + 1) ?_)
exact h_directed.sequence_le b a
-- Porting note: used `∘` below to deal with its reduced reducibility
calc
∫⁻ a, ⨅ b, f b a ∂μ
_ = ∫⁻ a, ⨅ n, (f ∘ h_directed.sequence f) n a ∂μ := by simp only [this, Function.comp_apply]
_ = ⨅ n, ∫⁻ a, (f ∘ h_directed.sequence f) n a ∂μ := by
rw [lintegral_iInf ?_ h_directed.sequence_anti]
· exact hf_int _
· exact fun n => hf _
_ = ⨅ b, ∫⁻ a, f b a ∂μ := by
refine le_antisymm (le_iInf fun b => ?_) (le_iInf fun n => ?_)
· exact iInf_le_of_le (Encodable.encode b + 1) (lintegral_mono <| h_directed.sequence_le b)
· exact iInf_le (fun b => ∫⁻ a, f b a ∂μ) _
|
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : ℕ}
section CommRing
variable [CommRing R] [IsDomain R] {p q : R[X]}
section Roots
open Multiset Finset
noncomputable def roots (p : R[X]) : Multiset R :=
haveI := Classical.decEq R
haveI := Classical.dec (p = 0)
if h : p = 0 then ∅ else Classical.choose (exists_multiset_roots h)
#align polynomial.roots Polynomial.roots
theorem roots_def [DecidableEq R] (p : R[X]) [Decidable (p = 0)] :
p.roots = if h : p = 0 then ∅ else Classical.choose (exists_multiset_roots h) := by
-- porting noteL `‹_›` doesn't work for instance arguments
rename_i iR ip0
obtain rfl := Subsingleton.elim iR (Classical.decEq R)
obtain rfl := Subsingleton.elim ip0 (Classical.dec (p = 0))
rfl
#align polynomial.roots_def Polynomial.roots_def
@[simp]
theorem roots_zero : (0 : R[X]).roots = 0 :=
dif_pos rfl
#align polynomial.roots_zero Polynomial.roots_zero
theorem card_roots (hp0 : p ≠ 0) : (Multiset.card (roots p) : WithBot ℕ) ≤ degree p := by
classical
unfold roots
rw [dif_neg hp0]
exact (Classical.choose_spec (exists_multiset_roots hp0)).1
#align polynomial.card_roots Polynomial.card_roots
theorem card_roots' (p : R[X]) : Multiset.card p.roots ≤ natDegree p := by
by_cases hp0 : p = 0
· simp [hp0]
exact WithBot.coe_le_coe.1 (le_trans (card_roots hp0) (le_of_eq <| degree_eq_natDegree hp0))
#align polynomial.card_roots' Polynomial.card_roots'
theorem card_roots_sub_C {p : R[X]} {a : R} (hp0 : 0 < degree p) :
(Multiset.card (p - C a).roots : WithBot ℕ) ≤ degree p :=
calc
(Multiset.card (p - C a).roots : WithBot ℕ) ≤ degree (p - C a) :=
card_roots <| mt sub_eq_zero.1 fun h => not_le_of_gt hp0 <| h.symm ▸ degree_C_le
_ = degree p := by rw [sub_eq_add_neg, ← C_neg]; exact degree_add_C hp0
set_option linter.uppercaseLean3 false in
#align polynomial.card_roots_sub_C Polynomial.card_roots_sub_C
theorem card_roots_sub_C' {p : R[X]} {a : R} (hp0 : 0 < degree p) :
Multiset.card (p - C a).roots ≤ natDegree p :=
WithBot.coe_le_coe.1
(le_trans (card_roots_sub_C hp0)
(le_of_eq <| degree_eq_natDegree fun h => by simp_all [lt_irrefl]))
set_option linter.uppercaseLean3 false in
#align polynomial.card_roots_sub_C' Polynomial.card_roots_sub_C'
@[simp]
theorem count_roots [DecidableEq R] (p : R[X]) : p.roots.count a = rootMultiplicity a p := by
classical
by_cases hp : p = 0
· simp [hp]
rw [roots_def, dif_neg hp]
exact (Classical.choose_spec (exists_multiset_roots hp)).2 a
#align polynomial.count_roots Polynomial.count_roots
@[simp]
theorem mem_roots' : a ∈ p.roots ↔ p ≠ 0 ∧ IsRoot p a := by
classical
rw [← count_pos, count_roots p, rootMultiplicity_pos']
#align polynomial.mem_roots' Polynomial.mem_roots'
theorem mem_roots (hp : p ≠ 0) : a ∈ p.roots ↔ IsRoot p a :=
mem_roots'.trans <| and_iff_right hp
#align polynomial.mem_roots Polynomial.mem_roots
theorem ne_zero_of_mem_roots (h : a ∈ p.roots) : p ≠ 0 :=
(mem_roots'.1 h).1
#align polynomial.ne_zero_of_mem_roots Polynomial.ne_zero_of_mem_roots
theorem isRoot_of_mem_roots (h : a ∈ p.roots) : IsRoot p a :=
(mem_roots'.1 h).2
#align polynomial.is_root_of_mem_roots Polynomial.isRoot_of_mem_roots
-- Porting note: added during port.
lemma mem_roots_iff_aeval_eq_zero {x : R} (w : p ≠ 0) : x ∈ roots p ↔ aeval x p = 0 := by
rw [mem_roots w, IsRoot.def, aeval_def, eval₂_eq_eval_map]
simp
theorem card_le_degree_of_subset_roots {p : R[X]} {Z : Finset R} (h : Z.val ⊆ p.roots) :
Z.card ≤ p.natDegree :=
(Multiset.card_le_card (Finset.val_le_iff_val_subset.2 h)).trans (Polynomial.card_roots' p)
#align polynomial.card_le_degree_of_subset_roots Polynomial.card_le_degree_of_subset_roots
theorem finite_setOf_isRoot {p : R[X]} (hp : p ≠ 0) : Set.Finite { x | IsRoot p x } := by
classical
simpa only [← Finset.setOf_mem, Multiset.mem_toFinset, mem_roots hp]
using p.roots.toFinset.finite_toSet
#align polynomial.finite_set_of_is_root Polynomial.finite_setOf_isRoot
theorem eq_zero_of_infinite_isRoot (p : R[X]) (h : Set.Infinite { x | IsRoot p x }) : p = 0 :=
not_imp_comm.mp finite_setOf_isRoot h
#align polynomial.eq_zero_of_infinite_is_root Polynomial.eq_zero_of_infinite_isRoot
theorem exists_max_root [LinearOrder R] (p : R[X]) (hp : p ≠ 0) : ∃ x₀, ∀ x, p.IsRoot x → x ≤ x₀ :=
Set.exists_upper_bound_image _ _ <| finite_setOf_isRoot hp
#align polynomial.exists_max_root Polynomial.exists_max_root
theorem exists_min_root [LinearOrder R] (p : R[X]) (hp : p ≠ 0) : ∃ x₀, ∀ x, p.IsRoot x → x₀ ≤ x :=
Set.exists_lower_bound_image _ _ <| finite_setOf_isRoot hp
#align polynomial.exists_min_root Polynomial.exists_min_root
theorem eq_of_infinite_eval_eq (p q : R[X]) (h : Set.Infinite { x | eval x p = eval x q }) :
p = q := by
rw [← sub_eq_zero]
apply eq_zero_of_infinite_isRoot
simpa only [IsRoot, eval_sub, sub_eq_zero]
#align polynomial.eq_of_infinite_eval_eq Polynomial.eq_of_infinite_eval_eq
theorem roots_mul {p q : R[X]} (hpq : p * q ≠ 0) : (p * q).roots = p.roots + q.roots := by
classical
exact Multiset.ext.mpr fun r => by
rw [count_add, count_roots, count_roots, count_roots, rootMultiplicity_mul hpq]
#align polynomial.roots_mul Polynomial.roots_mul
theorem roots.le_of_dvd (h : q ≠ 0) : p ∣ q → roots p ≤ roots q := by
rintro ⟨k, rfl⟩
exact Multiset.le_iff_exists_add.mpr ⟨k.roots, roots_mul h⟩
#align polynomial.roots.le_of_dvd Polynomial.roots.le_of_dvd
theorem mem_roots_sub_C' {p : R[X]} {a x : R} : x ∈ (p - C a).roots ↔ p ≠ C a ∧ p.eval x = a := by
rw [mem_roots', IsRoot.def, sub_ne_zero, eval_sub, sub_eq_zero, eval_C]
set_option linter.uppercaseLean3 false in
#align polynomial.mem_roots_sub_C' Polynomial.mem_roots_sub_C'
theorem mem_roots_sub_C {p : R[X]} {a x : R} (hp0 : 0 < degree p) :
x ∈ (p - C a).roots ↔ p.eval x = a :=
mem_roots_sub_C'.trans <| and_iff_right fun hp => hp0.not_le <| hp.symm ▸ degree_C_le
set_option linter.uppercaseLean3 false in
#align polynomial.mem_roots_sub_C Polynomial.mem_roots_sub_C
@[simp]
theorem roots_X_sub_C (r : R) : roots (X - C r) = {r} := by
classical
ext s
rw [count_roots, rootMultiplicity_X_sub_C, count_singleton]
set_option linter.uppercaseLean3 false in
#align polynomial.roots_X_sub_C Polynomial.roots_X_sub_C
@[simp]
theorem roots_X : roots (X : R[X]) = {0} := by rw [← roots_X_sub_C, C_0, sub_zero]
set_option linter.uppercaseLean3 false in
#align polynomial.roots_X Polynomial.roots_X
@[simp]
theorem roots_C (x : R) : (C x).roots = 0 := by
classical exact
if H : x = 0 then by rw [H, C_0, roots_zero]
else
Multiset.ext.mpr fun r => (by
rw [count_roots, count_zero, rootMultiplicity_eq_zero (not_isRoot_C _ _ H)])
set_option linter.uppercaseLean3 false in
#align polynomial.roots_C Polynomial.roots_C
@[simp]
theorem roots_one : (1 : R[X]).roots = ∅ :=
roots_C 1
#align polynomial.roots_one Polynomial.roots_one
@[simp]
theorem roots_C_mul (p : R[X]) (ha : a ≠ 0) : (C a * p).roots = p.roots := by
by_cases hp : p = 0 <;>
simp only [roots_mul, *, Ne, mul_eq_zero, C_eq_zero, or_self_iff, not_false_iff, roots_C,
zero_add, mul_zero]
set_option linter.uppercaseLean3 false in
#align polynomial.roots_C_mul Polynomial.roots_C_mul
@[simp]
theorem roots_smul_nonzero (p : R[X]) (ha : a ≠ 0) : (a • p).roots = p.roots := by
rw [smul_eq_C_mul, roots_C_mul _ ha]
#align polynomial.roots_smul_nonzero Polynomial.roots_smul_nonzero
@[simp]
lemma roots_neg (p : R[X]) : (-p).roots = p.roots := by
rw [← neg_one_smul R p, roots_smul_nonzero p (neg_ne_zero.mpr one_ne_zero)]
theorem roots_list_prod (L : List R[X]) :
(0 : R[X]) ∉ L → L.prod.roots = (L : Multiset R[X]).bind roots :=
List.recOn L (fun _ => roots_one) fun hd tl ih H => by
rw [List.mem_cons, not_or] at H
rw [List.prod_cons, roots_mul (mul_ne_zero (Ne.symm H.1) <| List.prod_ne_zero H.2), ←
Multiset.cons_coe, Multiset.cons_bind, ih H.2]
#align polynomial.roots_list_prod Polynomial.roots_list_prod
theorem roots_multiset_prod (m : Multiset R[X]) : (0 : R[X]) ∉ m → m.prod.roots = m.bind roots := by
rcases m with ⟨L⟩
simpa only [Multiset.prod_coe, quot_mk_to_coe''] using roots_list_prod L
#align polynomial.roots_multiset_prod Polynomial.roots_multiset_prod
theorem roots_prod {ι : Type*} (f : ι → R[X]) (s : Finset ι) :
s.prod f ≠ 0 → (s.prod f).roots = s.val.bind fun i => roots (f i) := by
rcases s with ⟨m, hm⟩
simpa [Multiset.prod_eq_zero_iff, Multiset.bind_map] using roots_multiset_prod (m.map f)
#align polynomial.roots_prod Polynomial.roots_prod
@[simp]
theorem roots_pow (p : R[X]) (n : ℕ) : (p ^ n).roots = n • p.roots := by
induction' n with n ihn
· rw [pow_zero, roots_one, zero_smul, empty_eq_zero]
· rcases eq_or_ne p 0 with (rfl | hp)
· rw [zero_pow n.succ_ne_zero, roots_zero, smul_zero]
· rw [pow_succ, roots_mul (mul_ne_zero (pow_ne_zero _ hp) hp), ihn, add_smul, one_smul]
#align polynomial.roots_pow Polynomial.roots_pow
theorem roots_X_pow (n : ℕ) : (X ^ n : R[X]).roots = n • ({0} : Multiset R) := by
rw [roots_pow, roots_X]
set_option linter.uppercaseLean3 false in
#align polynomial.roots_X_pow Polynomial.roots_X_pow
theorem roots_C_mul_X_pow (ha : a ≠ 0) (n : ℕ) :
Polynomial.roots (C a * X ^ n) = n • ({0} : Multiset R) := by
rw [roots_C_mul _ ha, roots_X_pow]
set_option linter.uppercaseLean3 false in
#align polynomial.roots_C_mul_X_pow Polynomial.roots_C_mul_X_pow
@[simp]
theorem roots_monomial (ha : a ≠ 0) (n : ℕ) : (monomial n a).roots = n • ({0} : Multiset R) := by
rw [← C_mul_X_pow_eq_monomial, roots_C_mul_X_pow ha]
#align polynomial.roots_monomial Polynomial.roots_monomial
theorem roots_prod_X_sub_C (s : Finset R) : (s.prod fun a => X - C a).roots = s.val := by
apply (roots_prod (fun a => X - C a) s ?_).trans
· simp_rw [roots_X_sub_C]
rw [Multiset.bind_singleton, Multiset.map_id']
· refine prod_ne_zero_iff.mpr (fun a _ => X_sub_C_ne_zero a)
set_option linter.uppercaseLean3 false in
#align polynomial.roots_prod_X_sub_C Polynomial.roots_prod_X_sub_C
@[simp]
theorem roots_multiset_prod_X_sub_C (s : Multiset R) : (s.map fun a => X - C a).prod.roots = s := by
rw [roots_multiset_prod, Multiset.bind_map]
· simp_rw [roots_X_sub_C]
rw [Multiset.bind_singleton, Multiset.map_id']
· rw [Multiset.mem_map]
rintro ⟨a, -, h⟩
exact X_sub_C_ne_zero a h
set_option linter.uppercaseLean3 false in
#align polynomial.roots_multiset_prod_X_sub_C Polynomial.roots_multiset_prod_X_sub_C
theorem card_roots_X_pow_sub_C {n : ℕ} (hn : 0 < n) (a : R) :
Multiset.card (roots ((X : R[X]) ^ n - C a)) ≤ n :=
WithBot.coe_le_coe.1 <|
calc
(Multiset.card (roots ((X : R[X]) ^ n - C a)) : WithBot ℕ) ≤ degree ((X : R[X]) ^ n - C a) :=
card_roots (X_pow_sub_C_ne_zero hn a)
_ = n := degree_X_pow_sub_C hn a
set_option linter.uppercaseLean3 false in
#align polynomial.card_roots_X_pow_sub_C Polynomial.card_roots_X_pow_sub_C
section NthRoots
def nthRoots (n : ℕ) (a : R) : Multiset R :=
roots ((X : R[X]) ^ n - C a)
#align polynomial.nth_roots Polynomial.nthRoots
@[simp]
theorem mem_nthRoots {n : ℕ} (hn : 0 < n) {a x : R} : x ∈ nthRoots n a ↔ x ^ n = a := by
rw [nthRoots, mem_roots (X_pow_sub_C_ne_zero hn a), IsRoot.def, eval_sub, eval_C, eval_pow,
eval_X, sub_eq_zero]
#align polynomial.mem_nth_roots Polynomial.mem_nthRoots
@[simp]
theorem nthRoots_zero (r : R) : nthRoots 0 r = 0 := by
simp only [empty_eq_zero, pow_zero, nthRoots, ← C_1, ← C_sub, roots_C]
#align polynomial.nth_roots_zero Polynomial.nthRoots_zero
@[simp]
theorem nthRoots_zero_right {R} [CommRing R] [IsDomain R] (n : ℕ) :
nthRoots n (0 : R) = Multiset.replicate n 0 := by
rw [nthRoots, C.map_zero, sub_zero, roots_pow, roots_X, Multiset.nsmul_singleton]
theorem card_nthRoots (n : ℕ) (a : R) : Multiset.card (nthRoots n a) ≤ n := by
classical exact
(if hn : n = 0 then
if h : (X : R[X]) ^ n - C a = 0 then by
simp [Nat.zero_le, nthRoots, roots, h, dif_pos rfl, empty_eq_zero, Multiset.card_zero]
else
WithBot.coe_le_coe.1
(le_trans (card_roots h)
(by
rw [hn, pow_zero, ← C_1, ← RingHom.map_sub]
exact degree_C_le))
else by
rw [← Nat.cast_le (α := WithBot ℕ)]
rw [← degree_X_pow_sub_C (Nat.pos_of_ne_zero hn) a]
exact card_roots (X_pow_sub_C_ne_zero (Nat.pos_of_ne_zero hn) a))
#align polynomial.card_nth_roots Polynomial.card_nthRoots
@[simp]
theorem nthRoots_two_eq_zero_iff {r : R} : nthRoots 2 r = 0 ↔ ¬IsSquare r := by
simp_rw [isSquare_iff_exists_sq, eq_zero_iff_forall_not_mem, mem_nthRoots (by norm_num : 0 < 2),
← not_exists, eq_comm]
#align polynomial.nth_roots_two_eq_zero_iff Polynomial.nthRoots_two_eq_zero_iff
def nthRootsFinset (n : ℕ) (R : Type*) [CommRing R] [IsDomain R] : Finset R :=
haveI := Classical.decEq R
Multiset.toFinset (nthRoots n (1 : R))
#align polynomial.nth_roots_finset Polynomial.nthRootsFinset
-- Porting note (#10756): new lemma
lemma nthRootsFinset_def (n : ℕ) (R : Type*) [CommRing R] [IsDomain R] [DecidableEq R] :
nthRootsFinset n R = Multiset.toFinset (nthRoots n (1 : R)) := by
unfold nthRootsFinset
convert rfl
@[simp]
theorem mem_nthRootsFinset {n : ℕ} (h : 0 < n) {x : R} :
x ∈ nthRootsFinset n R ↔ x ^ (n : ℕ) = 1 := by
classical
rw [nthRootsFinset_def, mem_toFinset, mem_nthRoots h]
#align polynomial.mem_nth_roots_finset Polynomial.mem_nthRootsFinset
@[simp]
theorem nthRootsFinset_zero : nthRootsFinset 0 R = ∅ := by classical simp [nthRootsFinset_def]
#align polynomial.nth_roots_finset_zero Polynomial.nthRootsFinset_zero
theorem mul_mem_nthRootsFinset
{η₁ η₂ : R} (hη₁ : η₁ ∈ nthRootsFinset n R) (hη₂ : η₂ ∈ nthRootsFinset n R) :
η₁ * η₂ ∈ nthRootsFinset n R := by
cases n with
| zero =>
simp only [Nat.zero_eq, nthRootsFinset_zero, not_mem_empty] at hη₁
| succ n =>
rw [mem_nthRootsFinset n.succ_pos] at hη₁ hη₂ ⊢
rw [mul_pow, hη₁, hη₂, one_mul]
theorem ne_zero_of_mem_nthRootsFinset {η : R} (hη : η ∈ nthRootsFinset n R) : η ≠ 0 := by
nontriviality R
rintro rfl
cases n with
| zero =>
simp only [Nat.zero_eq, nthRootsFinset_zero, not_mem_empty] at hη
| succ n =>
rw [mem_nthRootsFinset n.succ_pos, zero_pow n.succ_ne_zero] at hη
exact zero_ne_one hη
| Mathlib/Algebra/Polynomial/Roots.lean | 390 | 391 | theorem one_mem_nthRootsFinset (hn : 0 < n) : 1 ∈ nthRootsFinset n R := by |
rw [mem_nthRootsFinset hn, one_pow]
|
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 AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
variable {σ τ : Type*} {r : R} {e : ℕ} {n m : σ} {s : σ →₀ ℕ}
section CommSemiring
variable [CommSemiring R] {p q : MvPolynomial σ R}
section Degrees
def degrees (p : MvPolynomial σ R) : Multiset σ :=
letI := Classical.decEq σ
p.support.sup fun s : σ →₀ ℕ => toMultiset s
#align mv_polynomial.degrees MvPolynomial.degrees
theorem degrees_def [DecidableEq σ] (p : MvPolynomial σ R) :
p.degrees = p.support.sup fun s : σ →₀ ℕ => Finsupp.toMultiset s := by rw [degrees]; convert rfl
#align mv_polynomial.degrees_def MvPolynomial.degrees_def
theorem degrees_monomial (s : σ →₀ ℕ) (a : R) : degrees (monomial s a) ≤ toMultiset s := by
classical
refine (supDegree_single s a).trans_le ?_
split_ifs
exacts [bot_le, le_rfl]
#align mv_polynomial.degrees_monomial MvPolynomial.degrees_monomial
theorem degrees_monomial_eq (s : σ →₀ ℕ) (a : R) (ha : a ≠ 0) :
degrees (monomial s a) = toMultiset s := by
classical
exact (supDegree_single s a).trans (if_neg ha)
#align mv_polynomial.degrees_monomial_eq MvPolynomial.degrees_monomial_eq
theorem degrees_C (a : R) : degrees (C a : MvPolynomial σ R) = 0 :=
Multiset.le_zero.1 <| degrees_monomial _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.degrees_C MvPolynomial.degrees_C
theorem degrees_X' (n : σ) : degrees (X n : MvPolynomial σ R) ≤ {n} :=
le_trans (degrees_monomial _ _) <| le_of_eq <| toMultiset_single _ _
set_option linter.uppercaseLean3 false in
#align mv_polynomial.degrees_X' MvPolynomial.degrees_X'
@[simp]
theorem degrees_X [Nontrivial R] (n : σ) : degrees (X n : MvPolynomial σ R) = {n} :=
(degrees_monomial_eq _ (1 : R) one_ne_zero).trans (toMultiset_single _ _)
set_option linter.uppercaseLean3 false in
#align mv_polynomial.degrees_X MvPolynomial.degrees_X
@[simp]
theorem degrees_zero : degrees (0 : MvPolynomial σ R) = 0 := by
rw [← C_0]
exact degrees_C 0
#align mv_polynomial.degrees_zero MvPolynomial.degrees_zero
@[simp]
theorem degrees_one : degrees (1 : MvPolynomial σ R) = 0 :=
degrees_C 1
#align mv_polynomial.degrees_one MvPolynomial.degrees_one
theorem degrees_add [DecidableEq σ] (p q : MvPolynomial σ R) :
(p + q).degrees ≤ p.degrees ⊔ q.degrees := by
simp_rw [degrees_def]; exact supDegree_add_le
#align mv_polynomial.degrees_add MvPolynomial.degrees_add
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 133 | 135 | theorem degrees_sum {ι : Type*} [DecidableEq σ] (s : Finset ι) (f : ι → MvPolynomial σ R) :
(∑ i ∈ s, f i).degrees ≤ s.sup fun i => (f i).degrees := by |
simp_rw [degrees_def]; exact supDegree_sum_le
|
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.PNat.Defs
#align_import data.pnat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset Function PNat
namespace PNat
variable (a b : ℕ+)
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ+ := Subtype.instLocallyFiniteOrder _
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).subtype fun n : ℕ => 0 < n :=
rfl
#align pnat.Icc_eq_finset_subtype PNat.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).subtype fun n : ℕ => 0 < n :=
rfl
#align pnat.Ico_eq_finset_subtype PNat.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).subtype fun n : ℕ => 0 < n :=
rfl
#align pnat.Ioc_eq_finset_subtype PNat.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).subtype fun n : ℕ => 0 < n :=
rfl
#align pnat.Ioo_eq_finset_subtype PNat.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).subtype fun n : ℕ => 0 < n := rfl
#align pnat.uIcc_eq_finset_subtype PNat.uIcc_eq_finset_subtype
theorem map_subtype_embedding_Icc : (Icc a b).map (Embedding.subtype _) = Icc ↑a ↑b :=
Finset.map_subtype_embedding_Icc _ _ _ fun _c _ _x hx _ hc _ => hc.trans_le hx
#align pnat.map_subtype_embedding_Icc PNat.map_subtype_embedding_Icc
theorem map_subtype_embedding_Ico : (Ico a b).map (Embedding.subtype _) = Ico ↑a ↑b :=
Finset.map_subtype_embedding_Ico _ _ _ fun _c _ _x hx _ hc _ => hc.trans_le hx
#align pnat.map_subtype_embedding_Ico PNat.map_subtype_embedding_Ico
theorem map_subtype_embedding_Ioc : (Ioc a b).map (Embedding.subtype _) = Ioc ↑a ↑b :=
Finset.map_subtype_embedding_Ioc _ _ _ fun _c _ _x hx _ hc _ => hc.trans_le hx
#align pnat.map_subtype_embedding_Ioc PNat.map_subtype_embedding_Ioc
theorem map_subtype_embedding_Ioo : (Ioo a b).map (Embedding.subtype _) = Ioo ↑a ↑b :=
Finset.map_subtype_embedding_Ioo _ _ _ fun _c _ _x hx _ hc _ => hc.trans_le hx
#align pnat.map_subtype_embedding_Ioo PNat.map_subtype_embedding_Ioo
theorem map_subtype_embedding_uIcc : (uIcc a b).map (Embedding.subtype _) = uIcc ↑a ↑b :=
map_subtype_embedding_Icc _ _
#align pnat.map_subtype_embedding_uIcc PNat.map_subtype_embedding_uIcc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc]
-- Porting note: I had to change this to `erw` *and* provide the proof, yuck.
-- https://github.com/leanprover-community/mathlib4/issues/5164
erw [← Finset.map_subtype_embedding_Icc _ a b (fun c x _ hx _ hc _ => hc.trans_le hx)]
rw [card_map]
#align pnat.card_Icc PNat.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a := by
rw [← Nat.card_Ico]
-- Porting note: I had to change this to `erw` *and* provide the proof, yuck.
-- https://github.com/leanprover-community/mathlib4/issues/5164
erw [← Finset.map_subtype_embedding_Ico _ a b (fun c x _ hx _ hc _ => hc.trans_le hx)]
rw [card_map]
#align pnat.card_Ico PNat.card_Ico
@[simp]
| Mathlib/Data/PNat/Interval.lean | 85 | 90 | theorem card_Ioc : (Ioc a b).card = b - a := by |
rw [← Nat.card_Ioc]
-- Porting note: I had to change this to `erw` *and* provide the proof, yuck.
-- https://github.com/leanprover-community/mathlib4/issues/5164
erw [← Finset.map_subtype_embedding_Ioc _ a b (fun c x _ hx _ hc _ => hc.trans_le hx)]
rw [card_map]
|
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Order.Basic
import Mathlib.Order.Hom.Lattice
#align_import ring_theory.filtration from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open scoped Polynomial
@[ext]
structure Ideal.Filtration (M : Type u) [AddCommGroup M] [Module R M] where
N : ℕ → Submodule R M
mono : ∀ i, N (i + 1) ≤ N i
smul_le : ∀ i, I • N i ≤ N (i + 1)
#align ideal.filtration Ideal.Filtration
variable (F F' : I.Filtration M) {I}
namespace Ideal.Filtration
theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by
induction' i with _ ih
· simp
· rw [pow_succ', mul_smul, add_assoc, add_comm 1, ← add_assoc]
exact (smul_mono_right _ ih).trans (F.smul_le _)
#align ideal.filtration.pow_smul_le Ideal.Filtration.pow_smul_le
theorem pow_smul_le_pow_smul (i j k : ℕ) : I ^ (i + k) • F.N j ≤ I ^ k • F.N (i + j) := by
rw [add_comm, pow_add, mul_smul]
exact smul_mono_right _ (F.pow_smul_le i j)
#align ideal.filtration.pow_smul_le_pow_smul Ideal.Filtration.pow_smul_le_pow_smul
protected theorem antitone : Antitone F.N :=
antitone_nat_of_succ_le F.mono
#align ideal.filtration.antitone Ideal.Filtration.antitone
@[simps]
def _root_.Ideal.trivialFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N _ := N
mono _ := le_rfl
smul_le _ := Submodule.smul_le_right
#align ideal.trivial_filtration Ideal.trivialFiltration
instance : Sup (I.Filtration M) :=
⟨fun F F' =>
⟨F.N ⊔ F'.N, fun i => sup_le_sup (F.mono i) (F'.mono i), fun i =>
(Submodule.smul_sup _ _ _).trans_le <| sup_le_sup (F.smul_le i) (F'.smul_le i)⟩⟩
instance : SupSet (I.Filtration M) :=
⟨fun S =>
{ N := sSup (Ideal.Filtration.N '' S)
mono := fun i => by
apply sSup_le_sSup_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩
smul_le := fun i => by
rw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]
apply iSup_mono _
rintro ⟨_, F, hF, rfl⟩
exact F.smul_le i }⟩
instance : Inf (I.Filtration M) :=
⟨fun F F' =>
⟨F.N ⊓ F'.N, fun i => inf_le_inf (F.mono i) (F'.mono i), fun i =>
(smul_inf_le _ _ _).trans <| inf_le_inf (F.smul_le i) (F'.smul_le i)⟩⟩
instance : InfSet (I.Filtration M) :=
⟨fun S =>
{ N := sInf (Ideal.Filtration.N '' S)
mono := fun i => by
apply sInf_le_sInf_of_forall_exists_le _
rintro _ ⟨⟨_, F, hF, rfl⟩, rfl⟩
exact ⟨_, ⟨⟨_, F, hF, rfl⟩, rfl⟩, F.mono i⟩
smul_le := fun i => by
rw [sInf_eq_iInf', iInf_apply, iInf_apply]
refine smul_iInf_le.trans ?_
apply iInf_mono _
rintro ⟨_, F, hF, rfl⟩
exact F.smul_le i }⟩
instance : Top (I.Filtration M) :=
⟨I.trivialFiltration ⊤⟩
instance : Bot (I.Filtration M) :=
⟨I.trivialFiltration ⊥⟩
@[simp]
theorem sup_N : (F ⊔ F').N = F.N ⊔ F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.sup_N Ideal.Filtration.sup_N
@[simp]
theorem sSup_N (S : Set (I.Filtration M)) : (sSup S).N = sSup (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Sup_N Ideal.Filtration.sSup_N
@[simp]
theorem inf_N : (F ⊓ F').N = F.N ⊓ F'.N :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.inf_N Ideal.Filtration.inf_N
@[simp]
theorem sInf_N (S : Set (I.Filtration M)) : (sInf S).N = sInf (Ideal.Filtration.N '' S) :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.Inf_N Ideal.Filtration.sInf_N
@[simp]
theorem top_N : (⊤ : I.Filtration M).N = ⊤ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.top_N Ideal.Filtration.top_N
@[simp]
theorem bot_N : (⊥ : I.Filtration M).N = ⊥ :=
rfl
set_option linter.uppercaseLean3 false in
#align ideal.filtration.bot_N Ideal.Filtration.bot_N
@[simp]
theorem iSup_N {ι : Sort*} (f : ι → I.Filtration M) : (iSup f).N = ⨆ i, (f i).N :=
congr_arg sSup (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.supr_N Ideal.Filtration.iSup_N
@[simp]
theorem iInf_N {ι : Sort*} (f : ι → I.Filtration M) : (iInf f).N = ⨅ i, (f i).N :=
congr_arg sInf (Set.range_comp _ _).symm
set_option linter.uppercaseLean3 false in
#align ideal.filtration.infi_N Ideal.Filtration.iInf_N
instance : CompleteLattice (I.Filtration M) :=
Function.Injective.completeLattice Ideal.Filtration.N Ideal.Filtration.ext sup_N inf_N
(fun _ => sSup_image) (fun _ => sInf_image) top_N bot_N
instance : Inhabited (I.Filtration M) :=
⟨⊥⟩
def Stable : Prop :=
∃ n₀, ∀ n ≥ n₀, I • F.N n = F.N (n + 1)
#align ideal.filtration.stable Ideal.Filtration.Stable
@[simps]
def _root_.Ideal.stableFiltration (I : Ideal R) (N : Submodule R M) : I.Filtration M where
N i := I ^ i • N
mono i := by dsimp only; rw [add_comm, pow_add, mul_smul]; exact Submodule.smul_le_right
smul_le i := by dsimp only; rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration Ideal.stableFiltration
theorem _root_.Ideal.stableFiltration_stable (I : Ideal R) (N : Submodule R M) :
(I.stableFiltration N).Stable := by
use 0
intro n _
dsimp
rw [add_comm, pow_add, mul_smul, pow_one]
#align ideal.stable_filtration_stable Ideal.stableFiltration_stable
variable {F F'} (h : F.Stable)
theorem Stable.exists_pow_smul_eq : ∃ n₀, ∀ k, F.N (n₀ + k) = I ^ k • F.N n₀ := by
obtain ⟨n₀, hn⟩ := h
use n₀
intro k
induction' k with _ ih
· simp
· rw [← add_assoc, ← hn, ih, add_comm, pow_add, mul_smul, pow_one]
omega
#align ideal.filtration.stable.exists_pow_smul_eq Ideal.Filtration.Stable.exists_pow_smul_eq
theorem Stable.exists_pow_smul_eq_of_ge : ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by
obtain ⟨n₀, hn₀⟩ := h.exists_pow_smul_eq
use n₀
intro n hn
convert hn₀ (n - n₀)
rw [add_comm, tsub_add_cancel_of_le hn]
#align ideal.filtration.stable.exists_pow_smul_eq_of_ge Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge
theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by
refine ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => ?_⟩⟩
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by omega), smul_smul, ← pow_succ',
tsub_add_eq_add_tsub hn]
#align ideal.filtration.stable_iff_exists_pow_smul_eq_of_ge Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge
theorem Stable.exists_forall_le (h : F.Stable) (e : F.N 0 ≤ F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n := by
obtain ⟨n₀, hF⟩ := h
use n₀
intro n
induction' n with n hn
· refine (F.antitone ?_).trans e; simp
· rw [add_right_comm, ← hF]
· exact (smul_mono_right _ hn).trans (F'.smul_le _)
simp
#align ideal.filtration.stable.exists_forall_le Ideal.Filtration.Stable.exists_forall_le
theorem Stable.bounded_difference (h : F.Stable) (h' : F'.Stable) (e : F.N 0 = F'.N 0) :
∃ n₀, ∀ n, F.N (n + n₀) ≤ F'.N n ∧ F'.N (n + n₀) ≤ F.N n := by
obtain ⟨n₁, h₁⟩ := h.exists_forall_le (le_of_eq e)
obtain ⟨n₂, h₂⟩ := h'.exists_forall_le (le_of_eq e.symm)
use max n₁ n₂
intro n
refine ⟨(F.antitone ?_).trans (h₁ n), (F'.antitone ?_).trans (h₂ n)⟩ <;> simp
#align ideal.filtration.stable.bounded_difference Ideal.Filtration.Stable.bounded_difference
open PolynomialModule
variable (F F')
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | ∀ i, f i ∈ F.N i }
add_mem' hf hg i := Submodule.add_mem _ (hf i) (hg i)
zero_mem' i := Submodule.zero_mem _
smul_mem' r f hf i := by
rw [Subalgebra.smul_def, PolynomialModule.smul_apply]
apply Submodule.sum_mem
rintro ⟨j, k⟩ e
rw [Finset.mem_antidiagonal] at e
subst e
exact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))
#align ideal.filtration.submodule Ideal.Filtration.submodule
@[simp]
theorem mem_submodule (f : PolynomialModule R M) : f ∈ F.submodule ↔ ∀ i, f i ∈ F.N i :=
Iff.rfl
#align ideal.filtration.mem_submodule Ideal.Filtration.mem_submodule
theorem inf_submodule : (F ⊓ F').submodule = F.submodule ⊓ F'.submodule := by
ext
exact forall_and
#align ideal.filtration.inf_submodule Ideal.Filtration.inf_submodule
variable (I M)
def submoduleInfHom :
InfHom (I.Filtration M) (Submodule (reesAlgebra I) (PolynomialModule R M)) where
toFun := Ideal.Filtration.submodule
map_inf' := inf_submodule
#align ideal.filtration.submodule_inf_hom Ideal.Filtration.submoduleInfHom
variable {I M}
theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid := by
apply le_antisymm
· rw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]
rintro i _ ⟨m, hm, rfl⟩ j
rw [single_apply]
split_ifs with h
· rwa [← h]
· exact (F.N j).zero_mem
· intro f hf
rw [← f.sum_single]
apply AddSubmonoid.sum_mem _ _
rintro c -
exact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))
#align ideal.filtration.submodule_closure_single Ideal.Filtration.submodule_closure_single
theorem submodule_span_single :
Submodule.span (reesAlgebra I) (⋃ i, single R i '' (F.N i : Set M)) = F.submodule := by
rw [← Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]
exact Submodule.span_eq (Filtration.submodule F)
#align ideal.filtration.submodule_span_single Ideal.Filtration.submodule_span_single
theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) := by
rw [← submodule_span_single, ← LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]
swap; · exact Submodule.span_mono (Set.iUnion₂_subset_iUnion _ _)
constructor
· intro H n hn
refine (F.smul_le n).antisymm ?_
intro x hx
obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩)
replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl
dsimp only at hl
erw [Finsupp.single_eq_same] at hl
rw [← hl, Finsupp.total_apply, Finsupp.sum_apply]
apply Submodule.sum_mem _ _
rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ -
dsimp only [Subtype.coe_mk]
rw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' ≤ n + 1 by omega)]
have e : n' ≤ n := by omega
have := F.pow_smul_le_pow_smul (n - n') n' 1
rw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, ← add_tsub_assoc_of_le e, add_comm] at this
exact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)
· let F' := Submodule.span (reesAlgebra I) (⋃ i ≤ n₀, single R i '' (F.N i : Set M))
intro hF i
have : ∀ i ≤ n₀, single R i '' (F.N i : Set M) ⊆ F' := by
-- Porting note: Original proof was
-- `fun i hi => Set.Subset.trans (Set.subset_iUnion₂ i hi) Submodule.subset_span`
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_
exact hi
induction' i with j hj
· exact this _ (zero_le _)
by_cases hj' : j.succ ≤ n₀
· exact this _ hj'
simp only [not_le, Nat.lt_succ_iff] at hj'
rw [← hF _ hj']
rintro _ ⟨m, hm, rfl⟩
refine Submodule.smul_induction_on hm (fun r hr m' hm' => ?_) (fun x y hx hy => ?_)
· rw [add_comm, ← monomial_smul_single]
exact F'.smul_mem
⟨_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])⟩ (hj <| Set.mem_image_of_mem _ hm')
· rw [map_add]
exact F'.add_mem hx hy
#align ideal.filtration.submodule_eq_span_le_iff_stable_ge Ideal.Filtration.submodule_eq_span_le_iff_stable_ge
| Mathlib/RingTheory/Filtration.lean | 371 | 403 | theorem submodule_fg_iff_stable (hF' : ∀ i, (F.N i).FG) : F.submodule.FG ↔ F.Stable := by |
classical
delta Ideal.Filtration.Stable
simp_rw [← F.submodule_eq_span_le_iff_stable_ge]
constructor
· rintro H
refine H.stabilizes_of_iSup_eq
⟨fun n₀ => Submodule.span _ (⋃ (i : ℕ) (_ : i ≤ n₀), single R i '' ↑(F.N i)), ?_⟩ ?_
· intro n m e
rw [Submodule.span_le, Set.iUnion₂_subset_iff]
intro i hi
refine Set.Subset.trans ?_ Submodule.subset_span
refine @Set.subset_iUnion₂ _ _ _ (fun i => fun _ => ↑((single R i) '' ((N F i) : Set M))) i ?_
exact hi.trans e
· dsimp
rw [← Submodule.span_iUnion, ← submodule_span_single]
congr 1
ext
simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]
constructor
· rintro ⟨-, i, -, e⟩; exact ⟨i, e⟩
· rintro ⟨i, e⟩; exact ⟨i, i, le_refl i, e⟩
· rintro ⟨n, hn⟩
rw [hn]
simp_rw [Submodule.span_iUnion₂, ← Finset.mem_range_succ_iff, iSup_subtype']
apply Submodule.fg_iSup
rintro ⟨i, hi⟩
obtain ⟨s, hs⟩ := hF' i
have : Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =
Submodule.span _ (single R i '' (F.N i : Set M)) := by
rw [Finset.coe_image, ← Submodule.span_span_of_tower R, ← Submodule.map_span, hs]; rfl
rw [Subtype.coe_mk, ← this]
exact ⟨_, rfl⟩
|
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
#align_import topology.uniform_space.separation from "leanprover-community/mathlib"@"0c1f285a9f6e608ae2bdffa3f993eafb01eba829"
open Filter Set Function Topology Uniformity UniformSpace
open scoped Classical
noncomputable section
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
variable [UniformSpace α] [UniformSpace β] [UniformSpace γ]
instance (priority := 100) UniformSpace.to_regularSpace : RegularSpace α :=
.of_hasBasis
(fun _ ↦ nhds_basis_uniformity' uniformity_hasBasis_closed)
fun a _V hV ↦ isClosed_ball a hV.2
#align uniform_space.to_regular_space UniformSpace.to_regularSpace
#align separation_rel Inseparable
#noalign separated_equiv
#align separation_rel_iff_specializes specializes_iff_inseparable
#noalign separation_rel_iff_inseparable
theorem Filter.HasBasis.specializes_iff_uniformity {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
(h : (𝓤 α).HasBasis p s) {x y : α} : x ⤳ y ↔ ∀ i, p i → (x, y) ∈ s i :=
(nhds_basis_uniformity h).specializes_iff
theorem Filter.HasBasis.inseparable_iff_uniformity {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
(h : (𝓤 α).HasBasis p s) {x y : α} : Inseparable x y ↔ ∀ i, p i → (x, y) ∈ s i :=
specializes_iff_inseparable.symm.trans h.specializes_iff_uniformity
#align filter.has_basis.mem_separation_rel Filter.HasBasis.inseparable_iff_uniformity
theorem inseparable_iff_ker_uniformity {x y : α} : Inseparable x y ↔ (x, y) ∈ (𝓤 α).ker :=
(𝓤 α).basis_sets.inseparable_iff_uniformity
protected theorem Inseparable.nhds_le_uniformity {x y : α} (h : Inseparable x y) :
𝓝 (x, y) ≤ 𝓤 α := by
rw [h.prod rfl]
apply nhds_le_uniformity
theorem inseparable_iff_clusterPt_uniformity {x y : α} :
Inseparable x y ↔ ClusterPt (x, y) (𝓤 α) := by
refine ⟨fun h ↦ .of_nhds_le h.nhds_le_uniformity, fun h ↦ ?_⟩
simp_rw [uniformity_hasBasis_closed.inseparable_iff_uniformity, isClosed_iff_clusterPt]
exact fun U ⟨hU, hUc⟩ ↦ hUc _ <| h.mono <| le_principal_iff.2 hU
#align separated_space T0Space
theorem t0Space_iff_uniformity :
T0Space α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y := by
simp only [t0Space_iff_inseparable, inseparable_iff_ker_uniformity, mem_ker, id]
#align separated_def t0Space_iff_uniformity
theorem t0Space_iff_uniformity' :
T0Space α ↔ Pairwise fun x y ↦ ∃ r ∈ 𝓤 α, (x, y) ∉ r := by
simp [t0Space_iff_not_inseparable, inseparable_iff_ker_uniformity]
#align separated_def' t0Space_iff_uniformity'
theorem t0Space_iff_ker_uniformity : T0Space α ↔ (𝓤 α).ker = diagonal α := by
simp_rw [t0Space_iff_uniformity, subset_antisymm_iff, diagonal_subset_iff, subset_def,
Prod.forall, Filter.mem_ker, mem_diagonal_iff, iff_self_and]
exact fun _ x s hs ↦ refl_mem_uniformity hs
#align separated_space_iff t0Space_iff_ker_uniformity
theorem eq_of_uniformity {α : Type*} [UniformSpace α] [T0Space α] {x y : α}
(h : ∀ {V}, V ∈ 𝓤 α → (x, y) ∈ V) : x = y :=
t0Space_iff_uniformity.mp ‹T0Space α› x y @h
#align eq_of_uniformity eq_of_uniformity
theorem eq_of_uniformity_basis {α : Type*} [UniformSpace α] [T0Space α] {ι : Sort*}
{p : ι → Prop} {s : ι → Set (α × α)} (hs : (𝓤 α).HasBasis p s) {x y : α}
(h : ∀ {i}, p i → (x, y) ∈ s i) : x = y :=
(hs.inseparable_iff_uniformity.2 @h).eq
#align eq_of_uniformity_basis eq_of_uniformity_basis
theorem eq_of_forall_symmetric {α : Type*} [UniformSpace α] [T0Space α] {x y : α}
(h : ∀ {V}, V ∈ 𝓤 α → SymmetricRel V → (x, y) ∈ V) : x = y :=
eq_of_uniformity_basis hasBasis_symmetric (by simpa)
#align eq_of_forall_symmetric eq_of_forall_symmetric
theorem eq_of_clusterPt_uniformity [T0Space α] {x y : α} (h : ClusterPt (x, y) (𝓤 α)) : x = y :=
(inseparable_iff_clusterPt_uniformity.2 h).eq
#align eq_of_cluster_pt_uniformity eq_of_clusterPt_uniformity
theorem Filter.Tendsto.inseparable_iff_uniformity {l : Filter β} [NeBot l] {f g : β → α} {a b : α}
(ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) :
Inseparable a b ↔ Tendsto (fun x ↦ (f x, g x)) l (𝓤 α) := by
refine ⟨fun h ↦ (ha.prod_mk_nhds hb).mono_right h.nhds_le_uniformity, fun h ↦ ?_⟩
rw [inseparable_iff_clusterPt_uniformity]
exact (ClusterPt.of_le_nhds (ha.prod_mk_nhds hb)).mono h
#align id_rel_sub_separation_relation Inseparable.rfl
#align separation_rel_comap Inducing.inseparable_iff
#align filter.has_basis.separation_rel Filter.HasBasis.ker
#noalign separation_rel_eq_inter_closure
#align is_closed_separation_rel isClosed_setOf_inseparable
#align separated_iff_t2 R1Space.t2Space_iff_t0Space
#align separated_t3 RegularSpace.t3Space_iff_t0Space
#align subtype.separated_space Subtype.t0Space
theorem isClosed_of_spaced_out [T0Space α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α) {s : Set α}
(hs : s.Pairwise fun x y => (x, y) ∉ V₀) : IsClosed s := by
rcases comp_symm_mem_uniformity_sets V₀_in with ⟨V₁, V₁_in, V₁_symm, h_comp⟩
apply isClosed_of_closure_subset
intro x hx
rw [mem_closure_iff_ball] at hx
rcases hx V₁_in with ⟨y, hy, hy'⟩
suffices x = y by rwa [this]
apply eq_of_forall_symmetric
intro V V_in _
rcases hx (inter_mem V₁_in V_in) with ⟨z, hz, hz'⟩
obtain rfl : z = y := by
by_contra hzy
exact hs hz' hy' hzy (h_comp <| mem_comp_of_mem_ball V₁_symm (ball_inter_left x _ _ hz) hy)
exact ball_inter_right x _ _ hz
#align is_closed_of_spaced_out isClosed_of_spaced_out
theorem isClosed_range_of_spaced_out {ι} [T0Space α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
{f : ι → α} (hf : Pairwise fun x y => (f x, f y) ∉ V₀) : IsClosed (range f) :=
isClosed_of_spaced_out V₀_in <| by
rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ h
exact hf (ne_of_apply_ne f h)
#align is_closed_range_of_spaced_out isClosed_range_of_spaced_out
#align uniform_space.separation_setoid inseparableSetoid
namespace SeparationQuotient
theorem comap_map_mk_uniformity : comap (Prod.map mk mk) (map (Prod.map mk mk) (𝓤 α)) = 𝓤 α := by
refine le_antisymm ?_ le_comap_map
refine ((((𝓤 α).basis_sets.map _).comap _).le_basis_iff uniformity_hasBasis_open).2 fun U hU ↦ ?_
refine ⟨U, hU.1, fun (x₁, x₂) ⟨(y₁, y₂), hyU, hxy⟩ ↦ ?_⟩
simp only [Prod.map, Prod.ext_iff, mk_eq_mk] at hxy
exact ((hxy.1.prod hxy.2).mem_open_iff hU.2).1 hyU
instance instUniformSpace : UniformSpace (SeparationQuotient α) where
uniformity := map (Prod.map mk mk) (𝓤 α)
symm := tendsto_map' <| tendsto_map.comp tendsto_swap_uniformity
comp := fun t ht ↦ by
rcases comp_open_symm_mem_uniformity_sets ht with ⟨U, hU, hUo, -, hUt⟩
refine mem_of_superset (mem_lift' <| image_mem_map hU) ?_
simp only [subset_def, Prod.forall, mem_compRel, mem_image, Prod.ext_iff]
rintro _ _ ⟨_, ⟨⟨x, y⟩, hxyU, rfl, rfl⟩, ⟨⟨y', z⟩, hyzU, hy, rfl⟩⟩
have : y' ⤳ y := (mk_eq_mk.1 hy).specializes
exact @hUt (x, z) ⟨y', this.mem_open (UniformSpace.isOpen_ball _ hUo) hxyU, hyzU⟩
nhds_eq_comap_uniformity := surjective_mk.forall.2 fun x ↦ comap_injective surjective_mk <| by
conv_lhs => rw [comap_mk_nhds_mk, nhds_eq_comap_uniformity, ← comap_map_mk_uniformity]
simp only [Filter.comap_comap, Function.comp, Prod.map_apply]
theorem uniformity_eq : 𝓤 (SeparationQuotient α) = (𝓤 α).map (Prod.map mk mk) := rfl
#align uniform_space.uniformity_quotient SeparationQuotient.uniformity_eq
theorem uniformContinuous_mk : UniformContinuous (mk : α → SeparationQuotient α) :=
le_rfl
#align uniform_space.uniform_continuous_quotient_mk SeparationQuotient.uniformContinuous_mk
theorem uniformContinuous_dom {f : SeparationQuotient α → β} :
UniformContinuous f ↔ UniformContinuous (f ∘ mk) :=
.rfl
#align uniform_space.uniform_continuous_quotient SeparationQuotient.uniformContinuous_dom
| Mathlib/Topology/UniformSpace/Separation.lean | 267 | 271 | theorem uniformContinuous_dom₂ {f : SeparationQuotient α × SeparationQuotient β → γ} :
UniformContinuous f ↔ UniformContinuous fun p : α × β ↦ f (mk p.1, mk p.2) := by |
simp only [UniformContinuous, uniformity_prod_eq_prod, uniformity_eq, prod_map_map_eq,
tendsto_map'_iff]
rfl
|
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
#align_import algebraic_geometry.stalks from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
universe v u v' u'
open Opposite CategoryTheory CategoryTheory.Category CategoryTheory.Functor CategoryTheory.Limits
AlgebraicGeometry TopologicalSpace
variable {C : Type u} [Category.{v} C] [HasColimits C]
-- Porting note: no tidy tactic
-- attribute [local tidy] tactic.auto_cases_opens
-- this could be replaced by
-- attribute [local aesop safe cases (rule_sets := [CategoryTheory])] Opens
-- but it doesn't appear to be needed here.
open TopCat.Presheaf
namespace AlgebraicGeometry.PresheafedSpace
abbrev stalk (X : PresheafedSpace C) (x : X) : C :=
X.presheaf.stalk x
set_option linter.uppercaseLean3 false in
#align algebraic_geometry.PresheafedSpace.stalk AlgebraicGeometry.PresheafedSpace.stalk
def stalkMap {X Y : PresheafedSpace.{_, _, v} C} (α : X ⟶ Y) (x : X) :
Y.stalk (α.base x) ⟶ X.stalk x :=
(stalkFunctor C (α.base x)).map α.c ≫ X.presheaf.stalkPushforward C α.base x
set_option linter.uppercaseLean3 false in
#align algebraic_geometry.PresheafedSpace.stalk_map AlgebraicGeometry.PresheafedSpace.stalkMap
@[elementwise, reassoc]
theorem stalkMap_germ {X Y : PresheafedSpace.{_, _, v} C} (α : X ⟶ Y) (U : Opens Y)
(x : (Opens.map α.base).obj U) :
Y.presheaf.germ ⟨α.base x.1, x.2⟩ ≫ stalkMap α ↑x = α.c.app (op U) ≫ X.presheaf.germ x := by
rw [stalkMap, stalkFunctor_map_germ_assoc, stalkPushforward_germ]
set_option linter.uppercaseLean3 false in
#align algebraic_geometry.PresheafedSpace.stalk_map_germ AlgebraicGeometry.PresheafedSpace.stalkMap_germ
@[simp, elementwise, reassoc]
theorem stalkMap_germ' {X Y : PresheafedSpace.{_, _, v} C}
(α : X ⟶ Y) (U : Opens Y) (x : X) (hx : α.base x ∈ U) :
Y.presheaf.germ ⟨α.base x, hx⟩ ≫ stalkMap α x = α.c.app (op U) ≫
X.presheaf.germ (U := (Opens.map α.base).obj U) ⟨x, hx⟩ :=
PresheafedSpace.stalkMap_germ α U ⟨x, hx⟩
namespace stalkMap
@[simp]
theorem id (X : PresheafedSpace.{_, _, v} C) (x : X) :
stalkMap (𝟙 X) x = 𝟙 (X.stalk x) := by
dsimp [stalkMap]
simp only [stalkPushforward.id]
erw [← map_comp]
convert (stalkFunctor C x).map_id X.presheaf
ext
simp only [id_c, id_comp, Pushforward.id_hom_app, op_obj, eqToHom_refl, map_id]
rfl
set_option linter.uppercaseLean3 false in
#align algebraic_geometry.PresheafedSpace.stalk_map.id AlgebraicGeometry.PresheafedSpace.stalkMap.id
@[simp]
theorem comp {X Y Z : PresheafedSpace.{_, _, v} C} (α : X ⟶ Y) (β : Y ⟶ Z) (x : X) :
stalkMap (α ≫ β) x =
(stalkMap β (α.base x) : Z.stalk (β.base (α.base x)) ⟶ Y.stalk (α.base x)) ≫
(stalkMap α x : Y.stalk (α.base x) ⟶ X.stalk x) := by
dsimp [stalkMap, stalkFunctor, stalkPushforward]
-- We can't use `ext` here due to https://github.com/leanprover/std4/pull/159
refine colimit.hom_ext fun U => ?_
induction U with | h U => ?_
cases U
simp only [whiskeringLeft_obj_obj, comp_obj, op_obj, unop_op, OpenNhds.inclusion_obj,
ι_colimMap_assoc, pushforwardObj_obj, Opens.map_comp_obj, whiskerLeft_app, comp_c_app,
OpenNhds.map_obj, whiskerRight_app, NatTrans.id_app, map_id, colimit.ι_pre, id_comp, assoc,
colimit.ι_pre_assoc]
set_option linter.uppercaseLean3 false in
#align algebraic_geometry.PresheafedSpace.stalk_map.comp AlgebraicGeometry.PresheafedSpace.stalkMap.comp
theorem congr {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y)
(h₁ : α = β) (x x' : X) (h₂ : x = x') :
stalkMap α x ≫ eqToHom (show X.stalk x = X.stalk x' by rw [h₂]) =
eqToHom (show Y.stalk (α.base x) = Y.stalk (β.base x') by rw [h₁, h₂]) ≫ stalkMap β x' := by
ext
substs h₁ h₂
simp
set_option linter.uppercaseLean3 false in
#align algebraic_geometry.PresheafedSpace.stalk_map.congr AlgebraicGeometry.PresheafedSpace.stalkMap.congr
| Mathlib/Geometry/RingedSpace/Stalks.lean | 181 | 184 | theorem congr_hom {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y) (h : α = β) (x : X) :
stalkMap α x =
eqToHom (show Y.stalk (α.base x) = Y.stalk (β.base x) by rw [h]) ≫ stalkMap β x := by |
rw [← stalkMap.congr α β h x x rfl, eqToHom_refl, Category.comp_id]
|
import Mathlib.Probability.Kernel.Composition
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.integral_comp_prod from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
noncomputable section
open scoped Topology ENNReal MeasureTheory ProbabilityTheory
open Set Function Real ENNReal MeasureTheory Filter ProbabilityTheory ProbabilityTheory.kernel
variable {α β γ E : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
{mγ : MeasurableSpace γ} [NormedAddCommGroup E] {κ : kernel α β} [IsSFiniteKernel κ]
{η : kernel (α × β) γ} [IsSFiniteKernel η] {a : α}
namespace ProbabilityTheory
theorem hasFiniteIntegral_prod_mk_left (a : α) {s : Set (β × γ)} (h2s : (κ ⊗ₖ η) a s ≠ ∞) :
HasFiniteIntegral (fun b => (η (a, b) (Prod.mk b ⁻¹' s)).toReal) (κ a) := by
let t := toMeasurable ((κ ⊗ₖ η) a) s
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal toReal_nonneg]
calc
∫⁻ b, ENNReal.ofReal (η (a, b) (Prod.mk b ⁻¹' s)).toReal ∂κ a
_ ≤ ∫⁻ b, η (a, b) (Prod.mk b ⁻¹' t) ∂κ a := by
refine lintegral_mono_ae ?_
filter_upwards [ae_kernel_lt_top a h2s] with b hb
rw [ofReal_toReal hb.ne]
exact measure_mono (preimage_mono (subset_toMeasurable _ _))
_ ≤ (κ ⊗ₖ η) a t := le_compProd_apply _ _ _ _
_ = (κ ⊗ₖ η) a s := measure_toMeasurable s
_ < ⊤ := h2s.lt_top
#align probability_theory.has_finite_integral_prod_mk_left ProbabilityTheory.hasFiniteIntegral_prod_mk_left
theorem integrable_kernel_prod_mk_left (a : α) {s : Set (β × γ)} (hs : MeasurableSet s)
(h2s : (κ ⊗ₖ η) a s ≠ ∞) : Integrable (fun b => (η (a, b) (Prod.mk b ⁻¹' s)).toReal) (κ a) := by
constructor
· exact (measurable_kernel_prod_mk_left' hs a).ennreal_toReal.aestronglyMeasurable
· exact hasFiniteIntegral_prod_mk_left a h2s
#align probability_theory.integrable_kernel_prod_mk_left ProbabilityTheory.integrable_kernel_prod_mk_left
theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_kernel_compProd [NormedSpace ℝ E]
⦃f : β × γ → E⦄ (hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
AEStronglyMeasurable (fun x => ∫ y, f (x, y) ∂η (a, x)) (κ a) :=
⟨fun x => ∫ y, hf.mk f (x, y) ∂η (a, x), hf.stronglyMeasurable_mk.integral_kernel_prod_right'', by
filter_upwards [ae_ae_of_ae_compProd hf.ae_eq_mk] with _ hx using integral_congr_ae hx⟩
#align measure_theory.ae_strongly_measurable.integral_kernel_comp_prod MeasureTheory.AEStronglyMeasurable.integral_kernel_compProd
theorem _root_.MeasureTheory.AEStronglyMeasurable.compProd_mk_left {δ : Type*} [TopologicalSpace δ]
{f : β × γ → δ} (hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
∀ᵐ x ∂κ a, AEStronglyMeasurable (fun y => f (x, y)) (η (a, x)) := by
filter_upwards [ae_ae_of_ae_compProd hf.ae_eq_mk] with x hx using
⟨fun y => hf.mk f (x, y), hf.stronglyMeasurable_mk.comp_measurable measurable_prod_mk_left, hx⟩
#align measure_theory.ae_strongly_measurable.comp_prod_mk_left MeasureTheory.AEStronglyMeasurable.compProd_mk_left
theorem hasFiniteIntegral_compProd_iff ⦃f : β × γ → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
(∀ᵐ x ∂κ a, HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) := by
simp only [HasFiniteIntegral]
rw [kernel.lintegral_compProd _ _ _ h1f.ennnorm]
have : ∀ x, ∀ᵐ y ∂η (a, x), 0 ≤ ‖f (x, y)‖ := fun x => eventually_of_forall fun y => norm_nonneg _
simp_rw [integral_eq_lintegral_of_nonneg_ae (this _)
(h1f.norm.comp_measurable measurable_prod_mk_left).aestronglyMeasurable,
ennnorm_eq_ofReal toReal_nonneg, ofReal_norm_eq_coe_nnnorm]
have : ∀ {p q r : Prop} (_ : r → p), (r ↔ p ∧ q) ↔ p → (r ↔ q) := fun {p q r} h1 => by
rw [← and_congr_right_iff, and_iff_right_of_imp h1]
rw [this]
· intro h2f; rw [lintegral_congr_ae]
filter_upwards [h2f] with x hx
rw [ofReal_toReal]; rw [← lt_top_iff_ne_top]; exact hx
· intro h2f; refine ae_lt_top ?_ h2f.ne; exact h1f.ennnorm.lintegral_kernel_prod_right''
#align probability_theory.has_finite_integral_comp_prod_iff ProbabilityTheory.hasFiniteIntegral_compProd_iff
theorem hasFiniteIntegral_compProd_iff' ⦃f : β × γ → E⦄
(h1f : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
(∀ᵐ x ∂κ a, HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) := by
rw [hasFiniteIntegral_congr h1f.ae_eq_mk,
hasFiniteIntegral_compProd_iff h1f.stronglyMeasurable_mk]
apply and_congr
· apply eventually_congr
filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with x hx using
hasFiniteIntegral_congr hx
· apply hasFiniteIntegral_congr
filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with _ hx using
integral_congr_ae (EventuallyEq.fun_comp hx _)
#align probability_theory.has_finite_integral_comp_prod_iff' ProbabilityTheory.hasFiniteIntegral_compProd_iff'
| Mathlib/Probability/Kernel/IntegralCompProd.lean | 123 | 128 | theorem integrable_compProd_iff ⦃f : β × γ → E⦄ (hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
Integrable f ((κ ⊗ₖ η) a) ↔
(∀ᵐ x ∂κ a, Integrable (fun y => f (x, y)) (η (a, x))) ∧
Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) := by |
simp only [Integrable, hasFiniteIntegral_compProd_iff' hf, hf.norm.integral_kernel_compProd,
hf, hf.compProd_mk_left, eventually_and, true_and_iff]
|
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.ordinal.principal from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
universe u v w
noncomputable section
open Order
namespace Ordinal
-- Porting note: commented out, doesn't seem necessary
--local infixr:0 "^" => @pow Ordinal Ordinal Ordinal.hasPow
def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=
∀ ⦃a b⦄, a < o → b < o → op a b < o
#align ordinal.principal Ordinal.Principal
| Mathlib/SetTheory/Ordinal/Principal.lean | 52 | 54 | theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :
Principal op o ↔ Principal (Function.swap op) o := by |
constructor <;> exact fun h a b ha hb => h hb ha
|
import Mathlib.SetTheory.Game.Basic
import Mathlib.Tactic.NthRewrite
#align_import set_theory.game.impartial from "leanprover-community/mathlib"@"2e0975f6a25dd3fbfb9e41556a77f075f6269748"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def ImpartialAux : PGame → Prop
| G => (G ≈ -G) ∧ (∀ i, ImpartialAux (G.moveLeft i)) ∧ ∀ j, ImpartialAux (G.moveRight j)
termination_by G => G -- Porting note: Added `termination_by`
#align pgame.impartial_aux SetTheory.PGame.ImpartialAux
theorem impartialAux_def {G : PGame} :
G.ImpartialAux ↔
(G ≈ -G) ∧ (∀ i, ImpartialAux (G.moveLeft i)) ∧ ∀ j, ImpartialAux (G.moveRight j) := by
rw [ImpartialAux]
#align pgame.impartial_aux_def SetTheory.PGame.impartialAux_def
class Impartial (G : PGame) : Prop where
out : ImpartialAux G
#align pgame.impartial SetTheory.PGame.Impartial
theorem impartial_iff_aux {G : PGame} : G.Impartial ↔ G.ImpartialAux :=
⟨fun h => h.1, fun h => ⟨h⟩⟩
#align pgame.impartial_iff_aux SetTheory.PGame.impartial_iff_aux
theorem impartial_def {G : PGame} :
G.Impartial ↔ (G ≈ -G) ∧ (∀ i, Impartial (G.moveLeft i)) ∧ ∀ j, Impartial (G.moveRight j) := by
simpa only [impartial_iff_aux] using impartialAux_def
#align pgame.impartial_def SetTheory.PGame.impartial_def
namespace Impartial
instance impartial_zero : Impartial 0 := by rw [impartial_def]; dsimp; simp
#align pgame.impartial.impartial_zero SetTheory.PGame.Impartial.impartial_zero
instance impartial_star : Impartial star := by
rw [impartial_def]; simpa using Impartial.impartial_zero
#align pgame.impartial.impartial_star SetTheory.PGame.Impartial.impartial_star
theorem neg_equiv_self (G : PGame) [h : G.Impartial] : G ≈ -G :=
(impartial_def.1 h).1
#align pgame.impartial.neg_equiv_self SetTheory.PGame.Impartial.neg_equiv_self
-- Porting note: Changed `-⟦G⟧` to `-(⟦G⟧ : Quotient setoid)`
@[simp]
theorem mk'_neg_equiv_self (G : PGame) [G.Impartial] : -(⟦G⟧ : Quotient setoid) = ⟦G⟧ :=
Quot.sound (Equiv.symm (neg_equiv_self G))
#align pgame.impartial.mk_neg_equiv_self SetTheory.PGame.Impartial.mk'_neg_equiv_self
instance moveLeft_impartial {G : PGame} [h : G.Impartial] (i : G.LeftMoves) :
(G.moveLeft i).Impartial :=
(impartial_def.1 h).2.1 i
#align pgame.impartial.move_left_impartial SetTheory.PGame.Impartial.moveLeft_impartial
instance moveRight_impartial {G : PGame} [h : G.Impartial] (j : G.RightMoves) :
(G.moveRight j).Impartial :=
(impartial_def.1 h).2.2 j
#align pgame.impartial.move_right_impartial SetTheory.PGame.Impartial.moveRight_impartial
theorem impartial_congr : ∀ {G H : PGame} (_ : G ≡r H) [G.Impartial], H.Impartial
| G, H => fun e => by
intro h
exact impartial_def.2
⟨Equiv.trans e.symm.equiv (Equiv.trans (neg_equiv_self G) (neg_equiv_neg_iff.2 e.equiv)),
fun i => impartial_congr (e.moveLeftSymm i), fun j => impartial_congr (e.moveRightSymm j)⟩
termination_by G H => (G, H)
#align pgame.impartial.impartial_congr SetTheory.PGame.Impartial.impartial_congr
instance impartial_add : ∀ (G H : PGame) [G.Impartial] [H.Impartial], (G + H).Impartial
| G, H, _, _ => by
rw [impartial_def]
refine ⟨Equiv.trans (add_congr (neg_equiv_self G) (neg_equiv_self _))
(Equiv.symm (negAddRelabelling _ _).equiv), fun k => ?_, fun k => ?_⟩
· apply leftMoves_add_cases k
all_goals
intro i; simp only [add_moveLeft_inl, add_moveLeft_inr]
apply impartial_add
· apply rightMoves_add_cases k
all_goals
intro i; simp only [add_moveRight_inl, add_moveRight_inr]
apply impartial_add
termination_by G H => (G, H)
#align pgame.impartial.impartial_add SetTheory.PGame.Impartial.impartial_add
instance impartial_neg : ∀ (G : PGame) [G.Impartial], (-G).Impartial
| G, _ => by
rw [impartial_def]
refine ⟨?_, fun i => ?_, fun i => ?_⟩
· rw [neg_neg]
exact Equiv.symm (neg_equiv_self G)
· rw [moveLeft_neg']
apply impartial_neg
· rw [moveRight_neg']
apply impartial_neg
termination_by G => G
#align pgame.impartial.impartial_neg SetTheory.PGame.Impartial.impartial_neg
variable (G : PGame) [Impartial G]
theorem nonpos : ¬0 < G := fun h => by
have h' := neg_lt_neg_iff.2 h
rw [neg_zero, lt_congr_left (Equiv.symm (neg_equiv_self G))] at h'
exact (h.trans h').false
#align pgame.impartial.nonpos SetTheory.PGame.Impartial.nonpos
theorem nonneg : ¬G < 0 := fun h => by
have h' := neg_lt_neg_iff.2 h
rw [neg_zero, lt_congr_right (Equiv.symm (neg_equiv_self G))] at h'
exact (h.trans h').false
#align pgame.impartial.nonneg SetTheory.PGame.Impartial.nonneg
theorem equiv_or_fuzzy_zero : (G ≈ 0) ∨ G ‖ 0 := by
rcases lt_or_equiv_or_gt_or_fuzzy G 0 with (h | h | h | h)
· exact ((nonneg G) h).elim
· exact Or.inl h
· exact ((nonpos G) h).elim
· exact Or.inr h
#align pgame.impartial.equiv_or_fuzzy_zero SetTheory.PGame.Impartial.equiv_or_fuzzy_zero
@[simp]
theorem not_equiv_zero_iff : ¬(G ≈ 0) ↔ G ‖ 0 :=
⟨(equiv_or_fuzzy_zero G).resolve_left, Fuzzy.not_equiv⟩
#align pgame.impartial.not_equiv_zero_iff SetTheory.PGame.Impartial.not_equiv_zero_iff
@[simp]
theorem not_fuzzy_zero_iff : ¬G ‖ 0 ↔ (G ≈ 0) :=
⟨(equiv_or_fuzzy_zero G).resolve_right, Equiv.not_fuzzy⟩
#align pgame.impartial.not_fuzzy_zero_iff SetTheory.PGame.Impartial.not_fuzzy_zero_iff
theorem add_self : G + G ≈ 0 :=
Equiv.trans (add_congr_left (neg_equiv_self G)) (add_left_neg_equiv G)
#align pgame.impartial.add_self SetTheory.PGame.Impartial.add_self
-- Porting note: Changed `⟦G⟧` to `(⟦G⟧ : Quotient setoid)`
@[simp]
theorem mk'_add_self : (⟦G⟧ : Quotient setoid) + ⟦G⟧ = 0 :=
Quot.sound (add_self G)
#align pgame.impartial.mk_add_self SetTheory.PGame.Impartial.mk'_add_self
theorem equiv_iff_add_equiv_zero (H : PGame) : (H ≈ G) ↔ (H + G ≈ 0) := by
rw [Game.PGame.equiv_iff_game_eq, ← @add_right_cancel_iff _ _ _ ⟦G⟧, mk'_add_self, ← quot_add,
Game.PGame.equiv_iff_game_eq]
rfl
#align pgame.impartial.equiv_iff_add_equiv_zero SetTheory.PGame.Impartial.equiv_iff_add_equiv_zero
theorem equiv_iff_add_equiv_zero' (H : PGame) : (G ≈ H) ↔ (G + H ≈ 0) := by
rw [Game.PGame.equiv_iff_game_eq, ← @add_left_cancel_iff _ _ _ ⟦G⟧, mk'_add_self, ← quot_add,
Game.PGame.equiv_iff_game_eq]
exact ⟨Eq.symm, Eq.symm⟩
#align pgame.impartial.equiv_iff_add_equiv_zero' SetTheory.PGame.Impartial.equiv_iff_add_equiv_zero'
theorem le_zero_iff {G : PGame} [G.Impartial] : G ≤ 0 ↔ 0 ≤ G := by
rw [← zero_le_neg_iff, le_congr_right (neg_equiv_self G)]
#align pgame.impartial.le_zero_iff SetTheory.PGame.Impartial.le_zero_iff
theorem lf_zero_iff {G : PGame} [G.Impartial] : G ⧏ 0 ↔ 0 ⧏ G := by
rw [← zero_lf_neg_iff, lf_congr_right (neg_equiv_self G)]
#align pgame.impartial.lf_zero_iff SetTheory.PGame.Impartial.lf_zero_iff
theorem equiv_zero_iff_le : (G ≈ 0) ↔ G ≤ 0 :=
⟨And.left, fun h => ⟨h, le_zero_iff.1 h⟩⟩
#align pgame.impartial.equiv_zero_iff_le SetTheory.PGame.Impartial.equiv_zero_iff_le
theorem fuzzy_zero_iff_lf : G ‖ 0 ↔ G ⧏ 0 :=
⟨And.left, fun h => ⟨h, lf_zero_iff.1 h⟩⟩
#align pgame.impartial.fuzzy_zero_iff_lf SetTheory.PGame.Impartial.fuzzy_zero_iff_lf
theorem equiv_zero_iff_ge : (G ≈ 0) ↔ 0 ≤ G :=
⟨And.right, fun h => ⟨le_zero_iff.2 h, h⟩⟩
#align pgame.impartial.equiv_zero_iff_ge SetTheory.PGame.Impartial.equiv_zero_iff_ge
theorem fuzzy_zero_iff_gf : G ‖ 0 ↔ 0 ⧏ G :=
⟨And.right, fun h => ⟨lf_zero_iff.2 h, h⟩⟩
#align pgame.impartial.fuzzy_zero_iff_gf SetTheory.PGame.Impartial.fuzzy_zero_iff_gf
theorem forall_leftMoves_fuzzy_iff_equiv_zero : (∀ i, G.moveLeft i ‖ 0) ↔ (G ≈ 0) := by
refine ⟨fun hb => ?_, fun hp i => ?_⟩
· rw [equiv_zero_iff_le G, le_zero_lf]
exact fun i => (hb i).1
· rw [fuzzy_zero_iff_lf]
exact hp.1.moveLeft_lf i
#align pgame.impartial.forall_left_moves_fuzzy_iff_equiv_zero SetTheory.PGame.Impartial.forall_leftMoves_fuzzy_iff_equiv_zero
theorem forall_rightMoves_fuzzy_iff_equiv_zero : (∀ j, G.moveRight j ‖ 0) ↔ (G ≈ 0) := by
refine ⟨fun hb => ?_, fun hp i => ?_⟩
· rw [equiv_zero_iff_ge G, zero_le_lf]
exact fun i => (hb i).2
· rw [fuzzy_zero_iff_gf]
exact hp.2.lf_moveRight i
#align pgame.impartial.forall_right_moves_fuzzy_iff_equiv_zero SetTheory.PGame.Impartial.forall_rightMoves_fuzzy_iff_equiv_zero
theorem exists_left_move_equiv_iff_fuzzy_zero : (∃ i, G.moveLeft i ≈ 0) ↔ G ‖ 0 := by
refine ⟨fun ⟨i, hi⟩ => (fuzzy_zero_iff_gf G).2 (lf_of_le_moveLeft hi.2), fun hn => ?_⟩
rw [fuzzy_zero_iff_gf G, zero_lf_le] at hn
cases' hn with i hi
exact ⟨i, (equiv_zero_iff_ge _).2 hi⟩
#align pgame.impartial.exists_left_move_equiv_iff_fuzzy_zero SetTheory.PGame.Impartial.exists_left_move_equiv_iff_fuzzy_zero
| Mathlib/SetTheory/Game/Impartial.lean | 226 | 230 | theorem exists_right_move_equiv_iff_fuzzy_zero : (∃ j, G.moveRight j ≈ 0) ↔ G ‖ 0 := by |
refine ⟨fun ⟨i, hi⟩ => (fuzzy_zero_iff_lf G).2 (lf_of_moveRight_le hi.1), fun hn => ?_⟩
rw [fuzzy_zero_iff_lf G, lf_zero_le] at hn
cases' hn with i hi
exact ⟨i, (equiv_zero_iff_le _).2 hi⟩
|
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integral.partition.basic from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set Finset Function
open scoped Classical
open NNReal
noncomputable section
namespace BoxIntegral
variable {ι : Type*}
structure Prepartition (I : Box ι) where
boxes : Finset (Box ι)
le_of_mem' : ∀ J ∈ boxes, J ≤ I
pairwiseDisjoint : Set.Pairwise (↑boxes) (Disjoint on ((↑) : Box ι → Set (ι → ℝ)))
#align box_integral.prepartition BoxIntegral.Prepartition
namespace Prepartition
variable {I J J₁ J₂ : Box ι} (π : Prepartition I) {π₁ π₂ : Prepartition I} {x : ι → ℝ}
instance : Membership (Box ι) (Prepartition I) :=
⟨fun J π => J ∈ π.boxes⟩
@[simp]
theorem mem_boxes : J ∈ π.boxes ↔ J ∈ π := Iff.rfl
#align box_integral.prepartition.mem_boxes BoxIntegral.Prepartition.mem_boxes
@[simp]
theorem mem_mk {s h₁ h₂} : J ∈ (mk s h₁ h₂ : Prepartition I) ↔ J ∈ s := Iff.rfl
#align box_integral.prepartition.mem_mk BoxIntegral.Prepartition.mem_mk
theorem disjoint_coe_of_mem (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (h : J₁ ≠ J₂) :
Disjoint (J₁ : Set (ι → ℝ)) J₂ :=
π.pairwiseDisjoint h₁ h₂ h
#align box_integral.prepartition.disjoint_coe_of_mem BoxIntegral.Prepartition.disjoint_coe_of_mem
theorem eq_of_mem_of_mem (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hx₁ : x ∈ J₁) (hx₂ : x ∈ J₂) : J₁ = J₂ :=
by_contra fun H => (π.disjoint_coe_of_mem h₁ h₂ H).le_bot ⟨hx₁, hx₂⟩
#align box_integral.prepartition.eq_of_mem_of_mem BoxIntegral.Prepartition.eq_of_mem_of_mem
theorem eq_of_le_of_le (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hle₁ : J ≤ J₁) (hle₂ : J ≤ J₂) : J₁ = J₂ :=
π.eq_of_mem_of_mem h₁ h₂ (hle₁ J.upper_mem) (hle₂ J.upper_mem)
#align box_integral.prepartition.eq_of_le_of_le BoxIntegral.Prepartition.eq_of_le_of_le
theorem eq_of_le (h₁ : J₁ ∈ π) (h₂ : J₂ ∈ π) (hle : J₁ ≤ J₂) : J₁ = J₂ :=
π.eq_of_le_of_le h₁ h₂ le_rfl hle
#align box_integral.prepartition.eq_of_le BoxIntegral.Prepartition.eq_of_le
theorem le_of_mem (hJ : J ∈ π) : J ≤ I :=
π.le_of_mem' J hJ
#align box_integral.prepartition.le_of_mem BoxIntegral.Prepartition.le_of_mem
theorem lower_le_lower (hJ : J ∈ π) : I.lower ≤ J.lower :=
Box.antitone_lower (π.le_of_mem hJ)
#align box_integral.prepartition.lower_le_lower BoxIntegral.Prepartition.lower_le_lower
theorem upper_le_upper (hJ : J ∈ π) : J.upper ≤ I.upper :=
Box.monotone_upper (π.le_of_mem hJ)
#align box_integral.prepartition.upper_le_upper BoxIntegral.Prepartition.upper_le_upper
theorem injective_boxes : Function.Injective (boxes : Prepartition I → Finset (Box ι)) := by
rintro ⟨s₁, h₁, h₁'⟩ ⟨s₂, h₂, h₂'⟩ (rfl : s₁ = s₂)
rfl
#align box_integral.prepartition.injective_boxes BoxIntegral.Prepartition.injective_boxes
@[ext]
theorem ext (h : ∀ J, J ∈ π₁ ↔ J ∈ π₂) : π₁ = π₂ :=
injective_boxes <| Finset.ext h
#align box_integral.prepartition.ext BoxIntegral.Prepartition.ext
@[simps]
def single (I J : Box ι) (h : J ≤ I) : Prepartition I :=
⟨{J}, by simpa, by simp⟩
#align box_integral.prepartition.single BoxIntegral.Prepartition.single
@[simp]
theorem mem_single {J'} (h : J ≤ I) : J' ∈ single I J h ↔ J' = J :=
mem_singleton
#align box_integral.prepartition.mem_single BoxIntegral.Prepartition.mem_single
instance : LE (Prepartition I) :=
⟨fun π π' => ∀ ⦃I⦄, I ∈ π → ∃ I' ∈ π', I ≤ I'⟩
instance partialOrder : PartialOrder (Prepartition I) where
le := (· ≤ ·)
le_refl π I hI := ⟨I, hI, le_rfl⟩
le_trans π₁ π₂ π₃ h₁₂ h₂₃ I₁ hI₁ :=
let ⟨I₂, hI₂, hI₁₂⟩ := h₁₂ hI₁
let ⟨I₃, hI₃, hI₂₃⟩ := h₂₃ hI₂
⟨I₃, hI₃, hI₁₂.trans hI₂₃⟩
le_antisymm := by
suffices ∀ {π₁ π₂ : Prepartition I}, π₁ ≤ π₂ → π₂ ≤ π₁ → π₁.boxes ⊆ π₂.boxes from
fun π₁ π₂ h₁ h₂ => injective_boxes (Subset.antisymm (this h₁ h₂) (this h₂ h₁))
intro π₁ π₂ h₁ h₂ J hJ
rcases h₁ hJ with ⟨J', hJ', hle⟩; rcases h₂ hJ' with ⟨J'', hJ'', hle'⟩
obtain rfl : J = J'' := π₁.eq_of_le hJ hJ'' (hle.trans hle')
obtain rfl : J' = J := le_antisymm ‹_› ‹_›
assumption
instance : OrderTop (Prepartition I) where
top := single I I le_rfl
le_top π J hJ := ⟨I, by simp, π.le_of_mem hJ⟩
instance : OrderBot (Prepartition I) where
bot := ⟨∅,
fun _ hJ => (Finset.not_mem_empty _ hJ).elim,
fun _ hJ => (Set.not_mem_empty _ <| Finset.coe_empty ▸ hJ).elim⟩
bot_le _ _ hJ := (Finset.not_mem_empty _ hJ).elim
instance : Inhabited (Prepartition I) := ⟨⊤⟩
theorem le_def : π₁ ≤ π₂ ↔ ∀ J ∈ π₁, ∃ J' ∈ π₂, J ≤ J' := Iff.rfl
#align box_integral.prepartition.le_def BoxIntegral.Prepartition.le_def
@[simp]
theorem mem_top : J ∈ (⊤ : Prepartition I) ↔ J = I :=
mem_singleton
#align box_integral.prepartition.mem_top BoxIntegral.Prepartition.mem_top
@[simp]
theorem top_boxes : (⊤ : Prepartition I).boxes = {I} := rfl
#align box_integral.prepartition.top_boxes BoxIntegral.Prepartition.top_boxes
@[simp]
theorem not_mem_bot : J ∉ (⊥ : Prepartition I) :=
Finset.not_mem_empty _
#align box_integral.prepartition.not_mem_bot BoxIntegral.Prepartition.not_mem_bot
@[simp]
theorem bot_boxes : (⊥ : Prepartition I).boxes = ∅ := rfl
#align box_integral.prepartition.bot_boxes BoxIntegral.Prepartition.bot_boxes
theorem injOn_setOf_mem_Icc_setOf_lower_eq (x : ι → ℝ) :
InjOn (fun J : Box ι => { i | J.lower i = x i }) { J | J ∈ π ∧ x ∈ Box.Icc J } := by
rintro J₁ ⟨h₁, hx₁⟩ J₂ ⟨h₂, hx₂⟩ (H : { i | J₁.lower i = x i } = { i | J₂.lower i = x i })
suffices ∀ i, (Ioc (J₁.lower i) (J₁.upper i) ∩ Ioc (J₂.lower i) (J₂.upper i)).Nonempty by
choose y hy₁ hy₂ using this
exact π.eq_of_mem_of_mem h₁ h₂ hy₁ hy₂
intro i
simp only [Set.ext_iff, mem_setOf] at H
rcases (hx₁.1 i).eq_or_lt with hi₁ | hi₁
· have hi₂ : J₂.lower i = x i := (H _).1 hi₁
have H₁ : x i < J₁.upper i := by simpa only [hi₁] using J₁.lower_lt_upper i
have H₂ : x i < J₂.upper i := by simpa only [hi₂] using J₂.lower_lt_upper i
rw [Ioc_inter_Ioc, hi₁, hi₂, sup_idem, Set.nonempty_Ioc]
exact lt_min H₁ H₂
· have hi₂ : J₂.lower i < x i := (hx₂.1 i).lt_of_ne (mt (H _).2 hi₁.ne)
exact ⟨x i, ⟨hi₁, hx₁.2 i⟩, ⟨hi₂, hx₂.2 i⟩⟩
#align box_integral.prepartition.inj_on_set_of_mem_Icc_set_of_lower_eq BoxIntegral.Prepartition.injOn_setOf_mem_Icc_setOf_lower_eq
theorem card_filter_mem_Icc_le [Fintype ι] (x : ι → ℝ) :
(π.boxes.filter fun J : Box ι => x ∈ Box.Icc J).card ≤ 2 ^ Fintype.card ι := by
rw [← Fintype.card_set]
refine Finset.card_le_card_of_inj_on (fun J : Box ι => { i | J.lower i = x i })
(fun _ _ => Finset.mem_univ _) ?_
simpa only [Finset.mem_filter] using π.injOn_setOf_mem_Icc_setOf_lower_eq x
#align box_integral.prepartition.card_filter_mem_Icc_le BoxIntegral.Prepartition.card_filter_mem_Icc_le
protected def iUnion : Set (ι → ℝ) :=
⋃ J ∈ π, ↑J
#align box_integral.prepartition.Union BoxIntegral.Prepartition.iUnion
theorem iUnion_def : π.iUnion = ⋃ J ∈ π, ↑J := rfl
#align box_integral.prepartition.Union_def BoxIntegral.Prepartition.iUnion_def
theorem iUnion_def' : π.iUnion = ⋃ J ∈ π.boxes, ↑J := rfl
#align box_integral.prepartition.Union_def' BoxIntegral.Prepartition.iUnion_def'
-- Porting note: Previous proof was `:= Set.mem_iUnion₂`
@[simp]
theorem mem_iUnion : x ∈ π.iUnion ↔ ∃ J ∈ π, x ∈ J := by
convert Set.mem_iUnion₂
rw [Box.mem_coe, exists_prop]
#align box_integral.prepartition.mem_Union BoxIntegral.Prepartition.mem_iUnion
@[simp]
theorem iUnion_single (h : J ≤ I) : (single I J h).iUnion = J := by simp [iUnion_def]
#align box_integral.prepartition.Union_single BoxIntegral.Prepartition.iUnion_single
@[simp]
theorem iUnion_top : (⊤ : Prepartition I).iUnion = I := by simp [Prepartition.iUnion]
#align box_integral.prepartition.Union_top BoxIntegral.Prepartition.iUnion_top
@[simp]
theorem iUnion_eq_empty : π₁.iUnion = ∅ ↔ π₁ = ⊥ := by
simp [← injective_boxes.eq_iff, Finset.ext_iff, Prepartition.iUnion, imp_false]
#align box_integral.prepartition.Union_eq_empty BoxIntegral.Prepartition.iUnion_eq_empty
@[simp]
theorem iUnion_bot : (⊥ : Prepartition I).iUnion = ∅ :=
iUnion_eq_empty.2 rfl
#align box_integral.prepartition.Union_bot BoxIntegral.Prepartition.iUnion_bot
theorem subset_iUnion (h : J ∈ π) : ↑J ⊆ π.iUnion :=
subset_biUnion_of_mem h
#align box_integral.prepartition.subset_Union BoxIntegral.Prepartition.subset_iUnion
theorem iUnion_subset : π.iUnion ⊆ I :=
iUnion₂_subset π.le_of_mem'
#align box_integral.prepartition.Union_subset BoxIntegral.Prepartition.iUnion_subset
@[mono]
theorem iUnion_mono (h : π₁ ≤ π₂) : π₁.iUnion ⊆ π₂.iUnion := fun _ hx =>
let ⟨_, hJ₁, hx⟩ := π₁.mem_iUnion.1 hx
let ⟨J₂, hJ₂, hle⟩ := h hJ₁
π₂.mem_iUnion.2 ⟨J₂, hJ₂, hle hx⟩
#align box_integral.prepartition.Union_mono BoxIntegral.Prepartition.iUnion_mono
theorem disjoint_boxes_of_disjoint_iUnion (h : Disjoint π₁.iUnion π₂.iUnion) :
Disjoint π₁.boxes π₂.boxes :=
Finset.disjoint_left.2 fun J h₁ h₂ =>
Disjoint.le_bot (h.mono (π₁.subset_iUnion h₁) (π₂.subset_iUnion h₂)) ⟨J.upper_mem, J.upper_mem⟩
#align box_integral.prepartition.disjoint_boxes_of_disjoint_Union BoxIntegral.Prepartition.disjoint_boxes_of_disjoint_iUnion
theorem le_iff_nonempty_imp_le_and_iUnion_subset :
π₁ ≤ π₂ ↔
(∀ J ∈ π₁, ∀ J' ∈ π₂, (J ∩ J' : Set (ι → ℝ)).Nonempty → J ≤ J') ∧ π₁.iUnion ⊆ π₂.iUnion := by
constructor
· refine fun H => ⟨fun J hJ J' hJ' Hne => ?_, iUnion_mono H⟩
rcases H hJ with ⟨J'', hJ'', Hle⟩
rcases Hne with ⟨x, hx, hx'⟩
rwa [π₂.eq_of_mem_of_mem hJ' hJ'' hx' (Hle hx)]
· rintro ⟨H, HU⟩ J hJ
simp only [Set.subset_def, mem_iUnion] at HU
rcases HU J.upper ⟨J, hJ, J.upper_mem⟩ with ⟨J₂, hJ₂, hx⟩
exact ⟨J₂, hJ₂, H _ hJ _ hJ₂ ⟨_, J.upper_mem, hx⟩⟩
#align box_integral.prepartition.le_iff_nonempty_imp_le_and_Union_subset BoxIntegral.Prepartition.le_iff_nonempty_imp_le_and_iUnion_subset
theorem eq_of_boxes_subset_iUnion_superset (h₁ : π₁.boxes ⊆ π₂.boxes) (h₂ : π₂.iUnion ⊆ π₁.iUnion) :
π₁ = π₂ :=
le_antisymm (fun J hJ => ⟨J, h₁ hJ, le_rfl⟩) <|
le_iff_nonempty_imp_le_and_iUnion_subset.2
⟨fun _ hJ₁ _ hJ₂ Hne =>
(π₂.eq_of_mem_of_mem hJ₁ (h₁ hJ₂) Hne.choose_spec.1 Hne.choose_spec.2).le, h₂⟩
#align box_integral.prepartition.eq_of_boxes_subset_Union_superset BoxIntegral.Prepartition.eq_of_boxes_subset_iUnion_superset
@[simps]
def biUnion (πi : ∀ J : Box ι, Prepartition J) : Prepartition I where
boxes := π.boxes.biUnion fun J => (πi J).boxes
le_of_mem' J hJ := by
simp only [Finset.mem_biUnion, exists_prop, mem_boxes] at hJ
rcases hJ with ⟨J', hJ', hJ⟩
exact ((πi J').le_of_mem hJ).trans (π.le_of_mem hJ')
pairwiseDisjoint := by
simp only [Set.Pairwise, Finset.mem_coe, Finset.mem_biUnion]
rintro J₁' ⟨J₁, hJ₁, hJ₁'⟩ J₂' ⟨J₂, hJ₂, hJ₂'⟩ Hne
rw [Function.onFun, Set.disjoint_left]
rintro x hx₁ hx₂; apply Hne
obtain rfl : J₁ = J₂ :=
π.eq_of_mem_of_mem hJ₁ hJ₂ ((πi J₁).le_of_mem hJ₁' hx₁) ((πi J₂).le_of_mem hJ₂' hx₂)
exact (πi J₁).eq_of_mem_of_mem hJ₁' hJ₂' hx₁ hx₂
#align box_integral.prepartition.bUnion BoxIntegral.Prepartition.biUnion
variable {πi πi₁ πi₂ : ∀ J : Box ι, Prepartition J}
@[simp]
theorem mem_biUnion : J ∈ π.biUnion πi ↔ ∃ J' ∈ π, J ∈ πi J' := by simp [biUnion]
#align box_integral.prepartition.mem_bUnion BoxIntegral.Prepartition.mem_biUnion
theorem biUnion_le (πi : ∀ J, Prepartition J) : π.biUnion πi ≤ π := fun _ hJ =>
let ⟨J', hJ', hJ⟩ := π.mem_biUnion.1 hJ
⟨J', hJ', (πi J').le_of_mem hJ⟩
#align box_integral.prepartition.bUnion_le BoxIntegral.Prepartition.biUnion_le
@[simp]
theorem biUnion_top : (π.biUnion fun _ => ⊤) = π := by
ext
simp
#align box_integral.prepartition.bUnion_top BoxIntegral.Prepartition.biUnion_top
@[congr]
theorem biUnion_congr (h : π₁ = π₂) (hi : ∀ J ∈ π₁, πi₁ J = πi₂ J) :
π₁.biUnion πi₁ = π₂.biUnion πi₂ := by
subst π₂
ext J
simp only [mem_biUnion]
constructor <;> exact fun ⟨J', h₁, h₂⟩ => ⟨J', h₁, hi J' h₁ ▸ h₂⟩
#align box_integral.prepartition.bUnion_congr BoxIntegral.Prepartition.biUnion_congr
theorem biUnion_congr_of_le (h : π₁ = π₂) (hi : ∀ J ≤ I, πi₁ J = πi₂ J) :
π₁.biUnion πi₁ = π₂.biUnion πi₂ :=
biUnion_congr h fun J hJ => hi J (π₁.le_of_mem hJ)
#align box_integral.prepartition.bUnion_congr_of_le BoxIntegral.Prepartition.biUnion_congr_of_le
@[simp]
theorem iUnion_biUnion (πi : ∀ J : Box ι, Prepartition J) :
(π.biUnion πi).iUnion = ⋃ J ∈ π, (πi J).iUnion := by simp [Prepartition.iUnion]
#align box_integral.prepartition.Union_bUnion BoxIntegral.Prepartition.iUnion_biUnion
@[simp]
theorem sum_biUnion_boxes {M : Type*} [AddCommMonoid M] (π : Prepartition I)
(πi : ∀ J, Prepartition J) (f : Box ι → M) :
(∑ J ∈ π.boxes.biUnion fun J => (πi J).boxes, f J) =
∑ J ∈ π.boxes, ∑ J' ∈ (πi J).boxes, f J' := by
refine Finset.sum_biUnion fun J₁ h₁ J₂ h₂ hne => Finset.disjoint_left.2 fun J' h₁' h₂' => ?_
exact hne (π.eq_of_le_of_le h₁ h₂ ((πi J₁).le_of_mem h₁') ((πi J₂).le_of_mem h₂'))
#align box_integral.prepartition.sum_bUnion_boxes BoxIntegral.Prepartition.sum_biUnion_boxes
def biUnionIndex (πi : ∀ (J : Box ι), Prepartition J) (J : Box ι) : Box ι :=
if hJ : J ∈ π.biUnion πi then (π.mem_biUnion.1 hJ).choose else I
#align box_integral.prepartition.bUnion_index BoxIntegral.Prepartition.biUnionIndex
theorem biUnionIndex_mem (hJ : J ∈ π.biUnion πi) : π.biUnionIndex πi J ∈ π := by
rw [biUnionIndex, dif_pos hJ]
exact (π.mem_biUnion.1 hJ).choose_spec.1
#align box_integral.prepartition.bUnion_index_mem BoxIntegral.Prepartition.biUnionIndex_mem
theorem biUnionIndex_le (πi : ∀ J, Prepartition J) (J : Box ι) : π.biUnionIndex πi J ≤ I := by
by_cases hJ : J ∈ π.biUnion πi
· exact π.le_of_mem (π.biUnionIndex_mem hJ)
· rw [biUnionIndex, dif_neg hJ]
#align box_integral.prepartition.bUnion_index_le BoxIntegral.Prepartition.biUnionIndex_le
theorem mem_biUnionIndex (hJ : J ∈ π.biUnion πi) : J ∈ πi (π.biUnionIndex πi J) := by
convert (π.mem_biUnion.1 hJ).choose_spec.2 <;> exact dif_pos hJ
#align box_integral.prepartition.mem_bUnion_index BoxIntegral.Prepartition.mem_biUnionIndex
theorem le_biUnionIndex (hJ : J ∈ π.biUnion πi) : J ≤ π.biUnionIndex πi J :=
le_of_mem _ (π.mem_biUnionIndex hJ)
#align box_integral.prepartition.le_bUnion_index BoxIntegral.Prepartition.le_biUnionIndex
theorem biUnionIndex_of_mem (hJ : J ∈ π) {J'} (hJ' : J' ∈ πi J) : π.biUnionIndex πi J' = J :=
have : J' ∈ π.biUnion πi := π.mem_biUnion.2 ⟨J, hJ, hJ'⟩
π.eq_of_le_of_le (π.biUnionIndex_mem this) hJ (π.le_biUnionIndex this) (le_of_mem _ hJ')
#align box_integral.prepartition.bUnion_index_of_mem BoxIntegral.Prepartition.biUnionIndex_of_mem
theorem biUnion_assoc (πi : ∀ J, Prepartition J) (πi' : Box ι → ∀ J : Box ι, Prepartition J) :
(π.biUnion fun J => (πi J).biUnion (πi' J)) =
(π.biUnion πi).biUnion fun J => πi' (π.biUnionIndex πi J) J := by
ext J
simp only [mem_biUnion, exists_prop]
constructor
· rintro ⟨J₁, hJ₁, J₂, hJ₂, hJ⟩
refine ⟨J₂, ⟨J₁, hJ₁, hJ₂⟩, ?_⟩
rwa [π.biUnionIndex_of_mem hJ₁ hJ₂]
· rintro ⟨J₁, ⟨J₂, hJ₂, hJ₁⟩, hJ⟩
refine ⟨J₂, hJ₂, J₁, hJ₁, ?_⟩
rwa [π.biUnionIndex_of_mem hJ₂ hJ₁] at hJ
#align box_integral.prepartition.bUnion_assoc BoxIntegral.Prepartition.biUnion_assoc
def ofWithBot (boxes : Finset (WithBot (Box ι)))
(le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I)
(pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint) :
Prepartition I where
boxes := Finset.eraseNone boxes
le_of_mem' J hJ := by
rw [mem_eraseNone] at hJ
simpa only [WithBot.some_eq_coe, WithBot.coe_le_coe] using le_of_mem _ hJ
pairwiseDisjoint J₁ h₁ J₂ h₂ hne := by
simp only [mem_coe, mem_eraseNone] at h₁ h₂
exact Box.disjoint_coe.1 (pairwise_disjoint h₁ h₂ (mt Option.some_inj.1 hne))
#align box_integral.prepartition.of_with_bot BoxIntegral.Prepartition.ofWithBot
@[simp]
theorem mem_ofWithBot {boxes : Finset (WithBot (Box ι))} {h₁ h₂} :
J ∈ (ofWithBot boxes h₁ h₂ : Prepartition I) ↔ (J : WithBot (Box ι)) ∈ boxes :=
mem_eraseNone
#align box_integral.prepartition.mem_of_with_bot BoxIntegral.Prepartition.mem_ofWithBot
@[simp]
theorem iUnion_ofWithBot (boxes : Finset (WithBot (Box ι)))
(le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I)
(pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint) :
(ofWithBot boxes le_of_mem pairwise_disjoint).iUnion = ⋃ J ∈ boxes, ↑J := by
suffices ⋃ (J : Box ι) (_ : ↑J ∈ boxes), ↑J = ⋃ J ∈ boxes, (J : Set (ι → ℝ)) by
simpa [ofWithBot, Prepartition.iUnion]
simp only [← Box.biUnion_coe_eq_coe, @iUnion_comm _ _ (Box ι), @iUnion_comm _ _ (@Eq _ _ _),
iUnion_iUnion_eq_right]
#align box_integral.prepartition.Union_of_with_bot BoxIntegral.Prepartition.iUnion_ofWithBot
theorem ofWithBot_le {boxes : Finset (WithBot (Box ι))}
{le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I}
{pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint}
(H : ∀ J ∈ boxes, J ≠ ⊥ → ∃ J' ∈ π, J ≤ ↑J') :
ofWithBot boxes le_of_mem pairwise_disjoint ≤ π := by
have : ∀ J : Box ι, ↑J ∈ boxes → ∃ J' ∈ π, J ≤ J' := fun J hJ => by
simpa only [WithBot.coe_le_coe] using H J hJ WithBot.coe_ne_bot
simpa [ofWithBot, le_def]
#align box_integral.prepartition.of_with_bot_le BoxIntegral.Prepartition.ofWithBot_le
theorem le_ofWithBot {boxes : Finset (WithBot (Box ι))}
{le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I}
{pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint}
(H : ∀ J ∈ π, ∃ J' ∈ boxes, ↑J ≤ J') : π ≤ ofWithBot boxes le_of_mem pairwise_disjoint := by
intro J hJ
rcases H J hJ with ⟨J', J'mem, hle⟩
lift J' to Box ι using ne_bot_of_le_ne_bot WithBot.coe_ne_bot hle
exact ⟨J', mem_ofWithBot.2 J'mem, WithBot.coe_le_coe.1 hle⟩
#align box_integral.prepartition.le_of_with_bot BoxIntegral.Prepartition.le_ofWithBot
theorem ofWithBot_mono {boxes₁ : Finset (WithBot (Box ι))}
{le_of_mem₁ : ∀ J ∈ boxes₁, (J : WithBot (Box ι)) ≤ I}
{pairwise_disjoint₁ : Set.Pairwise (boxes₁ : Set (WithBot (Box ι))) Disjoint}
{boxes₂ : Finset (WithBot (Box ι))} {le_of_mem₂ : ∀ J ∈ boxes₂, (J : WithBot (Box ι)) ≤ I}
{pairwise_disjoint₂ : Set.Pairwise (boxes₂ : Set (WithBot (Box ι))) Disjoint}
(H : ∀ J ∈ boxes₁, J ≠ ⊥ → ∃ J' ∈ boxes₂, J ≤ J') :
ofWithBot boxes₁ le_of_mem₁ pairwise_disjoint₁ ≤
ofWithBot boxes₂ le_of_mem₂ pairwise_disjoint₂ :=
le_ofWithBot _ fun J hJ => H J (mem_ofWithBot.1 hJ) WithBot.coe_ne_bot
#align box_integral.prepartition.of_with_bot_mono BoxIntegral.Prepartition.ofWithBot_mono
theorem sum_ofWithBot {M : Type*} [AddCommMonoid M] (boxes : Finset (WithBot (Box ι)))
(le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I)
(pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint) (f : Box ι → M) :
(∑ J ∈ (ofWithBot boxes le_of_mem pairwise_disjoint).boxes, f J) =
∑ J ∈ boxes, Option.elim' 0 f J :=
Finset.sum_eraseNone _ _
#align box_integral.prepartition.sum_of_with_bot BoxIntegral.Prepartition.sum_ofWithBot
def restrict (π : Prepartition I) (J : Box ι) : Prepartition J :=
ofWithBot (π.boxes.image fun J' : Box ι => J ⊓ J')
(fun J' hJ' => by
rcases Finset.mem_image.1 hJ' with ⟨J', -, rfl⟩
exact inf_le_left)
(by
simp only [Set.Pairwise, onFun, Finset.mem_coe, Finset.mem_image]
rintro _ ⟨J₁, h₁, rfl⟩ _ ⟨J₂, h₂, rfl⟩ Hne
have : J₁ ≠ J₂ := by
rintro rfl
exact Hne rfl
exact ((Box.disjoint_coe.2 <| π.disjoint_coe_of_mem h₁ h₂ this).inf_left' _).inf_right' _)
#align box_integral.prepartition.restrict BoxIntegral.Prepartition.restrict
@[simp]
theorem mem_restrict : J₁ ∈ π.restrict J ↔ ∃ J' ∈ π, (J₁ : WithBot (Box ι)) = ↑J ⊓ ↑J' := by
simp [restrict, eq_comm]
#align box_integral.prepartition.mem_restrict BoxIntegral.Prepartition.mem_restrict
theorem mem_restrict' : J₁ ∈ π.restrict J ↔ ∃ J' ∈ π, (J₁ : Set (ι → ℝ)) = ↑J ∩ ↑J' := by
simp only [mem_restrict, ← Box.withBotCoe_inj, Box.coe_inf, Box.coe_coe]
#align box_integral.prepartition.mem_restrict' BoxIntegral.Prepartition.mem_restrict'
@[mono]
theorem restrict_mono {π₁ π₂ : Prepartition I} (Hle : π₁ ≤ π₂) : π₁.restrict J ≤ π₂.restrict J := by
refine ofWithBot_mono fun J₁ hJ₁ hne => ?_
rw [Finset.mem_image] at hJ₁; rcases hJ₁ with ⟨J₁, hJ₁, rfl⟩
rcases Hle hJ₁ with ⟨J₂, hJ₂, hle⟩
exact ⟨_, Finset.mem_image_of_mem _ hJ₂, inf_le_inf_left _ <| WithBot.coe_le_coe.2 hle⟩
#align box_integral.prepartition.restrict_mono BoxIntegral.Prepartition.restrict_mono
theorem monotone_restrict : Monotone fun π : Prepartition I => restrict π J :=
fun _ _ => restrict_mono
#align box_integral.prepartition.monotone_restrict BoxIntegral.Prepartition.monotone_restrict
theorem restrict_boxes_of_le (π : Prepartition I) (h : I ≤ J) : (π.restrict J).boxes = π.boxes := by
simp only [restrict, ofWithBot, eraseNone_eq_biUnion]
refine Finset.image_biUnion.trans ?_
refine (Finset.biUnion_congr rfl ?_).trans Finset.biUnion_singleton_eq_self
intro J' hJ'
rw [inf_of_le_right, ← WithBot.some_eq_coe, Option.toFinset_some]
exact WithBot.coe_le_coe.2 ((π.le_of_mem hJ').trans h)
#align box_integral.prepartition.restrict_boxes_of_le BoxIntegral.Prepartition.restrict_boxes_of_le
@[simp]
theorem restrict_self : π.restrict I = π :=
injective_boxes <| restrict_boxes_of_le π le_rfl
#align box_integral.prepartition.restrict_self BoxIntegral.Prepartition.restrict_self
@[simp]
theorem iUnion_restrict : (π.restrict J).iUnion = (J : Set (ι → ℝ)) ∩ (π.iUnion) := by
simp [restrict, ← inter_iUnion, ← iUnion_def]
#align box_integral.prepartition.Union_restrict BoxIntegral.Prepartition.iUnion_restrict
@[simp]
theorem restrict_biUnion (πi : ∀ J, Prepartition J) (hJ : J ∈ π) :
(π.biUnion πi).restrict J = πi J := by
refine (eq_of_boxes_subset_iUnion_superset (fun J₁ h₁ => ?_) ?_).symm
· refine (mem_restrict _).2 ⟨J₁, π.mem_biUnion.2 ⟨J, hJ, h₁⟩, (inf_of_le_right ?_).symm⟩
exact WithBot.coe_le_coe.2 (le_of_mem _ h₁)
· simp only [iUnion_restrict, iUnion_biUnion, Set.subset_def, Set.mem_inter_iff, Set.mem_iUnion]
rintro x ⟨hxJ, J₁, h₁, hx⟩
obtain rfl : J = J₁ := π.eq_of_mem_of_mem hJ h₁ hxJ (iUnion_subset _ hx)
exact hx
#align box_integral.prepartition.restrict_bUnion BoxIntegral.Prepartition.restrict_biUnion
theorem biUnion_le_iff {πi : ∀ J, Prepartition J} {π' : Prepartition I} :
π.biUnion πi ≤ π' ↔ ∀ J ∈ π, πi J ≤ π'.restrict J := by
constructor <;> intro H J hJ
· rw [← π.restrict_biUnion πi hJ]
exact restrict_mono H
· rw [mem_biUnion] at hJ
rcases hJ with ⟨J₁, h₁, hJ⟩
rcases H J₁ h₁ hJ with ⟨J₂, h₂, Hle⟩
rcases π'.mem_restrict.mp h₂ with ⟨J₃, h₃, H⟩
exact ⟨J₃, h₃, Hle.trans <| WithBot.coe_le_coe.1 <| H.trans_le inf_le_right⟩
#align box_integral.prepartition.bUnion_le_iff BoxIntegral.Prepartition.biUnion_le_iff
theorem le_biUnion_iff {πi : ∀ J, Prepartition J} {π' : Prepartition I} :
π' ≤ π.biUnion πi ↔ π' ≤ π ∧ ∀ J ∈ π, π'.restrict J ≤ πi J := by
refine ⟨fun H => ⟨H.trans (π.biUnion_le πi), fun J hJ => ?_⟩, ?_⟩
· rw [← π.restrict_biUnion πi hJ]
exact restrict_mono H
· rintro ⟨H, Hi⟩ J' hJ'
rcases H hJ' with ⟨J, hJ, hle⟩
have : J' ∈ π'.restrict J :=
π'.mem_restrict.2 ⟨J', hJ', (inf_of_le_right <| WithBot.coe_le_coe.2 hle).symm⟩
rcases Hi J hJ this with ⟨Ji, hJi, hlei⟩
exact ⟨Ji, π.mem_biUnion.2 ⟨J, hJ, hJi⟩, hlei⟩
#align box_integral.prepartition.le_bUnion_iff BoxIntegral.Prepartition.le_biUnion_iff
instance inf : Inf (Prepartition I) :=
⟨fun π₁ π₂ => π₁.biUnion fun J => π₂.restrict J⟩
theorem inf_def (π₁ π₂ : Prepartition I) : π₁ ⊓ π₂ = π₁.biUnion fun J => π₂.restrict J := rfl
#align box_integral.prepartition.inf_def BoxIntegral.Prepartition.inf_def
@[simp]
theorem mem_inf {π₁ π₂ : Prepartition I} :
J ∈ π₁ ⊓ π₂ ↔ ∃ J₁ ∈ π₁, ∃ J₂ ∈ π₂, (J : WithBot (Box ι)) = ↑J₁ ⊓ ↑J₂ := by
simp only [inf_def, mem_biUnion, mem_restrict]
#align box_integral.prepartition.mem_inf BoxIntegral.Prepartition.mem_inf
@[simp]
theorem iUnion_inf (π₁ π₂ : Prepartition I) : (π₁ ⊓ π₂).iUnion = π₁.iUnion ∩ π₂.iUnion := by
simp only [inf_def, iUnion_biUnion, iUnion_restrict, ← iUnion_inter, ← iUnion_def]
#align box_integral.prepartition.Union_inf BoxIntegral.Prepartition.iUnion_inf
instance : SemilatticeInf (Prepartition I) :=
{ Prepartition.inf,
Prepartition.partialOrder with
inf_le_left := fun π₁ _ => π₁.biUnion_le _
inf_le_right := fun _ _ => (biUnion_le_iff _).2 fun _ _ => le_rfl
le_inf := fun _ π₁ _ h₁ h₂ => π₁.le_biUnion_iff.2 ⟨h₁, fun _ _ => restrict_mono h₂⟩ }
@[simps]
def filter (π : Prepartition I) (p : Box ι → Prop) : Prepartition I where
boxes := π.boxes.filter p
le_of_mem' _ hJ := π.le_of_mem (mem_filter.1 hJ).1
pairwiseDisjoint _ h₁ _ h₂ := π.disjoint_coe_of_mem (mem_filter.1 h₁).1 (mem_filter.1 h₂).1
#align box_integral.prepartition.filter BoxIntegral.Prepartition.filter
@[simp]
theorem mem_filter {p : Box ι → Prop} : J ∈ π.filter p ↔ J ∈ π ∧ p J :=
Finset.mem_filter
#align box_integral.prepartition.mem_filter BoxIntegral.Prepartition.mem_filter
theorem filter_le (π : Prepartition I) (p : Box ι → Prop) : π.filter p ≤ π := fun J hJ =>
let ⟨hπ, _⟩ := π.mem_filter.1 hJ
⟨J, hπ, le_rfl⟩
#align box_integral.prepartition.filter_le BoxIntegral.Prepartition.filter_le
theorem filter_of_true {p : Box ι → Prop} (hp : ∀ J ∈ π, p J) : π.filter p = π := by
ext J
simpa using hp J
#align box_integral.prepartition.filter_of_true BoxIntegral.Prepartition.filter_of_true
@[simp]
theorem filter_true : (π.filter fun _ => True) = π :=
π.filter_of_true fun _ _ => trivial
#align box_integral.prepartition.filter_true BoxIntegral.Prepartition.filter_true
@[simp]
| Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 622 | 630 | theorem iUnion_filter_not (π : Prepartition I) (p : Box ι → Prop) :
(π.filter fun J => ¬p J).iUnion = π.iUnion \ (π.filter p).iUnion := by |
simp only [Prepartition.iUnion]
convert (@Set.biUnion_diff_biUnion_eq (ι → ℝ) (Box ι) π.boxes (π.filter p).boxes (↑) _).symm
· simp (config := { contextual := true })
· rw [Set.PairwiseDisjoint]
convert π.pairwiseDisjoint
rw [Set.union_eq_left, filter_boxes, coe_filter]
exact fun _ ⟨h, _⟩ => h
|
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/mathlib"@"4b92a463033b5587bb011657e25e4710bfca7364"
suppress_compilation
section KaehlerDifferential
open scoped TensorProduct
open Algebra
universe u v
variable (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S]
abbrev KaehlerDifferential.ideal : Ideal (S ⊗[R] S) :=
RingHom.ker (TensorProduct.lmul' R : S ⊗[R] S →ₐ[R] S)
#align kaehler_differential.ideal KaehlerDifferential.ideal
variable {S}
theorem KaehlerDifferential.one_smul_sub_smul_one_mem_ideal (a : S) :
(1 : S) ⊗ₜ[R] a - a ⊗ₜ[R] (1 : S) ∈ KaehlerDifferential.ideal R S := by simp [RingHom.mem_ker]
#align kaehler_differential.one_smul_sub_smul_one_mem_ideal KaehlerDifferential.one_smul_sub_smul_one_mem_ideal
variable {R}
variable {M : Type*} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M]
def Derivation.tensorProductTo (D : Derivation R S M) : S ⊗[R] S →ₗ[S] M :=
TensorProduct.AlgebraTensorModule.lift ((LinearMap.lsmul S (S →ₗ[R] M)).flip D.toLinearMap)
#align derivation.tensor_product_to Derivation.tensorProductTo
theorem Derivation.tensorProductTo_tmul (D : Derivation R S M) (s t : S) :
D.tensorProductTo (s ⊗ₜ t) = s • D t := rfl
#align derivation.tensor_product_to_tmul Derivation.tensorProductTo_tmul
theorem Derivation.tensorProductTo_mul (D : Derivation R S M) (x y : S ⊗[R] S) :
D.tensorProductTo (x * y) =
TensorProduct.lmul' (S := S) R x • D.tensorProductTo y +
TensorProduct.lmul' (S := S) R y • D.tensorProductTo x := by
refine TensorProduct.induction_on x ?_ ?_ ?_
· rw [zero_mul, map_zero, map_zero, zero_smul, smul_zero, add_zero]
swap
· intro x₁ y₁ h₁ h₂
rw [add_mul, map_add, map_add, map_add, add_smul, smul_add, h₁, h₂, add_add_add_comm]
intro x₁ x₂
refine TensorProduct.induction_on y ?_ ?_ ?_
· rw [mul_zero, map_zero, map_zero, zero_smul, smul_zero, add_zero]
swap
· intro x₁ y₁ h₁ h₂
rw [mul_add, map_add, map_add, map_add, add_smul, smul_add, h₁, h₂, add_add_add_comm]
intro x y
simp only [TensorProduct.tmul_mul_tmul, Derivation.tensorProductTo,
TensorProduct.AlgebraTensorModule.lift_apply, TensorProduct.lift.tmul',
TensorProduct.lmul'_apply_tmul]
dsimp
rw [D.leibniz]
simp only [smul_smul, smul_add, mul_comm (x * y) x₁, mul_right_comm x₁ x₂, ← mul_assoc]
#align derivation.tensor_product_to_mul Derivation.tensorProductTo_mul
variable (R S)
| 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
refine TensorProduct.induction_on x ?_ ?_ ?_
· rw [map_zero, TensorProduct.zero_tmul, sub_zero]; exact zero_mem _
· intro x y
have : x ⊗ₜ[R] y - (x * y) ⊗ₜ[R] (1 : S) = x • ((1 : S) ⊗ₜ y - y ⊗ₜ (1 : S)) := by
simp_rw [smul_sub, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
rw [TensorProduct.lmul'_apply_tmul, this]
refine Submodule.smul_mem _ x ?_
apply Submodule.subset_span
exact Set.mem_range_self y
· intro x y hx hy
rw [map_add, TensorProduct.add_tmul, ← sub_add_sub_comm]
exact add_mem hx hy
|
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Order.Interval.Finset.Nat
#align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bfb4330ddf6624f1028ba186117d82"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {A : Type z} {a b : R} {n : ℕ}
section Semiring
variable [Semiring R] {p q : R[X]}
def divX (p : R[X]) : R[X] :=
⟨AddMonoidAlgebra.divOf p.toFinsupp 1⟩
set_option linter.uppercaseLean3 false in
#align polynomial.div_X Polynomial.divX
@[simp]
theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) := by
rw [add_comm]; cases p; rfl
set_option linter.uppercaseLean3 false in
#align polynomial.coeff_div_X Polynomial.coeff_divX
theorem divX_mul_X_add (p : R[X]) : divX p * X + C (p.coeff 0) = p :=
ext <| by rintro ⟨_ | _⟩ <;> simp [coeff_C, Nat.succ_ne_zero, coeff_mul_X]
set_option linter.uppercaseLean3 false in
#align polynomial.div_X_mul_X_add Polynomial.divX_mul_X_add
@[simp]
theorem X_mul_divX_add (p : R[X]) : X * divX p + C (p.coeff 0) = p :=
ext <| by rintro ⟨_ | _⟩ <;> simp [coeff_C, Nat.succ_ne_zero, coeff_mul_X]
@[simp]
theorem divX_C (a : R) : divX (C a) = 0 :=
ext fun n => by simp [coeff_divX, coeff_C, Finsupp.single_eq_of_ne _]
set_option linter.uppercaseLean3 false in
#align polynomial.div_X_C Polynomial.divX_C
theorem divX_eq_zero_iff : divX p = 0 ↔ p = C (p.coeff 0) :=
⟨fun h => by simpa [eq_comm, h] using divX_mul_X_add p, fun h => by rw [h, divX_C]⟩
set_option linter.uppercaseLean3 false in
#align polynomial.div_X_eq_zero_iff Polynomial.divX_eq_zero_iff
theorem divX_add : divX (p + q) = divX p + divX q :=
ext <| by simp
set_option linter.uppercaseLean3 false in
#align polynomial.div_X_add Polynomial.divX_add
@[simp]
theorem divX_zero : divX (0 : R[X]) = 0 := leadingCoeff_eq_zero.mp rfl
@[simp]
theorem divX_one : divX (1 : R[X]) = 0 := by
ext
simpa only [coeff_divX, coeff_zero] using coeff_one
@[simp]
theorem divX_C_mul : divX (C a * p) = C a * divX p := by
ext
simp
theorem divX_X_pow : divX (X ^ n : R[X]) = if (n = 0) then 0 else X ^ (n - 1) := by
cases n
· simp
· ext n
simp [coeff_X_pow]
noncomputable
def divX_hom : R[X] →+ R[X] :=
{ toFun := divX
map_zero' := divX_zero
map_add' := fun _ _ => divX_add }
@[simp] theorem divX_hom_toFun : divX_hom p = divX p := rfl
theorem natDegree_divX_eq_natDegree_tsub_one : p.divX.natDegree = p.natDegree - 1 := by
apply map_natDegree_eq_sub (φ := divX_hom)
· intro f
simpa [divX_hom, divX_eq_zero_iff] using eq_C_of_natDegree_eq_zero
· intros n c c0
rw [← C_mul_X_pow_eq_monomial, divX_hom_toFun, divX_C_mul, divX_X_pow]
split_ifs with n0
· simp [n0]
· exact natDegree_C_mul_X_pow (n - 1) c c0
theorem natDegree_divX_le : p.divX.natDegree ≤ p.natDegree :=
natDegree_divX_eq_natDegree_tsub_one.trans_le (Nat.pred_le _)
theorem divX_C_mul_X_pow : divX (C a * X ^ n) = if n = 0 then 0 else C a * X ^ (n - 1) := by
simp only [divX_C_mul, divX_X_pow, mul_ite, mul_zero]
| Mathlib/Algebra/Polynomial/Inductions.lean | 119 | 143 | theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by |
haveI := Nontrivial.of_polynomial_ne hp0
calc
degree (divX p) < (divX p * X + C (p.coeff 0)).degree :=
if h : degree p ≤ 0 then by
have h' : C (p.coeff 0) ≠ 0 := by rwa [← eq_C_of_degree_le_zero h]
rw [eq_C_of_degree_le_zero h, divX_C, degree_zero, zero_mul, zero_add]
exact lt_of_le_of_ne bot_le (Ne.symm (mt degree_eq_bot.1 <| by simpa using h'))
else by
have hXp0 : divX p ≠ 0 := by
simpa [divX_eq_zero_iff, -not_le, degree_le_zero_iff] using h
have : leadingCoeff (divX p) * leadingCoeff X ≠ 0 := by simpa
have : degree (C (p.coeff 0)) < degree (divX p * X) :=
calc
degree (C (p.coeff 0)) ≤ 0 := degree_C_le
_ < 1 := by decide
_ = degree (X : R[X]) := degree_X.symm
_ ≤ degree (divX p * X) := by
rw [← zero_add (degree X), degree_mul' this]
exact add_le_add
(by rw [zero_le_degree_iff, Ne, divX_eq_zero_iff]
exact fun h0 => h (h0.symm ▸ degree_C_le))
le_rfl
rw [degree_add_eq_left_of_degree_lt this]; exact degree_lt_degree_mul_X hXp0
_ = degree p := congr_arg _ (divX_mul_X_add _)
|
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open CategoryTheory Category Limits HomologicalComplex
variable {ι : Type*}
variable {V : Type u} [Category.{v} V] [Preadditive V]
variable {c : ComplexShape ι} {C D E : HomologicalComplex V c}
variable (f g : C ⟶ D) (h k : D ⟶ E) (i : ι)
section
def dNext (i : ι) : (∀ i j, C.X i ⟶ D.X j) →+ (C.X i ⟶ D.X i) :=
AddMonoidHom.mk' (fun f => C.d i (c.next i) ≫ f (c.next i) i) fun _ _ =>
Preadditive.comp_add _ _ _ _ _ _
#align d_next dNext
def fromNext (i : ι) : (∀ i j, C.X i ⟶ D.X j) →+ (C.xNext i ⟶ D.X i) :=
AddMonoidHom.mk' (fun f => f (c.next i) i) fun _ _ => rfl
#align from_next fromNext
@[simp]
theorem dNext_eq_dFrom_fromNext (f : ∀ i j, C.X i ⟶ D.X j) (i : ι) :
dNext i f = C.dFrom i ≫ fromNext i f :=
rfl
#align d_next_eq_d_from_from_next dNext_eq_dFrom_fromNext
theorem dNext_eq (f : ∀ i j, C.X i ⟶ D.X j) {i i' : ι} (w : c.Rel i i') :
dNext i f = C.d i i' ≫ f i' i := by
obtain rfl := c.next_eq' w
rfl
#align d_next_eq dNext_eq
lemma dNext_eq_zero (f : ∀ i j, C.X i ⟶ D.X j) (i : ι) (hi : ¬ c.Rel i (c.next i)) :
dNext i f = 0 := by
dsimp [dNext]
rw [shape _ _ _ hi, zero_comp]
@[simp 1100]
theorem dNext_comp_left (f : C ⟶ D) (g : ∀ i j, D.X i ⟶ E.X j) (i : ι) :
(dNext i fun i j => f.f i ≫ g i j) = f.f i ≫ dNext i g :=
(f.comm_assoc _ _ _).symm
#align d_next_comp_left dNext_comp_left
@[simp 1100]
theorem dNext_comp_right (f : ∀ i j, C.X i ⟶ D.X j) (g : D ⟶ E) (i : ι) :
(dNext i fun i j => f i j ≫ g.f j) = dNext i f ≫ g.f i :=
(assoc _ _ _).symm
#align d_next_comp_right dNext_comp_right
def prevD (j : ι) : (∀ i j, C.X i ⟶ D.X j) →+ (C.X j ⟶ D.X j) :=
AddMonoidHom.mk' (fun f => f j (c.prev j) ≫ D.d (c.prev j) j) fun _ _ =>
Preadditive.add_comp _ _ _ _ _ _
#align prev_d prevD
lemma prevD_eq_zero (f : ∀ i j, C.X i ⟶ D.X j) (i : ι) (hi : ¬ c.Rel (c.prev i) i) :
prevD i f = 0 := by
dsimp [prevD]
rw [shape _ _ _ hi, comp_zero]
def toPrev (j : ι) : (∀ i j, C.X i ⟶ D.X j) →+ (C.X j ⟶ D.xPrev j) :=
AddMonoidHom.mk' (fun f => f j (c.prev j)) fun _ _ => rfl
#align to_prev toPrev
@[simp]
theorem prevD_eq_toPrev_dTo (f : ∀ i j, C.X i ⟶ D.X j) (j : ι) :
prevD j f = toPrev j f ≫ D.dTo j :=
rfl
#align prev_d_eq_to_prev_d_to prevD_eq_toPrev_dTo
theorem prevD_eq (f : ∀ i j, C.X i ⟶ D.X j) {j j' : ι} (w : c.Rel j' j) :
prevD j f = f j j' ≫ D.d j' j := by
obtain rfl := c.prev_eq' w
rfl
#align prev_d_eq prevD_eq
@[simp 1100]
theorem prevD_comp_left (f : C ⟶ D) (g : ∀ i j, D.X i ⟶ E.X j) (j : ι) :
(prevD j fun i j => f.f i ≫ g i j) = f.f j ≫ prevD j g :=
assoc _ _ _
#align prev_d_comp_left prevD_comp_left
@[simp 1100]
theorem prevD_comp_right (f : ∀ i j, C.X i ⟶ D.X j) (g : D ⟶ E) (j : ι) :
(prevD j fun i j => f i j ≫ g.f j) = prevD j f ≫ g.f j := by
dsimp [prevD]
simp only [assoc, g.comm]
#align prev_d_comp_right prevD_comp_right
theorem dNext_nat (C D : ChainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
dNext i f = C.d i (i - 1) ≫ f (i - 1) i := by
dsimp [dNext]
cases i
· simp only [shape, ChainComplex.next_nat_zero, ComplexShape.down_Rel, Nat.one_ne_zero,
not_false_iff, zero_comp]
· congr <;> simp
#align d_next_nat dNext_nat
theorem prevD_nat (C D : CochainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
prevD i f = f i (i - 1) ≫ D.d (i - 1) i := by
dsimp [prevD]
cases i
· simp only [shape, CochainComplex.prev_nat_zero, ComplexShape.up_Rel, Nat.one_ne_zero,
not_false_iff, comp_zero]
· congr <;> simp
#align prev_d_nat prevD_nat
-- Porting note(#5171): removed @[has_nonempty_instance]
@[ext]
structure Homotopy (f g : C ⟶ D) where
hom : ∀ i j, C.X i ⟶ D.X j
zero : ∀ i j, ¬c.Rel j i → hom i j = 0 := by aesop_cat
comm : ∀ i, f.f i = dNext i hom + prevD i hom + g.f i := by aesop_cat
#align homotopy Homotopy
variable {f g}
namespace Homotopy
def equivSubZero : Homotopy f g ≃ Homotopy (f - g) 0 where
toFun h :=
{ hom := fun i j => h.hom i j
zero := fun i j w => h.zero _ _ w
comm := fun i => by simp [h.comm] }
invFun h :=
{ hom := fun i j => h.hom i j
zero := fun i j w => h.zero _ _ w
comm := fun i => by simpa [sub_eq_iff_eq_add] using h.comm i }
left_inv := by aesop_cat
right_inv := by aesop_cat
#align homotopy.equiv_sub_zero Homotopy.equivSubZero
@[simps]
def ofEq (h : f = g) : Homotopy f g where
hom := 0
zero _ _ _ := rfl
#align homotopy.of_eq Homotopy.ofEq
@[simps!, refl]
def refl (f : C ⟶ D) : Homotopy f f :=
ofEq (rfl : f = f)
#align homotopy.refl Homotopy.refl
@[simps!, symm]
def symm {f g : C ⟶ D} (h : Homotopy f g) : Homotopy g f where
hom := -h.hom
zero i j w := by rw [Pi.neg_apply, Pi.neg_apply, h.zero i j w, neg_zero]
comm i := by
rw [AddMonoidHom.map_neg, AddMonoidHom.map_neg, h.comm, ← neg_add, ← add_assoc, neg_add_self,
zero_add]
#align homotopy.symm Homotopy.symm
@[simps!, trans]
def trans {e f g : C ⟶ D} (h : Homotopy e f) (k : Homotopy f g) : Homotopy e g where
hom := h.hom + k.hom
zero i j w := by rw [Pi.add_apply, Pi.add_apply, h.zero i j w, k.zero i j w, zero_add]
comm i := by
rw [AddMonoidHom.map_add, AddMonoidHom.map_add, h.comm, k.comm]
abel
#align homotopy.trans Homotopy.trans
@[simps!]
def add {f₁ g₁ f₂ g₂ : C ⟶ D} (h₁ : Homotopy f₁ g₁) (h₂ : Homotopy f₂ g₂) :
Homotopy (f₁ + f₂) (g₁ + g₂) where
hom := h₁.hom + h₂.hom
zero i j hij := by rw [Pi.add_apply, Pi.add_apply, h₁.zero i j hij, h₂.zero i j hij, add_zero]
comm i := by
simp only [HomologicalComplex.add_f_apply, h₁.comm, h₂.comm, AddMonoidHom.map_add]
abel
#align homotopy.add Homotopy.add
@[simps!]
def smul {R : Type*} [Semiring R] [Linear R V] (h : Homotopy f g) (a : R) :
Homotopy (a • f) (a • g) where
hom i j := a • h.hom i j
zero i j hij := by
dsimp
rw [h.zero i j hij, smul_zero]
comm i := by
dsimp
rw [h.comm]
dsimp [fromNext, toPrev]
simp only [smul_add, Linear.comp_smul, Linear.smul_comp]
@[simps]
def compRight {e f : C ⟶ D} (h : Homotopy e f) (g : D ⟶ E) : Homotopy (e ≫ g) (f ≫ g) where
hom i j := h.hom i j ≫ g.f j
zero i j w := by dsimp; rw [h.zero i j w, zero_comp]
comm i := by rw [comp_f, h.comm i, dNext_comp_right, prevD_comp_right, Preadditive.add_comp,
comp_f, Preadditive.add_comp]
#align homotopy.comp_right Homotopy.compRight
@[simps]
def compLeft {f g : D ⟶ E} (h : Homotopy f g) (e : C ⟶ D) : Homotopy (e ≫ f) (e ≫ g) where
hom i j := e.f i ≫ h.hom i j
zero i j w := by dsimp; rw [h.zero i j w, comp_zero]
comm i := by rw [comp_f, h.comm i, dNext_comp_left, prevD_comp_left, comp_f,
Preadditive.comp_add, Preadditive.comp_add]
#align homotopy.comp_left Homotopy.compLeft
@[simps!]
def comp {C₁ C₂ C₃ : HomologicalComplex V c} {f₁ g₁ : C₁ ⟶ C₂} {f₂ g₂ : C₂ ⟶ C₃}
(h₁ : Homotopy f₁ g₁) (h₂ : Homotopy f₂ g₂) : Homotopy (f₁ ≫ f₂) (g₁ ≫ g₂) :=
(h₁.compRight _).trans (h₂.compLeft _)
#align homotopy.comp Homotopy.comp
@[simps!]
def compRightId {f : C ⟶ C} (h : Homotopy f (𝟙 C)) (g : C ⟶ D) : Homotopy (f ≫ g) g :=
(h.compRight g).trans (ofEq <| id_comp _)
#align homotopy.comp_right_id Homotopy.compRightId
@[simps!]
def compLeftId {f : D ⟶ D} (h : Homotopy f (𝟙 D)) (g : C ⟶ D) : Homotopy (g ≫ f) g :=
(h.compLeft g).trans (ofEq <| comp_id _)
#align homotopy.comp_left_id Homotopy.compLeftId
def nullHomotopicMap (hom : ∀ i j, C.X i ⟶ D.X j) : C ⟶ D where
f i := dNext i hom + prevD i hom
comm' i j hij := by
have eq1 : prevD i hom ≫ D.d i j = 0 := by
simp only [prevD, AddMonoidHom.mk'_apply, assoc, d_comp_d, comp_zero]
have eq2 : C.d i j ≫ dNext j hom = 0 := by
simp only [dNext, AddMonoidHom.mk'_apply, d_comp_d_assoc, zero_comp]
dsimp only
rw [dNext_eq hom hij, prevD_eq hom hij, Preadditive.comp_add, Preadditive.add_comp, eq1, eq2,
add_zero, zero_add, assoc]
#align homotopy.null_homotopic_map Homotopy.nullHomotopicMap
def nullHomotopicMap' (h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) : C ⟶ D :=
nullHomotopicMap fun i j => dite (c.Rel j i) (h i j) fun _ => 0
#align homotopy.null_homotopic_map' Homotopy.nullHomotopicMap'
theorem nullHomotopicMap_comp (hom : ∀ i j, C.X i ⟶ D.X j) (g : D ⟶ E) :
nullHomotopicMap hom ≫ g = nullHomotopicMap fun i j => hom i j ≫ g.f j := by
ext n
dsimp [nullHomotopicMap, fromNext, toPrev, AddMonoidHom.mk'_apply]
simp only [Preadditive.add_comp, assoc, g.comm]
#align homotopy.null_homotopic_map_comp Homotopy.nullHomotopicMap_comp
theorem nullHomotopicMap'_comp (hom : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) (g : D ⟶ E) :
nullHomotopicMap' hom ≫ g = nullHomotopicMap' fun i j hij => hom i j hij ≫ g.f j := by
ext n
erw [nullHomotopicMap_comp]
congr
ext i j
split_ifs
· rfl
· rw [zero_comp]
#align homotopy.null_homotopic_map'_comp Homotopy.nullHomotopicMap'_comp
theorem comp_nullHomotopicMap (f : C ⟶ D) (hom : ∀ i j, D.X i ⟶ E.X j) :
f ≫ nullHomotopicMap hom = nullHomotopicMap fun i j => f.f i ≫ hom i j := by
ext n
dsimp [nullHomotopicMap, fromNext, toPrev, AddMonoidHom.mk'_apply]
simp only [Preadditive.comp_add, assoc, f.comm_assoc]
#align homotopy.comp_null_homotopic_map Homotopy.comp_nullHomotopicMap
theorem comp_nullHomotopicMap' (f : C ⟶ D) (hom : ∀ i j, c.Rel j i → (D.X i ⟶ E.X j)) :
f ≫ nullHomotopicMap' hom = nullHomotopicMap' fun i j hij => f.f i ≫ hom i j hij := by
ext n
erw [comp_nullHomotopicMap]
congr
ext i j
split_ifs
· rfl
· rw [comp_zero]
#align homotopy.comp_null_homotopic_map' Homotopy.comp_nullHomotopicMap'
theorem map_nullHomotopicMap {W : Type*} [Category W] [Preadditive W] (G : V ⥤ W) [G.Additive]
(hom : ∀ i j, C.X i ⟶ D.X j) :
(G.mapHomologicalComplex c).map (nullHomotopicMap hom) =
nullHomotopicMap (fun i j => by exact G.map (hom i j)) := by
ext i
dsimp [nullHomotopicMap, dNext, prevD]
simp only [G.map_comp, Functor.map_add]
#align homotopy.map_null_homotopic_map Homotopy.map_nullHomotopicMap
theorem map_nullHomotopicMap' {W : Type*} [Category W] [Preadditive W] (G : V ⥤ W) [G.Additive]
(hom : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(G.mapHomologicalComplex c).map (nullHomotopicMap' hom) =
nullHomotopicMap' fun i j hij => by exact G.map (hom i j hij) := by
ext n
erw [map_nullHomotopicMap]
congr
ext i j
split_ifs
· rfl
· rw [G.map_zero]
#align homotopy.map_null_homotopic_map' Homotopy.map_nullHomotopicMap'
@[simps]
def nullHomotopy (hom : ∀ i j, C.X i ⟶ D.X j) (zero : ∀ i j, ¬c.Rel j i → hom i j = 0) :
Homotopy (nullHomotopicMap hom) 0 :=
{ hom := hom
zero := zero
comm := by
intro i
rw [HomologicalComplex.zero_f_apply, add_zero]
rfl }
#align homotopy.null_homotopy Homotopy.nullHomotopy
@[simps!]
def nullHomotopy' (h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) : Homotopy (nullHomotopicMap' h) 0 := by
apply nullHomotopy fun i j => dite (c.Rel j i) (h i j) fun _ => 0
intro i j hij
rw [dite_eq_right_iff]
intro hij'
exfalso
exact hij hij'
#align homotopy.null_homotopy' Homotopy.nullHomotopy'
@[simp]
theorem nullHomotopicMap_f {k₂ k₁ k₀ : ι} (r₂₁ : c.Rel k₂ k₁) (r₁₀ : c.Rel k₁ k₀)
(hom : ∀ i j, C.X i ⟶ D.X j) :
(nullHomotopicMap hom).f k₁ = C.d k₁ k₀ ≫ hom k₀ k₁ + hom k₁ k₂ ≫ D.d k₂ k₁ := by
dsimp only [nullHomotopicMap]
rw [dNext_eq hom r₁₀, prevD_eq hom r₂₁]
#align homotopy.null_homotopic_map_f Homotopy.nullHomotopicMap_f
@[simp]
theorem nullHomotopicMap'_f {k₂ k₁ k₀ : ι} (r₂₁ : c.Rel k₂ k₁) (r₁₀ : c.Rel k₁ k₀)
(h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(nullHomotopicMap' h).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ D.d k₂ k₁ := by
simp only [nullHomotopicMap']
rw [nullHomotopicMap_f r₂₁ r₁₀]
split_ifs
rfl
#align homotopy.null_homotopic_map'_f Homotopy.nullHomotopicMap'_f
@[simp]
theorem nullHomotopicMap_f_of_not_rel_left {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₀ : ∀ l : ι, ¬c.Rel k₀ l) (hom : ∀ i j, C.X i ⟶ D.X j) :
(nullHomotopicMap hom).f k₀ = hom k₀ k₁ ≫ D.d k₁ k₀ := by
dsimp only [nullHomotopicMap]
rw [prevD_eq hom r₁₀, dNext, AddMonoidHom.mk'_apply, C.shape, zero_comp, zero_add]
exact hk₀ _
#align homotopy.null_homotopic_map_f_of_not_rel_left Homotopy.nullHomotopicMap_f_of_not_rel_left
@[simp]
theorem nullHomotopicMap'_f_of_not_rel_left {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₀ : ∀ l : ι, ¬c.Rel k₀ l) (h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(nullHomotopicMap' h).f k₀ = h k₀ k₁ r₁₀ ≫ D.d k₁ k₀ := by
simp only [nullHomotopicMap']
rw [nullHomotopicMap_f_of_not_rel_left r₁₀ hk₀]
split_ifs
rfl
#align homotopy.null_homotopic_map'_f_of_not_rel_left Homotopy.nullHomotopicMap'_f_of_not_rel_left
@[simp]
theorem nullHomotopicMap_f_of_not_rel_right {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₁ : ∀ l : ι, ¬c.Rel l k₁) (hom : ∀ i j, C.X i ⟶ D.X j) :
(nullHomotopicMap hom).f k₁ = C.d k₁ k₀ ≫ hom k₀ k₁ := by
dsimp only [nullHomotopicMap]
rw [dNext_eq hom r₁₀, prevD, AddMonoidHom.mk'_apply, D.shape, comp_zero, add_zero]
exact hk₁ _
#align homotopy.null_homotopic_map_f_of_not_rel_right Homotopy.nullHomotopicMap_f_of_not_rel_right
@[simp]
theorem nullHomotopicMap'_f_of_not_rel_right {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₁ : ∀ l : ι, ¬c.Rel l k₁) (h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(nullHomotopicMap' h).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ := by
simp only [nullHomotopicMap']
rw [nullHomotopicMap_f_of_not_rel_right r₁₀ hk₁]
split_ifs
rfl
#align homotopy.null_homotopic_map'_f_of_not_rel_right Homotopy.nullHomotopicMap'_f_of_not_rel_right
@[simp]
theorem nullHomotopicMap_f_eq_zero {k₀ : ι} (hk₀ : ∀ l : ι, ¬c.Rel k₀ l)
(hk₀' : ∀ l : ι, ¬c.Rel l k₀) (hom : ∀ i j, C.X i ⟶ D.X j) :
(nullHomotopicMap hom).f k₀ = 0 := by
dsimp [nullHomotopicMap, dNext, prevD]
rw [C.shape, D.shape, zero_comp, comp_zero, add_zero] <;> apply_assumption
#align homotopy.null_homotopic_map_f_eq_zero Homotopy.nullHomotopicMap_f_eq_zero
@[simp]
theorem nullHomotopicMap'_f_eq_zero {k₀ : ι} (hk₀ : ∀ l : ι, ¬c.Rel k₀ l)
(hk₀' : ∀ l : ι, ¬c.Rel l k₀) (h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(nullHomotopicMap' h).f k₀ = 0 := by
simp only [nullHomotopicMap']
apply nullHomotopicMap_f_eq_zero hk₀ hk₀'
#align homotopy.null_homotopic_map'_f_eq_zero Homotopy.nullHomotopicMap'_f_eq_zero
section MkInductive
variable {P Q : ChainComplex V ℕ}
@[simp 1100]
theorem prevD_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (j : ℕ) :
prevD j f = f j (j + 1) ≫ Q.d _ _ := by
dsimp [prevD]
have : (ComplexShape.down ℕ).prev j = j + 1 := ChainComplex.prev ℕ j
congr 2
#align homotopy.prev_d_chain_complex Homotopy.prevD_chainComplex
@[simp 1100]
theorem dNext_succ_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (i : ℕ) :
dNext (i + 1) f = P.d _ _ ≫ f i (i + 1) := by
dsimp [dNext]
have : (ComplexShape.down ℕ).next (i + 1) = i := ChainComplex.next_nat_succ _
congr 2
#align homotopy.d_next_succ_chain_complex Homotopy.dNext_succ_chainComplex
@[simp 1100]
| Mathlib/Algebra/Homology/Homotopy.lean | 493 | 496 | theorem dNext_zero_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) : dNext 0 f = 0 := by |
dsimp [dNext]
rw [P.shape, zero_comp]
rw [ChainComplex.next_nat_zero]; dsimp; decide
|
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b37058d0"
open Function Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*}
namespace Finset
variable [DecidableEq α'] [DecidableEq β'] [DecidableEq γ] [DecidableEq γ'] [DecidableEq δ]
[DecidableEq δ'] [DecidableEq ε] [DecidableEq ε'] {f f' : α → β → γ} {g g' : α → β → γ → δ}
{s s' : Finset α} {t t' : Finset β} {u u' : Finset γ} {a a' : α} {b b' : β} {c : γ}
def image₂ (f : α → β → γ) (s : Finset α) (t : Finset β) : Finset γ :=
(s ×ˢ t).image <| uncurry f
#align finset.image₂ Finset.image₂
@[simp]
theorem mem_image₂ : c ∈ image₂ f s t ↔ ∃ a ∈ s, ∃ b ∈ t, f a b = c := by
simp [image₂, and_assoc]
#align finset.mem_image₂ Finset.mem_image₂
@[simp, norm_cast]
theorem coe_image₂ (f : α → β → γ) (s : Finset α) (t : Finset β) :
(image₂ f s t : Set γ) = Set.image2 f s t :=
Set.ext fun _ => mem_image₂
#align finset.coe_image₂ Finset.coe_image₂
theorem card_image₂_le (f : α → β → γ) (s : Finset α) (t : Finset β) :
(image₂ f s t).card ≤ s.card * t.card :=
card_image_le.trans_eq <| card_product _ _
#align finset.card_image₂_le Finset.card_image₂_le
theorem card_image₂_iff :
(image₂ f s t).card = s.card * t.card ↔ (s ×ˢ t : Set (α × β)).InjOn fun x => f x.1 x.2 := by
rw [← card_product, ← coe_product]
exact card_image_iff
#align finset.card_image₂_iff Finset.card_image₂_iff
theorem card_image₂ (hf : Injective2 f) (s : Finset α) (t : Finset β) :
(image₂ f s t).card = s.card * t.card :=
(card_image_of_injective _ hf.uncurry).trans <| card_product _ _
#align finset.card_image₂ Finset.card_image₂
theorem mem_image₂_of_mem (ha : a ∈ s) (hb : b ∈ t) : f a b ∈ image₂ f s t :=
mem_image₂.2 ⟨a, ha, b, hb, rfl⟩
#align finset.mem_image₂_of_mem Finset.mem_image₂_of_mem
theorem mem_image₂_iff (hf : Injective2 f) : f a b ∈ image₂ f s t ↔ a ∈ s ∧ b ∈ t := by
rw [← mem_coe, coe_image₂, mem_image2_iff hf, mem_coe, mem_coe]
#align finset.mem_image₂_iff Finset.mem_image₂_iff
theorem image₂_subset (hs : s ⊆ s') (ht : t ⊆ t') : image₂ f s t ⊆ image₂ f s' t' := by
rw [← coe_subset, coe_image₂, coe_image₂]
exact image2_subset hs ht
#align finset.image₂_subset Finset.image₂_subset
theorem image₂_subset_left (ht : t ⊆ t') : image₂ f s t ⊆ image₂ f s t' :=
image₂_subset Subset.rfl ht
#align finset.image₂_subset_left Finset.image₂_subset_left
theorem image₂_subset_right (hs : s ⊆ s') : image₂ f s t ⊆ image₂ f s' t :=
image₂_subset hs Subset.rfl
#align finset.image₂_subset_right Finset.image₂_subset_right
theorem image_subset_image₂_left (hb : b ∈ t) : s.image (fun a => f a b) ⊆ image₂ f s t :=
image_subset_iff.2 fun _ ha => mem_image₂_of_mem ha hb
#align finset.image_subset_image₂_left Finset.image_subset_image₂_left
theorem image_subset_image₂_right (ha : a ∈ s) : t.image (fun b => f a b) ⊆ image₂ f s t :=
image_subset_iff.2 fun _ => mem_image₂_of_mem ha
#align finset.image_subset_image₂_right Finset.image_subset_image₂_right
theorem forall_image₂_iff {p : γ → Prop} :
(∀ z ∈ image₂ f s t, p z) ↔ ∀ x ∈ s, ∀ y ∈ t, p (f x y) := by
simp_rw [← mem_coe, coe_image₂, forall_image2_iff]
#align finset.forall_image₂_iff Finset.forall_image₂_iff
@[simp]
theorem image₂_subset_iff : image₂ f s t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, f x y ∈ u :=
forall_image₂_iff
#align finset.image₂_subset_iff Finset.image₂_subset_iff
theorem image₂_subset_iff_left : image₂ f s t ⊆ u ↔ ∀ a ∈ s, (t.image fun b => f a b) ⊆ u := by
simp_rw [image₂_subset_iff, image_subset_iff]
#align finset.image₂_subset_iff_left Finset.image₂_subset_iff_left
theorem image₂_subset_iff_right : image₂ f s t ⊆ u ↔ ∀ b ∈ t, (s.image fun a => f a b) ⊆ u := by
simp_rw [image₂_subset_iff, image_subset_iff, @forall₂_swap α]
#align finset.image₂_subset_iff_right Finset.image₂_subset_iff_right
@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
theorem image₂_nonempty_iff : (image₂ f s t).Nonempty ↔ s.Nonempty ∧ t.Nonempty := by
rw [← coe_nonempty, coe_image₂]
exact image2_nonempty_iff
#align finset.image₂_nonempty_iff Finset.image₂_nonempty_iff
theorem Nonempty.image₂ (hs : s.Nonempty) (ht : t.Nonempty) : (image₂ f s t).Nonempty :=
image₂_nonempty_iff.2 ⟨hs, ht⟩
#align finset.nonempty.image₂ Finset.Nonempty.image₂
theorem Nonempty.of_image₂_left (h : (s.image₂ f t).Nonempty) : s.Nonempty :=
(image₂_nonempty_iff.1 h).1
#align finset.nonempty.of_image₂_left Finset.Nonempty.of_image₂_left
theorem Nonempty.of_image₂_right (h : (s.image₂ f t).Nonempty) : t.Nonempty :=
(image₂_nonempty_iff.1 h).2
#align finset.nonempty.of_image₂_right Finset.Nonempty.of_image₂_right
@[simp]
theorem image₂_empty_left : image₂ f ∅ t = ∅ :=
coe_injective <| by simp
#align finset.image₂_empty_left Finset.image₂_empty_left
@[simp]
theorem image₂_empty_right : image₂ f s ∅ = ∅ :=
coe_injective <| by simp
#align finset.image₂_empty_right Finset.image₂_empty_right
@[simp]
theorem image₂_eq_empty_iff : image₂ f s t = ∅ ↔ s = ∅ ∨ t = ∅ := by
simp_rw [← not_nonempty_iff_eq_empty, image₂_nonempty_iff, not_and_or]
#align finset.image₂_eq_empty_iff Finset.image₂_eq_empty_iff
@[simp]
theorem image₂_singleton_left : image₂ f {a} t = t.image fun b => f a b :=
ext fun x => by simp
#align finset.image₂_singleton_left Finset.image₂_singleton_left
@[simp]
theorem image₂_singleton_right : image₂ f s {b} = s.image fun a => f a b :=
ext fun x => by simp
#align finset.image₂_singleton_right Finset.image₂_singleton_right
theorem image₂_singleton_left' : image₂ f {a} t = t.image (f a) :=
image₂_singleton_left
#align finset.image₂_singleton_left' Finset.image₂_singleton_left'
theorem image₂_singleton : image₂ f {a} {b} = {f a b} := by simp
#align finset.image₂_singleton Finset.image₂_singleton
theorem image₂_union_left [DecidableEq α] : image₂ f (s ∪ s') t = image₂ f s t ∪ image₂ f s' t :=
coe_injective <| by
push_cast
exact image2_union_left
#align finset.image₂_union_left Finset.image₂_union_left
theorem image₂_union_right [DecidableEq β] : image₂ f s (t ∪ t') = image₂ f s t ∪ image₂ f s t' :=
coe_injective <| by
push_cast
exact image2_union_right
#align finset.image₂_union_right Finset.image₂_union_right
@[simp]
theorem image₂_insert_left [DecidableEq α] :
image₂ f (insert a s) t = (t.image fun b => f a b) ∪ image₂ f s t :=
coe_injective <| by
push_cast
exact image2_insert_left
#align finset.image₂_insert_left Finset.image₂_insert_left
@[simp]
theorem image₂_insert_right [DecidableEq β] :
image₂ f s (insert b t) = (s.image fun a => f a b) ∪ image₂ f s t :=
coe_injective <| by
push_cast
exact image2_insert_right
#align finset.image₂_insert_right Finset.image₂_insert_right
theorem image₂_inter_left [DecidableEq α] (hf : Injective2 f) :
image₂ f (s ∩ s') t = image₂ f s t ∩ image₂ f s' t :=
coe_injective <| by
push_cast
exact image2_inter_left hf
#align finset.image₂_inter_left Finset.image₂_inter_left
theorem image₂_inter_right [DecidableEq β] (hf : Injective2 f) :
image₂ f s (t ∩ t') = image₂ f s t ∩ image₂ f s t' :=
coe_injective <| by
push_cast
exact image2_inter_right hf
#align finset.image₂_inter_right Finset.image₂_inter_right
theorem image₂_inter_subset_left [DecidableEq α] :
image₂ f (s ∩ s') t ⊆ image₂ f s t ∩ image₂ f s' t :=
coe_subset.1 <| by
push_cast
exact image2_inter_subset_left
#align finset.image₂_inter_subset_left Finset.image₂_inter_subset_left
theorem image₂_inter_subset_right [DecidableEq β] :
image₂ f s (t ∩ t') ⊆ image₂ f s t ∩ image₂ f s t' :=
coe_subset.1 <| by
push_cast
exact image2_inter_subset_right
#align finset.image₂_inter_subset_right Finset.image₂_inter_subset_right
theorem image₂_congr (h : ∀ a ∈ s, ∀ b ∈ t, f a b = f' a b) : image₂ f s t = image₂ f' s t :=
coe_injective <| by
push_cast
exact image2_congr h
#align finset.image₂_congr Finset.image₂_congr
theorem image₂_congr' (h : ∀ a b, f a b = f' a b) : image₂ f s t = image₂ f' s t :=
image₂_congr fun a _ b _ => h a b
#align finset.image₂_congr' Finset.image₂_congr'
variable (s t)
theorem card_image₂_singleton_left (hf : Injective (f a)) : (image₂ f {a} t).card = t.card := by
rw [image₂_singleton_left, card_image_of_injective _ hf]
#align finset.card_image₂_singleton_left Finset.card_image₂_singleton_left
theorem card_image₂_singleton_right (hf : Injective fun a => f a b) :
(image₂ f s {b}).card = s.card := by rw [image₂_singleton_right, card_image_of_injective _ hf]
#align finset.card_image₂_singleton_right Finset.card_image₂_singleton_right
theorem image₂_singleton_inter [DecidableEq β] (t₁ t₂ : Finset β) (hf : Injective (f a)) :
image₂ f {a} (t₁ ∩ t₂) = image₂ f {a} t₁ ∩ image₂ f {a} t₂ := by
simp_rw [image₂_singleton_left, image_inter _ _ hf]
#align finset.image₂_singleton_inter Finset.image₂_singleton_inter
theorem image₂_inter_singleton [DecidableEq α] (s₁ s₂ : Finset α) (hf : Injective fun a => f a b) :
image₂ f (s₁ ∩ s₂) {b} = image₂ f s₁ {b} ∩ image₂ f s₂ {b} := by
simp_rw [image₂_singleton_right, image_inter _ _ hf]
#align finset.image₂_inter_singleton Finset.image₂_inter_singleton
theorem card_le_card_image₂_left {s : Finset α} (hs : s.Nonempty) (hf : ∀ a, Injective (f a)) :
t.card ≤ (image₂ f s t).card := by
obtain ⟨a, ha⟩ := hs
rw [← card_image₂_singleton_left _ (hf a)]
exact card_le_card (image₂_subset_right <| singleton_subset_iff.2 ha)
#align finset.card_le_card_image₂_left Finset.card_le_card_image₂_left
theorem card_le_card_image₂_right {t : Finset β} (ht : t.Nonempty)
(hf : ∀ b, Injective fun a => f a b) : s.card ≤ (image₂ f s t).card := by
obtain ⟨b, hb⟩ := ht
rw [← card_image₂_singleton_right _ (hf b)]
exact card_le_card (image₂_subset_left <| singleton_subset_iff.2 hb)
#align finset.card_le_card_image₂_right Finset.card_le_card_image₂_right
variable {s t}
theorem biUnion_image_left : (s.biUnion fun a => t.image <| f a) = image₂ f s t :=
coe_injective <| by
push_cast
exact Set.iUnion_image_left _
#align finset.bUnion_image_left Finset.biUnion_image_left
theorem biUnion_image_right : (t.biUnion fun b => s.image fun a => f a b) = image₂ f s t :=
coe_injective <| by
push_cast
exact Set.iUnion_image_right _
#align finset.bUnion_image_right Finset.biUnion_image_right
theorem image_image₂ (f : α → β → γ) (g : γ → δ) :
(image₂ f s t).image g = image₂ (fun a b => g (f a b)) s t :=
coe_injective <| by
push_cast
exact image_image2 _ _
#align finset.image_image₂ Finset.image_image₂
theorem image₂_image_left (f : γ → β → δ) (g : α → γ) :
image₂ f (s.image g) t = image₂ (fun a b => f (g a) b) s t :=
coe_injective <| by
push_cast
exact image2_image_left _ _
#align finset.image₂_image_left Finset.image₂_image_left
theorem image₂_image_right (f : α → γ → δ) (g : β → γ) :
image₂ f s (t.image g) = image₂ (fun a b => f a (g b)) s t :=
coe_injective <| by
push_cast
exact image2_image_right _ _
#align finset.image₂_image_right Finset.image₂_image_right
@[simp]
theorem image₂_mk_eq_product [DecidableEq α] [DecidableEq β] (s : Finset α) (t : Finset β) :
image₂ Prod.mk s t = s ×ˢ t := by ext; simp [Prod.ext_iff]
#align finset.image₂_mk_eq_product Finset.image₂_mk_eq_product
@[simp]
theorem image₂_curry (f : α × β → γ) (s : Finset α) (t : Finset β) :
image₂ (curry f) s t = (s ×ˢ t).image f := rfl
#align finset.image₂_curry Finset.image₂_curry
@[simp]
theorem image_uncurry_product (f : α → β → γ) (s : Finset α) (t : Finset β) :
(s ×ˢ t).image (uncurry f) = image₂ f s t := rfl
#align finset.image_uncurry_product Finset.image_uncurry_product
theorem image₂_swap (f : α → β → γ) (s : Finset α) (t : Finset β) :
image₂ f s t = image₂ (fun a b => f b a) t s :=
coe_injective <| by
push_cast
exact image2_swap _ _ _
#align finset.image₂_swap Finset.image₂_swap
@[simp]
theorem image₂_left [DecidableEq α] (h : t.Nonempty) : image₂ (fun x _ => x) s t = s :=
coe_injective <| by
push_cast
exact image2_left h
#align finset.image₂_left Finset.image₂_left
@[simp]
theorem image₂_right [DecidableEq β] (h : s.Nonempty) : image₂ (fun _ y => y) s t = t :=
coe_injective <| by
push_cast
exact image2_right h
#align finset.image₂_right Finset.image₂_right
theorem image₂_assoc {γ : Type*} {u : Finset γ} {f : δ → γ → ε} {g : α → β → δ} {f' : α → ε' → ε}
{g' : β → γ → ε'} (h_assoc : ∀ a b c, f (g a b) c = f' a (g' b c)) :
image₂ f (image₂ g s t) u = image₂ f' s (image₂ g' t u) :=
coe_injective <| by
push_cast
exact image2_assoc h_assoc
#align finset.image₂_assoc Finset.image₂_assoc
theorem image₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : image₂ f s t = image₂ g t s :=
(image₂_swap _ _ _).trans <| by simp_rw [h_comm]
#align finset.image₂_comm Finset.image₂_comm
theorem image₂_left_comm {γ : Type*} {u : Finset γ} {f : α → δ → ε} {g : β → γ → δ}
{f' : α → γ → δ'} {g' : β → δ' → ε} (h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
image₂ f s (image₂ g t u) = image₂ g' t (image₂ f' s u) :=
coe_injective <| by
push_cast
exact image2_left_comm h_left_comm
#align finset.image₂_left_comm Finset.image₂_left_comm
theorem image₂_right_comm {γ : Type*} {u : Finset γ} {f : δ → γ → ε} {g : α → β → δ}
{f' : α → γ → δ'} {g' : δ' → β → ε} (h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
image₂ f (image₂ g s t) u = image₂ g' (image₂ f' s u) t :=
coe_injective <| by
push_cast
exact image2_right_comm h_right_comm
#align finset.image₂_right_comm Finset.image₂_right_comm
theorem image₂_image₂_image₂_comm {γ δ : Type*} {u : Finset γ} {v : Finset δ} [DecidableEq ζ]
[DecidableEq ζ'] [DecidableEq ν] {f : ε → ζ → ν} {g : α → β → ε} {h : γ → δ → ζ}
{f' : ε' → ζ' → ν} {g' : α → γ → ε'} {h' : β → δ → ζ'}
(h_comm : ∀ a b c d, f (g a b) (h c d) = f' (g' a c) (h' b d)) :
image₂ f (image₂ g s t) (image₂ h u v) = image₂ f' (image₂ g' s u) (image₂ h' t v) :=
coe_injective <| by
push_cast
exact image2_image2_image2_comm h_comm
#align finset.image₂_image₂_image₂_comm Finset.image₂_image₂_image₂_comm
theorem image_image₂_distrib {g : γ → δ} {f' : α' → β' → δ} {g₁ : α → α'} {g₂ : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' (g₁ a) (g₂ b)) :
(image₂ f s t).image g = image₂ f' (s.image g₁) (t.image g₂) :=
coe_injective <| by
push_cast
exact image_image2_distrib h_distrib
#align finset.image_image₂_distrib Finset.image_image₂_distrib
theorem image_image₂_distrib_left {g : γ → δ} {f' : α' → β → δ} {g' : α → α'}
(h_distrib : ∀ a b, g (f a b) = f' (g' a) b) :
(image₂ f s t).image g = image₂ f' (s.image g') t :=
coe_injective <| by
push_cast
exact image_image2_distrib_left h_distrib
#align finset.image_image₂_distrib_left Finset.image_image₂_distrib_left
theorem image_image₂_distrib_right {g : γ → δ} {f' : α → β' → δ} {g' : β → β'}
(h_distrib : ∀ a b, g (f a b) = f' a (g' b)) :
(image₂ f s t).image g = image₂ f' s (t.image g') :=
coe_injective <| by
push_cast
exact image_image2_distrib_right h_distrib
#align finset.image_image₂_distrib_right Finset.image_image₂_distrib_right
theorem image₂_image_left_comm {f : α' → β → γ} {g : α → α'} {f' : α → β → δ} {g' : δ → γ}
(h_left_comm : ∀ a b, f (g a) b = g' (f' a b)) :
image₂ f (s.image g) t = (image₂ f' s t).image g' :=
(image_image₂_distrib_left fun a b => (h_left_comm a b).symm).symm
#align finset.image₂_image_left_comm Finset.image₂_image_left_comm
theorem image_image₂_right_comm {f : α → β' → γ} {g : β → β'} {f' : α → β → δ} {g' : δ → γ}
(h_right_comm : ∀ a b, f a (g b) = g' (f' a b)) :
image₂ f s (t.image g) = (image₂ f' s t).image g' :=
(image_image₂_distrib_right fun a b => (h_right_comm a b).symm).symm
#align finset.image_image₂_right_comm Finset.image_image₂_right_comm
theorem image₂_distrib_subset_left {γ : Type*} {u : Finset γ} {f : α → δ → ε} {g : β → γ → δ}
{f₁ : α → β → β'} {f₂ : α → γ → γ'} {g' : β' → γ' → ε}
(h_distrib : ∀ a b c, f a (g b c) = g' (f₁ a b) (f₂ a c)) :
image₂ f s (image₂ g t u) ⊆ image₂ g' (image₂ f₁ s t) (image₂ f₂ s u) :=
coe_subset.1 <| by
push_cast
exact Set.image2_distrib_subset_left h_distrib
#align finset.image₂_distrib_subset_left Finset.image₂_distrib_subset_left
theorem image₂_distrib_subset_right {γ : Type*} {u : Finset γ} {f : δ → γ → ε} {g : α → β → δ}
{f₁ : α → γ → α'} {f₂ : β → γ → β'} {g' : α' → β' → ε}
(h_distrib : ∀ a b c, f (g a b) c = g' (f₁ a c) (f₂ b c)) :
image₂ f (image₂ g s t) u ⊆ image₂ g' (image₂ f₁ s u) (image₂ f₂ t u) :=
coe_subset.1 <| by
push_cast
exact Set.image2_distrib_subset_right h_distrib
#align finset.image₂_distrib_subset_right Finset.image₂_distrib_subset_right
theorem image_image₂_antidistrib {g : γ → δ} {f' : β' → α' → δ} {g₁ : β → β'} {g₂ : α → α'}
(h_antidistrib : ∀ a b, g (f a b) = f' (g₁ b) (g₂ a)) :
(image₂ f s t).image g = image₂ f' (t.image g₁) (s.image g₂) := by
rw [image₂_swap f]
exact image_image₂_distrib fun _ _ => h_antidistrib _ _
#align finset.image_image₂_antidistrib Finset.image_image₂_antidistrib
theorem image_image₂_antidistrib_left {g : γ → δ} {f' : β' → α → δ} {g' : β → β'}
(h_antidistrib : ∀ a b, g (f a b) = f' (g' b) a) :
(image₂ f s t).image g = image₂ f' (t.image g') s :=
coe_injective <| by
push_cast
exact image_image2_antidistrib_left h_antidistrib
#align finset.image_image₂_antidistrib_left Finset.image_image₂_antidistrib_left
theorem image_image₂_antidistrib_right {g : γ → δ} {f' : β → α' → δ} {g' : α → α'}
(h_antidistrib : ∀ a b, g (f a b) = f' b (g' a)) :
(image₂ f s t).image g = image₂ f' t (s.image g') :=
coe_injective <| by
push_cast
exact image_image2_antidistrib_right h_antidistrib
#align finset.image_image₂_antidistrib_right Finset.image_image₂_antidistrib_right
theorem image₂_image_left_anticomm {f : α' → β → γ} {g : α → α'} {f' : β → α → δ} {g' : δ → γ}
(h_left_anticomm : ∀ a b, f (g a) b = g' (f' b a)) :
image₂ f (s.image g) t = (image₂ f' t s).image g' :=
(image_image₂_antidistrib_left fun a b => (h_left_anticomm b a).symm).symm
#align finset.image₂_image_left_anticomm Finset.image₂_image_left_anticomm
theorem image_image₂_right_anticomm {f : α → β' → γ} {g : β → β'} {f' : β → α → δ} {g' : δ → γ}
(h_right_anticomm : ∀ a b, f a (g b) = g' (f' b a)) :
image₂ f s (t.image g) = (image₂ f' t s).image g' :=
(image_image₂_antidistrib_right fun a b => (h_right_anticomm b a).symm).symm
#align finset.image_image₂_right_anticomm Finset.image_image₂_right_anticomm
theorem image₂_left_identity {f : α → γ → γ} {a : α} (h : ∀ b, f a b = b) (t : Finset γ) :
image₂ f {a} t = t :=
coe_injective <| by rw [coe_image₂, coe_singleton, Set.image2_left_identity h]
#align finset.image₂_left_identity Finset.image₂_left_identity
| Mathlib/Data/Finset/NAry.lean | 494 | 495 | theorem image₂_right_identity {f : γ → β → γ} {b : β} (h : ∀ a, f a b = a) (s : Finset γ) :
image₂ f s {b} = s := by | rw [image₂_singleton_right, funext h, image_id']
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
set_option autoImplicit true
universe u v
open Function Order
noncomputable section
def NatOrdinal : Type _ :=
-- Porting note: used to derive LinearOrder & SuccOrder but need to manually define
Ordinal deriving Zero, Inhabited, One, WellFoundedRelation
#align nat_ordinal NatOrdinal
instance NatOrdinal.linearOrder : LinearOrder NatOrdinal := {Ordinal.linearOrder with}
instance NatOrdinal.succOrder : SuccOrder NatOrdinal := {Ordinal.succOrder with}
@[match_pattern]
def Ordinal.toNatOrdinal : Ordinal ≃o NatOrdinal :=
OrderIso.refl _
#align ordinal.to_nat_ordinal Ordinal.toNatOrdinal
@[match_pattern]
def NatOrdinal.toOrdinal : NatOrdinal ≃o Ordinal :=
OrderIso.refl _
#align nat_ordinal.to_ordinal NatOrdinal.toOrdinal
namespace Ordinal
variable {a b c : Ordinal.{u}}
@[simp]
theorem toNatOrdinal_symm_eq : toNatOrdinal.symm = NatOrdinal.toOrdinal :=
rfl
#align ordinal.to_nat_ordinal_symm_eq Ordinal.toNatOrdinal_symm_eq
@[simp]
theorem toNatOrdinal_toOrdinal (a : Ordinal) : NatOrdinal.toOrdinal (toNatOrdinal a) = a :=
rfl
#align ordinal.to_nat_ordinal_to_ordinal Ordinal.toNatOrdinal_toOrdinal
@[simp]
theorem toNatOrdinal_zero : toNatOrdinal 0 = 0 :=
rfl
#align ordinal.to_nat_ordinal_zero Ordinal.toNatOrdinal_zero
@[simp]
theorem toNatOrdinal_one : toNatOrdinal 1 = 1 :=
rfl
#align ordinal.to_nat_ordinal_one Ordinal.toNatOrdinal_one
@[simp]
theorem toNatOrdinal_eq_zero (a) : toNatOrdinal a = 0 ↔ a = 0 :=
Iff.rfl
#align ordinal.to_nat_ordinal_eq_zero Ordinal.toNatOrdinal_eq_zero
@[simp]
theorem toNatOrdinal_eq_one (a) : toNatOrdinal a = 1 ↔ a = 1 :=
Iff.rfl
#align ordinal.to_nat_ordinal_eq_one Ordinal.toNatOrdinal_eq_one
@[simp]
theorem toNatOrdinal_max (a b : Ordinal) :
toNatOrdinal (max a b) = max (toNatOrdinal a) (toNatOrdinal b) :=
rfl
#align ordinal.to_nat_ordinal_max Ordinal.toNatOrdinal_max
@[simp]
theorem toNatOrdinal_min (a b : Ordinal) :
toNatOrdinal (linearOrder.min a b) = linearOrder.min (toNatOrdinal a) (toNatOrdinal b) :=
rfl
#align ordinal.to_nat_ordinal_min Ordinal.toNatOrdinal_min
noncomputable def nadd : Ordinal → Ordinal → Ordinal
| a, b =>
max (blsub.{u, u} a fun a' _ => nadd a' b) (blsub.{u, u} b fun b' _ => nadd a b')
termination_by o₁ o₂ => (o₁, o₂)
#align ordinal.nadd Ordinal.nadd
@[inherit_doc]
scoped[NaturalOps] infixl:65 " ♯ " => Ordinal.nadd
open NaturalOps
noncomputable def nmul : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}
| a, b => sInf {c | ∀ a' < a, ∀ b' < b, nmul a' b ♯ nmul a b' < c ♯ nmul a' b'}
termination_by a b => (a, b)
#align ordinal.nmul Ordinal.nmul
@[inherit_doc]
scoped[NaturalOps] infixl:70 " ⨳ " => Ordinal.nmul
theorem nadd_def (a b : Ordinal) :
a ♯ b = max (blsub.{u, u} a fun a' _ => a' ♯ b) (blsub.{u, u} b fun b' _ => a ♯ b') := by
rw [nadd]
#align ordinal.nadd_def Ordinal.nadd_def
theorem lt_nadd_iff : a < b ♯ c ↔ (∃ b' < b, a ≤ b' ♯ c) ∨ ∃ c' < c, a ≤ b ♯ c' := by
rw [nadd_def]
simp [lt_blsub_iff]
#align ordinal.lt_nadd_iff Ordinal.lt_nadd_iff
theorem nadd_le_iff : b ♯ c ≤ a ↔ (∀ b' < b, b' ♯ c < a) ∧ ∀ c' < c, b ♯ c' < a := by
rw [nadd_def]
simp [blsub_le_iff]
#align ordinal.nadd_le_iff Ordinal.nadd_le_iff
theorem nadd_lt_nadd_left (h : b < c) (a) : a ♯ b < a ♯ c :=
lt_nadd_iff.2 (Or.inr ⟨b, h, le_rfl⟩)
#align ordinal.nadd_lt_nadd_left Ordinal.nadd_lt_nadd_left
theorem nadd_lt_nadd_right (h : b < c) (a) : b ♯ a < c ♯ a :=
lt_nadd_iff.2 (Or.inl ⟨b, h, le_rfl⟩)
#align ordinal.nadd_lt_nadd_right Ordinal.nadd_lt_nadd_right
theorem nadd_le_nadd_left (h : b ≤ c) (a) : a ♯ b ≤ a ♯ c := by
rcases lt_or_eq_of_le h with (h | rfl)
· exact (nadd_lt_nadd_left h a).le
· exact le_rfl
#align ordinal.nadd_le_nadd_left Ordinal.nadd_le_nadd_left
theorem nadd_le_nadd_right (h : b ≤ c) (a) : b ♯ a ≤ c ♯ a := by
rcases lt_or_eq_of_le h with (h | rfl)
· exact (nadd_lt_nadd_right h a).le
· exact le_rfl
#align ordinal.nadd_le_nadd_right Ordinal.nadd_le_nadd_right
variable (a b)
theorem nadd_comm : ∀ a b, a ♯ b = b ♯ a
| a, b => by
rw [nadd_def, nadd_def, max_comm]
congr <;> ext <;> apply nadd_comm
termination_by a b => (a,b)
#align ordinal.nadd_comm Ordinal.nadd_comm
theorem blsub_nadd_of_mono {f : ∀ c < a ♯ b, Ordinal.{max u v}}
(hf : ∀ {i j} (hi hj), i ≤ j → f i hi ≤ f j hj) :
-- Porting note: needed to add universe hint blsub.{u,v} in the line below
blsub.{u,v} _ f =
max (blsub.{u, v} a fun a' ha' => f (a' ♯ b) <| nadd_lt_nadd_right ha' b)
(blsub.{u, v} b fun b' hb' => f (a ♯ b') <| nadd_lt_nadd_left hb' a) := by
apply (blsub_le_iff.2 fun i h => _).antisymm (max_le _ _)
· intro i h
rcases lt_nadd_iff.1 h with (⟨a', ha', hi⟩ | ⟨b', hb', hi⟩)
· exact lt_max_of_lt_left ((hf h (nadd_lt_nadd_right ha' b) hi).trans_lt (lt_blsub _ _ ha'))
· exact lt_max_of_lt_right ((hf h (nadd_lt_nadd_left hb' a) hi).trans_lt (lt_blsub _ _ hb'))
all_goals
apply blsub_le_of_brange_subset.{u, u, v}
rintro c ⟨d, hd, rfl⟩
apply mem_brange_self
#align ordinal.blsub_nadd_of_mono Ordinal.blsub_nadd_of_mono
theorem nadd_assoc (a b c) : a ♯ b ♯ c = a ♯ (b ♯ c) := by
rw [nadd_def a (b ♯ c), nadd_def, blsub_nadd_of_mono, blsub_nadd_of_mono, max_assoc]
· congr <;> ext <;> apply nadd_assoc
· exact fun _ _ h => nadd_le_nadd_left h a
· exact fun _ _ h => nadd_le_nadd_right h c
termination_by (a, b, c)
#align ordinal.nadd_assoc Ordinal.nadd_assoc
@[simp]
theorem nadd_zero : a ♯ 0 = a := by
induction' a using Ordinal.induction with a IH
rw [nadd_def, blsub_zero, max_zero_right]
convert blsub_id a
rename_i hb
exact IH _ hb
#align ordinal.nadd_zero Ordinal.nadd_zero
@[simp]
theorem zero_nadd : 0 ♯ a = a := by rw [nadd_comm, nadd_zero]
#align ordinal.zero_nadd Ordinal.zero_nadd
@[simp]
theorem nadd_one : a ♯ 1 = succ a := by
induction' a using Ordinal.induction with a IH
rw [nadd_def, blsub_one, nadd_zero, max_eq_right_iff, blsub_le_iff]
intro i hi
rwa [IH i hi, succ_lt_succ_iff]
#align ordinal.nadd_one Ordinal.nadd_one
@[simp]
theorem one_nadd : 1 ♯ a = succ a := by rw [nadd_comm, nadd_one]
#align ordinal.one_nadd Ordinal.one_nadd
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 326 | 326 | theorem nadd_succ : a ♯ succ b = succ (a ♯ b) := by | rw [← nadd_one (a ♯ b), nadd_assoc, nadd_one]
|
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
namespace CategoryTheory
universe w v u
open Category Iso
-- intended to be used with explicit universe parameters
@[nolint checkUnivs]
class Bicategory (B : Type u) extends CategoryStruct.{v} B where
-- category structure on the collection of 1-morphisms:
homCategory : ∀ a b : B, Category.{w} (a ⟶ b) := by infer_instance
-- left whiskering:
whiskerLeft {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) : f ≫ g ⟶ f ≫ h
-- right whiskering:
whiskerRight {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) : f ≫ h ⟶ g ≫ h
-- associator:
associator {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : (f ≫ g) ≫ h ≅ f ≫ g ≫ h
-- left unitor:
leftUnitor {a b : B} (f : a ⟶ b) : 𝟙 a ≫ f ≅ f
-- right unitor:
rightUnitor {a b : B} (f : a ⟶ b) : f ≫ 𝟙 b ≅ f
-- axioms for left whiskering:
whiskerLeft_id : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerLeft f (𝟙 g) = 𝟙 (f ≫ g) := by
aesop_cat
whiskerLeft_comp :
∀ {a b c} (f : a ⟶ b) {g h i : b ⟶ c} (η : g ⟶ h) (θ : h ⟶ i),
whiskerLeft f (η ≫ θ) = whiskerLeft f η ≫ whiskerLeft f θ := by
aesop_cat
id_whiskerLeft :
∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
whiskerLeft (𝟙 a) η = (leftUnitor f).hom ≫ η ≫ (leftUnitor g).inv := by
aesop_cat
comp_whiskerLeft :
∀ {a b c d} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h'),
whiskerLeft (f ≫ g) η =
(associator f g h).hom ≫ whiskerLeft f (whiskerLeft g η) ≫ (associator f g h').inv := by
aesop_cat
-- axioms for right whiskering:
id_whiskerRight : ∀ {a b c} (f : a ⟶ b) (g : b ⟶ c), whiskerRight (𝟙 f) g = 𝟙 (f ≫ g) := by
aesop_cat
comp_whiskerRight :
∀ {a b c} {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) (i : b ⟶ c),
whiskerRight (η ≫ θ) i = whiskerRight η i ≫ whiskerRight θ i := by
aesop_cat
whiskerRight_id :
∀ {a b} {f g : a ⟶ b} (η : f ⟶ g),
whiskerRight η (𝟙 b) = (rightUnitor f).hom ≫ η ≫ (rightUnitor g).inv := by
aesop_cat
whiskerRight_comp :
∀ {a b c d} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d),
whiskerRight η (g ≫ h) =
(associator f g h).inv ≫ whiskerRight (whiskerRight η g) h ≫ (associator f' g h).hom := by
aesop_cat
-- associativity of whiskerings:
whisker_assoc :
∀ {a b c d} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d),
whiskerRight (whiskerLeft f η) h =
(associator f g h).hom ≫ whiskerLeft f (whiskerRight η h) ≫ (associator f g' h).inv := by
aesop_cat
-- exchange law of left and right whiskerings:
whisker_exchange :
∀ {a b c} {f g : a ⟶ b} {h i : b ⟶ c} (η : f ⟶ g) (θ : h ⟶ i),
whiskerLeft f θ ≫ whiskerRight η i = whiskerRight η h ≫ whiskerLeft g θ := by
aesop_cat
-- pentagon identity:
pentagon :
∀ {a b c d e} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e),
whiskerRight (associator f g h).hom i ≫
(associator f (g ≫ h) i).hom ≫ whiskerLeft f (associator g h i).hom =
(associator (f ≫ g) h i).hom ≫ (associator f g (h ≫ i)).hom := by
aesop_cat
-- triangle identity:
triangle :
∀ {a b c} (f : a ⟶ b) (g : b ⟶ c),
(associator f (𝟙 b) g).hom ≫ whiskerLeft f (leftUnitor g).hom
= whiskerRight (rightUnitor f).hom g := by
aesop_cat
#align category_theory.bicategory CategoryTheory.Bicategory
#align category_theory.bicategory.hom_category CategoryTheory.Bicategory.homCategory
#align category_theory.bicategory.whisker_left CategoryTheory.Bicategory.whiskerLeft
#align category_theory.bicategory.whisker_right CategoryTheory.Bicategory.whiskerRight
#align category_theory.bicategory.left_unitor CategoryTheory.Bicategory.leftUnitor
#align category_theory.bicategory.right_unitor CategoryTheory.Bicategory.rightUnitor
#align category_theory.bicategory.whisker_left_id' CategoryTheory.Bicategory.whiskerLeft_id
#align category_theory.bicategory.whisker_left_comp' CategoryTheory.Bicategory.whiskerLeft_comp
#align category_theory.bicategory.id_whisker_left' CategoryTheory.Bicategory.id_whiskerLeft
#align category_theory.bicategory.comp_whisker_left' CategoryTheory.Bicategory.comp_whiskerLeft
#align category_theory.bicategory.id_whisker_right' CategoryTheory.Bicategory.id_whiskerRight
#align category_theory.bicategory.comp_whisker_right' CategoryTheory.Bicategory.comp_whiskerRight
#align category_theory.bicategory.whisker_right_id' CategoryTheory.Bicategory.whiskerRight_id
#align category_theory.bicategory.whisker_right_comp' CategoryTheory.Bicategory.whiskerRight_comp
#align category_theory.bicategory.whisker_assoc' CategoryTheory.Bicategory.whisker_assoc
#align category_theory.bicategory.whisker_exchange' CategoryTheory.Bicategory.whisker_exchange
#align category_theory.bicategory.pentagon' CategoryTheory.Bicategory.pentagon
#align category_theory.bicategory.triangle' CategoryTheory.Bicategory.triangle
namespace Bicategory
scoped infixr:81 " ◁ " => Bicategory.whiskerLeft
scoped infixl:81 " ▷ " => Bicategory.whiskerRight
scoped notation "α_" => Bicategory.associator
scoped notation "λ_" => Bicategory.leftUnitor
scoped notation "ρ_" => Bicategory.rightUnitor
attribute [instance] homCategory
attribute [reassoc]
whiskerLeft_comp id_whiskerLeft comp_whiskerLeft comp_whiskerRight whiskerRight_id
whiskerRight_comp whisker_assoc whisker_exchange
attribute [reassoc (attr := simp)] pentagon triangle
attribute [simp]
whiskerLeft_id whiskerLeft_comp id_whiskerLeft comp_whiskerLeft id_whiskerRight comp_whiskerRight
whiskerRight_id whiskerRight_comp whisker_assoc
variable {B : Type u} [Bicategory.{w, v} B] {a b c d e : B}
@[reassoc (attr := simp)]
theorem whiskerLeft_hom_inv (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) :
f ◁ η.hom ≫ f ◁ η.inv = 𝟙 (f ≫ g) := by rw [← whiskerLeft_comp, hom_inv_id, whiskerLeft_id]
#align category_theory.bicategory.hom_inv_whisker_left CategoryTheory.Bicategory.whiskerLeft_hom_inv
@[reassoc (attr := simp)]
theorem hom_inv_whiskerRight {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) :
η.hom ▷ h ≫ η.inv ▷ h = 𝟙 (f ≫ h) := by rw [← comp_whiskerRight, hom_inv_id, id_whiskerRight]
#align category_theory.bicategory.hom_inv_whisker_right CategoryTheory.Bicategory.hom_inv_whiskerRight
@[reassoc (attr := simp)]
theorem whiskerLeft_inv_hom (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) :
f ◁ η.inv ≫ f ◁ η.hom = 𝟙 (f ≫ h) := by rw [← whiskerLeft_comp, inv_hom_id, whiskerLeft_id]
#align category_theory.bicategory.inv_hom_whisker_left CategoryTheory.Bicategory.whiskerLeft_inv_hom
@[reassoc (attr := simp)]
theorem inv_hom_whiskerRight {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) :
η.inv ▷ h ≫ η.hom ▷ h = 𝟙 (g ≫ h) := by rw [← comp_whiskerRight, inv_hom_id, id_whiskerRight]
#align category_theory.bicategory.inv_hom_whisker_right CategoryTheory.Bicategory.inv_hom_whiskerRight
@[simps]
def whiskerLeftIso (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) : f ≫ g ≅ f ≫ h where
hom := f ◁ η.hom
inv := f ◁ η.inv
#align category_theory.bicategory.whisker_left_iso CategoryTheory.Bicategory.whiskerLeftIso
instance whiskerLeft_isIso (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) [IsIso η] : IsIso (f ◁ η) :=
(whiskerLeftIso f (asIso η)).isIso_hom
#align category_theory.bicategory.whisker_left_is_iso CategoryTheory.Bicategory.whiskerLeft_isIso
@[simp]
theorem inv_whiskerLeft (f : a ⟶ b) {g h : b ⟶ c} (η : g ⟶ h) [IsIso η] :
inv (f ◁ η) = f ◁ inv η := by
apply IsIso.inv_eq_of_hom_inv_id
simp only [← whiskerLeft_comp, whiskerLeft_id, IsIso.hom_inv_id]
#align category_theory.bicategory.inv_whisker_left CategoryTheory.Bicategory.inv_whiskerLeft
@[simps!]
def whiskerRightIso {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) : f ≫ h ≅ g ≫ h where
hom := η.hom ▷ h
inv := η.inv ▷ h
#align category_theory.bicategory.whisker_right_iso CategoryTheory.Bicategory.whiskerRightIso
instance whiskerRight_isIso {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) [IsIso η] : IsIso (η ▷ h) :=
(whiskerRightIso (asIso η) h).isIso_hom
#align category_theory.bicategory.whisker_right_is_iso CategoryTheory.Bicategory.whiskerRight_isIso
@[simp]
theorem inv_whiskerRight {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) [IsIso η] :
inv (η ▷ h) = inv η ▷ h := by
apply IsIso.inv_eq_of_hom_inv_id
simp only [← comp_whiskerRight, id_whiskerRight, IsIso.hom_inv_id]
#align category_theory.bicategory.inv_whisker_right CategoryTheory.Bicategory.inv_whiskerRight
@[reassoc (attr := simp)]
theorem pentagon_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i =
(α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv :=
eq_of_inv_eq_inv (by simp)
#align category_theory.bicategory.pentagon_inv CategoryTheory.Bicategory.pentagon_inv
@[reassoc (attr := simp)]
theorem pentagon_inv_inv_hom_hom_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom =
f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv := by
rw [← cancel_epi (f ◁ (α_ g h i).inv), ← cancel_mono (α_ (f ≫ g) h i).inv]
simp
#align category_theory.bicategory.pentagon_inv_inv_hom_hom_inv CategoryTheory.Bicategory.pentagon_inv_inv_hom_hom_inv
@[reassoc (attr := simp)]
theorem pentagon_inv_hom_hom_hom_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i ≫ (α_ f (g ≫ h) i).hom =
(α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv :=
eq_of_inv_eq_inv (by simp)
#align category_theory.bicategory.pentagon_inv_hom_hom_hom_inv CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_inv
@[reassoc (attr := simp)]
theorem pentagon_hom_inv_inv_inv_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv =
(α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i := by
simp [← cancel_epi (f ◁ (α_ g h i).inv)]
#align category_theory.bicategory.pentagon_hom_inv_inv_inv_inv CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_inv
@[reassoc (attr := simp)]
theorem pentagon_hom_hom_inv_hom_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ (f ≫ g) h i).hom ≫ (α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv =
(α_ f g h).hom ▷ i ≫ (α_ f (g ≫ h) i).hom :=
eq_of_inv_eq_inv (by simp)
#align category_theory.bicategory.pentagon_hom_hom_inv_hom_hom CategoryTheory.Bicategory.pentagon_hom_hom_inv_hom_hom
@[reassoc (attr := simp)]
theorem pentagon_hom_inv_inv_inv_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv =
(α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i := by
rw [← cancel_epi (α_ f g (h ≫ i)).inv, ← cancel_mono ((α_ f g h).inv ▷ i)]
simp
#align category_theory.bicategory.pentagon_hom_inv_inv_inv_hom CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_hom
@[reassoc (attr := simp)]
theorem pentagon_hom_hom_inv_inv_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f (g ≫ h) i).hom ≫ f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv =
(α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom :=
eq_of_inv_eq_inv (by simp)
#align category_theory.bicategory.pentagon_hom_hom_inv_inv_hom CategoryTheory.Bicategory.pentagon_hom_hom_inv_inv_hom
@[reassoc (attr := simp)]
theorem pentagon_inv_hom_hom_hom_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom ≫ (α_ f g (h ≫ i)).hom =
(α_ f (g ≫ h) i).hom ≫ f ◁ (α_ g h i).hom := by
simp [← cancel_epi ((α_ f g h).hom ▷ i)]
#align category_theory.bicategory.pentagon_inv_hom_hom_hom_hom CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_hom
@[reassoc (attr := simp)]
theorem pentagon_inv_inv_hom_inv_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i =
f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv :=
eq_of_inv_eq_inv (by simp)
#align category_theory.bicategory.pentagon_inv_inv_hom_inv_inv CategoryTheory.Bicategory.pentagon_inv_inv_hom_inv_inv
theorem triangle_assoc_comp_left (f : a ⟶ b) (g : b ⟶ c) :
(α_ f (𝟙 b) g).hom ≫ f ◁ (λ_ g).hom = (ρ_ f).hom ▷ g :=
triangle f g
#align category_theory.bicategory.triangle_assoc_comp_left CategoryTheory.Bicategory.triangle_assoc_comp_left
@[reassoc (attr := simp)]
theorem triangle_assoc_comp_right (f : a ⟶ b) (g : b ⟶ c) :
(α_ f (𝟙 b) g).inv ≫ (ρ_ f).hom ▷ g = f ◁ (λ_ g).hom := by rw [← triangle, inv_hom_id_assoc]
#align category_theory.bicategory.triangle_assoc_comp_right CategoryTheory.Bicategory.triangle_assoc_comp_right
@[reassoc (attr := simp)]
theorem triangle_assoc_comp_right_inv (f : a ⟶ b) (g : b ⟶ c) :
(ρ_ f).inv ▷ g ≫ (α_ f (𝟙 b) g).hom = f ◁ (λ_ g).inv := by
simp [← cancel_mono (f ◁ (λ_ g).hom)]
#align category_theory.bicategory.triangle_assoc_comp_right_inv CategoryTheory.Bicategory.triangle_assoc_comp_right_inv
@[reassoc (attr := simp)]
theorem triangle_assoc_comp_left_inv (f : a ⟶ b) (g : b ⟶ c) :
f ◁ (λ_ g).inv ≫ (α_ f (𝟙 b) g).inv = (ρ_ f).inv ▷ g := by
simp [← cancel_mono ((ρ_ f).hom ▷ g)]
#align category_theory.bicategory.triangle_assoc_comp_left_inv CategoryTheory.Bicategory.triangle_assoc_comp_left_inv
@[reassoc]
theorem associator_naturality_left {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
η ▷ g ▷ h ≫ (α_ f' g h).hom = (α_ f g h).hom ≫ η ▷ (g ≫ h) := by simp
#align category_theory.bicategory.associator_naturality_left CategoryTheory.Bicategory.associator_naturality_left
@[reassoc]
theorem associator_inv_naturality_left {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
η ▷ (g ≫ h) ≫ (α_ f' g h).inv = (α_ f g h).inv ≫ η ▷ g ▷ h := by simp
#align category_theory.bicategory.associator_inv_naturality_left CategoryTheory.Bicategory.associator_inv_naturality_left
@[reassoc]
theorem whiskerRight_comp_symm {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
η ▷ g ▷ h = (α_ f g h).hom ≫ η ▷ (g ≫ h) ≫ (α_ f' g h).inv := by simp
#align category_theory.bicategory.whisker_right_comp_symm CategoryTheory.Bicategory.whiskerRight_comp_symm
@[reassoc]
theorem associator_naturality_middle (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
(f ◁ η) ▷ h ≫ (α_ f g' h).hom = (α_ f g h).hom ≫ f ◁ η ▷ h := by simp
#align category_theory.bicategory.associator_naturality_middle CategoryTheory.Bicategory.associator_naturality_middle
@[reassoc]
theorem associator_inv_naturality_middle (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
f ◁ η ▷ h ≫ (α_ f g' h).inv = (α_ f g h).inv ≫ (f ◁ η) ▷ h := by simp
#align category_theory.bicategory.associator_inv_naturality_middle CategoryTheory.Bicategory.associator_inv_naturality_middle
@[reassoc]
theorem whisker_assoc_symm (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
f ◁ η ▷ h = (α_ f g h).inv ≫ (f ◁ η) ▷ h ≫ (α_ f g' h).hom := by simp
#align category_theory.bicategory.whisker_assoc_symm CategoryTheory.Bicategory.whisker_assoc_symm
@[reassoc]
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 369 | 370 | theorem associator_naturality_right (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h') :
(f ≫ g) ◁ η ≫ (α_ f g h').hom = (α_ f g h).hom ≫ f ◁ g ◁ η := by | simp
|
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
-- @[simp]
theorem join_singleton (l : List α) : [l].join = l := by rw [join, join, append_nil]
#align list.join_singleton List.join_singleton
@[simp]
theorem join_eq_nil : ∀ {L : List (List α)}, join L = [] ↔ ∀ l ∈ L, l = []
| [] => iff_of_true rfl (forall_mem_nil _)
| l :: L => by simp only [join, append_eq_nil, join_eq_nil, forall_mem_cons]
#align list.join_eq_nil List.join_eq_nil
@[simp]
theorem join_append (L₁ L₂ : List (List α)) : join (L₁ ++ L₂) = join L₁ ++ join L₂ := by
induction L₁
· rfl
· simp [*]
#align list.join_append List.join_append
theorem join_concat (L : List (List α)) (l : List α) : join (L.concat l) = join L ++ l := by simp
#align list.join_concat List.join_concat
@[simp]
theorem join_filter_not_isEmpty :
∀ {L : List (List α)}, join (L.filter fun l => !l.isEmpty) = L.join
| [] => rfl
| [] :: L => by
simp [join_filter_not_isEmpty (L := L), isEmpty_iff_eq_nil]
| (a :: l) :: L => by
simp [join_filter_not_isEmpty (L := L)]
#align list.join_filter_empty_eq_ff List.join_filter_not_isEmpty
@[deprecated (since := "2024-02-25")] alias join_filter_isEmpty_eq_false := join_filter_not_isEmpty
@[simp]
theorem join_filter_ne_nil [DecidablePred fun l : List α => l ≠ []] {L : List (List α)} :
join (L.filter fun l => l ≠ []) = L.join := by
simp [join_filter_not_isEmpty, ← isEmpty_iff_eq_nil]
#align list.join_filter_ne_nil List.join_filter_ne_nil
theorem join_join (l : List (List (List α))) : l.join.join = (l.map join).join := by
induction l <;> simp [*]
#align list.join_join List.join_join
lemma length_join' (L : List (List α)) : length (join L) = Nat.sum (map length L) := by
induction L <;> [rfl; simp only [*, join, map, Nat.sum_cons, length_append]]
lemma countP_join' (p : α → Bool) :
∀ L : List (List α), countP p L.join = Nat.sum (L.map (countP p))
| [] => rfl
| a :: l => by rw [join, countP_append, map_cons, Nat.sum_cons, countP_join' _ l]
lemma count_join' [BEq α] (L : List (List α)) (a : α) :
L.join.count a = Nat.sum (L.map (count a)) := countP_join' _ _
lemma length_bind' (l : List α) (f : α → List β) :
length (l.bind f) = Nat.sum (map (length ∘ f) l) := by rw [List.bind, length_join', map_map]
lemma countP_bind' (p : β → Bool) (l : List α) (f : α → List β) :
countP p (l.bind f) = Nat.sum (map (countP p ∘ f) l) := by rw [List.bind, countP_join', map_map]
lemma count_bind' [BEq β] (l : List α) (f : α → List β) (x : β) :
count x (l.bind f) = Nat.sum (map (count x ∘ f) l) := countP_bind' _ _ _
@[simp]
theorem bind_eq_nil {l : List α} {f : α → List β} : List.bind l f = [] ↔ ∀ x ∈ l, f x = [] :=
join_eq_nil.trans <| by
simp only [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
#align list.bind_eq_nil List.bind_eq_nil
theorem take_sum_join' (L : List (List α)) (i : ℕ) :
L.join.take (Nat.sum ((L.map length).take i)) = (L.take i).join := by
induction L generalizing i
· simp
· cases i <;> simp [take_append, *]
| Mathlib/Data/List/Join.lean | 115 | 119 | theorem drop_sum_join' (L : List (List α)) (i : ℕ) :
L.join.drop (Nat.sum ((L.map length).take i)) = (L.drop i).join := by |
induction L generalizing i
· simp
· cases i <;> simp [drop_append, *]
|
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_theory.monoidal.braided from "leanprover-community/mathlib"@"2efd2423f8d25fa57cf7a179f5d8652ab4d0df44"
open CategoryTheory MonoidalCategory
universe v v₁ v₂ v₃ u u₁ u₂ u₃
namespace CategoryTheory
class BraidedCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] where
braiding : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X
braiding_naturality_right :
∀ (X : C) {Y Z : C} (f : Y ⟶ Z),
X ◁ f ≫ (braiding X Z).hom = (braiding X Y).hom ≫ f ▷ X := by
aesop_cat
braiding_naturality_left :
∀ {X Y : C} (f : X ⟶ Y) (Z : C),
f ▷ Z ≫ (braiding Y Z).hom = (braiding X Z).hom ≫ Z ◁ f := by
aesop_cat
hexagon_forward :
∀ X Y Z : C,
(α_ X Y Z).hom ≫ (braiding X (Y ⊗ Z)).hom ≫ (α_ Y Z X).hom =
((braiding X Y).hom ▷ Z) ≫ (α_ Y X Z).hom ≫ (Y ◁ (braiding X Z).hom) := by
aesop_cat
hexagon_reverse :
∀ X Y Z : C,
(α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv =
(X ◁ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ▷ Y) := by
aesop_cat
#align category_theory.braided_category CategoryTheory.BraidedCategory
attribute [reassoc (attr := simp)]
BraidedCategory.braiding_naturality_left
BraidedCategory.braiding_naturality_right
attribute [reassoc] BraidedCategory.hexagon_forward BraidedCategory.hexagon_reverse
open Category
open MonoidalCategory
open BraidedCategory
@[inherit_doc]
notation "β_" => BraidedCategory.braiding
def braidedCategoryOfFaithful {C D : Type*} [Category C] [Category D] [MonoidalCategory C]
[MonoidalCategory D] (F : MonoidalFunctor C D) [F.Faithful] [BraidedCategory D]
(β : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X)
(w : ∀ X Y, F.μ _ _ ≫ F.map (β X Y).hom = (β_ _ _).hom ≫ F.μ _ _) : BraidedCategory C where
braiding := β
braiding_naturality_left := by
intros
apply F.map_injective
refine (cancel_epi (F.μ ?_ ?_)).1 ?_
rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_left_assoc, w, Functor.map_comp,
reassoc_of% w, braiding_naturality_left_assoc, LaxMonoidalFunctor.μ_natural_right]
braiding_naturality_right := by
intros
apply F.map_injective
refine (cancel_epi (F.μ ?_ ?_)).1 ?_
rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_right_assoc, w, Functor.map_comp,
reassoc_of% w, braiding_naturality_right_assoc, LaxMonoidalFunctor.μ_natural_left]
hexagon_forward := by
intros
apply F.map_injective
refine (cancel_epi (F.μ _ _)).1 ?_
refine (cancel_epi (F.μ _ _ ▷ _)).1 ?_
rw [Functor.map_comp, Functor.map_comp, Functor.map_comp, Functor.map_comp, ←
LaxMonoidalFunctor.μ_natural_left_assoc, ← comp_whiskerRight_assoc, w,
comp_whiskerRight_assoc, LaxMonoidalFunctor.associativity_assoc,
LaxMonoidalFunctor.associativity_assoc, ← LaxMonoidalFunctor.μ_natural_right, ←
MonoidalCategory.whiskerLeft_comp_assoc, w, MonoidalCategory.whiskerLeft_comp_assoc,
reassoc_of% w, braiding_naturality_right_assoc,
LaxMonoidalFunctor.associativity, hexagon_forward_assoc]
hexagon_reverse := by
intros
apply F.toFunctor.map_injective
refine (cancel_epi (F.μ _ _)).1 ?_
refine (cancel_epi (_ ◁ F.μ _ _)).1 ?_
rw [Functor.map_comp, Functor.map_comp, Functor.map_comp, Functor.map_comp, ←
LaxMonoidalFunctor.μ_natural_right_assoc, ← MonoidalCategory.whiskerLeft_comp_assoc, w,
MonoidalCategory.whiskerLeft_comp_assoc, LaxMonoidalFunctor.associativity_inv_assoc,
LaxMonoidalFunctor.associativity_inv_assoc, ← LaxMonoidalFunctor.μ_natural_left,
← comp_whiskerRight_assoc, w, comp_whiskerRight_assoc, reassoc_of% w,
braiding_naturality_left_assoc, LaxMonoidalFunctor.associativity_inv, hexagon_reverse_assoc]
#align category_theory.braided_category_of_faithful CategoryTheory.braidedCategoryOfFaithful
noncomputable def braidedCategoryOfFullyFaithful {C D : Type*} [Category C] [Category D]
[MonoidalCategory C] [MonoidalCategory D] (F : MonoidalFunctor C D) [F.Full]
[F.Faithful] [BraidedCategory D] : BraidedCategory C :=
braidedCategoryOfFaithful F
(fun X Y => F.toFunctor.preimageIso
((asIso (F.μ _ _)).symm ≪≫ β_ (F.obj X) (F.obj Y) ≪≫ asIso (F.μ _ _)))
(by aesop_cat)
#align category_theory.braided_category_of_fully_faithful CategoryTheory.braidedCategoryOfFullyFaithful
section
variable (C : Type u₁) [Category.{v₁} C] [MonoidalCategory C] [BraidedCategory C]
theorem braiding_leftUnitor_aux₁ (X : C) :
(α_ (𝟙_ C) (𝟙_ C) X).hom ≫
(𝟙_ C ◁ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).hom ▷ _) =
((λ_ _).hom ▷ X) ≫ (β_ X (𝟙_ C)).inv := by
coherence
#align category_theory.braiding_left_unitor_aux₁ CategoryTheory.braiding_leftUnitor_aux₁
theorem braiding_leftUnitor_aux₂ (X : C) :
((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ ((λ_ X).hom ▷ 𝟙_ C) = (ρ_ X).hom ▷ 𝟙_ C :=
calc
((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ ((λ_ X).hom ▷ 𝟙_ C) =
((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ (α_ _ _ _).hom ≫ (α_ _ _ _).inv ≫ ((λ_ X).hom ▷ 𝟙_ C) := by
coherence
_ = ((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ (α_ _ _ _).hom ≫ (_ ◁ (β_ X _).hom) ≫
(_ ◁ (β_ X _).inv) ≫ (α_ _ _ _).inv ≫ ((λ_ X).hom ▷ 𝟙_ C) := by
simp
_ = (α_ _ _ _).hom ≫ (β_ _ _).hom ≫ (α_ _ _ _).hom ≫ (_ ◁ (β_ X _).inv) ≫ (α_ _ _ _).inv ≫
((λ_ X).hom ▷ 𝟙_ C) := by
(slice_lhs 1 3 => rw [← hexagon_forward]); simp only [assoc]
_ = (α_ _ _ _).hom ≫ (β_ _ _).hom ≫ ((λ_ _).hom ▷ X) ≫ (β_ X _).inv := by
rw [braiding_leftUnitor_aux₁]
_ = (α_ _ _ _).hom ≫ (_ ◁ (λ_ _).hom) ≫ (β_ _ _).hom ≫ (β_ X _).inv := by
(slice_lhs 2 3 => rw [← braiding_naturality_right]); simp only [assoc]
_ = (α_ _ _ _).hom ≫ (_ ◁ (λ_ _).hom) := by rw [Iso.hom_inv_id, comp_id]
_ = (ρ_ X).hom ▷ 𝟙_ C := by rw [triangle]
#align category_theory.braiding_left_unitor_aux₂ CategoryTheory.braiding_leftUnitor_aux₂
@[reassoc]
theorem braiding_leftUnitor (X : C) : (β_ X (𝟙_ C)).hom ≫ (λ_ X).hom = (ρ_ X).hom := by
rw [← whiskerRight_iff, comp_whiskerRight, braiding_leftUnitor_aux₂]
#align category_theory.braiding_left_unitor CategoryTheory.braiding_leftUnitor
theorem braiding_rightUnitor_aux₁ (X : C) :
(α_ X (𝟙_ C) (𝟙_ C)).inv ≫
((β_ (𝟙_ C) X).inv ▷ 𝟙_ C) ≫ (α_ _ X _).hom ≫ (_ ◁ (ρ_ X).hom) =
(X ◁ (ρ_ _).hom) ≫ (β_ (𝟙_ C) X).inv := by
coherence
#align category_theory.braiding_right_unitor_aux₁ CategoryTheory.braiding_rightUnitor_aux₁
theorem braiding_rightUnitor_aux₂ (X : C) :
(𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (𝟙_ C ◁ (ρ_ X).hom) = 𝟙_ C ◁ (λ_ X).hom :=
calc
(𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (𝟙_ C ◁ (ρ_ X).hom) =
(𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (α_ _ _ _).inv ≫ (α_ _ _ _).hom ≫ (𝟙_ C ◁ (ρ_ X).hom) := by
coherence
_ = (𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (α_ _ _ _).inv ≫ ((β_ _ X).hom ▷ _) ≫
((β_ _ X).inv ▷ _) ≫ (α_ _ _ _).hom ≫ (𝟙_ C ◁ (ρ_ X).hom) := by
simp
_ = (α_ _ _ _).inv ≫ (β_ _ _).hom ≫ (α_ _ _ _).inv ≫ ((β_ _ X).inv ▷ _) ≫ (α_ _ _ _).hom ≫
(𝟙_ C ◁ (ρ_ X).hom) := by
(slice_lhs 1 3 => rw [← hexagon_reverse]); simp only [assoc]
_ = (α_ _ _ _).inv ≫ (β_ _ _).hom ≫ (X ◁ (ρ_ _).hom) ≫ (β_ _ X).inv := by
rw [braiding_rightUnitor_aux₁]
_ = (α_ _ _ _).inv ≫ ((ρ_ _).hom ▷ _) ≫ (β_ _ X).hom ≫ (β_ _ _).inv := by
(slice_lhs 2 3 => rw [← braiding_naturality_left]); simp only [assoc]
_ = (α_ _ _ _).inv ≫ ((ρ_ _).hom ▷ _) := by rw [Iso.hom_inv_id, comp_id]
_ = 𝟙_ C ◁ (λ_ X).hom := by rw [triangle_assoc_comp_right]
#align category_theory.braiding_right_unitor_aux₂ CategoryTheory.braiding_rightUnitor_aux₂
@[reassoc]
theorem braiding_rightUnitor (X : C) : (β_ (𝟙_ C) X).hom ≫ (ρ_ X).hom = (λ_ X).hom := by
rw [← whiskerLeft_iff, MonoidalCategory.whiskerLeft_comp, braiding_rightUnitor_aux₂]
#align category_theory.braiding_right_unitor CategoryTheory.braiding_rightUnitor
@[reassoc, simp]
theorem braiding_tensorUnit_left (X : C) : (β_ (𝟙_ C) X).hom = (λ_ X).hom ≫ (ρ_ X).inv := by
simp [← braiding_rightUnitor]
@[reassoc, simp]
theorem braiding_inv_tensorUnit_left (X : C) : (β_ (𝟙_ C) X).inv = (ρ_ X).hom ≫ (λ_ X).inv := by
rw [Iso.inv_ext]
rw [braiding_tensorUnit_left]
coherence
@[reassoc]
| Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 342 | 343 | theorem leftUnitor_inv_braiding (X : C) : (λ_ X).inv ≫ (β_ (𝟙_ C) X).hom = (ρ_ X).inv := by |
simp
|
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
open Function Polynomial
class PerfectRing (R : Type*) (p : ℕ) [CommSemiring R] [ExpChar R p] : Prop where
bijective_frobenius : Bijective <| frobenius R p
section PerfectRing
variable (R : Type*) (p m n : ℕ) [CommSemiring R] [ExpChar R p]
lemma PerfectRing.ofSurjective (R : Type*) (p : ℕ) [CommRing R] [ExpChar R p]
[IsReduced R] (h : Surjective <| frobenius R p) : PerfectRing R p :=
⟨frobenius_inj R p, h⟩
#align perfect_ring.of_surjective PerfectRing.ofSurjective
instance PerfectRing.ofFiniteOfIsReduced (R : Type*) [CommRing R] [ExpChar R p]
[Finite R] [IsReduced R] : PerfectRing R p :=
ofSurjective _ _ <| Finite.surjective_of_injective (frobenius_inj R p)
variable [PerfectRing R p]
@[simp]
theorem bijective_frobenius : Bijective (frobenius R p) := PerfectRing.bijective_frobenius
theorem bijective_iterateFrobenius : Bijective (iterateFrobenius R p n) :=
coe_iterateFrobenius R p n ▸ (bijective_frobenius R p).iterate n
@[simp]
theorem injective_frobenius : Injective (frobenius R p) := (bijective_frobenius R p).1
@[simp]
theorem surjective_frobenius : Surjective (frobenius R p) := (bijective_frobenius R p).2
@[simps! apply]
noncomputable def frobeniusEquiv : R ≃+* R :=
RingEquiv.ofBijective (frobenius R p) PerfectRing.bijective_frobenius
#align frobenius_equiv frobeniusEquiv
@[simp]
theorem coe_frobeniusEquiv : ⇑(frobeniusEquiv R p) = frobenius R p := rfl
#align coe_frobenius_equiv coe_frobeniusEquiv
theorem frobeniusEquiv_def (x : R) : frobeniusEquiv R p x = x ^ p := rfl
@[simps! apply]
noncomputable def iterateFrobeniusEquiv : R ≃+* R :=
RingEquiv.ofBijective (iterateFrobenius R p n) (bijective_iterateFrobenius R p n)
@[simp]
theorem coe_iterateFrobeniusEquiv : ⇑(iterateFrobeniusEquiv R p n) = iterateFrobenius R p n := rfl
theorem iterateFrobeniusEquiv_def (x : R) : iterateFrobeniusEquiv R p n x = x ^ p ^ n := rfl
theorem iterateFrobeniusEquiv_add_apply (x : R) : iterateFrobeniusEquiv R p (m + n) x =
iterateFrobeniusEquiv R p m (iterateFrobeniusEquiv R p n x) :=
iterateFrobenius_add_apply R p m n x
theorem iterateFrobeniusEquiv_add : iterateFrobeniusEquiv R p (m + n) =
(iterateFrobeniusEquiv R p n).trans (iterateFrobeniusEquiv R p m) :=
RingEquiv.ext (iterateFrobeniusEquiv_add_apply R p m n)
theorem iterateFrobeniusEquiv_symm_add_apply (x : R) : (iterateFrobeniusEquiv R p (m + n)).symm x =
(iterateFrobeniusEquiv R p m).symm ((iterateFrobeniusEquiv R p n).symm x) :=
(iterateFrobeniusEquiv R p (m + n)).injective <| by rw [RingEquiv.apply_symm_apply, add_comm,
iterateFrobeniusEquiv_add_apply, RingEquiv.apply_symm_apply, RingEquiv.apply_symm_apply]
theorem iterateFrobeniusEquiv_symm_add : (iterateFrobeniusEquiv R p (m + n)).symm =
(iterateFrobeniusEquiv R p n).symm.trans (iterateFrobeniusEquiv R p m).symm :=
RingEquiv.ext (iterateFrobeniusEquiv_symm_add_apply R p m n)
theorem iterateFrobeniusEquiv_zero_apply (x : R) : iterateFrobeniusEquiv R p 0 x = x := by
rw [iterateFrobeniusEquiv_def, pow_zero, pow_one]
| Mathlib/FieldTheory/Perfect.lean | 116 | 117 | theorem iterateFrobeniusEquiv_one_apply (x : R) : iterateFrobeniusEquiv R p 1 x = x ^ p := by |
rw [iterateFrobeniusEquiv_def, pow_one]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Filter.Bases
#align_import order.filter.at_top_bot from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
set_option autoImplicit true
variable {ι ι' α β γ : Type*}
open Set
namespace Filter
def atTop [Preorder α] : Filter α :=
⨅ a, 𝓟 (Ici a)
#align filter.at_top Filter.atTop
def atBot [Preorder α] : Filter α :=
⨅ a, 𝓟 (Iic a)
#align filter.at_bot Filter.atBot
theorem mem_atTop [Preorder α] (a : α) : { b : α | a ≤ b } ∈ @atTop α _ :=
mem_iInf_of_mem a <| Subset.refl _
#align filter.mem_at_top Filter.mem_atTop
theorem Ici_mem_atTop [Preorder α] (a : α) : Ici a ∈ (atTop : Filter α) :=
mem_atTop a
#align filter.Ici_mem_at_top Filter.Ici_mem_atTop
theorem Ioi_mem_atTop [Preorder α] [NoMaxOrder α] (x : α) : Ioi x ∈ (atTop : Filter α) :=
let ⟨z, hz⟩ := exists_gt x
mem_of_superset (mem_atTop z) fun _ h => lt_of_lt_of_le hz h
#align filter.Ioi_mem_at_top Filter.Ioi_mem_atTop
theorem mem_atBot [Preorder α] (a : α) : { b : α | b ≤ a } ∈ @atBot α _ :=
mem_iInf_of_mem a <| Subset.refl _
#align filter.mem_at_bot Filter.mem_atBot
theorem Iic_mem_atBot [Preorder α] (a : α) : Iic a ∈ (atBot : Filter α) :=
mem_atBot a
#align filter.Iic_mem_at_bot Filter.Iic_mem_atBot
theorem Iio_mem_atBot [Preorder α] [NoMinOrder α] (x : α) : Iio x ∈ (atBot : Filter α) :=
let ⟨z, hz⟩ := exists_lt x
mem_of_superset (mem_atBot z) fun _ h => lt_of_le_of_lt h hz
#align filter.Iio_mem_at_bot Filter.Iio_mem_atBot
theorem disjoint_atBot_principal_Ioi [Preorder α] (x : α) : Disjoint atBot (𝓟 (Ioi x)) :=
disjoint_of_disjoint_of_mem (Iic_disjoint_Ioi le_rfl) (Iic_mem_atBot x) (mem_principal_self _)
#align filter.disjoint_at_bot_principal_Ioi Filter.disjoint_atBot_principal_Ioi
theorem disjoint_atTop_principal_Iio [Preorder α] (x : α) : Disjoint atTop (𝓟 (Iio x)) :=
@disjoint_atBot_principal_Ioi αᵒᵈ _ _
#align filter.disjoint_at_top_principal_Iio Filter.disjoint_atTop_principal_Iio
theorem disjoint_atTop_principal_Iic [Preorder α] [NoMaxOrder α] (x : α) :
Disjoint atTop (𝓟 (Iic x)) :=
disjoint_of_disjoint_of_mem (Iic_disjoint_Ioi le_rfl).symm (Ioi_mem_atTop x)
(mem_principal_self _)
#align filter.disjoint_at_top_principal_Iic Filter.disjoint_atTop_principal_Iic
theorem disjoint_atBot_principal_Ici [Preorder α] [NoMinOrder α] (x : α) :
Disjoint atBot (𝓟 (Ici x)) :=
@disjoint_atTop_principal_Iic αᵒᵈ _ _ _
#align filter.disjoint_at_bot_principal_Ici Filter.disjoint_atBot_principal_Ici
theorem disjoint_pure_atTop [Preorder α] [NoMaxOrder α] (x : α) : Disjoint (pure x) atTop :=
Disjoint.symm <| (disjoint_atTop_principal_Iic x).mono_right <| le_principal_iff.2 <|
mem_pure.2 right_mem_Iic
#align filter.disjoint_pure_at_top Filter.disjoint_pure_atTop
theorem disjoint_pure_atBot [Preorder α] [NoMinOrder α] (x : α) : Disjoint (pure x) atBot :=
@disjoint_pure_atTop αᵒᵈ _ _ _
#align filter.disjoint_pure_at_bot Filter.disjoint_pure_atBot
theorem not_tendsto_const_atTop [Preorder α] [NoMaxOrder α] (x : α) (l : Filter β) [l.NeBot] :
¬Tendsto (fun _ => x) l atTop :=
tendsto_const_pure.not_tendsto (disjoint_pure_atTop x)
#align filter.not_tendsto_const_at_top Filter.not_tendsto_const_atTop
theorem not_tendsto_const_atBot [Preorder α] [NoMinOrder α] (x : α) (l : Filter β) [l.NeBot] :
¬Tendsto (fun _ => x) l atBot :=
tendsto_const_pure.not_tendsto (disjoint_pure_atBot x)
#align filter.not_tendsto_const_at_bot Filter.not_tendsto_const_atBot
| Mathlib/Order/Filter/AtTopBot.lean | 118 | 125 | theorem disjoint_atBot_atTop [PartialOrder α] [Nontrivial α] :
Disjoint (atBot : Filter α) atTop := by |
rcases exists_pair_ne α with ⟨x, y, hne⟩
by_cases hle : x ≤ y
· refine disjoint_of_disjoint_of_mem ?_ (Iic_mem_atBot x) (Ici_mem_atTop y)
exact Iic_disjoint_Ici.2 (hle.lt_of_ne hne).not_le
· refine disjoint_of_disjoint_of_mem ?_ (Iic_mem_atBot y) (Ici_mem_atTop x)
exact Iic_disjoint_Ici.2 hle
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd : ∀ e : ℕ, e ∣ a → e ∣ b → e ∣ d) :
d = a.gcd b :=
(dvd_antisymm (hd _ (gcd_dvd_left a b) (gcd_dvd_right a b)) (dvd_gcd hda hdb)).symm
#align nat.gcd_greatest Nat.gcd_greatest
@[simp]
theorem gcd_add_mul_right_right (m n k : ℕ) : gcd m (n + k * m) = gcd m n := by
simp [gcd_rec m (n + k * m), gcd_rec m n]
#align nat.gcd_add_mul_right_right Nat.gcd_add_mul_right_right
@[simp]
theorem gcd_add_mul_left_right (m n k : ℕ) : gcd m (n + m * k) = gcd m n := by
simp [gcd_rec m (n + m * k), gcd_rec m n]
#align nat.gcd_add_mul_left_right Nat.gcd_add_mul_left_right
@[simp]
theorem gcd_mul_right_add_right (m n k : ℕ) : gcd m (k * m + n) = gcd m n := by simp [add_comm _ n]
#align nat.gcd_mul_right_add_right Nat.gcd_mul_right_add_right
@[simp]
theorem gcd_mul_left_add_right (m n k : ℕ) : gcd m (m * k + n) = gcd m n := by simp [add_comm _ n]
#align nat.gcd_mul_left_add_right Nat.gcd_mul_left_add_right
@[simp]
theorem gcd_add_mul_right_left (m n k : ℕ) : gcd (m + k * n) n = gcd m n := by
rw [gcd_comm, gcd_add_mul_right_right, gcd_comm]
#align nat.gcd_add_mul_right_left Nat.gcd_add_mul_right_left
@[simp]
theorem gcd_add_mul_left_left (m n k : ℕ) : gcd (m + n * k) n = gcd m n := by
rw [gcd_comm, gcd_add_mul_left_right, gcd_comm]
#align nat.gcd_add_mul_left_left Nat.gcd_add_mul_left_left
@[simp]
theorem gcd_mul_right_add_left (m n k : ℕ) : gcd (k * n + m) n = gcd m n := by
rw [gcd_comm, gcd_mul_right_add_right, gcd_comm]
#align nat.gcd_mul_right_add_left Nat.gcd_mul_right_add_left
@[simp]
theorem gcd_mul_left_add_left (m n k : ℕ) : gcd (n * k + m) n = gcd m n := by
rw [gcd_comm, gcd_mul_left_add_right, gcd_comm]
#align nat.gcd_mul_left_add_left Nat.gcd_mul_left_add_left
@[simp]
theorem gcd_add_self_right (m n : ℕ) : gcd m (n + m) = gcd m n :=
Eq.trans (by rw [one_mul]) (gcd_add_mul_right_right m n 1)
#align nat.gcd_add_self_right Nat.gcd_add_self_right
@[simp]
theorem gcd_add_self_left (m n : ℕ) : gcd (m + n) n = gcd m n := by
rw [gcd_comm, gcd_add_self_right, gcd_comm]
#align nat.gcd_add_self_left Nat.gcd_add_self_left
@[simp]
theorem gcd_self_add_left (m n : ℕ) : gcd (m + n) m = gcd n m := by rw [add_comm, gcd_add_self_left]
#align nat.gcd_self_add_left Nat.gcd_self_add_left
@[simp]
theorem gcd_self_add_right (m n : ℕ) : gcd m (m + n) = gcd m n := by
rw [add_comm, gcd_add_self_right]
#align nat.gcd_self_add_right Nat.gcd_self_add_right
@[simp]
theorem gcd_sub_self_left {m n : ℕ} (h : m ≤ n) : gcd (n - m) m = gcd n m := by
calc
gcd (n - m) m = gcd (n - m + m) m := by rw [← gcd_add_self_left (n - m) m]
_ = gcd n m := by rw [Nat.sub_add_cancel h]
@[simp]
theorem gcd_sub_self_right {m n : ℕ} (h : m ≤ n) : gcd m (n - m) = gcd m n := by
rw [gcd_comm, gcd_sub_self_left h, gcd_comm]
@[simp]
theorem gcd_self_sub_left {m n : ℕ} (h : m ≤ n) : gcd (n - m) n = gcd m n := by
have := Nat.sub_add_cancel h
rw [gcd_comm m n, ← this, gcd_add_self_left (n - m) m]
have : gcd (n - m) n = gcd (n - m) m := by
nth_rw 2 [← Nat.add_sub_cancel' h]
rw [gcd_add_self_right, gcd_comm]
convert this
@[simp]
theorem gcd_self_sub_right {m n : ℕ} (h : m ≤ n) : gcd n (n - m) = gcd n m := by
rw [gcd_comm, gcd_self_sub_left h, gcd_comm]
theorem lcm_dvd_mul (m n : ℕ) : lcm m n ∣ m * n :=
lcm_dvd (dvd_mul_right _ _) (dvd_mul_left _ _)
#align nat.lcm_dvd_mul Nat.lcm_dvd_mul
theorem lcm_dvd_iff {m n k : ℕ} : lcm m n ∣ k ↔ m ∣ k ∧ n ∣ k :=
⟨fun h => ⟨(dvd_lcm_left _ _).trans h, (dvd_lcm_right _ _).trans h⟩, and_imp.2 lcm_dvd⟩
#align nat.lcm_dvd_iff Nat.lcm_dvd_iff
theorem lcm_pos {m n : ℕ} : 0 < m → 0 < n → 0 < m.lcm n := by
simp_rw [pos_iff_ne_zero]
exact lcm_ne_zero
#align nat.lcm_pos Nat.lcm_pos
theorem lcm_mul_left {m n k : ℕ} : (m * n).lcm (m * k) = m * n.lcm k := by
apply dvd_antisymm
· exact lcm_dvd (mul_dvd_mul_left m (dvd_lcm_left n k)) (mul_dvd_mul_left m (dvd_lcm_right n k))
· have h : m ∣ lcm (m * n) (m * k) := (dvd_mul_right m n).trans (dvd_lcm_left (m * n) (m * k))
rw [← dvd_div_iff h, lcm_dvd_iff, dvd_div_iff h, dvd_div_iff h, ← lcm_dvd_iff]
theorem lcm_mul_right {m n k : ℕ} : (m * n).lcm (k * n) = m.lcm k * n := by
rw [mul_comm, mul_comm k n, lcm_mul_left, mul_comm]
instance (m n : ℕ) : Decidable (Coprime m n) := inferInstanceAs (Decidable (gcd m n = 1))
theorem Coprime.lcm_eq_mul {m n : ℕ} (h : Coprime m n) : lcm m n = m * n := by
rw [← one_mul (lcm m n), ← h.gcd_eq_one, gcd_mul_lcm]
#align nat.coprime.lcm_eq_mul Nat.Coprime.lcm_eq_mul
theorem Coprime.symmetric : Symmetric Coprime := fun _ _ => Coprime.symm
#align nat.coprime.symmetric Nat.Coprime.symmetric
theorem Coprime.dvd_mul_right {m n k : ℕ} (H : Coprime k n) : k ∣ m * n ↔ k ∣ m :=
⟨H.dvd_of_dvd_mul_right, fun h => dvd_mul_of_dvd_left h n⟩
#align nat.coprime.dvd_mul_right Nat.Coprime.dvd_mul_right
theorem Coprime.dvd_mul_left {m n k : ℕ} (H : Coprime k m) : k ∣ m * n ↔ k ∣ n :=
⟨H.dvd_of_dvd_mul_left, fun h => dvd_mul_of_dvd_right h m⟩
#align nat.coprime.dvd_mul_left Nat.Coprime.dvd_mul_left
@[simp]
theorem coprime_add_self_right {m n : ℕ} : Coprime m (n + m) ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_add_self_right]
#align nat.coprime_add_self_right Nat.coprime_add_self_right
@[simp]
theorem coprime_self_add_right {m n : ℕ} : Coprime m (m + n) ↔ Coprime m n := by
rw [add_comm, coprime_add_self_right]
#align nat.coprime_self_add_right Nat.coprime_self_add_right
@[simp]
theorem coprime_add_self_left {m n : ℕ} : Coprime (m + n) n ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_add_self_left]
#align nat.coprime_add_self_left Nat.coprime_add_self_left
@[simp]
theorem coprime_self_add_left {m n : ℕ} : Coprime (m + n) m ↔ Coprime n m := by
rw [Coprime, Coprime, gcd_self_add_left]
#align nat.coprime_self_add_left Nat.coprime_self_add_left
@[simp]
theorem coprime_add_mul_right_right (m n k : ℕ) : Coprime m (n + k * m) ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_add_mul_right_right]
#align nat.coprime_add_mul_right_right Nat.coprime_add_mul_right_right
@[simp]
theorem coprime_add_mul_left_right (m n k : ℕ) : Coprime m (n + m * k) ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_add_mul_left_right]
#align nat.coprime_add_mul_left_right Nat.coprime_add_mul_left_right
@[simp]
theorem coprime_mul_right_add_right (m n k : ℕ) : Coprime m (k * m + n) ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_mul_right_add_right]
#align nat.coprime_mul_right_add_right Nat.coprime_mul_right_add_right
@[simp]
theorem coprime_mul_left_add_right (m n k : ℕ) : Coprime m (m * k + n) ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_mul_left_add_right]
#align nat.coprime_mul_left_add_right Nat.coprime_mul_left_add_right
@[simp]
theorem coprime_add_mul_right_left (m n k : ℕ) : Coprime (m + k * n) n ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_add_mul_right_left]
#align nat.coprime_add_mul_right_left Nat.coprime_add_mul_right_left
@[simp]
theorem coprime_add_mul_left_left (m n k : ℕ) : Coprime (m + n * k) n ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_add_mul_left_left]
#align nat.coprime_add_mul_left_left Nat.coprime_add_mul_left_left
@[simp]
theorem coprime_mul_right_add_left (m n k : ℕ) : Coprime (k * n + m) n ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_mul_right_add_left]
#align nat.coprime_mul_right_add_left Nat.coprime_mul_right_add_left
@[simp]
theorem coprime_mul_left_add_left (m n k : ℕ) : Coprime (n * k + m) n ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_mul_left_add_left]
#align nat.coprime_mul_left_add_left Nat.coprime_mul_left_add_left
@[simp]
theorem coprime_sub_self_left {m n : ℕ} (h : m ≤ n) : Coprime (n - m) m ↔ Coprime n m := by
rw [Coprime, Coprime, gcd_sub_self_left h]
@[simp]
theorem coprime_sub_self_right {m n : ℕ} (h : m ≤ n) : Coprime m (n - m) ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_sub_self_right h]
@[simp]
theorem coprime_self_sub_left {m n : ℕ} (h : m ≤ n) : Coprime (n - m) n ↔ Coprime m n := by
rw [Coprime, Coprime, gcd_self_sub_left h]
@[simp]
theorem coprime_self_sub_right {m n : ℕ} (h : m ≤ n) : Coprime n (n - m) ↔ Coprime n m := by
rw [Coprime, Coprime, gcd_self_sub_right h]
@[simp]
theorem coprime_pow_left_iff {n : ℕ} (hn : 0 < n) (a b : ℕ) :
Nat.Coprime (a ^ n) b ↔ Nat.Coprime a b := by
obtain ⟨n, rfl⟩ := exists_eq_succ_of_ne_zero hn.ne'
rw [Nat.pow_succ, Nat.coprime_mul_iff_left]
exact ⟨And.right, fun hab => ⟨hab.pow_left _, hab⟩⟩
#align nat.coprime_pow_left_iff Nat.coprime_pow_left_iff
@[simp]
theorem coprime_pow_right_iff {n : ℕ} (hn : 0 < n) (a b : ℕ) :
Nat.Coprime a (b ^ n) ↔ Nat.Coprime a b := by
rw [Nat.coprime_comm, coprime_pow_left_iff hn, Nat.coprime_comm]
#align nat.coprime_pow_right_iff Nat.coprime_pow_right_iff
theorem not_coprime_zero_zero : ¬Coprime 0 0 := by simp
#align nat.not_coprime_zero_zero Nat.not_coprime_zero_zero
theorem coprime_one_left_iff (n : ℕ) : Coprime 1 n ↔ True := by simp [Coprime]
#align nat.coprime_one_left_iff Nat.coprime_one_left_iff
theorem coprime_one_right_iff (n : ℕ) : Coprime n 1 ↔ True := by simp [Coprime]
#align nat.coprime_one_right_iff Nat.coprime_one_right_iff
theorem gcd_mul_of_coprime_of_dvd {a b c : ℕ} (hac : Coprime a c) (b_dvd_c : b ∣ c) :
gcd (a * b) c = b := by
rcases exists_eq_mul_left_of_dvd b_dvd_c with ⟨d, rfl⟩
rw [gcd_mul_right]
convert one_mul b
exact Coprime.coprime_mul_right_right hac
#align nat.gcd_mul_of_coprime_of_dvd Nat.gcd_mul_of_coprime_of_dvd
theorem Coprime.eq_of_mul_eq_zero {m n : ℕ} (h : m.Coprime n) (hmn : m * n = 0) :
m = 0 ∧ n = 1 ∨ m = 1 ∧ n = 0 :=
(Nat.eq_zero_of_mul_eq_zero hmn).imp (fun hm => ⟨hm, n.coprime_zero_left.mp <| hm ▸ h⟩) fun hn =>
let eq := hn ▸ h.symm
⟨m.coprime_zero_left.mp <| eq, hn⟩
#align nat.coprime.eq_of_mul_eq_zero Nat.Coprime.eq_of_mul_eq_zero
def prodDvdAndDvdOfDvdProd {m n k : ℕ} (H : k ∣ m * n) :
{ d : { m' // m' ∣ m } × { n' // n' ∣ n } // k = d.1 * d.2 } := by
cases h0 : gcd k m with
| zero =>
obtain rfl : k = 0 := eq_zero_of_gcd_eq_zero_left h0
obtain rfl : m = 0 := eq_zero_of_gcd_eq_zero_right h0
exact ⟨⟨⟨0, dvd_refl 0⟩, ⟨n, dvd_refl n⟩⟩, (zero_mul n).symm⟩
| succ tmp =>
have hpos : 0 < gcd k m := h0.symm ▸ Nat.zero_lt_succ _; clear h0 tmp
have hd : gcd k m * (k / gcd k m) = k := Nat.mul_div_cancel' (gcd_dvd_left k m)
refine ⟨⟨⟨gcd k m, gcd_dvd_right k m⟩, ⟨k / gcd k m, ?_⟩⟩, hd.symm⟩
apply Nat.dvd_of_mul_dvd_mul_left hpos
rw [hd, ← gcd_mul_right]
exact dvd_gcd (dvd_mul_right _ _) H
#align nat.prod_dvd_and_dvd_of_dvd_prod Nat.prodDvdAndDvdOfDvdProd
theorem dvd_mul {x m n : ℕ} : x ∣ m * n ↔ ∃ y z, y ∣ m ∧ z ∣ n ∧ y * z = x := by
constructor
· intro h
obtain ⟨⟨⟨y, hy⟩, ⟨z, hz⟩⟩, rfl⟩ := prod_dvd_and_dvd_of_dvd_prod h
exact ⟨y, z, hy, hz, rfl⟩
· rintro ⟨y, z, hy, hz, rfl⟩
exact mul_dvd_mul hy hz
#align nat.dvd_mul Nat.dvd_mul
theorem pow_dvd_pow_iff {a b n : ℕ} (n0 : n ≠ 0) : a ^ n ∣ b ^ n ↔ a ∣ b := by
refine ⟨fun h => ?_, fun h => pow_dvd_pow_of_dvd h _⟩
rcases Nat.eq_zero_or_pos (gcd a b) with g0 | g0
· simp [eq_zero_of_gcd_eq_zero_right g0]
rcases exists_coprime' g0 with ⟨g, a', b', g0', co, rfl, rfl⟩
rw [mul_pow, mul_pow] at h
replace h := Nat.dvd_of_mul_dvd_mul_right (pow_pos g0' _) h
have := pow_dvd_pow a' <| Nat.pos_of_ne_zero n0
rw [pow_one, (co.pow n n).eq_one_of_dvd h] at this
simp [eq_one_of_dvd_one this]
#align nat.pow_dvd_pow_iff Nat.pow_dvd_pow_iff
theorem coprime_iff_isRelPrime {m n : ℕ} : m.Coprime n ↔ IsRelPrime m n := by
simp_rw [coprime_iff_gcd_eq_one, IsRelPrime, ← and_imp, ← dvd_gcd_iff, isUnit_iff_dvd_one]
exact ⟨fun h _ ↦ (h ▸ ·), (dvd_one.mp <| · dvd_rfl)⟩
theorem eq_one_of_dvd_coprimes {a b k : ℕ} (h_ab_coprime : Coprime a b) (hka : k ∣ a)
(hkb : k ∣ b) : k = 1 :=
dvd_one.mp (isUnit_iff_dvd_one.mp <| coprime_iff_isRelPrime.mp h_ab_coprime hka hkb)
#align nat.eq_one_of_dvd_coprimes Nat.eq_one_of_dvd_coprimes
theorem Coprime.mul_add_mul_ne_mul {m n a b : ℕ} (cop : Coprime m n) (ha : a ≠ 0) (hb : b ≠ 0) :
a * m + b * n ≠ m * n := by
intro h
obtain ⟨x, rfl⟩ : n ∣ a :=
cop.symm.dvd_of_dvd_mul_right
((Nat.dvd_add_iff_left (Nat.dvd_mul_left n b)).mpr
((congr_arg _ h).mpr (Nat.dvd_mul_left n m)))
obtain ⟨y, rfl⟩ : m ∣ b :=
cop.dvd_of_dvd_mul_right
((Nat.dvd_add_iff_right (Nat.dvd_mul_left m (n * x))).mpr
((congr_arg _ h).mpr (Nat.dvd_mul_right m n)))
rw [mul_comm, mul_ne_zero_iff, ← one_le_iff_ne_zero] at ha hb
refine mul_ne_zero hb.2 ha.2 (eq_zero_of_mul_eq_self_left (ne_of_gt (add_le_add ha.1 hb.1)) ?_)
rw [← mul_assoc, ← h, add_mul, add_mul, mul_comm _ n, ← mul_assoc, mul_comm y]
#align nat.coprime.mul_add_mul_ne_mul Nat.Coprime.mul_add_mul_ne_mul
variable {x n m : ℕ}
theorem dvd_gcd_mul_iff_dvd_mul : x ∣ gcd x n * m ↔ x ∣ n * m := by
refine ⟨(·.trans <| mul_dvd_mul_right (x.gcd_dvd_right n) m), fun ⟨y, hy⟩ ↦ ?_⟩
rw [← gcd_mul_right, hy, gcd_mul_left]
exact dvd_mul_right x (gcd m y)
theorem dvd_mul_gcd_iff_dvd_mul : x ∣ n * gcd x m ↔ x ∣ n * m := by
rw [mul_comm, dvd_gcd_mul_iff_dvd_mul, mul_comm]
theorem dvd_gcd_mul_gcd_iff_dvd_mul : x ∣ gcd x n * gcd x m ↔ x ∣ n * m := by
rw [dvd_gcd_mul_iff_dvd_mul, dvd_mul_gcd_iff_dvd_mul]
| Mathlib/Data/Nat/GCD/Basic.lean | 359 | 363 | theorem gcd_mul_gcd_eq_iff_dvd_mul_of_coprime (hcop : Coprime n m) :
gcd x n * gcd x m = x ↔ x ∣ n * m := by |
refine ⟨fun h ↦ ?_, (dvd_antisymm ?_ <| dvd_gcd_mul_gcd_iff_dvd_mul.mpr ·)⟩
refine h ▸ Nat.mul_dvd_mul ?_ ?_ <;> exact x.gcd_dvd_right _
refine (hcop.gcd_both x x).mul_dvd_of_dvd_of_dvd ?_ ?_ <;> exact x.gcd_dvd_left _
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.invertible from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
variable {α : Type*} [LinearOrderedSemiring α] {a : α}
@[simp]
| Mathlib/Algebra/Order/Invertible.lean | 19 | 21 | theorem invOf_pos [Invertible a] : 0 < ⅟ a ↔ 0 < a :=
haveI : 0 < a * ⅟ a := by | simp only [mul_invOf_self, zero_lt_one]
⟨fun h => pos_of_mul_pos_left this h.le, fun h => pos_of_mul_pos_right this h.le⟩
|
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6"
universe u₁ u₂
namespace Matrix
open Matrix
variable (n p : Type*) (R : Type u₂) {𝕜 : Type*} [Field 𝕜]
variable [DecidableEq n] [DecidableEq p]
variable [CommRing R]
section Transvection
variable {R n} (i j : n)
def transvection (c : R) : Matrix n n R :=
1 + Matrix.stdBasisMatrix i j c
#align matrix.transvection Matrix.transvection
@[simp]
theorem transvection_zero : transvection i j (0 : R) = 1 := by simp [transvection]
#align matrix.transvection_zero Matrix.transvection_zero
section
theorem updateRow_eq_transvection [Finite n] (c : R) :
updateRow (1 : Matrix n n R) i ((1 : Matrix n n R) i + c • (1 : Matrix n n R) j) =
transvection i j c := by
cases nonempty_fintype n
ext a b
by_cases ha : i = a
· by_cases hb : j = b
· simp only [updateRow_self, transvection, ha, hb, Pi.add_apply, StdBasisMatrix.apply_same,
one_apply_eq, Pi.smul_apply, mul_one, Algebra.id.smul_eq_mul, add_apply]
· simp only [updateRow_self, transvection, ha, hb, StdBasisMatrix.apply_of_ne, Pi.add_apply,
Ne, not_false_iff, Pi.smul_apply, and_false_iff, one_apply_ne, Algebra.id.smul_eq_mul,
mul_zero, add_apply]
· simp only [updateRow_ne, transvection, ha, Ne.symm ha, StdBasisMatrix.apply_of_ne, add_zero,
Algebra.id.smul_eq_mul, Ne, not_false_iff, DMatrix.add_apply, Pi.smul_apply,
mul_zero, false_and_iff, add_apply]
#align matrix.update_row_eq_transvection Matrix.updateRow_eq_transvection
variable [Fintype n]
theorem transvection_mul_transvection_same (h : i ≠ j) (c d : R) :
transvection i j c * transvection i j d = transvection i j (c + d) := by
simp [transvection, Matrix.add_mul, Matrix.mul_add, h, h.symm, add_smul, add_assoc,
stdBasisMatrix_add]
#align matrix.transvection_mul_transvection_same Matrix.transvection_mul_transvection_same
@[simp]
theorem transvection_mul_apply_same (b : n) (c : R) (M : Matrix n n R) :
(transvection i j c * M) i b = M i b + c * M j b := by simp [transvection, Matrix.add_mul]
#align matrix.transvection_mul_apply_same Matrix.transvection_mul_apply_same
@[simp]
theorem mul_transvection_apply_same (a : n) (c : R) (M : Matrix n n R) :
(M * transvection i j c) a j = M a j + c * M a i := by
simp [transvection, Matrix.mul_add, mul_comm]
#align matrix.mul_transvection_apply_same Matrix.mul_transvection_apply_same
@[simp]
theorem transvection_mul_apply_of_ne (a b : n) (ha : a ≠ i) (c : R) (M : Matrix n n R) :
(transvection i j c * M) a b = M a b := by simp [transvection, Matrix.add_mul, ha]
#align matrix.transvection_mul_apply_of_ne Matrix.transvection_mul_apply_of_ne
@[simp]
theorem mul_transvection_apply_of_ne (a b : n) (hb : b ≠ j) (c : R) (M : Matrix n n R) :
(M * transvection i j c) a b = M a b := by simp [transvection, Matrix.mul_add, hb]
#align matrix.mul_transvection_apply_of_ne Matrix.mul_transvection_apply_of_ne
@[simp]
theorem det_transvection_of_ne (h : i ≠ j) (c : R) : det (transvection i j c) = 1 := by
rw [← updateRow_eq_transvection i j, det_updateRow_add_smul_self _ h, det_one]
#align matrix.det_transvection_of_ne Matrix.det_transvection_of_ne
end
variable (R n)
-- porting note (#5171): removed @[nolint has_nonempty_instance]
structure TransvectionStruct where
(i j : n)
hij : i ≠ j
c : R
#align matrix.transvection_struct Matrix.TransvectionStruct
instance [Nontrivial n] : Nonempty (TransvectionStruct n R) := by
choose x y hxy using exists_pair_ne n
exact ⟨⟨x, y, hxy, 0⟩⟩
namespace TransvectionStruct
variable {R n}
def toMatrix (t : TransvectionStruct n R) : Matrix n n R :=
transvection t.i t.j t.c
#align matrix.transvection_struct.to_matrix Matrix.TransvectionStruct.toMatrix
@[simp]
theorem toMatrix_mk (i j : n) (hij : i ≠ j) (c : R) :
TransvectionStruct.toMatrix ⟨i, j, hij, c⟩ = transvection i j c :=
rfl
#align matrix.transvection_struct.to_matrix_mk Matrix.TransvectionStruct.toMatrix_mk
@[simp]
protected theorem det [Fintype n] (t : TransvectionStruct n R) : det t.toMatrix = 1 :=
det_transvection_of_ne _ _ t.hij _
#align matrix.transvection_struct.det Matrix.TransvectionStruct.det
@[simp]
theorem det_toMatrix_prod [Fintype n] (L : List (TransvectionStruct n 𝕜)) :
det (L.map toMatrix).prod = 1 := by
induction' L with t L IH
· simp
· simp [IH]
#align matrix.transvection_struct.det_to_matrix_prod Matrix.TransvectionStruct.det_toMatrix_prod
@[simps]
protected def inv (t : TransvectionStruct n R) : TransvectionStruct n R where
i := t.i
j := t.j
hij := t.hij
c := -t.c
#align matrix.transvection_struct.inv Matrix.TransvectionStruct.inv
section
variable [Fintype n]
theorem inv_mul (t : TransvectionStruct n R) : t.inv.toMatrix * t.toMatrix = 1 := by
rcases t with ⟨_, _, t_hij⟩
simp [toMatrix, transvection_mul_transvection_same, t_hij]
#align matrix.transvection_struct.inv_mul Matrix.TransvectionStruct.inv_mul
theorem mul_inv (t : TransvectionStruct n R) : t.toMatrix * t.inv.toMatrix = 1 := by
rcases t with ⟨_, _, t_hij⟩
simp [toMatrix, transvection_mul_transvection_same, t_hij]
#align matrix.transvection_struct.mul_inv Matrix.TransvectionStruct.mul_inv
theorem reverse_inv_prod_mul_prod (L : List (TransvectionStruct n R)) :
(L.reverse.map (toMatrix ∘ TransvectionStruct.inv)).prod * (L.map toMatrix).prod = 1 := by
induction' L with t L IH
· simp
· suffices
(L.reverse.map (toMatrix ∘ TransvectionStruct.inv)).prod * (t.inv.toMatrix * t.toMatrix) *
(L.map toMatrix).prod = 1
by simpa [Matrix.mul_assoc]
simpa [inv_mul] using IH
#align matrix.transvection_struct.reverse_inv_prod_mul_prod Matrix.TransvectionStruct.reverse_inv_prod_mul_prod
theorem prod_mul_reverse_inv_prod (L : List (TransvectionStruct n R)) :
(L.map toMatrix).prod * (L.reverse.map (toMatrix ∘ TransvectionStruct.inv)).prod = 1 := by
induction' L with t L IH
· simp
· suffices
t.toMatrix *
((L.map toMatrix).prod * (L.reverse.map (toMatrix ∘ TransvectionStruct.inv)).prod) *
t.inv.toMatrix = 1
by simpa [Matrix.mul_assoc]
simp_rw [IH, Matrix.mul_one, t.mul_inv]
#align matrix.transvection_struct.prod_mul_reverse_inv_prod Matrix.TransvectionStruct.prod_mul_reverse_inv_prod
theorem _root_.Matrix.mem_range_scalar_of_commute_transvectionStruct {M : Matrix n n R}
(hM : ∀ t : TransvectionStruct n R, Commute t.toMatrix M) :
M ∈ Set.range (Matrix.scalar n) := by
refine mem_range_scalar_of_commute_stdBasisMatrix ?_
intro i j hij
simpa [transvection, mul_add, add_mul] using (hM ⟨i, j, hij, 1⟩).eq
| Mathlib/LinearAlgebra/Matrix/Transvection.lean | 246 | 252 | theorem _root_.Matrix.mem_range_scalar_iff_commute_transvectionStruct {M : Matrix n n R} :
M ∈ Set.range (Matrix.scalar n) ↔ ∀ t : TransvectionStruct n R, Commute t.toMatrix M := by |
refine ⟨fun h t => ?_, mem_range_scalar_of_commute_transvectionStruct⟩
rw [mem_range_scalar_iff_commute_stdBasisMatrix] at h
refine (Commute.one_left M).add_left ?_
convert (h _ _ t.hij).smul_left t.c using 1
rw [smul_stdBasisMatrix, smul_eq_mul, mul_one]
|
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Basic
import Mathlib.Order.CompleteLatticeIntervals
#align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
variable {α β ι ι' : Type*}
namespace Finset
section Lattice
variable [Lattice α] [OrderBot α]
def SupIndep (s : Finset ι) (f : ι → α) : Prop :=
∀ ⦃t⦄, t ⊆ s → ∀ ⦃i⦄, i ∈ s → i ∉ t → Disjoint (f i) (t.sup f)
#align finset.sup_indep Finset.SupIndep
variable {s t : Finset ι} {f : ι → α} {i : ι}
instance [DecidableEq ι] [DecidableEq α] : Decidable (SupIndep s f) := by
refine @Finset.decidableForallOfDecidableSubsets _ _ _ (?_)
rintro t -
refine @Finset.decidableDforallFinset _ _ _ (?_)
rintro i -
have : Decidable (Disjoint (f i) (sup t f)) := decidable_of_iff' (_ = ⊥) disjoint_iff
infer_instance
theorem SupIndep.subset (ht : t.SupIndep f) (h : s ⊆ t) : s.SupIndep f := fun _ hu _ hi =>
ht (hu.trans h) (h hi)
#align finset.sup_indep.subset Finset.SupIndep.subset
@[simp]
theorem supIndep_empty (f : ι → α) : (∅ : Finset ι).SupIndep f := fun _ _ a ha =>
(not_mem_empty a ha).elim
#align finset.sup_indep_empty Finset.supIndep_empty
theorem supIndep_singleton (i : ι) (f : ι → α) : ({i} : Finset ι).SupIndep f :=
fun s hs j hji hj => by
rw [eq_empty_of_ssubset_singleton ⟨hs, fun h => hj (h hji)⟩, sup_empty]
exact disjoint_bot_right
#align finset.sup_indep_singleton Finset.supIndep_singleton
theorem SupIndep.pairwiseDisjoint (hs : s.SupIndep f) : (s : Set ι).PairwiseDisjoint f :=
fun _ ha _ hb hab =>
sup_singleton.subst <| hs (singleton_subset_iff.2 hb) ha <| not_mem_singleton.2 hab
#align finset.sup_indep.pairwise_disjoint Finset.SupIndep.pairwiseDisjoint
theorem SupIndep.le_sup_iff (hs : s.SupIndep f) (hts : t ⊆ s) (hi : i ∈ s) (hf : ∀ i, f i ≠ ⊥) :
f i ≤ t.sup f ↔ i ∈ t := by
refine ⟨fun h => ?_, le_sup⟩
by_contra hit
exact hf i (disjoint_self.1 <| (hs hts hi hit).mono_right h)
#align finset.sup_indep.le_sup_iff Finset.SupIndep.le_sup_iff
theorem supIndep_iff_disjoint_erase [DecidableEq ι] :
s.SupIndep f ↔ ∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f) :=
⟨fun hs _ hi => hs (erase_subset _ _) hi (not_mem_erase _ _), fun hs _ ht i hi hit =>
(hs i hi).mono_right (sup_mono fun _ hj => mem_erase.2 ⟨ne_of_mem_of_not_mem hj hit, ht hj⟩)⟩
#align finset.sup_indep_iff_disjoint_erase Finset.supIndep_iff_disjoint_erase
theorem SupIndep.image [DecidableEq ι] {s : Finset ι'} {g : ι' → ι} (hs : s.SupIndep (f ∘ g)) :
(s.image g).SupIndep f := by
intro t ht i hi hit
rw [mem_image] at hi
obtain ⟨i, hi, rfl⟩ := hi
haveI : DecidableEq ι' := Classical.decEq _
suffices hts : t ⊆ (s.erase i).image g by
refine (supIndep_iff_disjoint_erase.1 hs i hi).mono_right ((sup_mono hts).trans ?_)
rw [sup_image]
rintro j hjt
obtain ⟨j, hj, rfl⟩ := mem_image.1 (ht hjt)
exact mem_image_of_mem _ (mem_erase.2 ⟨ne_of_apply_ne g (ne_of_mem_of_not_mem hjt hit), hj⟩)
#align finset.sup_indep.image Finset.SupIndep.image
theorem supIndep_map {s : Finset ι'} {g : ι' ↪ ι} : (s.map g).SupIndep f ↔ s.SupIndep (f ∘ g) := by
refine ⟨fun hs t ht i hi hit => ?_, fun hs => ?_⟩
· rw [← sup_map]
exact hs (map_subset_map.2 ht) ((mem_map' _).2 hi) (by rwa [mem_map'])
· classical
rw [map_eq_image]
exact hs.image
#align finset.sup_indep_map Finset.supIndep_map
@[simp]
theorem supIndep_pair [DecidableEq ι] {i j : ι} (hij : i ≠ j) :
({i, j} : Finset ι).SupIndep f ↔ Disjoint (f i) (f j) :=
⟨fun h => h.pairwiseDisjoint (by simp) (by simp) hij,
fun h => by
rw [supIndep_iff_disjoint_erase]
intro k hk
rw [Finset.mem_insert, Finset.mem_singleton] at hk
obtain rfl | rfl := hk
· convert h using 1
rw [Finset.erase_insert, Finset.sup_singleton]
simpa using hij
· convert h.symm using 1
have : ({i, k} : Finset ι).erase k = {i} := by
ext
rw [mem_erase, mem_insert, mem_singleton, mem_singleton, and_or_left, Ne,
not_and_self_iff, or_false_iff, and_iff_right_of_imp]
rintro rfl
exact hij
rw [this, Finset.sup_singleton]⟩
#align finset.sup_indep_pair Finset.supIndep_pair
theorem supIndep_univ_bool (f : Bool → α) :
(Finset.univ : Finset Bool).SupIndep f ↔ Disjoint (f false) (f true) :=
haveI : true ≠ false := by simp only [Ne, not_false_iff]
(supIndep_pair this).trans disjoint_comm
#align finset.sup_indep_univ_bool Finset.supIndep_univ_bool
@[simp]
| Mathlib/Order/SupIndep.lean | 158 | 161 | theorem supIndep_univ_fin_two (f : Fin 2 → α) :
(Finset.univ : Finset (Fin 2)).SupIndep f ↔ Disjoint (f 0) (f 1) :=
haveI : (0 : Fin 2) ≠ 1 := by | simp
supIndep_pair this
|
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
assert_not_exists Absorbs
noncomputable section
namespace Complex
variable {z : ℂ}
open ComplexConjugate Topology Filter
instance : Norm ℂ :=
⟨abs⟩
@[simp]
theorem norm_eq_abs (z : ℂ) : ‖z‖ = abs z :=
rfl
#align complex.norm_eq_abs Complex.norm_eq_abs
lemma norm_I : ‖I‖ = 1 := abs_I
theorem norm_exp_ofReal_mul_I (t : ℝ) : ‖exp (t * I)‖ = 1 := by
simp only [norm_eq_abs, abs_exp_ofReal_mul_I]
set_option linter.uppercaseLean3 false in
#align complex.norm_exp_of_real_mul_I Complex.norm_exp_ofReal_mul_I
instance instNormedAddCommGroup : NormedAddCommGroup ℂ :=
AddGroupNorm.toNormedAddCommGroup
{ abs with
map_zero' := map_zero abs
neg' := abs.map_neg
eq_zero_of_map_eq_zero' := fun _ => abs.eq_zero.1 }
instance : NormedField ℂ where
dist_eq _ _ := rfl
norm_mul' := map_mul abs
instance : DenselyNormedField ℂ where
lt_norm_lt r₁ r₂ h₀ hr :=
let ⟨x, h⟩ := exists_between hr
⟨x, by rwa [norm_eq_abs, abs_ofReal, abs_of_pos (h₀.trans_lt h.1)]⟩
instance {R : Type*} [NormedField R] [NormedAlgebra R ℝ] : NormedAlgebra R ℂ where
norm_smul_le r x := by
rw [← algebraMap_smul ℝ r x, real_smul, norm_mul, norm_eq_abs, abs_ofReal, ← Real.norm_eq_abs,
norm_algebraMap']
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℂ E]
-- see Note [lower instance priority]
instance (priority := 900) _root_.NormedSpace.complexToReal : NormedSpace ℝ E :=
NormedSpace.restrictScalars ℝ ℂ E
#align normed_space.complex_to_real NormedSpace.complexToReal
-- see Note [lower instance priority]
instance (priority := 900) _root_.NormedAlgebra.complexToReal {A : Type*} [SeminormedRing A]
[NormedAlgebra ℂ A] : NormedAlgebra ℝ A :=
NormedAlgebra.restrictScalars ℝ ℂ A
theorem dist_eq (z w : ℂ) : dist z w = abs (z - w) :=
rfl
#align complex.dist_eq Complex.dist_eq
theorem dist_eq_re_im (z w : ℂ) : dist z w = √((z.re - w.re) ^ 2 + (z.im - w.im) ^ 2) := by
rw [sq, sq]
rfl
#align complex.dist_eq_re_im Complex.dist_eq_re_im
@[simp]
theorem dist_mk (x₁ y₁ x₂ y₂ : ℝ) :
dist (mk x₁ y₁) (mk x₂ y₂) = √((x₁ - x₂) ^ 2 + (y₁ - y₂) ^ 2) :=
dist_eq_re_im _ _
#align complex.dist_mk Complex.dist_mk
theorem dist_of_re_eq {z w : ℂ} (h : z.re = w.re) : dist z w = dist z.im w.im := by
rw [dist_eq_re_im, h, sub_self, zero_pow two_ne_zero, zero_add, Real.sqrt_sq_eq_abs, Real.dist_eq]
#align complex.dist_of_re_eq Complex.dist_of_re_eq
theorem nndist_of_re_eq {z w : ℂ} (h : z.re = w.re) : nndist z w = nndist z.im w.im :=
NNReal.eq <| dist_of_re_eq h
#align complex.nndist_of_re_eq Complex.nndist_of_re_eq
theorem edist_of_re_eq {z w : ℂ} (h : z.re = w.re) : edist z w = edist z.im w.im := by
rw [edist_nndist, edist_nndist, nndist_of_re_eq h]
#align complex.edist_of_re_eq Complex.edist_of_re_eq
theorem dist_of_im_eq {z w : ℂ} (h : z.im = w.im) : dist z w = dist z.re w.re := by
rw [dist_eq_re_im, h, sub_self, zero_pow two_ne_zero, add_zero, Real.sqrt_sq_eq_abs, Real.dist_eq]
#align complex.dist_of_im_eq Complex.dist_of_im_eq
theorem nndist_of_im_eq {z w : ℂ} (h : z.im = w.im) : nndist z w = nndist z.re w.re :=
NNReal.eq <| dist_of_im_eq h
#align complex.nndist_of_im_eq Complex.nndist_of_im_eq
theorem edist_of_im_eq {z w : ℂ} (h : z.im = w.im) : edist z w = edist z.re w.re := by
rw [edist_nndist, edist_nndist, nndist_of_im_eq h]
#align complex.edist_of_im_eq Complex.edist_of_im_eq
theorem dist_conj_self (z : ℂ) : dist (conj z) z = 2 * |z.im| := by
rw [dist_of_re_eq (conj_re z), conj_im, dist_comm, Real.dist_eq, sub_neg_eq_add, ← two_mul,
_root_.abs_mul, abs_of_pos (zero_lt_two' ℝ)]
#align complex.dist_conj_self Complex.dist_conj_self
theorem nndist_conj_self (z : ℂ) : nndist (conj z) z = 2 * Real.nnabs z.im :=
NNReal.eq <| by rw [← dist_nndist, NNReal.coe_mul, NNReal.coe_two, Real.coe_nnabs, dist_conj_self]
#align complex.nndist_conj_self Complex.nndist_conj_self
theorem dist_self_conj (z : ℂ) : dist z (conj z) = 2 * |z.im| := by rw [dist_comm, dist_conj_self]
#align complex.dist_self_conj Complex.dist_self_conj
theorem nndist_self_conj (z : ℂ) : nndist z (conj z) = 2 * Real.nnabs z.im := by
rw [nndist_comm, nndist_conj_self]
#align complex.nndist_self_conj Complex.nndist_self_conj
@[simp 1100]
theorem comap_abs_nhds_zero : comap abs (𝓝 0) = 𝓝 0 :=
comap_norm_nhds_zero
#align complex.comap_abs_nhds_zero Complex.comap_abs_nhds_zero
theorem norm_real (r : ℝ) : ‖(r : ℂ)‖ = ‖r‖ :=
abs_ofReal _
#align complex.norm_real Complex.norm_real
@[simp 1100]
theorem norm_rat (r : ℚ) : ‖(r : ℂ)‖ = |(r : ℝ)| := by
rw [← ofReal_ratCast]
exact norm_real _
#align complex.norm_rat Complex.norm_rat
@[simp 1100]
theorem norm_nat (n : ℕ) : ‖(n : ℂ)‖ = n :=
abs_natCast _
#align complex.norm_nat Complex.norm_nat
@[simp 1100]
lemma norm_int {n : ℤ} : ‖(n : ℂ)‖ = |(n : ℝ)| := abs_intCast n
#align complex.norm_int Complex.norm_int
theorem norm_int_of_nonneg {n : ℤ} (hn : 0 ≤ n) : ‖(n : ℂ)‖ = n := by
rw [norm_int, ← Int.cast_abs, _root_.abs_of_nonneg hn]
#align complex.norm_int_of_nonneg Complex.norm_int_of_nonneg
lemma normSq_eq_norm_sq (z : ℂ) : normSq z = ‖z‖ ^ 2 := by
rw [normSq_eq_abs, norm_eq_abs]
@[continuity]
theorem continuous_abs : Continuous abs :=
continuous_norm
#align complex.continuous_abs Complex.continuous_abs
@[continuity]
theorem continuous_normSq : Continuous normSq := by
simpa [← normSq_eq_abs] using continuous_abs.pow 2
#align complex.continuous_norm_sq Complex.continuous_normSq
@[simp, norm_cast]
theorem nnnorm_real (r : ℝ) : ‖(r : ℂ)‖₊ = ‖r‖₊ :=
Subtype.ext <| norm_real r
#align complex.nnnorm_real Complex.nnnorm_real
@[simp, norm_cast]
theorem nnnorm_nat (n : ℕ) : ‖(n : ℂ)‖₊ = n :=
Subtype.ext <| by simp
#align complex.nnnorm_nat Complex.nnnorm_nat
@[simp, norm_cast]
theorem nnnorm_int (n : ℤ) : ‖(n : ℂ)‖₊ = ‖n‖₊ :=
Subtype.ext norm_int
#align complex.nnnorm_int Complex.nnnorm_int
theorem nnnorm_eq_one_of_pow_eq_one {ζ : ℂ} {n : ℕ} (h : ζ ^ n = 1) (hn : n ≠ 0) : ‖ζ‖₊ = 1 :=
(pow_left_inj zero_le' zero_le' hn).1 <| by rw [← nnnorm_pow, h, nnnorm_one, one_pow]
#align complex.nnnorm_eq_one_of_pow_eq_one Complex.nnnorm_eq_one_of_pow_eq_one
theorem norm_eq_one_of_pow_eq_one {ζ : ℂ} {n : ℕ} (h : ζ ^ n = 1) (hn : n ≠ 0) : ‖ζ‖ = 1 :=
congr_arg Subtype.val (nnnorm_eq_one_of_pow_eq_one h hn)
#align complex.norm_eq_one_of_pow_eq_one Complex.norm_eq_one_of_pow_eq_one
theorem equivRealProd_apply_le (z : ℂ) : ‖equivRealProd z‖ ≤ abs z := by
simp [Prod.norm_def, abs_re_le_abs, abs_im_le_abs]
#align complex.equiv_real_prod_apply_le Complex.equivRealProd_apply_le
theorem equivRealProd_apply_le' (z : ℂ) : ‖equivRealProd z‖ ≤ 1 * abs z := by
simpa using equivRealProd_apply_le z
#align complex.equiv_real_prod_apply_le' Complex.equivRealProd_apply_le'
theorem lipschitz_equivRealProd : LipschitzWith 1 equivRealProd := by
simpa using AddMonoidHomClass.lipschitz_of_bound equivRealProdLm 1 equivRealProd_apply_le'
#align complex.lipschitz_equiv_real_prod Complex.lipschitz_equivRealProd
theorem antilipschitz_equivRealProd : AntilipschitzWith (NNReal.sqrt 2) equivRealProd :=
AddMonoidHomClass.antilipschitz_of_bound equivRealProdLm fun z ↦ by
simpa only [Real.coe_sqrt, NNReal.coe_ofNat] using abs_le_sqrt_two_mul_max z
#align complex.antilipschitz_equiv_real_prod Complex.antilipschitz_equivRealProd
theorem uniformEmbedding_equivRealProd : UniformEmbedding equivRealProd :=
antilipschitz_equivRealProd.uniformEmbedding lipschitz_equivRealProd.uniformContinuous
#align complex.uniform_embedding_equiv_real_prod Complex.uniformEmbedding_equivRealProd
instance : CompleteSpace ℂ :=
(completeSpace_congr uniformEmbedding_equivRealProd).mpr inferInstance
instance instT2Space : T2Space ℂ := TopologicalSpace.t2Space_of_metrizableSpace
@[simps! (config := { simpRhs := true }) apply symm_apply_re symm_apply_im]
def equivRealProdCLM : ℂ ≃L[ℝ] ℝ × ℝ :=
equivRealProdLm.toContinuousLinearEquivOfBounds 1 (√2) equivRealProd_apply_le' fun p =>
abs_le_sqrt_two_mul_max (equivRealProd.symm p)
#align complex.equiv_real_prod_clm Complex.equivRealProdCLM
theorem equivRealProdCLM_symm_apply (p : ℝ × ℝ) :
Complex.equivRealProdCLM.symm p = p.1 + p.2 * Complex.I := Complex.equivRealProd_symm_apply p
instance : ProperSpace ℂ :=
(id lipschitz_equivRealProd : LipschitzWith 1 equivRealProdCLM.toHomeomorph).properSpace
theorem tendsto_abs_cocompact_atTop : Tendsto abs (cocompact ℂ) atTop :=
tendsto_norm_cocompact_atTop
#align complex.tendsto_abs_cocompact_at_top Complex.tendsto_abs_cocompact_atTop
theorem tendsto_normSq_cocompact_atTop : Tendsto normSq (cocompact ℂ) atTop := by
simpa [mul_self_abs]
using tendsto_abs_cocompact_atTop.atTop_mul_atTop tendsto_abs_cocompact_atTop
#align complex.tendsto_norm_sq_cocompact_at_top Complex.tendsto_normSq_cocompact_atTop
open ContinuousLinearMap
def reCLM : ℂ →L[ℝ] ℝ :=
reLm.mkContinuous 1 fun x => by simp [abs_re_le_abs]
#align complex.re_clm Complex.reCLM
@[continuity, fun_prop]
theorem continuous_re : Continuous re :=
reCLM.continuous
#align complex.continuous_re Complex.continuous_re
@[simp]
theorem reCLM_coe : (reCLM : ℂ →ₗ[ℝ] ℝ) = reLm :=
rfl
#align complex.re_clm_coe Complex.reCLM_coe
@[simp]
theorem reCLM_apply (z : ℂ) : (reCLM : ℂ → ℝ) z = z.re :=
rfl
#align complex.re_clm_apply Complex.reCLM_apply
def imCLM : ℂ →L[ℝ] ℝ :=
imLm.mkContinuous 1 fun x => by simp [abs_im_le_abs]
#align complex.im_clm Complex.imCLM
@[continuity, fun_prop]
theorem continuous_im : Continuous im :=
imCLM.continuous
#align complex.continuous_im Complex.continuous_im
@[simp]
theorem imCLM_coe : (imCLM : ℂ →ₗ[ℝ] ℝ) = imLm :=
rfl
#align complex.im_clm_coe Complex.imCLM_coe
@[simp]
theorem imCLM_apply (z : ℂ) : (imCLM : ℂ → ℝ) z = z.im :=
rfl
#align complex.im_clm_apply Complex.imCLM_apply
theorem restrictScalars_one_smulRight' (x : E) :
ContinuousLinearMap.restrictScalars ℝ ((1 : ℂ →L[ℂ] ℂ).smulRight x : ℂ →L[ℂ] E) =
reCLM.smulRight x + I • imCLM.smulRight x := by
ext ⟨a, b⟩
simp [mk_eq_add_mul_I, mul_smul, smul_comm I b x]
#align complex.restrict_scalars_one_smul_right' Complex.restrictScalars_one_smulRight'
theorem restrictScalars_one_smulRight (x : ℂ) :
ContinuousLinearMap.restrictScalars ℝ ((1 : ℂ →L[ℂ] ℂ).smulRight x : ℂ →L[ℂ] ℂ) =
x • (1 : ℂ →L[ℝ] ℂ) := by
ext1 z
dsimp
apply mul_comm
#align complex.restrict_scalars_one_smul_right Complex.restrictScalars_one_smulRight
def conjLIE : ℂ ≃ₗᵢ[ℝ] ℂ :=
⟨conjAe.toLinearEquiv, abs_conj⟩
#align complex.conj_lie Complex.conjLIE
@[simp]
theorem conjLIE_apply (z : ℂ) : conjLIE z = conj z :=
rfl
#align complex.conj_lie_apply Complex.conjLIE_apply
@[simp]
theorem conjLIE_symm : conjLIE.symm = conjLIE :=
rfl
#align complex.conj_lie_symm Complex.conjLIE_symm
theorem isometry_conj : Isometry (conj : ℂ → ℂ) :=
conjLIE.isometry
#align complex.isometry_conj Complex.isometry_conj
@[simp]
theorem dist_conj_conj (z w : ℂ) : dist (conj z) (conj w) = dist z w :=
isometry_conj.dist_eq z w
#align complex.dist_conj_conj Complex.dist_conj_conj
@[simp]
theorem nndist_conj_conj (z w : ℂ) : nndist (conj z) (conj w) = nndist z w :=
isometry_conj.nndist_eq z w
#align complex.nndist_conj_conj Complex.nndist_conj_conj
| Mathlib/Analysis/Complex/Basic.lean | 353 | 354 | theorem dist_conj_comm (z w : ℂ) : dist (conj z) w = dist z (conj w) := by |
rw [← dist_conj_conj, conj_conj]
|
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/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
variable {𝓕 𝕜 α ι κ E F G : Type*}
open Filter Function Metric Bornology
open ENNReal Filter NNReal Uniformity Pointwise Topology
@[notation_class]
class Norm (E : Type*) where
norm : E → ℝ
#align has_norm Norm
@[notation_class]
class NNNorm (E : Type*) where
nnnorm : E → ℝ≥0
#align has_nnnorm NNNorm
export Norm (norm)
export NNNorm (nnnorm)
@[inherit_doc]
notation "‖" e "‖" => norm e
@[inherit_doc]
notation "‖" e "‖₊" => nnnorm e
class SeminormedAddGroup (E : Type*) extends Norm E, AddGroup E, PseudoMetricSpace E where
dist := fun x y => ‖x - y‖
dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
#align seminormed_add_group SeminormedAddGroup
@[to_additive]
class SeminormedGroup (E : Type*) extends Norm E, Group E, PseudoMetricSpace E where
dist := fun x y => ‖x / y‖
dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
#align seminormed_group SeminormedGroup
class NormedAddGroup (E : Type*) extends Norm E, AddGroup E, MetricSpace E where
dist := fun x y => ‖x - y‖
dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
#align normed_add_group NormedAddGroup
@[to_additive]
class NormedGroup (E : Type*) extends Norm E, Group E, MetricSpace E where
dist := fun x y => ‖x / y‖
dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
#align normed_group NormedGroup
class SeminormedAddCommGroup (E : Type*) extends Norm E, AddCommGroup E,
PseudoMetricSpace E where
dist := fun x y => ‖x - y‖
dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
#align seminormed_add_comm_group SeminormedAddCommGroup
@[to_additive]
class SeminormedCommGroup (E : Type*) extends Norm E, CommGroup E, PseudoMetricSpace E where
dist := fun x y => ‖x / y‖
dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
#align seminormed_comm_group SeminormedCommGroup
class NormedAddCommGroup (E : Type*) extends Norm E, AddCommGroup E, MetricSpace E where
dist := fun x y => ‖x - y‖
dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
#align normed_add_comm_group NormedAddCommGroup
@[to_additive]
class NormedCommGroup (E : Type*) extends Norm E, CommGroup E, MetricSpace E where
dist := fun x y => ‖x / y‖
dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
#align normed_comm_group NormedCommGroup
-- See note [lower instance priority]
@[to_additive]
instance (priority := 100) NormedGroup.toSeminormedGroup [NormedGroup E] : SeminormedGroup E :=
{ ‹NormedGroup E› with }
#align normed_group.to_seminormed_group NormedGroup.toSeminormedGroup
#align normed_add_group.to_seminormed_add_group NormedAddGroup.toSeminormedAddGroup
-- See note [lower instance priority]
@[to_additive]
instance (priority := 100) NormedCommGroup.toSeminormedCommGroup [NormedCommGroup E] :
SeminormedCommGroup E :=
{ ‹NormedCommGroup E› with }
#align normed_comm_group.to_seminormed_comm_group NormedCommGroup.toSeminormedCommGroup
#align normed_add_comm_group.to_seminormed_add_comm_group NormedAddCommGroup.toSeminormedAddCommGroup
-- See note [lower instance priority]
@[to_additive]
instance (priority := 100) SeminormedCommGroup.toSeminormedGroup [SeminormedCommGroup E] :
SeminormedGroup E :=
{ ‹SeminormedCommGroup E› with }
#align seminormed_comm_group.to_seminormed_group SeminormedCommGroup.toSeminormedGroup
#align seminormed_add_comm_group.to_seminormed_add_group SeminormedAddCommGroup.toSeminormedAddGroup
-- See note [lower instance priority]
@[to_additive]
instance (priority := 100) NormedCommGroup.toNormedGroup [NormedCommGroup E] : NormedGroup E :=
{ ‹NormedCommGroup E› with }
#align normed_comm_group.to_normed_group NormedCommGroup.toNormedGroup
#align normed_add_comm_group.to_normed_add_group NormedAddCommGroup.toNormedAddGroup
-- See note [reducible non-instances]
@[to_additive (attr := reducible) "Construct a `NormedAddGroup` from a `SeminormedAddGroup`
satisfying `∀ x, ‖x‖ = 0 → x = 0`. This avoids having to go back to the `(Pseudo)MetricSpace`
level when declaring a `NormedAddGroup` instance as a special case of a more general
`SeminormedAddGroup` instance."]
def NormedGroup.ofSeparation [SeminormedGroup E] (h : ∀ x : E, ‖x‖ = 0 → x = 1) :
NormedGroup E where
dist_eq := ‹SeminormedGroup E›.dist_eq
toMetricSpace :=
{ eq_of_dist_eq_zero := fun hxy =>
div_eq_one.1 <| h _ <| by exact (‹SeminormedGroup E›.dist_eq _ _).symm.trans hxy }
-- Porting note: the `rwa` no longer worked, but it was easy enough to provide the term.
-- however, notice that if you make `x` and `y` accessible, then the following does work:
-- `have := ‹SeminormedGroup E›.dist_eq x y; rwa [← this]`, so I'm not sure why the `rwa`
-- was broken.
#align normed_group.of_separation NormedGroup.ofSeparation
#align normed_add_group.of_separation NormedAddGroup.ofSeparation
-- See note [reducible non-instances]
@[to_additive (attr := reducible) "Construct a `NormedAddCommGroup` from a
`SeminormedAddCommGroup` satisfying `∀ x, ‖x‖ = 0 → x = 0`. This avoids having to go back to the
`(Pseudo)MetricSpace` level when declaring a `NormedAddCommGroup` instance as a special case
of a more general `SeminormedAddCommGroup` instance."]
def NormedCommGroup.ofSeparation [SeminormedCommGroup E] (h : ∀ x : E, ‖x‖ = 0 → x = 1) :
NormedCommGroup E :=
{ ‹SeminormedCommGroup E›, NormedGroup.ofSeparation h with }
#align normed_comm_group.of_separation NormedCommGroup.ofSeparation
#align normed_add_comm_group.of_separation NormedAddCommGroup.ofSeparation
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a translation-invariant distance."]
def SeminormedGroup.ofMulDist [Norm E] [Group E] [PseudoMetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
SeminormedGroup E where
dist_eq x y := by
rw [h₁]; apply le_antisymm
· simpa only [div_eq_mul_inv, ← mul_right_inv y] using h₂ _ _ _
· simpa only [div_mul_cancel, one_mul] using h₂ (x / y) 1 y
#align seminormed_group.of_mul_dist SeminormedGroup.ofMulDist
#align seminormed_add_group.of_add_dist SeminormedAddGroup.ofAddDist
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a translation-invariant pseudodistance."]
def SeminormedGroup.ofMulDist' [Norm E] [Group E] [PseudoMetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
SeminormedGroup E where
dist_eq x y := by
rw [h₁]; apply le_antisymm
· simpa only [div_mul_cancel, one_mul] using h₂ (x / y) 1 y
· simpa only [div_eq_mul_inv, ← mul_right_inv y] using h₂ _ _ _
#align seminormed_group.of_mul_dist' SeminormedGroup.ofMulDist'
#align seminormed_add_group.of_add_dist' SeminormedAddGroup.ofAddDist'
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a translation-invariant pseudodistance."]
def SeminormedCommGroup.ofMulDist [Norm E] [CommGroup E] [PseudoMetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
SeminormedCommGroup E :=
{ SeminormedGroup.ofMulDist h₁ h₂ with
mul_comm := mul_comm }
#align seminormed_comm_group.of_mul_dist SeminormedCommGroup.ofMulDist
#align seminormed_add_comm_group.of_add_dist SeminormedAddCommGroup.ofAddDist
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a translation-invariant pseudodistance."]
def SeminormedCommGroup.ofMulDist' [Norm E] [CommGroup E] [PseudoMetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
SeminormedCommGroup E :=
{ SeminormedGroup.ofMulDist' h₁ h₂ with
mul_comm := mul_comm }
#align seminormed_comm_group.of_mul_dist' SeminormedCommGroup.ofMulDist'
#align seminormed_add_comm_group.of_add_dist' SeminormedAddCommGroup.ofAddDist'
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a translation-invariant distance."]
def NormedGroup.ofMulDist [Norm E] [Group E] [MetricSpace E] (h₁ : ∀ x : E, ‖x‖ = dist x 1)
(h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) : NormedGroup E :=
{ SeminormedGroup.ofMulDist h₁ h₂ with
eq_of_dist_eq_zero := eq_of_dist_eq_zero }
#align normed_group.of_mul_dist NormedGroup.ofMulDist
#align normed_add_group.of_add_dist NormedAddGroup.ofAddDist
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a translation-invariant pseudodistance."]
def NormedGroup.ofMulDist' [Norm E] [Group E] [MetricSpace E] (h₁ : ∀ x : E, ‖x‖ = dist x 1)
(h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) : NormedGroup E :=
{ SeminormedGroup.ofMulDist' h₁ h₂ with
eq_of_dist_eq_zero := eq_of_dist_eq_zero }
#align normed_group.of_mul_dist' NormedGroup.ofMulDist'
#align normed_add_group.of_add_dist' NormedAddGroup.ofAddDist'
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a translation-invariant pseudodistance."]
def NormedCommGroup.ofMulDist [Norm E] [CommGroup E] [MetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
NormedCommGroup E :=
{ NormedGroup.ofMulDist h₁ h₂ with
mul_comm := mul_comm }
#align normed_comm_group.of_mul_dist NormedCommGroup.ofMulDist
#align normed_add_comm_group.of_add_dist NormedAddCommGroup.ofAddDist
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a translation-invariant pseudodistance."]
def NormedCommGroup.ofMulDist' [Norm E] [CommGroup E] [MetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
NormedCommGroup E :=
{ NormedGroup.ofMulDist' h₁ h₂ with
mul_comm := mul_comm }
#align normed_comm_group.of_mul_dist' NormedCommGroup.ofMulDist'
#align normed_add_comm_group.of_add_dist' NormedAddCommGroup.ofAddDist'
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a seminorm, i.e., registering the pseudodistance
and the pseudometric space structure from the seminorm properties. Note that in most cases this
instance creates bad definitional equalities (e.g., it does not take into account a possibly
existing `UniformSpace` instance on `E`)."]
def GroupSeminorm.toSeminormedGroup [Group E] (f : GroupSeminorm E) : SeminormedGroup E where
dist x y := f (x / y)
norm := f
dist_eq x y := rfl
dist_self x := by simp only [div_self', map_one_eq_zero]
dist_triangle := le_map_div_add_map_div f
dist_comm := map_div_rev f
edist_dist x y := by exact ENNReal.coe_nnreal_eq _
-- Porting note: how did `mathlib3` solve this automatically?
#align group_seminorm.to_seminormed_group GroupSeminorm.toSeminormedGroup
#align add_group_seminorm.to_seminormed_add_group AddGroupSeminorm.toSeminormedAddGroup
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a seminorm, i.e., registering the pseudodistance
and the pseudometric space structure from the seminorm properties. Note that in most cases this
instance creates bad definitional equalities (e.g., it does not take into account a possibly
existing `UniformSpace` instance on `E`)."]
def GroupSeminorm.toSeminormedCommGroup [CommGroup E] (f : GroupSeminorm E) :
SeminormedCommGroup E :=
{ f.toSeminormedGroup with
mul_comm := mul_comm }
#align group_seminorm.to_seminormed_comm_group GroupSeminorm.toSeminormedCommGroup
#align add_group_seminorm.to_seminormed_add_comm_group AddGroupSeminorm.toSeminormedAddCommGroup
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a norm, i.e., registering the distance and the metric
space structure from the norm properties. Note that in most cases this instance creates bad
definitional equalities (e.g., it does not take into account a possibly existing `UniformSpace`
instance on `E`)."]
def GroupNorm.toNormedGroup [Group E] (f : GroupNorm E) : NormedGroup E :=
{ f.toGroupSeminorm.toSeminormedGroup with
eq_of_dist_eq_zero := fun h => div_eq_one.1 <| eq_one_of_map_eq_zero f h }
#align group_norm.to_normed_group GroupNorm.toNormedGroup
#align add_group_norm.to_normed_add_group AddGroupNorm.toNormedAddGroup
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a norm, i.e., registering the distance and the metric
space structure from the norm properties. Note that in most cases this instance creates bad
definitional equalities (e.g., it does not take into account a possibly existing `UniformSpace`
instance on `E`)."]
def GroupNorm.toNormedCommGroup [CommGroup E] (f : GroupNorm E) : NormedCommGroup E :=
{ f.toNormedGroup with
mul_comm := mul_comm }
#align group_norm.to_normed_comm_group GroupNorm.toNormedCommGroup
#align add_group_norm.to_normed_add_comm_group AddGroupNorm.toNormedAddCommGroup
instance PUnit.normedAddCommGroup : NormedAddCommGroup PUnit where
norm := Function.const _ 0
dist_eq _ _ := rfl
@[simp]
theorem PUnit.norm_eq_zero (r : PUnit) : ‖r‖ = 0 :=
rfl
#align punit.norm_eq_zero PUnit.norm_eq_zero
section SeminormedGroup
variable [SeminormedGroup E] [SeminormedGroup F] [SeminormedGroup G] {s : Set E}
{a a₁ a₂ b b₁ b₂ : E} {r r₁ r₂ : ℝ}
@[to_additive]
theorem dist_eq_norm_div (a b : E) : dist a b = ‖a / b‖ :=
SeminormedGroup.dist_eq _ _
#align dist_eq_norm_div dist_eq_norm_div
#align dist_eq_norm_sub dist_eq_norm_sub
@[to_additive]
theorem dist_eq_norm_div' (a b : E) : dist a b = ‖b / a‖ := by rw [dist_comm, dist_eq_norm_div]
#align dist_eq_norm_div' dist_eq_norm_div'
#align dist_eq_norm_sub' dist_eq_norm_sub'
alias dist_eq_norm := dist_eq_norm_sub
#align dist_eq_norm dist_eq_norm
alias dist_eq_norm' := dist_eq_norm_sub'
#align dist_eq_norm' dist_eq_norm'
@[to_additive]
instance NormedGroup.to_isometricSMul_right : IsometricSMul Eᵐᵒᵖ E :=
⟨fun a => Isometry.of_dist_eq fun b c => by simp [dist_eq_norm_div]⟩
#align normed_group.to_has_isometric_smul_right NormedGroup.to_isometricSMul_right
#align normed_add_group.to_has_isometric_vadd_right NormedAddGroup.to_isometricVAdd_right
@[to_additive (attr := simp)]
theorem dist_one_right (a : E) : dist a 1 = ‖a‖ := by rw [dist_eq_norm_div, div_one]
#align dist_one_right dist_one_right
#align dist_zero_right dist_zero_right
@[to_additive]
theorem inseparable_one_iff_norm {a : E} : Inseparable a 1 ↔ ‖a‖ = 0 := by
rw [Metric.inseparable_iff, dist_one_right]
@[to_additive (attr := simp)]
theorem dist_one_left : dist (1 : E) = norm :=
funext fun a => by rw [dist_comm, dist_one_right]
#align dist_one_left dist_one_left
#align dist_zero_left dist_zero_left
@[to_additive]
theorem Isometry.norm_map_of_map_one {f : E → F} (hi : Isometry f) (h₁ : f 1 = 1) (x : E) :
‖f x‖ = ‖x‖ := by rw [← dist_one_right, ← h₁, hi.dist_eq, dist_one_right]
#align isometry.norm_map_of_map_one Isometry.norm_map_of_map_one
#align isometry.norm_map_of_map_zero Isometry.norm_map_of_map_zero
@[to_additive (attr := simp) comap_norm_atTop]
theorem comap_norm_atTop' : comap norm atTop = cobounded E := by
simpa only [dist_one_right] using comap_dist_right_atTop (1 : E)
@[to_additive Filter.HasBasis.cobounded_of_norm]
lemma Filter.HasBasis.cobounded_of_norm' {ι : Sort*} {p : ι → Prop} {s : ι → Set ℝ}
(h : HasBasis atTop p s) : HasBasis (cobounded E) p fun i ↦ norm ⁻¹' s i :=
comap_norm_atTop' (E := E) ▸ h.comap _
@[to_additive Filter.hasBasis_cobounded_norm]
lemma Filter.hasBasis_cobounded_norm' : HasBasis (cobounded E) (fun _ ↦ True) ({x | · ≤ ‖x‖}) :=
atTop_basis.cobounded_of_norm'
@[to_additive (attr := simp) tendsto_norm_atTop_iff_cobounded]
theorem tendsto_norm_atTop_iff_cobounded' {f : α → E} {l : Filter α} :
Tendsto (‖f ·‖) l atTop ↔ Tendsto f l (cobounded E) := by
rw [← comap_norm_atTop', tendsto_comap_iff]; rfl
@[to_additive tendsto_norm_cobounded_atTop]
theorem tendsto_norm_cobounded_atTop' : Tendsto norm (cobounded E) atTop :=
tendsto_norm_atTop_iff_cobounded'.2 tendsto_id
@[to_additive eventually_cobounded_le_norm]
lemma eventually_cobounded_le_norm' (a : ℝ) : ∀ᶠ x in cobounded E, a ≤ ‖x‖ :=
tendsto_norm_cobounded_atTop'.eventually_ge_atTop a
@[to_additive tendsto_norm_cocompact_atTop]
theorem tendsto_norm_cocompact_atTop' [ProperSpace E] : Tendsto norm (cocompact E) atTop :=
cobounded_eq_cocompact (α := E) ▸ tendsto_norm_cobounded_atTop'
#align tendsto_norm_cocompact_at_top' tendsto_norm_cocompact_atTop'
#align tendsto_norm_cocompact_at_top tendsto_norm_cocompact_atTop
@[to_additive]
theorem norm_div_rev (a b : E) : ‖a / b‖ = ‖b / a‖ := by
simpa only [dist_eq_norm_div] using dist_comm a b
#align norm_div_rev norm_div_rev
#align norm_sub_rev norm_sub_rev
@[to_additive (attr := simp) norm_neg]
theorem norm_inv' (a : E) : ‖a⁻¹‖ = ‖a‖ := by simpa using norm_div_rev 1 a
#align norm_inv' norm_inv'
#align norm_neg norm_neg
open scoped symmDiff in
@[to_additive]
theorem dist_mulIndicator (s t : Set α) (f : α → E) (x : α) :
dist (s.mulIndicator f x) (t.mulIndicator f x) = ‖(s ∆ t).mulIndicator f x‖ := by
rw [dist_eq_norm_div, Set.apply_mulIndicator_symmDiff norm_inv']
@[to_additive (attr := simp)]
theorem dist_mul_self_right (a b : E) : dist b (a * b) = ‖a‖ := by
rw [← dist_one_left, ← dist_mul_right 1 a b, one_mul]
#align dist_mul_self_right dist_mul_self_right
#align dist_add_self_right dist_add_self_right
@[to_additive (attr := simp)]
theorem dist_mul_self_left (a b : E) : dist (a * b) b = ‖a‖ := by
rw [dist_comm, dist_mul_self_right]
#align dist_mul_self_left dist_mul_self_left
#align dist_add_self_left dist_add_self_left
@[to_additive (attr := simp)]
theorem dist_div_eq_dist_mul_left (a b c : E) : dist (a / b) c = dist a (c * b) := by
rw [← dist_mul_right _ _ b, div_mul_cancel]
#align dist_div_eq_dist_mul_left dist_div_eq_dist_mul_left
#align dist_sub_eq_dist_add_left dist_sub_eq_dist_add_left
@[to_additive (attr := simp)]
theorem dist_div_eq_dist_mul_right (a b c : E) : dist a (b / c) = dist (a * c) b := by
rw [← dist_mul_right _ _ c, div_mul_cancel]
#align dist_div_eq_dist_mul_right dist_div_eq_dist_mul_right
#align dist_sub_eq_dist_add_right dist_sub_eq_dist_add_right
@[to_additive (attr := simp)]
lemma Filter.inv_cobounded : (cobounded E)⁻¹ = cobounded E := by
simp only [← comap_norm_atTop', ← Filter.comap_inv, comap_comap, (· ∘ ·), norm_inv']
@[to_additive "In a (semi)normed group, negation `x ↦ -x` tends to infinity at infinity."]
theorem Filter.tendsto_inv_cobounded : Tendsto Inv.inv (cobounded E) (cobounded E) :=
inv_cobounded.le
#align filter.tendsto_inv_cobounded Filter.tendsto_inv_cobounded
#align filter.tendsto_neg_cobounded Filter.tendsto_neg_cobounded
@[to_additive norm_add_le "**Triangle inequality** for the norm."]
theorem norm_mul_le' (a b : E) : ‖a * b‖ ≤ ‖a‖ + ‖b‖ := by
simpa [dist_eq_norm_div] using dist_triangle a 1 b⁻¹
#align norm_mul_le' norm_mul_le'
#align norm_add_le norm_add_le
@[to_additive]
theorem norm_mul_le_of_le (h₁ : ‖a₁‖ ≤ r₁) (h₂ : ‖a₂‖ ≤ r₂) : ‖a₁ * a₂‖ ≤ r₁ + r₂ :=
(norm_mul_le' a₁ a₂).trans <| add_le_add h₁ h₂
#align norm_mul_le_of_le norm_mul_le_of_le
#align norm_add_le_of_le norm_add_le_of_le
@[to_additive norm_add₃_le]
theorem norm_mul₃_le (a b c : E) : ‖a * b * c‖ ≤ ‖a‖ + ‖b‖ + ‖c‖ :=
norm_mul_le_of_le (norm_mul_le' _ _) le_rfl
#align norm_mul₃_le norm_mul₃_le
#align norm_add₃_le norm_add₃_le
@[to_additive]
lemma norm_div_le_norm_div_add_norm_div (a b c : E) : ‖a / c‖ ≤ ‖a / b‖ + ‖b / c‖ := by
simpa only [dist_eq_norm_div] using dist_triangle a b c
@[to_additive (attr := simp) norm_nonneg]
theorem norm_nonneg' (a : E) : 0 ≤ ‖a‖ := by
rw [← dist_one_right]
exact dist_nonneg
#align norm_nonneg' norm_nonneg'
#align norm_nonneg norm_nonneg
@[to_additive (attr := simp) abs_norm]
theorem abs_norm' (z : E) : |‖z‖| = ‖z‖ := abs_of_nonneg <| norm_nonneg' _
#align abs_norm abs_norm
@[to_additive (attr := simp) norm_zero]
theorem norm_one' : ‖(1 : E)‖ = 0 := by rw [← dist_one_right, dist_self]
#align norm_one' norm_one'
#align norm_zero norm_zero
@[to_additive]
theorem ne_one_of_norm_ne_zero : ‖a‖ ≠ 0 → a ≠ 1 :=
mt <| by
rintro rfl
exact norm_one'
#align ne_one_of_norm_ne_zero ne_one_of_norm_ne_zero
#align ne_zero_of_norm_ne_zero ne_zero_of_norm_ne_zero
@[to_additive (attr := nontriviality) norm_of_subsingleton]
theorem norm_of_subsingleton' [Subsingleton E] (a : E) : ‖a‖ = 0 := by
rw [Subsingleton.elim a 1, norm_one']
#align norm_of_subsingleton' norm_of_subsingleton'
#align norm_of_subsingleton norm_of_subsingleton
@[to_additive zero_lt_one_add_norm_sq]
theorem zero_lt_one_add_norm_sq' (x : E) : 0 < 1 + ‖x‖ ^ 2 := by
positivity
#align zero_lt_one_add_norm_sq' zero_lt_one_add_norm_sq'
#align zero_lt_one_add_norm_sq zero_lt_one_add_norm_sq
@[to_additive]
theorem norm_div_le (a b : E) : ‖a / b‖ ≤ ‖a‖ + ‖b‖ := by
simpa [dist_eq_norm_div] using dist_triangle a 1 b
#align norm_div_le norm_div_le
#align norm_sub_le norm_sub_le
@[to_additive]
theorem norm_div_le_of_le {r₁ r₂ : ℝ} (H₁ : ‖a₁‖ ≤ r₁) (H₂ : ‖a₂‖ ≤ r₂) : ‖a₁ / a₂‖ ≤ r₁ + r₂ :=
(norm_div_le a₁ a₂).trans <| add_le_add H₁ H₂
#align norm_div_le_of_le norm_div_le_of_le
#align norm_sub_le_of_le norm_sub_le_of_le
@[to_additive dist_le_norm_add_norm]
theorem dist_le_norm_add_norm' (a b : E) : dist a b ≤ ‖a‖ + ‖b‖ := by
rw [dist_eq_norm_div]
apply norm_div_le
#align dist_le_norm_add_norm' dist_le_norm_add_norm'
#align dist_le_norm_add_norm dist_le_norm_add_norm
@[to_additive abs_norm_sub_norm_le]
theorem abs_norm_sub_norm_le' (a b : E) : |‖a‖ - ‖b‖| ≤ ‖a / b‖ := by
simpa [dist_eq_norm_div] using abs_dist_sub_le a b 1
#align abs_norm_sub_norm_le' abs_norm_sub_norm_le'
#align abs_norm_sub_norm_le abs_norm_sub_norm_le
@[to_additive norm_sub_norm_le]
theorem norm_sub_norm_le' (a b : E) : ‖a‖ - ‖b‖ ≤ ‖a / b‖ :=
(le_abs_self _).trans (abs_norm_sub_norm_le' a b)
#align norm_sub_norm_le' norm_sub_norm_le'
#align norm_sub_norm_le norm_sub_norm_le
@[to_additive dist_norm_norm_le]
theorem dist_norm_norm_le' (a b : E) : dist ‖a‖ ‖b‖ ≤ ‖a / b‖ :=
abs_norm_sub_norm_le' a b
#align dist_norm_norm_le' dist_norm_norm_le'
#align dist_norm_norm_le dist_norm_norm_le
@[to_additive]
theorem norm_le_norm_add_norm_div' (u v : E) : ‖u‖ ≤ ‖v‖ + ‖u / v‖ := by
rw [add_comm]
refine (norm_mul_le' _ _).trans_eq' ?_
rw [div_mul_cancel]
#align norm_le_norm_add_norm_div' norm_le_norm_add_norm_div'
#align norm_le_norm_add_norm_sub' norm_le_norm_add_norm_sub'
@[to_additive]
theorem norm_le_norm_add_norm_div (u v : E) : ‖v‖ ≤ ‖u‖ + ‖u / v‖ := by
rw [norm_div_rev]
exact norm_le_norm_add_norm_div' v u
#align norm_le_norm_add_norm_div norm_le_norm_add_norm_div
#align norm_le_norm_add_norm_sub norm_le_norm_add_norm_sub
alias norm_le_insert' := norm_le_norm_add_norm_sub'
#align norm_le_insert' norm_le_insert'
alias norm_le_insert := norm_le_norm_add_norm_sub
#align norm_le_insert norm_le_insert
@[to_additive]
theorem norm_le_mul_norm_add (u v : E) : ‖u‖ ≤ ‖u * v‖ + ‖v‖ :=
calc
‖u‖ = ‖u * v / v‖ := by rw [mul_div_cancel_right]
_ ≤ ‖u * v‖ + ‖v‖ := norm_div_le _ _
#align norm_le_mul_norm_add norm_le_mul_norm_add
#align norm_le_add_norm_add norm_le_add_norm_add
@[to_additive ball_eq]
theorem ball_eq' (y : E) (ε : ℝ) : ball y ε = { x | ‖x / y‖ < ε } :=
Set.ext fun a => by simp [dist_eq_norm_div]
#align ball_eq' ball_eq'
#align ball_eq ball_eq
@[to_additive]
theorem ball_one_eq (r : ℝ) : ball (1 : E) r = { x | ‖x‖ < r } :=
Set.ext fun a => by simp
#align ball_one_eq ball_one_eq
#align ball_zero_eq ball_zero_eq
@[to_additive mem_ball_iff_norm]
theorem mem_ball_iff_norm'' : b ∈ ball a r ↔ ‖b / a‖ < r := by rw [mem_ball, dist_eq_norm_div]
#align mem_ball_iff_norm'' mem_ball_iff_norm''
#align mem_ball_iff_norm mem_ball_iff_norm
@[to_additive mem_ball_iff_norm']
theorem mem_ball_iff_norm''' : b ∈ ball a r ↔ ‖a / b‖ < r := by rw [mem_ball', dist_eq_norm_div]
#align mem_ball_iff_norm''' mem_ball_iff_norm'''
#align mem_ball_iff_norm' mem_ball_iff_norm'
@[to_additive] -- Porting note (#10618): `simp` can prove it
theorem mem_ball_one_iff : a ∈ ball (1 : E) r ↔ ‖a‖ < r := by rw [mem_ball, dist_one_right]
#align mem_ball_one_iff mem_ball_one_iff
#align mem_ball_zero_iff mem_ball_zero_iff
@[to_additive mem_closedBall_iff_norm]
theorem mem_closedBall_iff_norm'' : b ∈ closedBall a r ↔ ‖b / a‖ ≤ r := by
rw [mem_closedBall, dist_eq_norm_div]
#align mem_closed_ball_iff_norm'' mem_closedBall_iff_norm''
#align mem_closed_ball_iff_norm mem_closedBall_iff_norm
@[to_additive] -- Porting note (#10618): `simp` can prove it
theorem mem_closedBall_one_iff : a ∈ closedBall (1 : E) r ↔ ‖a‖ ≤ r := by
rw [mem_closedBall, dist_one_right]
#align mem_closed_ball_one_iff mem_closedBall_one_iff
#align mem_closed_ball_zero_iff mem_closedBall_zero_iff
@[to_additive mem_closedBall_iff_norm']
theorem mem_closedBall_iff_norm''' : b ∈ closedBall a r ↔ ‖a / b‖ ≤ r := by
rw [mem_closedBall', dist_eq_norm_div]
#align mem_closed_ball_iff_norm''' mem_closedBall_iff_norm'''
#align mem_closed_ball_iff_norm' mem_closedBall_iff_norm'
@[to_additive norm_le_of_mem_closedBall]
theorem norm_le_of_mem_closedBall' (h : b ∈ closedBall a r) : ‖b‖ ≤ ‖a‖ + r :=
(norm_le_norm_add_norm_div' _ _).trans <| add_le_add_left (by rwa [← dist_eq_norm_div]) _
#align norm_le_of_mem_closed_ball' norm_le_of_mem_closedBall'
#align norm_le_of_mem_closed_ball norm_le_of_mem_closedBall
@[to_additive norm_le_norm_add_const_of_dist_le]
theorem norm_le_norm_add_const_of_dist_le' : dist a b ≤ r → ‖a‖ ≤ ‖b‖ + r :=
norm_le_of_mem_closedBall'
#align norm_le_norm_add_const_of_dist_le' norm_le_norm_add_const_of_dist_le'
#align norm_le_norm_add_const_of_dist_le norm_le_norm_add_const_of_dist_le
@[to_additive norm_lt_of_mem_ball]
theorem norm_lt_of_mem_ball' (h : b ∈ ball a r) : ‖b‖ < ‖a‖ + r :=
(norm_le_norm_add_norm_div' _ _).trans_lt <| add_lt_add_left (by rwa [← dist_eq_norm_div]) _
#align norm_lt_of_mem_ball' norm_lt_of_mem_ball'
#align norm_lt_of_mem_ball norm_lt_of_mem_ball
@[to_additive]
theorem norm_div_sub_norm_div_le_norm_div (u v w : E) : ‖u / w‖ - ‖v / w‖ ≤ ‖u / v‖ := by
simpa only [div_div_div_cancel_right'] using norm_sub_norm_le' (u / w) (v / w)
#align norm_div_sub_norm_div_le_norm_div norm_div_sub_norm_div_le_norm_div
#align norm_sub_sub_norm_sub_le_norm_sub norm_sub_sub_norm_sub_le_norm_sub
@[to_additive isBounded_iff_forall_norm_le]
theorem isBounded_iff_forall_norm_le' : Bornology.IsBounded s ↔ ∃ C, ∀ x ∈ s, ‖x‖ ≤ C := by
simpa only [Set.subset_def, mem_closedBall_one_iff] using isBounded_iff_subset_closedBall (1 : E)
#align bounded_iff_forall_norm_le' isBounded_iff_forall_norm_le'
#align bounded_iff_forall_norm_le isBounded_iff_forall_norm_le
alias ⟨Bornology.IsBounded.exists_norm_le', _⟩ := isBounded_iff_forall_norm_le'
#align metric.bounded.exists_norm_le' Bornology.IsBounded.exists_norm_le'
alias ⟨Bornology.IsBounded.exists_norm_le, _⟩ := isBounded_iff_forall_norm_le
#align metric.bounded.exists_norm_le Bornology.IsBounded.exists_norm_le
attribute [to_additive existing exists_norm_le] Bornology.IsBounded.exists_norm_le'
@[to_additive exists_pos_norm_le]
theorem Bornology.IsBounded.exists_pos_norm_le' (hs : IsBounded s) : ∃ R > 0, ∀ x ∈ s, ‖x‖ ≤ R :=
let ⟨R₀, hR₀⟩ := hs.exists_norm_le'
⟨max R₀ 1, by positivity, fun x hx => (hR₀ x hx).trans <| le_max_left _ _⟩
#align metric.bounded.exists_pos_norm_le' Bornology.IsBounded.exists_pos_norm_le'
#align metric.bounded.exists_pos_norm_le Bornology.IsBounded.exists_pos_norm_le
@[to_additive Bornology.IsBounded.exists_pos_norm_lt]
theorem Bornology.IsBounded.exists_pos_norm_lt' (hs : IsBounded s) : ∃ R > 0, ∀ x ∈ s, ‖x‖ < R :=
let ⟨R, hR₀, hR⟩ := hs.exists_pos_norm_le'
⟨R + 1, by positivity, fun x hx ↦ (hR x hx).trans_lt (lt_add_one _)⟩
@[to_additive (attr := simp 1001) mem_sphere_iff_norm]
-- Porting note: increase priority so the left-hand side doesn't reduce
theorem mem_sphere_iff_norm' : b ∈ sphere a r ↔ ‖b / a‖ = r := by simp [dist_eq_norm_div]
#align mem_sphere_iff_norm' mem_sphere_iff_norm'
#align mem_sphere_iff_norm mem_sphere_iff_norm
@[to_additive] -- `simp` can prove this
theorem mem_sphere_one_iff_norm : a ∈ sphere (1 : E) r ↔ ‖a‖ = r := by simp [dist_eq_norm_div]
#align mem_sphere_one_iff_norm mem_sphere_one_iff_norm
#align mem_sphere_zero_iff_norm mem_sphere_zero_iff_norm
@[to_additive (attr := simp) norm_eq_of_mem_sphere]
theorem norm_eq_of_mem_sphere' (x : sphere (1 : E) r) : ‖(x : E)‖ = r :=
mem_sphere_one_iff_norm.mp x.2
#align norm_eq_of_mem_sphere' norm_eq_of_mem_sphere'
#align norm_eq_of_mem_sphere norm_eq_of_mem_sphere
@[to_additive]
theorem ne_one_of_mem_sphere (hr : r ≠ 0) (x : sphere (1 : E) r) : (x : E) ≠ 1 :=
ne_one_of_norm_ne_zero <| by rwa [norm_eq_of_mem_sphere' x]
#align ne_one_of_mem_sphere ne_one_of_mem_sphere
#align ne_zero_of_mem_sphere ne_zero_of_mem_sphere
@[to_additive ne_zero_of_mem_unit_sphere]
theorem ne_one_of_mem_unit_sphere (x : sphere (1 : E) 1) : (x : E) ≠ 1 :=
ne_one_of_mem_sphere one_ne_zero _
#align ne_one_of_mem_unit_sphere ne_one_of_mem_unit_sphere
#align ne_zero_of_mem_unit_sphere ne_zero_of_mem_unit_sphere
variable (E)
@[to_additive "The norm of a seminormed group as an additive group seminorm."]
def normGroupSeminorm : GroupSeminorm E :=
⟨norm, norm_one', norm_mul_le', norm_inv'⟩
#align norm_group_seminorm normGroupSeminorm
#align norm_add_group_seminorm normAddGroupSeminorm
@[to_additive (attr := simp)]
theorem coe_normGroupSeminorm : ⇑(normGroupSeminorm E) = norm :=
rfl
#align coe_norm_group_seminorm coe_normGroupSeminorm
#align coe_norm_add_group_seminorm coe_normAddGroupSeminorm
variable {E}
@[to_additive]
theorem NormedCommGroup.tendsto_nhds_one {f : α → E} {l : Filter α} :
Tendsto f l (𝓝 1) ↔ ∀ ε > 0, ∀ᶠ x in l, ‖f x‖ < ε :=
Metric.tendsto_nhds.trans <| by simp only [dist_one_right]
#align normed_comm_group.tendsto_nhds_one NormedCommGroup.tendsto_nhds_one
#align normed_add_comm_group.tendsto_nhds_zero NormedAddCommGroup.tendsto_nhds_zero
@[to_additive]
theorem NormedCommGroup.tendsto_nhds_nhds {f : E → F} {x : E} {y : F} :
Tendsto f (𝓝 x) (𝓝 y) ↔ ∀ ε > 0, ∃ δ > 0, ∀ x', ‖x' / x‖ < δ → ‖f x' / y‖ < ε := by
simp_rw [Metric.tendsto_nhds_nhds, dist_eq_norm_div]
#align normed_comm_group.tendsto_nhds_nhds NormedCommGroup.tendsto_nhds_nhds
#align normed_add_comm_group.tendsto_nhds_nhds NormedAddCommGroup.tendsto_nhds_nhds
@[to_additive]
theorem NormedCommGroup.cauchySeq_iff [Nonempty α] [SemilatticeSup α] {u : α → E} :
CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ m, N ≤ m → ∀ n, N ≤ n → ‖u m / u n‖ < ε := by
simp [Metric.cauchySeq_iff, dist_eq_norm_div]
#align normed_comm_group.cauchy_seq_iff NormedCommGroup.cauchySeq_iff
#align normed_add_comm_group.cauchy_seq_iff NormedAddCommGroup.cauchySeq_iff
@[to_additive]
theorem NormedCommGroup.nhds_basis_norm_lt (x : E) :
(𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { y | ‖y / x‖ < ε } := by
simp_rw [← ball_eq']
exact Metric.nhds_basis_ball
#align normed_comm_group.nhds_basis_norm_lt NormedCommGroup.nhds_basis_norm_lt
#align normed_add_comm_group.nhds_basis_norm_lt NormedAddCommGroup.nhds_basis_norm_lt
@[to_additive]
theorem NormedCommGroup.nhds_one_basis_norm_lt :
(𝓝 (1 : E)).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { y | ‖y‖ < ε } := by
convert NormedCommGroup.nhds_basis_norm_lt (1 : E)
simp
#align normed_comm_group.nhds_one_basis_norm_lt NormedCommGroup.nhds_one_basis_norm_lt
#align normed_add_comm_group.nhds_zero_basis_norm_lt NormedAddCommGroup.nhds_zero_basis_norm_lt
@[to_additive]
theorem NormedCommGroup.uniformity_basis_dist :
(𝓤 E).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : E × E | ‖p.fst / p.snd‖ < ε } := by
convert Metric.uniformity_basis_dist (α := E) using 1
simp [dist_eq_norm_div]
#align normed_comm_group.uniformity_basis_dist NormedCommGroup.uniformity_basis_dist
#align normed_add_comm_group.uniformity_basis_dist NormedAddCommGroup.uniformity_basis_dist
open Finset
variable [FunLike 𝓕 E F]
@[to_additive "A homomorphism `f` of seminormed groups is Lipschitz, if there exists a constant
`C` such that for all `x`, one has `‖f x‖ ≤ C * ‖x‖`. The analogous condition for a linear map of
(semi)normed spaces is in `Mathlib/Analysis/NormedSpace/OperatorNorm.lean`."]
theorem MonoidHomClass.lipschitz_of_bound [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ)
(h : ∀ x, ‖f x‖ ≤ C * ‖x‖) : LipschitzWith (Real.toNNReal C) f :=
LipschitzWith.of_dist_le' fun x y => by simpa only [dist_eq_norm_div, map_div] using h (x / y)
#align monoid_hom_class.lipschitz_of_bound MonoidHomClass.lipschitz_of_bound
#align add_monoid_hom_class.lipschitz_of_bound AddMonoidHomClass.lipschitz_of_bound
@[to_additive]
theorem lipschitzOnWith_iff_norm_div_le {f : E → F} {C : ℝ≥0} :
LipschitzOnWith C f s ↔ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ‖f x / f y‖ ≤ C * ‖x / y‖ := by
simp only [lipschitzOnWith_iff_dist_le_mul, dist_eq_norm_div]
#align lipschitz_on_with_iff_norm_div_le lipschitzOnWith_iff_norm_div_le
#align lipschitz_on_with_iff_norm_sub_le lipschitzOnWith_iff_norm_sub_le
alias ⟨LipschitzOnWith.norm_div_le, _⟩ := lipschitzOnWith_iff_norm_div_le
#align lipschitz_on_with.norm_div_le LipschitzOnWith.norm_div_le
attribute [to_additive] LipschitzOnWith.norm_div_le
@[to_additive]
theorem LipschitzOnWith.norm_div_le_of_le {f : E → F} {C : ℝ≥0} (h : LipschitzOnWith C f s)
(ha : a ∈ s) (hb : b ∈ s) (hr : ‖a / b‖ ≤ r) : ‖f a / f b‖ ≤ C * r :=
(h.norm_div_le ha hb).trans <| by gcongr
#align lipschitz_on_with.norm_div_le_of_le LipschitzOnWith.norm_div_le_of_le
#align lipschitz_on_with.norm_sub_le_of_le LipschitzOnWith.norm_sub_le_of_le
@[to_additive]
theorem lipschitzWith_iff_norm_div_le {f : E → F} {C : ℝ≥0} :
LipschitzWith C f ↔ ∀ x y, ‖f x / f y‖ ≤ C * ‖x / y‖ := by
simp only [lipschitzWith_iff_dist_le_mul, dist_eq_norm_div]
#align lipschitz_with_iff_norm_div_le lipschitzWith_iff_norm_div_le
#align lipschitz_with_iff_norm_sub_le lipschitzWith_iff_norm_sub_le
alias ⟨LipschitzWith.norm_div_le, _⟩ := lipschitzWith_iff_norm_div_le
#align lipschitz_with.norm_div_le LipschitzWith.norm_div_le
attribute [to_additive] LipschitzWith.norm_div_le
@[to_additive]
theorem LipschitzWith.norm_div_le_of_le {f : E → F} {C : ℝ≥0} (h : LipschitzWith C f)
(hr : ‖a / b‖ ≤ r) : ‖f a / f b‖ ≤ C * r :=
(h.norm_div_le _ _).trans <| by gcongr
#align lipschitz_with.norm_div_le_of_le LipschitzWith.norm_div_le_of_le
#align lipschitz_with.norm_sub_le_of_le LipschitzWith.norm_sub_le_of_le
@[to_additive "A homomorphism `f` of seminormed groups is continuous, if there exists a constant `C`
such that for all `x`, one has `‖f x‖ ≤ C * ‖x‖`"]
theorem MonoidHomClass.continuous_of_bound [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ)
(h : ∀ x, ‖f x‖ ≤ C * ‖x‖) : Continuous f :=
(MonoidHomClass.lipschitz_of_bound f C h).continuous
#align monoid_hom_class.continuous_of_bound MonoidHomClass.continuous_of_bound
#align add_monoid_hom_class.continuous_of_bound AddMonoidHomClass.continuous_of_bound
@[to_additive]
theorem MonoidHomClass.uniformContinuous_of_bound [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ)
(h : ∀ x, ‖f x‖ ≤ C * ‖x‖) : UniformContinuous f :=
(MonoidHomClass.lipschitz_of_bound f C h).uniformContinuous
#align monoid_hom_class.uniform_continuous_of_bound MonoidHomClass.uniformContinuous_of_bound
#align add_monoid_hom_class.uniform_continuous_of_bound AddMonoidHomClass.uniformContinuous_of_bound
@[to_additive IsCompact.exists_bound_of_continuousOn]
theorem IsCompact.exists_bound_of_continuousOn' [TopologicalSpace α] {s : Set α} (hs : IsCompact s)
{f : α → E} (hf : ContinuousOn f s) : ∃ C, ∀ x ∈ s, ‖f x‖ ≤ C :=
(isBounded_iff_forall_norm_le'.1 (hs.image_of_continuousOn hf).isBounded).imp fun _C hC _x hx =>
hC _ <| Set.mem_image_of_mem _ hx
#align is_compact.exists_bound_of_continuous_on' IsCompact.exists_bound_of_continuousOn'
#align is_compact.exists_bound_of_continuous_on IsCompact.exists_bound_of_continuousOn
@[to_additive]
theorem HasCompactMulSupport.exists_bound_of_continuous [TopologicalSpace α]
{f : α → E} (hf : HasCompactMulSupport f) (h'f : Continuous f) : ∃ C, ∀ x, ‖f x‖ ≤ C := by
simpa using (hf.isCompact_range h'f).isBounded.exists_norm_le'
@[to_additive]
theorem MonoidHomClass.isometry_iff_norm [MonoidHomClass 𝓕 E F] (f : 𝓕) :
Isometry f ↔ ∀ x, ‖f x‖ = ‖x‖ := by
simp only [isometry_iff_dist_eq, dist_eq_norm_div, ← map_div]
refine ⟨fun h x => ?_, fun h x y => h _⟩
simpa using h x 1
#align monoid_hom_class.isometry_iff_norm MonoidHomClass.isometry_iff_norm
#align add_monoid_hom_class.isometry_iff_norm AddMonoidHomClass.isometry_iff_norm
alias ⟨_, MonoidHomClass.isometry_of_norm⟩ := MonoidHomClass.isometry_iff_norm
#align monoid_hom_class.isometry_of_norm MonoidHomClass.isometry_of_norm
attribute [to_additive] MonoidHomClass.isometry_of_norm
@[to_additive]
theorem tendsto_iff_norm_div_tendsto_zero {f : α → E} {a : Filter α} {b : E} :
Tendsto f a (𝓝 b) ↔ Tendsto (fun e => ‖f e / b‖) a (𝓝 0) := by
simp only [← dist_eq_norm_div, ← tendsto_iff_dist_tendsto_zero]
#align tendsto_iff_norm_tendsto_one tendsto_iff_norm_div_tendsto_zero
#align tendsto_iff_norm_tendsto_zero tendsto_iff_norm_sub_tendsto_zero
@[to_additive]
theorem tendsto_one_iff_norm_tendsto_zero {f : α → E} {a : Filter α} :
Tendsto f a (𝓝 1) ↔ Tendsto (‖f ·‖) a (𝓝 0) :=
tendsto_iff_norm_div_tendsto_zero.trans <| by simp only [div_one]
#align tendsto_one_iff_norm_tendsto_one tendsto_one_iff_norm_tendsto_zero
#align tendsto_zero_iff_norm_tendsto_zero tendsto_zero_iff_norm_tendsto_zero
@[to_additive]
theorem comap_norm_nhds_one : comap norm (𝓝 0) = 𝓝 (1 : E) := by
simpa only [dist_one_right] using nhds_comap_dist (1 : E)
#align comap_norm_nhds_one comap_norm_nhds_one
#align comap_norm_nhds_zero comap_norm_nhds_zero
@[to_additive "Special case of the sandwich theorem: if the norm of `f` is eventually bounded by a
real function `a` which tends to `0`, then `f` tends to `0`. In this pair of lemmas
(`squeeze_zero_norm'` and `squeeze_zero_norm`), following a convention of similar lemmas in
`Topology.MetricSpace.PseudoMetric` and `Topology.Algebra.Order`, the `'` version is phrased using
\"eventually\" and the non-`'` version is phrased absolutely."]
theorem squeeze_one_norm' {f : α → E} {a : α → ℝ} {t₀ : Filter α} (h : ∀ᶠ n in t₀, ‖f n‖ ≤ a n)
(h' : Tendsto a t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 1) :=
tendsto_one_iff_norm_tendsto_zero.2 <|
squeeze_zero' (eventually_of_forall fun _n => norm_nonneg' _) h h'
#align squeeze_one_norm' squeeze_one_norm'
#align squeeze_zero_norm' squeeze_zero_norm'
@[to_additive "Special case of the sandwich theorem: if the norm of `f` is bounded by a real
function `a` which tends to `0`, then `f` tends to `0`."]
theorem squeeze_one_norm {f : α → E} {a : α → ℝ} {t₀ : Filter α} (h : ∀ n, ‖f n‖ ≤ a n) :
Tendsto a t₀ (𝓝 0) → Tendsto f t₀ (𝓝 1) :=
squeeze_one_norm' <| eventually_of_forall h
#align squeeze_one_norm squeeze_one_norm
#align squeeze_zero_norm squeeze_zero_norm
@[to_additive]
theorem tendsto_norm_div_self (x : E) : Tendsto (fun a => ‖a / x‖) (𝓝 x) (𝓝 0) := by
simpa [dist_eq_norm_div] using
tendsto_id.dist (tendsto_const_nhds : Tendsto (fun _a => (x : E)) (𝓝 x) _)
#align tendsto_norm_div_self tendsto_norm_div_self
#align tendsto_norm_sub_self tendsto_norm_sub_self
@[to_additive tendsto_norm]
theorem tendsto_norm' {x : E} : Tendsto (fun a => ‖a‖) (𝓝 x) (𝓝 ‖x‖) := by
simpa using tendsto_id.dist (tendsto_const_nhds : Tendsto (fun _a => (1 : E)) _ _)
#align tendsto_norm' tendsto_norm'
#align tendsto_norm tendsto_norm
@[to_additive]
| Mathlib/Analysis/Normed/Group/Basic.lean | 1,220 | 1,221 | theorem tendsto_norm_one : Tendsto (fun a : E => ‖a‖) (𝓝 1) (𝓝 0) := by |
simpa using tendsto_norm_div_self (1 : E)
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsWithin {a : α} {s : Set α} : ((𝓝 a).bind fun x => 𝓝[s] x) = 𝓝[s] a :=
bind_inf_principal.trans <| congr_arg₂ _ nhds_bind_nhds rfl
#align nhds_bind_nhds_within nhds_bind_nhdsWithin
@[simp]
theorem eventually_nhds_nhdsWithin {a : α} {s : Set α} {p : α → Prop} :
(∀ᶠ y in 𝓝 a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x :=
Filter.ext_iff.1 nhds_bind_nhdsWithin { x | p x }
#align eventually_nhds_nhds_within eventually_nhds_nhdsWithin
theorem eventually_nhdsWithin_iff {a : α} {s : Set α} {p : α → Prop} :
(∀ᶠ x in 𝓝[s] a, p x) ↔ ∀ᶠ x in 𝓝 a, x ∈ s → p x :=
eventually_inf_principal
#align eventually_nhds_within_iff eventually_nhdsWithin_iff
theorem frequently_nhdsWithin_iff {z : α} {s : Set α} {p : α → Prop} :
(∃ᶠ x in 𝓝[s] z, p x) ↔ ∃ᶠ x in 𝓝 z, p x ∧ x ∈ s :=
frequently_inf_principal.trans <| by simp only [and_comm]
#align frequently_nhds_within_iff frequently_nhdsWithin_iff
theorem mem_closure_ne_iff_frequently_within {z : α} {s : Set α} :
z ∈ closure (s \ {z}) ↔ ∃ᶠ x in 𝓝[≠] z, x ∈ s := by
simp [mem_closure_iff_frequently, frequently_nhdsWithin_iff]
#align mem_closure_ne_iff_frequently_within mem_closure_ne_iff_frequently_within
@[simp]
theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop} :
(∀ᶠ y in 𝓝[s] a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x := by
refine ⟨fun h => ?_, fun h => (eventually_nhds_nhdsWithin.2 h).filter_mono inf_le_left⟩
simp only [eventually_nhdsWithin_iff] at h ⊢
exact h.mono fun x hx hxs => (hx hxs).self_of_nhds hxs
#align eventually_nhds_within_nhds_within eventually_nhdsWithin_nhdsWithin
theorem nhdsWithin_eq (a : α) (s : Set α) :
𝓝[s] a = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (t ∩ s) :=
((nhds_basis_opens a).inf_principal s).eq_biInf
#align nhds_within_eq nhdsWithin_eq
theorem nhdsWithin_univ (a : α) : 𝓝[Set.univ] a = 𝓝 a := by
rw [nhdsWithin, principal_univ, inf_top_eq]
#align nhds_within_univ nhdsWithin_univ
theorem nhdsWithin_hasBasis {p : β → Prop} {s : β → Set α} {a : α} (h : (𝓝 a).HasBasis p s)
(t : Set α) : (𝓝[t] a).HasBasis p fun i => s i ∩ t :=
h.inf_principal t
#align nhds_within_has_basis nhdsWithin_hasBasis
theorem nhdsWithin_basis_open (a : α) (t : Set α) :
(𝓝[t] a).HasBasis (fun u => a ∈ u ∧ IsOpen u) fun u => u ∩ t :=
nhdsWithin_hasBasis (nhds_basis_opens a) t
#align nhds_within_basis_open nhdsWithin_basis_open
theorem mem_nhdsWithin {t : Set α} {a : α} {s : Set α} :
t ∈ 𝓝[s] a ↔ ∃ u, IsOpen u ∧ a ∈ u ∧ u ∩ s ⊆ t := by
simpa only [and_assoc, and_left_comm] using (nhdsWithin_basis_open a s).mem_iff
#align mem_nhds_within mem_nhdsWithin
theorem mem_nhdsWithin_iff_exists_mem_nhds_inter {t : Set α} {a : α} {s : Set α} :
t ∈ 𝓝[s] a ↔ ∃ u ∈ 𝓝 a, u ∩ s ⊆ t :=
(nhdsWithin_hasBasis (𝓝 a).basis_sets s).mem_iff
#align mem_nhds_within_iff_exists_mem_nhds_inter mem_nhdsWithin_iff_exists_mem_nhds_inter
theorem diff_mem_nhdsWithin_compl {x : α} {s : Set α} (hs : s ∈ 𝓝 x) (t : Set α) :
s \ t ∈ 𝓝[tᶜ] x :=
diff_mem_inf_principal_compl hs t
#align diff_mem_nhds_within_compl diff_mem_nhdsWithin_compl
theorem diff_mem_nhdsWithin_diff {x : α} {s t : Set α} (hs : s ∈ 𝓝[t] x) (t' : Set α) :
s \ t' ∈ 𝓝[t \ t'] x := by
rw [nhdsWithin, diff_eq, diff_eq, ← inf_principal, ← inf_assoc]
exact inter_mem_inf hs (mem_principal_self _)
#align diff_mem_nhds_within_diff diff_mem_nhdsWithin_diff
theorem nhds_of_nhdsWithin_of_nhds {s t : Set α} {a : α} (h1 : s ∈ 𝓝 a) (h2 : t ∈ 𝓝[s] a) :
t ∈ 𝓝 a := by
rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.mp h2 with ⟨_, Hw, hw⟩
exact (𝓝 a).sets_of_superset ((𝓝 a).inter_sets Hw h1) hw
#align nhds_of_nhds_within_of_nhds nhds_of_nhdsWithin_of_nhds
theorem mem_nhdsWithin_iff_eventually {s t : Set α} {x : α} :
t ∈ 𝓝[s] x ↔ ∀ᶠ y in 𝓝 x, y ∈ s → y ∈ t :=
eventually_inf_principal
#align mem_nhds_within_iff_eventually mem_nhdsWithin_iff_eventually
theorem mem_nhdsWithin_iff_eventuallyEq {s t : Set α} {x : α} :
t ∈ 𝓝[s] x ↔ s =ᶠ[𝓝 x] (s ∩ t : Set α) := by
simp_rw [mem_nhdsWithin_iff_eventually, eventuallyEq_set, mem_inter_iff, iff_self_and]
#align mem_nhds_within_iff_eventually_eq mem_nhdsWithin_iff_eventuallyEq
theorem nhdsWithin_eq_iff_eventuallyEq {s t : Set α} {x : α} : 𝓝[s] x = 𝓝[t] x ↔ s =ᶠ[𝓝 x] t :=
set_eventuallyEq_iff_inf_principal.symm
#align nhds_within_eq_iff_eventually_eq nhdsWithin_eq_iff_eventuallyEq
theorem nhdsWithin_le_iff {s t : Set α} {x : α} : 𝓝[s] x ≤ 𝓝[t] x ↔ t ∈ 𝓝[s] x :=
set_eventuallyLE_iff_inf_principal_le.symm.trans set_eventuallyLE_iff_mem_inf_principal
#align nhds_within_le_iff nhdsWithin_le_iff
-- Porting note: golfed, dropped an unneeded assumption
theorem preimage_nhdsWithin_coinduced' {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t)
(hs : s ∈ @nhds β (.coinduced (fun x : t => π x) inferInstance) (π a)) :
π ⁻¹' s ∈ 𝓝[t] a := by
lift a to t using h
replace hs : (fun x : t => π x) ⁻¹' s ∈ 𝓝 a := preimage_nhds_coinduced hs
rwa [← map_nhds_subtype_val, mem_map]
#align preimage_nhds_within_coinduced' preimage_nhdsWithin_coinduced'ₓ
theorem mem_nhdsWithin_of_mem_nhds {s t : Set α} {a : α} (h : s ∈ 𝓝 a) : s ∈ 𝓝[t] a :=
mem_inf_of_left h
#align mem_nhds_within_of_mem_nhds mem_nhdsWithin_of_mem_nhds
theorem self_mem_nhdsWithin {a : α} {s : Set α} : s ∈ 𝓝[s] a :=
mem_inf_of_right (mem_principal_self s)
#align self_mem_nhds_within self_mem_nhdsWithin
theorem eventually_mem_nhdsWithin {a : α} {s : Set α} : ∀ᶠ x in 𝓝[s] a, x ∈ s :=
self_mem_nhdsWithin
#align eventually_mem_nhds_within eventually_mem_nhdsWithin
theorem inter_mem_nhdsWithin (s : Set α) {t : Set α} {a : α} (h : t ∈ 𝓝 a) : s ∩ t ∈ 𝓝[s] a :=
inter_mem self_mem_nhdsWithin (mem_inf_of_left h)
#align inter_mem_nhds_within inter_mem_nhdsWithin
theorem nhdsWithin_mono (a : α) {s t : Set α} (h : s ⊆ t) : 𝓝[s] a ≤ 𝓝[t] a :=
inf_le_inf_left _ (principal_mono.mpr h)
#align nhds_within_mono nhdsWithin_mono
theorem pure_le_nhdsWithin {a : α} {s : Set α} (ha : a ∈ s) : pure a ≤ 𝓝[s] a :=
le_inf (pure_le_nhds a) (le_principal_iff.2 ha)
#align pure_le_nhds_within pure_le_nhdsWithin
theorem mem_of_mem_nhdsWithin {a : α} {s t : Set α} (ha : a ∈ s) (ht : t ∈ 𝓝[s] a) : a ∈ t :=
pure_le_nhdsWithin ha ht
#align mem_of_mem_nhds_within mem_of_mem_nhdsWithin
theorem Filter.Eventually.self_of_nhdsWithin {p : α → Prop} {s : Set α} {x : α}
(h : ∀ᶠ y in 𝓝[s] x, p y) (hx : x ∈ s) : p x :=
mem_of_mem_nhdsWithin hx h
#align filter.eventually.self_of_nhds_within Filter.Eventually.self_of_nhdsWithin
theorem tendsto_const_nhdsWithin {l : Filter β} {s : Set α} {a : α} (ha : a ∈ s) :
Tendsto (fun _ : β => a) l (𝓝[s] a) :=
tendsto_const_pure.mono_right <| pure_le_nhdsWithin ha
#align tendsto_const_nhds_within tendsto_const_nhdsWithin
theorem nhdsWithin_restrict'' {a : α} (s : Set α) {t : Set α} (h : t ∈ 𝓝[s] a) :
𝓝[s] a = 𝓝[s ∩ t] a :=
le_antisymm (le_inf inf_le_left (le_principal_iff.mpr (inter_mem self_mem_nhdsWithin h)))
(inf_le_inf_left _ (principal_mono.mpr Set.inter_subset_left))
#align nhds_within_restrict'' nhdsWithin_restrict''
theorem nhdsWithin_restrict' {a : α} (s : Set α) {t : Set α} (h : t ∈ 𝓝 a) : 𝓝[s] a = 𝓝[s ∩ t] a :=
nhdsWithin_restrict'' s <| mem_inf_of_left h
#align nhds_within_restrict' nhdsWithin_restrict'
theorem nhdsWithin_restrict {a : α} (s : Set α) {t : Set α} (h₀ : a ∈ t) (h₁ : IsOpen t) :
𝓝[s] a = 𝓝[s ∩ t] a :=
nhdsWithin_restrict' s (IsOpen.mem_nhds h₁ h₀)
#align nhds_within_restrict nhdsWithin_restrict
theorem nhdsWithin_le_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[t] a ≤ 𝓝[s] a :=
nhdsWithin_le_iff.mpr h
#align nhds_within_le_of_mem nhdsWithin_le_of_mem
theorem nhdsWithin_le_nhds {a : α} {s : Set α} : 𝓝[s] a ≤ 𝓝 a := by
rw [← nhdsWithin_univ]
apply nhdsWithin_le_of_mem
exact univ_mem
#align nhds_within_le_nhds nhdsWithin_le_nhds
theorem nhdsWithin_eq_nhdsWithin' {a : α} {s t u : Set α} (hs : s ∈ 𝓝 a) (h₂ : t ∩ s = u ∩ s) :
𝓝[t] a = 𝓝[u] a := by rw [nhdsWithin_restrict' t hs, nhdsWithin_restrict' u hs, h₂]
#align nhds_within_eq_nhds_within' nhdsWithin_eq_nhdsWithin'
theorem nhdsWithin_eq_nhdsWithin {a : α} {s t u : Set α} (h₀ : a ∈ s) (h₁ : IsOpen s)
(h₂ : t ∩ s = u ∩ s) : 𝓝[t] a = 𝓝[u] a := by
rw [nhdsWithin_restrict t h₀ h₁, nhdsWithin_restrict u h₀ h₁, h₂]
#align nhds_within_eq_nhds_within nhdsWithin_eq_nhdsWithin
@[simp] theorem nhdsWithin_eq_nhds {a : α} {s : Set α} : 𝓝[s] a = 𝓝 a ↔ s ∈ 𝓝 a :=
inf_eq_left.trans le_principal_iff
#align nhds_within_eq_nhds nhdsWithin_eq_nhds
theorem IsOpen.nhdsWithin_eq {a : α} {s : Set α} (h : IsOpen s) (ha : a ∈ s) : 𝓝[s] a = 𝓝 a :=
nhdsWithin_eq_nhds.2 <| h.mem_nhds ha
#align is_open.nhds_within_eq IsOpen.nhdsWithin_eq
theorem preimage_nhds_within_coinduced {π : α → β} {s : Set β} {t : Set α} {a : α} (h : a ∈ t)
(ht : IsOpen t)
(hs : s ∈ @nhds β (.coinduced (fun x : t => π x) inferInstance) (π a)) :
π ⁻¹' s ∈ 𝓝 a := by
rw [← ht.nhdsWithin_eq h]
exact preimage_nhdsWithin_coinduced' h hs
#align preimage_nhds_within_coinduced preimage_nhds_within_coinduced
@[simp]
theorem nhdsWithin_empty (a : α) : 𝓝[∅] a = ⊥ := by rw [nhdsWithin, principal_empty, inf_bot_eq]
#align nhds_within_empty nhdsWithin_empty
theorem nhdsWithin_union (a : α) (s t : Set α) : 𝓝[s ∪ t] a = 𝓝[s] a ⊔ 𝓝[t] a := by
delta nhdsWithin
rw [← inf_sup_left, sup_principal]
#align nhds_within_union nhdsWithin_union
theorem nhdsWithin_biUnion {ι} {I : Set ι} (hI : I.Finite) (s : ι → Set α) (a : α) :
𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a :=
Set.Finite.induction_on hI (by simp) fun _ _ hT ↦ by
simp only [hT, nhdsWithin_union, iSup_insert, biUnion_insert]
#align nhds_within_bUnion nhdsWithin_biUnion
theorem nhdsWithin_sUnion {S : Set (Set α)} (hS : S.Finite) (a : α) :
𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a := by
rw [sUnion_eq_biUnion, nhdsWithin_biUnion hS]
#align nhds_within_sUnion nhdsWithin_sUnion
theorem nhdsWithin_iUnion {ι} [Finite ι] (s : ι → Set α) (a : α) :
𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a := by
rw [← sUnion_range, nhdsWithin_sUnion (finite_range s), iSup_range]
#align nhds_within_Union nhdsWithin_iUnion
theorem nhdsWithin_inter (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a := by
delta nhdsWithin
rw [inf_left_comm, inf_assoc, inf_principal, ← inf_assoc, inf_idem]
#align nhds_within_inter nhdsWithin_inter
theorem nhdsWithin_inter' (a : α) (s t : Set α) : 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓟 t := by
delta nhdsWithin
rw [← inf_principal, inf_assoc]
#align nhds_within_inter' nhdsWithin_inter'
theorem nhdsWithin_inter_of_mem {a : α} {s t : Set α} (h : s ∈ 𝓝[t] a) : 𝓝[s ∩ t] a = 𝓝[t] a := by
rw [nhdsWithin_inter, inf_eq_right]
exact nhdsWithin_le_of_mem h
#align nhds_within_inter_of_mem nhdsWithin_inter_of_mem
theorem nhdsWithin_inter_of_mem' {a : α} {s t : Set α} (h : t ∈ 𝓝[s] a) : 𝓝[s ∩ t] a = 𝓝[s] a := by
rw [inter_comm, nhdsWithin_inter_of_mem h]
#align nhds_within_inter_of_mem' nhdsWithin_inter_of_mem'
@[simp]
theorem nhdsWithin_singleton (a : α) : 𝓝[{a}] a = pure a := by
rw [nhdsWithin, principal_singleton, inf_eq_right.2 (pure_le_nhds a)]
#align nhds_within_singleton nhdsWithin_singleton
@[simp]
theorem nhdsWithin_insert (a : α) (s : Set α) : 𝓝[insert a s] a = pure a ⊔ 𝓝[s] a := by
rw [← singleton_union, nhdsWithin_union, nhdsWithin_singleton]
#align nhds_within_insert nhdsWithin_insert
theorem mem_nhdsWithin_insert {a : α} {s t : Set α} : t ∈ 𝓝[insert a s] a ↔ a ∈ t ∧ t ∈ 𝓝[s] a := by
simp
#align mem_nhds_within_insert mem_nhdsWithin_insert
theorem insert_mem_nhdsWithin_insert {a : α} {s t : Set α} (h : t ∈ 𝓝[s] a) :
insert a t ∈ 𝓝[insert a s] a := by simp [mem_of_superset h]
#align insert_mem_nhds_within_insert insert_mem_nhdsWithin_insert
theorem insert_mem_nhds_iff {a : α} {s : Set α} : insert a s ∈ 𝓝 a ↔ s ∈ 𝓝[≠] a := by
simp only [nhdsWithin, mem_inf_principal, mem_compl_iff, mem_singleton_iff, or_iff_not_imp_left,
insert_def]
#align insert_mem_nhds_iff insert_mem_nhds_iff
@[simp]
theorem nhdsWithin_compl_singleton_sup_pure (a : α) : 𝓝[≠] a ⊔ pure a = 𝓝 a := by
rw [← nhdsWithin_singleton, ← nhdsWithin_union, compl_union_self, nhdsWithin_univ]
#align nhds_within_compl_singleton_sup_pure nhdsWithin_compl_singleton_sup_pure
theorem nhdsWithin_prod {α : Type*} [TopologicalSpace α] {β : Type*} [TopologicalSpace β]
{s u : Set α} {t v : Set β} {a : α} {b : β} (hu : u ∈ 𝓝[s] a) (hv : v ∈ 𝓝[t] b) :
u ×ˢ v ∈ 𝓝[s ×ˢ t] (a, b) := by
rw [nhdsWithin_prod_eq]
exact prod_mem_prod hu hv
#align nhds_within_prod nhdsWithin_prod
theorem nhdsWithin_pi_eq' {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
(hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
𝓝[pi I s] x = ⨅ i, comap (fun x => x i) (𝓝 (x i) ⊓ ⨅ (_ : i ∈ I), 𝓟 (s i)) := by
simp only [nhdsWithin, nhds_pi, Filter.pi, comap_inf, comap_iInf, pi_def, comap_principal, ←
iInf_principal_finite hI, ← iInf_inf_eq]
#align nhds_within_pi_eq' nhdsWithin_pi_eq'
theorem nhdsWithin_pi_eq {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
(hI : I.Finite) (s : ∀ i, Set (α i)) (x : ∀ i, α i) :
𝓝[pi I s] x =
(⨅ i ∈ I, comap (fun x => x i) (𝓝[s i] x i)) ⊓
⨅ (i) (_ : i ∉ I), comap (fun x => x i) (𝓝 (x i)) := by
simp only [nhdsWithin, nhds_pi, Filter.pi, pi_def, ← iInf_principal_finite hI, comap_inf,
comap_principal, eval]
rw [iInf_split _ fun i => i ∈ I, inf_right_comm]
simp only [iInf_inf_eq]
#align nhds_within_pi_eq nhdsWithin_pi_eq
theorem nhdsWithin_pi_univ_eq {ι : Type*} {α : ι → Type*} [Finite ι] [∀ i, TopologicalSpace (α i)]
(s : ∀ i, Set (α i)) (x : ∀ i, α i) :
𝓝[pi univ s] x = ⨅ i, comap (fun x => x i) (𝓝[s i] x i) := by
simpa [nhdsWithin] using nhdsWithin_pi_eq finite_univ s x
#align nhds_within_pi_univ_eq nhdsWithin_pi_univ_eq
theorem nhdsWithin_pi_eq_bot {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
{s : ∀ i, Set (α i)} {x : ∀ i, α i} : 𝓝[pi I s] x = ⊥ ↔ ∃ i ∈ I, 𝓝[s i] x i = ⊥ := by
simp only [nhdsWithin, nhds_pi, pi_inf_principal_pi_eq_bot]
#align nhds_within_pi_eq_bot nhdsWithin_pi_eq_bot
theorem nhdsWithin_pi_neBot {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
{s : ∀ i, Set (α i)} {x : ∀ i, α i} : (𝓝[pi I s] x).NeBot ↔ ∀ i ∈ I, (𝓝[s i] x i).NeBot := by
simp [neBot_iff, nhdsWithin_pi_eq_bot]
#align nhds_within_pi_ne_bot nhdsWithin_pi_neBot
theorem Filter.Tendsto.piecewise_nhdsWithin {f g : α → β} {t : Set α} [∀ x, Decidable (x ∈ t)]
{a : α} {s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ t] a) l)
(h₁ : Tendsto g (𝓝[s ∩ tᶜ] a) l) : Tendsto (piecewise t f g) (𝓝[s] a) l := by
apply Tendsto.piecewise <;> rwa [← nhdsWithin_inter']
#align filter.tendsto.piecewise_nhds_within Filter.Tendsto.piecewise_nhdsWithin
theorem Filter.Tendsto.if_nhdsWithin {f g : α → β} {p : α → Prop} [DecidablePred p] {a : α}
{s : Set α} {l : Filter β} (h₀ : Tendsto f (𝓝[s ∩ { x | p x }] a) l)
(h₁ : Tendsto g (𝓝[s ∩ { x | ¬p x }] a) l) :
Tendsto (fun x => if p x then f x else g x) (𝓝[s] a) l :=
h₀.piecewise_nhdsWithin h₁
#align filter.tendsto.if_nhds_within Filter.Tendsto.if_nhdsWithin
theorem map_nhdsWithin (f : α → β) (a : α) (s : Set α) :
map f (𝓝[s] a) = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (f '' (t ∩ s)) :=
((nhdsWithin_basis_open a s).map f).eq_biInf
#align map_nhds_within map_nhdsWithin
theorem tendsto_nhdsWithin_mono_left {f : α → β} {a : α} {s t : Set α} {l : Filter β} (hst : s ⊆ t)
(h : Tendsto f (𝓝[t] a) l) : Tendsto f (𝓝[s] a) l :=
h.mono_left <| nhdsWithin_mono a hst
#align tendsto_nhds_within_mono_left tendsto_nhdsWithin_mono_left
theorem tendsto_nhdsWithin_mono_right {f : β → α} {l : Filter β} {a : α} {s t : Set α} (hst : s ⊆ t)
(h : Tendsto f l (𝓝[s] a)) : Tendsto f l (𝓝[t] a) :=
h.mono_right (nhdsWithin_mono a hst)
#align tendsto_nhds_within_mono_right tendsto_nhdsWithin_mono_right
theorem tendsto_nhdsWithin_of_tendsto_nhds {f : α → β} {a : α} {s : Set α} {l : Filter β}
(h : Tendsto f (𝓝 a) l) : Tendsto f (𝓝[s] a) l :=
h.mono_left inf_le_left
#align tendsto_nhds_within_of_tendsto_nhds tendsto_nhdsWithin_of_tendsto_nhds
theorem eventually_mem_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β}
(h : Tendsto f l (𝓝[s] a)) : ∀ᶠ i in l, f i ∈ s := by
simp_rw [nhdsWithin_eq, tendsto_iInf, mem_setOf_eq, tendsto_principal, mem_inter_iff,
eventually_and] at h
exact (h univ ⟨mem_univ a, isOpen_univ⟩).2
#align eventually_mem_of_tendsto_nhds_within eventually_mem_of_tendsto_nhdsWithin
theorem tendsto_nhds_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β}
(h : Tendsto f l (𝓝[s] a)) : Tendsto f l (𝓝 a) :=
h.mono_right nhdsWithin_le_nhds
#align tendsto_nhds_of_tendsto_nhds_within tendsto_nhds_of_tendsto_nhdsWithin
theorem nhdsWithin_neBot_of_mem {s : Set α} {x : α} (hx : x ∈ s) : NeBot (𝓝[s] x) :=
mem_closure_iff_nhdsWithin_neBot.1 <| subset_closure hx
#align nhds_within_ne_bot_of_mem nhdsWithin_neBot_of_mem
theorem IsClosed.mem_of_nhdsWithin_neBot {s : Set α} (hs : IsClosed s) {x : α}
(hx : NeBot <| 𝓝[s] x) : x ∈ s :=
hs.closure_eq ▸ mem_closure_iff_nhdsWithin_neBot.2 hx
#align is_closed.mem_of_nhds_within_ne_bot IsClosed.mem_of_nhdsWithin_neBot
theorem DenseRange.nhdsWithin_neBot {ι : Type*} {f : ι → α} (h : DenseRange f) (x : α) :
NeBot (𝓝[range f] x) :=
mem_closure_iff_clusterPt.1 (h x)
#align dense_range.nhds_within_ne_bot DenseRange.nhdsWithin_neBot
theorem mem_closure_pi {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
{s : ∀ i, Set (α i)} {x : ∀ i, α i} : x ∈ closure (pi I s) ↔ ∀ i ∈ I, x i ∈ closure (s i) := by
simp only [mem_closure_iff_nhdsWithin_neBot, nhdsWithin_pi_neBot]
#align mem_closure_pi mem_closure_pi
theorem closure_pi_set {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] (I : Set ι)
(s : ∀ i, Set (α i)) : closure (pi I s) = pi I fun i => closure (s i) :=
Set.ext fun _ => mem_closure_pi
#align closure_pi_set closure_pi_set
theorem dense_pi {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {s : ∀ i, Set (α i)}
(I : Set ι) (hs : ∀ i ∈ I, Dense (s i)) : Dense (pi I s) := by
simp only [dense_iff_closure_eq, closure_pi_set, pi_congr rfl fun i hi => (hs i hi).closure_eq,
pi_univ]
#align dense_pi dense_pi
theorem eventuallyEq_nhdsWithin_iff {f g : α → β} {s : Set α} {a : α} :
f =ᶠ[𝓝[s] a] g ↔ ∀ᶠ x in 𝓝 a, x ∈ s → f x = g x :=
mem_inf_principal
#align eventually_eq_nhds_within_iff eventuallyEq_nhdsWithin_iff
theorem eventuallyEq_nhdsWithin_of_eqOn {f g : α → β} {s : Set α} {a : α} (h : EqOn f g s) :
f =ᶠ[𝓝[s] a] g :=
mem_inf_of_right h
#align eventually_eq_nhds_within_of_eq_on eventuallyEq_nhdsWithin_of_eqOn
theorem Set.EqOn.eventuallyEq_nhdsWithin {f g : α → β} {s : Set α} {a : α} (h : EqOn f g s) :
f =ᶠ[𝓝[s] a] g :=
eventuallyEq_nhdsWithin_of_eqOn h
#align set.eq_on.eventually_eq_nhds_within Set.EqOn.eventuallyEq_nhdsWithin
theorem tendsto_nhdsWithin_congr {f g : α → β} {s : Set α} {a : α} {l : Filter β}
(hfg : ∀ x ∈ s, f x = g x) (hf : Tendsto f (𝓝[s] a) l) : Tendsto g (𝓝[s] a) l :=
(tendsto_congr' <| eventuallyEq_nhdsWithin_of_eqOn hfg).1 hf
#align tendsto_nhds_within_congr tendsto_nhdsWithin_congr
theorem eventually_nhdsWithin_of_forall {s : Set α} {a : α} {p : α → Prop} (h : ∀ x ∈ s, p x) :
∀ᶠ x in 𝓝[s] a, p x :=
mem_inf_of_right h
#align eventually_nhds_within_of_forall eventually_nhdsWithin_of_forall
theorem tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within {a : α} {l : Filter β} {s : Set α}
(f : β → α) (h1 : Tendsto f l (𝓝 a)) (h2 : ∀ᶠ x in l, f x ∈ s) : Tendsto f l (𝓝[s] a) :=
tendsto_inf.2 ⟨h1, tendsto_principal.2 h2⟩
#align tendsto_nhds_within_of_tendsto_nhds_of_eventually_within tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
theorem tendsto_nhdsWithin_iff {a : α} {l : Filter β} {s : Set α} {f : β → α} :
Tendsto f l (𝓝[s] a) ↔ Tendsto f l (𝓝 a) ∧ ∀ᶠ n in l, f n ∈ s :=
⟨fun h => ⟨tendsto_nhds_of_tendsto_nhdsWithin h, eventually_mem_of_tendsto_nhdsWithin h⟩, fun h =>
tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ h.1 h.2⟩
#align tendsto_nhds_within_iff tendsto_nhdsWithin_iff
@[simp]
theorem tendsto_nhdsWithin_range {a : α} {l : Filter β} {f : β → α} :
Tendsto f l (𝓝[range f] a) ↔ Tendsto f l (𝓝 a) :=
⟨fun h => h.mono_right inf_le_left, fun h =>
tendsto_inf.2 ⟨h, tendsto_principal.2 <| eventually_of_forall mem_range_self⟩⟩
#align tendsto_nhds_within_range tendsto_nhdsWithin_range
theorem Filter.EventuallyEq.eq_of_nhdsWithin {s : Set α} {f g : α → β} {a : α} (h : f =ᶠ[𝓝[s] a] g)
(hmem : a ∈ s) : f a = g a :=
h.self_of_nhdsWithin hmem
#align filter.eventually_eq.eq_of_nhds_within Filter.EventuallyEq.eq_of_nhdsWithin
theorem eventually_nhdsWithin_of_eventually_nhds {α : Type*} [TopologicalSpace α] {s : Set α}
{a : α} {p : α → Prop} (h : ∀ᶠ x in 𝓝 a, p x) : ∀ᶠ x in 𝓝[s] a, p x :=
mem_nhdsWithin_of_mem_nhds h
#align eventually_nhds_within_of_eventually_nhds eventually_nhdsWithin_of_eventually_nhds
theorem mem_nhdsWithin_subtype {s : Set α} {a : { x // x ∈ s }} {t u : Set { x // x ∈ s }} :
t ∈ 𝓝[u] a ↔ t ∈ comap ((↑) : s → α) (𝓝[(↑) '' u] a) := by
rw [nhdsWithin, nhds_subtype, principal_subtype, ← comap_inf, ← nhdsWithin]
#align mem_nhds_within_subtype mem_nhdsWithin_subtype
theorem nhdsWithin_subtype (s : Set α) (a : { x // x ∈ s }) (t : Set { x // x ∈ s }) :
𝓝[t] a = comap ((↑) : s → α) (𝓝[(↑) '' t] a) :=
Filter.ext fun _ => mem_nhdsWithin_subtype
#align nhds_within_subtype nhdsWithin_subtype
theorem nhdsWithin_eq_map_subtype_coe {s : Set α} {a : α} (h : a ∈ s) :
𝓝[s] a = map ((↑) : s → α) (𝓝 ⟨a, h⟩) :=
(map_nhds_subtype_val ⟨a, h⟩).symm
#align nhds_within_eq_map_subtype_coe nhdsWithin_eq_map_subtype_coe
theorem mem_nhds_subtype_iff_nhdsWithin {s : Set α} {a : s} {t : Set s} :
t ∈ 𝓝 a ↔ (↑) '' t ∈ 𝓝[s] (a : α) := by
rw [← map_nhds_subtype_val, image_mem_map_iff Subtype.val_injective]
#align mem_nhds_subtype_iff_nhds_within mem_nhds_subtype_iff_nhdsWithin
theorem preimage_coe_mem_nhds_subtype {s t : Set α} {a : s} : (↑) ⁻¹' t ∈ 𝓝 a ↔ t ∈ 𝓝[s] ↑a := by
rw [← map_nhds_subtype_val, mem_map]
#align preimage_coe_mem_nhds_subtype preimage_coe_mem_nhds_subtype
theorem eventually_nhds_subtype_iff (s : Set α) (a : s) (P : α → Prop) :
(∀ᶠ x : s in 𝓝 a, P x) ↔ ∀ᶠ x in 𝓝[s] a, P x :=
preimage_coe_mem_nhds_subtype
theorem frequently_nhds_subtype_iff (s : Set α) (a : s) (P : α → Prop) :
(∃ᶠ x : s in 𝓝 a, P x) ↔ ∃ᶠ x in 𝓝[s] a, P x :=
eventually_nhds_subtype_iff s a (¬ P ·) |>.not
theorem tendsto_nhdsWithin_iff_subtype {s : Set α} {a : α} (h : a ∈ s) (f : α → β) (l : Filter β) :
Tendsto f (𝓝[s] a) l ↔ Tendsto (s.restrict f) (𝓝 ⟨a, h⟩) l := by
rw [nhdsWithin_eq_map_subtype_coe h, tendsto_map'_iff]; rfl
#align tendsto_nhds_within_iff_subtype tendsto_nhdsWithin_iff_subtype
variable [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
theorem ContinuousWithinAt.tendsto {f : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) :
Tendsto f (𝓝[s] x) (𝓝 (f x)) :=
h
#align continuous_within_at.tendsto ContinuousWithinAt.tendsto
theorem ContinuousOn.continuousWithinAt {f : α → β} {s : Set α} {x : α} (hf : ContinuousOn f s)
(hx : x ∈ s) : ContinuousWithinAt f s x :=
hf x hx
#align continuous_on.continuous_within_at ContinuousOn.continuousWithinAt
theorem continuousWithinAt_univ (f : α → β) (x : α) :
ContinuousWithinAt f Set.univ x ↔ ContinuousAt f x := by
rw [ContinuousAt, ContinuousWithinAt, nhdsWithin_univ]
#align continuous_within_at_univ continuousWithinAt_univ
theorem continuous_iff_continuousOn_univ {f : α → β} : Continuous f ↔ ContinuousOn f univ := by
simp [continuous_iff_continuousAt, ContinuousOn, ContinuousAt, ContinuousWithinAt,
nhdsWithin_univ]
#align continuous_iff_continuous_on_univ continuous_iff_continuousOn_univ
theorem continuousWithinAt_iff_continuousAt_restrict (f : α → β) {x : α} {s : Set α} (h : x ∈ s) :
ContinuousWithinAt f s x ↔ ContinuousAt (s.restrict f) ⟨x, h⟩ :=
tendsto_nhdsWithin_iff_subtype h f _
#align continuous_within_at_iff_continuous_at_restrict continuousWithinAt_iff_continuousAt_restrict
theorem ContinuousWithinAt.tendsto_nhdsWithin {f : α → β} {x : α} {s : Set α} {t : Set β}
(h : ContinuousWithinAt f s x) (ht : MapsTo f s t) : Tendsto f (𝓝[s] x) (𝓝[t] f x) :=
tendsto_inf.2 ⟨h, tendsto_principal.2 <| mem_inf_of_right <| mem_principal.2 <| ht⟩
#align continuous_within_at.tendsto_nhds_within ContinuousWithinAt.tendsto_nhdsWithin
theorem ContinuousWithinAt.tendsto_nhdsWithin_image {f : α → β} {x : α} {s : Set α}
(h : ContinuousWithinAt f s x) : Tendsto f (𝓝[s] x) (𝓝[f '' s] f x) :=
h.tendsto_nhdsWithin (mapsTo_image _ _)
#align continuous_within_at.tendsto_nhds_within_image ContinuousWithinAt.tendsto_nhdsWithin_image
theorem ContinuousWithinAt.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t : Set β} {x : α} {y : β}
(hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g t y) :
ContinuousWithinAt (Prod.map f g) (s ×ˢ t) (x, y) := by
unfold ContinuousWithinAt at *
rw [nhdsWithin_prod_eq, Prod.map, nhds_prod_eq]
exact hf.prod_map hg
#align continuous_within_at.prod_map ContinuousWithinAt.prod_map
theorem continuousWithinAt_prod_of_discrete_left [DiscreteTopology α]
{f : α × β → γ} {s : Set (α × β)} {x : α × β} :
ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ⟨x.1, ·⟩) {b | (x.1, b) ∈ s} x.2 := by
rw [← x.eta]; simp_rw [ContinuousWithinAt, nhdsWithin, nhds_prod_eq, nhds_discrete, pure_prod,
← map_inf_principal_preimage]; rfl
theorem continuousWithinAt_prod_of_discrete_right [DiscreteTopology β]
{f : α × β → γ} {s : Set (α × β)} {x : α × β} :
ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ⟨·, x.2⟩) {a | (a, x.2) ∈ s} x.1 := by
rw [← x.eta]; simp_rw [ContinuousWithinAt, nhdsWithin, nhds_prod_eq, nhds_discrete, prod_pure,
← map_inf_principal_preimage]; rfl
theorem continuousAt_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} {x : α × β} :
ContinuousAt f x ↔ ContinuousAt (f ⟨x.1, ·⟩) x.2 := by
simp_rw [← continuousWithinAt_univ]; exact continuousWithinAt_prod_of_discrete_left
theorem continuousAt_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} {x : α × β} :
ContinuousAt f x ↔ ContinuousAt (f ⟨·, x.2⟩) x.1 := by
simp_rw [← continuousWithinAt_univ]; exact continuousWithinAt_prod_of_discrete_right
theorem continuousOn_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} {s : Set (α × β)} :
ContinuousOn f s ↔ ∀ a, ContinuousOn (f ⟨a, ·⟩) {b | (a, b) ∈ s} := by
simp_rw [ContinuousOn, Prod.forall, continuousWithinAt_prod_of_discrete_left]; rfl
theorem continuousOn_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} {s : Set (α × β)} :
ContinuousOn f s ↔ ∀ b, ContinuousOn (f ⟨·, b⟩) {a | (a, b) ∈ s} := by
simp_rw [ContinuousOn, Prod.forall, continuousWithinAt_prod_of_discrete_right]; apply forall_swap
theorem continuous_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} :
Continuous f ↔ ∀ a, Continuous (f ⟨a, ·⟩) := by
simp_rw [continuous_iff_continuousOn_univ]; exact continuousOn_prod_of_discrete_left
theorem continuous_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} :
Continuous f ↔ ∀ b, Continuous (f ⟨·, b⟩) := by
simp_rw [continuous_iff_continuousOn_univ]; exact continuousOn_prod_of_discrete_right
theorem isOpenMap_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} :
IsOpenMap f ↔ ∀ a, IsOpenMap (f ⟨a, ·⟩) := by
simp_rw [isOpenMap_iff_nhds_le, Prod.forall, nhds_prod_eq, nhds_discrete, pure_prod, map_map]
rfl
theorem isOpenMap_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} :
IsOpenMap f ↔ ∀ b, IsOpenMap (f ⟨·, b⟩) := by
simp_rw [isOpenMap_iff_nhds_le, Prod.forall, forall_swap (α := α) (β := β), nhds_prod_eq,
nhds_discrete, prod_pure, map_map]; rfl
theorem continuousWithinAt_pi {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
{f : α → ∀ i, π i} {s : Set α} {x : α} :
ContinuousWithinAt f s x ↔ ∀ i, ContinuousWithinAt (fun y => f y i) s x :=
tendsto_pi_nhds
#align continuous_within_at_pi continuousWithinAt_pi
theorem continuousOn_pi {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
{f : α → ∀ i, π i} {s : Set α} : ContinuousOn f s ↔ ∀ i, ContinuousOn (fun y => f y i) s :=
⟨fun h i x hx => tendsto_pi_nhds.1 (h x hx) i, fun h x hx => tendsto_pi_nhds.2 fun i => h i x hx⟩
#align continuous_on_pi continuousOn_pi
@[fun_prop]
theorem continuousOn_pi' {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
{f : α → ∀ i, π i} {s : Set α} (hf : ∀ i, ContinuousOn (fun y => f y i) s) :
ContinuousOn f s :=
continuousOn_pi.2 hf
theorem ContinuousWithinAt.fin_insertNth {n} {π : Fin (n + 1) → Type*}
[∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {a : α} {s : Set α}
(hf : ContinuousWithinAt f s a) {g : α → ∀ j : Fin n, π (i.succAbove j)}
(hg : ContinuousWithinAt g s a) : ContinuousWithinAt (fun a => i.insertNth (f a) (g a)) s a :=
hf.tendsto.fin_insertNth i hg
#align continuous_within_at.fin_insert_nth ContinuousWithinAt.fin_insertNth
nonrec theorem ContinuousOn.fin_insertNth {n} {π : Fin (n + 1) → Type*}
[∀ i, TopologicalSpace (π i)] (i : Fin (n + 1)) {f : α → π i} {s : Set α}
(hf : ContinuousOn f s) {g : α → ∀ j : Fin n, π (i.succAbove j)} (hg : ContinuousOn g s) :
ContinuousOn (fun a => i.insertNth (f a) (g a)) s := fun a ha =>
(hf a ha).fin_insertNth i (hg a ha)
#align continuous_on.fin_insert_nth ContinuousOn.fin_insertNth
theorem continuousOn_iff {f : α → β} {s : Set α} :
ContinuousOn f s ↔
∀ x ∈ s, ∀ t : Set β, IsOpen t → f x ∈ t → ∃ u, IsOpen u ∧ x ∈ u ∧ u ∩ s ⊆ f ⁻¹' t := by
simp only [ContinuousOn, ContinuousWithinAt, tendsto_nhds, mem_nhdsWithin]
#align continuous_on_iff continuousOn_iff
theorem continuousOn_iff_continuous_restrict {f : α → β} {s : Set α} :
ContinuousOn f s ↔ Continuous (s.restrict f) := by
rw [ContinuousOn, continuous_iff_continuousAt]; constructor
· rintro h ⟨x, xs⟩
exact (continuousWithinAt_iff_continuousAt_restrict f xs).mp (h x xs)
intro h x xs
exact (continuousWithinAt_iff_continuousAt_restrict f xs).mpr (h ⟨x, xs⟩)
#align continuous_on_iff_continuous_restrict continuousOn_iff_continuous_restrict
-- Porting note: 2 new lemmas
alias ⟨ContinuousOn.restrict, _⟩ := continuousOn_iff_continuous_restrict
theorem ContinuousOn.restrict_mapsTo {f : α → β} {s : Set α} {t : Set β} (hf : ContinuousOn f s)
(ht : MapsTo f s t) : Continuous (ht.restrict f s t) :=
hf.restrict.codRestrict _
theorem continuousOn_iff' {f : α → β} {s : Set α} :
ContinuousOn f s ↔ ∀ t : Set β, IsOpen t → ∃ u, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s := by
have : ∀ t, IsOpen (s.restrict f ⁻¹' t) ↔ ∃ u : Set α, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s := by
intro t
rw [isOpen_induced_iff, Set.restrict_eq, Set.preimage_comp]
simp only [Subtype.preimage_coe_eq_preimage_coe_iff]
constructor <;>
· rintro ⟨u, ou, useq⟩
exact ⟨u, ou, by simpa only [Set.inter_comm, eq_comm] using useq⟩
rw [continuousOn_iff_continuous_restrict, continuous_def]; simp only [this]
#align continuous_on_iff' continuousOn_iff'
theorem ContinuousOn.mono_dom {α β : Type*} {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β}
(h₁ : t₂ ≤ t₁) {s : Set α} {f : α → β} (h₂ : @ContinuousOn α β t₁ t₃ f s) :
@ContinuousOn α β t₂ t₃ f s := fun x hx _u hu =>
map_mono (inf_le_inf_right _ <| nhds_mono h₁) (h₂ x hx hu)
#align continuous_on.mono_dom ContinuousOn.mono_dom
theorem ContinuousOn.mono_rng {α β : Type*} {t₁ : TopologicalSpace α} {t₂ t₃ : TopologicalSpace β}
(h₁ : t₂ ≤ t₃) {s : Set α} {f : α → β} (h₂ : @ContinuousOn α β t₁ t₂ f s) :
@ContinuousOn α β t₁ t₃ f s := fun x hx _u hu =>
h₂ x hx <| nhds_mono h₁ hu
#align continuous_on.mono_rng ContinuousOn.mono_rng
theorem continuousOn_iff_isClosed {f : α → β} {s : Set α} :
ContinuousOn f s ↔ ∀ t : Set β, IsClosed t → ∃ u, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s := by
have : ∀ t, IsClosed (s.restrict f ⁻¹' t) ↔ ∃ u : Set α, IsClosed u ∧ f ⁻¹' t ∩ s = u ∩ s := by
intro t
rw [isClosed_induced_iff, Set.restrict_eq, Set.preimage_comp]
simp only [Subtype.preimage_coe_eq_preimage_coe_iff, eq_comm, Set.inter_comm s]
rw [continuousOn_iff_continuous_restrict, continuous_iff_isClosed]; simp only [this]
#align continuous_on_iff_is_closed continuousOn_iff_isClosed
theorem ContinuousOn.prod_map {f : α → γ} {g : β → δ} {s : Set α} {t : Set β}
(hf : ContinuousOn f s) (hg : ContinuousOn g t) : ContinuousOn (Prod.map f g) (s ×ˢ t) :=
fun ⟨x, y⟩ ⟨hx, hy⟩ => ContinuousWithinAt.prod_map (hf x hx) (hg y hy)
#align continuous_on.prod_map ContinuousOn.prod_map
theorem continuous_of_cover_nhds {ι : Sort*} {f : α → β} {s : ι → Set α}
(hs : ∀ x : α, ∃ i, s i ∈ 𝓝 x) (hf : ∀ i, ContinuousOn f (s i)) :
Continuous f :=
continuous_iff_continuousAt.mpr fun x ↦ let ⟨i, hi⟩ := hs x; by
rw [ContinuousAt, ← nhdsWithin_eq_nhds.2 hi]
exact hf _ _ (mem_of_mem_nhds hi)
#align continuous_of_cover_nhds continuous_of_cover_nhds
theorem continuousOn_empty (f : α → β) : ContinuousOn f ∅ := fun _ => False.elim
#align continuous_on_empty continuousOn_empty
@[simp]
theorem continuousOn_singleton (f : α → β) (a : α) : ContinuousOn f {a} :=
forall_eq.2 <| by
simpa only [ContinuousWithinAt, nhdsWithin_singleton, tendsto_pure_left] using fun s =>
mem_of_mem_nhds
#align continuous_on_singleton continuousOn_singleton
theorem Set.Subsingleton.continuousOn {s : Set α} (hs : s.Subsingleton) (f : α → β) :
ContinuousOn f s :=
hs.induction_on (continuousOn_empty f) (continuousOn_singleton f)
#align set.subsingleton.continuous_on Set.Subsingleton.continuousOn
theorem nhdsWithin_le_comap {x : α} {s : Set α} {f : α → β} (ctsf : ContinuousWithinAt f s x) :
𝓝[s] x ≤ comap f (𝓝[f '' s] f x) :=
ctsf.tendsto_nhdsWithin_image.le_comap
#align nhds_within_le_comap nhdsWithin_le_comap
@[simp]
theorem comap_nhdsWithin_range {α} (f : α → β) (y : β) : comap f (𝓝[range f] y) = comap f (𝓝 y) :=
comap_inf_principal_range
#align comap_nhds_within_range comap_nhdsWithin_range
theorem ContinuousWithinAt.mono {f : α → β} {s t : Set α} {x : α} (h : ContinuousWithinAt f t x)
(hs : s ⊆ t) : ContinuousWithinAt f s x :=
h.mono_left (nhdsWithin_mono x hs)
#align continuous_within_at.mono ContinuousWithinAt.mono
theorem ContinuousWithinAt.mono_of_mem {f : α → β} {s t : Set α} {x : α}
(h : ContinuousWithinAt f t x) (hs : t ∈ 𝓝[s] x) : ContinuousWithinAt f s x :=
h.mono_left (nhdsWithin_le_of_mem hs)
#align continuous_within_at.mono_of_mem ContinuousWithinAt.mono_of_mem
theorem continuousWithinAt_congr_nhds {f : α → β} {s t : Set α} {x : α} (h : 𝓝[s] x = 𝓝[t] x) :
ContinuousWithinAt f s x ↔ ContinuousWithinAt f t x := by
simp only [ContinuousWithinAt, h]
theorem continuousWithinAt_inter' {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝[s] x) :
ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x := by
simp [ContinuousWithinAt, nhdsWithin_restrict'' s h]
#align continuous_within_at_inter' continuousWithinAt_inter'
theorem continuousWithinAt_inter {f : α → β} {s t : Set α} {x : α} (h : t ∈ 𝓝 x) :
ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x := by
simp [ContinuousWithinAt, nhdsWithin_restrict' s h]
#align continuous_within_at_inter continuousWithinAt_inter
theorem continuousWithinAt_union {f : α → β} {s t : Set α} {x : α} :
ContinuousWithinAt f (s ∪ t) x ↔ ContinuousWithinAt f s x ∧ ContinuousWithinAt f t x := by
simp only [ContinuousWithinAt, nhdsWithin_union, tendsto_sup]
#align continuous_within_at_union continuousWithinAt_union
theorem ContinuousWithinAt.union {f : α → β} {s t : Set α} {x : α} (hs : ContinuousWithinAt f s x)
(ht : ContinuousWithinAt f t x) : ContinuousWithinAt f (s ∪ t) x :=
continuousWithinAt_union.2 ⟨hs, ht⟩
#align continuous_within_at.union ContinuousWithinAt.union
theorem ContinuousWithinAt.mem_closure_image {f : α → β} {s : Set α} {x : α}
(h : ContinuousWithinAt f s x) (hx : x ∈ closure s) : f x ∈ closure (f '' s) :=
haveI := mem_closure_iff_nhdsWithin_neBot.1 hx
mem_closure_of_tendsto h <| mem_of_superset self_mem_nhdsWithin (subset_preimage_image f s)
#align continuous_within_at.mem_closure_image ContinuousWithinAt.mem_closure_image
theorem ContinuousWithinAt.mem_closure {f : α → β} {s : Set α} {x : α} {A : Set β}
(h : ContinuousWithinAt f s x) (hx : x ∈ closure s) (hA : MapsTo f s A) : f x ∈ closure A :=
closure_mono (image_subset_iff.2 hA) (h.mem_closure_image hx)
#align continuous_within_at.mem_closure ContinuousWithinAt.mem_closure
theorem Set.MapsTo.closure_of_continuousWithinAt {f : α → β} {s : Set α} {t : Set β}
(h : MapsTo f s t) (hc : ∀ x ∈ closure s, ContinuousWithinAt f s x) :
MapsTo f (closure s) (closure t) := fun x hx => (hc x hx).mem_closure hx h
#align set.maps_to.closure_of_continuous_within_at Set.MapsTo.closure_of_continuousWithinAt
theorem Set.MapsTo.closure_of_continuousOn {f : α → β} {s : Set α} {t : Set β} (h : MapsTo f s t)
(hc : ContinuousOn f (closure s)) : MapsTo f (closure s) (closure t) :=
h.closure_of_continuousWithinAt fun x hx => (hc x hx).mono subset_closure
#align set.maps_to.closure_of_continuous_on Set.MapsTo.closure_of_continuousOn
theorem ContinuousWithinAt.image_closure {f : α → β} {s : Set α}
(hf : ∀ x ∈ closure s, ContinuousWithinAt f s x) : f '' closure s ⊆ closure (f '' s) :=
((mapsTo_image f s).closure_of_continuousWithinAt hf).image_subset
#align continuous_within_at.image_closure ContinuousWithinAt.image_closure
theorem ContinuousOn.image_closure {f : α → β} {s : Set α} (hf : ContinuousOn f (closure s)) :
f '' closure s ⊆ closure (f '' s) :=
ContinuousWithinAt.image_closure fun x hx => (hf x hx).mono subset_closure
#align continuous_on.image_closure ContinuousOn.image_closure
@[simp]
theorem continuousWithinAt_singleton {f : α → β} {x : α} : ContinuousWithinAt f {x} x := by
simp only [ContinuousWithinAt, nhdsWithin_singleton, tendsto_pure_nhds]
#align continuous_within_at_singleton continuousWithinAt_singleton
@[simp]
theorem continuousWithinAt_insert_self {f : α → β} {x : α} {s : Set α} :
ContinuousWithinAt f (insert x s) x ↔ ContinuousWithinAt f s x := by
simp only [← singleton_union, continuousWithinAt_union, continuousWithinAt_singleton,
true_and_iff]
#align continuous_within_at_insert_self continuousWithinAt_insert_self
alias ⟨_, ContinuousWithinAt.insert_self⟩ := continuousWithinAt_insert_self
#align continuous_within_at.insert_self ContinuousWithinAt.insert_self
theorem ContinuousWithinAt.diff_iff {f : α → β} {s t : Set α} {x : α}
(ht : ContinuousWithinAt f t x) : ContinuousWithinAt f (s \ t) x ↔ ContinuousWithinAt f s x :=
⟨fun h => (h.union ht).mono <| by simp only [diff_union_self, subset_union_left], fun h =>
h.mono diff_subset⟩
#align continuous_within_at.diff_iff ContinuousWithinAt.diff_iff
@[simp]
theorem continuousWithinAt_diff_self {f : α → β} {s : Set α} {x : α} :
ContinuousWithinAt f (s \ {x}) x ↔ ContinuousWithinAt f s x :=
continuousWithinAt_singleton.diff_iff
#align continuous_within_at_diff_self continuousWithinAt_diff_self
@[simp]
theorem continuousWithinAt_compl_self {f : α → β} {a : α} :
ContinuousWithinAt f {a}ᶜ a ↔ ContinuousAt f a := by
rw [compl_eq_univ_diff, continuousWithinAt_diff_self, continuousWithinAt_univ]
#align continuous_within_at_compl_self continuousWithinAt_compl_self
@[simp]
theorem continuousWithinAt_update_same [DecidableEq α] {f : α → β} {s : Set α} {x : α} {y : β} :
ContinuousWithinAt (update f x y) s x ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y) :=
calc
ContinuousWithinAt (update f x y) s x ↔ Tendsto (update f x y) (𝓝[s \ {x}] x) (𝓝 y) := by
{ rw [← continuousWithinAt_diff_self, ContinuousWithinAt, update_same] }
_ ↔ Tendsto f (𝓝[s \ {x}] x) (𝓝 y) :=
tendsto_congr' <| eventually_nhdsWithin_iff.2 <| eventually_of_forall
fun z hz => update_noteq hz.2 _ _
#align continuous_within_at_update_same continuousWithinAt_update_same
@[simp]
theorem continuousAt_update_same [DecidableEq α] {f : α → β} {x : α} {y : β} :
ContinuousAt (Function.update f x y) x ↔ Tendsto f (𝓝[≠] x) (𝓝 y) := by
rw [← continuousWithinAt_univ, continuousWithinAt_update_same, compl_eq_univ_diff]
#align continuous_at_update_same continuousAt_update_same
theorem IsOpenMap.continuousOn_image_of_leftInvOn {f : α → β} {s : Set α}
(h : IsOpenMap (s.restrict f)) {finv : β → α} (hleft : LeftInvOn finv f s) :
ContinuousOn finv (f '' s) := by
refine continuousOn_iff'.2 fun t ht => ⟨f '' (t ∩ s), ?_, ?_⟩
· rw [← image_restrict]
exact h _ (ht.preimage continuous_subtype_val)
· rw [inter_eq_self_of_subset_left (image_subset f inter_subset_right), hleft.image_inter']
#align is_open_map.continuous_on_image_of_left_inv_on IsOpenMap.continuousOn_image_of_leftInvOn
theorem IsOpenMap.continuousOn_range_of_leftInverse {f : α → β} (hf : IsOpenMap f) {finv : β → α}
(hleft : Function.LeftInverse finv f) : ContinuousOn finv (range f) := by
rw [← image_univ]
exact (hf.restrict isOpen_univ).continuousOn_image_of_leftInvOn fun x _ => hleft x
#align is_open_map.continuous_on_range_of_left_inverse IsOpenMap.continuousOn_range_of_leftInverse
theorem ContinuousOn.congr_mono {f g : α → β} {s s₁ : Set α} (h : ContinuousOn f s)
(h' : EqOn g f s₁) (h₁ : s₁ ⊆ s) : ContinuousOn g s₁ := by
intro x hx
unfold ContinuousWithinAt
have A := (h x (h₁ hx)).mono h₁
unfold ContinuousWithinAt at A
rw [← h' hx] at A
exact A.congr' h'.eventuallyEq_nhdsWithin.symm
#align continuous_on.congr_mono ContinuousOn.congr_mono
theorem ContinuousOn.congr {f g : α → β} {s : Set α} (h : ContinuousOn f s) (h' : EqOn g f s) :
ContinuousOn g s :=
h.congr_mono h' (Subset.refl _)
#align continuous_on.congr ContinuousOn.congr
theorem continuousOn_congr {f g : α → β} {s : Set α} (h' : EqOn g f s) :
ContinuousOn g s ↔ ContinuousOn f s :=
⟨fun h => ContinuousOn.congr h h'.symm, fun h => h.congr h'⟩
#align continuous_on_congr continuousOn_congr
theorem ContinuousAt.continuousWithinAt {f : α → β} {s : Set α} {x : α} (h : ContinuousAt f x) :
ContinuousWithinAt f s x :=
ContinuousWithinAt.mono ((continuousWithinAt_univ f x).2 h) (subset_univ _)
#align continuous_at.continuous_within_at ContinuousAt.continuousWithinAt
theorem continuousWithinAt_iff_continuousAt {f : α → β} {s : Set α} {x : α} (h : s ∈ 𝓝 x) :
ContinuousWithinAt f s x ↔ ContinuousAt f x := by
rw [← univ_inter s, continuousWithinAt_inter h, continuousWithinAt_univ]
#align continuous_within_at_iff_continuous_at continuousWithinAt_iff_continuousAt
theorem ContinuousWithinAt.continuousAt {f : α → β} {s : Set α} {x : α}
(h : ContinuousWithinAt f s x) (hs : s ∈ 𝓝 x) : ContinuousAt f x :=
(continuousWithinAt_iff_continuousAt hs).mp h
#align continuous_within_at.continuous_at ContinuousWithinAt.continuousAt
theorem IsOpen.continuousOn_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
ContinuousOn f s ↔ ∀ ⦃a⦄, a ∈ s → ContinuousAt f a :=
forall₂_congr fun _ => continuousWithinAt_iff_continuousAt ∘ hs.mem_nhds
#align is_open.continuous_on_iff IsOpen.continuousOn_iff
theorem ContinuousOn.continuousAt {f : α → β} {s : Set α} {x : α} (h : ContinuousOn f s)
(hx : s ∈ 𝓝 x) : ContinuousAt f x :=
(h x (mem_of_mem_nhds hx)).continuousAt hx
#align continuous_on.continuous_at ContinuousOn.continuousAt
theorem ContinuousAt.continuousOn {f : α → β} {s : Set α} (hcont : ∀ x ∈ s, ContinuousAt f x) :
ContinuousOn f s := fun x hx => (hcont x hx).continuousWithinAt
#align continuous_at.continuous_on ContinuousAt.continuousOn
theorem ContinuousWithinAt.comp {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α}
(hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) (h : MapsTo f s t) :
ContinuousWithinAt (g ∘ f) s x :=
hg.tendsto.comp (hf.tendsto_nhdsWithin h)
#align continuous_within_at.comp ContinuousWithinAt.comp
theorem ContinuousWithinAt.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α}
(hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) :
ContinuousWithinAt (g ∘ f) (s ∩ f ⁻¹' t) x :=
hg.comp (hf.mono inter_subset_left) inter_subset_right
#align continuous_within_at.comp' ContinuousWithinAt.comp'
theorem ContinuousAt.comp_continuousWithinAt {g : β → γ} {f : α → β} {s : Set α} {x : α}
(hg : ContinuousAt g (f x)) (hf : ContinuousWithinAt f s x) : ContinuousWithinAt (g ∘ f) s x :=
hg.continuousWithinAt.comp hf (mapsTo_univ _ _)
#align continuous_at.comp_continuous_within_at ContinuousAt.comp_continuousWithinAt
theorem ContinuousOn.comp {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t)
(hf : ContinuousOn f s) (h : MapsTo f s t) : ContinuousOn (g ∘ f) s := fun x hx =>
ContinuousWithinAt.comp (hg _ (h hx)) (hf x hx) h
#align continuous_on.comp ContinuousOn.comp
@[fun_prop]
theorem ContinuousOn.comp'' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t)
(hf : ContinuousOn f s) (h : Set.MapsTo f s t) : ContinuousOn (fun x => g (f x)) s :=
ContinuousOn.comp hg hf h
theorem ContinuousOn.mono {f : α → β} {s t : Set α} (hf : ContinuousOn f s) (h : t ⊆ s) :
ContinuousOn f t := fun x hx => (hf x (h hx)).mono_left (nhdsWithin_mono _ h)
#align continuous_on.mono ContinuousOn.mono
theorem antitone_continuousOn {f : α → β} : Antitone (ContinuousOn f) := fun _s _t hst hf =>
hf.mono hst
#align antitone_continuous_on antitone_continuousOn
@[fun_prop]
theorem ContinuousOn.comp' {g : β → γ} {f : α → β} {s : Set α} {t : Set β} (hg : ContinuousOn g t)
(hf : ContinuousOn f s) : ContinuousOn (g ∘ f) (s ∩ f ⁻¹' t) :=
hg.comp (hf.mono inter_subset_left) inter_subset_right
#align continuous_on.comp' ContinuousOn.comp'
@[fun_prop]
theorem Continuous.continuousOn {f : α → β} {s : Set α} (h : Continuous f) : ContinuousOn f s := by
rw [continuous_iff_continuousOn_univ] at h
exact h.mono (subset_univ _)
#align continuous.continuous_on Continuous.continuousOn
theorem Continuous.continuousWithinAt {f : α → β} {s : Set α} {x : α} (h : Continuous f) :
ContinuousWithinAt f s x :=
h.continuousAt.continuousWithinAt
#align continuous.continuous_within_at Continuous.continuousWithinAt
theorem Continuous.comp_continuousOn {g : β → γ} {f : α → β} {s : Set α} (hg : Continuous g)
(hf : ContinuousOn f s) : ContinuousOn (g ∘ f) s :=
hg.continuousOn.comp hf (mapsTo_univ _ _)
#align continuous.comp_continuous_on Continuous.comp_continuousOn
@[fun_prop]
theorem Continuous.comp_continuousOn'
{α β γ : Type*} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ}
{f : α → β} {s : Set α} (hg : Continuous g) (hf : ContinuousOn f s) :
ContinuousOn (fun x ↦ g (f x)) s :=
hg.comp_continuousOn hf
theorem ContinuousOn.comp_continuous {g : β → γ} {f : α → β} {s : Set β} (hg : ContinuousOn g s)
(hf : Continuous f) (hs : ∀ x, f x ∈ s) : Continuous (g ∘ f) := by
rw [continuous_iff_continuousOn_univ] at *
exact hg.comp hf fun x _ => hs x
#align continuous_on.comp_continuous ContinuousOn.comp_continuous
@[fun_prop]
theorem continuousOn_apply {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
(i : ι) (s) : ContinuousOn (fun p : ∀ i, π i => p i) s :=
Continuous.continuousOn (continuous_apply i)
theorem ContinuousWithinAt.preimage_mem_nhdsWithin {f : α → β} {x : α} {s : Set α} {t : Set β}
(h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝 (f x)) : f ⁻¹' t ∈ 𝓝[s] x :=
h ht
#align continuous_within_at.preimage_mem_nhds_within ContinuousWithinAt.preimage_mem_nhdsWithin
theorem Set.LeftInvOn.map_nhdsWithin_eq {f : α → β} {g : β → α} {x : β} {s : Set β}
(h : LeftInvOn f g s) (hx : f (g x) = x) (hf : ContinuousWithinAt f (g '' s) (g x))
(hg : ContinuousWithinAt g s x) : map g (𝓝[s] x) = 𝓝[g '' s] g x := by
apply le_antisymm
· exact hg.tendsto_nhdsWithin (mapsTo_image _ _)
· have A : g ∘ f =ᶠ[𝓝[g '' s] g x] id :=
h.rightInvOn_image.eqOn.eventuallyEq_of_mem self_mem_nhdsWithin
refine le_map_of_right_inverse A ?_
simpa only [hx] using hf.tendsto_nhdsWithin (h.mapsTo (surjOn_image _ _))
#align set.left_inv_on.map_nhds_within_eq Set.LeftInvOn.map_nhdsWithin_eq
theorem Function.LeftInverse.map_nhds_eq {f : α → β} {g : β → α} {x : β}
(h : Function.LeftInverse f g) (hf : ContinuousWithinAt f (range g) (g x))
(hg : ContinuousAt g x) : map g (𝓝 x) = 𝓝[range g] g x := by
simpa only [nhdsWithin_univ, image_univ] using
(h.leftInvOn univ).map_nhdsWithin_eq (h x) (by rwa [image_univ]) hg.continuousWithinAt
#align function.left_inverse.map_nhds_eq Function.LeftInverse.map_nhds_eq
theorem ContinuousWithinAt.preimage_mem_nhdsWithin' {f : α → β} {x : α} {s : Set α} {t : Set β}
(h : ContinuousWithinAt f s x) (ht : t ∈ 𝓝[f '' s] f x) : f ⁻¹' t ∈ 𝓝[s] x :=
h.tendsto_nhdsWithin (mapsTo_image _ _) ht
#align continuous_within_at.preimage_mem_nhds_within' ContinuousWithinAt.preimage_mem_nhdsWithin'
theorem ContinuousWithinAt.preimage_mem_nhdsWithin''
{f : α → β} {x : α} {y : β} {s t : Set β}
(h : ContinuousWithinAt f (f ⁻¹' s) x) (ht : t ∈ 𝓝[s] y) (hxy : y = f x) :
f ⁻¹' t ∈ 𝓝[f ⁻¹' s] x := by
rw [hxy] at ht
exact h.preimage_mem_nhdsWithin' (nhdsWithin_mono _ (image_preimage_subset f s) ht)
theorem Filter.EventuallyEq.congr_continuousWithinAt {f g : α → β} {s : Set α} {x : α}
(h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) :
ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x := by
rw [ContinuousWithinAt, hx, tendsto_congr' h, ContinuousWithinAt]
#align filter.eventually_eq.congr_continuous_within_at Filter.EventuallyEq.congr_continuousWithinAt
theorem ContinuousWithinAt.congr_of_eventuallyEq {f f₁ : α → β} {s : Set α} {x : α}
(h : ContinuousWithinAt f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
ContinuousWithinAt f₁ s x :=
(h₁.congr_continuousWithinAt hx).2 h
#align continuous_within_at.congr_of_eventually_eq ContinuousWithinAt.congr_of_eventuallyEq
theorem ContinuousWithinAt.congr {f f₁ : α → β} {s : Set α} {x : α} (h : ContinuousWithinAt f s x)
(h₁ : ∀ y ∈ s, f₁ y = f y) (hx : f₁ x = f x) : ContinuousWithinAt f₁ s x :=
h.congr_of_eventuallyEq (mem_of_superset self_mem_nhdsWithin h₁) hx
#align continuous_within_at.congr ContinuousWithinAt.congr
theorem ContinuousWithinAt.congr_mono {f g : α → β} {s s₁ : Set α} {x : α}
(h : ContinuousWithinAt f s x) (h' : EqOn g f s₁) (h₁ : s₁ ⊆ s) (hx : g x = f x) :
ContinuousWithinAt g s₁ x :=
(h.mono h₁).congr h' hx
#align continuous_within_at.congr_mono ContinuousWithinAt.congr_mono
@[fun_prop]
theorem continuousOn_const {s : Set α} {c : β} : ContinuousOn (fun _ => c) s :=
continuous_const.continuousOn
#align continuous_on_const continuousOn_const
theorem continuousWithinAt_const {b : β} {s : Set α} {x : α} :
ContinuousWithinAt (fun _ : α => b) s x :=
continuous_const.continuousWithinAt
#align continuous_within_at_const continuousWithinAt_const
theorem continuousOn_id {s : Set α} : ContinuousOn id s :=
continuous_id.continuousOn
#align continuous_on_id continuousOn_id
@[fun_prop]
theorem continuousOn_id' (s : Set α) : ContinuousOn (fun x : α => x) s := continuousOn_id
theorem continuousWithinAt_id {s : Set α} {x : α} : ContinuousWithinAt id s x :=
continuous_id.continuousWithinAt
#align continuous_within_at_id continuousWithinAt_id
theorem continuousOn_open_iff {f : α → β} {s : Set α} (hs : IsOpen s) :
ContinuousOn f s ↔ ∀ t, IsOpen t → IsOpen (s ∩ f ⁻¹' t) := by
rw [continuousOn_iff']
constructor
· intro h t ht
rcases h t ht with ⟨u, u_open, hu⟩
rw [inter_comm, hu]
apply IsOpen.inter u_open hs
· intro h t ht
refine ⟨s ∩ f ⁻¹' t, h t ht, ?_⟩
rw [@inter_comm _ s (f ⁻¹' t), inter_assoc, inter_self]
#align continuous_on_open_iff continuousOn_open_iff
theorem ContinuousOn.isOpen_inter_preimage {f : α → β} {s : Set α} {t : Set β}
(hf : ContinuousOn f s) (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ∩ f ⁻¹' t) :=
(continuousOn_open_iff hs).1 hf t ht
#align continuous_on.preimage_open_of_open ContinuousOn.isOpen_inter_preimage
theorem ContinuousOn.isOpen_preimage {f : α → β} {s : Set α} {t : Set β} (h : ContinuousOn f s)
(hs : IsOpen s) (hp : f ⁻¹' t ⊆ s) (ht : IsOpen t) : IsOpen (f ⁻¹' t) := by
convert (continuousOn_open_iff hs).mp h t ht
rw [inter_comm, inter_eq_self_of_subset_left hp]
#align continuous_on.is_open_preimage ContinuousOn.isOpen_preimage
theorem ContinuousOn.preimage_isClosed_of_isClosed {f : α → β} {s : Set α} {t : Set β}
(hf : ContinuousOn f s) (hs : IsClosed s) (ht : IsClosed t) : IsClosed (s ∩ f ⁻¹' t) := by
rcases continuousOn_iff_isClosed.1 hf t ht with ⟨u, hu⟩
rw [inter_comm, hu.2]
apply IsClosed.inter hu.1 hs
#align continuous_on.preimage_closed_of_closed ContinuousOn.preimage_isClosed_of_isClosed
theorem ContinuousOn.preimage_interior_subset_interior_preimage {f : α → β} {s : Set α} {t : Set β}
(hf : ContinuousOn f s) (hs : IsOpen s) : s ∩ f ⁻¹' interior t ⊆ s ∩ interior (f ⁻¹' t) :=
calc
s ∩ f ⁻¹' interior t ⊆ interior (s ∩ f ⁻¹' t) :=
interior_maximal (inter_subset_inter (Subset.refl _) (preimage_mono interior_subset))
(hf.isOpen_inter_preimage hs isOpen_interior)
_ = s ∩ interior (f ⁻¹' t) := by rw [interior_inter, hs.interior_eq]
#align continuous_on.preimage_interior_subset_interior_preimage ContinuousOn.preimage_interior_subset_interior_preimage
theorem continuousOn_of_locally_continuousOn {f : α → β} {s : Set α}
(h : ∀ x ∈ s, ∃ t, IsOpen t ∧ x ∈ t ∧ ContinuousOn f (s ∩ t)) : ContinuousOn f s := by
intro x xs
rcases h x xs with ⟨t, open_t, xt, ct⟩
have := ct x ⟨xs, xt⟩
rwa [ContinuousWithinAt, ← nhdsWithin_restrict _ xt open_t] at this
#align continuous_on_of_locally_continuous_on continuousOn_of_locally_continuousOn
-- Porting note (#10756): new lemma
theorem continuousOn_to_generateFrom_iff {s : Set α} {T : Set (Set β)} {f : α → β} :
@ContinuousOn α β _ (.generateFrom T) f s ↔ ∀ x ∈ s, ∀ t ∈ T, f x ∈ t → f ⁻¹' t ∈ 𝓝[s] x :=
forall₂_congr fun x _ => by
delta ContinuousWithinAt
simp only [TopologicalSpace.nhds_generateFrom, tendsto_iInf, tendsto_principal, mem_setOf_eq,
and_imp]
exact forall_congr' fun t => forall_swap
-- Porting note: dropped an unneeded assumption
theorem continuousOn_isOpen_of_generateFrom {β : Type*} {s : Set α} {T : Set (Set β)} {f : α → β}
(h : ∀ t ∈ T, IsOpen (s ∩ f ⁻¹' t)) :
@ContinuousOn α β _ (.generateFrom T) f s :=
continuousOn_to_generateFrom_iff.2 fun _x hx t ht hxt => mem_nhdsWithin.2
⟨_, h t ht, ⟨hx, hxt⟩, fun _y hy => hy.1.2⟩
#align continuous_on_open_of_generate_from continuousOn_isOpen_of_generateFromₓ
theorem ContinuousWithinAt.prod {f : α → β} {g : α → γ} {s : Set α} {x : α}
(hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (fun x => (f x, g x)) s x :=
hf.prod_mk_nhds hg
#align continuous_within_at.prod ContinuousWithinAt.prod
@[fun_prop]
theorem ContinuousOn.prod {f : α → β} {g : α → γ} {s : Set α} (hf : ContinuousOn f s)
(hg : ContinuousOn g s) : ContinuousOn (fun x => (f x, g x)) s := fun x hx =>
ContinuousWithinAt.prod (hf x hx) (hg x hx)
#align continuous_on.prod ContinuousOn.prod
theorem ContinuousAt.comp₂_continuousWithinAt {f : β × γ → δ} {g : α → β} {h : α → γ} {x : α}
{s : Set α} (hf : ContinuousAt f (g x, h x)) (hg : ContinuousWithinAt g s x)
(hh : ContinuousWithinAt h s x) :
ContinuousWithinAt (fun x ↦ f (g x, h x)) s x :=
ContinuousAt.comp_continuousWithinAt hf (hg.prod hh)
theorem ContinuousAt.comp₂_continuousWithinAt_of_eq {f : β × γ → δ} {g : α → β}
{h : α → γ} {x : α} {s : Set α} {y : β × γ} (hf : ContinuousAt f y)
(hg : ContinuousWithinAt g s x) (hh : ContinuousWithinAt h s x) (e : (g x, h x) = y) :
ContinuousWithinAt (fun x ↦ f (g x, h x)) s x := by
rw [← e] at hf
exact hf.comp₂_continuousWithinAt hg hh
theorem Inducing.continuousWithinAt_iff {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α}
{x : α} : ContinuousWithinAt f s x ↔ ContinuousWithinAt (g ∘ f) s x := by
simp_rw [ContinuousWithinAt, Inducing.tendsto_nhds_iff hg]; rfl
#align inducing.continuous_within_at_iff Inducing.continuousWithinAt_iff
theorem Inducing.continuousOn_iff {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α} :
ContinuousOn f s ↔ ContinuousOn (g ∘ f) s := by
simp_rw [ContinuousOn, hg.continuousWithinAt_iff]
#align inducing.continuous_on_iff Inducing.continuousOn_iff
theorem Embedding.continuousOn_iff {f : α → β} {g : β → γ} (hg : Embedding g) {s : Set α} :
ContinuousOn f s ↔ ContinuousOn (g ∘ f) s :=
Inducing.continuousOn_iff hg.1
#align embedding.continuous_on_iff Embedding.continuousOn_iff
theorem Embedding.map_nhdsWithin_eq {f : α → β} (hf : Embedding f) (s : Set α) (x : α) :
map f (𝓝[s] x) = 𝓝[f '' s] f x := by
rw [nhdsWithin, Filter.map_inf hf.inj, hf.map_nhds_eq, map_principal, ← nhdsWithin_inter',
inter_eq_self_of_subset_right (image_subset_range _ _)]
#align embedding.map_nhds_within_eq Embedding.map_nhdsWithin_eq
theorem OpenEmbedding.map_nhdsWithin_preimage_eq {f : α → β} (hf : OpenEmbedding f) (s : Set β)
(x : α) : map f (𝓝[f ⁻¹' s] x) = 𝓝[s] f x := by
rw [hf.toEmbedding.map_nhdsWithin_eq, image_preimage_eq_inter_range]
apply nhdsWithin_eq_nhdsWithin (mem_range_self _) hf.isOpen_range
rw [inter_assoc, inter_self]
#align open_embedding.map_nhds_within_preimage_eq OpenEmbedding.map_nhdsWithin_preimage_eq
theorem continuousWithinAt_of_not_mem_closure {f : α → β} {s : Set α} {x : α} (hx : x ∉ closure s) :
ContinuousWithinAt f s x := by
rw [mem_closure_iff_nhdsWithin_neBot, not_neBot] at hx
rw [ContinuousWithinAt, hx]
exact tendsto_bot
#align continuous_within_at_of_not_mem_closure continuousWithinAt_of_not_mem_closure
theorem ContinuousOn.if' {s : Set α} {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
(hpf : ∀ a ∈ s ∩ frontier { a | p a },
Tendsto f (𝓝[s ∩ { a | p a }] a) (𝓝 <| if p a then f a else g a))
(hpg :
∀ a ∈ s ∩ frontier { a | p a },
Tendsto g (𝓝[s ∩ { a | ¬p a }] a) (𝓝 <| if p a then f a else g a))
(hf : ContinuousOn f <| s ∩ { a | p a }) (hg : ContinuousOn g <| s ∩ { a | ¬p a }) :
ContinuousOn (fun a => if p a then f a else g a) s := by
intro x hx
by_cases hx' : x ∈ frontier { a | p a }
· exact (hpf x ⟨hx, hx'⟩).piecewise_nhdsWithin (hpg x ⟨hx, hx'⟩)
· rw [← inter_univ s, ← union_compl_self { a | p a }, inter_union_distrib_left] at hx ⊢
cases' hx with hx hx
· apply ContinuousWithinAt.union
· exact (hf x hx).congr (fun y hy => if_pos hy.2) (if_pos hx.2)
· have : x ∉ closure { a | p a }ᶜ := fun h => hx' ⟨subset_closure hx.2, by
rwa [closure_compl] at h⟩
exact continuousWithinAt_of_not_mem_closure fun h =>
this (closure_inter_subset_inter_closure _ _ h).2
· apply ContinuousWithinAt.union
· have : x ∉ closure { a | p a } := fun h =>
hx' ⟨h, fun h' : x ∈ interior { a | p a } => hx.2 (interior_subset h')⟩
exact continuousWithinAt_of_not_mem_closure fun h =>
this (closure_inter_subset_inter_closure _ _ h).2
· exact (hg x hx).congr (fun y hy => if_neg hy.2) (if_neg hx.2)
#align continuous_on.if' ContinuousOn.if'
theorem ContinuousOn.piecewise' {s t : Set α} {f g : α → β} [∀ a, Decidable (a ∈ t)]
(hpf : ∀ a ∈ s ∩ frontier t, Tendsto f (𝓝[s ∩ t] a) (𝓝 (piecewise t f g a)))
(hpg : ∀ a ∈ s ∩ frontier t, Tendsto g (𝓝[s ∩ tᶜ] a) (𝓝 (piecewise t f g a)))
(hf : ContinuousOn f <| s ∩ t) (hg : ContinuousOn g <| s ∩ tᶜ) :
ContinuousOn (piecewise t f g) s :=
hf.if' hpf hpg hg
#align continuous_on.piecewise' ContinuousOn.piecewise'
theorem ContinuousOn.if {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {p : α → Prop}
[∀ a, Decidable (p a)] {s : Set α} {f g : α → β}
(hp : ∀ a ∈ s ∩ frontier { a | p a }, f a = g a)
(hf : ContinuousOn f <| s ∩ closure { a | p a })
(hg : ContinuousOn g <| s ∩ closure { a | ¬p a }) :
ContinuousOn (fun a => if p a then f a else g a) s := by
apply ContinuousOn.if'
· rintro a ha
simp only [← hp a ha, ite_self]
apply tendsto_nhdsWithin_mono_left (inter_subset_inter_right s subset_closure)
exact hf a ⟨ha.1, ha.2.1⟩
· rintro a ha
simp only [hp a ha, ite_self]
apply tendsto_nhdsWithin_mono_left (inter_subset_inter_right s subset_closure)
rcases ha with ⟨has, ⟨_, ha⟩⟩
rw [← mem_compl_iff, ← closure_compl] at ha
apply hg a ⟨has, ha⟩
· exact hf.mono (inter_subset_inter_right s subset_closure)
· exact hg.mono (inter_subset_inter_right s subset_closure)
#align continuous_on.if ContinuousOn.if
theorem ContinuousOn.piecewise {s t : Set α} {f g : α → β} [∀ a, Decidable (a ∈ t)]
(ht : ∀ a ∈ s ∩ frontier t, f a = g a) (hf : ContinuousOn f <| s ∩ closure t)
(hg : ContinuousOn g <| s ∩ closure tᶜ) : ContinuousOn (piecewise t f g) s :=
hf.if ht hg
#align continuous_on.piecewise ContinuousOn.piecewise
theorem continuous_if' {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
(hpf : ∀ a ∈ frontier { x | p x }, Tendsto f (𝓝[{ x | p x }] a) (𝓝 <| ite (p a) (f a) (g a)))
(hpg : ∀ a ∈ frontier { x | p x }, Tendsto g (𝓝[{ x | ¬p x }] a) (𝓝 <| ite (p a) (f a) (g a)))
(hf : ContinuousOn f { x | p x }) (hg : ContinuousOn g { x | ¬p x }) :
Continuous fun a => ite (p a) (f a) (g a) := by
rw [continuous_iff_continuousOn_univ]
apply ContinuousOn.if' <;> simp [*] <;> assumption
#align continuous_if' continuous_if'
theorem continuous_if {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
(hp : ∀ a ∈ frontier { x | p x }, f a = g a) (hf : ContinuousOn f (closure { x | p x }))
(hg : ContinuousOn g (closure { x | ¬p x })) :
Continuous fun a => if p a then f a else g a := by
rw [continuous_iff_continuousOn_univ]
apply ContinuousOn.if <;> simp <;> assumption
#align continuous_if continuous_if
theorem Continuous.if {p : α → Prop} {f g : α → β} [∀ a, Decidable (p a)]
(hp : ∀ a ∈ frontier { x | p x }, f a = g a) (hf : Continuous f) (hg : Continuous g) :
Continuous fun a => if p a then f a else g a :=
continuous_if hp hf.continuousOn hg.continuousOn
#align continuous.if Continuous.if
theorem continuous_if_const (p : Prop) {f g : α → β} [Decidable p] (hf : p → Continuous f)
(hg : ¬p → Continuous g) : Continuous fun a => if p then f a else g a := by
split_ifs with h
exacts [hf h, hg h]
#align continuous_if_const continuous_if_const
theorem Continuous.if_const (p : Prop) {f g : α → β} [Decidable p] (hf : Continuous f)
(hg : Continuous g) : Continuous fun a => if p then f a else g a :=
continuous_if_const p (fun _ => hf) fun _ => hg
#align continuous.if_const Continuous.if_const
theorem continuous_piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a ∈ s)]
(hs : ∀ a ∈ frontier s, f a = g a) (hf : ContinuousOn f (closure s))
(hg : ContinuousOn g (closure sᶜ)) : Continuous (piecewise s f g) :=
continuous_if hs hf hg
#align continuous_piecewise continuous_piecewise
theorem Continuous.piecewise {s : Set α} {f g : α → β} [∀ a, Decidable (a ∈ s)]
(hs : ∀ a ∈ frontier s, f a = g a) (hf : Continuous f) (hg : Continuous g) :
Continuous (piecewise s f g) :=
hf.if hs hg
#align continuous.piecewise Continuous.piecewise
theorem IsOpen.ite' {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
(ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s') : IsOpen (t.ite s s') := by
classical
simp only [isOpen_iff_continuous_mem, Set.ite] at *
convert continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuousOn hs'.continuousOn
rename_i x
by_cases hx : x ∈ t <;> simp [hx]
#align is_open.ite' IsOpen.ite'
theorem IsOpen.ite {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
(ht : s ∩ frontier t = s' ∩ frontier t) : IsOpen (t.ite s s') :=
hs.ite' hs' fun x hx => by simpa [hx] using ext_iff.1 ht x
#align is_open.ite IsOpen.ite
theorem ite_inter_closure_eq_of_inter_frontier_eq {s s' t : Set α}
(ht : s ∩ frontier t = s' ∩ frontier t) : t.ite s s' ∩ closure t = s ∩ closure t := by
rw [closure_eq_self_union_frontier, inter_union_distrib_left, inter_union_distrib_left,
ite_inter_self, ite_inter_of_inter_eq _ ht]
#align ite_inter_closure_eq_of_inter_frontier_eq ite_inter_closure_eq_of_inter_frontier_eq
theorem ite_inter_closure_compl_eq_of_inter_frontier_eq {s s' t : Set α}
(ht : s ∩ frontier t = s' ∩ frontier t) : t.ite s s' ∩ closure tᶜ = s' ∩ closure tᶜ := by
rw [← ite_compl, ite_inter_closure_eq_of_inter_frontier_eq]
rwa [frontier_compl, eq_comm]
#align ite_inter_closure_compl_eq_of_inter_frontier_eq ite_inter_closure_compl_eq_of_inter_frontier_eq
| Mathlib/Topology/ContinuousOn.lean | 1,344 | 1,351 | theorem continuousOn_piecewise_ite' {s s' t : Set α} {f f' : α → β} [∀ x, Decidable (x ∈ t)]
(h : ContinuousOn f (s ∩ closure t)) (h' : ContinuousOn f' (s' ∩ closure tᶜ))
(H : s ∩ frontier t = s' ∩ frontier t) (Heq : EqOn f f' (s ∩ frontier t)) :
ContinuousOn (t.piecewise f f') (t.ite s s') := by |
apply ContinuousOn.piecewise
· rwa [ite_inter_of_inter_eq _ H]
· rwa [ite_inter_closure_eq_of_inter_frontier_eq H]
· rwa [ite_inter_closure_compl_eq_of_inter_frontier_eq H]
|
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.normed_space.star.basic from "leanprover-community/mathlib"@"aa6669832974f87406a3d9d70fc5707a60546207"
open Topology
local postfix:max "⋆" => star
class NormedStarGroup (E : Type*) [SeminormedAddCommGroup E] [StarAddMonoid E] : Prop where
norm_star : ∀ x : E, ‖x⋆‖ = ‖x‖
#align normed_star_group NormedStarGroup
export NormedStarGroup (norm_star)
attribute [simp] norm_star
variable {𝕜 E α : Type*}
instance RingHomIsometric.starRingEnd [NormedCommRing E] [StarRing E] [NormedStarGroup E] :
RingHomIsometric (starRingEnd E) :=
⟨@norm_star _ _ _ _⟩
#align ring_hom_isometric.star_ring_end RingHomIsometric.starRingEnd
class CstarRing (E : Type*) [NonUnitalNormedRing E] [StarRing E] : Prop where
norm_star_mul_self : ∀ {x : E}, ‖x⋆ * x‖ = ‖x‖ * ‖x‖
#align cstar_ring CstarRing
instance : CstarRing ℝ where norm_star_mul_self {x} := by simp only [star, id, norm_mul]
namespace CstarRing
section Unital
variable [NormedRing E] [StarRing E] [CstarRing E]
@[simp, nolint simpNF] -- Porting note (#10959): simp cannot prove this
theorem norm_one [Nontrivial E] : ‖(1 : E)‖ = 1 := by
have : 0 < ‖(1 : E)‖ := norm_pos_iff.mpr one_ne_zero
rw [← mul_left_inj' this.ne', ← norm_star_mul_self, mul_one, star_one, one_mul]
#align cstar_ring.norm_one CstarRing.norm_one
-- see Note [lower instance priority]
instance (priority := 100) [Nontrivial E] : NormOneClass E :=
⟨norm_one⟩
| Mathlib/Analysis/NormedSpace/Star/Basic.lean | 212 | 214 | theorem norm_coe_unitary [Nontrivial E] (U : unitary E) : ‖(U : E)‖ = 1 := by |
rw [← sq_eq_sq (norm_nonneg _) zero_le_one, one_pow 2, sq, ← CstarRing.norm_star_mul_self,
unitary.coe_star_mul_self, CstarRing.norm_one]
|
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Data.ZMod.Algebra
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
import Mathlib.FieldTheory.Perfect
#align_import ring_theory.witt_vector.frobenius from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"
namespace WittVector
variable {p : ℕ} {R S : Type*} [hp : Fact p.Prime] [CommRing R] [CommRing S]
local notation "𝕎" => WittVector p -- type as `\bbW`
noncomputable section
open MvPolynomial Finset
variable (p)
def frobeniusPolyRat (n : ℕ) : MvPolynomial ℕ ℚ :=
bind₁ (wittPolynomial p ℚ ∘ fun n => n + 1) (xInTermsOfW p ℚ n)
#align witt_vector.frobenius_poly_rat WittVector.frobeniusPolyRat
theorem bind₁_frobeniusPolyRat_wittPolynomial (n : ℕ) :
bind₁ (frobeniusPolyRat p) (wittPolynomial p ℚ n) = wittPolynomial p ℚ (n + 1) := by
delta frobeniusPolyRat
rw [← bind₁_bind₁, bind₁_xInTermsOfW_wittPolynomial, bind₁_X_right, Function.comp_apply]
#align witt_vector.bind₁_frobenius_poly_rat_witt_polynomial WittVector.bind₁_frobeniusPolyRat_wittPolynomial
private def pnat_multiplicity (n : ℕ+) : ℕ :=
(multiplicity p n).get <| multiplicity.finite_nat_iff.mpr <| ⟨ne_of_gt hp.1.one_lt, n.2⟩
local notation "v" => pnat_multiplicity
noncomputable def frobeniusPolyAux : ℕ → MvPolynomial ℕ ℤ
| n => X (n + 1) - ∑ i : Fin n, have _ := i.is_lt
∑ j ∈ range (p ^ (n - i)),
(((X (i : ℕ) ^ p) ^ (p ^ (n - (i : ℕ)) - (j + 1)) : MvPolynomial ℕ ℤ) *
(frobeniusPolyAux i) ^ (j + 1)) *
C (((p ^ (n - i)).choose (j + 1) / (p ^ (n - i - v p ⟨j + 1, Nat.succ_pos j⟩))
* ↑p ^ (j - v p ⟨j + 1, Nat.succ_pos j⟩) : ℕ) : ℤ)
#align witt_vector.frobenius_poly_aux WittVector.frobeniusPolyAux
theorem frobeniusPolyAux_eq (n : ℕ) :
frobeniusPolyAux p n =
X (n + 1) - ∑ i ∈ range n,
∑ j ∈ range (p ^ (n - i)),
(X i ^ p) ^ (p ^ (n - i) - (j + 1)) * frobeniusPolyAux p i ^ (j + 1) *
C ↑((p ^ (n - i)).choose (j + 1) / p ^ (n - i - v p ⟨j + 1, Nat.succ_pos j⟩) *
↑p ^ (j - v p ⟨j + 1, Nat.succ_pos j⟩) : ℕ) := by
rw [frobeniusPolyAux, ← Fin.sum_univ_eq_sum_range]
#align witt_vector.frobenius_poly_aux_eq WittVector.frobeniusPolyAux_eq
def frobeniusPoly (n : ℕ) : MvPolynomial ℕ ℤ :=
X n ^ p + C (p : ℤ) * frobeniusPolyAux p n
#align witt_vector.frobenius_poly WittVector.frobeniusPoly
theorem map_frobeniusPoly.key₁ (n j : ℕ) (hj : j < p ^ n) :
p ^ (n - v p ⟨j + 1, j.succ_pos⟩) ∣ (p ^ n).choose (j + 1) := by
apply multiplicity.pow_dvd_of_le_multiplicity
rw [hp.out.multiplicity_choose_prime_pow hj j.succ_ne_zero]
rfl
#align witt_vector.map_frobenius_poly.key₁ WittVector.map_frobeniusPoly.key₁
theorem map_frobeniusPoly.key₂ {n i j : ℕ} (hi : i ≤ n) (hj : j < p ^ (n - i)) :
j - v p ⟨j + 1, j.succ_pos⟩ + n = i + j + (n - i - v p ⟨j + 1, j.succ_pos⟩) := by
generalize h : v p ⟨j + 1, j.succ_pos⟩ = m
rsuffices ⟨h₁, h₂⟩ : m ≤ n - i ∧ m ≤ j
· rw [tsub_add_eq_add_tsub h₂, add_comm i j, add_tsub_assoc_of_le (h₁.trans (Nat.sub_le n i)),
add_assoc, tsub_right_comm, add_comm i,
tsub_add_cancel_of_le (le_tsub_of_add_le_right ((le_tsub_iff_left hi).mp h₁))]
have hle : p ^ m ≤ j + 1 := h ▸ Nat.le_of_dvd j.succ_pos (multiplicity.pow_multiplicity_dvd _)
exact ⟨(pow_le_pow_iff_right hp.1.one_lt).1 (hle.trans hj),
Nat.le_of_lt_succ ((Nat.lt_pow_self hp.1.one_lt m).trans_le hle)⟩
#align witt_vector.map_frobenius_poly.key₂ WittVector.map_frobeniusPoly.key₂
theorem map_frobeniusPoly (n : ℕ) :
MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n := by
rw [frobeniusPoly, RingHom.map_add, RingHom.map_mul, RingHom.map_pow, map_C, map_X, eq_intCast,
Int.cast_natCast, frobeniusPolyRat]
refine Nat.strong_induction_on n ?_; clear n
intro n IH
rw [xInTermsOfW_eq]
simp only [AlgHom.map_sum, AlgHom.map_sub, AlgHom.map_mul, AlgHom.map_pow, bind₁_C_right]
have h1 : (p : ℚ) ^ n * ⅟ (p : ℚ) ^ n = 1 := by rw [← mul_pow, mul_invOf_self, one_pow]
rw [bind₁_X_right, Function.comp_apply, wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ,
sum_range_succ, tsub_self, add_tsub_cancel_left, pow_zero, pow_one, pow_one, sub_mul, add_mul,
add_mul, mul_right_comm, mul_right_comm (C ((p : ℚ) ^ (n + 1))), ← C_mul, ← C_mul, pow_succ',
mul_assoc (p : ℚ) ((p : ℚ) ^ n), h1, mul_one, C_1, one_mul, add_comm _ (X n ^ p), add_assoc,
← add_sub, add_right_inj, frobeniusPolyAux_eq, RingHom.map_sub, map_X, mul_sub, sub_eq_add_neg,
add_comm _ (C (p : ℚ) * X (n + 1)), ← add_sub,
add_right_inj, neg_eq_iff_eq_neg, neg_sub, eq_comm]
simp only [map_sum, mul_sum, sum_mul, ← sum_sub_distrib]
apply sum_congr rfl
intro i hi
rw [mem_range] at hi
rw [← IH i hi]
clear IH
rw [add_comm (X i ^ p), add_pow, sum_range_succ', pow_zero, tsub_zero, Nat.choose_zero_right,
one_mul, Nat.cast_one, mul_one, mul_add, add_mul, Nat.succ_sub (le_of_lt hi),
Nat.succ_eq_add_one (n - i), pow_succ', pow_mul, add_sub_cancel_right, mul_sum, sum_mul]
apply sum_congr rfl
intro j hj
rw [mem_range] at hj
rw [RingHom.map_mul, RingHom.map_mul, RingHom.map_pow, RingHom.map_pow, RingHom.map_pow,
RingHom.map_pow, RingHom.map_pow, map_C, map_X, mul_pow]
rw [mul_comm (C (p : ℚ) ^ i), mul_comm _ ((X i ^ p) ^ _), mul_comm (C (p : ℚ) ^ (j + 1)),
mul_comm (C (p : ℚ))]
simp only [mul_assoc]
apply congr_arg
apply congr_arg
rw [← C_eq_coe_nat]
simp only [← RingHom.map_pow, ← C_mul]
rw [C_inj]
simp only [invOf_eq_inv, eq_intCast, inv_pow, Int.cast_natCast, Nat.cast_mul, Int.cast_mul]
rw [Rat.natCast_div _ _ (map_frobeniusPoly.key₁ p (n - i) j hj)]
simp only [Nat.cast_pow, pow_add, pow_one]
suffices
(((p ^ (n - i)).choose (j + 1): ℚ) * (p : ℚ) ^ (j - v p ⟨j + 1, j.succ_pos⟩) * ↑p * (p ^ n : ℚ))
= (p : ℚ) ^ j * p * ↑((p ^ (n - i)).choose (j + 1) * p ^ i) *
(p : ℚ) ^ (n - i - v p ⟨j + 1, j.succ_pos⟩) by
have aux : ∀ k : ℕ, (p : ℚ)^ k ≠ 0 := by
intro; apply pow_ne_zero; exact mod_cast hp.1.ne_zero
simpa [aux, -one_div, -pow_eq_zero_iff', field_simps] using this.symm
rw [mul_comm _ (p : ℚ), mul_assoc, mul_assoc, ← pow_add,
map_frobeniusPoly.key₂ p hi.le hj, Nat.cast_mul, Nat.cast_pow]
ring
#align witt_vector.map_frobenius_poly WittVector.map_frobeniusPoly
theorem frobeniusPoly_zmod (n : ℕ) :
MvPolynomial.map (Int.castRingHom (ZMod p)) (frobeniusPoly p n) = X n ^ p := by
rw [frobeniusPoly, RingHom.map_add, RingHom.map_pow, RingHom.map_mul, map_X, map_C]
simp only [Int.cast_natCast, add_zero, eq_intCast, ZMod.natCast_self, zero_mul, C_0]
#align witt_vector.frobenius_poly_zmod WittVector.frobeniusPoly_zmod
@[simp]
| Mathlib/RingTheory/WittVector/Frobenius.lean | 203 | 207 | theorem bind₁_frobeniusPoly_wittPolynomial (n : ℕ) :
bind₁ (frobeniusPoly p) (wittPolynomial p ℤ n) = wittPolynomial p ℤ (n + 1) := by |
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
simp only [map_bind₁, map_frobeniusPoly, bind₁_frobeniusPolyRat_wittPolynomial,
map_wittPolynomial]
|
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Init.Data.Bool.Lemmas
#align_import analysis.box_integral.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical Topology NNReal Filter Uniformity BoxIntegral
open Set Finset Function Filter Metric BoxIntegral.IntegrationParams
noncomputable section
namespace BoxIntegral
universe u v w
variable {ι : Type u} {E : Type v} {F : Type w} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] {I J : Box ι} {π : TaggedPrepartition I}
open TaggedPrepartition
local notation "ℝⁿ" => ι → ℝ
def integralSum (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) (π : TaggedPrepartition I) : F :=
∑ J ∈ π.boxes, vol J (f (π.tag J))
#align box_integral.integral_sum BoxIntegral.integralSum
theorem integralSum_biUnionTagged (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) (π : Prepartition I)
(πi : ∀ J, TaggedPrepartition J) :
integralSum f vol (π.biUnionTagged πi) = ∑ J ∈ π.boxes, integralSum f vol (πi J) := by
refine (π.sum_biUnion_boxes _ _).trans <| sum_congr rfl fun J hJ => sum_congr rfl fun J' hJ' => ?_
rw [π.tag_biUnionTagged hJ hJ']
#align box_integral.integral_sum_bUnion_tagged BoxIntegral.integralSum_biUnionTagged
theorem integralSum_biUnion_partition (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F)
(π : TaggedPrepartition I) (πi : ∀ J, Prepartition J) (hπi : ∀ J ∈ π, (πi J).IsPartition) :
integralSum f vol (π.biUnionPrepartition πi) = integralSum f vol π := by
refine (π.sum_biUnion_boxes _ _).trans (sum_congr rfl fun J hJ => ?_)
calc
(∑ J' ∈ (πi J).boxes, vol J' (f (π.tag <| π.toPrepartition.biUnionIndex πi J'))) =
∑ J' ∈ (πi J).boxes, vol J' (f (π.tag J)) :=
sum_congr rfl fun J' hJ' => by rw [Prepartition.biUnionIndex_of_mem _ hJ hJ']
_ = vol J (f (π.tag J)) :=
(vol.map ⟨⟨fun g : E →L[ℝ] F => g (f (π.tag J)), rfl⟩, fun _ _ => rfl⟩).sum_partition_boxes
le_top (hπi J hJ)
#align box_integral.integral_sum_bUnion_partition BoxIntegral.integralSum_biUnion_partition
theorem integralSum_inf_partition (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) (π : TaggedPrepartition I)
{π' : Prepartition I} (h : π'.IsPartition) :
integralSum f vol (π.infPrepartition π') = integralSum f vol π :=
integralSum_biUnion_partition f vol π _ fun _J hJ => h.restrict (Prepartition.le_of_mem _ hJ)
#align box_integral.integral_sum_inf_partition BoxIntegral.integralSum_inf_partition
theorem integralSum_fiberwise {α} (g : Box ι → α) (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F)
(π : TaggedPrepartition I) :
(∑ y ∈ π.boxes.image g, integralSum f vol (π.filter (g · = y))) = integralSum f vol π :=
π.sum_fiberwise g fun J => vol J (f <| π.tag J)
#align box_integral.integral_sum_fiberwise BoxIntegral.integralSum_fiberwise
theorem integralSum_sub_partitions (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F)
{π₁ π₂ : TaggedPrepartition I} (h₁ : π₁.IsPartition) (h₂ : π₂.IsPartition) :
integralSum f vol π₁ - integralSum f vol π₂ =
∑ J ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,
(vol J (f <| (π₁.infPrepartition π₂.toPrepartition).tag J) -
vol J (f <| (π₂.infPrepartition π₁.toPrepartition).tag J)) := by
rw [← integralSum_inf_partition f vol π₁ h₂, ← integralSum_inf_partition f vol π₂ h₁,
integralSum, integralSum, Finset.sum_sub_distrib]
simp only [infPrepartition_toPrepartition, inf_comm]
#align box_integral.integral_sum_sub_partitions BoxIntegral.integralSum_sub_partitions
@[simp]
theorem integralSum_disjUnion (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) {π₁ π₂ : TaggedPrepartition I}
(h : Disjoint π₁.iUnion π₂.iUnion) :
integralSum f vol (π₁.disjUnion π₂ h) = integralSum f vol π₁ + integralSum f vol π₂ := by
refine (Prepartition.sum_disj_union_boxes h _).trans
(congr_arg₂ (· + ·) (sum_congr rfl fun J hJ => ?_) (sum_congr rfl fun J hJ => ?_))
· rw [disjUnion_tag_of_mem_left _ hJ]
· rw [disjUnion_tag_of_mem_right _ hJ]
#align box_integral.integral_sum_disj_union BoxIntegral.integralSum_disjUnion
@[simp]
theorem integralSum_add (f g : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) (π : TaggedPrepartition I) :
integralSum (f + g) vol π = integralSum f vol π + integralSum g vol π := by
simp only [integralSum, Pi.add_apply, (vol _).map_add, Finset.sum_add_distrib]
#align box_integral.integral_sum_add BoxIntegral.integralSum_add
@[simp]
theorem integralSum_neg (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) (π : TaggedPrepartition I) :
integralSum (-f) vol π = -integralSum f vol π := by
simp only [integralSum, Pi.neg_apply, (vol _).map_neg, Finset.sum_neg_distrib]
#align box_integral.integral_sum_neg BoxIntegral.integralSum_neg
@[simp]
| Mathlib/Analysis/BoxIntegral/Basic.lean | 149 | 151 | theorem integralSum_smul (c : ℝ) (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) (π : TaggedPrepartition I) :
integralSum (c • f) vol π = c • integralSum f vol π := by |
simp only [integralSum, Finset.smul_sum, Pi.smul_apply, ContinuousLinearMap.map_smul]
|
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) : Measure α :=
Measure.ofMeasurable (fun s _ => ∫⁻ a in s, f a ∂μ) (by simp) fun s hs hd =>
lintegral_iUnion hs hd _
#align measure_theory.measure.with_density MeasureTheory.Measure.withDensity
@[simp]
theorem withDensity_apply (f : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) :
μ.withDensity f s = ∫⁻ a in s, f a ∂μ :=
Measure.ofMeasurable_apply s hs
#align measure_theory.with_density_apply MeasureTheory.withDensity_apply
theorem withDensity_apply_le (f : α → ℝ≥0∞) (s : Set α) :
∫⁻ a in s, f a ∂μ ≤ μ.withDensity f s := by
let t := toMeasurable (μ.withDensity f) s
calc
∫⁻ a in s, f a ∂μ ≤ ∫⁻ a in t, f a ∂μ :=
lintegral_mono_set (subset_toMeasurable (withDensity μ f) s)
_ = μ.withDensity f t :=
(withDensity_apply f (measurableSet_toMeasurable (withDensity μ f) s)).symm
_ = μ.withDensity f s := measure_toMeasurable s
theorem withDensity_apply' [SFinite μ] (f : α → ℝ≥0∞) (s : Set α) :
μ.withDensity f s = ∫⁻ a in s, f a ∂μ := by
apply le_antisymm ?_ (withDensity_apply_le f s)
let t := toMeasurable μ s
calc
μ.withDensity f s ≤ μ.withDensity f t := measure_mono (subset_toMeasurable μ s)
_ = ∫⁻ a in t, f a ∂μ := withDensity_apply f (measurableSet_toMeasurable μ s)
_ = ∫⁻ a in s, f a ∂μ := by congr 1; exact restrict_toMeasurable_of_sFinite s
@[simp]
lemma withDensity_zero_left (f : α → ℝ≥0∞) : (0 : Measure α).withDensity f = 0 := by
ext s hs
rw [withDensity_apply _ hs]
simp
theorem withDensity_congr_ae {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) :
μ.withDensity f = μ.withDensity g := by
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, withDensity_apply _ hs]
exact lintegral_congr_ae (ae_restrict_of_ae h)
#align measure_theory.with_density_congr_ae MeasureTheory.withDensity_congr_ae
lemma withDensity_mono {f g : α → ℝ≥0∞} (hfg : f ≤ᵐ[μ] g) :
μ.withDensity f ≤ μ.withDensity g := by
refine le_iff.2 fun s hs ↦ ?_
rw [withDensity_apply _ hs, withDensity_apply _ hs]
refine set_lintegral_mono_ae' hs ?_
filter_upwards [hfg] with x h_le using fun _ ↦ h_le
theorem withDensity_add_left {f : α → ℝ≥0∞} (hf : Measurable f) (g : α → ℝ≥0∞) :
μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g := by
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.add_apply, withDensity_apply _ hs, withDensity_apply _ hs,
← lintegral_add_left hf]
simp only [Pi.add_apply]
#align measure_theory.with_density_add_left MeasureTheory.withDensity_add_left
theorem withDensity_add_right (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : Measurable g) :
μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g := by
simpa only [add_comm] using withDensity_add_left hg f
#align measure_theory.with_density_add_right MeasureTheory.withDensity_add_right
theorem withDensity_add_measure {m : MeasurableSpace α} (μ ν : Measure α) (f : α → ℝ≥0∞) :
(μ + ν).withDensity f = μ.withDensity f + ν.withDensity f := by
ext1 s hs
simp only [withDensity_apply f hs, restrict_add, lintegral_add_measure, Measure.add_apply]
#align measure_theory.with_density_add_measure MeasureTheory.withDensity_add_measure
theorem withDensity_sum {ι : Type*} {m : MeasurableSpace α} (μ : ι → Measure α) (f : α → ℝ≥0∞) :
(sum μ).withDensity f = sum fun n => (μ n).withDensity f := by
ext1 s hs
simp_rw [sum_apply _ hs, withDensity_apply f hs, restrict_sum μ hs, lintegral_sum_measure]
#align measure_theory.with_density_sum MeasureTheory.withDensity_sum
theorem withDensity_smul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
μ.withDensity (r • f) = r • μ.withDensity f := by
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, ← lintegral_const_mul r hf]
simp only [Pi.smul_apply, smul_eq_mul]
#align measure_theory.with_density_smul MeasureTheory.withDensity_smul
theorem withDensity_smul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
μ.withDensity (r • f) = r • μ.withDensity f := by
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, ← lintegral_const_mul' r f hr]
simp only [Pi.smul_apply, smul_eq_mul]
#align measure_theory.with_density_smul' MeasureTheory.withDensity_smul'
theorem withDensity_smul_measure (r : ℝ≥0∞) (f : α → ℝ≥0∞) :
(r • μ).withDensity f = r • μ.withDensity f := by
ext s hs
rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, set_lintegral_smul_measure]
theorem isFiniteMeasure_withDensity {f : α → ℝ≥0∞} (hf : ∫⁻ a, f a ∂μ ≠ ∞) :
IsFiniteMeasure (μ.withDensity f) :=
{ measure_univ_lt_top := by
rwa [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ, lt_top_iff_ne_top] }
#align measure_theory.is_finite_measure_with_density MeasureTheory.isFiniteMeasure_withDensity
theorem withDensity_absolutelyContinuous {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) :
μ.withDensity f ≪ μ := by
refine AbsolutelyContinuous.mk fun s hs₁ hs₂ => ?_
rw [withDensity_apply _ hs₁]
exact set_lintegral_measure_zero _ _ hs₂
#align measure_theory.with_density_absolutely_continuous MeasureTheory.withDensity_absolutelyContinuous
@[simp]
theorem withDensity_zero : μ.withDensity 0 = 0 := by
ext1 s hs
simp [withDensity_apply _ hs]
#align measure_theory.with_density_zero MeasureTheory.withDensity_zero
@[simp]
theorem withDensity_one : μ.withDensity 1 = μ := by
ext1 s hs
simp [withDensity_apply _ hs]
#align measure_theory.with_density_one MeasureTheory.withDensity_one
@[simp]
theorem withDensity_const (c : ℝ≥0∞) : μ.withDensity (fun _ ↦ c) = c • μ := by
ext1 s hs
simp [withDensity_apply _ hs]
theorem withDensity_tsum {f : ℕ → α → ℝ≥0∞} (h : ∀ i, Measurable (f i)) :
μ.withDensity (∑' n, f n) = sum fun n => μ.withDensity (f n) := by
ext1 s hs
simp_rw [sum_apply _ hs, withDensity_apply _ hs]
change ∫⁻ x in s, (∑' n, f n) x ∂μ = ∑' i : ℕ, ∫⁻ x, f i x ∂μ.restrict s
rw [← lintegral_tsum fun i => (h i).aemeasurable]
exact lintegral_congr fun x => tsum_apply (Pi.summable.2 fun _ => ENNReal.summable)
#align measure_theory.with_density_tsum MeasureTheory.withDensity_tsum
theorem withDensity_indicator {s : Set α} (hs : MeasurableSet s) (f : α → ℝ≥0∞) :
μ.withDensity (s.indicator f) = (μ.restrict s).withDensity f := by
ext1 t ht
rw [withDensity_apply _ ht, lintegral_indicator _ hs, restrict_comm hs, ←
withDensity_apply _ ht]
#align measure_theory.with_density_indicator MeasureTheory.withDensity_indicator
theorem withDensity_indicator_one {s : Set α} (hs : MeasurableSet s) :
μ.withDensity (s.indicator 1) = μ.restrict s := by
rw [withDensity_indicator hs, withDensity_one]
#align measure_theory.with_density_indicator_one MeasureTheory.withDensity_indicator_one
| Mathlib/MeasureTheory/Measure/WithDensity.lean | 195 | 206 | theorem withDensity_ofReal_mutuallySingular {f : α → ℝ} (hf : Measurable f) :
(μ.withDensity fun x => ENNReal.ofReal <| f x) ⟂ₘ
μ.withDensity fun x => ENNReal.ofReal <| -f x := by |
set S : Set α := { x | f x < 0 }
have hS : MeasurableSet S := measurableSet_lt hf measurable_const
refine ⟨S, hS, ?_, ?_⟩
· rw [withDensity_apply _ hS, lintegral_eq_zero_iff hf.ennreal_ofReal, EventuallyEq]
exact (ae_restrict_mem hS).mono fun x hx => ENNReal.ofReal_eq_zero.2 (le_of_lt hx)
· rw [withDensity_apply _ hS.compl, lintegral_eq_zero_iff hf.neg.ennreal_ofReal, EventuallyEq]
exact
(ae_restrict_mem hS.compl).mono fun x hx =>
ENNReal.ofReal_eq_zero.2 (not_lt.1 <| mt neg_pos.1 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/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
variable {𝓕 𝕜 α ι κ E F G : Type*}
open Filter Function Metric Bornology
open ENNReal Filter NNReal Uniformity Pointwise Topology
@[notation_class]
class Norm (E : Type*) where
norm : E → ℝ
#align has_norm Norm
@[notation_class]
class NNNorm (E : Type*) where
nnnorm : E → ℝ≥0
#align has_nnnorm NNNorm
export Norm (norm)
export NNNorm (nnnorm)
@[inherit_doc]
notation "‖" e "‖" => norm e
@[inherit_doc]
notation "‖" e "‖₊" => nnnorm e
class SeminormedAddGroup (E : Type*) extends Norm E, AddGroup E, PseudoMetricSpace E where
dist := fun x y => ‖x - y‖
dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
#align seminormed_add_group SeminormedAddGroup
@[to_additive]
class SeminormedGroup (E : Type*) extends Norm E, Group E, PseudoMetricSpace E where
dist := fun x y => ‖x / y‖
dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
#align seminormed_group SeminormedGroup
class NormedAddGroup (E : Type*) extends Norm E, AddGroup E, MetricSpace E where
dist := fun x y => ‖x - y‖
dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
#align normed_add_group NormedAddGroup
@[to_additive]
class NormedGroup (E : Type*) extends Norm E, Group E, MetricSpace E where
dist := fun x y => ‖x / y‖
dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
#align normed_group NormedGroup
class SeminormedAddCommGroup (E : Type*) extends Norm E, AddCommGroup E,
PseudoMetricSpace E where
dist := fun x y => ‖x - y‖
dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
#align seminormed_add_comm_group SeminormedAddCommGroup
@[to_additive]
class SeminormedCommGroup (E : Type*) extends Norm E, CommGroup E, PseudoMetricSpace E where
dist := fun x y => ‖x / y‖
dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
#align seminormed_comm_group SeminormedCommGroup
class NormedAddCommGroup (E : Type*) extends Norm E, AddCommGroup E, MetricSpace E where
dist := fun x y => ‖x - y‖
dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
#align normed_add_comm_group NormedAddCommGroup
@[to_additive]
class NormedCommGroup (E : Type*) extends Norm E, CommGroup E, MetricSpace E where
dist := fun x y => ‖x / y‖
dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
#align normed_comm_group NormedCommGroup
-- See note [lower instance priority]
@[to_additive]
instance (priority := 100) NormedGroup.toSeminormedGroup [NormedGroup E] : SeminormedGroup E :=
{ ‹NormedGroup E› with }
#align normed_group.to_seminormed_group NormedGroup.toSeminormedGroup
#align normed_add_group.to_seminormed_add_group NormedAddGroup.toSeminormedAddGroup
-- See note [lower instance priority]
@[to_additive]
instance (priority := 100) NormedCommGroup.toSeminormedCommGroup [NormedCommGroup E] :
SeminormedCommGroup E :=
{ ‹NormedCommGroup E› with }
#align normed_comm_group.to_seminormed_comm_group NormedCommGroup.toSeminormedCommGroup
#align normed_add_comm_group.to_seminormed_add_comm_group NormedAddCommGroup.toSeminormedAddCommGroup
-- See note [lower instance priority]
@[to_additive]
instance (priority := 100) SeminormedCommGroup.toSeminormedGroup [SeminormedCommGroup E] :
SeminormedGroup E :=
{ ‹SeminormedCommGroup E› with }
#align seminormed_comm_group.to_seminormed_group SeminormedCommGroup.toSeminormedGroup
#align seminormed_add_comm_group.to_seminormed_add_group SeminormedAddCommGroup.toSeminormedAddGroup
-- See note [lower instance priority]
@[to_additive]
instance (priority := 100) NormedCommGroup.toNormedGroup [NormedCommGroup E] : NormedGroup E :=
{ ‹NormedCommGroup E› with }
#align normed_comm_group.to_normed_group NormedCommGroup.toNormedGroup
#align normed_add_comm_group.to_normed_add_group NormedAddCommGroup.toNormedAddGroup
-- See note [reducible non-instances]
@[to_additive (attr := reducible) "Construct a `NormedAddGroup` from a `SeminormedAddGroup`
satisfying `∀ x, ‖x‖ = 0 → x = 0`. This avoids having to go back to the `(Pseudo)MetricSpace`
level when declaring a `NormedAddGroup` instance as a special case of a more general
`SeminormedAddGroup` instance."]
def NormedGroup.ofSeparation [SeminormedGroup E] (h : ∀ x : E, ‖x‖ = 0 → x = 1) :
NormedGroup E where
dist_eq := ‹SeminormedGroup E›.dist_eq
toMetricSpace :=
{ eq_of_dist_eq_zero := fun hxy =>
div_eq_one.1 <| h _ <| by exact (‹SeminormedGroup E›.dist_eq _ _).symm.trans hxy }
-- Porting note: the `rwa` no longer worked, but it was easy enough to provide the term.
-- however, notice that if you make `x` and `y` accessible, then the following does work:
-- `have := ‹SeminormedGroup E›.dist_eq x y; rwa [← this]`, so I'm not sure why the `rwa`
-- was broken.
#align normed_group.of_separation NormedGroup.ofSeparation
#align normed_add_group.of_separation NormedAddGroup.ofSeparation
-- See note [reducible non-instances]
@[to_additive (attr := reducible) "Construct a `NormedAddCommGroup` from a
`SeminormedAddCommGroup` satisfying `∀ x, ‖x‖ = 0 → x = 0`. This avoids having to go back to the
`(Pseudo)MetricSpace` level when declaring a `NormedAddCommGroup` instance as a special case
of a more general `SeminormedAddCommGroup` instance."]
def NormedCommGroup.ofSeparation [SeminormedCommGroup E] (h : ∀ x : E, ‖x‖ = 0 → x = 1) :
NormedCommGroup E :=
{ ‹SeminormedCommGroup E›, NormedGroup.ofSeparation h with }
#align normed_comm_group.of_separation NormedCommGroup.ofSeparation
#align normed_add_comm_group.of_separation NormedAddCommGroup.ofSeparation
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a translation-invariant distance."]
def SeminormedGroup.ofMulDist [Norm E] [Group E] [PseudoMetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
SeminormedGroup E where
dist_eq x y := by
rw [h₁]; apply le_antisymm
· simpa only [div_eq_mul_inv, ← mul_right_inv y] using h₂ _ _ _
· simpa only [div_mul_cancel, one_mul] using h₂ (x / y) 1 y
#align seminormed_group.of_mul_dist SeminormedGroup.ofMulDist
#align seminormed_add_group.of_add_dist SeminormedAddGroup.ofAddDist
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a translation-invariant pseudodistance."]
def SeminormedGroup.ofMulDist' [Norm E] [Group E] [PseudoMetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
SeminormedGroup E where
dist_eq x y := by
rw [h₁]; apply le_antisymm
· simpa only [div_mul_cancel, one_mul] using h₂ (x / y) 1 y
· simpa only [div_eq_mul_inv, ← mul_right_inv y] using h₂ _ _ _
#align seminormed_group.of_mul_dist' SeminormedGroup.ofMulDist'
#align seminormed_add_group.of_add_dist' SeminormedAddGroup.ofAddDist'
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a translation-invariant pseudodistance."]
def SeminormedCommGroup.ofMulDist [Norm E] [CommGroup E] [PseudoMetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
SeminormedCommGroup E :=
{ SeminormedGroup.ofMulDist h₁ h₂ with
mul_comm := mul_comm }
#align seminormed_comm_group.of_mul_dist SeminormedCommGroup.ofMulDist
#align seminormed_add_comm_group.of_add_dist SeminormedAddCommGroup.ofAddDist
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a translation-invariant pseudodistance."]
def SeminormedCommGroup.ofMulDist' [Norm E] [CommGroup E] [PseudoMetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
SeminormedCommGroup E :=
{ SeminormedGroup.ofMulDist' h₁ h₂ with
mul_comm := mul_comm }
#align seminormed_comm_group.of_mul_dist' SeminormedCommGroup.ofMulDist'
#align seminormed_add_comm_group.of_add_dist' SeminormedAddCommGroup.ofAddDist'
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a translation-invariant distance."]
def NormedGroup.ofMulDist [Norm E] [Group E] [MetricSpace E] (h₁ : ∀ x : E, ‖x‖ = dist x 1)
(h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) : NormedGroup E :=
{ SeminormedGroup.ofMulDist h₁ h₂ with
eq_of_dist_eq_zero := eq_of_dist_eq_zero }
#align normed_group.of_mul_dist NormedGroup.ofMulDist
#align normed_add_group.of_add_dist NormedAddGroup.ofAddDist
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a translation-invariant pseudodistance."]
def NormedGroup.ofMulDist' [Norm E] [Group E] [MetricSpace E] (h₁ : ∀ x : E, ‖x‖ = dist x 1)
(h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) : NormedGroup E :=
{ SeminormedGroup.ofMulDist' h₁ h₂ with
eq_of_dist_eq_zero := eq_of_dist_eq_zero }
#align normed_group.of_mul_dist' NormedGroup.ofMulDist'
#align normed_add_group.of_add_dist' NormedAddGroup.ofAddDist'
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a translation-invariant pseudodistance."]
def NormedCommGroup.ofMulDist [Norm E] [CommGroup E] [MetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
NormedCommGroup E :=
{ NormedGroup.ofMulDist h₁ h₂ with
mul_comm := mul_comm }
#align normed_comm_group.of_mul_dist NormedCommGroup.ofMulDist
#align normed_add_comm_group.of_add_dist NormedAddCommGroup.ofAddDist
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a translation-invariant pseudodistance."]
def NormedCommGroup.ofMulDist' [Norm E] [CommGroup E] [MetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
NormedCommGroup E :=
{ NormedGroup.ofMulDist' h₁ h₂ with
mul_comm := mul_comm }
#align normed_comm_group.of_mul_dist' NormedCommGroup.ofMulDist'
#align normed_add_comm_group.of_add_dist' NormedAddCommGroup.ofAddDist'
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a seminorm, i.e., registering the pseudodistance
and the pseudometric space structure from the seminorm properties. Note that in most cases this
instance creates bad definitional equalities (e.g., it does not take into account a possibly
existing `UniformSpace` instance on `E`)."]
def GroupSeminorm.toSeminormedGroup [Group E] (f : GroupSeminorm E) : SeminormedGroup E where
dist x y := f (x / y)
norm := f
dist_eq x y := rfl
dist_self x := by simp only [div_self', map_one_eq_zero]
dist_triangle := le_map_div_add_map_div f
dist_comm := map_div_rev f
edist_dist x y := by exact ENNReal.coe_nnreal_eq _
-- Porting note: how did `mathlib3` solve this automatically?
#align group_seminorm.to_seminormed_group GroupSeminorm.toSeminormedGroup
#align add_group_seminorm.to_seminormed_add_group AddGroupSeminorm.toSeminormedAddGroup
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a seminorm, i.e., registering the pseudodistance
and the pseudometric space structure from the seminorm properties. Note that in most cases this
instance creates bad definitional equalities (e.g., it does not take into account a possibly
existing `UniformSpace` instance on `E`)."]
def GroupSeminorm.toSeminormedCommGroup [CommGroup E] (f : GroupSeminorm E) :
SeminormedCommGroup E :=
{ f.toSeminormedGroup with
mul_comm := mul_comm }
#align group_seminorm.to_seminormed_comm_group GroupSeminorm.toSeminormedCommGroup
#align add_group_seminorm.to_seminormed_add_comm_group AddGroupSeminorm.toSeminormedAddCommGroup
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a norm, i.e., registering the distance and the metric
space structure from the norm properties. Note that in most cases this instance creates bad
definitional equalities (e.g., it does not take into account a possibly existing `UniformSpace`
instance on `E`)."]
def GroupNorm.toNormedGroup [Group E] (f : GroupNorm E) : NormedGroup E :=
{ f.toGroupSeminorm.toSeminormedGroup with
eq_of_dist_eq_zero := fun h => div_eq_one.1 <| eq_one_of_map_eq_zero f h }
#align group_norm.to_normed_group GroupNorm.toNormedGroup
#align add_group_norm.to_normed_add_group AddGroupNorm.toNormedAddGroup
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a norm, i.e., registering the distance and the metric
space structure from the norm properties. Note that in most cases this instance creates bad
definitional equalities (e.g., it does not take into account a possibly existing `UniformSpace`
instance on `E`)."]
def GroupNorm.toNormedCommGroup [CommGroup E] (f : GroupNorm E) : NormedCommGroup E :=
{ f.toNormedGroup with
mul_comm := mul_comm }
#align group_norm.to_normed_comm_group GroupNorm.toNormedCommGroup
#align add_group_norm.to_normed_add_comm_group AddGroupNorm.toNormedAddCommGroup
instance PUnit.normedAddCommGroup : NormedAddCommGroup PUnit where
norm := Function.const _ 0
dist_eq _ _ := rfl
@[simp]
theorem PUnit.norm_eq_zero (r : PUnit) : ‖r‖ = 0 :=
rfl
#align punit.norm_eq_zero PUnit.norm_eq_zero
section SeminormedGroup
variable [SeminormedGroup E] [SeminormedGroup F] [SeminormedGroup G] {s : Set E}
{a a₁ a₂ b b₁ b₂ : E} {r r₁ r₂ : ℝ}
@[to_additive]
theorem dist_eq_norm_div (a b : E) : dist a b = ‖a / b‖ :=
SeminormedGroup.dist_eq _ _
#align dist_eq_norm_div dist_eq_norm_div
#align dist_eq_norm_sub dist_eq_norm_sub
@[to_additive]
theorem dist_eq_norm_div' (a b : E) : dist a b = ‖b / a‖ := by rw [dist_comm, dist_eq_norm_div]
#align dist_eq_norm_div' dist_eq_norm_div'
#align dist_eq_norm_sub' dist_eq_norm_sub'
alias dist_eq_norm := dist_eq_norm_sub
#align dist_eq_norm dist_eq_norm
alias dist_eq_norm' := dist_eq_norm_sub'
#align dist_eq_norm' dist_eq_norm'
@[to_additive]
instance NormedGroup.to_isometricSMul_right : IsometricSMul Eᵐᵒᵖ E :=
⟨fun a => Isometry.of_dist_eq fun b c => by simp [dist_eq_norm_div]⟩
#align normed_group.to_has_isometric_smul_right NormedGroup.to_isometricSMul_right
#align normed_add_group.to_has_isometric_vadd_right NormedAddGroup.to_isometricVAdd_right
@[to_additive (attr := simp)]
theorem dist_one_right (a : E) : dist a 1 = ‖a‖ := by rw [dist_eq_norm_div, div_one]
#align dist_one_right dist_one_right
#align dist_zero_right dist_zero_right
@[to_additive]
theorem inseparable_one_iff_norm {a : E} : Inseparable a 1 ↔ ‖a‖ = 0 := by
rw [Metric.inseparable_iff, dist_one_right]
@[to_additive (attr := simp)]
theorem dist_one_left : dist (1 : E) = norm :=
funext fun a => by rw [dist_comm, dist_one_right]
#align dist_one_left dist_one_left
#align dist_zero_left dist_zero_left
@[to_additive]
| Mathlib/Analysis/Normed/Group/Basic.lean | 439 | 440 | theorem Isometry.norm_map_of_map_one {f : E → F} (hi : Isometry f) (h₁ : f 1 = 1) (x : E) :
‖f x‖ = ‖x‖ := by | rw [← dist_one_right, ← h₁, hi.dist_eq, dist_one_right]
|
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Order.Interval.Finset.Nat
#align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bfb4330ddf6624f1028ba186117d82"
noncomputable section
open Polynomial
open Finset
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Type v} {T : Type w} {A : Type z} {a b : R} {n : ℕ}
section Semiring
variable [Semiring R] {p q : R[X]}
def divX (p : R[X]) : R[X] :=
⟨AddMonoidAlgebra.divOf p.toFinsupp 1⟩
set_option linter.uppercaseLean3 false in
#align polynomial.div_X Polynomial.divX
@[simp]
theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) := by
rw [add_comm]; cases p; rfl
set_option linter.uppercaseLean3 false in
#align polynomial.coeff_div_X Polynomial.coeff_divX
theorem divX_mul_X_add (p : R[X]) : divX p * X + C (p.coeff 0) = p :=
ext <| by rintro ⟨_ | _⟩ <;> simp [coeff_C, Nat.succ_ne_zero, coeff_mul_X]
set_option linter.uppercaseLean3 false in
#align polynomial.div_X_mul_X_add Polynomial.divX_mul_X_add
@[simp]
theorem X_mul_divX_add (p : R[X]) : X * divX p + C (p.coeff 0) = p :=
ext <| by rintro ⟨_ | _⟩ <;> simp [coeff_C, Nat.succ_ne_zero, coeff_mul_X]
@[simp]
theorem divX_C (a : R) : divX (C a) = 0 :=
ext fun n => by simp [coeff_divX, coeff_C, Finsupp.single_eq_of_ne _]
set_option linter.uppercaseLean3 false in
#align polynomial.div_X_C Polynomial.divX_C
theorem divX_eq_zero_iff : divX p = 0 ↔ p = C (p.coeff 0) :=
⟨fun h => by simpa [eq_comm, h] using divX_mul_X_add p, fun h => by rw [h, divX_C]⟩
set_option linter.uppercaseLean3 false in
#align polynomial.div_X_eq_zero_iff Polynomial.divX_eq_zero_iff
theorem divX_add : divX (p + q) = divX p + divX q :=
ext <| by simp
set_option linter.uppercaseLean3 false in
#align polynomial.div_X_add Polynomial.divX_add
@[simp]
theorem divX_zero : divX (0 : R[X]) = 0 := leadingCoeff_eq_zero.mp rfl
@[simp]
theorem divX_one : divX (1 : R[X]) = 0 := by
ext
simpa only [coeff_divX, coeff_zero] using coeff_one
@[simp]
theorem divX_C_mul : divX (C a * p) = C a * divX p := by
ext
simp
| Mathlib/Algebra/Polynomial/Inductions.lean | 88 | 92 | theorem divX_X_pow : divX (X ^ n : R[X]) = if (n = 0) then 0 else X ^ (n - 1) := by |
cases n
· simp
· ext n
simp [coeff_X_pow]
|
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics ContinuousLinearMap Filter
open scoped Classical Topology NNReal
theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b))
-- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
(hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by
change Icc a b ⊆ { x | f x ≤ B x }
set s := { x | f x ≤ B x } ∩ Icc a b
have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
have : IsClosed s := by
simp only [s, inter_comm]
exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
apply this.Icc_subset_of_forall_exists_gt ha
rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy
cases' hxB.lt_or_eq with hxB hxB
· -- If `f x < B x`, then all we need is continuity of both sides
refine nonempty_of_mem (inter_mem ?_ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))
have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x :=
A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB)
have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this
exact this.mono fun y => le_of_lt
· rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩
specialize hf' x xab r hfr
have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z :=
(hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici
(Ioi_mem_nhds hrB)
obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y :=
(hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists
refine ⟨z, ?_, hz⟩
have := (hfz.trans hzB).le
rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB,
sub_le_sub_iff_right] at this
#align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary'
theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b))
-- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x)
(bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha
(fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound
#align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary
theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
(hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
-- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x`
(bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) :
∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by
have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by
apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound
· rwa [sub_self, mul_zero, add_zero]
· exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const))
· intro x hx
exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r)
· intro x _ _
rw [mul_one]
exact (lt_add_iff_pos_right _).2 hr
exact hx
intro x hx
have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 :=
continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const)
convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp
#align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary
theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
(hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
image_le_of_liminf_slope_right_lt_deriv_boundary' hf
(fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound
#align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary'
theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x)
(bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha
(fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound
#align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary
theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
(hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr =>
(hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr)
#align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary
section
variable {f : ℝ → E} {a b : ℝ}
theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*}
[NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b))
-- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r)
{B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b))
(hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x :=
image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB
hB' bound
#align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary
theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E}
(hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b))
(hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x :=
image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf
(fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound
#align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary'
theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E}
(hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x)
(bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x :=
image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha
(fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound
#align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary
theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E}
(hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b))
(hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x :=
image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB'
fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr)
#align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary'
theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E}
(hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
{B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x)
(bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x :=
image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha
(fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound
#align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary
theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ}
(hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by
let g x := f x - f a
have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const
have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by
intro x hx
simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _)
let B x := C * (x - a)
have hB : ∀ x, HasDerivAt B C x := by
intro x
simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a))
convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound
simp only [g, B]; rw [sub_self, norm_zero, sub_self, mul_zero]
#align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment
theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ}
(hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x)
(bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by
refine
norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt)
(fun x hx => ?_) bound
exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx)
#align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment'
theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b))
(bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) :
∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by
refine norm_image_sub_le_of_norm_deriv_le_segment' ?_ bound
exact fun x hx => (hf x hx).hasDerivWithinAt
#align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment
theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ}
(hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x)
(bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by
simpa only [sub_zero, mul_one] using
norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one)
#align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01'
theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ}
(hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1))
(bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by
simpa only [sub_zero, mul_one] using
norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one)
#align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01
theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b))
(hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by
have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx =>
norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx
simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this
#align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero
theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b))
(hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by
have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by
simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx
simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx =>
norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx
#align constant_of_deriv_within_zero constant_of_derivWithin_zero
variable {f' g : ℝ → E}
theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
(derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b))
(gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by
simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢
exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by
simpa only [sub_self] using (derivf y hy).sub (derivg y hy)
#align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq
| Mathlib/Analysis/Calculus/MeanValue.lean | 423 | 433 | theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b))
(gdiff : DifferentiableOn ℝ g (Icc a b))
(hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) :
∀ y ∈ Icc a b, f y = g y := by |
have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy =>
(fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy)
have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy =>
(gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy)
exact
eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn
gdiff.continuousOn hi
|
import Mathlib.FieldTheory.Extension
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.GroupTheory.Solvable
#align_import field_theory.normal from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
noncomputable section
open scoped Classical Polynomial
open Polynomial IsScalarTower
variable (F K : Type*) [Field F] [Field K] [Algebra F K]
class Normal extends Algebra.IsAlgebraic F K : Prop where
splits' (x : K) : Splits (algebraMap F K) (minpoly F x)
#align normal Normal
variable {F K}
theorem Normal.isIntegral (_ : Normal F K) (x : K) : IsIntegral F x :=
Algebra.IsIntegral.isIntegral x
#align normal.is_integral Normal.isIntegral
theorem Normal.splits (_ : Normal F K) (x : K) : Splits (algebraMap F K) (minpoly F x) :=
Normal.splits' x
#align normal.splits Normal.splits
theorem normal_iff : Normal F K ↔ ∀ x : K, IsIntegral F x ∧ Splits (algebraMap F K) (minpoly F x) :=
⟨fun h x => ⟨h.isIntegral x, h.splits x⟩, fun h =>
{ isAlgebraic := fun x => (h x).1.isAlgebraic
splits' := fun x => (h x).2 }⟩
#align normal_iff normal_iff
theorem Normal.out : Normal F K → ∀ x : K, IsIntegral F x ∧ Splits (algebraMap F K) (minpoly F x) :=
normal_iff.1
#align normal.out Normal.out
variable (F K)
instance normal_self : Normal F F where
isAlgebraic := fun _ => isIntegral_algebraMap.isAlgebraic
splits' := fun x => (minpoly.eq_X_sub_C' x).symm ▸ splits_X_sub_C _
#align normal_self normal_self
theorem Normal.exists_isSplittingField [h : Normal F K] [FiniteDimensional F K] :
∃ p : F[X], IsSplittingField F K p := by
let s := Basis.ofVectorSpace F K
refine
⟨∏ x, minpoly F (s x), splits_prod _ fun x _ => h.splits (s x),
Subalgebra.toSubmodule.injective ?_⟩
rw [Algebra.top_toSubmodule, eq_top_iff, ← s.span_eq, Submodule.span_le, Set.range_subset_iff]
refine fun x =>
Algebra.subset_adjoin
(Multiset.mem_toFinset.mpr <|
(mem_roots <|
mt (Polynomial.map_eq_zero <| algebraMap F K).1 <|
Finset.prod_ne_zero_iff.2 fun x _ => ?_).2 ?_)
· exact minpoly.ne_zero (h.isIntegral (s x))
rw [IsRoot.def, eval_map, ← aeval_def, AlgHom.map_prod]
exact Finset.prod_eq_zero (Finset.mem_univ _) (minpoly.aeval _ _)
#align normal.exists_is_splitting_field Normal.exists_isSplittingField
section NormalTower
variable (E : Type*) [Field E] [Algebra F E] [Algebra K E] [IsScalarTower F K E]
theorem Normal.tower_top_of_normal [h : Normal F E] : Normal K E :=
normal_iff.2 fun x => by
cases' h.out x with hx hhx
rw [algebraMap_eq F K E] at hhx
exact
⟨hx.tower_top,
Polynomial.splits_of_splits_of_dvd (algebraMap K E)
(Polynomial.map_ne_zero (minpoly.ne_zero hx))
((Polynomial.splits_map_iff (algebraMap F K) (algebraMap K E)).mpr hhx)
(minpoly.dvd_map_of_isScalarTower F K x)⟩
#align normal.tower_top_of_normal Normal.tower_top_of_normal
theorem AlgHom.normal_bijective [h : Normal F E] (ϕ : E →ₐ[F] K) : Function.Bijective ϕ :=
h.toIsAlgebraic.bijective_of_isScalarTower' ϕ
#align alg_hom.normal_bijective AlgHom.normal_bijective
-- Porting note: `[Field F] [Field E] [Algebra F E]` added by hand.
variable {F E} {E' : Type*} [Field F] [Field E] [Algebra F E] [Field E'] [Algebra F E']
| Mathlib/FieldTheory/Normal.lean | 107 | 111 | theorem Normal.of_algEquiv [h : Normal F E] (f : E ≃ₐ[F] E') : Normal F E' := by |
rw [normal_iff] at h ⊢
intro x; specialize h (f.symm x)
rw [← f.apply_symm_apply x, minpoly.algEquiv_eq, ← f.toAlgHom.comp_algebraMap]
exact ⟨h.1.map f, splits_comp_of_splits _ _ h.2⟩
|
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104e7bbb471381592"
variable {ι α β γ : Type*} {π : ι → Type*}
namespace Set
def WellFoundedOn (s : Set α) (r : α → α → Prop) : Prop :=
WellFounded fun a b : s => r a b
#align set.well_founded_on Set.WellFoundedOn
@[simp]
theorem wellFoundedOn_empty (r : α → α → Prop) : WellFoundedOn ∅ r :=
wellFounded_of_isEmpty _
#align set.well_founded_on_empty Set.wellFoundedOn_empty
def PartiallyWellOrderedOn (s : Set α) (r : α → α → Prop) : Prop :=
∀ f : ℕ → α, (∀ n, f n ∈ s) → ∃ m n : ℕ, m < n ∧ r (f m) (f n)
#align set.partially_well_ordered_on Set.PartiallyWellOrderedOn
section PartiallyWellOrderedOn
variable {r : α → α → Prop} {r' : β → β → Prop} {f : α → β} {s : Set α} {t : Set α} {a : α}
theorem PartiallyWellOrderedOn.mono (ht : t.PartiallyWellOrderedOn r) (h : s ⊆ t) :
s.PartiallyWellOrderedOn r := fun f hf => ht f fun n => h <| hf n
#align set.partially_well_ordered_on.mono Set.PartiallyWellOrderedOn.mono
@[simp]
theorem partiallyWellOrderedOn_empty (r : α → α → Prop) : PartiallyWellOrderedOn ∅ r := fun _ h =>
(h 0).elim
#align set.partially_well_ordered_on_empty Set.partiallyWellOrderedOn_empty
theorem PartiallyWellOrderedOn.union (hs : s.PartiallyWellOrderedOn r)
(ht : t.PartiallyWellOrderedOn r) : (s ∪ t).PartiallyWellOrderedOn r := by
rintro f hf
rcases Nat.exists_subseq_of_forall_mem_union f hf with ⟨g, hgs | hgt⟩
· rcases hs _ hgs with ⟨m, n, hlt, hr⟩
exact ⟨g m, g n, g.strictMono hlt, hr⟩
· rcases ht _ hgt with ⟨m, n, hlt, hr⟩
exact ⟨g m, g n, g.strictMono hlt, hr⟩
#align set.partially_well_ordered_on.union Set.PartiallyWellOrderedOn.union
@[simp]
theorem partiallyWellOrderedOn_union :
(s ∪ t).PartiallyWellOrderedOn r ↔ s.PartiallyWellOrderedOn r ∧ t.PartiallyWellOrderedOn r :=
⟨fun h => ⟨h.mono subset_union_left, h.mono subset_union_right⟩, fun h =>
h.1.union h.2⟩
#align set.partially_well_ordered_on_union Set.partiallyWellOrderedOn_union
theorem PartiallyWellOrderedOn.image_of_monotone_on (hs : s.PartiallyWellOrderedOn r)
(hf : ∀ a₁ ∈ s, ∀ a₂ ∈ s, r a₁ a₂ → r' (f a₁) (f a₂)) : (f '' s).PartiallyWellOrderedOn r' := by
intro g' hg'
choose g hgs heq using hg'
obtain rfl : f ∘ g = g' := funext heq
obtain ⟨m, n, hlt, hmn⟩ := hs g hgs
exact ⟨m, n, hlt, hf _ (hgs m) _ (hgs n) hmn⟩
#align set.partially_well_ordered_on.image_of_monotone_on Set.PartiallyWellOrderedOn.image_of_monotone_on
| Mathlib/Order/WellFoundedSet.lean | 312 | 317 | theorem _root_.IsAntichain.finite_of_partiallyWellOrderedOn (ha : IsAntichain r s)
(hp : s.PartiallyWellOrderedOn r) : s.Finite := by |
refine not_infinite.1 fun hi => ?_
obtain ⟨m, n, hmn, h⟩ := hp (fun n => hi.natEmbedding _ n) fun n => (hi.natEmbedding _ n).2
exact hmn.ne ((hi.natEmbedding _).injective <| Subtype.val_injective <|
ha.eq (hi.natEmbedding _ m).2 (hi.natEmbedding _ n).2 h)
|
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Measure.Haar.Quotient
import Mathlib.MeasureTheory.Constructions.Polish
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Topology.Algebra.Order.Floor
#align_import measure_theory.integral.periodic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Function MeasureTheory MeasureTheory.Measure TopologicalSpace AddSubgroup intervalIntegral
open scoped MeasureTheory NNReal ENNReal
@[measurability]
protected theorem AddCircle.measurable_mk' {a : ℝ} :
Measurable (β := AddCircle a) ((↑) : ℝ → AddCircle a) :=
Continuous.measurable <| AddCircle.continuous_mk' a
#align add_circle.measurable_mk' AddCircle.measurable_mk'
theorem isAddFundamentalDomain_Ioc {T : ℝ} (hT : 0 < T) (t : ℝ)
(μ : Measure ℝ := by volume_tac) :
IsAddFundamentalDomain (AddSubgroup.zmultiples T) (Ioc t (t + T)) μ := by
refine IsAddFundamentalDomain.mk' measurableSet_Ioc.nullMeasurableSet fun x => ?_
have : Bijective (codRestrict (fun n : ℤ => n • T) (AddSubgroup.zmultiples T) _) :=
(Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_strictMono_left hT).injective).bijective
refine this.existsUnique_iff.2 ?_
simpa only [add_comm x] using existsUnique_add_zsmul_mem_Ioc hT x t
#align is_add_fundamental_domain_Ioc isAddFundamentalDomain_Ioc
theorem isAddFundamentalDomain_Ioc' {T : ℝ} (hT : 0 < T) (t : ℝ) (μ : Measure ℝ := by volume_tac) :
IsAddFundamentalDomain (AddSubgroup.op <| .zmultiples T) (Ioc t (t + T)) μ := by
refine IsAddFundamentalDomain.mk' measurableSet_Ioc.nullMeasurableSet fun x => ?_
have : Bijective (codRestrict (fun n : ℤ => n • T) (AddSubgroup.zmultiples T) _) :=
(Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_strictMono_left hT).injective).bijective
refine (AddSubgroup.equivOp _).bijective.comp this |>.existsUnique_iff.2 ?_
simpa using existsUnique_add_zsmul_mem_Ioc hT x t
#align is_add_fundamental_domain_Ioc' isAddFundamentalDomain_Ioc'
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
namespace Function
namespace Periodic
variable {f : ℝ → E} {T : ℝ}
theorem intervalIntegral_add_eq_of_pos (hf : Periodic f T) (hT : 0 < T) (t s : ℝ) :
∫ x in t..t + T, f x = ∫ x in s..s + T, f x := by
simp only [integral_of_le, hT.le, le_add_iff_nonneg_right]
haveI : VAddInvariantMeasure (AddSubgroup.zmultiples T) ℝ volume :=
⟨fun c s _ => measure_preimage_add _ _ _⟩
apply IsAddFundamentalDomain.setIntegral_eq (G := AddSubgroup.zmultiples T)
exacts [isAddFundamentalDomain_Ioc hT t, isAddFundamentalDomain_Ioc hT s, hf.map_vadd_zmultiples]
#align function.periodic.interval_integral_add_eq_of_pos Function.Periodic.intervalIntegral_add_eq_of_pos
theorem intervalIntegral_add_eq (hf : Periodic f T) (t s : ℝ) :
∫ x in t..t + T, f x = ∫ x in s..s + T, f x := by
rcases lt_trichotomy (0 : ℝ) T with (hT | rfl | hT)
· exact hf.intervalIntegral_add_eq_of_pos hT t s
· simp
· rw [← neg_inj, ← integral_symm, ← integral_symm]
simpa only [← sub_eq_add_neg, add_sub_cancel_right] using
hf.neg.intervalIntegral_add_eq_of_pos (neg_pos.2 hT) (t + T) (s + T)
#align function.periodic.interval_integral_add_eq Function.Periodic.intervalIntegral_add_eq
theorem intervalIntegral_add_eq_add (hf : Periodic f T) (t s : ℝ)
(h_int : ∀ t₁ t₂, IntervalIntegrable f MeasureSpace.volume t₁ t₂) :
∫ x in t..s + T, f x = (∫ x in t..s, f x) + ∫ x in t..t + T, f x := by
rw [hf.intervalIntegral_add_eq t s, integral_add_adjacent_intervals (h_int t s) (h_int s _)]
#align function.periodic.interval_integral_add_eq_add Function.Periodic.intervalIntegral_add_eq_add
theorem intervalIntegral_add_zsmul_eq (hf : Periodic f T) (n : ℤ) (t : ℝ)
(h_int : ∀ t₁ t₂, IntervalIntegrable f MeasureSpace.volume t₁ t₂) :
∫ x in t..t + n • T, f x = n • ∫ x in t..t + T, f x := by
-- Reduce to the case `b = 0`
suffices (∫ x in (0)..(n • T), f x) = n • ∫ x in (0)..T, f x by
simp only [hf.intervalIntegral_add_eq t 0, (hf.zsmul n).intervalIntegral_add_eq t 0, zero_add,
this]
-- First prove it for natural numbers
have : ∀ m : ℕ, (∫ x in (0)..m • T, f x) = m • ∫ x in (0)..T, f x := fun m ↦ by
induction' m with m ih
· simp
· simp only [succ_nsmul, hf.intervalIntegral_add_eq_add 0 (m • T) h_int, ih, zero_add]
-- Then prove it for all integers
cases' n with n n
· simp [← this n]
· conv_rhs => rw [negSucc_zsmul]
have h₀ : Int.negSucc n • T + (n + 1) • T = 0 := by simp; linarith
rw [integral_symm, ← (hf.nsmul (n + 1)).funext, neg_inj]
simp_rw [integral_comp_add_right, h₀, zero_add, this (n + 1), add_comm T,
hf.intervalIntegral_add_eq ((n + 1) • T) 0, zero_add]
#align function.periodic.interval_integral_add_zsmul_eq Function.Periodic.intervalIntegral_add_zsmul_eq
section RealValued
open Filter
variable {g : ℝ → ℝ}
variable (hg : Periodic g T) (h_int : ∀ t₁ t₂, IntervalIntegrable g MeasureSpace.volume t₁ t₂)
theorem sInf_add_zsmul_le_integral_of_pos (hT : 0 < T) (t : ℝ) :
(sInf ((fun t => ∫ x in (0)..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ x in (0)..T, g x) ≤
∫ x in (0)..t, g x := by
let ε := Int.fract (t / T) * T
conv_rhs =>
rw [← Int.fract_div_mul_self_add_zsmul_eq T t (by linarith), ←
integral_add_adjacent_intervals (h_int 0 ε) (h_int _ _)]
rw [hg.intervalIntegral_add_zsmul_eq ⌊t / T⌋ ε h_int, hg.intervalIntegral_add_eq ε 0, zero_add,
add_le_add_iff_right]
exact (continuous_primitive h_int 0).continuousOn.sInf_image_Icc_le <|
mem_Icc_of_Ico (Int.fract_div_mul_self_mem_Ico T t hT)
#align function.periodic.Inf_add_zsmul_le_integral_of_pos Function.Periodic.sInf_add_zsmul_le_integral_of_pos
| Mathlib/MeasureTheory/Integral/Periodic.lean | 335 | 345 | theorem integral_le_sSup_add_zsmul_of_pos (hT : 0 < T) (t : ℝ) :
(∫ x in (0)..t, g x) ≤
sSup ((fun t => ∫ x in (0)..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ x in (0)..T, g x := by |
let ε := Int.fract (t / T) * T
conv_lhs =>
rw [← Int.fract_div_mul_self_add_zsmul_eq T t (by linarith), ←
integral_add_adjacent_intervals (h_int 0 ε) (h_int _ _)]
rw [hg.intervalIntegral_add_zsmul_eq ⌊t / T⌋ ε h_int, hg.intervalIntegral_add_eq ε 0, zero_add,
add_le_add_iff_right]
exact (continuous_primitive h_int 0).continuousOn.le_sSup_image_Icc
(mem_Icc_of_Ico (Int.fract_div_mul_self_mem_Ico T t hT))
|
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.fderiv_symmetric from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Asymptotics Set
open scoped Topology
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F]
[NormedSpace ℝ F] {s : Set E} (s_conv : Convex ℝ s) {f : E → F} {f' : E → E →L[ℝ] F}
{f'' : E →L[ℝ] E →L[ℝ] F} (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
(hx : HasFDerivWithinAt f' f'' (interior s) x)
theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
(hw : x + v + w ∈ interior s) :
(fun h : ℝ => f (x + h • v + h • w)
- f (x + h • v) - h • f' x w - h ^ 2 • f'' v w - (h ^ 2 / 2) • f'' w w) =o[𝓝[>] 0]
fun h => h ^ 2 := by
-- it suffices to check that the expression is bounded by `ε * ((‖v‖ + ‖w‖) * ‖w‖) * h^2` for
-- small enough `h`, for any positive `ε`.
refine IsLittleO.trans_isBigO
(isLittleO_iff.2 fun ε εpos => ?_) (isBigO_const_mul_self ((‖v‖ + ‖w‖) * ‖w‖) _ _)
-- consider a ball of radius `δ` around `x` in which the Taylor approximation for `f''` is
-- good up to `δ`.
rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO, isLittleO_iff] at hx
rcases Metric.mem_nhdsWithin_iff.1 (hx εpos) with ⟨δ, δpos, sδ⟩
have E1 : ∀ᶠ h in 𝓝[>] (0 : ℝ), h * (‖v‖ + ‖w‖) < δ := by
have : Filter.Tendsto (fun h => h * (‖v‖ + ‖w‖)) (𝓝[>] (0 : ℝ)) (𝓝 (0 * (‖v‖ + ‖w‖))) :=
(continuous_id.mul continuous_const).continuousWithinAt
apply (tendsto_order.1 this).2 δ
simpa only [zero_mul] using δpos
have E2 : ∀ᶠ h in 𝓝[>] (0 : ℝ), (h : ℝ) < 1 :=
mem_nhdsWithin_Ioi_iff_exists_Ioo_subset.2
⟨(1 : ℝ), by simp only [mem_Ioi, zero_lt_one], fun x hx => hx.2⟩
filter_upwards [E1, E2, self_mem_nhdsWithin] with h hδ h_lt_1 hpos
-- we consider `h` small enough that all points under consideration belong to this ball,
-- and also with `0 < h < 1`.
replace hpos : 0 < h := hpos
have xt_mem : ∀ t ∈ Icc (0 : ℝ) 1, x + h • v + (t * h) • w ∈ interior s := by
intro t ht
have : x + h • v ∈ interior s := s_conv.add_smul_mem_interior xs hv ⟨hpos, h_lt_1.le⟩
rw [← smul_smul]
apply s_conv.interior.add_smul_mem this _ ht
rw [add_assoc] at hw
rw [add_assoc, ← smul_add]
exact s_conv.add_smul_mem_interior xs hw ⟨hpos, h_lt_1.le⟩
-- define a function `g` on `[0,1]` (identified with `[v, v + w]`) such that `g 1 - g 0` is the
-- quantity to be estimated. We will check that its derivative is given by an explicit
-- expression `g'`, that we can bound. Then the desired bound for `g 1 - g 0` follows from the
-- mean value inequality.
let g t :=
f (x + h • v + (t * h) • w) - (t * h) • f' x w - (t * h ^ 2) • f'' v w -
((t * h) ^ 2 / 2) • f'' w w
set g' := fun t =>
f' (x + h • v + (t * h) • w) (h • w) - h • f' x w - h ^ 2 • f'' v w - (t * h ^ 2) • f'' w w
with hg'
-- check that `g'` is the derivative of `g`, by a straightforward computation
have g_deriv : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt g (g' t) (Icc 0 1) t := by
intro t ht
apply_rules [HasDerivWithinAt.sub, HasDerivWithinAt.add]
· refine (hf _ ?_).comp_hasDerivWithinAt _ ?_
· exact xt_mem t ht
apply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.const_add, HasDerivAt.smul_const,
hasDerivAt_mul_const]
· apply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.smul_const, hasDerivAt_mul_const]
· apply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.smul_const, hasDerivAt_mul_const]
· suffices H : HasDerivWithinAt (fun u => ((u * h) ^ 2 / 2) • f'' w w)
((((2 : ℕ) : ℝ) * (t * h) ^ (2 - 1) * (1 * h) / 2) • f'' w w) (Icc 0 1) t by
convert H using 2
ring
apply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.smul_const, hasDerivAt_id',
HasDerivAt.pow, HasDerivAt.mul_const]
-- check that `g'` is uniformly bounded, with a suitable bound `ε * ((‖v‖ + ‖w‖) * ‖w‖) * h^2`.
have g'_bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖g' t‖ ≤ ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2 := by
intro t ht
have I : ‖h • v + (t * h) • w‖ ≤ h * (‖v‖ + ‖w‖) :=
calc
‖h • v + (t * h) • w‖ ≤ ‖h • v‖ + ‖(t * h) • w‖ := norm_add_le _ _
_ = h * ‖v‖ + t * (h * ‖w‖) := by
simp only [norm_smul, Real.norm_eq_abs, hpos.le, abs_of_nonneg, abs_mul, ht.left,
mul_assoc]
_ ≤ h * ‖v‖ + 1 * (h * ‖w‖) := by gcongr; exact ht.2.le
_ = h * (‖v‖ + ‖w‖) := by ring
calc
‖g' t‖ = ‖(f' (x + h • v + (t * h) • w) - f' x - f'' (h • v + (t * h) • w)) (h • w)‖ := by
rw [hg']
have : h * (t * h) = t * (h * h) := by ring
simp only [ContinuousLinearMap.coe_sub', ContinuousLinearMap.map_add, pow_two,
ContinuousLinearMap.add_apply, Pi.smul_apply, smul_sub, smul_add, smul_smul, ← sub_sub,
ContinuousLinearMap.coe_smul', Pi.sub_apply, ContinuousLinearMap.map_smul, this]
_ ≤ ‖f' (x + h • v + (t * h) • w) - f' x - f'' (h • v + (t * h) • w)‖ * ‖h • w‖ :=
(ContinuousLinearMap.le_opNorm _ _)
_ ≤ ε * ‖h • v + (t * h) • w‖ * ‖h • w‖ := by
apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
have H : x + h • v + (t * h) • w ∈ Metric.ball x δ ∩ interior s := by
refine ⟨?_, xt_mem t ⟨ht.1, ht.2.le⟩⟩
rw [add_assoc, add_mem_ball_iff_norm]
exact I.trans_lt hδ
simpa only [mem_setOf_eq, add_assoc x, add_sub_cancel_left] using sδ H
_ ≤ ε * (‖h • v‖ + ‖h • w‖) * ‖h • w‖ := by
gcongr
apply (norm_add_le _ _).trans
gcongr
simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, ht.1, hpos.le, mul_assoc]
exact mul_le_of_le_one_left (mul_nonneg hpos.le (norm_nonneg _)) ht.2.le
_ = ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2 := by
simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, hpos.le]; ring
-- conclude using the mean value inequality
have I : ‖g 1 - g 0‖ ≤ ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2 := by
simpa only [mul_one, sub_zero] using
norm_image_sub_le_of_norm_deriv_le_segment' g_deriv g'_bound 1 (right_mem_Icc.2 zero_le_one)
convert I using 1
· congr 1
simp only [g, Nat.one_ne_zero, add_zero, one_mul, zero_div, zero_mul, sub_zero,
zero_smul, Ne, not_false_iff, bit0_eq_zero, zero_pow]
abel
· simp only [Real.norm_eq_abs, abs_mul, add_nonneg (norm_nonneg v) (norm_nonneg w), abs_of_nonneg,
hpos.le, mul_assoc, norm_nonneg, abs_pow]
#align convex.taylor_approx_two_segment Convex.taylor_approx_two_segment
theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) • v ∈ interior s)
(h4w : x + (4 : ℝ) • w ∈ interior s) :
(fun h : ℝ => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w))
- f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • f'' v w) =o[𝓝[>] 0]
fun h => h ^ 2 := by
have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
have B : (1 : ℝ) / 2 ∈ Icc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
have C : ∀ w : E, (2 : ℝ) • w = 2 • w := fun w => by simp only [two_smul]
have h2v2w : x + (2 : ℝ) • v + (2 : ℝ) • w ∈ interior s := by
convert s_conv.interior.add_smul_sub_mem h4v h4w B using 1
simp only [smul_sub, smul_smul, one_div, add_sub_add_left_eq_sub, mul_add, add_smul]
norm_num
simp only [show (4 : ℝ) = (2 : ℝ) + (2 : ℝ) by norm_num, _root_.add_smul]
abel
have h2vww : x + (2 • v + w) + w ∈ interior s := by
convert h2v2w using 1
simp only [two_smul]
abel
have h2v : x + (2 : ℝ) • v ∈ interior s := by
convert s_conv.add_smul_sub_mem_interior xs h4v A using 1
simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj]
norm_num
have h2w : x + (2 : ℝ) • w ∈ interior s := by
convert s_conv.add_smul_sub_mem_interior xs h4w A using 1
simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj]
norm_num
have hvw : x + (v + w) ∈ interior s := by
convert s_conv.add_smul_sub_mem_interior xs h2v2w A using 1
simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj, smul_add, smul_sub]
norm_num
abel
have h2vw : x + (2 • v + w) ∈ interior s := by
convert s_conv.interior.add_smul_sub_mem h2v h2v2w B using 1
simp only [smul_add, smul_sub, smul_smul, ← C]
norm_num
abel
have hvww : x + (v + w) + w ∈ interior s := by
convert s_conv.interior.add_smul_sub_mem h2w h2v2w B using 1
rw [one_div, add_sub_add_right_eq_sub, add_sub_cancel_left, inv_smul_smul₀ two_ne_zero,
two_smul]
abel
have TA1 := s_conv.taylor_approx_two_segment hf xs hx h2vw h2vww
have TA2 := s_conv.taylor_approx_two_segment hf xs hx hvw hvww
convert TA1.sub TA2 using 1
ext h
simp only [two_smul, smul_add, ← add_assoc, ContinuousLinearMap.map_add,
ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul',
ContinuousLinearMap.map_smul]
abel
#align convex.is_o_alternate_sum_square Convex.isLittleO_alternate_sum_square
theorem Convex.second_derivative_within_at_symmetric_of_mem_interior {v w : E}
(h4v : x + (4 : ℝ) • v ∈ interior s) (h4w : x + (4 : ℝ) • w ∈ interior s) :
f'' w v = f'' v w := by
have A : (fun h : ℝ => h ^ 2 • (f'' w v - f'' v w)) =o[𝓝[>] 0] fun h => h ^ 2 := by
convert (s_conv.isLittleO_alternate_sum_square hf xs hx h4v h4w).sub
(s_conv.isLittleO_alternate_sum_square hf xs hx h4w h4v) using 1
ext h
simp only [add_comm, smul_add, smul_sub]
abel
have B : (fun _ : ℝ => f'' w v - f'' v w) =o[𝓝[>] 0] fun _ => (1 : ℝ) := by
have : (fun h : ℝ => 1 / h ^ 2) =O[𝓝[>] 0] fun h => 1 / h ^ 2 := isBigO_refl _ _
have C := this.smul_isLittleO A
apply C.congr' _ _
· filter_upwards [self_mem_nhdsWithin]
intro h (hpos : 0 < h)
rw [← one_smul ℝ (f'' w v - f'' v w), smul_smul, smul_smul]
congr 1
field_simp [LT.lt.ne' hpos]
· filter_upwards [self_mem_nhdsWithin] with h (hpos : 0 < h)
field_simp [LT.lt.ne' hpos, SMul.smul]
simpa only [sub_eq_zero] using isLittleO_const_const_iff.1 B
#align convex.second_derivative_within_at_symmetric_of_mem_interior Convex.second_derivative_within_at_symmetric_of_mem_interior
| Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean | 259 | 304 | theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
(hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
(hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
(hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v := by |
/- we work around a point `x + 4 z` in the interior of `s`. For any vector `m`,
then `x + 4 (z + t m)` also belongs to the interior of `s` for small enough `t`. This means that
we will be able to apply `second_derivative_within_at_symmetric_of_mem_interior` to show
that `f''` is symmetric, after cancelling all the contributions due to `z`. -/
rcases hne with ⟨y, hy⟩
obtain ⟨z, hz⟩ : ∃ z, z = ((1 : ℝ) / 4) • (y - x) := ⟨((1 : ℝ) / 4) • (y - x), rfl⟩
have A : ∀ m : E, Filter.Tendsto (fun t : ℝ => x + (4 : ℝ) • (z + t • m)) (𝓝 0) (𝓝 y) := by
intro m
have : x + (4 : ℝ) • (z + (0 : ℝ) • m) = y := by simp [hz]
rw [← this]
refine tendsto_const_nhds.add <| tendsto_const_nhds.smul <| tendsto_const_nhds.add ?_
exact continuousAt_id.smul continuousAt_const
have B : ∀ m : E, ∀ᶠ t in 𝓝[>] (0 : ℝ), x + (4 : ℝ) • (z + t • m) ∈ interior s := by
intro m
apply nhdsWithin_le_nhds
apply A m
rw [mem_interior_iff_mem_nhds] at hy
exact interior_mem_nhds.2 hy
-- we choose `t m > 0` such that `x + 4 (z + (t m) m)` belongs to the interior of `s`, for any
-- vector `m`.
choose t ts tpos using fun m => ((B m).and self_mem_nhdsWithin).exists
-- applying `second_derivative_within_at_symmetric_of_mem_interior` to the vectors `z`
-- and `z + (t m) m`, we deduce that `f'' m z = f'' z m` for all `m`.
have C : ∀ m : E, f'' m z = f'' z m := by
intro m
have : f'' (z + t m • m) (z + t 0 • (0 : E)) = f'' (z + t 0 • (0 : E)) (z + t m • m) :=
s_conv.second_derivative_within_at_symmetric_of_mem_interior hf xs hx (ts 0) (ts m)
simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, add_right_inj,
ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul', add_zero,
ContinuousLinearMap.zero_apply, smul_zero, ContinuousLinearMap.map_zero] at this
exact smul_right_injective F (tpos m).ne' this
-- applying `second_derivative_within_at_symmetric_of_mem_interior` to the vectors `z + (t v) v`
-- and `z + (t w) w`, we deduce that `f'' v w = f'' w v`. Cross terms involving `z` can be
-- eliminated thanks to the fact proved above that `f'' m z = f'' z m`.
have : f'' (z + t v • v) (z + t w • w) = f'' (z + t w • w) (z + t v • v) :=
s_conv.second_derivative_within_at_symmetric_of_mem_interior hf xs hx (ts w) (ts v)
simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, smul_add, smul_smul,
ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul', C] at this
rw [add_assoc, add_assoc, add_right_inj, add_left_comm, add_right_inj, add_right_inj, mul_comm]
at this
apply smul_right_injective F _ this
simp [(tpos v).ne', (tpos w).ne']
|
import Batteries.Data.Array.Lemmas
namespace ByteArray
@[ext] theorem ext : {a b : ByteArray} → a.data = b.data → a = b
| ⟨_⟩, ⟨_⟩, rfl => rfl
theorem getElem_eq_data_getElem (a : ByteArray) (h : i < a.size) : a[i] = a.data[i] := rfl
@[simp] theorem uset_eq_set (a : ByteArray) {i : USize} (h : i.toNat < a.size) (v : UInt8) :
a.uset i v h = a.set ⟨i.toNat, h⟩ v := rfl
@[simp] theorem mkEmpty_data (cap) : (mkEmpty cap).data = #[] := rfl
@[simp] theorem empty_data : empty.data = #[] := rfl
@[simp] theorem size_empty : empty.size = 0 := rfl
@[simp] theorem push_data (a : ByteArray) (b : UInt8) : (a.push b).data = a.data.push b := rfl
@[simp] theorem size_push (a : ByteArray) (b : UInt8) : (a.push b).size = a.size + 1 :=
Array.size_push ..
@[simp] theorem get_push_eq (a : ByteArray) (x : UInt8) : (a.push x)[a.size] = x :=
Array.get_push_eq ..
theorem get_push_lt (a : ByteArray) (x : UInt8) (i : Nat) (h : i < a.size) :
(a.push x)[i]'(size_push .. ▸ Nat.lt_succ_of_lt h) = a[i] :=
Array.get_push_lt ..
@[simp] theorem set_data (a : ByteArray) (i : Fin a.size) (v : UInt8) :
(a.set i v).data = a.data.set i v := rfl
@[simp] theorem size_set (a : ByteArray) (i : Fin a.size) (v : UInt8) :
(a.set i v).size = a.size :=
Array.size_set ..
@[simp] theorem get_set_eq (a : ByteArray) (i : Fin a.size) (v : UInt8) : (a.set i v)[i.val] = v :=
Array.get_set_eq ..
theorem get_set_ne (a : ByteArray) (i : Fin a.size) (v : UInt8) (hj : j < a.size) (h : i.val ≠ j) :
(a.set i v)[j]'(a.size_set .. ▸ hj) = a[j] :=
Array.get_set_ne (h:=h) ..
theorem set_set (a : ByteArray) (i : Fin a.size) (v v' : UInt8) :
(a.set i v).set ⟨i, by simp [i.2]⟩ v' = a.set i v' :=
ByteArray.ext <| Array.set_set ..
@[simp] theorem copySlice_data (a i b j len exact) :
(copySlice a i b j len exact).data = b.data.extract 0 j ++ a.data.extract i (i + len)
++ b.data.extract (j + min len (a.data.size - i)) b.data.size := rfl
@[simp] theorem append_eq (a b) : ByteArray.append a b = a ++ b := rfl
@[simp] theorem append_data (a b : ByteArray) : (a ++ b).data = a.data ++ b.data := by
rw [←append_eq]; simp [ByteArray.append, size]
rw [Array.extract_empty_of_stop_le_start (h:=Nat.le_add_right ..), Array.append_nil]
theorem size_append (a b : ByteArray) : (a ++ b).size = a.size + b.size := by
simp only [size, append_eq, append_data]; exact Array.size_append ..
| .lake/packages/batteries/Batteries/Data/ByteArray.lean | 79 | 82 | theorem get_append_left {a b : ByteArray} (hlt : i < a.size)
(h : i < (a ++ b).size := size_append .. ▸ Nat.lt_of_lt_of_le hlt (Nat.le_add_right ..)) :
(a ++ b)[i] = a[i] := by |
simp [getElem_eq_data_getElem]; exact Array.get_append_left hlt
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"84771a9f5f0bd5e5d6218811556508ddf476dcbd"
open Function OrderDual
variable {ι α β : Type*}
section LinearOrderedSemifield
variable [LinearOrderedSemifield α] {a b c d e : α} {m n : ℤ}
@[simps! (config := { simpRhs := true })]
def OrderIso.mulLeft₀ (a : α) (ha : 0 < a) : α ≃o α :=
{ Equiv.mulLeft₀ a ha.ne' with map_rel_iff' := @fun _ _ => mul_le_mul_left ha }
#align order_iso.mul_left₀ OrderIso.mulLeft₀
#align order_iso.mul_left₀_symm_apply OrderIso.mulLeft₀_symm_apply
#align order_iso.mul_left₀_apply OrderIso.mulLeft₀_apply
@[simps! (config := { simpRhs := true })]
def OrderIso.mulRight₀ (a : α) (ha : 0 < a) : α ≃o α :=
{ Equiv.mulRight₀ a ha.ne' with map_rel_iff' := @fun _ _ => mul_le_mul_right ha }
#align order_iso.mul_right₀ OrderIso.mulRight₀
#align order_iso.mul_right₀_symm_apply OrderIso.mulRight₀_symm_apply
#align order_iso.mul_right₀_apply OrderIso.mulRight₀_apply
theorem le_div_iff (hc : 0 < c) : a ≤ b / c ↔ a * c ≤ b :=
⟨fun h => div_mul_cancel₀ b (ne_of_lt hc).symm ▸ mul_le_mul_of_nonneg_right h hc.le, fun h =>
calc
a = a * c * (1 / c) := mul_mul_div a (ne_of_lt hc).symm
_ ≤ b * (1 / c) := mul_le_mul_of_nonneg_right h (one_div_pos.2 hc).le
_ = b / c := (div_eq_mul_one_div b c).symm
⟩
#align le_div_iff le_div_iff
theorem le_div_iff' (hc : 0 < c) : a ≤ b / c ↔ c * a ≤ b := by rw [mul_comm, le_div_iff hc]
#align le_div_iff' le_div_iff'
theorem div_le_iff (hb : 0 < b) : a / b ≤ c ↔ a ≤ c * b :=
⟨fun h =>
calc
a = a / b * b := by rw [div_mul_cancel₀ _ (ne_of_lt hb).symm]
_ ≤ c * b := mul_le_mul_of_nonneg_right h hb.le
,
fun h =>
calc
a / b = a * (1 / b) := div_eq_mul_one_div a b
_ ≤ c * b * (1 / b) := mul_le_mul_of_nonneg_right h (one_div_pos.2 hb).le
_ = c * b / b := (div_eq_mul_one_div (c * b) b).symm
_ = c := by refine (div_eq_iff (ne_of_gt hb)).mpr rfl
⟩
#align div_le_iff div_le_iff
theorem div_le_iff' (hb : 0 < b) : a / b ≤ c ↔ a ≤ b * c := by rw [mul_comm, div_le_iff hb]
#align div_le_iff' div_le_iff'
lemma div_le_comm₀ (hb : 0 < b) (hc : 0 < c) : a / b ≤ c ↔ a / c ≤ b := by
rw [div_le_iff hb, div_le_iff' hc]
theorem lt_div_iff (hc : 0 < c) : a < b / c ↔ a * c < b :=
lt_iff_lt_of_le_iff_le <| div_le_iff hc
#align lt_div_iff lt_div_iff
theorem lt_div_iff' (hc : 0 < c) : a < b / c ↔ c * a < b := by rw [mul_comm, lt_div_iff hc]
#align lt_div_iff' lt_div_iff'
theorem div_lt_iff (hc : 0 < c) : b / c < a ↔ b < a * c :=
lt_iff_lt_of_le_iff_le (le_div_iff hc)
#align div_lt_iff div_lt_iff
theorem div_lt_iff' (hc : 0 < c) : b / c < a ↔ b < c * a := by rw [mul_comm, div_lt_iff hc]
#align div_lt_iff' div_lt_iff'
lemma div_lt_comm₀ (hb : 0 < b) (hc : 0 < c) : a / b < c ↔ a / c < b := by
rw [div_lt_iff hb, div_lt_iff' hc]
theorem inv_mul_le_iff (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ b * c := by
rw [inv_eq_one_div, mul_comm, ← div_eq_mul_one_div]
exact div_le_iff' h
#align inv_mul_le_iff inv_mul_le_iff
theorem inv_mul_le_iff' (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ c * b := by rw [inv_mul_le_iff h, mul_comm]
#align inv_mul_le_iff' inv_mul_le_iff'
theorem mul_inv_le_iff (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ b * c := by rw [mul_comm, inv_mul_le_iff h]
#align mul_inv_le_iff mul_inv_le_iff
theorem mul_inv_le_iff' (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ c * b := by rw [mul_comm, inv_mul_le_iff' h]
#align mul_inv_le_iff' mul_inv_le_iff'
theorem div_self_le_one (a : α) : a / a ≤ 1 :=
if h : a = 0 then by simp [h] else by simp [h]
#align div_self_le_one div_self_le_one
theorem inv_mul_lt_iff (h : 0 < b) : b⁻¹ * a < c ↔ a < b * c := by
rw [inv_eq_one_div, mul_comm, ← div_eq_mul_one_div]
exact div_lt_iff' h
#align inv_mul_lt_iff inv_mul_lt_iff
theorem inv_mul_lt_iff' (h : 0 < b) : b⁻¹ * a < c ↔ a < c * b := by rw [inv_mul_lt_iff h, mul_comm]
#align inv_mul_lt_iff' inv_mul_lt_iff'
theorem mul_inv_lt_iff (h : 0 < b) : a * b⁻¹ < c ↔ a < b * c := by rw [mul_comm, inv_mul_lt_iff h]
#align mul_inv_lt_iff mul_inv_lt_iff
theorem mul_inv_lt_iff' (h : 0 < b) : a * b⁻¹ < c ↔ a < c * b := by rw [mul_comm, inv_mul_lt_iff' h]
#align mul_inv_lt_iff' mul_inv_lt_iff'
theorem inv_pos_le_iff_one_le_mul (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ b * a := by
rw [inv_eq_one_div]
exact div_le_iff ha
#align inv_pos_le_iff_one_le_mul inv_pos_le_iff_one_le_mul
theorem inv_pos_le_iff_one_le_mul' (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ a * b := by
rw [inv_eq_one_div]
exact div_le_iff' ha
#align inv_pos_le_iff_one_le_mul' inv_pos_le_iff_one_le_mul'
theorem inv_pos_lt_iff_one_lt_mul (ha : 0 < a) : a⁻¹ < b ↔ 1 < b * a := by
rw [inv_eq_one_div]
exact div_lt_iff ha
#align inv_pos_lt_iff_one_lt_mul inv_pos_lt_iff_one_lt_mul
theorem inv_pos_lt_iff_one_lt_mul' (ha : 0 < a) : a⁻¹ < b ↔ 1 < a * b := by
rw [inv_eq_one_div]
exact div_lt_iff' ha
#align inv_pos_lt_iff_one_lt_mul' inv_pos_lt_iff_one_lt_mul'
theorem div_le_of_nonneg_of_le_mul (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b) : a / b ≤ c := by
rcases eq_or_lt_of_le hb with (rfl | hb')
· simp only [div_zero, hc]
· rwa [div_le_iff hb']
#align div_le_of_nonneg_of_le_mul div_le_of_nonneg_of_le_mul
lemma mul_le_of_nonneg_of_le_div (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ b / c) : a * c ≤ b := by
obtain rfl | hc := hc.eq_or_lt
· simpa using hb
· rwa [le_div_iff hc] at h
#align mul_le_of_nonneg_of_le_div mul_le_of_nonneg_of_le_div
theorem div_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : a / b ≤ 1 :=
div_le_of_nonneg_of_le_mul hb zero_le_one <| by rwa [one_mul]
#align div_le_one_of_le div_le_one_of_le
lemma mul_inv_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : a * b⁻¹ ≤ 1 := by
simpa only [← div_eq_mul_inv] using div_le_one_of_le h hb
lemma inv_mul_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : b⁻¹ * a ≤ 1 := by
simpa only [← div_eq_inv_mul] using div_le_one_of_le h hb
@[gcongr]
theorem inv_le_inv_of_le (ha : 0 < a) (h : a ≤ b) : b⁻¹ ≤ a⁻¹ := by
rwa [← one_div a, le_div_iff' ha, ← div_eq_mul_inv, div_le_iff (ha.trans_le h), one_mul]
#align inv_le_inv_of_le inv_le_inv_of_le
theorem inv_le_inv (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a := by
rw [← one_div, div_le_iff ha, ← div_eq_inv_mul, le_div_iff hb, one_mul]
#align inv_le_inv inv_le_inv
theorem inv_le (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a := by
rw [← inv_le_inv hb (inv_pos.2 ha), inv_inv]
#align inv_le inv_le
theorem inv_le_of_inv_le (ha : 0 < a) (h : a⁻¹ ≤ b) : b⁻¹ ≤ a :=
(inv_le ha ((inv_pos.2 ha).trans_le h)).1 h
#align inv_le_of_inv_le inv_le_of_inv_le
theorem le_inv (ha : 0 < a) (hb : 0 < b) : a ≤ b⁻¹ ↔ b ≤ a⁻¹ := by
rw [← inv_le_inv (inv_pos.2 hb) ha, inv_inv]
#align le_inv le_inv
theorem inv_lt_inv (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b⁻¹ ↔ b < a :=
lt_iff_lt_of_le_iff_le (inv_le_inv hb ha)
#align inv_lt_inv inv_lt_inv
@[gcongr]
theorem inv_lt_inv_of_lt (hb : 0 < b) (h : b < a) : a⁻¹ < b⁻¹ :=
(inv_lt_inv (hb.trans h) hb).2 h
#align inv_lt_inv_of_lt inv_lt_inv_of_lt
theorem inv_lt (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b ↔ b⁻¹ < a :=
lt_iff_lt_of_le_iff_le (le_inv hb ha)
#align inv_lt inv_lt
theorem inv_lt_of_inv_lt (ha : 0 < a) (h : a⁻¹ < b) : b⁻¹ < a :=
(inv_lt ha ((inv_pos.2 ha).trans h)).1 h
#align inv_lt_of_inv_lt inv_lt_of_inv_lt
theorem lt_inv (ha : 0 < a) (hb : 0 < b) : a < b⁻¹ ↔ b < a⁻¹ :=
lt_iff_lt_of_le_iff_le (inv_le hb ha)
#align lt_inv lt_inv
theorem inv_lt_one (ha : 1 < a) : a⁻¹ < 1 := by
rwa [inv_lt (zero_lt_one.trans ha) zero_lt_one, inv_one]
#align inv_lt_one inv_lt_one
theorem one_lt_inv (h₁ : 0 < a) (h₂ : a < 1) : 1 < a⁻¹ := by
rwa [lt_inv (@zero_lt_one α _ _ _ _ _) h₁, inv_one]
#align one_lt_inv one_lt_inv
theorem inv_le_one (ha : 1 ≤ a) : a⁻¹ ≤ 1 := by
rwa [inv_le (zero_lt_one.trans_le ha) zero_lt_one, inv_one]
#align inv_le_one inv_le_one
theorem one_le_inv (h₁ : 0 < a) (h₂ : a ≤ 1) : 1 ≤ a⁻¹ := by
rwa [le_inv (@zero_lt_one α _ _ _ _ _) h₁, inv_one]
#align one_le_inv one_le_inv
theorem inv_lt_one_iff_of_pos (h₀ : 0 < a) : a⁻¹ < 1 ↔ 1 < a :=
⟨fun h₁ => inv_inv a ▸ one_lt_inv (inv_pos.2 h₀) h₁, inv_lt_one⟩
#align inv_lt_one_iff_of_pos inv_lt_one_iff_of_pos
theorem inv_lt_one_iff : a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a := by
rcases le_or_lt a 0 with ha | ha
· simp [ha, (inv_nonpos.2 ha).trans_lt zero_lt_one]
· simp only [ha.not_le, false_or_iff, inv_lt_one_iff_of_pos ha]
#align inv_lt_one_iff inv_lt_one_iff
theorem one_lt_inv_iff : 1 < a⁻¹ ↔ 0 < a ∧ a < 1 :=
⟨fun h => ⟨inv_pos.1 (zero_lt_one.trans h), inv_inv a ▸ inv_lt_one h⟩, and_imp.2 one_lt_inv⟩
#align one_lt_inv_iff one_lt_inv_iff
theorem inv_le_one_iff : a⁻¹ ≤ 1 ↔ a ≤ 0 ∨ 1 ≤ a := by
rcases em (a = 1) with (rfl | ha)
· simp [le_rfl]
· simp only [Ne.le_iff_lt (Ne.symm ha), Ne.le_iff_lt (mt inv_eq_one.1 ha), inv_lt_one_iff]
#align inv_le_one_iff inv_le_one_iff
theorem one_le_inv_iff : 1 ≤ a⁻¹ ↔ 0 < a ∧ a ≤ 1 :=
⟨fun h => ⟨inv_pos.1 (zero_lt_one.trans_le h), inv_inv a ▸ inv_le_one h⟩, and_imp.2 one_le_inv⟩
#align one_le_inv_iff one_le_inv_iff
@[mono, gcongr]
lemma div_le_div_of_nonneg_right (hab : a ≤ b) (hc : 0 ≤ c) : a / c ≤ b / c := by
rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c]
exact mul_le_mul_of_nonneg_right hab (one_div_nonneg.2 hc)
#align div_le_div_of_le_of_nonneg div_le_div_of_nonneg_right
@[gcongr]
lemma div_lt_div_of_pos_right (h : a < b) (hc : 0 < c) : a / c < b / c := by
rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c]
exact mul_lt_mul_of_pos_right h (one_div_pos.2 hc)
#align div_lt_div_of_lt div_lt_div_of_pos_right
-- Not a `mono` lemma b/c `div_le_div` is strictly more general
@[gcongr]
lemma div_le_div_of_nonneg_left (ha : 0 ≤ a) (hc : 0 < c) (h : c ≤ b) : a / b ≤ a / c := by
rw [div_eq_mul_inv, div_eq_mul_inv]
exact mul_le_mul_of_nonneg_left ((inv_le_inv (hc.trans_le h) hc).mpr h) ha
#align div_le_div_of_le_left div_le_div_of_nonneg_left
@[gcongr]
lemma div_lt_div_of_pos_left (ha : 0 < a) (hc : 0 < c) (h : c < b) : a / b < a / c := by
simpa only [div_eq_mul_inv, mul_lt_mul_left ha, inv_lt_inv (hc.trans h) hc]
#align div_lt_div_of_lt_left div_lt_div_of_pos_left
-- 2024-02-16
@[deprecated] alias div_le_div_of_le_of_nonneg := div_le_div_of_nonneg_right
@[deprecated] alias div_lt_div_of_lt := div_lt_div_of_pos_right
@[deprecated] alias div_le_div_of_le_left := div_le_div_of_nonneg_left
@[deprecated] alias div_lt_div_of_lt_left := div_lt_div_of_pos_left
@[deprecated div_le_div_of_nonneg_right (since := "2024-02-16")]
lemma div_le_div_of_le (hc : 0 ≤ c) (hab : a ≤ b) : a / c ≤ b / c :=
div_le_div_of_nonneg_right hab hc
#align div_le_div_of_le div_le_div_of_le
theorem div_le_div_right (hc : 0 < c) : a / c ≤ b / c ↔ a ≤ b :=
⟨le_imp_le_of_lt_imp_lt fun hab ↦ div_lt_div_of_pos_right hab hc,
fun hab ↦ div_le_div_of_nonneg_right hab hc.le⟩
#align div_le_div_right div_le_div_right
theorem div_lt_div_right (hc : 0 < c) : a / c < b / c ↔ a < b :=
lt_iff_lt_of_le_iff_le <| div_le_div_right hc
#align div_lt_div_right div_lt_div_right
theorem div_lt_div_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b < a / c ↔ c < b := by
simp only [div_eq_mul_inv, mul_lt_mul_left ha, inv_lt_inv hb hc]
#align div_lt_div_left div_lt_div_left
theorem div_le_div_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b ≤ a / c ↔ c ≤ b :=
le_iff_le_iff_lt_iff_lt.2 (div_lt_div_left ha hc hb)
#align div_le_div_left div_le_div_left
theorem div_lt_div_iff (b0 : 0 < b) (d0 : 0 < d) : a / b < c / d ↔ a * d < c * b := by
rw [lt_div_iff d0, div_mul_eq_mul_div, div_lt_iff b0]
#align div_lt_div_iff div_lt_div_iff
theorem div_le_div_iff (b0 : 0 < b) (d0 : 0 < d) : a / b ≤ c / d ↔ a * d ≤ c * b := by
rw [le_div_iff d0, div_mul_eq_mul_div, div_le_iff b0]
#align div_le_div_iff div_le_div_iff
@[mono, gcongr]
theorem div_le_div (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hbd : d ≤ b) : a / b ≤ c / d := by
rw [div_le_div_iff (hd.trans_le hbd) hd]
exact mul_le_mul hac hbd hd.le hc
#align div_le_div div_le_div
@[gcongr]
theorem div_lt_div (hac : a < c) (hbd : d ≤ b) (c0 : 0 ≤ c) (d0 : 0 < d) : a / b < c / d :=
(div_lt_div_iff (d0.trans_le hbd) d0).2 (mul_lt_mul hac hbd d0 c0)
#align div_lt_div div_lt_div
theorem div_lt_div' (hac : a ≤ c) (hbd : d < b) (c0 : 0 < c) (d0 : 0 < d) : a / b < c / d :=
(div_lt_div_iff (d0.trans hbd) d0).2 (mul_lt_mul' hac hbd d0.le c0)
#align div_lt_div' div_lt_div'
theorem div_le_self (ha : 0 ≤ a) (hb : 1 ≤ b) : a / b ≤ a := by
simpa only [div_one] using div_le_div_of_nonneg_left ha zero_lt_one hb
#align div_le_self div_le_self
theorem div_lt_self (ha : 0 < a) (hb : 1 < b) : a / b < a := by
simpa only [div_one] using div_lt_div_of_pos_left ha zero_lt_one hb
#align div_lt_self div_lt_self
theorem le_div_self (ha : 0 ≤ a) (hb₀ : 0 < b) (hb₁ : b ≤ 1) : a ≤ a / b := by
simpa only [div_one] using div_le_div_of_nonneg_left ha hb₀ hb₁
#align le_div_self le_div_self
| Mathlib/Algebra/Order/Field/Basic.lean | 364 | 364 | theorem one_le_div (hb : 0 < b) : 1 ≤ a / b ↔ b ≤ a := by | rw [le_div_iff hb, one_mul]
|
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/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
variable {𝓕 𝕜 α ι κ E F G : Type*}
open Filter Function Metric Bornology
open ENNReal Filter NNReal Uniformity Pointwise Topology
@[notation_class]
class Norm (E : Type*) where
norm : E → ℝ
#align has_norm Norm
@[notation_class]
class NNNorm (E : Type*) where
nnnorm : E → ℝ≥0
#align has_nnnorm NNNorm
export Norm (norm)
export NNNorm (nnnorm)
@[inherit_doc]
notation "‖" e "‖" => norm e
@[inherit_doc]
notation "‖" e "‖₊" => nnnorm e
class SeminormedAddGroup (E : Type*) extends Norm E, AddGroup E, PseudoMetricSpace E where
dist := fun x y => ‖x - y‖
dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
#align seminormed_add_group SeminormedAddGroup
@[to_additive]
class SeminormedGroup (E : Type*) extends Norm E, Group E, PseudoMetricSpace E where
dist := fun x y => ‖x / y‖
dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
#align seminormed_group SeminormedGroup
class NormedAddGroup (E : Type*) extends Norm E, AddGroup E, MetricSpace E where
dist := fun x y => ‖x - y‖
dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
#align normed_add_group NormedAddGroup
@[to_additive]
class NormedGroup (E : Type*) extends Norm E, Group E, MetricSpace E where
dist := fun x y => ‖x / y‖
dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
#align normed_group NormedGroup
class SeminormedAddCommGroup (E : Type*) extends Norm E, AddCommGroup E,
PseudoMetricSpace E where
dist := fun x y => ‖x - y‖
dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
#align seminormed_add_comm_group SeminormedAddCommGroup
@[to_additive]
class SeminormedCommGroup (E : Type*) extends Norm E, CommGroup E, PseudoMetricSpace E where
dist := fun x y => ‖x / y‖
dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
#align seminormed_comm_group SeminormedCommGroup
class NormedAddCommGroup (E : Type*) extends Norm E, AddCommGroup E, MetricSpace E where
dist := fun x y => ‖x - y‖
dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
#align normed_add_comm_group NormedAddCommGroup
@[to_additive]
class NormedCommGroup (E : Type*) extends Norm E, CommGroup E, MetricSpace E where
dist := fun x y => ‖x / y‖
dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
#align normed_comm_group NormedCommGroup
-- See note [lower instance priority]
@[to_additive]
instance (priority := 100) NormedGroup.toSeminormedGroup [NormedGroup E] : SeminormedGroup E :=
{ ‹NormedGroup E› with }
#align normed_group.to_seminormed_group NormedGroup.toSeminormedGroup
#align normed_add_group.to_seminormed_add_group NormedAddGroup.toSeminormedAddGroup
-- See note [lower instance priority]
@[to_additive]
instance (priority := 100) NormedCommGroup.toSeminormedCommGroup [NormedCommGroup E] :
SeminormedCommGroup E :=
{ ‹NormedCommGroup E› with }
#align normed_comm_group.to_seminormed_comm_group NormedCommGroup.toSeminormedCommGroup
#align normed_add_comm_group.to_seminormed_add_comm_group NormedAddCommGroup.toSeminormedAddCommGroup
-- See note [lower instance priority]
@[to_additive]
instance (priority := 100) SeminormedCommGroup.toSeminormedGroup [SeminormedCommGroup E] :
SeminormedGroup E :=
{ ‹SeminormedCommGroup E› with }
#align seminormed_comm_group.to_seminormed_group SeminormedCommGroup.toSeminormedGroup
#align seminormed_add_comm_group.to_seminormed_add_group SeminormedAddCommGroup.toSeminormedAddGroup
-- See note [lower instance priority]
@[to_additive]
instance (priority := 100) NormedCommGroup.toNormedGroup [NormedCommGroup E] : NormedGroup E :=
{ ‹NormedCommGroup E› with }
#align normed_comm_group.to_normed_group NormedCommGroup.toNormedGroup
#align normed_add_comm_group.to_normed_add_group NormedAddCommGroup.toNormedAddGroup
-- See note [reducible non-instances]
@[to_additive (attr := reducible) "Construct a `NormedAddGroup` from a `SeminormedAddGroup`
satisfying `∀ x, ‖x‖ = 0 → x = 0`. This avoids having to go back to the `(Pseudo)MetricSpace`
level when declaring a `NormedAddGroup` instance as a special case of a more general
`SeminormedAddGroup` instance."]
def NormedGroup.ofSeparation [SeminormedGroup E] (h : ∀ x : E, ‖x‖ = 0 → x = 1) :
NormedGroup E where
dist_eq := ‹SeminormedGroup E›.dist_eq
toMetricSpace :=
{ eq_of_dist_eq_zero := fun hxy =>
div_eq_one.1 <| h _ <| by exact (‹SeminormedGroup E›.dist_eq _ _).symm.trans hxy }
-- Porting note: the `rwa` no longer worked, but it was easy enough to provide the term.
-- however, notice that if you make `x` and `y` accessible, then the following does work:
-- `have := ‹SeminormedGroup E›.dist_eq x y; rwa [← this]`, so I'm not sure why the `rwa`
-- was broken.
#align normed_group.of_separation NormedGroup.ofSeparation
#align normed_add_group.of_separation NormedAddGroup.ofSeparation
-- See note [reducible non-instances]
@[to_additive (attr := reducible) "Construct a `NormedAddCommGroup` from a
`SeminormedAddCommGroup` satisfying `∀ x, ‖x‖ = 0 → x = 0`. This avoids having to go back to the
`(Pseudo)MetricSpace` level when declaring a `NormedAddCommGroup` instance as a special case
of a more general `SeminormedAddCommGroup` instance."]
def NormedCommGroup.ofSeparation [SeminormedCommGroup E] (h : ∀ x : E, ‖x‖ = 0 → x = 1) :
NormedCommGroup E :=
{ ‹SeminormedCommGroup E›, NormedGroup.ofSeparation h with }
#align normed_comm_group.of_separation NormedCommGroup.ofSeparation
#align normed_add_comm_group.of_separation NormedAddCommGroup.ofSeparation
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a translation-invariant distance."]
def SeminormedGroup.ofMulDist [Norm E] [Group E] [PseudoMetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
SeminormedGroup E where
dist_eq x y := by
rw [h₁]; apply le_antisymm
· simpa only [div_eq_mul_inv, ← mul_right_inv y] using h₂ _ _ _
· simpa only [div_mul_cancel, one_mul] using h₂ (x / y) 1 y
#align seminormed_group.of_mul_dist SeminormedGroup.ofMulDist
#align seminormed_add_group.of_add_dist SeminormedAddGroup.ofAddDist
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a translation-invariant pseudodistance."]
def SeminormedGroup.ofMulDist' [Norm E] [Group E] [PseudoMetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
SeminormedGroup E where
dist_eq x y := by
rw [h₁]; apply le_antisymm
· simpa only [div_mul_cancel, one_mul] using h₂ (x / y) 1 y
· simpa only [div_eq_mul_inv, ← mul_right_inv y] using h₂ _ _ _
#align seminormed_group.of_mul_dist' SeminormedGroup.ofMulDist'
#align seminormed_add_group.of_add_dist' SeminormedAddGroup.ofAddDist'
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a translation-invariant pseudodistance."]
def SeminormedCommGroup.ofMulDist [Norm E] [CommGroup E] [PseudoMetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
SeminormedCommGroup E :=
{ SeminormedGroup.ofMulDist h₁ h₂ with
mul_comm := mul_comm }
#align seminormed_comm_group.of_mul_dist SeminormedCommGroup.ofMulDist
#align seminormed_add_comm_group.of_add_dist SeminormedAddCommGroup.ofAddDist
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a translation-invariant pseudodistance."]
def SeminormedCommGroup.ofMulDist' [Norm E] [CommGroup E] [PseudoMetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
SeminormedCommGroup E :=
{ SeminormedGroup.ofMulDist' h₁ h₂ with
mul_comm := mul_comm }
#align seminormed_comm_group.of_mul_dist' SeminormedCommGroup.ofMulDist'
#align seminormed_add_comm_group.of_add_dist' SeminormedAddCommGroup.ofAddDist'
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a translation-invariant distance."]
def NormedGroup.ofMulDist [Norm E] [Group E] [MetricSpace E] (h₁ : ∀ x : E, ‖x‖ = dist x 1)
(h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) : NormedGroup E :=
{ SeminormedGroup.ofMulDist h₁ h₂ with
eq_of_dist_eq_zero := eq_of_dist_eq_zero }
#align normed_group.of_mul_dist NormedGroup.ofMulDist
#align normed_add_group.of_add_dist NormedAddGroup.ofAddDist
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a translation-invariant pseudodistance."]
def NormedGroup.ofMulDist' [Norm E] [Group E] [MetricSpace E] (h₁ : ∀ x : E, ‖x‖ = dist x 1)
(h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) : NormedGroup E :=
{ SeminormedGroup.ofMulDist' h₁ h₂ with
eq_of_dist_eq_zero := eq_of_dist_eq_zero }
#align normed_group.of_mul_dist' NormedGroup.ofMulDist'
#align normed_add_group.of_add_dist' NormedAddGroup.ofAddDist'
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a translation-invariant pseudodistance."]
def NormedCommGroup.ofMulDist [Norm E] [CommGroup E] [MetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist x y ≤ dist (x * z) (y * z)) :
NormedCommGroup E :=
{ NormedGroup.ofMulDist h₁ h₂ with
mul_comm := mul_comm }
#align normed_comm_group.of_mul_dist NormedCommGroup.ofMulDist
#align normed_add_comm_group.of_add_dist NormedAddCommGroup.ofAddDist
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a translation-invariant pseudodistance."]
def NormedCommGroup.ofMulDist' [Norm E] [CommGroup E] [MetricSpace E]
(h₁ : ∀ x : E, ‖x‖ = dist x 1) (h₂ : ∀ x y z : E, dist (x * z) (y * z) ≤ dist x y) :
NormedCommGroup E :=
{ NormedGroup.ofMulDist' h₁ h₂ with
mul_comm := mul_comm }
#align normed_comm_group.of_mul_dist' NormedCommGroup.ofMulDist'
#align normed_add_comm_group.of_add_dist' NormedAddCommGroup.ofAddDist'
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a seminorm, i.e., registering the pseudodistance
and the pseudometric space structure from the seminorm properties. Note that in most cases this
instance creates bad definitional equalities (e.g., it does not take into account a possibly
existing `UniformSpace` instance on `E`)."]
def GroupSeminorm.toSeminormedGroup [Group E] (f : GroupSeminorm E) : SeminormedGroup E where
dist x y := f (x / y)
norm := f
dist_eq x y := rfl
dist_self x := by simp only [div_self', map_one_eq_zero]
dist_triangle := le_map_div_add_map_div f
dist_comm := map_div_rev f
edist_dist x y := by exact ENNReal.coe_nnreal_eq _
-- Porting note: how did `mathlib3` solve this automatically?
#align group_seminorm.to_seminormed_group GroupSeminorm.toSeminormedGroup
#align add_group_seminorm.to_seminormed_add_group AddGroupSeminorm.toSeminormedAddGroup
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a seminormed group from a seminorm, i.e., registering the pseudodistance
and the pseudometric space structure from the seminorm properties. Note that in most cases this
instance creates bad definitional equalities (e.g., it does not take into account a possibly
existing `UniformSpace` instance on `E`)."]
def GroupSeminorm.toSeminormedCommGroup [CommGroup E] (f : GroupSeminorm E) :
SeminormedCommGroup E :=
{ f.toSeminormedGroup with
mul_comm := mul_comm }
#align group_seminorm.to_seminormed_comm_group GroupSeminorm.toSeminormedCommGroup
#align add_group_seminorm.to_seminormed_add_comm_group AddGroupSeminorm.toSeminormedAddCommGroup
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a norm, i.e., registering the distance and the metric
space structure from the norm properties. Note that in most cases this instance creates bad
definitional equalities (e.g., it does not take into account a possibly existing `UniformSpace`
instance on `E`)."]
def GroupNorm.toNormedGroup [Group E] (f : GroupNorm E) : NormedGroup E :=
{ f.toGroupSeminorm.toSeminormedGroup with
eq_of_dist_eq_zero := fun h => div_eq_one.1 <| eq_one_of_map_eq_zero f h }
#align group_norm.to_normed_group GroupNorm.toNormedGroup
#align add_group_norm.to_normed_add_group AddGroupNorm.toNormedAddGroup
-- See note [reducible non-instances]
@[to_additive (attr := reducible)
"Construct a normed group from a norm, i.e., registering the distance and the metric
space structure from the norm properties. Note that in most cases this instance creates bad
definitional equalities (e.g., it does not take into account a possibly existing `UniformSpace`
instance on `E`)."]
def GroupNorm.toNormedCommGroup [CommGroup E] (f : GroupNorm E) : NormedCommGroup E :=
{ f.toNormedGroup with
mul_comm := mul_comm }
#align group_norm.to_normed_comm_group GroupNorm.toNormedCommGroup
#align add_group_norm.to_normed_add_comm_group AddGroupNorm.toNormedAddCommGroup
instance PUnit.normedAddCommGroup : NormedAddCommGroup PUnit where
norm := Function.const _ 0
dist_eq _ _ := rfl
@[simp]
theorem PUnit.norm_eq_zero (r : PUnit) : ‖r‖ = 0 :=
rfl
#align punit.norm_eq_zero PUnit.norm_eq_zero
section SeminormedGroup
variable [SeminormedGroup E] [SeminormedGroup F] [SeminormedGroup G] {s : Set E}
{a a₁ a₂ b b₁ b₂ : E} {r r₁ r₂ : ℝ}
@[to_additive]
theorem dist_eq_norm_div (a b : E) : dist a b = ‖a / b‖ :=
SeminormedGroup.dist_eq _ _
#align dist_eq_norm_div dist_eq_norm_div
#align dist_eq_norm_sub dist_eq_norm_sub
@[to_additive]
theorem dist_eq_norm_div' (a b : E) : dist a b = ‖b / a‖ := by rw [dist_comm, dist_eq_norm_div]
#align dist_eq_norm_div' dist_eq_norm_div'
#align dist_eq_norm_sub' dist_eq_norm_sub'
alias dist_eq_norm := dist_eq_norm_sub
#align dist_eq_norm dist_eq_norm
alias dist_eq_norm' := dist_eq_norm_sub'
#align dist_eq_norm' dist_eq_norm'
@[to_additive]
instance NormedGroup.to_isometricSMul_right : IsometricSMul Eᵐᵒᵖ E :=
⟨fun a => Isometry.of_dist_eq fun b c => by simp [dist_eq_norm_div]⟩
#align normed_group.to_has_isometric_smul_right NormedGroup.to_isometricSMul_right
#align normed_add_group.to_has_isometric_vadd_right NormedAddGroup.to_isometricVAdd_right
@[to_additive (attr := simp)]
theorem dist_one_right (a : E) : dist a 1 = ‖a‖ := by rw [dist_eq_norm_div, div_one]
#align dist_one_right dist_one_right
#align dist_zero_right dist_zero_right
@[to_additive]
theorem inseparable_one_iff_norm {a : E} : Inseparable a 1 ↔ ‖a‖ = 0 := by
rw [Metric.inseparable_iff, dist_one_right]
@[to_additive (attr := simp)]
theorem dist_one_left : dist (1 : E) = norm :=
funext fun a => by rw [dist_comm, dist_one_right]
#align dist_one_left dist_one_left
#align dist_zero_left dist_zero_left
@[to_additive]
theorem Isometry.norm_map_of_map_one {f : E → F} (hi : Isometry f) (h₁ : f 1 = 1) (x : E) :
‖f x‖ = ‖x‖ := by rw [← dist_one_right, ← h₁, hi.dist_eq, dist_one_right]
#align isometry.norm_map_of_map_one Isometry.norm_map_of_map_one
#align isometry.norm_map_of_map_zero Isometry.norm_map_of_map_zero
@[to_additive (attr := simp) comap_norm_atTop]
theorem comap_norm_atTop' : comap norm atTop = cobounded E := by
simpa only [dist_one_right] using comap_dist_right_atTop (1 : E)
@[to_additive Filter.HasBasis.cobounded_of_norm]
lemma Filter.HasBasis.cobounded_of_norm' {ι : Sort*} {p : ι → Prop} {s : ι → Set ℝ}
(h : HasBasis atTop p s) : HasBasis (cobounded E) p fun i ↦ norm ⁻¹' s i :=
comap_norm_atTop' (E := E) ▸ h.comap _
@[to_additive Filter.hasBasis_cobounded_norm]
lemma Filter.hasBasis_cobounded_norm' : HasBasis (cobounded E) (fun _ ↦ True) ({x | · ≤ ‖x‖}) :=
atTop_basis.cobounded_of_norm'
@[to_additive (attr := simp) tendsto_norm_atTop_iff_cobounded]
theorem tendsto_norm_atTop_iff_cobounded' {f : α → E} {l : Filter α} :
Tendsto (‖f ·‖) l atTop ↔ Tendsto f l (cobounded E) := by
rw [← comap_norm_atTop', tendsto_comap_iff]; rfl
@[to_additive tendsto_norm_cobounded_atTop]
theorem tendsto_norm_cobounded_atTop' : Tendsto norm (cobounded E) atTop :=
tendsto_norm_atTop_iff_cobounded'.2 tendsto_id
@[to_additive eventually_cobounded_le_norm]
lemma eventually_cobounded_le_norm' (a : ℝ) : ∀ᶠ x in cobounded E, a ≤ ‖x‖ :=
tendsto_norm_cobounded_atTop'.eventually_ge_atTop a
@[to_additive tendsto_norm_cocompact_atTop]
theorem tendsto_norm_cocompact_atTop' [ProperSpace E] : Tendsto norm (cocompact E) atTop :=
cobounded_eq_cocompact (α := E) ▸ tendsto_norm_cobounded_atTop'
#align tendsto_norm_cocompact_at_top' tendsto_norm_cocompact_atTop'
#align tendsto_norm_cocompact_at_top tendsto_norm_cocompact_atTop
@[to_additive]
theorem norm_div_rev (a b : E) : ‖a / b‖ = ‖b / a‖ := by
simpa only [dist_eq_norm_div] using dist_comm a b
#align norm_div_rev norm_div_rev
#align norm_sub_rev norm_sub_rev
@[to_additive (attr := simp) norm_neg]
theorem norm_inv' (a : E) : ‖a⁻¹‖ = ‖a‖ := by simpa using norm_div_rev 1 a
#align norm_inv' norm_inv'
#align norm_neg norm_neg
open scoped symmDiff in
@[to_additive]
theorem dist_mulIndicator (s t : Set α) (f : α → E) (x : α) :
dist (s.mulIndicator f x) (t.mulIndicator f x) = ‖(s ∆ t).mulIndicator f x‖ := by
rw [dist_eq_norm_div, Set.apply_mulIndicator_symmDiff norm_inv']
@[to_additive (attr := simp)]
theorem dist_mul_self_right (a b : E) : dist b (a * b) = ‖a‖ := by
rw [← dist_one_left, ← dist_mul_right 1 a b, one_mul]
#align dist_mul_self_right dist_mul_self_right
#align dist_add_self_right dist_add_self_right
@[to_additive (attr := simp)]
theorem dist_mul_self_left (a b : E) : dist (a * b) b = ‖a‖ := by
rw [dist_comm, dist_mul_self_right]
#align dist_mul_self_left dist_mul_self_left
#align dist_add_self_left dist_add_self_left
@[to_additive (attr := simp)]
theorem dist_div_eq_dist_mul_left (a b c : E) : dist (a / b) c = dist a (c * b) := by
rw [← dist_mul_right _ _ b, div_mul_cancel]
#align dist_div_eq_dist_mul_left dist_div_eq_dist_mul_left
#align dist_sub_eq_dist_add_left dist_sub_eq_dist_add_left
@[to_additive (attr := simp)]
theorem dist_div_eq_dist_mul_right (a b c : E) : dist a (b / c) = dist (a * c) b := by
rw [← dist_mul_right _ _ c, div_mul_cancel]
#align dist_div_eq_dist_mul_right dist_div_eq_dist_mul_right
#align dist_sub_eq_dist_add_right dist_sub_eq_dist_add_right
@[to_additive (attr := simp)]
lemma Filter.inv_cobounded : (cobounded E)⁻¹ = cobounded E := by
simp only [← comap_norm_atTop', ← Filter.comap_inv, comap_comap, (· ∘ ·), norm_inv']
@[to_additive "In a (semi)normed group, negation `x ↦ -x` tends to infinity at infinity."]
theorem Filter.tendsto_inv_cobounded : Tendsto Inv.inv (cobounded E) (cobounded E) :=
inv_cobounded.le
#align filter.tendsto_inv_cobounded Filter.tendsto_inv_cobounded
#align filter.tendsto_neg_cobounded Filter.tendsto_neg_cobounded
@[to_additive norm_add_le "**Triangle inequality** for the norm."]
theorem norm_mul_le' (a b : E) : ‖a * b‖ ≤ ‖a‖ + ‖b‖ := by
simpa [dist_eq_norm_div] using dist_triangle a 1 b⁻¹
#align norm_mul_le' norm_mul_le'
#align norm_add_le norm_add_le
@[to_additive]
theorem norm_mul_le_of_le (h₁ : ‖a₁‖ ≤ r₁) (h₂ : ‖a₂‖ ≤ r₂) : ‖a₁ * a₂‖ ≤ r₁ + r₂ :=
(norm_mul_le' a₁ a₂).trans <| add_le_add h₁ h₂
#align norm_mul_le_of_le norm_mul_le_of_le
#align norm_add_le_of_le norm_add_le_of_le
@[to_additive norm_add₃_le]
theorem norm_mul₃_le (a b c : E) : ‖a * b * c‖ ≤ ‖a‖ + ‖b‖ + ‖c‖ :=
norm_mul_le_of_le (norm_mul_le' _ _) le_rfl
#align norm_mul₃_le norm_mul₃_le
#align norm_add₃_le norm_add₃_le
@[to_additive]
lemma norm_div_le_norm_div_add_norm_div (a b c : E) : ‖a / c‖ ≤ ‖a / b‖ + ‖b / c‖ := by
simpa only [dist_eq_norm_div] using dist_triangle a b c
@[to_additive (attr := simp) norm_nonneg]
theorem norm_nonneg' (a : E) : 0 ≤ ‖a‖ := by
rw [← dist_one_right]
exact dist_nonneg
#align norm_nonneg' norm_nonneg'
#align norm_nonneg norm_nonneg
@[to_additive (attr := simp) abs_norm]
theorem abs_norm' (z : E) : |‖z‖| = ‖z‖ := abs_of_nonneg <| norm_nonneg' _
#align abs_norm abs_norm
@[to_additive (attr := simp) norm_zero]
theorem norm_one' : ‖(1 : E)‖ = 0 := by rw [← dist_one_right, dist_self]
#align norm_one' norm_one'
#align norm_zero norm_zero
@[to_additive]
theorem ne_one_of_norm_ne_zero : ‖a‖ ≠ 0 → a ≠ 1 :=
mt <| by
rintro rfl
exact norm_one'
#align ne_one_of_norm_ne_zero ne_one_of_norm_ne_zero
#align ne_zero_of_norm_ne_zero ne_zero_of_norm_ne_zero
@[to_additive (attr := nontriviality) norm_of_subsingleton]
theorem norm_of_subsingleton' [Subsingleton E] (a : E) : ‖a‖ = 0 := by
rw [Subsingleton.elim a 1, norm_one']
#align norm_of_subsingleton' norm_of_subsingleton'
#align norm_of_subsingleton norm_of_subsingleton
@[to_additive zero_lt_one_add_norm_sq]
theorem zero_lt_one_add_norm_sq' (x : E) : 0 < 1 + ‖x‖ ^ 2 := by
positivity
#align zero_lt_one_add_norm_sq' zero_lt_one_add_norm_sq'
#align zero_lt_one_add_norm_sq zero_lt_one_add_norm_sq
@[to_additive]
theorem norm_div_le (a b : E) : ‖a / b‖ ≤ ‖a‖ + ‖b‖ := by
simpa [dist_eq_norm_div] using dist_triangle a 1 b
#align norm_div_le norm_div_le
#align norm_sub_le norm_sub_le
@[to_additive]
theorem norm_div_le_of_le {r₁ r₂ : ℝ} (H₁ : ‖a₁‖ ≤ r₁) (H₂ : ‖a₂‖ ≤ r₂) : ‖a₁ / a₂‖ ≤ r₁ + r₂ :=
(norm_div_le a₁ a₂).trans <| add_le_add H₁ H₂
#align norm_div_le_of_le norm_div_le_of_le
#align norm_sub_le_of_le norm_sub_le_of_le
@[to_additive dist_le_norm_add_norm]
theorem dist_le_norm_add_norm' (a b : E) : dist a b ≤ ‖a‖ + ‖b‖ := by
rw [dist_eq_norm_div]
apply norm_div_le
#align dist_le_norm_add_norm' dist_le_norm_add_norm'
#align dist_le_norm_add_norm dist_le_norm_add_norm
@[to_additive abs_norm_sub_norm_le]
| Mathlib/Analysis/Normed/Group/Basic.lean | 634 | 635 | theorem abs_norm_sub_norm_le' (a b : E) : |‖a‖ - ‖b‖| ≤ ‖a / b‖ := by |
simpa [dist_eq_norm_div] using abs_dist_sub_le a b 1
|
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
#align_import group_theory.perm.sign from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u v
open Equiv Function Fintype Finset
variable {α : Type u} {β : Type v}
-- An example on how to determine the order of an element of a finite group.
example : orderOf (-1 : ℤˣ) = 2 :=
orderOf_eq_prime (Int.units_sq _) (by decide)
namespace Equiv.Perm
| Mathlib/GroupTheory/Perm/Finite.lean | 57 | 65 | theorem perm_inv_on_of_perm_on_finset {s : Finset α} {f : Perm α} (h : ∀ x ∈ s, f x ∈ s) {y : α}
(hy : y ∈ s) : f⁻¹ y ∈ s := by |
have h0 : ∀ y ∈ s, ∃ (x : _) (hx : x ∈ s), y = (fun i (_ : i ∈ s) => f i) x hx :=
Finset.surj_on_of_inj_on_of_card_le (fun x hx => (fun i _ => f i) x hx) (fun a ha => h a ha)
(fun a₁ a₂ ha₁ ha₂ heq => (Equiv.apply_eq_iff_eq f).mp heq) rfl.ge
obtain ⟨y2, hy2, heq⟩ := h0 y hy
convert hy2
rw [heq]
simp only [inv_apply_self]
|
import Batteries.Tactic.Alias
import Batteries.Data.Nat.Basic
namespace Nat
@[simp] theorem recAux_zero {motive : Nat → Sort _} (zero : motive 0)
(succ : ∀ n, motive n → motive (n+1)) :
Nat.recAux zero succ 0 = zero := rfl
theorem recAux_succ {motive : Nat → Sort _} (zero : motive 0)
(succ : ∀ n, motive n → motive (n+1)) (n) :
Nat.recAux zero succ (n+1) = succ n (Nat.recAux zero succ n) := rfl
@[simp] theorem recAuxOn_zero {motive : Nat → Sort _} (zero : motive 0)
(succ : ∀ n, motive n → motive (n+1)) :
Nat.recAuxOn 0 zero succ = zero := rfl
theorem recAuxOn_succ {motive : Nat → Sort _} (zero : motive 0)
(succ : ∀ n, motive n → motive (n+1)) (n) :
Nat.recAuxOn (n+1) zero succ = succ n (Nat.recAuxOn n zero succ) := rfl
@[simp] theorem casesAuxOn_zero {motive : Nat → Sort _} (zero : motive 0)
(succ : ∀ n, motive (n+1)) :
Nat.casesAuxOn 0 zero succ = zero := rfl
theorem casesAuxOn_succ {motive : Nat → Sort _} (zero : motive 0)
(succ : ∀ n, motive (n+1)) (n) :
Nat.casesAuxOn (n+1) zero succ = succ n := rfl
| .lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean | 44 | 46 | theorem strongRec_eq {motive : Nat → Sort _} (ind : ∀ n, (∀ m, m < n → motive m) → motive n)
(t : Nat) : Nat.strongRec ind t = ind t fun m _ => Nat.strongRec ind m := by |
conv => lhs; unfold Nat.strongRec
|
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open CategoryTheory.Limits
open CategoryTheory.MonoidalCategory
variable (C : Type*) [Category C] [Preadditive C] [MonoidalCategory C]
class MonoidalPreadditive : Prop where
whiskerLeft_zero : ∀ {X Y Z : C}, X ◁ (0 : Y ⟶ Z) = 0 := by aesop_cat
zero_whiskerRight : ∀ {X Y Z : C}, (0 : Y ⟶ Z) ▷ X = 0 := by aesop_cat
whiskerLeft_add : ∀ {X Y Z : C} (f g : Y ⟶ Z), X ◁ (f + g) = X ◁ f + X ◁ g := by aesop_cat
add_whiskerRight : ∀ {X Y Z : C} (f g : Y ⟶ Z), (f + g) ▷ X = f ▷ X + g ▷ X := by aesop_cat
#align category_theory.monoidal_preadditive CategoryTheory.MonoidalPreadditive
attribute [simp] MonoidalPreadditive.whiskerLeft_zero MonoidalPreadditive.zero_whiskerRight
attribute [simp] MonoidalPreadditive.whiskerLeft_add MonoidalPreadditive.add_whiskerRight
variable {C}
variable [MonoidalPreadditive C]
instance tensorLeft_additive (X : C) : (tensorLeft X).Additive where
#align category_theory.tensor_left_additive CategoryTheory.tensorLeft_additive
instance tensorRight_additive (X : C) : (tensorRight X).Additive where
#align category_theory.tensor_right_additive CategoryTheory.tensorRight_additive
instance tensoringLeft_additive (X : C) : ((tensoringLeft C).obj X).Additive where
#align category_theory.tensoring_left_additive CategoryTheory.tensoringLeft_additive
instance tensoringRight_additive (X : C) : ((tensoringRight C).obj X).Additive where
#align category_theory.tensoring_right_additive CategoryTheory.tensoringRight_additive
theorem monoidalPreadditive_of_faithful {D} [Category D] [Preadditive D] [MonoidalCategory D]
(F : MonoidalFunctor D C) [F.Faithful] [F.Additive] :
MonoidalPreadditive D :=
{ whiskerLeft_zero := by
intros
apply F.toFunctor.map_injective
simp [F.map_whiskerLeft]
zero_whiskerRight := by
intros
apply F.toFunctor.map_injective
simp [F.map_whiskerRight]
whiskerLeft_add := by
intros
apply F.toFunctor.map_injective
simp only [F.map_whiskerLeft, Functor.map_add, Preadditive.comp_add, Preadditive.add_comp,
MonoidalPreadditive.whiskerLeft_add]
add_whiskerRight := by
intros
apply F.toFunctor.map_injective
simp only [F.map_whiskerRight, Functor.map_add, Preadditive.comp_add, Preadditive.add_comp,
MonoidalPreadditive.add_whiskerRight] }
#align category_theory.monoidal_preadditive_of_faithful CategoryTheory.monoidalPreadditive_of_faithful
theorem whiskerLeft_sum (P : C) {Q R : C} {J : Type*} (s : Finset J) (g : J → (Q ⟶ R)) :
P ◁ ∑ j ∈ s, g j = ∑ j ∈ s, P ◁ g j :=
map_sum ((tensoringLeft C).obj P).mapAddHom g s
theorem sum_whiskerRight {Q R : C} {J : Type*} (s : Finset J) (g : J → (Q ⟶ R)) (P : C) :
(∑ j ∈ s, g j) ▷ P = ∑ j ∈ s, g j ▷ P :=
map_sum ((tensoringRight C).obj P).mapAddHom g s
theorem tensor_sum {P Q R S : C} {J : Type*} (s : Finset J) (f : P ⟶ Q) (g : J → (R ⟶ S)) :
(f ⊗ ∑ j ∈ s, g j) = ∑ j ∈ s, f ⊗ g j := by
simp only [tensorHom_def, whiskerLeft_sum, Preadditive.comp_sum]
#align category_theory.tensor_sum CategoryTheory.tensor_sum
theorem sum_tensor {P Q R S : C} {J : Type*} (s : Finset J) (f : P ⟶ Q) (g : J → (R ⟶ S)) :
(∑ j ∈ s, g j) ⊗ f = ∑ j ∈ s, g j ⊗ f := by
simp only [tensorHom_def, sum_whiskerRight, Preadditive.sum_comp]
#align category_theory.sum_tensor CategoryTheory.sum_tensor
-- In a closed monoidal category, this would hold because
-- `tensorLeft X` is a left adjoint and hence preserves all colimits.
-- In any case it is true in any preadditive category.
instance (X : C) : PreservesFiniteBiproducts (tensorLeft X) where
preserves {J} :=
{ preserves := fun {f} =>
{ preserves := fun {b} i => isBilimitOfTotal _ (by
dsimp
simp_rw [← id_tensorHom]
simp only [← tensor_comp, Category.comp_id, ← tensor_sum, ← tensor_id,
IsBilimit.total i]) } }
instance (X : C) : PreservesFiniteBiproducts (tensorRight X) where
preserves {J} :=
{ preserves := fun {f} =>
{ preserves := fun {b} i => isBilimitOfTotal _ (by
dsimp
simp_rw [← tensorHom_id]
simp only [← tensor_comp, Category.comp_id, ← sum_tensor, ← tensor_id,
IsBilimit.total i]) } }
variable [HasFiniteBiproducts C]
def leftDistributor {J : Type} [Fintype J] (X : C) (f : J → C) : X ⊗ ⨁ f ≅ ⨁ fun j => X ⊗ f j :=
(tensorLeft X).mapBiproduct f
#align category_theory.left_distributor CategoryTheory.leftDistributor
| Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 151 | 158 | theorem leftDistributor_hom {J : Type} [Fintype J] (X : C) (f : J → C) :
(leftDistributor X f).hom =
∑ j : J, (X ◁ biproduct.π f j) ≫ biproduct.ι (fun j => X ⊗ f j) j := by |
ext
dsimp [leftDistributor, Functor.mapBiproduct, Functor.mapBicone]
erw [biproduct.lift_π]
simp only [Preadditive.sum_comp, Category.assoc, biproduct.ι_π, comp_dite, comp_zero,
Finset.sum_dite_eq', Finset.mem_univ, ite_true, eqToHom_refl, Category.comp_id]
|
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"
noncomputable section
namespace Zspan
open MeasureTheory MeasurableSet Submodule Bornology
variable {E ι : Type*}
section NormedLatticeField
variable {K : Type*} [NormedLinearOrderedField K]
variable [NormedAddCommGroup E] [NormedSpace K E]
variable (b : Basis ι K E)
theorem span_top : span K (span ℤ (Set.range b) : Set E) = ⊤ := by simp [span_span_of_tower]
def fundamentalDomain : Set E := {m | ∀ i, b.repr m i ∈ Set.Ico (0 : K) 1}
#align zspan.fundamental_domain Zspan.fundamentalDomain
@[simp]
theorem mem_fundamentalDomain {m : E} :
m ∈ fundamentalDomain b ↔ ∀ i, b.repr m i ∈ Set.Ico (0 : K) 1 := Iff.rfl
#align zspan.mem_fundamental_domain Zspan.mem_fundamentalDomain
theorem map_fundamentalDomain {F : Type*} [NormedAddCommGroup F] [NormedSpace K F] (f : E ≃ₗ[K] F) :
f '' (fundamentalDomain b) = fundamentalDomain (b.map f) := by
ext x
rw [mem_fundamentalDomain, Basis.map_repr, LinearEquiv.trans_apply, ← mem_fundamentalDomain,
show f.symm x = f.toEquiv.symm x by rfl, ← Set.mem_image_equiv]
rfl
@[simp]
theorem fundamentalDomain_reindex {ι' : Type*} (e : ι ≃ ι') :
fundamentalDomain (b.reindex e) = fundamentalDomain b := by
ext
simp_rw [mem_fundamentalDomain, Basis.repr_reindex_apply]
rw [Equiv.forall_congr' e]
simp_rw [implies_true]
lemma fundamentalDomain_pi_basisFun [Fintype ι] :
fundamentalDomain (Pi.basisFun ℝ ι) = Set.pi Set.univ fun _ : ι ↦ Set.Ico (0 : ℝ) 1 := by
ext; simp
variable [FloorRing K]
section Fintype
variable [Fintype ι]
def floor (m : E) : span ℤ (Set.range b) := ∑ i, ⌊b.repr m i⌋ • b.restrictScalars ℤ i
#align zspan.floor Zspan.floor
def ceil (m : E) : span ℤ (Set.range b) := ∑ i, ⌈b.repr m i⌉ • b.restrictScalars ℤ i
#align zspan.ceil Zspan.ceil
@[simp]
theorem repr_floor_apply (m : E) (i : ι) : b.repr (floor b m) i = ⌊b.repr m i⌋ := by
classical simp only [floor, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,
Finset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,
Finset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, map_sum]
#align zspan.repr_floor_apply Zspan.repr_floor_apply
@[simp]
theorem repr_ceil_apply (m : E) (i : ι) : b.repr (ceil b m) i = ⌈b.repr m i⌉ := by
classical simp only [ceil, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,
Finset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,
Finset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, map_sum]
#align zspan.repr_ceil_apply Zspan.repr_ceil_apply
@[simp]
theorem floor_eq_self_of_mem (m : E) (h : m ∈ span ℤ (Set.range b)) : (floor b m : E) = m := by
apply b.ext_elem
simp_rw [repr_floor_apply b]
intro i
obtain ⟨z, hz⟩ := (b.mem_span_iff_repr_mem ℤ _).mp h i
rw [← hz]
exact congr_arg (Int.cast : ℤ → K) (Int.floor_intCast z)
#align zspan.floor_eq_self_of_mem Zspan.floor_eq_self_of_mem
@[simp]
theorem ceil_eq_self_of_mem (m : E) (h : m ∈ span ℤ (Set.range b)) : (ceil b m : E) = m := by
apply b.ext_elem
simp_rw [repr_ceil_apply b]
intro i
obtain ⟨z, hz⟩ := (b.mem_span_iff_repr_mem ℤ _).mp h i
rw [← hz]
exact congr_arg (Int.cast : ℤ → K) (Int.ceil_intCast z)
#align zspan.ceil_eq_self_of_mem Zspan.ceil_eq_self_of_mem
def fract (m : E) : E := m - floor b m
#align zspan.fract Zspan.fract
theorem fract_apply (m : E) : fract b m = m - floor b m := rfl
#align zspan.fract_apply Zspan.fract_apply
@[simp]
theorem repr_fract_apply (m : E) (i : ι) : b.repr (fract b m) i = Int.fract (b.repr m i) := by
rw [fract, map_sub, Finsupp.coe_sub, Pi.sub_apply, repr_floor_apply, Int.fract]
#align zspan.repr_fract_apply Zspan.repr_fract_apply
@[simp]
theorem fract_fract (m : E) : fract b (fract b m) = fract b m :=
Basis.ext_elem b fun _ => by classical simp only [repr_fract_apply, Int.fract_fract]
#align zspan.fract_fract Zspan.fract_fract
@[simp]
theorem fract_zspan_add (m : E) {v : E} (h : v ∈ span ℤ (Set.range b)) :
fract b (v + m) = fract b m := by
classical
refine (Basis.ext_elem_iff b).mpr fun i => ?_
simp_rw [repr_fract_apply, Int.fract_eq_fract]
use (b.restrictScalars ℤ).repr ⟨v, h⟩ i
rw [map_add, Finsupp.coe_add, Pi.add_apply, add_tsub_cancel_right,
← eq_intCast (algebraMap ℤ K) _, Basis.restrictScalars_repr_apply, coe_mk]
#align zspan.fract_zspan_add Zspan.fract_zspan_add
@[simp]
theorem fract_add_zspan (m : E) {v : E} (h : v ∈ span ℤ (Set.range b)) :
fract b (m + v) = fract b m := by rw [add_comm, fract_zspan_add b m h]
#align zspan.fract_add_zspan Zspan.fract_add_zspan
variable {b}
theorem fract_eq_self {x : E} : fract b x = x ↔ x ∈ fundamentalDomain b := by
classical simp only [Basis.ext_elem_iff b, repr_fract_apply, Int.fract_eq_self,
mem_fundamentalDomain, Set.mem_Ico]
#align zspan.fract_eq_self Zspan.fract_eq_self
variable (b)
theorem fract_mem_fundamentalDomain (x : E) : fract b x ∈ fundamentalDomain b :=
fract_eq_self.mp (fract_fract b _)
#align zspan.fract_mem_fundamental_domain Zspan.fract_mem_fundamentalDomain
def fractRestrict (x : E) : fundamentalDomain b := ⟨fract b x, fract_mem_fundamentalDomain b x⟩
theorem fractRestrict_surjective : Function.Surjective (fractRestrict b) :=
fun x => ⟨↑x, Subtype.eq (fract_eq_self.mpr (Subtype.mem x))⟩
@[simp]
theorem fractRestrict_apply (x : E) : (fractRestrict b x : E) = fract b x := rfl
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]
#align zspan.fract_eq_fract Zspan.fract_eq_fract
theorem norm_fract_le [HasSolidNorm K] (m : E) : ‖fract b m‖ ≤ ∑ i, ‖b i‖ := by
classical
calc
‖fract b m‖ = ‖∑ i, b.repr (fract b m) i • b i‖ := by rw [b.sum_repr]
_ = ‖∑ i, Int.fract (b.repr m i) • b i‖ := by simp_rw [repr_fract_apply]
_ ≤ ∑ i, ‖Int.fract (b.repr m i) • b i‖ := norm_sum_le _ _
_ = ∑ i, ‖Int.fract (b.repr m i)‖ * ‖b i‖ := by simp_rw [norm_smul]
_ ≤ ∑ i, ‖b i‖ := Finset.sum_le_sum fun i _ => ?_
suffices ‖Int.fract ((b.repr m) i)‖ ≤ 1 by
convert mul_le_mul_of_nonneg_right this (norm_nonneg _ : 0 ≤ ‖b i‖)
exact (one_mul _).symm
rw [(norm_one.symm : 1 = ‖(1 : K)‖)]
apply norm_le_norm_of_abs_le_abs
rw [abs_one, Int.abs_fract]
exact le_of_lt (Int.fract_lt_one _)
#align zspan.norm_fract_le Zspan.norm_fract_le
| Mathlib/Algebra/Module/Zlattice/Basic.lean | 235 | 240 | theorem fundamentalDomain_isBounded [Finite ι] [HasSolidNorm K] :
IsBounded (fundamentalDomain b) := by |
cases nonempty_fintype ι
refine isBounded_iff_forall_norm_le.2 ⟨∑ j, ‖b j‖, fun x hx ↦ ?_⟩
rw [← fract_eq_self.mpr hx]
apply norm_fract_le
|
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scoped Pointwise
universe u v w x
namespace QuotientGroup
variable {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [Group H]
{M : Type x} [Monoid M]
@[to_additive "The additive congruence relation generated by a normal additive subgroup."]
protected def con : Con G where
toSetoid := leftRel N
mul' := @fun a b c d hab hcd => by
rw [leftRel_eq] at hab hcd ⊢
dsimp only
calc
(a * c)⁻¹ * (b * d) = c⁻¹ * (a⁻¹ * b) * c⁻¹⁻¹ * (c⁻¹ * d) := by
simp only [mul_inv_rev, mul_assoc, inv_mul_cancel_left]
_ ∈ N := N.mul_mem (nN.conj_mem _ hab _) hcd
#align quotient_group.con QuotientGroup.con
#align quotient_add_group.con QuotientAddGroup.con
@[to_additive]
instance Quotient.group : Group (G ⧸ N) :=
(QuotientGroup.con N).group
#align quotient_group.quotient.group QuotientGroup.Quotient.group
#align quotient_add_group.quotient.add_group QuotientAddGroup.Quotient.addGroup
@[to_additive "The additive group homomorphism from `G` to `G/N`."]
def mk' : G →* G ⧸ N :=
MonoidHom.mk' QuotientGroup.mk fun _ _ => rfl
#align quotient_group.mk' QuotientGroup.mk'
#align quotient_add_group.mk' QuotientAddGroup.mk'
@[to_additive (attr := simp)]
theorem coe_mk' : (mk' N : G → G ⧸ N) = mk :=
rfl
#align quotient_group.coe_mk' QuotientGroup.coe_mk'
#align quotient_add_group.coe_mk' QuotientAddGroup.coe_mk'
@[to_additive (attr := simp)]
theorem mk'_apply (x : G) : mk' N x = x :=
rfl
#align quotient_group.mk'_apply QuotientGroup.mk'_apply
#align quotient_add_group.mk'_apply QuotientAddGroup.mk'_apply
@[to_additive]
theorem mk'_surjective : Surjective <| mk' N :=
@mk_surjective _ _ N
#align quotient_group.mk'_surjective QuotientGroup.mk'_surjective
#align quotient_add_group.mk'_surjective QuotientAddGroup.mk'_surjective
@[to_additive]
theorem mk'_eq_mk' {x y : G} : mk' N x = mk' N y ↔ ∃ z ∈ N, x * z = y :=
QuotientGroup.eq'.trans <| by
simp only [← _root_.eq_inv_mul_iff_mul_eq, exists_prop, exists_eq_right]
#align quotient_group.mk'_eq_mk' QuotientGroup.mk'_eq_mk'
#align quotient_add_group.mk'_eq_mk' QuotientAddGroup.mk'_eq_mk'
open scoped Pointwise in
@[to_additive]
theorem sound (U : Set (G ⧸ N)) (g : N.op) :
g • (mk' N) ⁻¹' U = (mk' N) ⁻¹' U := by
ext x
simp only [Set.mem_preimage, Set.mem_smul_set_iff_inv_smul_mem]
congr! 1
exact Quotient.sound ⟨g⁻¹, rfl⟩
@[to_additive (attr := ext 1100) "Two `AddMonoidHom`s from an additive quotient group are equal if
their compositions with `AddQuotientGroup.mk'` are equal.
See note [partially-applied ext lemmas]. "]
theorem monoidHom_ext ⦃f g : G ⧸ N →* M⦄ (h : f.comp (mk' N) = g.comp (mk' N)) : f = g :=
MonoidHom.ext fun x => QuotientGroup.induction_on x <| (DFunLike.congr_fun h : _)
#align quotient_group.monoid_hom_ext QuotientGroup.monoidHom_ext
#align quotient_add_group.add_monoid_hom_ext QuotientAddGroup.addMonoidHom_ext
@[to_additive (attr := simp)]
theorem eq_one_iff {N : Subgroup G} [nN : N.Normal] (x : G) : (x : G ⧸ N) = 1 ↔ x ∈ N := by
refine QuotientGroup.eq.trans ?_
rw [mul_one, Subgroup.inv_mem_iff]
#align quotient_group.eq_one_iff QuotientGroup.eq_one_iff
#align quotient_add_group.eq_zero_iff QuotientAddGroup.eq_zero_iff
@[to_additive]
theorem ker_le_range_iff {I : Type w} [Group I] (f : G →* H) [f.range.Normal] (g : H →* I) :
g.ker ≤ f.range ↔ (mk' f.range).comp g.ker.subtype = 1 :=
⟨fun h => MonoidHom.ext fun ⟨_, hx⟩ => (eq_one_iff _).mpr <| h hx,
fun h x hx => (eq_one_iff _).mp <| by exact DFunLike.congr_fun h ⟨x, hx⟩⟩
@[to_additive (attr := simp)]
theorem ker_mk' : MonoidHom.ker (QuotientGroup.mk' N : G →* G ⧸ N) = N :=
Subgroup.ext eq_one_iff
#align quotient_group.ker_mk QuotientGroup.ker_mk'
#align quotient_add_group.ker_mk QuotientAddGroup.ker_mk'
-- Porting note: I think this is misnamed without the prime
@[to_additive]
theorem eq_iff_div_mem {N : Subgroup G} [nN : N.Normal] {x y : G} :
(x : G ⧸ N) = y ↔ x / y ∈ N := by
refine eq_comm.trans (QuotientGroup.eq.trans ?_)
rw [nN.mem_comm_iff, div_eq_mul_inv]
#align quotient_group.eq_iff_div_mem QuotientGroup.eq_iff_div_mem
#align quotient_add_group.eq_iff_sub_mem QuotientAddGroup.eq_iff_sub_mem
-- for commutative groups we don't need normality assumption
@[to_additive]
instance Quotient.commGroup {G : Type*} [CommGroup G] (N : Subgroup G) : CommGroup (G ⧸ N) :=
{ toGroup := @QuotientGroup.Quotient.group _ _ N N.normal_of_comm
mul_comm := fun a b => Quotient.inductionOn₂' a b fun a b => congr_arg mk (mul_comm a b) }
#align quotient_group.quotient.comm_group QuotientGroup.Quotient.commGroup
#align quotient_add_group.quotient.add_comm_group QuotientAddGroup.Quotient.addCommGroup
local notation " Q " => G ⧸ N
@[to_additive (attr := simp)]
theorem mk_one : ((1 : G) : Q) = 1 :=
rfl
#align quotient_group.coe_one QuotientGroup.mk_one
#align quotient_add_group.coe_zero QuotientAddGroup.mk_zero
@[to_additive (attr := simp)]
theorem mk_mul (a b : G) : ((a * b : G) : Q) = a * b :=
rfl
#align quotient_group.coe_mul QuotientGroup.mk_mul
#align quotient_add_group.coe_add QuotientAddGroup.mk_add
@[to_additive (attr := simp)]
theorem mk_inv (a : G) : ((a⁻¹ : G) : Q) = (a : Q)⁻¹ :=
rfl
#align quotient_group.coe_inv QuotientGroup.mk_inv
#align quotient_add_group.coe_neg QuotientAddGroup.mk_neg
@[to_additive (attr := simp)]
theorem mk_div (a b : G) : ((a / b : G) : Q) = a / b :=
rfl
#align quotient_group.coe_div QuotientGroup.mk_div
#align quotient_add_group.coe_sub QuotientAddGroup.mk_sub
@[to_additive (attr := simp)]
theorem mk_pow (a : G) (n : ℕ) : ((a ^ n : G) : Q) = (a : Q) ^ n :=
rfl
#align quotient_group.coe_pow QuotientGroup.mk_pow
#align quotient_add_group.coe_nsmul QuotientAddGroup.mk_nsmul
@[to_additive (attr := simp)]
theorem mk_zpow (a : G) (n : ℤ) : ((a ^ n : G) : Q) = (a : Q) ^ n :=
rfl
#align quotient_group.coe_zpow QuotientGroup.mk_zpow
#align quotient_add_group.coe_zsmul QuotientAddGroup.mk_zsmul
@[to_additive (attr := simp)]
theorem mk_prod {G ι : Type*} [CommGroup G] (N : Subgroup G) (s : Finset ι) {f : ι → G} :
((Finset.prod s f : G) : G ⧸ N) = Finset.prod s (fun i => (f i : G ⧸ N)) :=
map_prod (QuotientGroup.mk' N) _ _
@[to_additive (attr := simp)] lemma map_mk'_self : N.map (mk' N) = ⊥ := by aesop
@[to_additive "An `AddGroup` homomorphism `φ : G →+ M` with `N ⊆ ker(φ)` descends (i.e. `lift`s)
to a group homomorphism `G/N →* M`."]
def lift (φ : G →* M) (HN : N ≤ φ.ker) : Q →* M :=
(QuotientGroup.con N).lift φ fun x y h => by
simp only [QuotientGroup.con, leftRel_apply, Con.rel_mk] at h
rw [Con.ker_rel]
calc
φ x = φ (y * (x⁻¹ * y)⁻¹) := by rw [mul_inv_rev, inv_inv, mul_inv_cancel_left]
_ = φ y := by rw [φ.map_mul, HN (N.inv_mem h), mul_one]
#align quotient_group.lift QuotientGroup.lift
#align quotient_add_group.lift QuotientAddGroup.lift
@[to_additive (attr := simp)]
theorem lift_mk {φ : G →* M} (HN : N ≤ φ.ker) (g : G) : lift N φ HN (g : Q) = φ g :=
rfl
#align quotient_group.lift_mk QuotientGroup.lift_mk
#align quotient_add_group.lift_mk QuotientAddGroup.lift_mk
@[to_additive (attr := simp)]
theorem lift_mk' {φ : G →* M} (HN : N ≤ φ.ker) (g : G) : lift N φ HN (mk g : Q) = φ g :=
rfl
-- TODO: replace `mk` with `mk'`)
#align quotient_group.lift_mk' QuotientGroup.lift_mk'
#align quotient_add_group.lift_mk' QuotientAddGroup.lift_mk'
@[to_additive (attr := simp)]
theorem lift_quot_mk {φ : G →* M} (HN : N ≤ φ.ker) (g : G) :
lift N φ HN (Quot.mk _ g : Q) = φ g :=
rfl
#align quotient_group.lift_quot_mk QuotientGroup.lift_quot_mk
#align quotient_add_group.lift_quot_mk QuotientAddGroup.lift_quot_mk
@[to_additive
"An `AddGroup` homomorphism `f : G →+ H` induces a map `G/N →+ H/M` if `N ⊆ f⁻¹(M)`."]
def map (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ M.comap f) : G ⧸ N →* H ⧸ M := by
refine QuotientGroup.lift N ((mk' M).comp f) ?_
intro x hx
refine QuotientGroup.eq.2 ?_
rw [mul_one, Subgroup.inv_mem_iff]
exact h hx
#align quotient_group.map QuotientGroup.map
#align quotient_add_group.map QuotientAddGroup.map
@[to_additive (attr := simp)]
theorem map_mk (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ M.comap f) (x : G) :
map N M f h ↑x = ↑(f x) :=
rfl
#align quotient_group.map_coe QuotientGroup.map_mk
#align quotient_add_group.map_coe QuotientAddGroup.map_mk
@[to_additive]
theorem map_mk' (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ M.comap f) (x : G) :
map N M f h (mk' _ x) = ↑(f x) :=
rfl
#align quotient_group.map_mk' QuotientGroup.map_mk'
#align quotient_add_group.map_mk' QuotientAddGroup.map_mk'
@[to_additive]
theorem map_id_apply (h : N ≤ Subgroup.comap (MonoidHom.id _) N := (Subgroup.comap_id N).le) (x) :
map N N (MonoidHom.id _) h x = x :=
induction_on' x fun _x => rfl
#align quotient_group.map_id_apply QuotientGroup.map_id_apply
#align quotient_add_group.map_id_apply QuotientAddGroup.map_id_apply
@[to_additive (attr := simp)]
theorem map_id (h : N ≤ Subgroup.comap (MonoidHom.id _) N := (Subgroup.comap_id N).le) :
map N N (MonoidHom.id _) h = MonoidHom.id _ :=
MonoidHom.ext (map_id_apply N h)
#align quotient_group.map_id QuotientGroup.map_id
#align quotient_add_group.map_id QuotientAddGroup.map_id
@[to_additive (attr := simp)]
theorem map_map {I : Type*} [Group I] (M : Subgroup H) (O : Subgroup I) [M.Normal] [O.Normal]
(f : G →* H) (g : H →* I) (hf : N ≤ Subgroup.comap f M) (hg : M ≤ Subgroup.comap g O)
(hgf : N ≤ Subgroup.comap (g.comp f) O :=
hf.trans ((Subgroup.comap_mono hg).trans_eq (Subgroup.comap_comap _ _ _)))
(x : G ⧸ N) : map M O g hg (map N M f hf x) = map N O (g.comp f) hgf x := by
refine induction_on' x fun x => ?_
simp only [map_mk, MonoidHom.comp_apply]
#align quotient_group.map_map QuotientGroup.map_map
#align quotient_add_group.map_map QuotientAddGroup.map_map
@[to_additive (attr := simp)]
theorem map_comp_map {I : Type*} [Group I] (M : Subgroup H) (O : Subgroup I) [M.Normal] [O.Normal]
(f : G →* H) (g : H →* I) (hf : N ≤ Subgroup.comap f M) (hg : M ≤ Subgroup.comap g O)
(hgf : N ≤ Subgroup.comap (g.comp f) O :=
hf.trans ((Subgroup.comap_mono hg).trans_eq (Subgroup.comap_comap _ _ _))) :
(map M O g hg).comp (map N M f hf) = map N O (g.comp f) hgf :=
MonoidHom.ext (map_map N M O f g hf hg hgf)
#align quotient_group.map_comp_map QuotientGroup.map_comp_map
#align quotient_add_group.map_comp_map QuotientAddGroup.map_comp_map
variable (φ : G →* H)
open MonoidHom
@[to_additive "The induced map from the quotient by the kernel to the codomain."]
def kerLift : G ⧸ ker φ →* H :=
lift _ φ fun _g => φ.mem_ker.mp
#align quotient_group.ker_lift QuotientGroup.kerLift
#align quotient_add_group.ker_lift QuotientAddGroup.kerLift
@[to_additive (attr := simp)]
theorem kerLift_mk (g : G) : (kerLift φ) g = φ g :=
lift_mk _ _ _
#align quotient_group.ker_lift_mk QuotientGroup.kerLift_mk
#align quotient_add_group.ker_lift_mk QuotientAddGroup.kerLift_mk
@[to_additive (attr := simp)]
theorem kerLift_mk' (g : G) : (kerLift φ) (mk g) = φ g :=
lift_mk' _ _ _
#align quotient_group.ker_lift_mk' QuotientGroup.kerLift_mk'
#align quotient_add_group.ker_lift_mk' QuotientAddGroup.kerLift_mk'
@[to_additive]
theorem kerLift_injective : Injective (kerLift φ) := fun a b =>
Quotient.inductionOn₂' a b fun a b (h : φ a = φ b) =>
Quotient.sound' <| by rw [leftRel_apply, mem_ker, φ.map_mul, ← h, φ.map_inv, inv_mul_self]
#align quotient_group.ker_lift_injective QuotientGroup.kerLift_injective
#align quotient_add_group.ker_lift_injective QuotientAddGroup.kerLift_injective
-- Note that `ker φ` isn't definitionally `ker (φ.rangeRestrict)`
-- so there is a bit of annoying code duplication here
@[to_additive "The induced map from the quotient by the kernel to the range."]
def rangeKerLift : G ⧸ ker φ →* φ.range :=
lift _ φ.rangeRestrict fun g hg => (mem_ker _).mp <| by rwa [ker_rangeRestrict]
#align quotient_group.range_ker_lift QuotientGroup.rangeKerLift
#align quotient_add_group.range_ker_lift QuotientAddGroup.rangeKerLift
@[to_additive]
theorem rangeKerLift_injective : Injective (rangeKerLift φ) := fun a b =>
Quotient.inductionOn₂' a b fun a b (h : φ.rangeRestrict a = φ.rangeRestrict b) =>
Quotient.sound' <| by
rw [leftRel_apply, ← ker_rangeRestrict, mem_ker, φ.rangeRestrict.map_mul, ← h,
φ.rangeRestrict.map_inv, inv_mul_self]
#align quotient_group.range_ker_lift_injective QuotientGroup.rangeKerLift_injective
#align quotient_add_group.range_ker_lift_injective QuotientAddGroup.rangeKerLift_injective
@[to_additive]
theorem rangeKerLift_surjective : Surjective (rangeKerLift φ) := by
rintro ⟨_, g, rfl⟩
use mk g
rfl
#align quotient_group.range_ker_lift_surjective QuotientGroup.rangeKerLift_surjective
#align quotient_add_group.range_ker_lift_surjective QuotientAddGroup.rangeKerLift_surjective
@[to_additive "The first isomorphism theorem (a definition): the canonical isomorphism between
`G/(ker φ)` to `range φ`."]
noncomputable def quotientKerEquivRange : G ⧸ ker φ ≃* range φ :=
MulEquiv.ofBijective (rangeKerLift φ) ⟨rangeKerLift_injective φ, rangeKerLift_surjective φ⟩
#align quotient_group.quotient_ker_equiv_range QuotientGroup.quotientKerEquivRange
#align quotient_add_group.quotient_ker_equiv_range QuotientAddGroup.quotientKerEquivRange
@[to_additive (attr := simps) "The canonical isomorphism `G/(ker φ) ≃+ H` induced by a homomorphism
`φ : G →+ H` with a right inverse `ψ : H → G`."]
def quotientKerEquivOfRightInverse (ψ : H → G) (hφ : RightInverse ψ φ) : G ⧸ ker φ ≃* H :=
{ kerLift φ with
toFun := kerLift φ
invFun := mk ∘ ψ
left_inv := fun x => kerLift_injective φ (by rw [Function.comp_apply, kerLift_mk', hφ])
right_inv := hφ }
#align quotient_group.quotient_ker_equiv_of_right_inverse QuotientGroup.quotientKerEquivOfRightInverse
#align quotient_add_group.quotient_ker_equiv_of_right_inverse QuotientAddGroup.quotientKerEquivOfRightInverse
@[to_additive (attr := simps!) "The canonical isomorphism `G/⊥ ≃+ G`."]
def quotientBot : G ⧸ (⊥ : Subgroup G) ≃* G :=
quotientKerEquivOfRightInverse (MonoidHom.id G) id fun _x => rfl
#align quotient_group.quotient_bot QuotientGroup.quotientBot
#align quotient_add_group.quotient_bot QuotientAddGroup.quotientBot
@[to_additive "The canonical isomorphism `G/(ker φ) ≃+ H` induced by a surjection `φ : G →+ H`.
For a `computable` version, see `QuotientAddGroup.quotientKerEquivOfRightInverse`."]
noncomputable def quotientKerEquivOfSurjective (hφ : Surjective φ) : G ⧸ ker φ ≃* H :=
quotientKerEquivOfRightInverse φ _ hφ.hasRightInverse.choose_spec
#align quotient_group.quotient_ker_equiv_of_surjective QuotientGroup.quotientKerEquivOfSurjective
#align quotient_add_group.quotient_ker_equiv_of_surjective QuotientAddGroup.quotientKerEquivOfSurjective
@[to_additive "If two normal subgroups `M` and `N` of `G` are the same, their quotient groups are
isomorphic."]
def quotientMulEquivOfEq {M N : Subgroup G} [M.Normal] [N.Normal] (h : M = N) : G ⧸ M ≃* G ⧸ N :=
{ Subgroup.quotientEquivOfEq h with
map_mul' := fun q r => Quotient.inductionOn₂' q r fun _g _h => rfl }
#align quotient_group.quotient_mul_equiv_of_eq QuotientGroup.quotientMulEquivOfEq
#align quotient_add_group.quotient_add_equiv_of_eq QuotientAddGroup.quotientAddEquivOfEq
@[to_additive (attr := simp)]
theorem quotientMulEquivOfEq_mk {M N : Subgroup G} [M.Normal] [N.Normal] (h : M = N) (x : G) :
QuotientGroup.quotientMulEquivOfEq h (QuotientGroup.mk x) = QuotientGroup.mk x :=
rfl
#align quotient_group.quotient_mul_equiv_of_eq_mk QuotientGroup.quotientMulEquivOfEq_mk
#align quotient_add_group.quotient_add_equiv_of_eq_mk QuotientAddGroup.quotientAddEquivOfEq_mk
@[to_additive "Let `A', A, B', B` be subgroups of `G`. If `A' ≤ B'` and `A ≤ B`, then there is a map
`A / (A' ⊓ A) →+ B / (B' ⊓ B)` induced by the inclusions."]
def quotientMapSubgroupOfOfLe {A' A B' B : Subgroup G} [_hAN : (A'.subgroupOf A).Normal]
[_hBN : (B'.subgroupOf B).Normal] (h' : A' ≤ B') (h : A ≤ B) :
A ⧸ A'.subgroupOf A →* B ⧸ B'.subgroupOf B :=
map _ _ (Subgroup.inclusion h) <| Subgroup.comap_mono h'
#align quotient_group.quotient_map_subgroup_of_of_le QuotientGroup.quotientMapSubgroupOfOfLe
#align quotient_add_group.quotient_map_add_subgroup_of_of_le QuotientAddGroup.quotientMapAddSubgroupOfOfLe
@[to_additive (attr := simp)]
theorem quotientMapSubgroupOfOfLe_mk {A' A B' B : Subgroup G} [_hAN : (A'.subgroupOf A).Normal]
[_hBN : (B'.subgroupOf B).Normal] (h' : A' ≤ B') (h : A ≤ B) (x : A) :
quotientMapSubgroupOfOfLe h' h x = ↑(Subgroup.inclusion h x : B) :=
rfl
#align quotient_group.quotient_map_subgroup_of_of_le_coe QuotientGroup.quotientMapSubgroupOfOfLe_mk
#align quotient_add_group.quotient_map_add_subgroup_of_of_le_coe QuotientAddGroup.quotientMapAddSubgroupOfOfLe_mk
@[to_additive "Let `A', A, B', B` be subgroups of `G`. If `A' = B'` and `A = B`, then the quotients
`A / (A' ⊓ A)` and `B / (B' ⊓ B)` are isomorphic. Applying this equiv is nicer than rewriting along
the equalities, since the type of `(A'.addSubgroupOf A : AddSubgroup A)` depends on on `A`. "]
def equivQuotientSubgroupOfOfEq {A' A B' B : Subgroup G} [hAN : (A'.subgroupOf A).Normal]
[hBN : (B'.subgroupOf B).Normal] (h' : A' = B') (h : A = B) :
A ⧸ A'.subgroupOf A ≃* B ⧸ B'.subgroupOf B :=
MonoidHom.toMulEquiv (quotientMapSubgroupOfOfLe h'.le h.le) (quotientMapSubgroupOfOfLe h'.ge h.ge)
(by ext ⟨x, hx⟩; rfl)
(by ext ⟨x, hx⟩; rfl)
#align quotient_group.equiv_quotient_subgroup_of_of_eq QuotientGroup.equivQuotientSubgroupOfOfEq
#align quotient_add_group.equiv_quotient_add_subgroup_of_of_eq QuotientAddGroup.equivQuotientAddSubgroupOfOfEq
section trivial
@[to_additive]
| Mathlib/GroupTheory/QuotientGroup.lean | 704 | 707 | theorem subsingleton_quotient_top : Subsingleton (G ⧸ (⊤ : Subgroup G)) := by |
dsimp [HasQuotient.Quotient, QuotientGroup.instHasQuotientSubgroup, Quotient]
rw [leftRel_eq]
exact Trunc.instSubsingletonTrunc
|
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
open Function
open OrderDual (toDual ofDual)
variable {α β : Type*}
namespace Set
section Preorder
variable [Preorder α] {a a₁ a₂ b b₁ b₂ c x : α}
def Ioo (a b : α) :=
{ x | a < x ∧ x < b }
#align set.Ioo Set.Ioo
def Ico (a b : α) :=
{ x | a ≤ x ∧ x < b }
#align set.Ico Set.Ico
def Iio (a : α) :=
{ x | x < a }
#align set.Iio Set.Iio
def Icc (a b : α) :=
{ x | a ≤ x ∧ x ≤ b }
#align set.Icc Set.Icc
def Iic (b : α) :=
{ x | x ≤ b }
#align set.Iic Set.Iic
def Ioc (a b : α) :=
{ x | a < x ∧ x ≤ b }
#align set.Ioc Set.Ioc
def Ici (a : α) :=
{ x | a ≤ x }
#align set.Ici Set.Ici
def Ioi (a : α) :=
{ x | a < x }
#align set.Ioi Set.Ioi
theorem Ioo_def (a b : α) : { x | a < x ∧ x < b } = Ioo a b :=
rfl
#align set.Ioo_def Set.Ioo_def
theorem Ico_def (a b : α) : { x | a ≤ x ∧ x < b } = Ico a b :=
rfl
#align set.Ico_def Set.Ico_def
theorem Iio_def (a : α) : { x | x < a } = Iio a :=
rfl
#align set.Iio_def Set.Iio_def
theorem Icc_def (a b : α) : { x | a ≤ x ∧ x ≤ b } = Icc a b :=
rfl
#align set.Icc_def Set.Icc_def
theorem Iic_def (b : α) : { x | x ≤ b } = Iic b :=
rfl
#align set.Iic_def Set.Iic_def
theorem Ioc_def (a b : α) : { x | a < x ∧ x ≤ b } = Ioc a b :=
rfl
#align set.Ioc_def Set.Ioc_def
theorem Ici_def (a : α) : { x | a ≤ x } = Ici a :=
rfl
#align set.Ici_def Set.Ici_def
theorem Ioi_def (a : α) : { x | a < x } = Ioi a :=
rfl
#align set.Ioi_def Set.Ioi_def
@[simp]
theorem mem_Ioo : x ∈ Ioo a b ↔ a < x ∧ x < b :=
Iff.rfl
#align set.mem_Ioo Set.mem_Ioo
@[simp]
theorem mem_Ico : x ∈ Ico a b ↔ a ≤ x ∧ x < b :=
Iff.rfl
#align set.mem_Ico Set.mem_Ico
@[simp]
theorem mem_Iio : x ∈ Iio b ↔ x < b :=
Iff.rfl
#align set.mem_Iio Set.mem_Iio
@[simp]
theorem mem_Icc : x ∈ Icc a b ↔ a ≤ x ∧ x ≤ b :=
Iff.rfl
#align set.mem_Icc Set.mem_Icc
@[simp]
theorem mem_Iic : x ∈ Iic b ↔ x ≤ b :=
Iff.rfl
#align set.mem_Iic Set.mem_Iic
@[simp]
theorem mem_Ioc : x ∈ Ioc a b ↔ a < x ∧ x ≤ b :=
Iff.rfl
#align set.mem_Ioc Set.mem_Ioc
@[simp]
theorem mem_Ici : x ∈ Ici a ↔ a ≤ x :=
Iff.rfl
#align set.mem_Ici Set.mem_Ici
@[simp]
theorem mem_Ioi : x ∈ Ioi a ↔ a < x :=
Iff.rfl
#align set.mem_Ioi Set.mem_Ioi
instance decidableMemIoo [Decidable (a < x ∧ x < b)] : Decidable (x ∈ Ioo a b) := by assumption
#align set.decidable_mem_Ioo Set.decidableMemIoo
instance decidableMemIco [Decidable (a ≤ x ∧ x < b)] : Decidable (x ∈ Ico a b) := by assumption
#align set.decidable_mem_Ico Set.decidableMemIco
instance decidableMemIio [Decidable (x < b)] : Decidable (x ∈ Iio b) := by assumption
#align set.decidable_mem_Iio Set.decidableMemIio
instance decidableMemIcc [Decidable (a ≤ x ∧ x ≤ b)] : Decidable (x ∈ Icc a b) := by assumption
#align set.decidable_mem_Icc Set.decidableMemIcc
instance decidableMemIic [Decidable (x ≤ b)] : Decidable (x ∈ Iic b) := by assumption
#align set.decidable_mem_Iic Set.decidableMemIic
instance decidableMemIoc [Decidable (a < x ∧ x ≤ b)] : Decidable (x ∈ Ioc a b) := by assumption
#align set.decidable_mem_Ioc Set.decidableMemIoc
instance decidableMemIci [Decidable (a ≤ x)] : Decidable (x ∈ Ici a) := by assumption
#align set.decidable_mem_Ici Set.decidableMemIci
instance decidableMemIoi [Decidable (a < x)] : Decidable (x ∈ Ioi a) := by assumption
#align set.decidable_mem_Ioi Set.decidableMemIoi
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem left_mem_Ioo : a ∈ Ioo a b ↔ False := by simp [lt_irrefl]
#align set.left_mem_Ioo Set.left_mem_Ioo
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by simp [le_refl]
#align set.left_mem_Ico Set.left_mem_Ico
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by simp [le_refl]
#align set.left_mem_Icc Set.left_mem_Icc
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem left_mem_Ioc : a ∈ Ioc a b ↔ False := by simp [lt_irrefl]
#align set.left_mem_Ioc Set.left_mem_Ioc
theorem left_mem_Ici : a ∈ Ici a := by simp
#align set.left_mem_Ici Set.left_mem_Ici
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem right_mem_Ioo : b ∈ Ioo a b ↔ False := by simp [lt_irrefl]
#align set.right_mem_Ioo Set.right_mem_Ioo
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem right_mem_Ico : b ∈ Ico a b ↔ False := by simp [lt_irrefl]
#align set.right_mem_Ico Set.right_mem_Ico
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by simp [le_refl]
#align set.right_mem_Icc Set.right_mem_Icc
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b := by simp [le_refl]
#align set.right_mem_Ioc Set.right_mem_Ioc
theorem right_mem_Iic : a ∈ Iic a := by simp
#align set.right_mem_Iic Set.right_mem_Iic
@[simp]
theorem dual_Ici : Ici (toDual a) = ofDual ⁻¹' Iic a :=
rfl
#align set.dual_Ici Set.dual_Ici
@[simp]
theorem dual_Iic : Iic (toDual a) = ofDual ⁻¹' Ici a :=
rfl
#align set.dual_Iic Set.dual_Iic
@[simp]
theorem dual_Ioi : Ioi (toDual a) = ofDual ⁻¹' Iio a :=
rfl
#align set.dual_Ioi Set.dual_Ioi
@[simp]
theorem dual_Iio : Iio (toDual a) = ofDual ⁻¹' Ioi a :=
rfl
#align set.dual_Iio Set.dual_Iio
@[simp]
theorem dual_Icc : Icc (toDual a) (toDual b) = ofDual ⁻¹' Icc b a :=
Set.ext fun _ => and_comm
#align set.dual_Icc Set.dual_Icc
@[simp]
theorem dual_Ioc : Ioc (toDual a) (toDual b) = ofDual ⁻¹' Ico b a :=
Set.ext fun _ => and_comm
#align set.dual_Ioc Set.dual_Ioc
@[simp]
theorem dual_Ico : Ico (toDual a) (toDual b) = ofDual ⁻¹' Ioc b a :=
Set.ext fun _ => and_comm
#align set.dual_Ico Set.dual_Ico
@[simp]
theorem dual_Ioo : Ioo (toDual a) (toDual b) = ofDual ⁻¹' Ioo b a :=
Set.ext fun _ => and_comm
#align set.dual_Ioo Set.dual_Ioo
@[simp]
theorem nonempty_Icc : (Icc a b).Nonempty ↔ a ≤ b :=
⟨fun ⟨_, hx⟩ => hx.1.trans hx.2, fun h => ⟨a, left_mem_Icc.2 h⟩⟩
#align set.nonempty_Icc Set.nonempty_Icc
@[simp]
theorem nonempty_Ico : (Ico a b).Nonempty ↔ a < b :=
⟨fun ⟨_, hx⟩ => hx.1.trans_lt hx.2, fun h => ⟨a, left_mem_Ico.2 h⟩⟩
#align set.nonempty_Ico Set.nonempty_Ico
@[simp]
theorem nonempty_Ioc : (Ioc a b).Nonempty ↔ a < b :=
⟨fun ⟨_, hx⟩ => hx.1.trans_le hx.2, fun h => ⟨b, right_mem_Ioc.2 h⟩⟩
#align set.nonempty_Ioc Set.nonempty_Ioc
@[simp]
theorem nonempty_Ici : (Ici a).Nonempty :=
⟨a, left_mem_Ici⟩
#align set.nonempty_Ici Set.nonempty_Ici
@[simp]
theorem nonempty_Iic : (Iic a).Nonempty :=
⟨a, right_mem_Iic⟩
#align set.nonempty_Iic Set.nonempty_Iic
@[simp]
theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b :=
⟨fun ⟨_, ha, hb⟩ => ha.trans hb, exists_between⟩
#align set.nonempty_Ioo Set.nonempty_Ioo
@[simp]
theorem nonempty_Ioi [NoMaxOrder α] : (Ioi a).Nonempty :=
exists_gt a
#align set.nonempty_Ioi Set.nonempty_Ioi
@[simp]
theorem nonempty_Iio [NoMinOrder α] : (Iio a).Nonempty :=
exists_lt a
#align set.nonempty_Iio Set.nonempty_Iio
theorem nonempty_Icc_subtype (h : a ≤ b) : Nonempty (Icc a b) :=
Nonempty.to_subtype (nonempty_Icc.mpr h)
#align set.nonempty_Icc_subtype Set.nonempty_Icc_subtype
theorem nonempty_Ico_subtype (h : a < b) : Nonempty (Ico a b) :=
Nonempty.to_subtype (nonempty_Ico.mpr h)
#align set.nonempty_Ico_subtype Set.nonempty_Ico_subtype
theorem nonempty_Ioc_subtype (h : a < b) : Nonempty (Ioc a b) :=
Nonempty.to_subtype (nonempty_Ioc.mpr h)
#align set.nonempty_Ioc_subtype Set.nonempty_Ioc_subtype
instance nonempty_Ici_subtype : Nonempty (Ici a) :=
Nonempty.to_subtype nonempty_Ici
#align set.nonempty_Ici_subtype Set.nonempty_Ici_subtype
instance nonempty_Iic_subtype : Nonempty (Iic a) :=
Nonempty.to_subtype nonempty_Iic
#align set.nonempty_Iic_subtype Set.nonempty_Iic_subtype
theorem nonempty_Ioo_subtype [DenselyOrdered α] (h : a < b) : Nonempty (Ioo a b) :=
Nonempty.to_subtype (nonempty_Ioo.mpr h)
#align set.nonempty_Ioo_subtype Set.nonempty_Ioo_subtype
instance nonempty_Ioi_subtype [NoMaxOrder α] : Nonempty (Ioi a) :=
Nonempty.to_subtype nonempty_Ioi
#align set.nonempty_Ioi_subtype Set.nonempty_Ioi_subtype
instance nonempty_Iio_subtype [NoMinOrder α] : Nonempty (Iio a) :=
Nonempty.to_subtype nonempty_Iio
#align set.nonempty_Iio_subtype Set.nonempty_Iio_subtype
instance [NoMinOrder α] : NoMinOrder (Iio a) :=
⟨fun a =>
let ⟨b, hb⟩ := exists_lt (a : α)
⟨⟨b, lt_trans hb a.2⟩, hb⟩⟩
instance [NoMinOrder α] : NoMinOrder (Iic a) :=
⟨fun a =>
let ⟨b, hb⟩ := exists_lt (a : α)
⟨⟨b, hb.le.trans a.2⟩, hb⟩⟩
instance [NoMaxOrder α] : NoMaxOrder (Ioi a) :=
OrderDual.noMaxOrder (α := Iio (toDual a))
instance [NoMaxOrder α] : NoMaxOrder (Ici a) :=
OrderDual.noMaxOrder (α := Iic (toDual a))
@[simp]
theorem Icc_eq_empty (h : ¬a ≤ b) : Icc a b = ∅ :=
eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans hb)
#align set.Icc_eq_empty Set.Icc_eq_empty
@[simp]
theorem Ico_eq_empty (h : ¬a < b) : Ico a b = ∅ :=
eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans_lt hb)
#align set.Ico_eq_empty Set.Ico_eq_empty
@[simp]
theorem Ioc_eq_empty (h : ¬a < b) : Ioc a b = ∅ :=
eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans_le hb)
#align set.Ioc_eq_empty Set.Ioc_eq_empty
@[simp]
theorem Ioo_eq_empty (h : ¬a < b) : Ioo a b = ∅ :=
eq_empty_iff_forall_not_mem.2 fun _ ⟨ha, hb⟩ => h (ha.trans hb)
#align set.Ioo_eq_empty Set.Ioo_eq_empty
@[simp]
theorem Icc_eq_empty_of_lt (h : b < a) : Icc a b = ∅ :=
Icc_eq_empty h.not_le
#align set.Icc_eq_empty_of_lt Set.Icc_eq_empty_of_lt
@[simp]
theorem Ico_eq_empty_of_le (h : b ≤ a) : Ico a b = ∅ :=
Ico_eq_empty h.not_lt
#align set.Ico_eq_empty_of_le Set.Ico_eq_empty_of_le
@[simp]
theorem Ioc_eq_empty_of_le (h : b ≤ a) : Ioc a b = ∅ :=
Ioc_eq_empty h.not_lt
#align set.Ioc_eq_empty_of_le Set.Ioc_eq_empty_of_le
@[simp]
theorem Ioo_eq_empty_of_le (h : b ≤ a) : Ioo a b = ∅ :=
Ioo_eq_empty h.not_lt
#align set.Ioo_eq_empty_of_le Set.Ioo_eq_empty_of_le
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem Ico_self (a : α) : Ico a a = ∅ :=
Ico_eq_empty <| lt_irrefl _
#align set.Ico_self Set.Ico_self
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem Ioc_self (a : α) : Ioc a a = ∅ :=
Ioc_eq_empty <| lt_irrefl _
#align set.Ioc_self Set.Ioc_self
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem Ioo_self (a : α) : Ioo a a = ∅ :=
Ioo_eq_empty <| lt_irrefl _
#align set.Ioo_self Set.Ioo_self
theorem Ici_subset_Ici : Ici a ⊆ Ici b ↔ b ≤ a :=
⟨fun h => h <| left_mem_Ici, fun h _ hx => h.trans hx⟩
#align set.Ici_subset_Ici Set.Ici_subset_Ici
@[gcongr] alias ⟨_, _root_.GCongr.Ici_subset_Ici_of_le⟩ := Ici_subset_Ici
theorem Iic_subset_Iic : Iic a ⊆ Iic b ↔ a ≤ b :=
@Ici_subset_Ici αᵒᵈ _ _ _
#align set.Iic_subset_Iic Set.Iic_subset_Iic
@[gcongr] alias ⟨_, _root_.GCongr.Iic_subset_Iic_of_le⟩ := Iic_subset_Iic
theorem Ici_subset_Ioi : Ici a ⊆ Ioi b ↔ b < a :=
⟨fun h => h left_mem_Ici, fun h _ hx => h.trans_le hx⟩
#align set.Ici_subset_Ioi Set.Ici_subset_Ioi
theorem Iic_subset_Iio : Iic a ⊆ Iio b ↔ a < b :=
⟨fun h => h right_mem_Iic, fun h _ hx => lt_of_le_of_lt hx h⟩
#align set.Iic_subset_Iio Set.Iic_subset_Iio
@[gcongr]
theorem Ioo_subset_Ioo (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ =>
⟨h₁.trans_lt hx₁, hx₂.trans_le h₂⟩
#align set.Ioo_subset_Ioo Set.Ioo_subset_Ioo
@[gcongr]
theorem Ioo_subset_Ioo_left (h : a₁ ≤ a₂) : Ioo a₂ b ⊆ Ioo a₁ b :=
Ioo_subset_Ioo h le_rfl
#align set.Ioo_subset_Ioo_left Set.Ioo_subset_Ioo_left
@[gcongr]
theorem Ioo_subset_Ioo_right (h : b₁ ≤ b₂) : Ioo a b₁ ⊆ Ioo a b₂ :=
Ioo_subset_Ioo le_rfl h
#align set.Ioo_subset_Ioo_right Set.Ioo_subset_Ioo_right
@[gcongr]
theorem Ico_subset_Ico (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ico a₁ b₁ ⊆ Ico a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ =>
⟨h₁.trans hx₁, hx₂.trans_le h₂⟩
#align set.Ico_subset_Ico Set.Ico_subset_Ico
@[gcongr]
theorem Ico_subset_Ico_left (h : a₁ ≤ a₂) : Ico a₂ b ⊆ Ico a₁ b :=
Ico_subset_Ico h le_rfl
#align set.Ico_subset_Ico_left Set.Ico_subset_Ico_left
@[gcongr]
theorem Ico_subset_Ico_right (h : b₁ ≤ b₂) : Ico a b₁ ⊆ Ico a b₂ :=
Ico_subset_Ico le_rfl h
#align set.Ico_subset_Ico_right Set.Ico_subset_Ico_right
@[gcongr]
theorem Icc_subset_Icc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Icc a₁ b₁ ⊆ Icc a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ =>
⟨h₁.trans hx₁, le_trans hx₂ h₂⟩
#align set.Icc_subset_Icc Set.Icc_subset_Icc
@[gcongr]
theorem Icc_subset_Icc_left (h : a₁ ≤ a₂) : Icc a₂ b ⊆ Icc a₁ b :=
Icc_subset_Icc h le_rfl
#align set.Icc_subset_Icc_left Set.Icc_subset_Icc_left
@[gcongr]
theorem Icc_subset_Icc_right (h : b₁ ≤ b₂) : Icc a b₁ ⊆ Icc a b₂ :=
Icc_subset_Icc le_rfl h
#align set.Icc_subset_Icc_right Set.Icc_subset_Icc_right
theorem Icc_subset_Ioo (ha : a₂ < a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ := fun _ hx =>
⟨ha.trans_le hx.1, hx.2.trans_lt hb⟩
#align set.Icc_subset_Ioo Set.Icc_subset_Ioo
theorem Icc_subset_Ici_self : Icc a b ⊆ Ici a := fun _ => And.left
#align set.Icc_subset_Ici_self Set.Icc_subset_Ici_self
theorem Icc_subset_Iic_self : Icc a b ⊆ Iic b := fun _ => And.right
#align set.Icc_subset_Iic_self Set.Icc_subset_Iic_self
theorem Ioc_subset_Iic_self : Ioc a b ⊆ Iic b := fun _ => And.right
#align set.Ioc_subset_Iic_self Set.Ioc_subset_Iic_self
@[gcongr]
theorem Ioc_subset_Ioc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ := fun _ ⟨hx₁, hx₂⟩ =>
⟨h₁.trans_lt hx₁, hx₂.trans h₂⟩
#align set.Ioc_subset_Ioc Set.Ioc_subset_Ioc
@[gcongr]
theorem Ioc_subset_Ioc_left (h : a₁ ≤ a₂) : Ioc a₂ b ⊆ Ioc a₁ b :=
Ioc_subset_Ioc h le_rfl
#align set.Ioc_subset_Ioc_left Set.Ioc_subset_Ioc_left
@[gcongr]
theorem Ioc_subset_Ioc_right (h : b₁ ≤ b₂) : Ioc a b₁ ⊆ Ioc a b₂ :=
Ioc_subset_Ioc le_rfl h
#align set.Ioc_subset_Ioc_right Set.Ioc_subset_Ioc_right
theorem Ico_subset_Ioo_left (h₁ : a₁ < a₂) : Ico a₂ b ⊆ Ioo a₁ b := fun _ =>
And.imp_left h₁.trans_le
#align set.Ico_subset_Ioo_left Set.Ico_subset_Ioo_left
theorem Ioc_subset_Ioo_right (h : b₁ < b₂) : Ioc a b₁ ⊆ Ioo a b₂ := fun _ =>
And.imp_right fun h' => h'.trans_lt h
#align set.Ioc_subset_Ioo_right Set.Ioc_subset_Ioo_right
theorem Icc_subset_Ico_right (h₁ : b₁ < b₂) : Icc a b₁ ⊆ Ico a b₂ := fun _ =>
And.imp_right fun h₂ => h₂.trans_lt h₁
#align set.Icc_subset_Ico_right Set.Icc_subset_Ico_right
theorem Ioo_subset_Ico_self : Ioo a b ⊆ Ico a b := fun _ => And.imp_left le_of_lt
#align set.Ioo_subset_Ico_self Set.Ioo_subset_Ico_self
theorem Ioo_subset_Ioc_self : Ioo a b ⊆ Ioc a b := fun _ => And.imp_right le_of_lt
#align set.Ioo_subset_Ioc_self Set.Ioo_subset_Ioc_self
theorem Ico_subset_Icc_self : Ico a b ⊆ Icc a b := fun _ => And.imp_right le_of_lt
#align set.Ico_subset_Icc_self Set.Ico_subset_Icc_self
theorem Ioc_subset_Icc_self : Ioc a b ⊆ Icc a b := fun _ => And.imp_left le_of_lt
#align set.Ioc_subset_Icc_self Set.Ioc_subset_Icc_self
theorem Ioo_subset_Icc_self : Ioo a b ⊆ Icc a b :=
Subset.trans Ioo_subset_Ico_self Ico_subset_Icc_self
#align set.Ioo_subset_Icc_self Set.Ioo_subset_Icc_self
theorem Ico_subset_Iio_self : Ico a b ⊆ Iio b := fun _ => And.right
#align set.Ico_subset_Iio_self Set.Ico_subset_Iio_self
theorem Ioo_subset_Iio_self : Ioo a b ⊆ Iio b := fun _ => And.right
#align set.Ioo_subset_Iio_self Set.Ioo_subset_Iio_self
theorem Ioc_subset_Ioi_self : Ioc a b ⊆ Ioi a := fun _ => And.left
#align set.Ioc_subset_Ioi_self Set.Ioc_subset_Ioi_self
theorem Ioo_subset_Ioi_self : Ioo a b ⊆ Ioi a := fun _ => And.left
#align set.Ioo_subset_Ioi_self Set.Ioo_subset_Ioi_self
theorem Ioi_subset_Ici_self : Ioi a ⊆ Ici a := fun _ hx => le_of_lt hx
#align set.Ioi_subset_Ici_self Set.Ioi_subset_Ici_self
theorem Iio_subset_Iic_self : Iio a ⊆ Iic a := fun _ hx => le_of_lt hx
#align set.Iio_subset_Iic_self Set.Iio_subset_Iic_self
theorem Ico_subset_Ici_self : Ico a b ⊆ Ici a := fun _ => And.left
#align set.Ico_subset_Ici_self Set.Ico_subset_Ici_self
theorem Ioi_ssubset_Ici_self : Ioi a ⊂ Ici a :=
⟨Ioi_subset_Ici_self, fun h => lt_irrefl a (h le_rfl)⟩
#align set.Ioi_ssubset_Ici_self Set.Ioi_ssubset_Ici_self
theorem Iio_ssubset_Iic_self : Iio a ⊂ Iic a :=
@Ioi_ssubset_Ici_self αᵒᵈ _ _
#align set.Iio_ssubset_Iic_self Set.Iio_ssubset_Iic_self
theorem Icc_subset_Icc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ :=
⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ =>
⟨h.trans hx, hx'.trans h'⟩⟩
#align set.Icc_subset_Icc_iff Set.Icc_subset_Icc_iff
theorem Icc_subset_Ioo_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂ :=
⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ =>
⟨h.trans_le hx, hx'.trans_lt h'⟩⟩
#align set.Icc_subset_Ioo_iff Set.Icc_subset_Ioo_iff
theorem Icc_subset_Ico_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ < b₂ :=
⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ =>
⟨h.trans hx, hx'.trans_lt h'⟩⟩
#align set.Icc_subset_Ico_iff Set.Icc_subset_Ico_iff
theorem Icc_subset_Ioc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ a₂ < a₁ ∧ b₁ ≤ b₂ :=
⟨fun h => ⟨(h ⟨le_rfl, h₁⟩).1, (h ⟨h₁, le_rfl⟩).2⟩, fun ⟨h, h'⟩ _ ⟨hx, hx'⟩ =>
⟨h.trans_le hx, hx'.trans h'⟩⟩
#align set.Icc_subset_Ioc_iff Set.Icc_subset_Ioc_iff
theorem Icc_subset_Iio_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iio b₂ ↔ b₁ < b₂ :=
⟨fun h => h ⟨h₁, le_rfl⟩, fun h _ ⟨_, hx'⟩ => hx'.trans_lt h⟩
#align set.Icc_subset_Iio_iff Set.Icc_subset_Iio_iff
theorem Icc_subset_Ioi_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioi a₂ ↔ a₂ < a₁ :=
⟨fun h => h ⟨le_rfl, h₁⟩, fun h _ ⟨hx, _⟩ => h.trans_le hx⟩
#align set.Icc_subset_Ioi_iff Set.Icc_subset_Ioi_iff
theorem Icc_subset_Iic_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iic b₂ ↔ b₁ ≤ b₂ :=
⟨fun h => h ⟨h₁, le_rfl⟩, fun h _ ⟨_, hx'⟩ => hx'.trans h⟩
#align set.Icc_subset_Iic_iff Set.Icc_subset_Iic_iff
theorem Icc_subset_Ici_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ici a₂ ↔ a₂ ≤ a₁ :=
⟨fun h => h ⟨le_rfl, h₁⟩, fun h _ ⟨hx, _⟩ => h.trans hx⟩
#align set.Icc_subset_Ici_iff Set.Icc_subset_Ici_iff
theorem Icc_ssubset_Icc_left (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ :=
(ssubset_iff_of_subset (Icc_subset_Icc (le_of_lt ha) hb)).mpr
⟨a₂, left_mem_Icc.mpr hI, not_and.mpr fun f _ => lt_irrefl a₂ (ha.trans_le f)⟩
#align set.Icc_ssubset_Icc_left Set.Icc_ssubset_Icc_left
theorem Icc_ssubset_Icc_right (hI : a₂ ≤ b₂) (ha : a₂ ≤ a₁) (hb : b₁ < b₂) :
Icc a₁ b₁ ⊂ Icc a₂ b₂ :=
(ssubset_iff_of_subset (Icc_subset_Icc ha (le_of_lt hb))).mpr
⟨b₂, right_mem_Icc.mpr hI, fun f => lt_irrefl b₁ (hb.trans_le f.2)⟩
#align set.Icc_ssubset_Icc_right Set.Icc_ssubset_Icc_right
@[gcongr]
theorem Ioi_subset_Ioi (h : a ≤ b) : Ioi b ⊆ Ioi a := fun _ hx => h.trans_lt hx
#align set.Ioi_subset_Ioi Set.Ioi_subset_Ioi
theorem Ioi_subset_Ici (h : a ≤ b) : Ioi b ⊆ Ici a :=
Subset.trans (Ioi_subset_Ioi h) Ioi_subset_Ici_self
#align set.Ioi_subset_Ici Set.Ioi_subset_Ici
@[gcongr]
theorem Iio_subset_Iio (h : a ≤ b) : Iio a ⊆ Iio b := fun _ hx => lt_of_lt_of_le hx h
#align set.Iio_subset_Iio Set.Iio_subset_Iio
theorem Iio_subset_Iic (h : a ≤ b) : Iio a ⊆ Iic b :=
Subset.trans (Iio_subset_Iio h) Iio_subset_Iic_self
#align set.Iio_subset_Iic Set.Iio_subset_Iic
theorem Ici_inter_Iic : Ici a ∩ Iic b = Icc a b :=
rfl
#align set.Ici_inter_Iic Set.Ici_inter_Iic
theorem Ici_inter_Iio : Ici a ∩ Iio b = Ico a b :=
rfl
#align set.Ici_inter_Iio Set.Ici_inter_Iio
theorem Ioi_inter_Iic : Ioi a ∩ Iic b = Ioc a b :=
rfl
#align set.Ioi_inter_Iic Set.Ioi_inter_Iic
theorem Ioi_inter_Iio : Ioi a ∩ Iio b = Ioo a b :=
rfl
#align set.Ioi_inter_Iio Set.Ioi_inter_Iio
theorem Iic_inter_Ici : Iic a ∩ Ici b = Icc b a :=
inter_comm _ _
#align set.Iic_inter_Ici Set.Iic_inter_Ici
theorem Iio_inter_Ici : Iio a ∩ Ici b = Ico b a :=
inter_comm _ _
#align set.Iio_inter_Ici Set.Iio_inter_Ici
theorem Iic_inter_Ioi : Iic a ∩ Ioi b = Ioc b a :=
inter_comm _ _
#align set.Iic_inter_Ioi Set.Iic_inter_Ioi
theorem Iio_inter_Ioi : Iio a ∩ Ioi b = Ioo b a :=
inter_comm _ _
#align set.Iio_inter_Ioi Set.Iio_inter_Ioi
theorem mem_Icc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Icc a b :=
Ioo_subset_Icc_self h
#align set.mem_Icc_of_Ioo Set.mem_Icc_of_Ioo
theorem mem_Ico_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ico a b :=
Ioo_subset_Ico_self h
#align set.mem_Ico_of_Ioo Set.mem_Ico_of_Ioo
theorem mem_Ioc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ioc a b :=
Ioo_subset_Ioc_self h
#align set.mem_Ioc_of_Ioo Set.mem_Ioc_of_Ioo
theorem mem_Icc_of_Ico (h : x ∈ Ico a b) : x ∈ Icc a b :=
Ico_subset_Icc_self h
#align set.mem_Icc_of_Ico Set.mem_Icc_of_Ico
theorem mem_Icc_of_Ioc (h : x ∈ Ioc a b) : x ∈ Icc a b :=
Ioc_subset_Icc_self h
#align set.mem_Icc_of_Ioc Set.mem_Icc_of_Ioc
theorem mem_Ici_of_Ioi (h : x ∈ Ioi a) : x ∈ Ici a :=
Ioi_subset_Ici_self h
#align set.mem_Ici_of_Ioi Set.mem_Ici_of_Ioi
theorem mem_Iic_of_Iio (h : x ∈ Iio a) : x ∈ Iic a :=
Iio_subset_Iic_self h
#align set.mem_Iic_of_Iio Set.mem_Iic_of_Iio
theorem Icc_eq_empty_iff : Icc a b = ∅ ↔ ¬a ≤ b := by
rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Icc]
#align set.Icc_eq_empty_iff Set.Icc_eq_empty_iff
| Mathlib/Order/Interval/Set/Basic.lean | 700 | 701 | theorem Ico_eq_empty_iff : Ico a b = ∅ ↔ ¬a < b := by |
rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ico]
|
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"
open Function
universe u v w x
namespace Set
variable {α : Type u} {s t : Set α}
instance instBooleanAlgebraSet : BooleanAlgebra (Set α) :=
{ (inferInstance : BooleanAlgebra (α → Prop)) with
sup := (· ∪ ·),
le := (· ≤ ·),
lt := fun s t => s ⊆ t ∧ ¬t ⊆ s,
inf := (· ∩ ·),
bot := ∅,
compl := (·ᶜ),
top := univ,
sdiff := (· \ ·) }
instance : HasSSubset (Set α) :=
⟨(· < ·)⟩
@[simp]
theorem top_eq_univ : (⊤ : Set α) = univ :=
rfl
#align set.top_eq_univ Set.top_eq_univ
@[simp]
theorem bot_eq_empty : (⊥ : Set α) = ∅ :=
rfl
#align set.bot_eq_empty Set.bot_eq_empty
@[simp]
theorem sup_eq_union : ((· ⊔ ·) : Set α → Set α → Set α) = (· ∪ ·) :=
rfl
#align set.sup_eq_union Set.sup_eq_union
@[simp]
theorem inf_eq_inter : ((· ⊓ ·) : Set α → Set α → Set α) = (· ∩ ·) :=
rfl
#align set.inf_eq_inter Set.inf_eq_inter
@[simp]
theorem le_eq_subset : ((· ≤ ·) : Set α → Set α → Prop) = (· ⊆ ·) :=
rfl
#align set.le_eq_subset Set.le_eq_subset
@[simp]
theorem lt_eq_ssubset : ((· < ·) : Set α → Set α → Prop) = (· ⊂ ·) :=
rfl
#align set.lt_eq_ssubset Set.lt_eq_ssubset
theorem le_iff_subset : s ≤ t ↔ s ⊆ t :=
Iff.rfl
#align set.le_iff_subset Set.le_iff_subset
theorem lt_iff_ssubset : s < t ↔ s ⊂ t :=
Iff.rfl
#align set.lt_iff_ssubset Set.lt_iff_ssubset
alias ⟨_root_.LE.le.subset, _root_.HasSubset.Subset.le⟩ := le_iff_subset
#align has_subset.subset.le HasSubset.Subset.le
alias ⟨_root_.LT.lt.ssubset, _root_.HasSSubset.SSubset.lt⟩ := lt_iff_ssubset
#align has_ssubset.ssubset.lt HasSSubset.SSubset.lt
instance PiSetCoe.canLift (ι : Type u) (α : ι → Type v) [∀ i, Nonempty (α i)] (s : Set ι) :
CanLift (∀ i : s, α i) (∀ i, α i) (fun f i => f i) fun _ => True :=
PiSubtype.canLift ι α s
#align set.pi_set_coe.can_lift Set.PiSetCoe.canLift
instance PiSetCoe.canLift' (ι : Type u) (α : Type v) [Nonempty α] (s : Set ι) :
CanLift (s → α) (ι → α) (fun f i => f i) fun _ => True :=
PiSetCoe.canLift ι (fun _ => α) s
#align set.pi_set_coe.can_lift' Set.PiSetCoe.canLift'
end Set
theorem Subtype.mem {α : Type*} {s : Set α} (p : s) : (p : α) ∈ s :=
p.prop
#align subtype.mem Subtype.mem
theorem Eq.subset {α} {s t : Set α} : s = t → s ⊆ t :=
fun h₁ _ h₂ => by rw [← h₁]; exact h₂
#align eq.subset Eq.subset
namespace Set
variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x} {a b : α} {s s₁ s₂ t t₁ t₂ u : Set α}
instance : Inhabited (Set α) :=
⟨∅⟩
theorem ext_iff {s t : Set α} : s = t ↔ ∀ x, x ∈ s ↔ x ∈ t :=
⟨fun h x => by rw [h], ext⟩
#align set.ext_iff Set.ext_iff
@[trans]
theorem mem_of_mem_of_subset {x : α} {s t : Set α} (hx : x ∈ s) (h : s ⊆ t) : x ∈ t :=
h hx
#align set.mem_of_mem_of_subset Set.mem_of_mem_of_subset
theorem forall_in_swap {p : α → β → Prop} : (∀ a ∈ s, ∀ (b), p a b) ↔ ∀ (b), ∀ a ∈ s, p a b := by
tauto
#align set.forall_in_swap Set.forall_in_swap
theorem mem_setOf {a : α} {p : α → Prop} : a ∈ { x | p x } ↔ p a :=
Iff.rfl
#align set.mem_set_of Set.mem_setOf
theorem _root_.Membership.mem.out {p : α → Prop} {a : α} (h : a ∈ { x | p x }) : p a :=
h
#align has_mem.mem.out Membership.mem.out
theorem nmem_setOf_iff {a : α} {p : α → Prop} : a ∉ { x | p x } ↔ ¬p a :=
Iff.rfl
#align set.nmem_set_of_iff Set.nmem_setOf_iff
@[simp]
theorem setOf_mem_eq {s : Set α} : { x | x ∈ s } = s :=
rfl
#align set.set_of_mem_eq Set.setOf_mem_eq
theorem setOf_set {s : Set α} : setOf s = s :=
rfl
#align set.set_of_set Set.setOf_set
theorem setOf_app_iff {p : α → Prop} {x : α} : { x | p x } x ↔ p x :=
Iff.rfl
#align set.set_of_app_iff Set.setOf_app_iff
theorem mem_def {a : α} {s : Set α} : a ∈ s ↔ s a :=
Iff.rfl
#align set.mem_def Set.mem_def
theorem setOf_bijective : Bijective (setOf : (α → Prop) → Set α) :=
bijective_id
#align set.set_of_bijective Set.setOf_bijective
theorem subset_setOf {p : α → Prop} {s : Set α} : s ⊆ setOf p ↔ ∀ x, x ∈ s → p x :=
Iff.rfl
theorem setOf_subset {p : α → Prop} {s : Set α} : setOf p ⊆ s ↔ ∀ x, p x → x ∈ s :=
Iff.rfl
@[simp]
theorem setOf_subset_setOf {p q : α → Prop} : { a | p a } ⊆ { a | q a } ↔ ∀ a, p a → q a :=
Iff.rfl
#align set.set_of_subset_set_of Set.setOf_subset_setOf
theorem setOf_and {p q : α → Prop} : { a | p a ∧ q a } = { a | p a } ∩ { a | q a } :=
rfl
#align set.set_of_and Set.setOf_and
theorem setOf_or {p q : α → Prop} : { a | p a ∨ q a } = { a | p a } ∪ { a | q a } :=
rfl
#align set.set_of_or Set.setOf_or
instance : IsRefl (Set α) (· ⊆ ·) :=
show IsRefl (Set α) (· ≤ ·) by infer_instance
instance : IsTrans (Set α) (· ⊆ ·) :=
show IsTrans (Set α) (· ≤ ·) by infer_instance
instance : Trans ((· ⊆ ·) : Set α → Set α → Prop) (· ⊆ ·) (· ⊆ ·) :=
show Trans (· ≤ ·) (· ≤ ·) (· ≤ ·) by infer_instance
instance : IsAntisymm (Set α) (· ⊆ ·) :=
show IsAntisymm (Set α) (· ≤ ·) by infer_instance
instance : IsIrrefl (Set α) (· ⊂ ·) :=
show IsIrrefl (Set α) (· < ·) by infer_instance
instance : IsTrans (Set α) (· ⊂ ·) :=
show IsTrans (Set α) (· < ·) by infer_instance
instance : Trans ((· ⊂ ·) : Set α → Set α → Prop) (· ⊂ ·) (· ⊂ ·) :=
show Trans (· < ·) (· < ·) (· < ·) by infer_instance
instance : Trans ((· ⊂ ·) : Set α → Set α → Prop) (· ⊆ ·) (· ⊂ ·) :=
show Trans (· < ·) (· ≤ ·) (· < ·) by infer_instance
instance : Trans ((· ⊆ ·) : Set α → Set α → Prop) (· ⊂ ·) (· ⊂ ·) :=
show Trans (· ≤ ·) (· < ·) (· < ·) by infer_instance
instance : IsAsymm (Set α) (· ⊂ ·) :=
show IsAsymm (Set α) (· < ·) by infer_instance
instance : IsNonstrictStrictOrder (Set α) (· ⊆ ·) (· ⊂ ·) :=
⟨fun _ _ => Iff.rfl⟩
-- TODO(Jeremy): write a tactic to unfold specific instances of generic notation?
theorem subset_def : (s ⊆ t) = ∀ x, x ∈ s → x ∈ t :=
rfl
#align set.subset_def Set.subset_def
theorem ssubset_def : (s ⊂ t) = (s ⊆ t ∧ ¬t ⊆ s) :=
rfl
#align set.ssubset_def Set.ssubset_def
@[refl]
theorem Subset.refl (a : Set α) : a ⊆ a := fun _ => id
#align set.subset.refl Set.Subset.refl
theorem Subset.rfl {s : Set α} : s ⊆ s :=
Subset.refl s
#align set.subset.rfl Set.Subset.rfl
@[trans]
theorem Subset.trans {a b c : Set α} (ab : a ⊆ b) (bc : b ⊆ c) : a ⊆ c := fun _ h => bc <| ab h
#align set.subset.trans Set.Subset.trans
@[trans]
theorem mem_of_eq_of_mem {x y : α} {s : Set α} (hx : x = y) (h : y ∈ s) : x ∈ s :=
hx.symm ▸ h
#align set.mem_of_eq_of_mem Set.mem_of_eq_of_mem
theorem Subset.antisymm {a b : Set α} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b :=
Set.ext fun _ => ⟨@h₁ _, @h₂ _⟩
#align set.subset.antisymm Set.Subset.antisymm
theorem Subset.antisymm_iff {a b : Set α} : a = b ↔ a ⊆ b ∧ b ⊆ a :=
⟨fun e => ⟨e.subset, e.symm.subset⟩, fun ⟨h₁, h₂⟩ => Subset.antisymm h₁ h₂⟩
#align set.subset.antisymm_iff Set.Subset.antisymm_iff
-- an alternative name
theorem eq_of_subset_of_subset {a b : Set α} : a ⊆ b → b ⊆ a → a = b :=
Subset.antisymm
#align set.eq_of_subset_of_subset Set.eq_of_subset_of_subset
theorem mem_of_subset_of_mem {s₁ s₂ : Set α} {a : α} (h : s₁ ⊆ s₂) : a ∈ s₁ → a ∈ s₂ :=
@h _
#align set.mem_of_subset_of_mem Set.mem_of_subset_of_mem
theorem not_mem_subset (h : s ⊆ t) : a ∉ t → a ∉ s :=
mt <| mem_of_subset_of_mem h
#align set.not_mem_subset Set.not_mem_subset
theorem not_subset : ¬s ⊆ t ↔ ∃ a ∈ s, a ∉ t := by
simp only [subset_def, not_forall, exists_prop]
#align set.not_subset Set.not_subset
lemma eq_of_forall_subset_iff (h : ∀ u, s ⊆ u ↔ t ⊆ u) : s = t := eq_of_forall_ge_iff h
protected theorem eq_or_ssubset_of_subset (h : s ⊆ t) : s = t ∨ s ⊂ t :=
eq_or_lt_of_le h
#align set.eq_or_ssubset_of_subset Set.eq_or_ssubset_of_subset
theorem exists_of_ssubset {s t : Set α} (h : s ⊂ t) : ∃ x ∈ t, x ∉ s :=
not_subset.1 h.2
#align set.exists_of_ssubset Set.exists_of_ssubset
protected theorem ssubset_iff_subset_ne {s t : Set α} : s ⊂ t ↔ s ⊆ t ∧ s ≠ t :=
@lt_iff_le_and_ne (Set α) _ s t
#align set.ssubset_iff_subset_ne Set.ssubset_iff_subset_ne
theorem ssubset_iff_of_subset {s t : Set α} (h : s ⊆ t) : s ⊂ t ↔ ∃ x ∈ t, x ∉ s :=
⟨exists_of_ssubset, fun ⟨_, hxt, hxs⟩ => ⟨h, fun h => hxs <| h hxt⟩⟩
#align set.ssubset_iff_of_subset Set.ssubset_iff_of_subset
protected theorem ssubset_of_ssubset_of_subset {s₁ s₂ s₃ : Set α} (hs₁s₂ : s₁ ⊂ s₂)
(hs₂s₃ : s₂ ⊆ s₃) : s₁ ⊂ s₃ :=
⟨Subset.trans hs₁s₂.1 hs₂s₃, fun hs₃s₁ => hs₁s₂.2 (Subset.trans hs₂s₃ hs₃s₁)⟩
#align set.ssubset_of_ssubset_of_subset Set.ssubset_of_ssubset_of_subset
protected theorem ssubset_of_subset_of_ssubset {s₁ s₂ s₃ : Set α} (hs₁s₂ : s₁ ⊆ s₂)
(hs₂s₃ : s₂ ⊂ s₃) : s₁ ⊂ s₃ :=
⟨Subset.trans hs₁s₂ hs₂s₃.1, fun hs₃s₁ => hs₂s₃.2 (Subset.trans hs₃s₁ hs₁s₂)⟩
#align set.ssubset_of_subset_of_ssubset Set.ssubset_of_subset_of_ssubset
theorem not_mem_empty (x : α) : ¬x ∈ (∅ : Set α) :=
id
#align set.not_mem_empty Set.not_mem_empty
-- Porting note (#10618): removed `simp` because `simp` can prove it
theorem not_not_mem : ¬a ∉ s ↔ a ∈ s :=
not_not
#align set.not_not_mem Set.not_not_mem
-- Porting note: we seem to need parentheses at `(↥s)`,
-- even if we increase the right precedence of `↥` in `Mathlib.Tactic.Coe`.
-- Porting note: removed `simp` as it is competing with `nonempty_subtype`.
-- @[simp]
theorem nonempty_coe_sort {s : Set α} : Nonempty (↥s) ↔ s.Nonempty :=
nonempty_subtype
#align set.nonempty_coe_sort Set.nonempty_coe_sort
alias ⟨_, Nonempty.coe_sort⟩ := nonempty_coe_sort
#align set.nonempty.coe_sort Set.Nonempty.coe_sort
theorem nonempty_def : s.Nonempty ↔ ∃ x, x ∈ s :=
Iff.rfl
#align set.nonempty_def Set.nonempty_def
theorem nonempty_of_mem {x} (h : x ∈ s) : s.Nonempty :=
⟨x, h⟩
#align set.nonempty_of_mem Set.nonempty_of_mem
theorem Nonempty.not_subset_empty : s.Nonempty → ¬s ⊆ ∅
| ⟨_, hx⟩, hs => hs hx
#align set.nonempty.not_subset_empty Set.Nonempty.not_subset_empty
protected noncomputable def Nonempty.some (h : s.Nonempty) : α :=
Classical.choose h
#align set.nonempty.some Set.Nonempty.some
protected theorem Nonempty.some_mem (h : s.Nonempty) : h.some ∈ s :=
Classical.choose_spec h
#align set.nonempty.some_mem Set.Nonempty.some_mem
theorem Nonempty.mono (ht : s ⊆ t) (hs : s.Nonempty) : t.Nonempty :=
hs.imp ht
#align set.nonempty.mono Set.Nonempty.mono
theorem nonempty_of_not_subset (h : ¬s ⊆ t) : (s \ t).Nonempty :=
let ⟨x, xs, xt⟩ := not_subset.1 h
⟨x, xs, xt⟩
#align set.nonempty_of_not_subset Set.nonempty_of_not_subset
theorem nonempty_of_ssubset (ht : s ⊂ t) : (t \ s).Nonempty :=
nonempty_of_not_subset ht.2
#align set.nonempty_of_ssubset Set.nonempty_of_ssubset
theorem Nonempty.of_diff (h : (s \ t).Nonempty) : s.Nonempty :=
h.imp fun _ => And.left
#align set.nonempty.of_diff Set.Nonempty.of_diff
theorem nonempty_of_ssubset' (ht : s ⊂ t) : t.Nonempty :=
(nonempty_of_ssubset ht).of_diff
#align set.nonempty_of_ssubset' Set.nonempty_of_ssubset'
theorem Nonempty.inl (hs : s.Nonempty) : (s ∪ t).Nonempty :=
hs.imp fun _ => Or.inl
#align set.nonempty.inl Set.Nonempty.inl
theorem Nonempty.inr (ht : t.Nonempty) : (s ∪ t).Nonempty :=
ht.imp fun _ => Or.inr
#align set.nonempty.inr Set.Nonempty.inr
@[simp]
theorem union_nonempty : (s ∪ t).Nonempty ↔ s.Nonempty ∨ t.Nonempty :=
exists_or
#align set.union_nonempty Set.union_nonempty
theorem Nonempty.left (h : (s ∩ t).Nonempty) : s.Nonempty :=
h.imp fun _ => And.left
#align set.nonempty.left Set.Nonempty.left
theorem Nonempty.right (h : (s ∩ t).Nonempty) : t.Nonempty :=
h.imp fun _ => And.right
#align set.nonempty.right Set.Nonempty.right
theorem inter_nonempty : (s ∩ t).Nonempty ↔ ∃ x, x ∈ s ∧ x ∈ t :=
Iff.rfl
#align set.inter_nonempty Set.inter_nonempty
theorem inter_nonempty_iff_exists_left : (s ∩ t).Nonempty ↔ ∃ x ∈ s, x ∈ t := by
simp_rw [inter_nonempty]
#align set.inter_nonempty_iff_exists_left Set.inter_nonempty_iff_exists_left
theorem inter_nonempty_iff_exists_right : (s ∩ t).Nonempty ↔ ∃ x ∈ t, x ∈ s := by
simp_rw [inter_nonempty, and_comm]
#align set.inter_nonempty_iff_exists_right Set.inter_nonempty_iff_exists_right
theorem nonempty_iff_univ_nonempty : Nonempty α ↔ (univ : Set α).Nonempty :=
⟨fun ⟨x⟩ => ⟨x, trivial⟩, fun ⟨x, _⟩ => ⟨x⟩⟩
#align set.nonempty_iff_univ_nonempty Set.nonempty_iff_univ_nonempty
@[simp]
theorem univ_nonempty : ∀ [Nonempty α], (univ : Set α).Nonempty
| ⟨x⟩ => ⟨x, trivial⟩
#align set.univ_nonempty Set.univ_nonempty
theorem Nonempty.to_subtype : s.Nonempty → Nonempty (↥s) :=
nonempty_subtype.2
#align set.nonempty.to_subtype Set.Nonempty.to_subtype
theorem Nonempty.to_type : s.Nonempty → Nonempty α := fun ⟨x, _⟩ => ⟨x⟩
#align set.nonempty.to_type Set.Nonempty.to_type
instance univ.nonempty [Nonempty α] : Nonempty (↥(Set.univ : Set α)) :=
Set.univ_nonempty.to_subtype
#align set.univ.nonempty Set.univ.nonempty
theorem nonempty_of_nonempty_subtype [Nonempty (↥s)] : s.Nonempty :=
nonempty_subtype.mp ‹_›
#align set.nonempty_of_nonempty_subtype Set.nonempty_of_nonempty_subtype
theorem empty_def : (∅ : Set α) = { _x : α | False } :=
rfl
#align set.empty_def Set.empty_def
@[simp]
theorem mem_empty_iff_false (x : α) : x ∈ (∅ : Set α) ↔ False :=
Iff.rfl
#align set.mem_empty_iff_false Set.mem_empty_iff_false
@[simp]
theorem setOf_false : { _a : α | False } = ∅ :=
rfl
#align set.set_of_false Set.setOf_false
@[simp] theorem setOf_bot : { _x : α | ⊥ } = ∅ := rfl
@[simp]
theorem empty_subset (s : Set α) : ∅ ⊆ s :=
nofun
#align set.empty_subset Set.empty_subset
theorem subset_empty_iff {s : Set α} : s ⊆ ∅ ↔ s = ∅ :=
(Subset.antisymm_iff.trans <| and_iff_left (empty_subset _)).symm
#align set.subset_empty_iff Set.subset_empty_iff
theorem eq_empty_iff_forall_not_mem {s : Set α} : s = ∅ ↔ ∀ x, x ∉ s :=
subset_empty_iff.symm
#align set.eq_empty_iff_forall_not_mem Set.eq_empty_iff_forall_not_mem
theorem eq_empty_of_forall_not_mem (h : ∀ x, x ∉ s) : s = ∅ :=
subset_empty_iff.1 h
#align set.eq_empty_of_forall_not_mem Set.eq_empty_of_forall_not_mem
theorem eq_empty_of_subset_empty {s : Set α} : s ⊆ ∅ → s = ∅ :=
subset_empty_iff.1
#align set.eq_empty_of_subset_empty Set.eq_empty_of_subset_empty
theorem eq_empty_of_isEmpty [IsEmpty α] (s : Set α) : s = ∅ :=
eq_empty_of_subset_empty fun x _ => isEmptyElim x
#align set.eq_empty_of_is_empty Set.eq_empty_of_isEmpty
instance uniqueEmpty [IsEmpty α] : Unique (Set α) where
default := ∅
uniq := eq_empty_of_isEmpty
#align set.unique_empty Set.uniqueEmpty
theorem not_nonempty_iff_eq_empty {s : Set α} : ¬s.Nonempty ↔ s = ∅ := by
simp only [Set.Nonempty, not_exists, eq_empty_iff_forall_not_mem]
#align set.not_nonempty_iff_eq_empty Set.not_nonempty_iff_eq_empty
theorem nonempty_iff_ne_empty : s.Nonempty ↔ s ≠ ∅ :=
not_nonempty_iff_eq_empty.not_right
#align set.nonempty_iff_ne_empty Set.nonempty_iff_ne_empty
theorem not_nonempty_iff_eq_empty' : ¬Nonempty s ↔ s = ∅ := by
rw [nonempty_subtype, not_exists, eq_empty_iff_forall_not_mem]
theorem nonempty_iff_ne_empty' : Nonempty s ↔ s ≠ ∅ :=
not_nonempty_iff_eq_empty'.not_right
alias ⟨Nonempty.ne_empty, _⟩ := nonempty_iff_ne_empty
#align set.nonempty.ne_empty Set.Nonempty.ne_empty
@[simp]
theorem not_nonempty_empty : ¬(∅ : Set α).Nonempty := fun ⟨_, hx⟩ => hx
#align set.not_nonempty_empty Set.not_nonempty_empty
-- Porting note: removing `@[simp]` as it is competing with `isEmpty_subtype`.
-- @[simp]
theorem isEmpty_coe_sort {s : Set α} : IsEmpty (↥s) ↔ s = ∅ :=
not_iff_not.1 <| by simpa using nonempty_iff_ne_empty
#align set.is_empty_coe_sort Set.isEmpty_coe_sort
theorem eq_empty_or_nonempty (s : Set α) : s = ∅ ∨ s.Nonempty :=
or_iff_not_imp_left.2 nonempty_iff_ne_empty.2
#align set.eq_empty_or_nonempty Set.eq_empty_or_nonempty
theorem subset_eq_empty {s t : Set α} (h : t ⊆ s) (e : s = ∅) : t = ∅ :=
subset_empty_iff.1 <| e ▸ h
#align set.subset_eq_empty Set.subset_eq_empty
theorem forall_mem_empty {p : α → Prop} : (∀ x ∈ (∅ : Set α), p x) ↔ True :=
iff_true_intro fun _ => False.elim
#align set.ball_empty_iff Set.forall_mem_empty
@[deprecated (since := "2024-03-23")] alias ball_empty_iff := forall_mem_empty
instance (α : Type u) : IsEmpty.{u + 1} (↥(∅ : Set α)) :=
⟨fun x => x.2⟩
@[simp]
theorem empty_ssubset : ∅ ⊂ s ↔ s.Nonempty :=
(@bot_lt_iff_ne_bot (Set α) _ _ _).trans nonempty_iff_ne_empty.symm
#align set.empty_ssubset Set.empty_ssubset
alias ⟨_, Nonempty.empty_ssubset⟩ := empty_ssubset
#align set.nonempty.empty_ssubset Set.Nonempty.empty_ssubset
@[simp]
theorem setOf_true : { _x : α | True } = univ :=
rfl
#align set.set_of_true Set.setOf_true
@[simp] theorem setOf_top : { _x : α | ⊤ } = univ := rfl
@[simp]
theorem univ_eq_empty_iff : (univ : Set α) = ∅ ↔ IsEmpty α :=
eq_empty_iff_forall_not_mem.trans
⟨fun H => ⟨fun x => H x trivial⟩, fun H x _ => @IsEmpty.false α H x⟩
#align set.univ_eq_empty_iff Set.univ_eq_empty_iff
theorem empty_ne_univ [Nonempty α] : (∅ : Set α) ≠ univ := fun e =>
not_isEmpty_of_nonempty α <| univ_eq_empty_iff.1 e.symm
#align set.empty_ne_univ Set.empty_ne_univ
@[simp]
theorem subset_univ (s : Set α) : s ⊆ univ := fun _ _ => trivial
#align set.subset_univ Set.subset_univ
@[simp]
theorem univ_subset_iff {s : Set α} : univ ⊆ s ↔ s = univ :=
@top_le_iff _ _ _ s
#align set.univ_subset_iff Set.univ_subset_iff
alias ⟨eq_univ_of_univ_subset, _⟩ := univ_subset_iff
#align set.eq_univ_of_univ_subset Set.eq_univ_of_univ_subset
theorem eq_univ_iff_forall {s : Set α} : s = univ ↔ ∀ x, x ∈ s :=
univ_subset_iff.symm.trans <| forall_congr' fun _ => imp_iff_right trivial
#align set.eq_univ_iff_forall Set.eq_univ_iff_forall
theorem eq_univ_of_forall {s : Set α} : (∀ x, x ∈ s) → s = univ :=
eq_univ_iff_forall.2
#align set.eq_univ_of_forall Set.eq_univ_of_forall
theorem Nonempty.eq_univ [Subsingleton α] : s.Nonempty → s = univ := by
rintro ⟨x, hx⟩
exact eq_univ_of_forall fun y => by rwa [Subsingleton.elim y x]
#align set.nonempty.eq_univ Set.Nonempty.eq_univ
theorem eq_univ_of_subset {s t : Set α} (h : s ⊆ t) (hs : s = univ) : t = univ :=
eq_univ_of_univ_subset <| (hs ▸ h : univ ⊆ t)
#align set.eq_univ_of_subset Set.eq_univ_of_subset
theorem exists_mem_of_nonempty (α) : ∀ [Nonempty α], ∃ x : α, x ∈ (univ : Set α)
| ⟨x⟩ => ⟨x, trivial⟩
#align set.exists_mem_of_nonempty Set.exists_mem_of_nonempty
theorem ne_univ_iff_exists_not_mem {α : Type*} (s : Set α) : s ≠ univ ↔ ∃ a, a ∉ s := by
rw [← not_forall, ← eq_univ_iff_forall]
#align set.ne_univ_iff_exists_not_mem Set.ne_univ_iff_exists_not_mem
theorem not_subset_iff_exists_mem_not_mem {α : Type*} {s t : Set α} :
¬s ⊆ t ↔ ∃ x, x ∈ s ∧ x ∉ t := by simp [subset_def]
#align set.not_subset_iff_exists_mem_not_mem Set.not_subset_iff_exists_mem_not_mem
theorem univ_unique [Unique α] : @Set.univ α = {default} :=
Set.ext fun x => iff_of_true trivial <| Subsingleton.elim x default
#align set.univ_unique Set.univ_unique
theorem ssubset_univ_iff : s ⊂ univ ↔ s ≠ univ :=
lt_top_iff_ne_top
#align set.ssubset_univ_iff Set.ssubset_univ_iff
instance nontrivial_of_nonempty [Nonempty α] : Nontrivial (Set α) :=
⟨⟨∅, univ, empty_ne_univ⟩⟩
#align set.nontrivial_of_nonempty Set.nontrivial_of_nonempty
theorem union_def {s₁ s₂ : Set α} : s₁ ∪ s₂ = { a | a ∈ s₁ ∨ a ∈ s₂ } :=
rfl
#align set.union_def Set.union_def
theorem mem_union_left {x : α} {a : Set α} (b : Set α) : x ∈ a → x ∈ a ∪ b :=
Or.inl
#align set.mem_union_left Set.mem_union_left
theorem mem_union_right {x : α} {b : Set α} (a : Set α) : x ∈ b → x ∈ a ∪ b :=
Or.inr
#align set.mem_union_right Set.mem_union_right
theorem mem_or_mem_of_mem_union {x : α} {a b : Set α} (H : x ∈ a ∪ b) : x ∈ a ∨ x ∈ b :=
H
#align set.mem_or_mem_of_mem_union Set.mem_or_mem_of_mem_union
theorem MemUnion.elim {x : α} {a b : Set α} {P : Prop} (H₁ : x ∈ a ∪ b) (H₂ : x ∈ a → P)
(H₃ : x ∈ b → P) : P :=
Or.elim H₁ H₂ H₃
#align set.mem_union.elim Set.MemUnion.elim
@[simp]
theorem mem_union (x : α) (a b : Set α) : x ∈ a ∪ b ↔ x ∈ a ∨ x ∈ b :=
Iff.rfl
#align set.mem_union Set.mem_union
@[simp]
theorem union_self (a : Set α) : a ∪ a = a :=
ext fun _ => or_self_iff
#align set.union_self Set.union_self
@[simp]
theorem union_empty (a : Set α) : a ∪ ∅ = a :=
ext fun _ => or_false_iff _
#align set.union_empty Set.union_empty
@[simp]
theorem empty_union (a : Set α) : ∅ ∪ a = a :=
ext fun _ => false_or_iff _
#align set.empty_union Set.empty_union
theorem union_comm (a b : Set α) : a ∪ b = b ∪ a :=
ext fun _ => or_comm
#align set.union_comm Set.union_comm
theorem union_assoc (a b c : Set α) : a ∪ b ∪ c = a ∪ (b ∪ c) :=
ext fun _ => or_assoc
#align set.union_assoc Set.union_assoc
instance union_isAssoc : Std.Associative (α := Set α) (· ∪ ·) :=
⟨union_assoc⟩
#align set.union_is_assoc Set.union_isAssoc
instance union_isComm : Std.Commutative (α := Set α) (· ∪ ·) :=
⟨union_comm⟩
#align set.union_is_comm Set.union_isComm
theorem union_left_comm (s₁ s₂ s₃ : Set α) : s₁ ∪ (s₂ ∪ s₃) = s₂ ∪ (s₁ ∪ s₃) :=
ext fun _ => or_left_comm
#align set.union_left_comm Set.union_left_comm
theorem union_right_comm (s₁ s₂ s₃ : Set α) : s₁ ∪ s₂ ∪ s₃ = s₁ ∪ s₃ ∪ s₂ :=
ext fun _ => or_right_comm
#align set.union_right_comm Set.union_right_comm
@[simp]
theorem union_eq_left {s t : Set α} : s ∪ t = s ↔ t ⊆ s :=
sup_eq_left
#align set.union_eq_left_iff_subset Set.union_eq_left
@[simp]
theorem union_eq_right {s t : Set α} : s ∪ t = t ↔ s ⊆ t :=
sup_eq_right
#align set.union_eq_right_iff_subset Set.union_eq_right
theorem union_eq_self_of_subset_left {s t : Set α} (h : s ⊆ t) : s ∪ t = t :=
union_eq_right.mpr h
#align set.union_eq_self_of_subset_left Set.union_eq_self_of_subset_left
theorem union_eq_self_of_subset_right {s t : Set α} (h : t ⊆ s) : s ∪ t = s :=
union_eq_left.mpr h
#align set.union_eq_self_of_subset_right Set.union_eq_self_of_subset_right
@[simp]
theorem subset_union_left {s t : Set α} : s ⊆ s ∪ t := fun _ => Or.inl
#align set.subset_union_left Set.subset_union_left
@[simp]
theorem subset_union_right {s t : Set α} : t ⊆ s ∪ t := fun _ => Or.inr
#align set.subset_union_right Set.subset_union_right
theorem union_subset {s t r : Set α} (sr : s ⊆ r) (tr : t ⊆ r) : s ∪ t ⊆ r := fun _ =>
Or.rec (@sr _) (@tr _)
#align set.union_subset Set.union_subset
@[simp]
theorem union_subset_iff {s t u : Set α} : s ∪ t ⊆ u ↔ s ⊆ u ∧ t ⊆ u :=
(forall_congr' fun _ => or_imp).trans forall_and
#align set.union_subset_iff Set.union_subset_iff
@[gcongr]
theorem union_subset_union {s₁ s₂ t₁ t₂ : Set α} (h₁ : s₁ ⊆ s₂) (h₂ : t₁ ⊆ t₂) :
s₁ ∪ t₁ ⊆ s₂ ∪ t₂ := fun _ => Or.imp (@h₁ _) (@h₂ _)
#align set.union_subset_union Set.union_subset_union
@[gcongr]
theorem union_subset_union_left {s₁ s₂ : Set α} (t) (h : s₁ ⊆ s₂) : s₁ ∪ t ⊆ s₂ ∪ t :=
union_subset_union h Subset.rfl
#align set.union_subset_union_left Set.union_subset_union_left
@[gcongr]
theorem union_subset_union_right (s) {t₁ t₂ : Set α} (h : t₁ ⊆ t₂) : s ∪ t₁ ⊆ s ∪ t₂ :=
union_subset_union Subset.rfl h
#align set.union_subset_union_right Set.union_subset_union_right
theorem subset_union_of_subset_left {s t : Set α} (h : s ⊆ t) (u : Set α) : s ⊆ t ∪ u :=
h.trans subset_union_left
#align set.subset_union_of_subset_left Set.subset_union_of_subset_left
theorem subset_union_of_subset_right {s u : Set α} (h : s ⊆ u) (t : Set α) : s ⊆ t ∪ u :=
h.trans subset_union_right
#align set.subset_union_of_subset_right Set.subset_union_of_subset_right
-- Porting note: replaced `⊔` in RHS
theorem union_congr_left (ht : t ⊆ s ∪ u) (hu : u ⊆ s ∪ t) : s ∪ t = s ∪ u :=
sup_congr_left ht hu
#align set.union_congr_left Set.union_congr_left
theorem union_congr_right (hs : s ⊆ t ∪ u) (ht : t ⊆ s ∪ u) : s ∪ u = t ∪ u :=
sup_congr_right hs ht
#align set.union_congr_right Set.union_congr_right
theorem union_eq_union_iff_left : s ∪ t = s ∪ u ↔ t ⊆ s ∪ u ∧ u ⊆ s ∪ t :=
sup_eq_sup_iff_left
#align set.union_eq_union_iff_left Set.union_eq_union_iff_left
theorem union_eq_union_iff_right : s ∪ u = t ∪ u ↔ s ⊆ t ∪ u ∧ t ⊆ s ∪ u :=
sup_eq_sup_iff_right
#align set.union_eq_union_iff_right Set.union_eq_union_iff_right
@[simp]
theorem union_empty_iff {s t : Set α} : s ∪ t = ∅ ↔ s = ∅ ∧ t = ∅ := by
simp only [← subset_empty_iff]
exact union_subset_iff
#align set.union_empty_iff Set.union_empty_iff
@[simp]
theorem union_univ (s : Set α) : s ∪ univ = univ := sup_top_eq _
#align set.union_univ Set.union_univ
@[simp]
theorem univ_union (s : Set α) : univ ∪ s = univ := top_sup_eq _
#align set.univ_union Set.univ_union
theorem inter_def {s₁ s₂ : Set α} : s₁ ∩ s₂ = { a | a ∈ s₁ ∧ a ∈ s₂ } :=
rfl
#align set.inter_def Set.inter_def
@[simp, mfld_simps]
theorem mem_inter_iff (x : α) (a b : Set α) : x ∈ a ∩ b ↔ x ∈ a ∧ x ∈ b :=
Iff.rfl
#align set.mem_inter_iff Set.mem_inter_iff
theorem mem_inter {x : α} {a b : Set α} (ha : x ∈ a) (hb : x ∈ b) : x ∈ a ∩ b :=
⟨ha, hb⟩
#align set.mem_inter Set.mem_inter
theorem mem_of_mem_inter_left {x : α} {a b : Set α} (h : x ∈ a ∩ b) : x ∈ a :=
h.left
#align set.mem_of_mem_inter_left Set.mem_of_mem_inter_left
theorem mem_of_mem_inter_right {x : α} {a b : Set α} (h : x ∈ a ∩ b) : x ∈ b :=
h.right
#align set.mem_of_mem_inter_right Set.mem_of_mem_inter_right
@[simp]
theorem inter_self (a : Set α) : a ∩ a = a :=
ext fun _ => and_self_iff
#align set.inter_self Set.inter_self
@[simp]
theorem inter_empty (a : Set α) : a ∩ ∅ = ∅ :=
ext fun _ => and_false_iff _
#align set.inter_empty Set.inter_empty
@[simp]
theorem empty_inter (a : Set α) : ∅ ∩ a = ∅ :=
ext fun _ => false_and_iff _
#align set.empty_inter Set.empty_inter
theorem inter_comm (a b : Set α) : a ∩ b = b ∩ a :=
ext fun _ => and_comm
#align set.inter_comm Set.inter_comm
theorem inter_assoc (a b c : Set α) : a ∩ b ∩ c = a ∩ (b ∩ c) :=
ext fun _ => and_assoc
#align set.inter_assoc Set.inter_assoc
instance inter_isAssoc : Std.Associative (α := Set α) (· ∩ ·) :=
⟨inter_assoc⟩
#align set.inter_is_assoc Set.inter_isAssoc
instance inter_isComm : Std.Commutative (α := Set α) (· ∩ ·) :=
⟨inter_comm⟩
#align set.inter_is_comm Set.inter_isComm
theorem inter_left_comm (s₁ s₂ s₃ : Set α) : s₁ ∩ (s₂ ∩ s₃) = s₂ ∩ (s₁ ∩ s₃) :=
ext fun _ => and_left_comm
#align set.inter_left_comm Set.inter_left_comm
theorem inter_right_comm (s₁ s₂ s₃ : Set α) : s₁ ∩ s₂ ∩ s₃ = s₁ ∩ s₃ ∩ s₂ :=
ext fun _ => and_right_comm
#align set.inter_right_comm Set.inter_right_comm
@[simp, mfld_simps]
theorem inter_subset_left {s t : Set α} : s ∩ t ⊆ s := fun _ => And.left
#align set.inter_subset_left Set.inter_subset_left
@[simp]
theorem inter_subset_right {s t : Set α} : s ∩ t ⊆ t := fun _ => And.right
#align set.inter_subset_right Set.inter_subset_right
theorem subset_inter {s t r : Set α} (rs : r ⊆ s) (rt : r ⊆ t) : r ⊆ s ∩ t := fun _ h =>
⟨rs h, rt h⟩
#align set.subset_inter Set.subset_inter
@[simp]
theorem subset_inter_iff {s t r : Set α} : r ⊆ s ∩ t ↔ r ⊆ s ∧ r ⊆ t :=
(forall_congr' fun _ => imp_and).trans forall_and
#align set.subset_inter_iff Set.subset_inter_iff
@[simp] lemma inter_eq_left : s ∩ t = s ↔ s ⊆ t := inf_eq_left
#align set.inter_eq_left_iff_subset Set.inter_eq_left
@[simp] lemma inter_eq_right : s ∩ t = t ↔ t ⊆ s := inf_eq_right
#align set.inter_eq_right_iff_subset Set.inter_eq_right
@[simp] lemma left_eq_inter : s = s ∩ t ↔ s ⊆ t := left_eq_inf
@[simp] lemma right_eq_inter : t = s ∩ t ↔ t ⊆ s := right_eq_inf
theorem inter_eq_self_of_subset_left {s t : Set α} : s ⊆ t → s ∩ t = s :=
inter_eq_left.mpr
#align set.inter_eq_self_of_subset_left Set.inter_eq_self_of_subset_left
theorem inter_eq_self_of_subset_right {s t : Set α} : t ⊆ s → s ∩ t = t :=
inter_eq_right.mpr
#align set.inter_eq_self_of_subset_right Set.inter_eq_self_of_subset_right
theorem inter_congr_left (ht : s ∩ u ⊆ t) (hu : s ∩ t ⊆ u) : s ∩ t = s ∩ u :=
inf_congr_left ht hu
#align set.inter_congr_left Set.inter_congr_left
theorem inter_congr_right (hs : t ∩ u ⊆ s) (ht : s ∩ u ⊆ t) : s ∩ u = t ∩ u :=
inf_congr_right hs ht
#align set.inter_congr_right Set.inter_congr_right
theorem inter_eq_inter_iff_left : s ∩ t = s ∩ u ↔ s ∩ u ⊆ t ∧ s ∩ t ⊆ u :=
inf_eq_inf_iff_left
#align set.inter_eq_inter_iff_left Set.inter_eq_inter_iff_left
theorem inter_eq_inter_iff_right : s ∩ u = t ∩ u ↔ t ∩ u ⊆ s ∧ s ∩ u ⊆ t :=
inf_eq_inf_iff_right
#align set.inter_eq_inter_iff_right Set.inter_eq_inter_iff_right
@[simp, mfld_simps]
theorem inter_univ (a : Set α) : a ∩ univ = a := inf_top_eq _
#align set.inter_univ Set.inter_univ
@[simp, mfld_simps]
theorem univ_inter (a : Set α) : univ ∩ a = a := top_inf_eq _
#align set.univ_inter Set.univ_inter
@[gcongr]
theorem inter_subset_inter {s₁ s₂ t₁ t₂ : Set α} (h₁ : s₁ ⊆ t₁) (h₂ : s₂ ⊆ t₂) :
s₁ ∩ s₂ ⊆ t₁ ∩ t₂ := fun _ => And.imp (@h₁ _) (@h₂ _)
#align set.inter_subset_inter Set.inter_subset_inter
@[gcongr]
theorem inter_subset_inter_left {s t : Set α} (u : Set α) (H : s ⊆ t) : s ∩ u ⊆ t ∩ u :=
inter_subset_inter H Subset.rfl
#align set.inter_subset_inter_left Set.inter_subset_inter_left
@[gcongr]
theorem inter_subset_inter_right {s t : Set α} (u : Set α) (H : s ⊆ t) : u ∩ s ⊆ u ∩ t :=
inter_subset_inter Subset.rfl H
#align set.inter_subset_inter_right Set.inter_subset_inter_right
theorem union_inter_cancel_left {s t : Set α} : (s ∪ t) ∩ s = s :=
inter_eq_self_of_subset_right subset_union_left
#align set.union_inter_cancel_left Set.union_inter_cancel_left
theorem union_inter_cancel_right {s t : Set α} : (s ∪ t) ∩ t = t :=
inter_eq_self_of_subset_right subset_union_right
#align set.union_inter_cancel_right Set.union_inter_cancel_right
theorem inter_setOf_eq_sep (s : Set α) (p : α → Prop) : s ∩ {a | p a} = {a ∈ s | p a} :=
rfl
#align set.inter_set_of_eq_sep Set.inter_setOf_eq_sep
theorem setOf_inter_eq_sep (p : α → Prop) (s : Set α) : {a | p a} ∩ s = {a ∈ s | p a} :=
inter_comm _ _
#align set.set_of_inter_eq_sep Set.setOf_inter_eq_sep
theorem inter_union_distrib_left (s t u : Set α) : s ∩ (t ∪ u) = s ∩ t ∪ s ∩ u :=
inf_sup_left _ _ _
#align set.inter_distrib_left Set.inter_union_distrib_left
theorem union_inter_distrib_right (s t u : Set α) : (s ∪ t) ∩ u = s ∩ u ∪ t ∩ u :=
inf_sup_right _ _ _
#align set.inter_distrib_right Set.union_inter_distrib_right
theorem union_inter_distrib_left (s t u : Set α) : s ∪ t ∩ u = (s ∪ t) ∩ (s ∪ u) :=
sup_inf_left _ _ _
#align set.union_distrib_left Set.union_inter_distrib_left
theorem inter_union_distrib_right (s t u : Set α) : s ∩ t ∪ u = (s ∪ u) ∩ (t ∪ u) :=
sup_inf_right _ _ _
#align set.union_distrib_right Set.inter_union_distrib_right
-- 2024-03-22
@[deprecated] alias inter_distrib_left := inter_union_distrib_left
@[deprecated] alias inter_distrib_right := union_inter_distrib_right
@[deprecated] alias union_distrib_left := union_inter_distrib_left
@[deprecated] alias union_distrib_right := inter_union_distrib_right
theorem union_union_distrib_left (s t u : Set α) : s ∪ (t ∪ u) = s ∪ t ∪ (s ∪ u) :=
sup_sup_distrib_left _ _ _
#align set.union_union_distrib_left Set.union_union_distrib_left
theorem union_union_distrib_right (s t u : Set α) : s ∪ t ∪ u = s ∪ u ∪ (t ∪ u) :=
sup_sup_distrib_right _ _ _
#align set.union_union_distrib_right Set.union_union_distrib_right
theorem inter_inter_distrib_left (s t u : Set α) : s ∩ (t ∩ u) = s ∩ t ∩ (s ∩ u) :=
inf_inf_distrib_left _ _ _
#align set.inter_inter_distrib_left Set.inter_inter_distrib_left
theorem inter_inter_distrib_right (s t u : Set α) : s ∩ t ∩ u = s ∩ u ∩ (t ∩ u) :=
inf_inf_distrib_right _ _ _
#align set.inter_inter_distrib_right Set.inter_inter_distrib_right
theorem union_union_union_comm (s t u v : Set α) : s ∪ t ∪ (u ∪ v) = s ∪ u ∪ (t ∪ v) :=
sup_sup_sup_comm _ _ _ _
#align set.union_union_union_comm Set.union_union_union_comm
theorem inter_inter_inter_comm (s t u v : Set α) : s ∩ t ∩ (u ∩ v) = s ∩ u ∩ (t ∩ v) :=
inf_inf_inf_comm _ _ _ _
#align set.inter_inter_inter_comm Set.inter_inter_inter_comm
theorem insert_def (x : α) (s : Set α) : insert x s = { y | y = x ∨ y ∈ s } :=
rfl
#align set.insert_def Set.insert_def
@[simp]
theorem subset_insert (x : α) (s : Set α) : s ⊆ insert x s := fun _ => Or.inr
#align set.subset_insert Set.subset_insert
theorem mem_insert (x : α) (s : Set α) : x ∈ insert x s :=
Or.inl rfl
#align set.mem_insert Set.mem_insert
theorem mem_insert_of_mem {x : α} {s : Set α} (y : α) : x ∈ s → x ∈ insert y s :=
Or.inr
#align set.mem_insert_of_mem Set.mem_insert_of_mem
theorem eq_or_mem_of_mem_insert {x a : α} {s : Set α} : x ∈ insert a s → x = a ∨ x ∈ s :=
id
#align set.eq_or_mem_of_mem_insert Set.eq_or_mem_of_mem_insert
theorem mem_of_mem_insert_of_ne : b ∈ insert a s → b ≠ a → b ∈ s :=
Or.resolve_left
#align set.mem_of_mem_insert_of_ne Set.mem_of_mem_insert_of_ne
theorem eq_of_not_mem_of_mem_insert : b ∈ insert a s → b ∉ s → b = a :=
Or.resolve_right
#align set.eq_of_not_mem_of_mem_insert Set.eq_of_not_mem_of_mem_insert
@[simp]
theorem mem_insert_iff {x a : α} {s : Set α} : x ∈ insert a s ↔ x = a ∨ x ∈ s :=
Iff.rfl
#align set.mem_insert_iff Set.mem_insert_iff
@[simp]
theorem insert_eq_of_mem {a : α} {s : Set α} (h : a ∈ s) : insert a s = s :=
ext fun _ => or_iff_right_of_imp fun e => e.symm ▸ h
#align set.insert_eq_of_mem Set.insert_eq_of_mem
theorem ne_insert_of_not_mem {s : Set α} (t : Set α) {a : α} : a ∉ s → s ≠ insert a t :=
mt fun e => e.symm ▸ mem_insert _ _
#align set.ne_insert_of_not_mem Set.ne_insert_of_not_mem
@[simp]
theorem insert_eq_self : insert a s = s ↔ a ∈ s :=
⟨fun h => h ▸ mem_insert _ _, insert_eq_of_mem⟩
#align set.insert_eq_self Set.insert_eq_self
theorem insert_ne_self : insert a s ≠ s ↔ a ∉ s :=
insert_eq_self.not
#align set.insert_ne_self Set.insert_ne_self
theorem insert_subset_iff : insert a s ⊆ t ↔ a ∈ t ∧ s ⊆ t := by
simp only [subset_def, mem_insert_iff, or_imp, forall_and, forall_eq]
#align set.insert_subset Set.insert_subset_iff
theorem insert_subset (ha : a ∈ t) (hs : s ⊆ t) : insert a s ⊆ t :=
insert_subset_iff.mpr ⟨ha, hs⟩
theorem insert_subset_insert (h : s ⊆ t) : insert a s ⊆ insert a t := fun _ => Or.imp_right (@h _)
#align set.insert_subset_insert Set.insert_subset_insert
@[simp] theorem insert_subset_insert_iff (ha : a ∉ s) : insert a s ⊆ insert a t ↔ s ⊆ t := by
refine ⟨fun h x hx => ?_, insert_subset_insert⟩
rcases h (subset_insert _ _ hx) with (rfl | hxt)
exacts [(ha hx).elim, hxt]
#align set.insert_subset_insert_iff Set.insert_subset_insert_iff
theorem subset_insert_iff_of_not_mem (ha : a ∉ s) : s ⊆ insert a t ↔ s ⊆ t :=
forall₂_congr fun _ hb => or_iff_right <| ne_of_mem_of_not_mem hb ha
#align set.subset_insert_iff_of_not_mem Set.subset_insert_iff_of_not_mem
theorem ssubset_iff_insert {s t : Set α} : s ⊂ t ↔ ∃ a ∉ s, insert a s ⊆ t := by
simp only [insert_subset_iff, exists_and_right, ssubset_def, not_subset]
aesop
#align set.ssubset_iff_insert Set.ssubset_iff_insert
theorem ssubset_insert {s : Set α} {a : α} (h : a ∉ s) : s ⊂ insert a s :=
ssubset_iff_insert.2 ⟨a, h, Subset.rfl⟩
#align set.ssubset_insert Set.ssubset_insert
theorem insert_comm (a b : α) (s : Set α) : insert a (insert b s) = insert b (insert a s) :=
ext fun _ => or_left_comm
#align set.insert_comm Set.insert_comm
-- Porting note (#10618): removing `simp` attribute because `simp` can prove it
theorem insert_idem (a : α) (s : Set α) : insert a (insert a s) = insert a s :=
insert_eq_of_mem <| mem_insert _ _
#align set.insert_idem Set.insert_idem
theorem insert_union : insert a s ∪ t = insert a (s ∪ t) :=
ext fun _ => or_assoc
#align set.insert_union Set.insert_union
@[simp]
theorem union_insert : s ∪ insert a t = insert a (s ∪ t) :=
ext fun _ => or_left_comm
#align set.union_insert Set.union_insert
@[simp]
theorem insert_nonempty (a : α) (s : Set α) : (insert a s).Nonempty :=
⟨a, mem_insert a s⟩
#align set.insert_nonempty Set.insert_nonempty
instance (a : α) (s : Set α) : Nonempty (insert a s : Set α) :=
(insert_nonempty a s).to_subtype
theorem insert_inter_distrib (a : α) (s t : Set α) : insert a (s ∩ t) = insert a s ∩ insert a t :=
ext fun _ => or_and_left
#align set.insert_inter_distrib Set.insert_inter_distrib
theorem insert_union_distrib (a : α) (s t : Set α) : insert a (s ∪ t) = insert a s ∪ insert a t :=
ext fun _ => or_or_distrib_left
#align set.insert_union_distrib Set.insert_union_distrib
theorem insert_inj (ha : a ∉ s) : insert a s = insert b s ↔ a = b :=
⟨fun h => eq_of_not_mem_of_mem_insert (h.subst <| mem_insert a s) ha,
congr_arg (fun x => insert x s)⟩
#align set.insert_inj Set.insert_inj
-- useful in proofs by induction
theorem forall_of_forall_insert {P : α → Prop} {a : α} {s : Set α} (H : ∀ x, x ∈ insert a s → P x)
(x) (h : x ∈ s) : P x :=
H _ (Or.inr h)
#align set.forall_of_forall_insert Set.forall_of_forall_insert
theorem forall_insert_of_forall {P : α → Prop} {a : α} {s : Set α} (H : ∀ x, x ∈ s → P x) (ha : P a)
(x) (h : x ∈ insert a s) : P x :=
h.elim (fun e => e.symm ▸ ha) (H _)
#align set.forall_insert_of_forall Set.forall_insert_of_forall
theorem exists_mem_insert {P : α → Prop} {a : α} {s : Set α} :
(∃ x ∈ insert a s, P x) ↔ (P a ∨ ∃ x ∈ s, P x) := by
simp [mem_insert_iff, or_and_right, exists_and_left, exists_or]
#align set.bex_insert_iff Set.exists_mem_insert
@[deprecated (since := "2024-03-23")] alias bex_insert_iff := exists_mem_insert
theorem forall_mem_insert {P : α → Prop} {a : α} {s : Set α} :
(∀ x ∈ insert a s, P x) ↔ P a ∧ ∀ x ∈ s, P x :=
forall₂_or_left.trans <| and_congr_left' forall_eq
#align set.ball_insert_iff Set.forall_mem_insert
@[deprecated (since := "2024-03-23")] alias ball_insert_iff := forall_mem_insert
instance : LawfulSingleton α (Set α) :=
⟨fun x => Set.ext fun a => by
simp only [mem_empty_iff_false, mem_insert_iff, or_false]
exact Iff.rfl⟩
theorem singleton_def (a : α) : ({a} : Set α) = insert a ∅ :=
(insert_emptyc_eq a).symm
#align set.singleton_def Set.singleton_def
@[simp]
theorem mem_singleton_iff {a b : α} : a ∈ ({b} : Set α) ↔ a = b :=
Iff.rfl
#align set.mem_singleton_iff Set.mem_singleton_iff
@[simp]
theorem setOf_eq_eq_singleton {a : α} : { n | n = a } = {a} :=
rfl
#align set.set_of_eq_eq_singleton Set.setOf_eq_eq_singleton
@[simp]
theorem setOf_eq_eq_singleton' {a : α} : { x | a = x } = {a} :=
ext fun _ => eq_comm
#align set.set_of_eq_eq_singleton' Set.setOf_eq_eq_singleton'
-- TODO: again, annotation needed
--Porting note (#11119): removed `simp` attribute
theorem mem_singleton (a : α) : a ∈ ({a} : Set α) :=
@rfl _ _
#align set.mem_singleton Set.mem_singleton
theorem eq_of_mem_singleton {x y : α} (h : x ∈ ({y} : Set α)) : x = y :=
h
#align set.eq_of_mem_singleton Set.eq_of_mem_singleton
@[simp]
theorem singleton_eq_singleton_iff {x y : α} : {x} = ({y} : Set α) ↔ x = y :=
ext_iff.trans eq_iff_eq_cancel_left
#align set.singleton_eq_singleton_iff Set.singleton_eq_singleton_iff
theorem singleton_injective : Injective (singleton : α → Set α) := fun _ _ =>
singleton_eq_singleton_iff.mp
#align set.singleton_injective Set.singleton_injective
theorem mem_singleton_of_eq {x y : α} (H : x = y) : x ∈ ({y} : Set α) :=
H
#align set.mem_singleton_of_eq Set.mem_singleton_of_eq
theorem insert_eq (x : α) (s : Set α) : insert x s = ({x} : Set α) ∪ s :=
rfl
#align set.insert_eq Set.insert_eq
@[simp]
theorem singleton_nonempty (a : α) : ({a} : Set α).Nonempty :=
⟨a, rfl⟩
#align set.singleton_nonempty Set.singleton_nonempty
@[simp]
theorem singleton_ne_empty (a : α) : ({a} : Set α) ≠ ∅ :=
(singleton_nonempty _).ne_empty
#align set.singleton_ne_empty Set.singleton_ne_empty
--Porting note (#10618): removed `simp` attribute because `simp` can prove it
theorem empty_ssubset_singleton : (∅ : Set α) ⊂ {a} :=
(singleton_nonempty _).empty_ssubset
#align set.empty_ssubset_singleton Set.empty_ssubset_singleton
@[simp]
theorem singleton_subset_iff {a : α} {s : Set α} : {a} ⊆ s ↔ a ∈ s :=
forall_eq
#align set.singleton_subset_iff Set.singleton_subset_iff
theorem singleton_subset_singleton : ({a} : Set α) ⊆ {b} ↔ a = b := by simp
#align set.singleton_subset_singleton Set.singleton_subset_singleton
theorem set_compr_eq_eq_singleton {a : α} : { b | b = a } = {a} :=
rfl
#align set.set_compr_eq_eq_singleton Set.set_compr_eq_eq_singleton
@[simp]
theorem singleton_union : {a} ∪ s = insert a s :=
rfl
#align set.singleton_union Set.singleton_union
@[simp]
theorem union_singleton : s ∪ {a} = insert a s :=
union_comm _ _
#align set.union_singleton Set.union_singleton
@[simp]
theorem singleton_inter_nonempty : ({a} ∩ s).Nonempty ↔ a ∈ s := by
simp only [Set.Nonempty, mem_inter_iff, mem_singleton_iff, exists_eq_left]
#align set.singleton_inter_nonempty Set.singleton_inter_nonempty
@[simp]
theorem inter_singleton_nonempty : (s ∩ {a}).Nonempty ↔ a ∈ s := by
rw [inter_comm, singleton_inter_nonempty]
#align set.inter_singleton_nonempty Set.inter_singleton_nonempty
@[simp]
theorem singleton_inter_eq_empty : {a} ∩ s = ∅ ↔ a ∉ s :=
not_nonempty_iff_eq_empty.symm.trans singleton_inter_nonempty.not
#align set.singleton_inter_eq_empty Set.singleton_inter_eq_empty
@[simp]
theorem inter_singleton_eq_empty : s ∩ {a} = ∅ ↔ a ∉ s := by
rw [inter_comm, singleton_inter_eq_empty]
#align set.inter_singleton_eq_empty Set.inter_singleton_eq_empty
theorem nmem_singleton_empty {s : Set α} : s ∉ ({∅} : Set (Set α)) ↔ s.Nonempty :=
nonempty_iff_ne_empty.symm
#align set.nmem_singleton_empty Set.nmem_singleton_empty
instance uniqueSingleton (a : α) : Unique (↥({a} : Set α)) :=
⟨⟨⟨a, mem_singleton a⟩⟩, fun ⟨_, h⟩ => Subtype.eq h⟩
#align set.unique_singleton Set.uniqueSingleton
theorem eq_singleton_iff_unique_mem : s = {a} ↔ a ∈ s ∧ ∀ x ∈ s, x = a :=
Subset.antisymm_iff.trans <| and_comm.trans <| and_congr_left' singleton_subset_iff
#align set.eq_singleton_iff_unique_mem Set.eq_singleton_iff_unique_mem
theorem eq_singleton_iff_nonempty_unique_mem : s = {a} ↔ s.Nonempty ∧ ∀ x ∈ s, x = a :=
eq_singleton_iff_unique_mem.trans <|
and_congr_left fun H => ⟨fun h' => ⟨_, h'⟩, fun ⟨x, h⟩ => H x h ▸ h⟩
#align set.eq_singleton_iff_nonempty_unique_mem Set.eq_singleton_iff_nonempty_unique_mem
set_option backward.synthInstance.canonInstances false in -- See https://github.com/leanprover-community/mathlib4/issues/12532
-- while `simp` is capable of proving this, it is not capable of turning the LHS into the RHS.
@[simp]
theorem default_coe_singleton (x : α) : (default : ({x} : Set α)) = ⟨x, rfl⟩ :=
rfl
#align set.default_coe_singleton Set.default_coe_singleton
@[simp]
theorem subset_singleton_iff {α : Type*} {s : Set α} {x : α} : s ⊆ {x} ↔ ∀ y ∈ s, y = x :=
Iff.rfl
#align set.subset_singleton_iff Set.subset_singleton_iff
theorem subset_singleton_iff_eq {s : Set α} {x : α} : s ⊆ {x} ↔ s = ∅ ∨ s = {x} := by
obtain rfl | hs := s.eq_empty_or_nonempty
· exact ⟨fun _ => Or.inl rfl, fun _ => empty_subset _⟩
· simp [eq_singleton_iff_nonempty_unique_mem, hs, hs.ne_empty]
#align set.subset_singleton_iff_eq Set.subset_singleton_iff_eq
theorem Nonempty.subset_singleton_iff (h : s.Nonempty) : s ⊆ {a} ↔ s = {a} :=
subset_singleton_iff_eq.trans <| or_iff_right h.ne_empty
#align set.nonempty.subset_singleton_iff Set.Nonempty.subset_singleton_iff
theorem ssubset_singleton_iff {s : Set α} {x : α} : s ⊂ {x} ↔ s = ∅ := by
rw [ssubset_iff_subset_ne, subset_singleton_iff_eq, or_and_right, and_not_self_iff, or_false_iff,
and_iff_left_iff_imp]
exact fun h => h ▸ (singleton_ne_empty _).symm
#align set.ssubset_singleton_iff Set.ssubset_singleton_iff
theorem eq_empty_of_ssubset_singleton {s : Set α} {x : α} (hs : s ⊂ {x}) : s = ∅ :=
ssubset_singleton_iff.1 hs
#align set.eq_empty_of_ssubset_singleton Set.eq_empty_of_ssubset_singleton
theorem eq_of_nonempty_of_subsingleton {α} [Subsingleton α] (s t : Set α) [Nonempty s]
[Nonempty t] : s = t :=
nonempty_of_nonempty_subtype.eq_univ.trans nonempty_of_nonempty_subtype.eq_univ.symm
theorem eq_of_nonempty_of_subsingleton' {α} [Subsingleton α] {s : Set α} (t : Set α)
(hs : s.Nonempty) [Nonempty t] : s = t :=
have := hs.to_subtype; eq_of_nonempty_of_subsingleton s t
set_option backward.synthInstance.canonInstances false in -- See https://github.com/leanprover-community/mathlib4/issues/12532
theorem Nonempty.eq_zero [Subsingleton α] [Zero α] {s : Set α} (h : s.Nonempty) :
s = {0} := eq_of_nonempty_of_subsingleton' {0} h
set_option backward.synthInstance.canonInstances false in -- See https://github.com/leanprover-community/mathlib4/issues/12532
theorem Nonempty.eq_one [Subsingleton α] [One α] {s : Set α} (h : s.Nonempty) :
s = {1} := eq_of_nonempty_of_subsingleton' {1} h
protected theorem disjoint_iff : Disjoint s t ↔ s ∩ t ⊆ ∅ :=
disjoint_iff_inf_le
#align set.disjoint_iff Set.disjoint_iff
theorem disjoint_iff_inter_eq_empty : Disjoint s t ↔ s ∩ t = ∅ :=
disjoint_iff
#align set.disjoint_iff_inter_eq_empty Set.disjoint_iff_inter_eq_empty
theorem _root_.Disjoint.inter_eq : Disjoint s t → s ∩ t = ∅ :=
Disjoint.eq_bot
#align disjoint.inter_eq Disjoint.inter_eq
theorem disjoint_left : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ s → a ∉ t :=
disjoint_iff_inf_le.trans <| forall_congr' fun _ => not_and
#align set.disjoint_left Set.disjoint_left
theorem disjoint_right : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ t → a ∉ s := by rw [disjoint_comm, disjoint_left]
#align set.disjoint_right Set.disjoint_right
lemma not_disjoint_iff : ¬Disjoint s t ↔ ∃ x, x ∈ s ∧ x ∈ t :=
Set.disjoint_iff.not.trans <| not_forall.trans <| exists_congr fun _ ↦ not_not
#align set.not_disjoint_iff Set.not_disjoint_iff
lemma not_disjoint_iff_nonempty_inter : ¬ Disjoint s t ↔ (s ∩ t).Nonempty := not_disjoint_iff
#align set.not_disjoint_iff_nonempty_inter Set.not_disjoint_iff_nonempty_inter
alias ⟨_, Nonempty.not_disjoint⟩ := not_disjoint_iff_nonempty_inter
#align set.nonempty.not_disjoint Set.Nonempty.not_disjoint
lemma disjoint_or_nonempty_inter (s t : Set α) : Disjoint s t ∨ (s ∩ t).Nonempty :=
(em _).imp_right not_disjoint_iff_nonempty_inter.1
#align set.disjoint_or_nonempty_inter Set.disjoint_or_nonempty_inter
lemma disjoint_iff_forall_ne : Disjoint s t ↔ ∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ t → a ≠ b := by
simp only [Ne, disjoint_left, @imp_not_comm _ (_ = _), forall_eq']
#align set.disjoint_iff_forall_ne Set.disjoint_iff_forall_ne
alias ⟨_root_.Disjoint.ne_of_mem, _⟩ := disjoint_iff_forall_ne
#align disjoint.ne_of_mem Disjoint.ne_of_mem
lemma disjoint_of_subset_left (h : s ⊆ u) (d : Disjoint u t) : Disjoint s t := d.mono_left h
#align set.disjoint_of_subset_left Set.disjoint_of_subset_left
lemma disjoint_of_subset_right (h : t ⊆ u) (d : Disjoint s u) : Disjoint s t := d.mono_right h
#align set.disjoint_of_subset_right Set.disjoint_of_subset_right
lemma disjoint_of_subset (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) (h : Disjoint s₂ t₂) : Disjoint s₁ t₁ :=
h.mono hs ht
#align set.disjoint_of_subset Set.disjoint_of_subset
@[simp]
lemma disjoint_union_left : Disjoint (s ∪ t) u ↔ Disjoint s u ∧ Disjoint t u := disjoint_sup_left
#align set.disjoint_union_left Set.disjoint_union_left
@[simp]
lemma disjoint_union_right : Disjoint s (t ∪ u) ↔ Disjoint s t ∧ Disjoint s u := disjoint_sup_right
#align set.disjoint_union_right Set.disjoint_union_right
@[simp] lemma disjoint_empty (s : Set α) : Disjoint s ∅ := disjoint_bot_right
#align set.disjoint_empty Set.disjoint_empty
@[simp] lemma empty_disjoint (s : Set α) : Disjoint ∅ s := disjoint_bot_left
#align set.empty_disjoint Set.empty_disjoint
@[simp] lemma univ_disjoint : Disjoint univ s ↔ s = ∅ := top_disjoint
#align set.univ_disjoint Set.univ_disjoint
@[simp] lemma disjoint_univ : Disjoint s univ ↔ s = ∅ := disjoint_top
#align set.disjoint_univ Set.disjoint_univ
lemma disjoint_sdiff_left : Disjoint (t \ s) s := disjoint_sdiff_self_left
#align set.disjoint_sdiff_left Set.disjoint_sdiff_left
lemma disjoint_sdiff_right : Disjoint s (t \ s) := disjoint_sdiff_self_right
#align set.disjoint_sdiff_right Set.disjoint_sdiff_right
-- TODO: prove this in terms of a lattice lemma
theorem disjoint_sdiff_inter : Disjoint (s \ t) (s ∩ t) :=
disjoint_of_subset_right inter_subset_right disjoint_sdiff_left
#align set.disjoint_sdiff_inter Set.disjoint_sdiff_inter
theorem diff_union_diff_cancel (hts : t ⊆ s) (hut : u ⊆ t) : s \ t ∪ t \ u = s \ u :=
sdiff_sup_sdiff_cancel hts hut
#align set.diff_union_diff_cancel Set.diff_union_diff_cancel
theorem diff_diff_eq_sdiff_union (h : u ⊆ s) : s \ (t \ u) = s \ t ∪ u := sdiff_sdiff_eq_sdiff_sup h
#align set.diff_diff_eq_sdiff_union Set.diff_diff_eq_sdiff_union
@[simp default+1]
lemma disjoint_singleton_left : Disjoint {a} s ↔ a ∉ s := by simp [Set.disjoint_iff, subset_def]
#align set.disjoint_singleton_left Set.disjoint_singleton_left
@[simp]
lemma disjoint_singleton_right : Disjoint s {a} ↔ a ∉ s :=
disjoint_comm.trans disjoint_singleton_left
#align set.disjoint_singleton_right Set.disjoint_singleton_right
lemma disjoint_singleton : Disjoint ({a} : Set α) {b} ↔ a ≠ b := by
simp
#align set.disjoint_singleton Set.disjoint_singleton
lemma subset_diff : s ⊆ t \ u ↔ s ⊆ t ∧ Disjoint s u := le_iff_subset.symm.trans le_sdiff
#align set.subset_diff Set.subset_diff
lemma ssubset_iff_sdiff_singleton : s ⊂ t ↔ ∃ a ∈ t, s ⊆ t \ {a} := by
simp [ssubset_iff_insert, subset_diff, insert_subset_iff]; aesop
theorem inter_diff_distrib_left (s t u : Set α) : s ∩ (t \ u) = (s ∩ t) \ (s ∩ u) :=
inf_sdiff_distrib_left _ _ _
#align set.inter_diff_distrib_left Set.inter_diff_distrib_left
theorem inter_diff_distrib_right (s t u : Set α) : s \ t ∩ u = (s ∩ u) \ (t ∩ u) :=
inf_sdiff_distrib_right _ _ _
#align set.inter_diff_distrib_right Set.inter_diff_distrib_right
theorem compl_def (s : Set α) : sᶜ = { x | x ∉ s } :=
rfl
#align set.compl_def Set.compl_def
theorem mem_compl {s : Set α} {x : α} (h : x ∉ s) : x ∈ sᶜ :=
h
#align set.mem_compl Set.mem_compl
theorem compl_setOf {α} (p : α → Prop) : { a | p a }ᶜ = { a | ¬p a } :=
rfl
#align set.compl_set_of Set.compl_setOf
theorem not_mem_of_mem_compl {s : Set α} {x : α} (h : x ∈ sᶜ) : x ∉ s :=
h
#align set.not_mem_of_mem_compl Set.not_mem_of_mem_compl
theorem not_mem_compl_iff {x : α} : x ∉ sᶜ ↔ x ∈ s :=
not_not
#align set.not_mem_compl_iff Set.not_mem_compl_iff
@[simp]
theorem inter_compl_self (s : Set α) : s ∩ sᶜ = ∅ :=
inf_compl_eq_bot
#align set.inter_compl_self Set.inter_compl_self
@[simp]
theorem compl_inter_self (s : Set α) : sᶜ ∩ s = ∅ :=
compl_inf_eq_bot
#align set.compl_inter_self Set.compl_inter_self
@[simp]
theorem compl_empty : (∅ : Set α)ᶜ = univ :=
compl_bot
#align set.compl_empty Set.compl_empty
@[simp]
theorem compl_union (s t : Set α) : (s ∪ t)ᶜ = sᶜ ∩ tᶜ :=
compl_sup
#align set.compl_union Set.compl_union
theorem compl_inter (s t : Set α) : (s ∩ t)ᶜ = sᶜ ∪ tᶜ :=
compl_inf
#align set.compl_inter Set.compl_inter
@[simp]
theorem compl_univ : (univ : Set α)ᶜ = ∅ :=
compl_top
#align set.compl_univ Set.compl_univ
@[simp]
theorem compl_empty_iff {s : Set α} : sᶜ = ∅ ↔ s = univ :=
compl_eq_bot
#align set.compl_empty_iff Set.compl_empty_iff
@[simp]
theorem compl_univ_iff {s : Set α} : sᶜ = univ ↔ s = ∅ :=
compl_eq_top
#align set.compl_univ_iff Set.compl_univ_iff
theorem compl_ne_univ : sᶜ ≠ univ ↔ s.Nonempty :=
compl_univ_iff.not.trans nonempty_iff_ne_empty.symm
#align set.compl_ne_univ Set.compl_ne_univ
theorem nonempty_compl : sᶜ.Nonempty ↔ s ≠ univ :=
(ne_univ_iff_exists_not_mem s).symm
#align set.nonempty_compl Set.nonempty_compl
@[simp] lemma nonempty_compl_of_nontrivial [Nontrivial α] (x : α) : Set.Nonempty {x}ᶜ := by
obtain ⟨y, hy⟩ := exists_ne x
exact ⟨y, by simp [hy]⟩
theorem mem_compl_singleton_iff {a x : α} : x ∈ ({a} : Set α)ᶜ ↔ x ≠ a :=
Iff.rfl
#align set.mem_compl_singleton_iff Set.mem_compl_singleton_iff
theorem compl_singleton_eq (a : α) : ({a} : Set α)ᶜ = { x | x ≠ a } :=
rfl
#align set.compl_singleton_eq Set.compl_singleton_eq
@[simp]
theorem compl_ne_eq_singleton (a : α) : ({ x | x ≠ a } : Set α)ᶜ = {a} :=
compl_compl _
#align set.compl_ne_eq_singleton Set.compl_ne_eq_singleton
theorem union_eq_compl_compl_inter_compl (s t : Set α) : s ∪ t = (sᶜ ∩ tᶜ)ᶜ :=
ext fun _ => or_iff_not_and_not
#align set.union_eq_compl_compl_inter_compl Set.union_eq_compl_compl_inter_compl
theorem inter_eq_compl_compl_union_compl (s t : Set α) : s ∩ t = (sᶜ ∪ tᶜ)ᶜ :=
ext fun _ => and_iff_not_or_not
#align set.inter_eq_compl_compl_union_compl Set.inter_eq_compl_compl_union_compl
@[simp]
theorem union_compl_self (s : Set α) : s ∪ sᶜ = univ :=
eq_univ_iff_forall.2 fun _ => em _
#align set.union_compl_self Set.union_compl_self
@[simp]
theorem compl_union_self (s : Set α) : sᶜ ∪ s = univ := by rw [union_comm, union_compl_self]
#align set.compl_union_self Set.compl_union_self
theorem compl_subset_comm : sᶜ ⊆ t ↔ tᶜ ⊆ s :=
@compl_le_iff_compl_le _ s _ _
#align set.compl_subset_comm Set.compl_subset_comm
theorem subset_compl_comm : s ⊆ tᶜ ↔ t ⊆ sᶜ :=
@le_compl_iff_le_compl _ _ _ t
#align set.subset_compl_comm Set.subset_compl_comm
@[simp]
theorem compl_subset_compl : sᶜ ⊆ tᶜ ↔ t ⊆ s :=
@compl_le_compl_iff_le (Set α) _ _ _
#align set.compl_subset_compl Set.compl_subset_compl
@[gcongr] theorem compl_subset_compl_of_subset (h : t ⊆ s) : sᶜ ⊆ tᶜ := compl_subset_compl.2 h
theorem subset_compl_iff_disjoint_left : s ⊆ tᶜ ↔ Disjoint t s :=
@le_compl_iff_disjoint_left (Set α) _ _ _
#align set.subset_compl_iff_disjoint_left Set.subset_compl_iff_disjoint_left
theorem subset_compl_iff_disjoint_right : s ⊆ tᶜ ↔ Disjoint s t :=
@le_compl_iff_disjoint_right (Set α) _ _ _
#align set.subset_compl_iff_disjoint_right Set.subset_compl_iff_disjoint_right
theorem disjoint_compl_left_iff_subset : Disjoint sᶜ t ↔ t ⊆ s :=
disjoint_compl_left_iff
#align set.disjoint_compl_left_iff_subset Set.disjoint_compl_left_iff_subset
theorem disjoint_compl_right_iff_subset : Disjoint s tᶜ ↔ s ⊆ t :=
disjoint_compl_right_iff
#align set.disjoint_compl_right_iff_subset Set.disjoint_compl_right_iff_subset
alias ⟨_, _root_.Disjoint.subset_compl_right⟩ := subset_compl_iff_disjoint_right
#align disjoint.subset_compl_right Disjoint.subset_compl_right
alias ⟨_, _root_.Disjoint.subset_compl_left⟩ := subset_compl_iff_disjoint_left
#align disjoint.subset_compl_left Disjoint.subset_compl_left
alias ⟨_, _root_.HasSubset.Subset.disjoint_compl_left⟩ := disjoint_compl_left_iff_subset
#align has_subset.subset.disjoint_compl_left HasSubset.Subset.disjoint_compl_left
alias ⟨_, _root_.HasSubset.Subset.disjoint_compl_right⟩ := disjoint_compl_right_iff_subset
#align has_subset.subset.disjoint_compl_right HasSubset.Subset.disjoint_compl_right
theorem subset_union_compl_iff_inter_subset {s t u : Set α} : s ⊆ t ∪ uᶜ ↔ s ∩ u ⊆ t :=
(@isCompl_compl _ u _).le_sup_right_iff_inf_left_le
#align set.subset_union_compl_iff_inter_subset Set.subset_union_compl_iff_inter_subset
theorem compl_subset_iff_union {s t : Set α} : sᶜ ⊆ t ↔ s ∪ t = univ :=
Iff.symm <| eq_univ_iff_forall.trans <| forall_congr' fun _ => or_iff_not_imp_left
#align set.compl_subset_iff_union Set.compl_subset_iff_union
@[simp]
theorem subset_compl_singleton_iff {a : α} {s : Set α} : s ⊆ {a}ᶜ ↔ a ∉ s :=
subset_compl_comm.trans singleton_subset_iff
#align set.subset_compl_singleton_iff Set.subset_compl_singleton_iff
theorem inter_subset (a b c : Set α) : a ∩ b ⊆ c ↔ a ⊆ bᶜ ∪ c :=
forall_congr' fun _ => and_imp.trans <| imp_congr_right fun _ => imp_iff_not_or
#align set.inter_subset Set.inter_subset
theorem inter_compl_nonempty_iff {s t : Set α} : (s ∩ tᶜ).Nonempty ↔ ¬s ⊆ t :=
(not_subset.trans <| exists_congr fun x => by simp [mem_compl]).symm
#align set.inter_compl_nonempty_iff Set.inter_compl_nonempty_iff
theorem not_mem_diff_of_mem {s t : Set α} {x : α} (hx : x ∈ t) : x ∉ s \ t := fun h => h.2 hx
#align set.not_mem_diff_of_mem Set.not_mem_diff_of_mem
theorem mem_of_mem_diff {s t : Set α} {x : α} (h : x ∈ s \ t) : x ∈ s :=
h.left
#align set.mem_of_mem_diff Set.mem_of_mem_diff
theorem not_mem_of_mem_diff {s t : Set α} {x : α} (h : x ∈ s \ t) : x ∉ t :=
h.right
#align set.not_mem_of_mem_diff Set.not_mem_of_mem_diff
theorem diff_eq_compl_inter {s t : Set α} : s \ t = tᶜ ∩ s := by rw [diff_eq, inter_comm]
#align set.diff_eq_compl_inter Set.diff_eq_compl_inter
theorem nonempty_diff {s t : Set α} : (s \ t).Nonempty ↔ ¬s ⊆ t :=
inter_compl_nonempty_iff
#align set.nonempty_diff Set.nonempty_diff
theorem diff_subset {s t : Set α} : s \ t ⊆ s := show s \ t ≤ s from sdiff_le
#align set.diff_subset Set.diff_subset
theorem diff_subset_compl (s t : Set α) : s \ t ⊆ tᶜ :=
diff_eq_compl_inter ▸ inter_subset_left
theorem union_diff_cancel' {s t u : Set α} (h₁ : s ⊆ t) (h₂ : t ⊆ u) : t ∪ u \ s = u :=
sup_sdiff_cancel' h₁ h₂
#align set.union_diff_cancel' Set.union_diff_cancel'
theorem union_diff_cancel {s t : Set α} (h : s ⊆ t) : s ∪ t \ s = t :=
sup_sdiff_cancel_right h
#align set.union_diff_cancel Set.union_diff_cancel
theorem union_diff_cancel_left {s t : Set α} (h : s ∩ t ⊆ ∅) : (s ∪ t) \ s = t :=
Disjoint.sup_sdiff_cancel_left <| disjoint_iff_inf_le.2 h
#align set.union_diff_cancel_left Set.union_diff_cancel_left
theorem union_diff_cancel_right {s t : Set α} (h : s ∩ t ⊆ ∅) : (s ∪ t) \ t = s :=
Disjoint.sup_sdiff_cancel_right <| disjoint_iff_inf_le.2 h
#align set.union_diff_cancel_right Set.union_diff_cancel_right
@[simp]
theorem union_diff_left {s t : Set α} : (s ∪ t) \ s = t \ s :=
sup_sdiff_left_self
#align set.union_diff_left Set.union_diff_left
@[simp]
theorem union_diff_right {s t : Set α} : (s ∪ t) \ t = s \ t :=
sup_sdiff_right_self
#align set.union_diff_right Set.union_diff_right
theorem union_diff_distrib {s t u : Set α} : (s ∪ t) \ u = s \ u ∪ t \ u :=
sup_sdiff
#align set.union_diff_distrib Set.union_diff_distrib
theorem inter_diff_assoc (a b c : Set α) : (a ∩ b) \ c = a ∩ (b \ c) :=
inf_sdiff_assoc
#align set.inter_diff_assoc Set.inter_diff_assoc
@[simp]
theorem inter_diff_self (a b : Set α) : a ∩ (b \ a) = ∅ :=
inf_sdiff_self_right
#align set.inter_diff_self Set.inter_diff_self
@[simp]
theorem inter_union_diff (s t : Set α) : s ∩ t ∪ s \ t = s :=
sup_inf_sdiff s t
#align set.inter_union_diff Set.inter_union_diff
@[simp]
theorem diff_union_inter (s t : Set α) : s \ t ∪ s ∩ t = s := by
rw [union_comm]
exact sup_inf_sdiff _ _
#align set.diff_union_inter Set.diff_union_inter
@[simp]
theorem inter_union_compl (s t : Set α) : s ∩ t ∪ s ∩ tᶜ = s :=
inter_union_diff _ _
#align set.inter_union_compl Set.inter_union_compl
@[gcongr]
theorem diff_subset_diff {s₁ s₂ t₁ t₂ : Set α} : s₁ ⊆ s₂ → t₂ ⊆ t₁ → s₁ \ t₁ ⊆ s₂ \ t₂ :=
show s₁ ≤ s₂ → t₂ ≤ t₁ → s₁ \ t₁ ≤ s₂ \ t₂ from sdiff_le_sdiff
#align set.diff_subset_diff Set.diff_subset_diff
@[gcongr]
theorem diff_subset_diff_left {s₁ s₂ t : Set α} (h : s₁ ⊆ s₂) : s₁ \ t ⊆ s₂ \ t :=
sdiff_le_sdiff_right ‹s₁ ≤ s₂›
#align set.diff_subset_diff_left Set.diff_subset_diff_left
@[gcongr]
theorem diff_subset_diff_right {s t u : Set α} (h : t ⊆ u) : s \ u ⊆ s \ t :=
sdiff_le_sdiff_left ‹t ≤ u›
#align set.diff_subset_diff_right Set.diff_subset_diff_right
theorem compl_eq_univ_diff (s : Set α) : sᶜ = univ \ s :=
top_sdiff.symm
#align set.compl_eq_univ_diff Set.compl_eq_univ_diff
@[simp]
theorem empty_diff (s : Set α) : (∅ \ s : Set α) = ∅ :=
bot_sdiff
#align set.empty_diff Set.empty_diff
theorem diff_eq_empty {s t : Set α} : s \ t = ∅ ↔ s ⊆ t :=
sdiff_eq_bot_iff
#align set.diff_eq_empty Set.diff_eq_empty
@[simp]
theorem diff_empty {s : Set α} : s \ ∅ = s :=
sdiff_bot
#align set.diff_empty Set.diff_empty
@[simp]
theorem diff_univ (s : Set α) : s \ univ = ∅ :=
diff_eq_empty.2 (subset_univ s)
#align set.diff_univ Set.diff_univ
theorem diff_diff {u : Set α} : (s \ t) \ u = s \ (t ∪ u) :=
sdiff_sdiff_left
#align set.diff_diff Set.diff_diff
-- the following statement contains parentheses to help the reader
theorem diff_diff_comm {s t u : Set α} : (s \ t) \ u = (s \ u) \ t :=
sdiff_sdiff_comm
#align set.diff_diff_comm Set.diff_diff_comm
theorem diff_subset_iff {s t u : Set α} : s \ t ⊆ u ↔ s ⊆ t ∪ u :=
show s \ t ≤ u ↔ s ≤ t ∪ u from sdiff_le_iff
#align set.diff_subset_iff Set.diff_subset_iff
theorem subset_diff_union (s t : Set α) : s ⊆ s \ t ∪ t :=
show s ≤ s \ t ∪ t from le_sdiff_sup
#align set.subset_diff_union Set.subset_diff_union
theorem diff_union_of_subset {s t : Set α} (h : t ⊆ s) : s \ t ∪ t = s :=
Subset.antisymm (union_subset diff_subset h) (subset_diff_union _ _)
#align set.diff_union_of_subset Set.diff_union_of_subset
@[simp]
theorem diff_singleton_subset_iff {x : α} {s t : Set α} : s \ {x} ⊆ t ↔ s ⊆ insert x t := by
rw [← union_singleton, union_comm]
apply diff_subset_iff
#align set.diff_singleton_subset_iff Set.diff_singleton_subset_iff
theorem subset_diff_singleton {x : α} {s t : Set α} (h : s ⊆ t) (hx : x ∉ s) : s ⊆ t \ {x} :=
subset_inter h <| subset_compl_comm.1 <| singleton_subset_iff.2 hx
#align set.subset_diff_singleton Set.subset_diff_singleton
theorem subset_insert_diff_singleton (x : α) (s : Set α) : s ⊆ insert x (s \ {x}) := by
rw [← diff_singleton_subset_iff]
#align set.subset_insert_diff_singleton Set.subset_insert_diff_singleton
theorem diff_subset_comm {s t u : Set α} : s \ t ⊆ u ↔ s \ u ⊆ t :=
show s \ t ≤ u ↔ s \ u ≤ t from sdiff_le_comm
#align set.diff_subset_comm Set.diff_subset_comm
theorem diff_inter {s t u : Set α} : s \ (t ∩ u) = s \ t ∪ s \ u :=
sdiff_inf
#align set.diff_inter Set.diff_inter
theorem diff_inter_diff {s t u : Set α} : s \ t ∩ (s \ u) = s \ (t ∪ u) :=
sdiff_sup.symm
#align set.diff_inter_diff Set.diff_inter_diff
theorem diff_compl : s \ tᶜ = s ∩ t :=
sdiff_compl
#align set.diff_compl Set.diff_compl
theorem diff_diff_right {s t u : Set α} : s \ (t \ u) = s \ t ∪ s ∩ u :=
sdiff_sdiff_right'
#align set.diff_diff_right Set.diff_diff_right
@[simp]
theorem insert_diff_of_mem (s) (h : a ∈ t) : insert a s \ t = s \ t := by
ext
constructor <;> simp (config := { contextual := true }) [or_imp, h]
#align set.insert_diff_of_mem Set.insert_diff_of_mem
theorem insert_diff_of_not_mem (s) (h : a ∉ t) : insert a s \ t = insert a (s \ t) := by
classical
ext x
by_cases h' : x ∈ t
· have : x ≠ a := by
intro H
rw [H] at h'
exact h h'
simp [h, h', this]
· simp [h, h']
#align set.insert_diff_of_not_mem Set.insert_diff_of_not_mem
theorem insert_diff_self_of_not_mem {a : α} {s : Set α} (h : a ∉ s) : insert a s \ {a} = s := by
ext x
simp [and_iff_left_of_imp fun hx : x ∈ s => show x ≠ a from fun hxa => h <| hxa ▸ hx]
#align set.insert_diff_self_of_not_mem Set.insert_diff_self_of_not_mem
@[simp]
theorem insert_diff_eq_singleton {a : α} {s : Set α} (h : a ∉ s) : insert a s \ s = {a} := by
ext
rw [Set.mem_diff, Set.mem_insert_iff, Set.mem_singleton_iff, or_and_right, and_not_self_iff,
or_false_iff, and_iff_left_iff_imp]
rintro rfl
exact h
#align set.insert_diff_eq_singleton Set.insert_diff_eq_singleton
theorem inter_insert_of_mem (h : a ∈ s) : s ∩ insert a t = insert a (s ∩ t) := by
rw [insert_inter_distrib, insert_eq_of_mem h]
#align set.inter_insert_of_mem Set.inter_insert_of_mem
theorem insert_inter_of_mem (h : a ∈ t) : insert a s ∩ t = insert a (s ∩ t) := by
rw [insert_inter_distrib, insert_eq_of_mem h]
#align set.insert_inter_of_mem Set.insert_inter_of_mem
theorem inter_insert_of_not_mem (h : a ∉ s) : s ∩ insert a t = s ∩ t :=
ext fun _ => and_congr_right fun hx => or_iff_right <| ne_of_mem_of_not_mem hx h
#align set.inter_insert_of_not_mem Set.inter_insert_of_not_mem
theorem insert_inter_of_not_mem (h : a ∉ t) : insert a s ∩ t = s ∩ t :=
ext fun _ => and_congr_left fun hx => or_iff_right <| ne_of_mem_of_not_mem hx h
#align set.insert_inter_of_not_mem Set.insert_inter_of_not_mem
@[simp]
theorem union_diff_self {s t : Set α} : s ∪ t \ s = s ∪ t :=
sup_sdiff_self _ _
#align set.union_diff_self Set.union_diff_self
@[simp]
theorem diff_union_self {s t : Set α} : s \ t ∪ t = s ∪ t :=
sdiff_sup_self _ _
#align set.diff_union_self Set.diff_union_self
@[simp]
theorem diff_inter_self {a b : Set α} : b \ a ∩ a = ∅ :=
inf_sdiff_self_left
#align set.diff_inter_self Set.diff_inter_self
@[simp]
theorem diff_inter_self_eq_diff {s t : Set α} : s \ (t ∩ s) = s \ t :=
sdiff_inf_self_right _ _
#align set.diff_inter_self_eq_diff Set.diff_inter_self_eq_diff
@[simp]
theorem diff_self_inter {s t : Set α} : s \ (s ∩ t) = s \ t :=
sdiff_inf_self_left _ _
#align set.diff_self_inter Set.diff_self_inter
@[simp]
theorem diff_singleton_eq_self {a : α} {s : Set α} (h : a ∉ s) : s \ {a} = s :=
sdiff_eq_self_iff_disjoint.2 <| by simp [h]
#align set.diff_singleton_eq_self Set.diff_singleton_eq_self
@[simp]
theorem diff_singleton_sSubset {s : Set α} {a : α} : s \ {a} ⊂ s ↔ a ∈ s :=
sdiff_le.lt_iff_ne.trans <| sdiff_eq_left.not.trans <| by simp
#align set.diff_singleton_ssubset Set.diff_singleton_sSubset
@[simp]
theorem insert_diff_singleton {a : α} {s : Set α} : insert a (s \ {a}) = insert a s := by
simp [insert_eq, union_diff_self, -union_singleton, -singleton_union]
#align set.insert_diff_singleton Set.insert_diff_singleton
theorem insert_diff_singleton_comm (hab : a ≠ b) (s : Set α) :
insert a (s \ {b}) = insert a s \ {b} := by
simp_rw [← union_singleton, union_diff_distrib,
diff_singleton_eq_self (mem_singleton_iff.not.2 hab.symm)]
#align set.insert_diff_singleton_comm Set.insert_diff_singleton_comm
--Porting note (#10618): removed `simp` attribute because `simp` can prove it
theorem diff_self {s : Set α} : s \ s = ∅ :=
sdiff_self
#align set.diff_self Set.diff_self
theorem diff_diff_right_self (s t : Set α) : s \ (s \ t) = s ∩ t :=
sdiff_sdiff_right_self
#align set.diff_diff_right_self Set.diff_diff_right_self
theorem diff_diff_cancel_left {s t : Set α} (h : s ⊆ t) : t \ (t \ s) = s :=
sdiff_sdiff_eq_self h
#align set.diff_diff_cancel_left Set.diff_diff_cancel_left
theorem mem_diff_singleton {x y : α} {s : Set α} : x ∈ s \ {y} ↔ x ∈ s ∧ x ≠ y :=
Iff.rfl
#align set.mem_diff_singleton Set.mem_diff_singleton
theorem mem_diff_singleton_empty {t : Set (Set α)} : s ∈ t \ {∅} ↔ s ∈ t ∧ s.Nonempty :=
mem_diff_singleton.trans <| and_congr_right' nonempty_iff_ne_empty.symm
#align set.mem_diff_singleton_empty Set.mem_diff_singleton_empty
theorem subset_insert_iff {s t : Set α} {x : α} :
s ⊆ insert x t ↔ s ⊆ t ∨ (x ∈ s ∧ s \ {x} ⊆ t) := by
rw [← diff_singleton_subset_iff]
by_cases hx : x ∈ s
· rw [and_iff_right hx, or_iff_right_of_imp diff_subset.trans]
rw [diff_singleton_eq_self hx, or_iff_left_of_imp And.right]
theorem union_eq_diff_union_diff_union_inter (s t : Set α) : s ∪ t = s \ t ∪ t \ s ∪ s ∩ t :=
sup_eq_sdiff_sup_sdiff_sup_inf
#align set.union_eq_diff_union_diff_union_inter Set.union_eq_diff_union_diff_union_inter
--Porting note (#10618): removed `simp` attribute because `simp` can prove it
theorem pair_eq_singleton (a : α) : ({a, a} : Set α) = {a} :=
union_self _
#align set.pair_eq_singleton Set.pair_eq_singleton
theorem pair_comm (a b : α) : ({a, b} : Set α) = {b, a} :=
union_comm _ _
#align set.pair_comm Set.pair_comm
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
#align set.pair_eq_pair_iff Set.pair_eq_pair_iff
theorem pair_diff_left (hne : a ≠ b) : ({a, b} : Set α) \ {a} = {b} := by
rw [insert_diff_of_mem _ (mem_singleton a), diff_singleton_eq_self (by simpa)]
theorem pair_diff_right (hne : a ≠ b) : ({a, b} : Set α) \ {b} = {a} := by
rw [pair_comm, pair_diff_left hne.symm]
theorem pair_subset_iff : {a, b} ⊆ s ↔ a ∈ s ∧ b ∈ s := by
rw [insert_subset_iff, singleton_subset_iff]
theorem pair_subset (ha : a ∈ s) (hb : b ∈ s) : {a, b} ⊆ s :=
pair_subset_iff.2 ⟨ha,hb⟩
theorem subset_pair_iff : s ⊆ {a, b} ↔ ∀ x ∈ s, x = a ∨ x = b := by
simp [subset_def]
theorem subset_pair_iff_eq {x y : α} : s ⊆ {x, y} ↔ s = ∅ ∨ s = {x} ∨ s = {y} ∨ s = {x, y} := by
refine ⟨?_, by rintro (rfl | rfl | rfl | rfl) <;> simp [pair_subset_iff]⟩
rw [subset_insert_iff, subset_singleton_iff_eq, subset_singleton_iff_eq,
← subset_empty_iff (s := s \ {x}), diff_subset_iff, union_empty, subset_singleton_iff_eq]
have h : x ∈ s → {y} = s \ {x} → s = {x,y} := fun h₁ h₂ ↦ by simp [h₁, h₂]
tauto
theorem Nonempty.subset_pair_iff_eq (hs : s.Nonempty) :
s ⊆ {a, b} ↔ s = {a} ∨ s = {b} ∨ s = {a, b} := by
rw [Set.subset_pair_iff_eq, or_iff_right]; exact hs.ne_empty
section
open scoped symmDiff
theorem mem_symmDiff : a ∈ s ∆ t ↔ a ∈ s ∧ a ∉ t ∨ a ∈ t ∧ a ∉ s :=
Iff.rfl
#align set.mem_symm_diff Set.mem_symmDiff
protected theorem symmDiff_def (s t : Set α) : s ∆ t = s \ t ∪ t \ s :=
rfl
#align set.symm_diff_def Set.symmDiff_def
theorem symmDiff_subset_union : s ∆ t ⊆ s ∪ t :=
@symmDiff_le_sup (Set α) _ _ _
#align set.symm_diff_subset_union Set.symmDiff_subset_union
@[simp]
theorem symmDiff_eq_empty : s ∆ t = ∅ ↔ s = t :=
symmDiff_eq_bot
#align set.symm_diff_eq_empty Set.symmDiff_eq_empty
@[simp]
theorem symmDiff_nonempty : (s ∆ t).Nonempty ↔ s ≠ t :=
nonempty_iff_ne_empty.trans symmDiff_eq_empty.not
#align set.symm_diff_nonempty Set.symmDiff_nonempty
theorem inter_symmDiff_distrib_left (s t u : Set α) : s ∩ t ∆ u = (s ∩ t) ∆ (s ∩ u) :=
inf_symmDiff_distrib_left _ _ _
#align set.inter_symm_diff_distrib_left Set.inter_symmDiff_distrib_left
theorem inter_symmDiff_distrib_right (s t u : Set α) : s ∆ t ∩ u = (s ∩ u) ∆ (t ∩ u) :=
inf_symmDiff_distrib_right _ _ _
#align set.inter_symm_diff_distrib_right Set.inter_symmDiff_distrib_right
theorem subset_symmDiff_union_symmDiff_left (h : Disjoint s t) : u ⊆ s ∆ u ∪ t ∆ u :=
h.le_symmDiff_sup_symmDiff_left
#align set.subset_symm_diff_union_symm_diff_left Set.subset_symmDiff_union_symmDiff_left
theorem subset_symmDiff_union_symmDiff_right (h : Disjoint t u) : s ⊆ s ∆ t ∪ s ∆ u :=
h.le_symmDiff_sup_symmDiff_right
#align set.subset_symm_diff_union_symm_diff_right Set.subset_symmDiff_union_symmDiff_right
end
#align set.powerset Set.powerset
theorem mem_powerset {x s : Set α} (h : x ⊆ s) : x ∈ 𝒫 s := @h
#align set.mem_powerset Set.mem_powerset
theorem subset_of_mem_powerset {x s : Set α} (h : x ∈ 𝒫 s) : x ⊆ s := @h
#align set.subset_of_mem_powerset Set.subset_of_mem_powerset
@[simp]
theorem mem_powerset_iff (x s : Set α) : x ∈ 𝒫 s ↔ x ⊆ s :=
Iff.rfl
#align set.mem_powerset_iff Set.mem_powerset_iff
theorem powerset_inter (s t : Set α) : 𝒫(s ∩ t) = 𝒫 s ∩ 𝒫 t :=
ext fun _ => subset_inter_iff
#align set.powerset_inter Set.powerset_inter
@[simp]
theorem powerset_mono : 𝒫 s ⊆ 𝒫 t ↔ s ⊆ t :=
⟨fun h => @h _ (fun _ h => h), fun h _ hu _ ha => h (hu ha)⟩
#align set.powerset_mono Set.powerset_mono
theorem monotone_powerset : Monotone (powerset : Set α → Set (Set α)) := fun _ _ => powerset_mono.2
#align set.monotone_powerset Set.monotone_powerset
@[simp]
theorem powerset_nonempty : (𝒫 s).Nonempty :=
⟨∅, fun _ h => empty_subset s h⟩
#align set.powerset_nonempty Set.powerset_nonempty
@[simp]
theorem powerset_empty : 𝒫(∅ : Set α) = {∅} :=
ext fun _ => subset_empty_iff
#align set.powerset_empty Set.powerset_empty
@[simp]
theorem powerset_univ : 𝒫(univ : Set α) = univ :=
eq_univ_of_forall subset_univ
#align set.powerset_univ Set.powerset_univ
theorem powerset_singleton (x : α) : 𝒫({x} : Set α) = {∅, {x}} := by
ext y
rw [mem_powerset_iff, subset_singleton_iff_eq, mem_insert_iff, mem_singleton_iff]
#align set.powerset_singleton Set.powerset_singleton
theorem mem_dite (p : Prop) [Decidable p] (s : p → Set α) (t : ¬ p → Set α) (x : α) :
(x ∈ if h : p then s h else t h) ↔ (∀ h : p, x ∈ s h) ∧ ∀ h : ¬p, x ∈ t h := by
split_ifs with hp
· exact ⟨fun hx => ⟨fun _ => hx, fun hnp => (hnp hp).elim⟩, fun hx => hx.1 hp⟩
· exact ⟨fun hx => ⟨fun h => (hp h).elim, fun _ => hx⟩, fun hx => hx.2 hp⟩
theorem mem_dite_univ_right (p : Prop) [Decidable p] (t : p → Set α) (x : α) :
(x ∈ if h : p then t h else univ) ↔ ∀ h : p, x ∈ t h := by
split_ifs <;> simp_all
#align set.mem_dite_univ_right Set.mem_dite_univ_right
@[simp]
theorem mem_ite_univ_right (p : Prop) [Decidable p] (t : Set α) (x : α) :
x ∈ ite p t Set.univ ↔ p → x ∈ t :=
mem_dite_univ_right p (fun _ => t) x
#align set.mem_ite_univ_right Set.mem_ite_univ_right
theorem mem_dite_univ_left (p : Prop) [Decidable p] (t : ¬p → Set α) (x : α) :
(x ∈ if h : p then univ else t h) ↔ ∀ h : ¬p, x ∈ t h := by
split_ifs <;> simp_all
#align set.mem_dite_univ_left Set.mem_dite_univ_left
@[simp]
theorem mem_ite_univ_left (p : Prop) [Decidable p] (t : Set α) (x : α) :
x ∈ ite p Set.univ t ↔ ¬p → x ∈ t :=
mem_dite_univ_left p (fun _ => t) x
#align set.mem_ite_univ_left Set.mem_ite_univ_left
theorem mem_dite_empty_right (p : Prop) [Decidable p] (t : p → Set α) (x : α) :
(x ∈ if h : p then t h else ∅) ↔ ∃ h : p, x ∈ t h := by
simp only [mem_dite, mem_empty_iff_false, imp_false, not_not]
exact ⟨fun h => ⟨h.2, h.1 h.2⟩, fun ⟨h₁, h₂⟩ => ⟨fun _ => h₂, h₁⟩⟩
#align set.mem_dite_empty_right Set.mem_dite_empty_right
@[simp]
theorem mem_ite_empty_right (p : Prop) [Decidable p] (t : Set α) (x : α) :
x ∈ ite p t ∅ ↔ p ∧ x ∈ t :=
(mem_dite_empty_right p (fun _ => t) x).trans (by simp)
#align set.mem_ite_empty_right Set.mem_ite_empty_right
theorem mem_dite_empty_left (p : Prop) [Decidable p] (t : ¬p → Set α) (x : α) :
(x ∈ if h : p then ∅ else t h) ↔ ∃ h : ¬p, x ∈ t h := by
simp only [mem_dite, mem_empty_iff_false, imp_false]
exact ⟨fun h => ⟨h.1, h.2 h.1⟩, fun ⟨h₁, h₂⟩ => ⟨fun h => h₁ h, fun _ => h₂⟩⟩
#align set.mem_dite_empty_left Set.mem_dite_empty_left
@[simp]
theorem mem_ite_empty_left (p : Prop) [Decidable p] (t : Set α) (x : α) :
x ∈ ite p ∅ t ↔ ¬p ∧ x ∈ t :=
(mem_dite_empty_left p (fun _ => t) x).trans (by simp)
#align set.mem_ite_empty_left Set.mem_ite_empty_left
protected def ite (t s s' : Set α) : Set α :=
s ∩ t ∪ s' \ t
#align set.ite Set.ite
@[simp]
theorem ite_inter_self (t s s' : Set α) : t.ite s s' ∩ t = s ∩ t := by
rw [Set.ite, union_inter_distrib_right, diff_inter_self, inter_assoc, inter_self, union_empty]
#align set.ite_inter_self Set.ite_inter_self
@[simp]
theorem ite_compl (t s s' : Set α) : tᶜ.ite s s' = t.ite s' s := by
rw [Set.ite, Set.ite, diff_compl, union_comm, diff_eq]
#align set.ite_compl Set.ite_compl
@[simp]
theorem ite_inter_compl_self (t s s' : Set α) : t.ite s s' ∩ tᶜ = s' ∩ tᶜ := by
rw [← ite_compl, ite_inter_self]
#align set.ite_inter_compl_self Set.ite_inter_compl_self
@[simp]
theorem ite_diff_self (t s s' : Set α) : t.ite s s' \ t = s' \ t :=
ite_inter_compl_self t s s'
#align set.ite_diff_self Set.ite_diff_self
@[simp]
theorem ite_same (t s : Set α) : t.ite s s = s :=
inter_union_diff _ _
#align set.ite_same Set.ite_same
@[simp]
theorem ite_left (s t : Set α) : s.ite s t = s ∪ t := by simp [Set.ite]
#align set.ite_left Set.ite_left
@[simp]
theorem ite_right (s t : Set α) : s.ite t s = t ∩ s := by simp [Set.ite]
#align set.ite_right Set.ite_right
@[simp]
theorem ite_empty (s s' : Set α) : Set.ite ∅ s s' = s' := by simp [Set.ite]
#align set.ite_empty Set.ite_empty
@[simp]
theorem ite_univ (s s' : Set α) : Set.ite univ s s' = s := by simp [Set.ite]
#align set.ite_univ Set.ite_univ
@[simp]
theorem ite_empty_left (t s : Set α) : t.ite ∅ s = s \ t := by simp [Set.ite]
#align set.ite_empty_left Set.ite_empty_left
@[simp]
theorem ite_empty_right (t s : Set α) : t.ite s ∅ = s ∩ t := by simp [Set.ite]
#align set.ite_empty_right Set.ite_empty_right
theorem ite_mono (t : Set α) {s₁ s₁' s₂ s₂' : Set α} (h : s₁ ⊆ s₂) (h' : s₁' ⊆ s₂') :
t.ite s₁ s₁' ⊆ t.ite s₂ s₂' :=
union_subset_union (inter_subset_inter_left _ h) (inter_subset_inter_left _ h')
#align set.ite_mono Set.ite_mono
theorem ite_subset_union (t s s' : Set α) : t.ite s s' ⊆ s ∪ s' :=
union_subset_union inter_subset_left diff_subset
#align set.ite_subset_union Set.ite_subset_union
theorem inter_subset_ite (t s s' : Set α) : s ∩ s' ⊆ t.ite s s' :=
ite_same t (s ∩ s') ▸ ite_mono _ inter_subset_left inter_subset_right
#align set.inter_subset_ite Set.inter_subset_ite
| Mathlib/Data/Set/Basic.lean | 2,327 | 2,331 | theorem ite_inter_inter (t s₁ s₂ s₁' s₂' : Set α) :
t.ite (s₁ ∩ s₂) (s₁' ∩ s₂') = t.ite s₁ s₁' ∩ t.ite s₂ s₂' := by |
ext x
simp only [Set.ite, Set.mem_inter_iff, Set.mem_diff, Set.mem_union]
tauto
|
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.Tactic.Positivity
#align_import combinatorics.catalan from "leanprover-community/mathlib"@"26b40791e4a5772a4e53d0e28e4df092119dc7da"
open Finset
open Finset.antidiagonal (fst_le snd_le)
def catalan : ℕ → ℕ
| 0 => 1
| n + 1 =>
∑ i : Fin n.succ,
catalan i * catalan (n - i)
#align catalan catalan
@[simp]
theorem catalan_zero : catalan 0 = 1 := by rw [catalan]
#align catalan_zero catalan_zero
theorem catalan_succ (n : ℕ) : catalan (n + 1) = ∑ i : Fin n.succ, catalan i * catalan (n - i) := by
rw [catalan]
#align catalan_succ catalan_succ
| Mathlib/Combinatorics/Enumerative/Catalan.lean | 72 | 75 | theorem catalan_succ' (n : ℕ) :
catalan (n + 1) = ∑ ij ∈ antidiagonal n, catalan ij.1 * catalan ij.2 := by |
rw [catalan_succ, Nat.sum_antidiagonal_eq_sum_range_succ (fun x y => catalan x * catalan y) n,
sum_range]
|
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.integer from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
variable {R : Type*} [CommSemiring R] {M : Submonoid R} {S : Type*} [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
open Function
namespace IsLocalization
section
variable (R)
-- TODO: define a subalgebra of `IsInteger`s
def IsInteger (a : S) : Prop :=
a ∈ (algebraMap R S).rangeS
#align is_localization.is_integer IsLocalization.IsInteger
end
theorem isInteger_zero : IsInteger R (0 : S) :=
Subsemiring.zero_mem _
#align is_localization.is_integer_zero IsLocalization.isInteger_zero
theorem isInteger_one : IsInteger R (1 : S) :=
Subsemiring.one_mem _
#align is_localization.is_integer_one IsLocalization.isInteger_one
theorem isInteger_add {a b : S} (ha : IsInteger R a) (hb : IsInteger R b) : IsInteger R (a + b) :=
Subsemiring.add_mem _ ha hb
#align is_localization.is_integer_add IsLocalization.isInteger_add
theorem isInteger_mul {a b : S} (ha : IsInteger R a) (hb : IsInteger R b) : IsInteger R (a * b) :=
Subsemiring.mul_mem _ ha hb
#align is_localization.is_integer_mul IsLocalization.isInteger_mul
theorem isInteger_smul {a : R} {b : S} (hb : IsInteger R b) : IsInteger R (a • b) := by
rcases hb with ⟨b', hb⟩
use a * b'
rw [← hb, (algebraMap R S).map_mul, Algebra.smul_def]
#align is_localization.is_integer_smul IsLocalization.isInteger_smul
variable (M)
variable [IsLocalization M S]
theorem exists_integer_multiple' (a : S) : ∃ b : M, IsInteger R (a * algebraMap R S b) :=
let ⟨⟨Num, denom⟩, h⟩ := IsLocalization.surj _ a
⟨denom, Set.mem_range.mpr ⟨Num, h.symm⟩⟩
#align is_localization.exists_integer_multiple' IsLocalization.exists_integer_multiple'
theorem exists_integer_multiple (a : S) : ∃ b : M, IsInteger R ((b : R) • a) := by
simp_rw [Algebra.smul_def, mul_comm _ a]
apply exists_integer_multiple'
#align is_localization.exists_integer_multiple IsLocalization.exists_integer_multiple
theorem exist_integer_multiples {ι : Type*} (s : Finset ι) (f : ι → S) :
∃ b : M, ∀ i ∈ s, IsLocalization.IsInteger R ((b : R) • f i) := by
haveI := Classical.propDecidable
refine ⟨∏ i ∈ s, (sec M (f i)).2, fun i hi => ⟨?_, ?_⟩⟩
· exact (∏ j ∈ s.erase i, (sec M (f j)).2) * (sec M (f i)).1
rw [RingHom.map_mul, sec_spec', ← mul_assoc, ← (algebraMap R S).map_mul, ← Algebra.smul_def]
congr 2
refine _root_.trans ?_ (map_prod (Submonoid.subtype M) _ _).symm
rw [mul_comm,Submonoid.coe_finset_prod,
-- Porting note: explicitly supplied `f`
← Finset.prod_insert (f := fun i => ((sec M (f i)).snd : R)) (s.not_mem_erase i),
Finset.insert_erase hi]
rfl
#align is_localization.exist_integer_multiples IsLocalization.exist_integer_multiples
| Mathlib/RingTheory/Localization/Integer.lean | 107 | 111 | theorem exist_integer_multiples_of_finite {ι : Type*} [Finite ι] (f : ι → S) :
∃ b : M, ∀ i, IsLocalization.IsInteger R ((b : R) • f i) := by |
cases nonempty_fintype ι
obtain ⟨b, hb⟩ := exist_integer_multiples M Finset.univ f
exact ⟨b, fun i => hb i (Finset.mem_univ _)⟩
|
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.monoidal.center from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
open CategoryTheory
open CategoryTheory.MonoidalCategory
universe v v₁ v₂ v₃ u u₁ u₂ u₃
noncomputable section
namespace CategoryTheory
variable {C : Type u₁} [Category.{v₁} C] [MonoidalCategory C]
-- @[nolint has_nonempty_instance] -- Porting note(#5171): This linter does not exist yet.
structure HalfBraiding (X : C) where
β : ∀ U, X ⊗ U ≅ U ⊗ X
monoidal : ∀ U U', (β (U ⊗ U')).hom =
(α_ _ _ _).inv ≫
((β U).hom ▷ U') ≫ (α_ _ _ _).hom ≫ (U ◁ (β U').hom) ≫ (α_ _ _ _).inv := by
aesop_cat
naturality : ∀ {U U'} (f : U ⟶ U'), (X ◁ f) ≫ (β U').hom = (β U).hom ≫ (f ▷ X) := by
aesop_cat
#align category_theory.half_braiding CategoryTheory.HalfBraiding
attribute [reassoc, simp] HalfBraiding.monoidal -- the reassoc lemma is redundant as a simp lemma
attribute [simp, reassoc] HalfBraiding.naturality
variable (C)
-- @[nolint has_nonempty_instance] -- Porting note(#5171): This linter does not exist yet.
def Center :=
Σ X : C, HalfBraiding X
#align category_theory.center CategoryTheory.Center
namespace Center
variable {C}
-- Porting note(#5171): linter not ported yet
@[ext] -- @[nolint has_nonempty_instance]
structure Hom (X Y : Center C) where
f : X.1 ⟶ Y.1
comm : ∀ U, (f ▷ U) ≫ (Y.2.β U).hom = (X.2.β U).hom ≫ (U ◁ f) := by aesop_cat
#align category_theory.center.hom CategoryTheory.Center.Hom
attribute [reassoc (attr := simp)] Hom.comm
instance : Quiver (Center C) where
Hom := Hom
@[ext]
theorem ext {X Y : Center C} (f g : X ⟶ Y) (w : f.f = g.f) : f = g := by
cases f; cases g; congr
#align category_theory.center.ext CategoryTheory.Center.ext
instance : Category (Center C) where
id X := { f := 𝟙 X.1 }
comp f g := { f := f.f ≫ g.f }
@[simp]
theorem id_f (X : Center C) : Hom.f (𝟙 X) = 𝟙 X.1 :=
rfl
#align category_theory.center.id_f CategoryTheory.Center.id_f
@[simp]
theorem comp_f {X Y Z : Center C} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g).f = f.f ≫ g.f :=
rfl
#align category_theory.center.comp_f CategoryTheory.Center.comp_f
@[simps]
def isoMk {X Y : Center C} (f : X ⟶ Y) [IsIso f.f] : X ≅ Y where
hom := f
inv := ⟨inv f.f,
fun U => by simp [← cancel_epi (f.f ▷ U), ← comp_whiskerRight_assoc,
← MonoidalCategory.whiskerLeft_comp] ⟩
#align category_theory.center.iso_mk CategoryTheory.Center.isoMk
instance isIso_of_f_isIso {X Y : Center C} (f : X ⟶ Y) [IsIso f.f] : IsIso f := by
change IsIso (isoMk f).hom
infer_instance
#align category_theory.center.is_iso_of_f_is_iso CategoryTheory.Center.isIso_of_f_isIso
@[simps]
def tensorObj (X Y : Center C) : Center C :=
⟨X.1 ⊗ Y.1,
{ β := fun U =>
α_ _ _ _ ≪≫
(whiskerLeftIso X.1 (Y.2.β U)) ≪≫ (α_ _ _ _).symm ≪≫
(whiskerRightIso (X.2.β U) Y.1) ≪≫ α_ _ _ _
monoidal := fun U U' => by
dsimp only [Iso.trans_hom, whiskerLeftIso_hom, Iso.symm_hom, whiskerRightIso_hom]
simp only [HalfBraiding.monoidal]
-- We'd like to commute `X.1 ◁ U ◁ (HalfBraiding.β Y.2 U').hom`
-- and `((HalfBraiding.β X.2 U).hom ▷ U' ▷ Y.1)` past each other.
-- We do this with the help of the monoidal composition `⊗≫` and the `coherence` tactic.
calc
_ = 𝟙 _ ⊗≫
X.1 ◁ (HalfBraiding.β Y.2 U).hom ▷ U' ⊗≫
(_ ◁ (HalfBraiding.β Y.2 U').hom ≫
(HalfBraiding.β X.2 U).hom ▷ _) ⊗≫
U ◁ (HalfBraiding.β X.2 U').hom ▷ Y.1 ⊗≫ 𝟙 _ := by coherence
_ = _ := by rw [whisker_exchange]; coherence
naturality := fun {U U'} f => by
dsimp only [Iso.trans_hom, whiskerLeftIso_hom, Iso.symm_hom, whiskerRightIso_hom]
calc
_ = 𝟙 _ ⊗≫
(X.1 ◁ (Y.1 ◁ f ≫ (HalfBraiding.β Y.2 U').hom)) ⊗≫
(HalfBraiding.β X.2 U').hom ▷ Y.1 ⊗≫ 𝟙 _ := by coherence
_ = 𝟙 _ ⊗≫
X.1 ◁ (HalfBraiding.β Y.2 U).hom ⊗≫
(X.1 ◁ f ≫ (HalfBraiding.β X.2 U').hom) ▷ Y.1 ⊗≫ 𝟙 _ := by
rw [HalfBraiding.naturality]; coherence
_ = _ := by rw [HalfBraiding.naturality]; coherence }⟩
#align category_theory.center.tensor_obj CategoryTheory.Center.tensorObj
@[reassoc]
| Mathlib/CategoryTheory/Monoidal/Center.lean | 166 | 179 | theorem whiskerLeft_comm (X : Center C) {Y₁ Y₂ : Center C} (f : Y₁ ⟶ Y₂) (U : C) :
(X.1 ◁ f.f) ▷ U ≫ ((tensorObj X Y₂).2.β U).hom =
((tensorObj X Y₁).2.β U).hom ≫ U ◁ X.1 ◁ f.f := by |
dsimp only [tensorObj_fst, tensorObj_snd_β, Iso.trans_hom, whiskerLeftIso_hom,
Iso.symm_hom, whiskerRightIso_hom]
calc
_ = 𝟙 _ ⊗≫
X.fst ◁ (f.f ▷ U ≫ (HalfBraiding.β Y₂.snd U).hom) ⊗≫
(HalfBraiding.β X.snd U).hom ▷ Y₂.fst ⊗≫ 𝟙 _ := by coherence
_ = 𝟙 _ ⊗≫
X.fst ◁ (HalfBraiding.β Y₁.snd U).hom ⊗≫
((X.fst ⊗ U) ◁ f.f ≫ (HalfBraiding.β X.snd U).hom ▷ Y₂.fst) ⊗≫ 𝟙 _ := by
rw [f.comm]; coherence
_ = _ := by rw [whisker_exchange]; coherence
|
import Mathlib.Topology.ContinuousOn
#align_import topology.algebra.order.left_right from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Topology
section TopologicalSpace
variable {α β : Type*} [TopologicalSpace α] [LinearOrder α] [TopologicalSpace β]
theorem nhds_left_sup_nhds_right (a : α) : 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a := by
rw [← nhdsWithin_union, Iic_union_Ici, nhdsWithin_univ]
#align nhds_left_sup_nhds_right nhds_left_sup_nhds_right
theorem nhds_left'_sup_nhds_right (a : α) : 𝓝[<] a ⊔ 𝓝[≥] a = 𝓝 a := by
rw [← nhdsWithin_union, Iio_union_Ici, nhdsWithin_univ]
#align nhds_left'_sup_nhds_right nhds_left'_sup_nhds_right
| Mathlib/Topology/Order/LeftRight.lean | 119 | 120 | theorem nhds_left_sup_nhds_right' (a : α) : 𝓝[≤] a ⊔ 𝓝[>] a = 𝓝 a := by |
rw [← nhdsWithin_union, Iic_union_Ioi, nhdsWithin_univ]
|
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 false
open TopologicalSpace
open CategoryTheory
open CategoryTheory.Limits
universe v u w
noncomputable section
namespace TopCat
variable {J : Type v} [SmallCategory J]
section Pullback
variable {X Y Z : TopCat.{u}}
abbrev pullbackFst (f : X ⟶ Z) (g : Y ⟶ Z) : TopCat.of { p : X × Y // f p.1 = g p.2 } ⟶ X :=
⟨Prod.fst ∘ Subtype.val, by
apply Continuous.comp <;> set_option tactic.skipAssignedInstances false in continuity⟩
#align Top.pullback_fst TopCat.pullbackFst
lemma pullbackFst_apply (f : X ⟶ Z) (g : Y ⟶ Z) (x) : pullbackFst f g x = x.1.1 := rfl
abbrev pullbackSnd (f : X ⟶ Z) (g : Y ⟶ Z) : TopCat.of { p : X × Y // f p.1 = g p.2 } ⟶ Y :=
⟨Prod.snd ∘ Subtype.val, by
apply Continuous.comp <;> set_option tactic.skipAssignedInstances false in continuity⟩
#align Top.pullback_snd TopCat.pullbackSnd
lemma pullbackSnd_apply (f : X ⟶ Z) (g : Y ⟶ Z) (x) : pullbackSnd f g x = x.1.2 := rfl
def pullbackCone (f : X ⟶ Z) (g : Y ⟶ Z) : PullbackCone f g :=
PullbackCone.mk (pullbackFst f g) (pullbackSnd f g)
(by
dsimp [pullbackFst, pullbackSnd, Function.comp_def]
ext ⟨x, h⟩
-- Next 2 lines were
-- `rw [comp_apply, ContinuousMap.coe_mk, comp_apply, ContinuousMap.coe_mk]`
-- `exact h` before leanprover/lean4#2644
rw [comp_apply, comp_apply]
congr!)
#align Top.pullback_cone TopCat.pullbackCone
def pullbackConeIsLimit (f : X ⟶ Z) (g : Y ⟶ Z) : IsLimit (pullbackCone f g) :=
PullbackCone.isLimitAux' _
(by
intro S
constructor; swap
· exact
{ toFun := fun x =>
⟨⟨S.fst x, S.snd x⟩, by simpa using ConcreteCategory.congr_hom S.condition x⟩
continuous_toFun := by
apply Continuous.subtype_mk <| Continuous.prod_mk ?_ ?_
· exact (PullbackCone.fst S)|>.continuous_toFun
· exact (PullbackCone.snd S)|>.continuous_toFun
}
refine ⟨?_, ?_, ?_⟩
· delta pullbackCone
ext a
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [comp_apply, ContinuousMap.coe_mk]
· delta pullbackCone
ext a
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [comp_apply, ContinuousMap.coe_mk]
· intro m h₁ h₂
-- Porting note: used to be ext x
apply ContinuousMap.ext; intro x
apply Subtype.ext
apply Prod.ext
· simpa using ConcreteCategory.congr_hom h₁ x
· simpa using ConcreteCategory.congr_hom h₂ x)
#align Top.pullback_cone_is_limit TopCat.pullbackConeIsLimit
def pullbackIsoProdSubtype (f : X ⟶ Z) (g : Y ⟶ Z) :
pullback f g ≅ TopCat.of { p : X × Y // f p.1 = g p.2 } :=
(limit.isLimit _).conePointUniqueUpToIso (pullbackConeIsLimit f g)
#align Top.pullback_iso_prod_subtype TopCat.pullbackIsoProdSubtype
@[reassoc (attr := simp)]
theorem pullbackIsoProdSubtype_inv_fst (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullbackIsoProdSubtype f g).inv ≫ pullback.fst = pullbackFst f g := by
simp [pullbackCone, pullbackIsoProdSubtype]
#align Top.pullback_iso_prod_subtype_inv_fst TopCat.pullbackIsoProdSubtype_inv_fst
theorem pullbackIsoProdSubtype_inv_fst_apply (f : X ⟶ Z) (g : Y ⟶ Z)
(x : { p : X × Y // f p.1 = g p.2 }) :
(pullback.fst : pullback f g ⟶ _) ((pullbackIsoProdSubtype f g).inv x) = (x : X × Y).fst :=
ConcreteCategory.congr_hom (pullbackIsoProdSubtype_inv_fst f g) x
#align Top.pullback_iso_prod_subtype_inv_fst_apply TopCat.pullbackIsoProdSubtype_inv_fst_apply
@[reassoc (attr := simp)]
theorem pullbackIsoProdSubtype_inv_snd (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullbackIsoProdSubtype f g).inv ≫ pullback.snd = pullbackSnd f g := by
simp [pullbackCone, pullbackIsoProdSubtype]
#align Top.pullback_iso_prod_subtype_inv_snd TopCat.pullbackIsoProdSubtype_inv_snd
theorem pullbackIsoProdSubtype_inv_snd_apply (f : X ⟶ Z) (g : Y ⟶ Z)
(x : { p : X × Y // f p.1 = g p.2 }) :
(pullback.snd : pullback f g ⟶ _) ((pullbackIsoProdSubtype f g).inv x) = (x : X × Y).snd :=
ConcreteCategory.congr_hom (pullbackIsoProdSubtype_inv_snd f g) x
#align Top.pullback_iso_prod_subtype_inv_snd_apply TopCat.pullbackIsoProdSubtype_inv_snd_apply
theorem pullbackIsoProdSubtype_hom_fst (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullbackIsoProdSubtype f g).hom ≫ pullbackFst f g = pullback.fst := by
rw [← Iso.eq_inv_comp, pullbackIsoProdSubtype_inv_fst]
#align Top.pullback_iso_prod_subtype_hom_fst TopCat.pullbackIsoProdSubtype_hom_fst
theorem pullbackIsoProdSubtype_hom_snd (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullbackIsoProdSubtype f g).hom ≫ pullbackSnd f g = pullback.snd := by
rw [← Iso.eq_inv_comp, pullbackIsoProdSubtype_inv_snd]
#align Top.pullback_iso_prod_subtype_hom_snd TopCat.pullbackIsoProdSubtype_hom_snd
-- Porting note: why do I need to tell Lean to coerce pullback to a type
theorem pullbackIsoProdSubtype_hom_apply {f : X ⟶ Z} {g : Y ⟶ Z}
(x : ConcreteCategory.forget.obj (pullback f g)) :
(pullbackIsoProdSubtype f g).hom x =
⟨⟨(pullback.fst : pullback f g ⟶ _) x, (pullback.snd : pullback f g ⟶ _) x⟩, by
simpa using ConcreteCategory.congr_hom pullback.condition x⟩ := by
apply Subtype.ext; apply Prod.ext
exacts [ConcreteCategory.congr_hom (pullbackIsoProdSubtype_hom_fst f g) x,
ConcreteCategory.congr_hom (pullbackIsoProdSubtype_hom_snd f g) x]
#align Top.pullback_iso_prod_subtype_hom_apply TopCat.pullbackIsoProdSubtype_hom_apply
theorem pullback_topology {X Y Z : TopCat.{u}} (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullback f g).str =
induced (pullback.fst : pullback f g ⟶ _) X.str ⊓
induced (pullback.snd : pullback f g ⟶ _) Y.str := by
let homeo := homeoOfIso (pullbackIsoProdSubtype f g)
refine homeo.inducing.induced.trans ?_
change induced homeo (induced _ ( (induced Prod.fst X.str) ⊓ (induced Prod.snd Y.str))) = _
simp only [induced_compose, induced_inf]
congr
#align Top.pullback_topology TopCat.pullback_topology
theorem range_pullback_to_prod {X Y Z : TopCat} (f : X ⟶ Z) (g : Y ⟶ Z) :
Set.range (prod.lift pullback.fst pullback.snd : pullback f g ⟶ X ⨯ Y) =
{ x | (Limits.prod.fst ≫ f) x = (Limits.prod.snd ≫ g) x } := by
ext x
constructor
· rintro ⟨y, rfl⟩
change (_ ≫ _ ≫ f) _ = (_ ≫ _ ≫ g) _ -- new `change` after #13170
simp [pullback.condition]
· rintro (h : f (_, _).1 = g (_, _).2)
use (pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, h⟩
change (forget TopCat).map _ _ = _ -- new `change` after #13170
apply Concrete.limit_ext
rintro ⟨⟨⟩⟩ <;>
erw [← comp_apply, ← comp_apply, limit.lift_π] <;> -- now `erw` after #13170
-- This used to be `simp` before leanprover/lean4#2644
aesop_cat
#align Top.range_pullback_to_prod TopCat.range_pullback_to_prod
noncomputable
def pullbackHomeoPreimage
{X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
(f : X → Z) (hf : Continuous f) (g : Y → Z) (hg : Embedding g) :
{ p : X × Y // f p.1 = g p.2 } ≃ₜ f ⁻¹' Set.range g where
toFun := fun x ↦ ⟨x.1.1, _, x.2.symm⟩
invFun := fun x ↦ ⟨⟨x.1, Exists.choose x.2⟩, (Exists.choose_spec x.2).symm⟩
left_inv := by
intro x
ext <;> dsimp
apply hg.inj
convert x.prop
exact Exists.choose_spec (p := fun y ↦ g y = f (↑x : X × Y).1) _
right_inv := fun x ↦ rfl
continuous_toFun := by
apply Continuous.subtype_mk
exact continuous_fst.comp continuous_subtype_val
continuous_invFun := by
apply Continuous.subtype_mk
refine continuous_prod_mk.mpr ⟨continuous_subtype_val, hg.toInducing.continuous_iff.mpr ?_⟩
convert hf.comp continuous_subtype_val
ext x
exact Exists.choose_spec x.2
theorem inducing_pullback_to_prod {X Y Z : TopCat.{u}} (f : X ⟶ Z) (g : Y ⟶ Z) :
Inducing <| ⇑(prod.lift pullback.fst pullback.snd : pullback f g ⟶ X ⨯ Y) :=
⟨by simp [topologicalSpace_coe, prod_topology, pullback_topology, induced_compose, ← coe_comp]⟩
#align Top.inducing_pullback_to_prod TopCat.inducing_pullback_to_prod
theorem embedding_pullback_to_prod {X Y Z : TopCat.{u}} (f : X ⟶ Z) (g : Y ⟶ Z) :
Embedding <| ⇑(prod.lift pullback.fst pullback.snd : pullback f g ⟶ X ⨯ Y) :=
⟨inducing_pullback_to_prod f g, (TopCat.mono_iff_injective _).mp inferInstance⟩
#align Top.embedding_pullback_to_prod TopCat.embedding_pullback_to_prod
theorem range_pullback_map {W X Y Z S T : TopCat} (f₁ : W ⟶ S) (f₂ : X ⟶ S) (g₁ : Y ⟶ T)
(g₂ : Z ⟶ T) (i₁ : W ⟶ Y) (i₂ : X ⟶ Z) (i₃ : S ⟶ T) [H₃ : Mono i₃] (eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁)
(eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) :
Set.range (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) =
(pullback.fst : pullback g₁ g₂ ⟶ _) ⁻¹' Set.range i₁ ∩
(pullback.snd : pullback g₁ g₂ ⟶ _) ⁻¹' Set.range i₂ := by
ext
constructor
· rintro ⟨y, rfl⟩
simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_range]
erw [← comp_apply, ← comp_apply] -- now `erw` after #13170
simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app, comp_apply]
exact ⟨exists_apply_eq_apply _ _, exists_apply_eq_apply _ _⟩
rintro ⟨⟨x₁, hx₁⟩, ⟨x₂, hx₂⟩⟩
have : f₁ x₁ = f₂ x₂ := by
apply (TopCat.mono_iff_injective _).mp H₃
erw [← comp_apply, eq₁, ← comp_apply, eq₂, -- now `erw` after #13170
comp_apply, comp_apply, hx₁, hx₂, ← comp_apply, pullback.condition]
rfl -- `rfl` was not needed before #13170
use (pullbackIsoProdSubtype f₁ f₂).inv ⟨⟨x₁, x₂⟩, this⟩
change (forget TopCat).map _ _ = _
apply Concrete.limit_ext
rintro (_ | _ | _) <;>
erw [← comp_apply, ← comp_apply] -- now `erw` after #13170
simp only [Category.assoc, limit.lift_π, PullbackCone.mk_π_app_one]
· simp only [cospan_one, pullbackIsoProdSubtype_inv_fst_assoc, comp_apply]
erw [pullbackFst_apply, hx₁]
rw [← limit.w _ WalkingCospan.Hom.inl, cospan_map_inl, comp_apply (g := g₁)]
rfl -- `rfl` was not needed before #13170
· simp only [cospan_left, limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app,
pullbackIsoProdSubtype_inv_fst_assoc, comp_apply]
erw [hx₁] -- now `erw` after #13170
rfl -- `rfl` was not needed before #13170
· simp only [cospan_right, limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app,
pullbackIsoProdSubtype_inv_snd_assoc, comp_apply]
erw [hx₂] -- now `erw` after #13170
rfl -- `rfl` was not needed before #13170
#align Top.range_pullback_map TopCat.range_pullback_map
theorem pullback_fst_range {X Y S : TopCat} (f : X ⟶ S) (g : Y ⟶ S) :
Set.range (pullback.fst : pullback f g ⟶ _) = { x : X | ∃ y : Y, f x = g y } := by
ext x
constructor
· rintro ⟨(y : (forget TopCat).obj _), rfl⟩
use (pullback.snd : pullback f g ⟶ _) y
exact ConcreteCategory.congr_hom pullback.condition y
· rintro ⟨y, eq⟩
use (TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨x, y⟩, eq⟩
rw [pullbackIsoProdSubtype_inv_fst_apply]
#align Top.pullback_fst_range TopCat.pullback_fst_range
theorem pullback_snd_range {X Y S : TopCat} (f : X ⟶ S) (g : Y ⟶ S) :
Set.range (pullback.snd : pullback f g ⟶ _) = { y : Y | ∃ x : X, f x = g y } := by
ext y
constructor
· rintro ⟨(x : (forget TopCat).obj _), rfl⟩
use (pullback.fst : pullback f g ⟶ _) x
exact ConcreteCategory.congr_hom pullback.condition x
· rintro ⟨x, eq⟩
use (TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨x, y⟩, eq⟩
rw [pullbackIsoProdSubtype_inv_snd_apply]
#align Top.pullback_snd_range TopCat.pullback_snd_range
theorem pullback_map_embedding_of_embeddings {W X Y Z S T : TopCat.{u}} (f₁ : W ⟶ S) (f₂ : X ⟶ S)
(g₁ : Y ⟶ T) (g₂ : Z ⟶ T) {i₁ : W ⟶ Y} {i₂ : X ⟶ Z} (H₁ : Embedding i₁) (H₂ : Embedding i₂)
(i₃ : S ⟶ T) (eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁) (eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) :
Embedding (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) := by
refine
embedding_of_embedding_compose (ContinuousMap.continuous_toFun _)
(show Continuous (prod.lift pullback.fst pullback.snd : pullback g₁ g₂ ⟶ Y ⨯ Z) from
ContinuousMap.continuous_toFun _)
?_
suffices
Embedding (prod.lift pullback.fst pullback.snd ≫ Limits.prod.map i₁ i₂ : pullback f₁ f₂ ⟶ _) by
simpa [← coe_comp] using this
rw [coe_comp]
exact Embedding.comp (embedding_prod_map H₁ H₂) (embedding_pullback_to_prod _ _)
#align Top.pullback_map_embedding_of_embeddings TopCat.pullback_map_embedding_of_embeddings
| Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 308 | 320 | theorem pullback_map_openEmbedding_of_open_embeddings {W X Y Z S T : TopCat.{u}} (f₁ : W ⟶ S)
(f₂ : X ⟶ S) (g₁ : Y ⟶ T) (g₂ : Z ⟶ T) {i₁ : W ⟶ Y} {i₂ : X ⟶ Z} (H₁ : OpenEmbedding i₁)
(H₂ : OpenEmbedding i₂) (i₃ : S ⟶ T) [H₃ : Mono i₃] (eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁)
(eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) : OpenEmbedding (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) := by |
constructor
· apply
pullback_map_embedding_of_embeddings f₁ f₂ g₁ g₂ H₁.toEmbedding H₂.toEmbedding i₃ eq₁ eq₂
· rw [range_pullback_map]
apply IsOpen.inter <;> apply Continuous.isOpen_preimage
· apply ContinuousMap.continuous_toFun
· exact H₁.isOpen_range
· apply ContinuousMap.continuous_toFun
· exact H₂.isOpen_range
|
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace groupCohomology
section IsMulCocycle
section
variable {G M : Type*} [Mul G] [CommGroup M] [SMul G M]
def IsMulOneCocycle (f : G → M) : Prop := ∀ g h : G, f (g * h) = g • f h * f g
def IsMulTwoCocycle (f : G × G → M) : Prop :=
∀ g h j : G, f (g * h, j) * f (g, h) = g • (f (h, j)) * f (g, h * j)
end
section
variable {G M : Type*} [Monoid G] [CommGroup M] [MulAction G M]
| Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 524 | 526 | theorem map_one_of_isMulOneCocycle {f : G → M} (hf : IsMulOneCocycle f) :
f 1 = 1 := by |
simpa only [mul_one, one_smul, self_eq_mul_right] using hf 1 1
|
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [TopologicalSpace (X × X)] :
𝓝ˢ (diagonal X) = ⨆ (x : X), 𝓝 (x, x) := by
rw [nhdsSet, ← range_diag, ← range_comp]
rfl
#align nhds_set_diagonal nhdsSet_diagonal
theorem mem_nhdsSet_iff_forall : s ∈ 𝓝ˢ t ↔ ∀ x : X, x ∈ t → s ∈ 𝓝 x := by
simp_rw [nhdsSet, Filter.mem_sSup, forall_mem_image]
#align mem_nhds_set_iff_forall mem_nhdsSet_iff_forall
lemma nhdsSet_le : 𝓝ˢ s ≤ f ↔ ∀ x ∈ s, 𝓝 x ≤ f := by simp [nhdsSet]
theorem bUnion_mem_nhdsSet {t : X → Set X} (h : ∀ x ∈ s, t x ∈ 𝓝 x) : (⋃ x ∈ s, t x) ∈ 𝓝ˢ s :=
mem_nhdsSet_iff_forall.2 fun x hx => mem_of_superset (h x hx) <|
subset_iUnion₂ (s := fun x _ => t x) x hx -- Porting note: fails to find `s`
#align bUnion_mem_nhds_set bUnion_mem_nhdsSet
theorem subset_interior_iff_mem_nhdsSet : s ⊆ interior t ↔ t ∈ 𝓝ˢ s := by
simp_rw [mem_nhdsSet_iff_forall, subset_interior_iff_nhds]
#align subset_interior_iff_mem_nhds_set subset_interior_iff_mem_nhdsSet
theorem disjoint_principal_nhdsSet : Disjoint (𝓟 s) (𝓝ˢ t) ↔ Disjoint (closure s) t := by
rw [disjoint_principal_left, ← subset_interior_iff_mem_nhdsSet, interior_compl,
subset_compl_iff_disjoint_left]
theorem disjoint_nhdsSet_principal : Disjoint (𝓝ˢ s) (𝓟 t) ↔ Disjoint s (closure t) := by
rw [disjoint_comm, disjoint_principal_nhdsSet, disjoint_comm]
theorem mem_nhdsSet_iff_exists : s ∈ 𝓝ˢ t ↔ ∃ U : Set X, IsOpen U ∧ t ⊆ U ∧ U ⊆ s := by
rw [← subset_interior_iff_mem_nhdsSet, subset_interior_iff]
#align mem_nhds_set_iff_exists mem_nhdsSet_iff_exists
theorem eventually_nhdsSet_iff_exists {p : X → Prop} :
(∀ᶠ x in 𝓝ˢ s, p x) ↔ ∃ t, IsOpen t ∧ s ⊆ t ∧ ∀ x, x ∈ t → p x :=
mem_nhdsSet_iff_exists
theorem eventually_nhdsSet_iff_forall {p : X → Prop} :
(∀ᶠ x in 𝓝ˢ s, p x) ↔ ∀ x, x ∈ s → ∀ᶠ y in 𝓝 x, p y :=
mem_nhdsSet_iff_forall
theorem hasBasis_nhdsSet (s : Set X) : (𝓝ˢ s).HasBasis (fun U => IsOpen U ∧ s ⊆ U) fun U => U :=
⟨fun t => by simp [mem_nhdsSet_iff_exists, and_assoc]⟩
#align has_basis_nhds_set hasBasis_nhdsSet
@[simp]
lemma lift'_nhdsSet_interior (s : Set X) : (𝓝ˢ s).lift' interior = 𝓝ˢ s :=
(hasBasis_nhdsSet s).lift'_interior_eq_self fun _ ↦ And.left
lemma Filter.HasBasis.nhdsSet_interior {ι : Sort*} {p : ι → Prop} {s : ι → Set X} {t : Set X}
(h : (𝓝ˢ t).HasBasis p s) : (𝓝ˢ t).HasBasis p (interior <| s ·) :=
lift'_nhdsSet_interior t ▸ h.lift'_interior
theorem IsOpen.mem_nhdsSet (hU : IsOpen s) : s ∈ 𝓝ˢ t ↔ t ⊆ s := by
rw [← subset_interior_iff_mem_nhdsSet, hU.interior_eq]
#align is_open.mem_nhds_set IsOpen.mem_nhdsSet
theorem IsOpen.mem_nhdsSet_self (ho : IsOpen s) : s ∈ 𝓝ˢ s := ho.mem_nhdsSet.mpr Subset.rfl
theorem principal_le_nhdsSet : 𝓟 s ≤ 𝓝ˢ s := fun _s hs =>
(subset_interior_iff_mem_nhdsSet.mpr hs).trans interior_subset
#align principal_le_nhds_set principal_le_nhdsSet
theorem subset_of_mem_nhdsSet (h : t ∈ 𝓝ˢ s) : s ⊆ t := principal_le_nhdsSet h
theorem Filter.Eventually.self_of_nhdsSet {p : X → Prop} (h : ∀ᶠ x in 𝓝ˢ s, p x) : ∀ x ∈ s, p x :=
principal_le_nhdsSet h
nonrec theorem Filter.EventuallyEq.self_of_nhdsSet {f g : X → Y} (h : f =ᶠ[𝓝ˢ s] g) : EqOn f g s :=
h.self_of_nhdsSet
@[simp]
theorem nhdsSet_eq_principal_iff : 𝓝ˢ s = 𝓟 s ↔ IsOpen s := by
rw [← principal_le_nhdsSet.le_iff_eq, le_principal_iff, mem_nhdsSet_iff_forall,
isOpen_iff_mem_nhds]
#align nhds_set_eq_principal_iff nhdsSet_eq_principal_iff
alias ⟨_, IsOpen.nhdsSet_eq⟩ := nhdsSet_eq_principal_iff
#align is_open.nhds_set_eq IsOpen.nhdsSet_eq
@[simp]
theorem nhdsSet_interior : 𝓝ˢ (interior s) = 𝓟 (interior s) :=
isOpen_interior.nhdsSet_eq
#align nhds_set_interior nhdsSet_interior
@[simp]
| Mathlib/Topology/NhdsSet.lean | 124 | 124 | theorem nhdsSet_singleton : 𝓝ˢ {x} = 𝓝 x := by | simp [nhdsSet]
|
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
open scoped Real
open scoped RealInnerProductSpace
namespace Orientation
open FiniteDimensional
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
variable [hd2 : Fact (finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2))
theorem oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arccos (‖x‖ / ‖x + y‖) := by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arccos_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)]
#align orientation.oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two Orientation.oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two
theorem oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arccos (‖y‖ / ‖x + y‖) := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two h
#align orientation.oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two Orientation.oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two
theorem oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arcsin (‖y‖ / ‖x + y‖) := by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arcsin_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_two h))]
#align orientation.oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two Orientation.oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two
theorem oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arcsin (‖x‖ / ‖x + y‖) := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two h
#align orientation.oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two Orientation.oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two
theorem oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arctan (‖y‖ / ‖x‖) := by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arctan_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h) (o.left_ne_zero_of_oangle_eq_pi_div_two h)]
#align orientation.oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two Orientation.oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two
theorem oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arctan (‖x‖ / ‖y‖) := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two h
#align orientation.oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two Orientation.oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two
theorem cos_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.cos (o.oangle x (x + y)) = ‖x‖ / ‖x + y‖ := by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
InnerProductGeometry.cos_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)]
#align orientation.cos_oangle_add_right_of_oangle_eq_pi_div_two Orientation.cos_oangle_add_right_of_oangle_eq_pi_div_two
theorem cos_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.cos (o.oangle (x + y) y) = ‖y‖ / ‖x + y‖ := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).cos_oangle_add_right_of_oangle_eq_pi_div_two h
#align orientation.cos_oangle_add_left_of_oangle_eq_pi_div_two Orientation.cos_oangle_add_left_of_oangle_eq_pi_div_two
theorem sin_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.sin (o.oangle x (x + y)) = ‖y‖ / ‖x + y‖ := by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
InnerProductGeometry.sin_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_two h))]
#align orientation.sin_oangle_add_right_of_oangle_eq_pi_div_two Orientation.sin_oangle_add_right_of_oangle_eq_pi_div_two
theorem sin_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.sin (o.oangle (x + y) y) = ‖x‖ / ‖x + y‖ := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).sin_oangle_add_right_of_oangle_eq_pi_div_two h
#align orientation.sin_oangle_add_left_of_oangle_eq_pi_div_two Orientation.sin_oangle_add_left_of_oangle_eq_pi_div_two
theorem tan_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.tan (o.oangle x (x + y)) = ‖y‖ / ‖x‖ := by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe,
InnerProductGeometry.tan_angle_add_of_inner_eq_zero (o.inner_eq_zero_of_oangle_eq_pi_div_two h)]
#align orientation.tan_oangle_add_right_of_oangle_eq_pi_div_two Orientation.tan_oangle_add_right_of_oangle_eq_pi_div_two
theorem tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.tan (o.oangle (x + y) y) = ‖x‖ / ‖y‖ := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).tan_oangle_add_right_of_oangle_eq_pi_div_two h
#align orientation.tan_oangle_add_left_of_oangle_eq_pi_div_two Orientation.tan_oangle_add_left_of_oangle_eq_pi_div_two
theorem cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle x (x + y)) * ‖x + y‖ = ‖x‖ := by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
InnerProductGeometry.cos_angle_add_mul_norm_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)]
#align orientation.cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two Orientation.cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two
theorem cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle (x + y) y) * ‖x + y‖ = ‖y‖ := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h
#align orientation.cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two Orientation.cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two
theorem sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle x (x + y)) * ‖x + y‖ = ‖y‖ := by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
InnerProductGeometry.sin_angle_add_mul_norm_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)]
#align orientation.sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two Orientation.sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two
theorem sin_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle (x + y) y) * ‖x + y‖ = ‖x‖ := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h
#align orientation.sin_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two Orientation.sin_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two
theorem tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle x (x + y)) * ‖x‖ = ‖y‖ := by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe,
InnerProductGeometry.tan_angle_add_mul_norm_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_two h))]
#align orientation.tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two Orientation.tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two
theorem tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.tan (o.oangle (x + y) y) * ‖y‖ = ‖x‖ := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two h
#align orientation.tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two Orientation.tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two
theorem norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.cos (o.oangle x (x + y)) = ‖x + y‖ := by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
InnerProductGeometry.norm_div_cos_angle_add_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_two h))]
#align orientation.norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two Orientation.norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two
theorem norm_div_cos_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.cos (o.oangle (x + y) y) = ‖x + y‖ := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two h
#align orientation.norm_div_cos_oangle_add_left_of_oangle_eq_pi_div_two Orientation.norm_div_cos_oangle_add_left_of_oangle_eq_pi_div_two
theorem norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.sin (o.oangle x (x + y)) = ‖x + y‖ := by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
InnerProductGeometry.norm_div_sin_angle_add_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inr (o.right_ne_zero_of_oangle_eq_pi_div_two h))]
#align orientation.norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two Orientation.norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two
theorem norm_div_sin_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.sin (o.oangle (x + y) y) = ‖x + y‖ := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two h
#align orientation.norm_div_sin_oangle_add_left_of_oangle_eq_pi_div_two Orientation.norm_div_sin_oangle_add_left_of_oangle_eq_pi_div_two
theorem norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.tan (o.oangle x (x + y)) = ‖x‖ := by
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe,
InnerProductGeometry.norm_div_tan_angle_add_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inr (o.right_ne_zero_of_oangle_eq_pi_div_two h))]
#align orientation.norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two Orientation.norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two
theorem norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.tan (o.oangle (x + y) y) = ‖y‖ := by
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two h
#align orientation.norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two Orientation.norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two
| Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 267 | 273 | theorem oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle y (y - x) = Real.arccos (‖y‖ / ‖y - x‖) := by |
have hs : (o.oangle y (y - x)).sign = 1 := by
rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_sub_eq_arccos_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)]
|
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import topology.category.Top.limits.products from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 false
open TopologicalSpace
open CategoryTheory
open CategoryTheory.Limits
universe v u w
noncomputable section
namespace TopCat
variable {J : Type v} [SmallCategory J]
abbrev piπ {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) : TopCat.of (∀ i, α i) ⟶ α i :=
⟨fun f => f i, continuous_apply i⟩
#align Top.pi_π TopCat.piπ
@[simps! pt π_app]
def piFan {ι : Type v} (α : ι → TopCat.{max v u}) : Fan α :=
Fan.mk (TopCat.of (∀ i, α i)) (piπ.{v,u} α)
#align Top.pi_fan TopCat.piFan
def piFanIsLimit {ι : Type v} (α : ι → TopCat.{max v u}) : IsLimit (piFan α) where
lift S :=
{ toFun := fun s i => S.π.app ⟨i⟩ s
continuous_toFun := continuous_pi (fun i => (S.π.app ⟨i⟩).2) }
uniq := by
intro S m h
apply ContinuousMap.ext; intro x
funext i
set_option tactic.skipAssignedInstances false in
dsimp
rw [ContinuousMap.coe_mk, ← h ⟨i⟩]
rfl
fac s j := rfl
#align Top.pi_fan_is_limit TopCat.piFanIsLimit
def piIsoPi {ι : Type v} (α : ι → TopCat.{max v u}) : ∏ᶜ α ≅ TopCat.of (∀ i, α i) :=
(limit.isLimit _).conePointUniqueUpToIso (piFanIsLimit.{v, u} α)
-- Specifying the universes in `piFanIsLimit` wasn't necessary when we had `TopCatMax`
#align Top.pi_iso_pi TopCat.piIsoPi
@[reassoc (attr := simp)]
theorem piIsoPi_inv_π {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) :
(piIsoPi α).inv ≫ Pi.π α i = piπ α i := by simp [piIsoPi]
#align Top.pi_iso_pi_inv_π TopCat.piIsoPi_inv_π
theorem piIsoPi_inv_π_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) (x : ∀ i, α i) :
(Pi.π α i : _) ((piIsoPi α).inv x) = x i :=
ConcreteCategory.congr_hom (piIsoPi_inv_π α i) x
#align Top.pi_iso_pi_inv_π_apply TopCat.piIsoPi_inv_π_apply
-- Porting note: needing the type ascription on `∏ᶜ α : TopCat.{max v u}` is unfortunate.
theorem piIsoPi_hom_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι)
(x : (∏ᶜ α : TopCat.{max v u})) : (piIsoPi α).hom x i = (Pi.π α i : _) x := by
have := piIsoPi_inv_π α i
rw [Iso.inv_comp_eq] at this
exact ConcreteCategory.congr_hom this x
#align Top.pi_iso_pi_hom_apply TopCat.piIsoPi_hom_apply
-- Porting note: Lean doesn't automatically reduce TopCat.of X|>.α to X now
abbrev sigmaι {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) : α i ⟶ TopCat.of (Σi, α i) := by
refine ContinuousMap.mk ?_ ?_
· dsimp
apply Sigma.mk i
· dsimp; continuity
#align Top.sigma_ι TopCat.sigmaι
@[simps! pt ι_app]
def sigmaCofan {ι : Type v} (α : ι → TopCat.{max v u}) : Cofan α :=
Cofan.mk (TopCat.of (Σi, α i)) (sigmaι α)
#align Top.sigma_cofan TopCat.sigmaCofan
def sigmaCofanIsColimit {ι : Type v} (β : ι → TopCat.{max v u}) : IsColimit (sigmaCofan β) where
desc S :=
{ toFun := fun (s : of (Σ i, β i)) => S.ι.app ⟨s.1⟩ s.2
continuous_toFun := continuous_sigma fun i => (S.ι.app ⟨i⟩).continuous_toFun }
uniq := by
intro S m h
ext ⟨i, x⟩
simp only [hom_apply, ← h]
congr
fac s j := by
cases j
aesop_cat
#align Top.sigma_cofan_is_colimit TopCat.sigmaCofanIsColimit
def sigmaIsoSigma {ι : Type v} (α : ι → TopCat.{max v u}) : ∐ α ≅ TopCat.of (Σi, α i) :=
(colimit.isColimit _).coconePointUniqueUpToIso (sigmaCofanIsColimit.{v, u} α)
-- Specifying the universes in `sigmaCofanIsColimit` wasn't necessary when we had `TopCatMax`
#align Top.sigma_iso_sigma TopCat.sigmaIsoSigma
@[reassoc (attr := simp)]
theorem sigmaIsoSigma_hom_ι {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) :
Sigma.ι α i ≫ (sigmaIsoSigma α).hom = sigmaι α i := by simp [sigmaIsoSigma]
#align Top.sigma_iso_sigma_hom_ι TopCat.sigmaIsoSigma_hom_ι
theorem sigmaIsoSigma_hom_ι_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) (x : α i) :
(sigmaIsoSigma α).hom ((Sigma.ι α i : _) x) = Sigma.mk i x :=
ConcreteCategory.congr_hom (sigmaIsoSigma_hom_ι α i) x
#align Top.sigma_iso_sigma_hom_ι_apply TopCat.sigmaIsoSigma_hom_ι_apply
theorem sigmaIsoSigma_inv_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) (x : α i) :
(sigmaIsoSigma α).inv ⟨i, x⟩ = (Sigma.ι α i : _) x := by
rw [← sigmaIsoSigma_hom_ι_apply, ← comp_app, ← comp_app, Iso.hom_inv_id,
Category.comp_id]
#align Top.sigma_iso_sigma_inv_apply TopCat.sigmaIsoSigma_inv_apply
-- Porting note: cannot use .topologicalSpace in place .str
theorem induced_of_isLimit {F : J ⥤ TopCat.{max v u}} (C : Cone F) (hC : IsLimit C) :
C.pt.str = ⨅ j, (F.obj j).str.induced (C.π.app j) := by
let homeo := homeoOfIso (hC.conePointUniqueUpToIso (limitConeInfiIsLimit F))
refine homeo.inducing.induced.trans ?_
change induced homeo (⨅ j : J, _) = _
simp [induced_iInf, induced_compose]
rfl
#align Top.induced_of_is_limit TopCat.induced_of_isLimit
theorem limit_topology (F : J ⥤ TopCat.{max v u}) :
(limit F).str = ⨅ j, (F.obj j).str.induced (limit.π F j) :=
induced_of_isLimit _ (limit.isLimit F)
#align Top.limit_topology TopCat.limit_topology
section Prod
-- Porting note: why is autoParam not firing?
abbrev prodFst {X Y : TopCat.{u}} : TopCat.of (X × Y) ⟶ X :=
⟨Prod.fst, by continuity⟩
#align Top.prod_fst TopCat.prodFst
abbrev prodSnd {X Y : TopCat.{u}} : TopCat.of (X × Y) ⟶ Y :=
⟨Prod.snd, by continuity⟩
#align Top.prod_snd TopCat.prodSnd
def prodBinaryFan (X Y : TopCat.{u}) : BinaryFan X Y :=
BinaryFan.mk prodFst prodSnd
#align Top.prod_binary_fan TopCat.prodBinaryFan
def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where
lift := fun S : BinaryFan X Y => {
toFun := fun s => (S.fst s, S.snd s)
-- Porting note: continuity failed again here. Lean cannot infer
-- ContinuousMapClass (X ⟶ Y) X Y for X Y : TopCat which may be one of the problems
continuous_toFun := Continuous.prod_mk
(BinaryFan.fst S).continuous_toFun (BinaryFan.snd S).continuous_toFun }
fac := by
rintro S (_ | _) <;> {dsimp; ext; rfl}
uniq := by
intro S m h
-- Porting note: used to be `ext x`
refine ContinuousMap.ext (fun (x : ↥(S.pt)) => Prod.ext ?_ ?_)
· specialize h ⟨WalkingPair.left⟩
apply_fun fun e => e x at h
exact h
· specialize h ⟨WalkingPair.right⟩
apply_fun fun e => e x at h
exact h
#align Top.prod_binary_fan_is_limit TopCat.prodBinaryFanIsLimit
def prodIsoProd (X Y : TopCat.{u}) : X ⨯ Y ≅ TopCat.of (X × Y) :=
(limit.isLimit _).conePointUniqueUpToIso (prodBinaryFanIsLimit X Y)
#align Top.prod_iso_prod TopCat.prodIsoProd
@[reassoc (attr := simp)]
theorem prodIsoProd_hom_fst (X Y : TopCat.{u}) :
(prodIsoProd X Y).hom ≫ prodFst = Limits.prod.fst := by
simp [← Iso.eq_inv_comp, prodIsoProd]
rfl
#align Top.prod_iso_prod_hom_fst TopCat.prodIsoProd_hom_fst
@[reassoc (attr := simp)]
theorem prodIsoProd_hom_snd (X Y : TopCat.{u}) :
(prodIsoProd X Y).hom ≫ prodSnd = Limits.prod.snd := by
simp [← Iso.eq_inv_comp, prodIsoProd]
rfl
#align Top.prod_iso_prod_hom_snd TopCat.prodIsoProd_hom_snd
-- Porting note: need to force Lean to coerce X × Y to a type
theorem prodIsoProd_hom_apply {X Y : TopCat.{u}} (x : ↑ (X ⨯ Y)) :
(prodIsoProd X Y).hom x = ((Limits.prod.fst : X ⨯ Y ⟶ _) x,
(Limits.prod.snd : X ⨯ Y ⟶ _) x) := by
-- Porting note: ext didn't pick this up
apply Prod.ext
· exact ConcreteCategory.congr_hom (prodIsoProd_hom_fst X Y) x
· exact ConcreteCategory.congr_hom (prodIsoProd_hom_snd X Y) x
#align Top.prod_iso_prod_hom_apply TopCat.prodIsoProd_hom_apply
@[reassoc (attr := simp), elementwise]
theorem prodIsoProd_inv_fst (X Y : TopCat.{u}) :
(prodIsoProd X Y).inv ≫ Limits.prod.fst = prodFst := by simp [Iso.inv_comp_eq]
#align Top.prod_iso_prod_inv_fst TopCat.prodIsoProd_inv_fst
@[reassoc (attr := simp), elementwise]
theorem prodIsoProd_inv_snd (X Y : TopCat.{u}) :
(prodIsoProd X Y).inv ≫ Limits.prod.snd = prodSnd := by simp [Iso.inv_comp_eq]
#align Top.prod_iso_prod_inv_snd TopCat.prodIsoProd_inv_snd
theorem prod_topology {X Y : TopCat.{u}} :
(X ⨯ Y).str =
induced (Limits.prod.fst : X ⨯ Y ⟶ _) X.str ⊓
induced (Limits.prod.snd : X ⨯ Y ⟶ _) Y.str := by
let homeo := homeoOfIso (prodIsoProd X Y)
refine homeo.inducing.induced.trans ?_
change induced homeo (_ ⊓ _) = _
simp [induced_compose]
rfl
#align Top.prod_topology TopCat.prod_topology
| Mathlib/Topology/Category/TopCat/Limits/Products.lean | 249 | 276 | theorem range_prod_map {W X Y Z : TopCat.{u}} (f : W ⟶ Y) (g : X ⟶ Z) :
Set.range (Limits.prod.map f g) =
(Limits.prod.fst : Y ⨯ Z ⟶ _) ⁻¹' Set.range f ∩
(Limits.prod.snd : Y ⨯ Z ⟶ _) ⁻¹' Set.range g := by |
ext x
constructor
· rintro ⟨y, rfl⟩
simp_rw [Set.mem_inter_iff, Set.mem_preimage, Set.mem_range]
-- sizable changes in this proof after #13170
erw [← comp_apply, ← comp_apply]
simp_rw [Limits.prod.map_fst,
Limits.prod.map_snd, comp_apply]
exact ⟨exists_apply_eq_apply _ _, exists_apply_eq_apply _ _⟩
· rintro ⟨⟨x₁, hx₁⟩, ⟨x₂, hx₂⟩⟩
use (prodIsoProd W X).inv (x₁, x₂)
change (forget TopCat).map _ _ = _
apply Concrete.limit_ext
rintro ⟨⟨⟩⟩
· change limit.π (pair Y Z) _ ((prod.map f g) _) = _
erw [← comp_apply, Limits.prod.map_fst]
change (_ ≫ _ ≫ f) _ = _
erw [TopCat.prodIsoProd_inv_fst_assoc,TopCat.comp_app]
exact hx₁
· change limit.π (pair Y Z) _ ((prod.map f g) _) = _
erw [← comp_apply, Limits.prod.map_snd]
change (_ ≫ _ ≫ g) _ = _
erw [TopCat.prodIsoProd_inv_snd_assoc,TopCat.comp_app]
exact hx₂
|
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Real
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*}
[NormedAddCommGroup F] [NormedSpace ℝ F]
namespace SameRay
variable {x y : E}
theorem norm_add (h : SameRay ℝ x y) : ‖x + y‖ = ‖x‖ + ‖y‖ := by
rcases h.exists_eq_smul with ⟨u, a, b, ha, hb, -, rfl, rfl⟩
rw [← add_smul, norm_smul_of_nonneg (add_nonneg ha hb), norm_smul_of_nonneg ha,
norm_smul_of_nonneg hb, add_mul]
#align same_ray.norm_add SameRay.norm_add
| Mathlib/Analysis/NormedSpace/Ray.lean | 38 | 46 | theorem norm_sub (h : SameRay ℝ x y) : ‖x - y‖ = |‖x‖ - ‖y‖| := by |
rcases h.exists_eq_smul with ⟨u, a, b, ha, hb, -, rfl, rfl⟩
wlog hab : b ≤ a generalizing a b with H
· rw [SameRay.sameRay_comm] at h
rw [norm_sub_rev, abs_sub_comm]
exact H b a hb ha h (le_of_not_le hab)
rw [← sub_nonneg] at hab
rw [← sub_smul, norm_smul_of_nonneg hab, norm_smul_of_nonneg ha, norm_smul_of_nonneg hb, ←
sub_mul, abs_of_nonneg (mul_nonneg hab (norm_nonneg _))]
|
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fintype (α : Type*) where
elems : Finset α
complete : ∀ x : α, x ∈ elems
#align fintype Fintype
namespace Finset
variable [Fintype α] {s t : Finset α}
def univ : Finset α :=
@Fintype.elems α _
#align finset.univ Finset.univ
@[simp]
theorem mem_univ (x : α) : x ∈ (univ : Finset α) :=
Fintype.complete x
#align finset.mem_univ Finset.mem_univ
-- Porting note: removing @[simp], simp can prove it
theorem mem_univ_val : ∀ x, x ∈ (univ : Finset α).1 :=
mem_univ
#align finset.mem_univ_val Finset.mem_univ_val
theorem eq_univ_iff_forall : s = univ ↔ ∀ x, x ∈ s := by simp [ext_iff]
#align finset.eq_univ_iff_forall Finset.eq_univ_iff_forall
theorem eq_univ_of_forall : (∀ x, x ∈ s) → s = univ :=
eq_univ_iff_forall.2
#align finset.eq_univ_of_forall Finset.eq_univ_of_forall
@[simp, norm_cast]
theorem coe_univ : ↑(univ : Finset α) = (Set.univ : Set α) := by ext; simp
#align finset.coe_univ Finset.coe_univ
@[simp, norm_cast]
theorem coe_eq_univ : (s : Set α) = Set.univ ↔ s = univ := by rw [← coe_univ, coe_inj]
#align finset.coe_eq_univ Finset.coe_eq_univ
| Mathlib/Data/Fintype/Basic.lean | 99 | 101 | theorem Nonempty.eq_univ [Subsingleton α] : s.Nonempty → s = univ := by |
rintro ⟨x, hx⟩
exact eq_univ_of_forall fun y => by rwa [Subsingleton.elim y x]
|
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 : ∅ ∈ T)
(sInter_mem : ∀ A, A ⊆ T → ⋂₀ A ∈ T)
(union_mem : ∀ A, A ∈ T → ∀ B, B ∈ T → A ∪ B ∈ T) : TopologicalSpace X where
IsOpen X := Xᶜ ∈ T
isOpen_univ := by simp [empty_mem]
isOpen_inter s t hs ht := by simpa only [compl_inter] using union_mem sᶜ hs tᶜ ht
isOpen_sUnion s hs := by
simp only [Set.compl_sUnion]
exact sInter_mem (compl '' s) fun z ⟨y, hy, hz⟩ => hz ▸ hs y hy
#align topological_space.of_closed TopologicalSpace.ofClosed
section TopologicalSpace
variable {X : Type u} {Y : Type v} {ι : Sort w} {α β : Type*}
{x : X} {s s₁ s₂ t : Set X} {p p₁ p₂ : X → Prop}
open Topology
lemma isOpen_mk {p h₁ h₂ h₃} : IsOpen[⟨p, h₁, h₂, h₃⟩] s ↔ p s := Iff.rfl
#align is_open_mk isOpen_mk
@[ext]
protected theorem TopologicalSpace.ext :
∀ {f g : TopologicalSpace X}, IsOpen[f] = IsOpen[g] → f = g
| ⟨_, _, _, _⟩, ⟨_, _, _, _⟩, rfl => rfl
#align topological_space_eq TopologicalSpace.ext
section
variable [TopologicalSpace X]
end
protected theorem TopologicalSpace.ext_iff {t t' : TopologicalSpace X} :
t = t' ↔ ∀ s, IsOpen[t] s ↔ IsOpen[t'] s :=
⟨fun h s => h ▸ Iff.rfl, fun h => by ext; exact h _⟩
#align topological_space_eq_iff TopologicalSpace.ext_iff
theorem isOpen_fold {t : TopologicalSpace X} : t.IsOpen s = IsOpen[t] s :=
rfl
#align is_open_fold isOpen_fold
variable [TopologicalSpace X]
theorem isOpen_iUnion {f : ι → Set X} (h : ∀ i, IsOpen (f i)) : IsOpen (⋃ i, f i) :=
isOpen_sUnion (forall_mem_range.2 h)
#align is_open_Union isOpen_iUnion
theorem isOpen_biUnion {s : Set α} {f : α → Set X} (h : ∀ i ∈ s, IsOpen (f i)) :
IsOpen (⋃ i ∈ s, f i) :=
isOpen_iUnion fun i => isOpen_iUnion fun hi => h i hi
#align is_open_bUnion isOpen_biUnion
theorem IsOpen.union (h₁ : IsOpen s₁) (h₂ : IsOpen s₂) : IsOpen (s₁ ∪ s₂) := by
rw [union_eq_iUnion]; exact isOpen_iUnion (Bool.forall_bool.2 ⟨h₂, h₁⟩)
#align is_open.union IsOpen.union
lemma isOpen_iff_of_cover {f : α → Set X} (ho : ∀ i, IsOpen (f i)) (hU : (⋃ i, f i) = univ) :
IsOpen s ↔ ∀ i, IsOpen (f i ∩ s) := by
refine ⟨fun h i ↦ (ho i).inter h, fun h ↦ ?_⟩
rw [← s.inter_univ, inter_comm, ← hU, iUnion_inter]
exact isOpen_iUnion fun i ↦ h i
@[simp] theorem isOpen_empty : IsOpen (∅ : Set X) := by
rw [← sUnion_empty]; exact isOpen_sUnion fun a => False.elim
#align is_open_empty isOpen_empty
theorem Set.Finite.isOpen_sInter {s : Set (Set X)} (hs : s.Finite) :
(∀ t ∈ s, IsOpen t) → IsOpen (⋂₀ s) :=
Finite.induction_on hs (fun _ => by rw [sInter_empty]; exact isOpen_univ) fun _ _ ih h => by
simp only [sInter_insert, forall_mem_insert] at h ⊢
exact h.1.inter (ih h.2)
#align is_open_sInter Set.Finite.isOpen_sInter
theorem Set.Finite.isOpen_biInter {s : Set α} {f : α → Set X} (hs : s.Finite)
(h : ∀ i ∈ s, IsOpen (f i)) :
IsOpen (⋂ i ∈ s, f i) :=
sInter_image f s ▸ (hs.image _).isOpen_sInter (forall_mem_image.2 h)
#align is_open_bInter Set.Finite.isOpen_biInter
theorem isOpen_iInter_of_finite [Finite ι] {s : ι → Set X} (h : ∀ i, IsOpen (s i)) :
IsOpen (⋂ i, s i) :=
(finite_range _).isOpen_sInter (forall_mem_range.2 h)
#align is_open_Inter isOpen_iInter_of_finite
theorem isOpen_biInter_finset {s : Finset α} {f : α → Set X} (h : ∀ i ∈ s, IsOpen (f i)) :
IsOpen (⋂ i ∈ s, f i) :=
s.finite_toSet.isOpen_biInter h
#align is_open_bInter_finset isOpen_biInter_finset
@[simp] -- Porting note: added `simp`
theorem isOpen_const {p : Prop} : IsOpen { _x : X | p } := by by_cases p <;> simp [*]
#align is_open_const isOpen_const
theorem IsOpen.and : IsOpen { x | p₁ x } → IsOpen { x | p₂ x } → IsOpen { x | p₁ x ∧ p₂ x } :=
IsOpen.inter
#align is_open.and IsOpen.and
@[simp] theorem isOpen_compl_iff : IsOpen sᶜ ↔ IsClosed s :=
⟨fun h => ⟨h⟩, fun h => h.isOpen_compl⟩
#align is_open_compl_iff isOpen_compl_iff
theorem TopologicalSpace.ext_iff_isClosed {t₁ t₂ : TopologicalSpace X} :
t₁ = t₂ ↔ ∀ s, IsClosed[t₁] s ↔ IsClosed[t₂] s := by
rw [TopologicalSpace.ext_iff, compl_surjective.forall]
simp only [@isOpen_compl_iff _ _ t₁, @isOpen_compl_iff _ _ t₂]
alias ⟨_, TopologicalSpace.ext_isClosed⟩ := TopologicalSpace.ext_iff_isClosed
-- Porting note (#10756): new lemma
theorem isClosed_const {p : Prop} : IsClosed { _x : X | p } := ⟨isOpen_const (p := ¬p)⟩
@[simp] theorem isClosed_empty : IsClosed (∅ : Set X) := isClosed_const
#align is_closed_empty isClosed_empty
@[simp] theorem isClosed_univ : IsClosed (univ : Set X) := isClosed_const
#align is_closed_univ isClosed_univ
theorem IsClosed.union : IsClosed s₁ → IsClosed s₂ → IsClosed (s₁ ∪ s₂) := by
simpa only [← isOpen_compl_iff, compl_union] using IsOpen.inter
#align is_closed.union IsClosed.union
theorem isClosed_sInter {s : Set (Set X)} : (∀ t ∈ s, IsClosed t) → IsClosed (⋂₀ s) := by
simpa only [← isOpen_compl_iff, compl_sInter, sUnion_image] using isOpen_biUnion
#align is_closed_sInter isClosed_sInter
theorem isClosed_iInter {f : ι → Set X} (h : ∀ i, IsClosed (f i)) : IsClosed (⋂ i, f i) :=
isClosed_sInter <| forall_mem_range.2 h
#align is_closed_Inter isClosed_iInter
theorem isClosed_biInter {s : Set α} {f : α → Set X} (h : ∀ i ∈ s, IsClosed (f i)) :
IsClosed (⋂ i ∈ s, f i) :=
isClosed_iInter fun i => isClosed_iInter <| h i
#align is_closed_bInter isClosed_biInter
@[simp]
theorem isClosed_compl_iff {s : Set X} : IsClosed sᶜ ↔ IsOpen s := by
rw [← isOpen_compl_iff, compl_compl]
#align is_closed_compl_iff isClosed_compl_iff
alias ⟨_, IsOpen.isClosed_compl⟩ := isClosed_compl_iff
#align is_open.is_closed_compl IsOpen.isClosed_compl
theorem IsOpen.sdiff (h₁ : IsOpen s) (h₂ : IsClosed t) : IsOpen (s \ t) :=
IsOpen.inter h₁ h₂.isOpen_compl
#align is_open.sdiff IsOpen.sdiff
theorem IsClosed.inter (h₁ : IsClosed s₁) (h₂ : IsClosed s₂) : IsClosed (s₁ ∩ s₂) := by
rw [← isOpen_compl_iff] at *
rw [compl_inter]
exact IsOpen.union h₁ h₂
#align is_closed.inter IsClosed.inter
theorem IsClosed.sdiff (h₁ : IsClosed s) (h₂ : IsOpen t) : IsClosed (s \ t) :=
IsClosed.inter h₁ (isClosed_compl_iff.mpr h₂)
#align is_closed.sdiff IsClosed.sdiff
theorem Set.Finite.isClosed_biUnion {s : Set α} {f : α → Set X} (hs : s.Finite)
(h : ∀ i ∈ s, IsClosed (f i)) :
IsClosed (⋃ i ∈ s, f i) := by
simp only [← isOpen_compl_iff, compl_iUnion] at *
exact hs.isOpen_biInter h
#align is_closed_bUnion Set.Finite.isClosed_biUnion
lemma isClosed_biUnion_finset {s : Finset α} {f : α → Set X} (h : ∀ i ∈ s, IsClosed (f i)) :
IsClosed (⋃ i ∈ s, f i) :=
s.finite_toSet.isClosed_biUnion h
theorem isClosed_iUnion_of_finite [Finite ι] {s : ι → Set X} (h : ∀ i, IsClosed (s i)) :
IsClosed (⋃ i, s i) := by
simp only [← isOpen_compl_iff, compl_iUnion] at *
exact isOpen_iInter_of_finite h
#align is_closed_Union isClosed_iUnion_of_finite
theorem isClosed_imp {p q : X → Prop} (hp : IsOpen { x | p x }) (hq : IsClosed { x | q x }) :
IsClosed { x | p x → q x } := by
simpa only [imp_iff_not_or] using hp.isClosed_compl.union hq
#align is_closed_imp isClosed_imp
theorem IsClosed.not : IsClosed { a | p a } → IsOpen { a | ¬p a } :=
isOpen_compl_iff.mpr
#align is_closed.not IsClosed.not
theorem mem_interior : x ∈ interior s ↔ ∃ t ⊆ s, IsOpen t ∧ x ∈ t := by
simp only [interior, mem_sUnion, mem_setOf_eq, and_assoc, and_left_comm]
#align mem_interior mem_interiorₓ
@[simp]
theorem isOpen_interior : IsOpen (interior s) :=
isOpen_sUnion fun _ => And.left
#align is_open_interior isOpen_interior
theorem interior_subset : interior s ⊆ s :=
sUnion_subset fun _ => And.right
#align interior_subset interior_subset
theorem interior_maximal (h₁ : t ⊆ s) (h₂ : IsOpen t) : t ⊆ interior s :=
subset_sUnion_of_mem ⟨h₂, h₁⟩
#align interior_maximal interior_maximal
theorem IsOpen.interior_eq (h : IsOpen s) : interior s = s :=
interior_subset.antisymm (interior_maximal (Subset.refl s) h)
#align is_open.interior_eq IsOpen.interior_eq
theorem interior_eq_iff_isOpen : interior s = s ↔ IsOpen s :=
⟨fun h => h ▸ isOpen_interior, IsOpen.interior_eq⟩
#align interior_eq_iff_is_open interior_eq_iff_isOpen
theorem subset_interior_iff_isOpen : s ⊆ interior s ↔ IsOpen s := by
simp only [interior_eq_iff_isOpen.symm, Subset.antisymm_iff, interior_subset, true_and]
#align subset_interior_iff_is_open subset_interior_iff_isOpen
theorem IsOpen.subset_interior_iff (h₁ : IsOpen s) : s ⊆ interior t ↔ s ⊆ t :=
⟨fun h => Subset.trans h interior_subset, fun h₂ => interior_maximal h₂ h₁⟩
#align is_open.subset_interior_iff IsOpen.subset_interior_iff
theorem subset_interior_iff : t ⊆ interior s ↔ ∃ U, IsOpen U ∧ t ⊆ U ∧ U ⊆ s :=
⟨fun h => ⟨interior s, isOpen_interior, h, interior_subset⟩, fun ⟨_U, hU, htU, hUs⟩ =>
htU.trans (interior_maximal hUs hU)⟩
#align subset_interior_iff subset_interior_iff
lemma interior_subset_iff : interior s ⊆ t ↔ ∀ U, IsOpen U → U ⊆ s → U ⊆ t := by
simp [interior]
@[mono, gcongr]
theorem interior_mono (h : s ⊆ t) : interior s ⊆ interior t :=
interior_maximal (Subset.trans interior_subset h) isOpen_interior
#align interior_mono interior_mono
@[simp]
theorem interior_empty : interior (∅ : Set X) = ∅ :=
isOpen_empty.interior_eq
#align interior_empty interior_empty
@[simp]
theorem interior_univ : interior (univ : Set X) = univ :=
isOpen_univ.interior_eq
#align interior_univ interior_univ
@[simp]
theorem interior_eq_univ : interior s = univ ↔ s = univ :=
⟨fun h => univ_subset_iff.mp <| h.symm.trans_le interior_subset, fun h => h.symm ▸ interior_univ⟩
#align interior_eq_univ interior_eq_univ
@[simp]
theorem interior_interior : interior (interior s) = interior s :=
isOpen_interior.interior_eq
#align interior_interior interior_interior
@[simp]
theorem interior_inter : interior (s ∩ t) = interior s ∩ interior t :=
(Monotone.map_inf_le (fun _ _ ↦ interior_mono) s t).antisymm <|
interior_maximal (inter_subset_inter interior_subset interior_subset) <|
isOpen_interior.inter isOpen_interior
#align interior_inter interior_inter
theorem Set.Finite.interior_biInter {ι : Type*} {s : Set ι} (hs : s.Finite) (f : ι → Set X) :
interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) :=
hs.induction_on (by simp) <| by intros; simp [*]
theorem Set.Finite.interior_sInter {S : Set (Set X)} (hS : S.Finite) :
interior (⋂₀ S) = ⋂ s ∈ S, interior s := by
rw [sInter_eq_biInter, hS.interior_biInter]
@[simp]
theorem Finset.interior_iInter {ι : Type*} (s : Finset ι) (f : ι → Set X) :
interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) :=
s.finite_toSet.interior_biInter f
#align finset.interior_Inter Finset.interior_iInter
@[simp]
theorem interior_iInter_of_finite [Finite ι] (f : ι → Set X) :
interior (⋂ i, f i) = ⋂ i, interior (f i) := by
rw [← sInter_range, (finite_range f).interior_sInter, biInter_range]
#align interior_Inter interior_iInter_of_finite
theorem interior_union_isClosed_of_interior_empty (h₁ : IsClosed s)
(h₂ : interior t = ∅) : interior (s ∪ t) = interior s :=
have : interior (s ∪ t) ⊆ s := fun x ⟨u, ⟨(hu₁ : IsOpen u), (hu₂ : u ⊆ s ∪ t)⟩, (hx₁ : x ∈ u)⟩ =>
by_contradiction fun hx₂ : x ∉ s =>
have : u \ s ⊆ t := fun x ⟨h₁, h₂⟩ => Or.resolve_left (hu₂ h₁) h₂
have : u \ s ⊆ interior t := by rwa [(IsOpen.sdiff hu₁ h₁).subset_interior_iff]
have : u \ s ⊆ ∅ := by rwa [h₂] at this
this ⟨hx₁, hx₂⟩
Subset.antisymm (interior_maximal this isOpen_interior) (interior_mono subset_union_left)
#align interior_union_is_closed_of_interior_empty interior_union_isClosed_of_interior_empty
theorem isOpen_iff_forall_mem_open : IsOpen s ↔ ∀ x ∈ s, ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := by
rw [← subset_interior_iff_isOpen]
simp only [subset_def, mem_interior]
#align is_open_iff_forall_mem_open isOpen_iff_forall_mem_open
theorem interior_iInter_subset (s : ι → Set X) : interior (⋂ i, s i) ⊆ ⋂ i, interior (s i) :=
subset_iInter fun _ => interior_mono <| iInter_subset _ _
#align interior_Inter_subset interior_iInter_subset
theorem interior_iInter₂_subset (p : ι → Sort*) (s : ∀ i, p i → Set X) :
interior (⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), interior (s i j) :=
(interior_iInter_subset _).trans <| iInter_mono fun _ => interior_iInter_subset _
#align interior_Inter₂_subset interior_iInter₂_subset
theorem interior_sInter_subset (S : Set (Set X)) : interior (⋂₀ S) ⊆ ⋂ s ∈ S, interior s :=
calc
interior (⋂₀ S) = interior (⋂ s ∈ S, s) := by rw [sInter_eq_biInter]
_ ⊆ ⋂ s ∈ S, interior s := interior_iInter₂_subset _ _
#align interior_sInter_subset interior_sInter_subset
theorem Filter.HasBasis.lift'_interior {l : Filter X} {p : ι → Prop} {s : ι → Set X}
(h : l.HasBasis p s) : (l.lift' interior).HasBasis p fun i => interior (s i) :=
h.lift' fun _ _ ↦ interior_mono
theorem Filter.lift'_interior_le (l : Filter X) : l.lift' interior ≤ l := fun _s hs ↦
mem_of_superset (mem_lift' hs) interior_subset
theorem Filter.HasBasis.lift'_interior_eq_self {l : Filter X} {p : ι → Prop} {s : ι → Set X}
(h : l.HasBasis p s) (ho : ∀ i, p i → IsOpen (s i)) : l.lift' interior = l :=
le_antisymm l.lift'_interior_le <| h.lift'_interior.ge_iff.2 fun i hi ↦ by
simpa only [(ho i hi).interior_eq] using h.mem_of_mem hi
@[simp]
theorem isClosed_closure : IsClosed (closure s) :=
isClosed_sInter fun _ => And.left
#align is_closed_closure isClosed_closure
theorem subset_closure : s ⊆ closure s :=
subset_sInter fun _ => And.right
#align subset_closure subset_closure
theorem not_mem_of_not_mem_closure {P : X} (hP : P ∉ closure s) : P ∉ s := fun h =>
hP (subset_closure h)
#align not_mem_of_not_mem_closure not_mem_of_not_mem_closure
theorem closure_minimal (h₁ : s ⊆ t) (h₂ : IsClosed t) : closure s ⊆ t :=
sInter_subset_of_mem ⟨h₂, h₁⟩
#align closure_minimal closure_minimal
theorem Disjoint.closure_left (hd : Disjoint s t) (ht : IsOpen t) :
Disjoint (closure s) t :=
disjoint_compl_left.mono_left <| closure_minimal hd.subset_compl_right ht.isClosed_compl
#align disjoint.closure_left Disjoint.closure_left
theorem Disjoint.closure_right (hd : Disjoint s t) (hs : IsOpen s) :
Disjoint s (closure t) :=
(hd.symm.closure_left hs).symm
#align disjoint.closure_right Disjoint.closure_right
theorem IsClosed.closure_eq (h : IsClosed s) : closure s = s :=
Subset.antisymm (closure_minimal (Subset.refl s) h) subset_closure
#align is_closed.closure_eq IsClosed.closure_eq
theorem IsClosed.closure_subset (hs : IsClosed s) : closure s ⊆ s :=
closure_minimal (Subset.refl _) hs
#align is_closed.closure_subset IsClosed.closure_subset
theorem IsClosed.closure_subset_iff (h₁ : IsClosed t) : closure s ⊆ t ↔ s ⊆ t :=
⟨Subset.trans subset_closure, fun h => closure_minimal h h₁⟩
#align is_closed.closure_subset_iff IsClosed.closure_subset_iff
theorem IsClosed.mem_iff_closure_subset (hs : IsClosed s) :
x ∈ s ↔ closure ({x} : Set X) ⊆ s :=
(hs.closure_subset_iff.trans Set.singleton_subset_iff).symm
#align is_closed.mem_iff_closure_subset IsClosed.mem_iff_closure_subset
@[mono, gcongr]
theorem closure_mono (h : s ⊆ t) : closure s ⊆ closure t :=
closure_minimal (Subset.trans h subset_closure) isClosed_closure
#align closure_mono closure_mono
theorem monotone_closure (X : Type*) [TopologicalSpace X] : Monotone (@closure X _) := fun _ _ =>
closure_mono
#align monotone_closure monotone_closure
theorem diff_subset_closure_iff : s \ t ⊆ closure t ↔ s ⊆ closure t := by
rw [diff_subset_iff, union_eq_self_of_subset_left subset_closure]
#align diff_subset_closure_iff diff_subset_closure_iff
theorem closure_inter_subset_inter_closure (s t : Set X) :
closure (s ∩ t) ⊆ closure s ∩ closure t :=
(monotone_closure X).map_inf_le s t
#align closure_inter_subset_inter_closure closure_inter_subset_inter_closure
theorem isClosed_of_closure_subset (h : closure s ⊆ s) : IsClosed s := by
rw [subset_closure.antisymm h]; exact isClosed_closure
#align is_closed_of_closure_subset isClosed_of_closure_subset
theorem closure_eq_iff_isClosed : closure s = s ↔ IsClosed s :=
⟨fun h => h ▸ isClosed_closure, IsClosed.closure_eq⟩
#align closure_eq_iff_is_closed closure_eq_iff_isClosed
theorem closure_subset_iff_isClosed : closure s ⊆ s ↔ IsClosed s :=
⟨isClosed_of_closure_subset, IsClosed.closure_subset⟩
#align closure_subset_iff_is_closed closure_subset_iff_isClosed
@[simp]
theorem closure_empty : closure (∅ : Set X) = ∅ :=
isClosed_empty.closure_eq
#align closure_empty closure_empty
@[simp]
theorem closure_empty_iff (s : Set X) : closure s = ∅ ↔ s = ∅ :=
⟨subset_eq_empty subset_closure, fun h => h.symm ▸ closure_empty⟩
#align closure_empty_iff closure_empty_iff
@[simp]
theorem closure_nonempty_iff : (closure s).Nonempty ↔ s.Nonempty := by
simp only [nonempty_iff_ne_empty, Ne, closure_empty_iff]
#align closure_nonempty_iff closure_nonempty_iff
alias ⟨Set.Nonempty.of_closure, Set.Nonempty.closure⟩ := closure_nonempty_iff
#align set.nonempty.of_closure Set.Nonempty.of_closure
#align set.nonempty.closure Set.Nonempty.closure
@[simp]
theorem closure_univ : closure (univ : Set X) = univ :=
isClosed_univ.closure_eq
#align closure_univ closure_univ
@[simp]
theorem closure_closure : closure (closure s) = closure s :=
isClosed_closure.closure_eq
#align closure_closure closure_closure
theorem closure_eq_compl_interior_compl : closure s = (interior sᶜ)ᶜ := by
rw [interior, closure, compl_sUnion, compl_image_set_of]
simp only [compl_subset_compl, isOpen_compl_iff]
#align closure_eq_compl_interior_compl closure_eq_compl_interior_compl
@[simp]
theorem closure_union : closure (s ∪ t) = closure s ∪ closure t := by
simp [closure_eq_compl_interior_compl, compl_inter]
#align closure_union closure_union
theorem Set.Finite.closure_biUnion {ι : Type*} {s : Set ι} (hs : s.Finite) (f : ι → Set X) :
closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) := by
simp [closure_eq_compl_interior_compl, hs.interior_biInter]
theorem Set.Finite.closure_sUnion {S : Set (Set X)} (hS : S.Finite) :
closure (⋃₀ S) = ⋃ s ∈ S, closure s := by
rw [sUnion_eq_biUnion, hS.closure_biUnion]
@[simp]
theorem Finset.closure_biUnion {ι : Type*} (s : Finset ι) (f : ι → Set X) :
closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) :=
s.finite_toSet.closure_biUnion f
#align finset.closure_bUnion Finset.closure_biUnion
@[simp]
theorem closure_iUnion_of_finite [Finite ι] (f : ι → Set X) :
closure (⋃ i, f i) = ⋃ i, closure (f i) := by
rw [← sUnion_range, (finite_range _).closure_sUnion, biUnion_range]
#align closure_Union closure_iUnion_of_finite
theorem interior_subset_closure : interior s ⊆ closure s :=
Subset.trans interior_subset subset_closure
#align interior_subset_closure interior_subset_closure
@[simp]
theorem interior_compl : interior sᶜ = (closure s)ᶜ := by
simp [closure_eq_compl_interior_compl]
#align interior_compl interior_compl
@[simp]
theorem closure_compl : closure sᶜ = (interior s)ᶜ := by
simp [closure_eq_compl_interior_compl]
#align closure_compl closure_compl
theorem mem_closure_iff :
x ∈ closure s ↔ ∀ o, IsOpen o → x ∈ o → (o ∩ s).Nonempty :=
⟨fun h o oo ao =>
by_contradiction fun os =>
have : s ⊆ oᶜ := fun x xs xo => os ⟨x, xo, xs⟩
closure_minimal this (isClosed_compl_iff.2 oo) h ao,
fun H _ ⟨h₁, h₂⟩ =>
by_contradiction fun nc =>
let ⟨_, hc, hs⟩ := H _ h₁.isOpen_compl nc
hc (h₂ hs)⟩
#align mem_closure_iff mem_closure_iff
theorem closure_inter_open_nonempty_iff (h : IsOpen t) :
(closure s ∩ t).Nonempty ↔ (s ∩ t).Nonempty :=
⟨fun ⟨_x, hxcs, hxt⟩ => inter_comm t s ▸ mem_closure_iff.1 hxcs t h hxt, fun h =>
h.mono <| inf_le_inf_right t subset_closure⟩
#align closure_inter_open_nonempty_iff closure_inter_open_nonempty_iff
theorem Filter.le_lift'_closure (l : Filter X) : l ≤ l.lift' closure :=
le_lift'.2 fun _ h => mem_of_superset h subset_closure
#align filter.le_lift'_closure Filter.le_lift'_closure
theorem Filter.HasBasis.lift'_closure {l : Filter X} {p : ι → Prop} {s : ι → Set X}
(h : l.HasBasis p s) : (l.lift' closure).HasBasis p fun i => closure (s i) :=
h.lift' (monotone_closure X)
#align filter.has_basis.lift'_closure Filter.HasBasis.lift'_closure
theorem Filter.HasBasis.lift'_closure_eq_self {l : Filter X} {p : ι → Prop} {s : ι → Set X}
(h : l.HasBasis p s) (hc : ∀ i, p i → IsClosed (s i)) : l.lift' closure = l :=
le_antisymm (h.ge_iff.2 fun i hi => (hc i hi).closure_eq ▸ mem_lift' (h.mem_of_mem hi))
l.le_lift'_closure
#align filter.has_basis.lift'_closure_eq_self Filter.HasBasis.lift'_closure_eq_self
@[simp]
theorem Filter.lift'_closure_eq_bot {l : Filter X} : l.lift' closure = ⊥ ↔ l = ⊥ :=
⟨fun h => bot_unique <| h ▸ l.le_lift'_closure, fun h =>
h.symm ▸ by rw [lift'_bot (monotone_closure _), closure_empty, principal_empty]⟩
#align filter.lift'_closure_eq_bot Filter.lift'_closure_eq_bot
theorem dense_iff_closure_eq : Dense s ↔ closure s = univ :=
eq_univ_iff_forall.symm
#align dense_iff_closure_eq dense_iff_closure_eq
alias ⟨Dense.closure_eq, _⟩ := dense_iff_closure_eq
#align dense.closure_eq Dense.closure_eq
theorem interior_eq_empty_iff_dense_compl : interior s = ∅ ↔ Dense sᶜ := by
rw [dense_iff_closure_eq, closure_compl, compl_univ_iff]
#align interior_eq_empty_iff_dense_compl interior_eq_empty_iff_dense_compl
theorem Dense.interior_compl (h : Dense s) : interior sᶜ = ∅ :=
interior_eq_empty_iff_dense_compl.2 <| by rwa [compl_compl]
#align dense.interior_compl Dense.interior_compl
@[simp]
theorem dense_closure : Dense (closure s) ↔ Dense s := by
rw [Dense, Dense, closure_closure]
#align dense_closure dense_closure
protected alias ⟨_, Dense.closure⟩ := dense_closure
alias ⟨Dense.of_closure, _⟩ := dense_closure
#align dense.of_closure Dense.of_closure
#align dense.closure Dense.closure
@[simp]
theorem dense_univ : Dense (univ : Set X) := fun _ => subset_closure trivial
#align dense_univ dense_univ
theorem dense_iff_inter_open :
Dense s ↔ ∀ U, IsOpen U → U.Nonempty → (U ∩ s).Nonempty := by
constructor <;> intro h
· rintro U U_op ⟨x, x_in⟩
exact mem_closure_iff.1 (h _) U U_op x_in
· intro x
rw [mem_closure_iff]
intro U U_op x_in
exact h U U_op ⟨_, x_in⟩
#align dense_iff_inter_open dense_iff_inter_open
alias ⟨Dense.inter_open_nonempty, _⟩ := dense_iff_inter_open
#align dense.inter_open_nonempty Dense.inter_open_nonempty
theorem Dense.exists_mem_open (hs : Dense s) {U : Set X} (ho : IsOpen U)
(hne : U.Nonempty) : ∃ x ∈ s, x ∈ U :=
let ⟨x, hx⟩ := hs.inter_open_nonempty U ho hne
⟨x, hx.2, hx.1⟩
#align dense.exists_mem_open Dense.exists_mem_open
theorem Dense.nonempty_iff (hs : Dense s) : s.Nonempty ↔ Nonempty X :=
⟨fun ⟨x, _⟩ => ⟨x⟩, fun ⟨x⟩ =>
let ⟨y, hy⟩ := hs.inter_open_nonempty _ isOpen_univ ⟨x, trivial⟩
⟨y, hy.2⟩⟩
#align dense.nonempty_iff Dense.nonempty_iff
theorem Dense.nonempty [h : Nonempty X] (hs : Dense s) : s.Nonempty :=
hs.nonempty_iff.2 h
#align dense.nonempty Dense.nonempty
@[mono]
theorem Dense.mono (h : s₁ ⊆ s₂) (hd : Dense s₁) : Dense s₂ := fun x =>
closure_mono h (hd x)
#align dense.mono Dense.mono
theorem dense_compl_singleton_iff_not_open :
Dense ({x}ᶜ : Set X) ↔ ¬IsOpen ({x} : Set X) := by
constructor
· intro hd ho
exact (hd.inter_open_nonempty _ ho (singleton_nonempty _)).ne_empty (inter_compl_self _)
· refine fun ho => dense_iff_inter_open.2 fun U hU hne => inter_compl_nonempty_iff.2 fun hUx => ?_
obtain rfl : U = {x} := eq_singleton_iff_nonempty_unique_mem.2 ⟨hne, hUx⟩
exact ho hU
#align dense_compl_singleton_iff_not_open dense_compl_singleton_iff_not_open
@[simp]
theorem closure_diff_interior (s : Set X) : closure s \ interior s = frontier s :=
rfl
#align closure_diff_interior closure_diff_interior
lemma disjoint_interior_frontier : Disjoint (interior s) (frontier s) := by
rw [disjoint_iff_inter_eq_empty, ← closure_diff_interior, diff_eq,
← inter_assoc, inter_comm, ← inter_assoc, compl_inter_self, empty_inter]
@[simp]
| Mathlib/Topology/Basic.lean | 667 | 668 | theorem closure_diff_frontier (s : Set X) : closure s \ frontier s = interior s := by |
rw [frontier, diff_diff_right_self, inter_eq_self_of_subset_right interior_subset_closure]
|
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Subtype
import Mathlib.Order.Notation
#align_import algebra.ring.idempotents from "leanprover-community/mathlib"@"655994e298904d7e5bbd1e18c95defd7b543eb94"
variable {M N S M₀ M₁ R G G₀ : Type*}
variable [Mul M] [Monoid N] [Semigroup S] [MulZeroClass M₀] [MulOneClass M₁] [NonAssocRing R]
[Group G] [CancelMonoidWithZero G₀]
def IsIdempotentElem (p : M) : Prop :=
p * p = p
#align is_idempotent_elem IsIdempotentElem
namespace IsIdempotentElem
theorem of_isIdempotent [Std.IdempotentOp (α := M) (· * ·)] (a : M) : IsIdempotentElem a :=
Std.IdempotentOp.idempotent a
#align is_idempotent_elem.of_is_idempotent IsIdempotentElem.of_isIdempotent
theorem eq {p : M} (h : IsIdempotentElem p) : p * p = p :=
h
#align is_idempotent_elem.eq IsIdempotentElem.eq
theorem mul_of_commute {p q : S} (h : Commute p q) (h₁ : IsIdempotentElem p)
(h₂ : IsIdempotentElem q) : IsIdempotentElem (p * q) := by
rw [IsIdempotentElem, mul_assoc, ← mul_assoc q, ← h.eq, mul_assoc p, h₂.eq, ← mul_assoc, h₁.eq]
#align is_idempotent_elem.mul_of_commute IsIdempotentElem.mul_of_commute
theorem zero : IsIdempotentElem (0 : M₀) :=
mul_zero _
#align is_idempotent_elem.zero IsIdempotentElem.zero
theorem one : IsIdempotentElem (1 : M₁) :=
mul_one _
#align is_idempotent_elem.one IsIdempotentElem.one
theorem one_sub {p : R} (h : IsIdempotentElem p) : IsIdempotentElem (1 - p) := by
rw [IsIdempotentElem, mul_sub, mul_one, sub_mul, one_mul, h.eq, sub_self, sub_zero]
#align is_idempotent_elem.one_sub IsIdempotentElem.one_sub
@[simp]
theorem one_sub_iff {p : R} : IsIdempotentElem (1 - p) ↔ IsIdempotentElem p :=
⟨fun h => sub_sub_cancel 1 p ▸ h.one_sub, IsIdempotentElem.one_sub⟩
#align is_idempotent_elem.one_sub_iff IsIdempotentElem.one_sub_iff
theorem pow {p : N} (n : ℕ) (h : IsIdempotentElem p) : IsIdempotentElem (p ^ n) :=
Nat.recOn n ((pow_zero p).symm ▸ one) fun n _ =>
show p ^ n.succ * p ^ n.succ = p ^ n.succ by
conv_rhs => rw [← h.eq] -- Porting note: was `nth_rw 3 [← h.eq]`
rw [← sq, ← sq, ← pow_mul, ← pow_mul']
#align is_idempotent_elem.pow IsIdempotentElem.pow
theorem pow_succ_eq {p : N} (n : ℕ) (h : IsIdempotentElem p) : p ^ (n + 1) = p :=
Nat.recOn n ((Nat.zero_add 1).symm ▸ pow_one p) fun n ih => by rw [pow_succ, ih, h.eq]
#align is_idempotent_elem.pow_succ_eq IsIdempotentElem.pow_succ_eq
@[simp]
theorem iff_eq_one {p : G} : IsIdempotentElem p ↔ p = 1 :=
Iff.intro (fun h => mul_left_cancel ((mul_one p).symm ▸ h.eq : p * p = p * 1)) fun h =>
h.symm ▸ one
#align is_idempotent_elem.iff_eq_one IsIdempotentElem.iff_eq_one
@[simp]
| Mathlib/Algebra/Ring/Idempotents.lean | 93 | 97 | theorem iff_eq_zero_or_one {p : G₀} : IsIdempotentElem p ↔ p = 0 ∨ p = 1 := by |
refine
Iff.intro (fun h => or_iff_not_imp_left.mpr fun hp => ?_) fun h =>
h.elim (fun hp => hp.symm ▸ zero) fun hp => hp.symm ▸ one
exact mul_left_cancel₀ hp (h.trans (mul_one p).symm)
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
open Nat
namespace List
@[simp]
theorem sublists'_nil : sublists' (@nil α) = [[]] :=
rfl
#align list.sublists'_nil List.sublists'_nil
@[simp]
theorem sublists'_singleton (a : α) : sublists' [a] = [[], [a]] :=
rfl
#align list.sublists'_singleton List.sublists'_singleton
#noalign list.map_sublists'_aux
#noalign list.sublists'_aux_append
#noalign list.sublists'_aux_eq_sublists'
-- Porting note: Not the same as `sublists'_aux` from Lean3
def sublists'Aux (a : α) (r₁ r₂ : List (List α)) : List (List α) :=
r₁.foldl (init := r₂) fun r l => r ++ [a :: l]
#align list.sublists'_aux List.sublists'Aux
theorem sublists'Aux_eq_array_foldl (a : α) : ∀ (r₁ r₂ : List (List α)),
sublists'Aux a r₁ r₂ = ((r₁.toArray).foldl (init := r₂.toArray)
(fun r l => r.push (a :: l))).toList := by
intro r₁ r₂
rw [sublists'Aux, Array.foldl_eq_foldl_data]
have := List.foldl_hom Array.toList (fun r l => r.push (a :: l))
(fun r l => r ++ [a :: l]) r₁ r₂.toArray (by simp)
simpa using this
theorem sublists'_eq_sublists'Aux (l : List α) :
sublists' l = l.foldr (fun a r => sublists'Aux a r r) [[]] := by
simp only [sublists', sublists'Aux_eq_array_foldl]
rw [← List.foldr_hom Array.toList]
· rfl
· intros _ _; congr <;> simp
theorem sublists'Aux_eq_map (a : α) (r₁ : List (List α)) : ∀ (r₂ : List (List α)),
sublists'Aux a r₁ r₂ = r₂ ++ map (cons a) r₁ :=
List.reverseRecOn r₁ (fun _ => by simp [sublists'Aux]) fun r₁ l ih r₂ => by
rw [map_append, map_singleton, ← append_assoc, ← ih, sublists'Aux, foldl_append, foldl]
simp [sublists'Aux]
-- Porting note: simp can prove `sublists'_singleton`
@[simp 900]
theorem sublists'_cons (a : α) (l : List α) :
sublists' (a :: l) = sublists' l ++ map (cons a) (sublists' l) := by
simp [sublists'_eq_sublists'Aux, foldr_cons, sublists'Aux_eq_map]
#align list.sublists'_cons List.sublists'_cons
@[simp]
theorem mem_sublists' {s t : List α} : s ∈ sublists' t ↔ s <+ t := by
induction' t with a t IH generalizing s
· simp only [sublists'_nil, mem_singleton]
exact ⟨fun h => by rw [h], eq_nil_of_sublist_nil⟩
simp only [sublists'_cons, mem_append, IH, mem_map]
constructor <;> intro h
· rcases h with (h | ⟨s, h, rfl⟩)
· exact sublist_cons_of_sublist _ h
· exact h.cons_cons _
· cases' h with _ _ _ h s _ _ h
· exact Or.inl h
· exact Or.inr ⟨s, h, rfl⟩
#align list.mem_sublists' List.mem_sublists'
@[simp]
theorem length_sublists' : ∀ l : List α, length (sublists' l) = 2 ^ length l
| [] => rfl
| a :: l => by
simp_arith only [sublists'_cons, length_append, length_sublists' l,
length_map, length, Nat.pow_succ']
#align list.length_sublists' List.length_sublists'
@[simp]
theorem sublists_nil : sublists (@nil α) = [[]] :=
rfl
#align list.sublists_nil List.sublists_nil
@[simp]
theorem sublists_singleton (a : α) : sublists [a] = [[], [a]] :=
rfl
#align list.sublists_singleton List.sublists_singleton
-- Porting note: Not the same as `sublists_aux` from Lean3
def sublistsAux (a : α) (r : List (List α)) : List (List α) :=
r.foldl (init := []) fun r l => r ++ [l, a :: l]
#align list.sublists_aux List.sublistsAux
theorem sublistsAux_eq_array_foldl :
sublistsAux = fun (a : α) (r : List (List α)) =>
(r.toArray.foldl (init := #[])
fun r l => (r.push l).push (a :: l)).toList := by
funext a r
simp only [sublistsAux, Array.foldl_eq_foldl_data, Array.mkEmpty]
have := foldl_hom Array.toList (fun r l => (r.push l).push (a :: l))
(fun (r : List (List α)) l => r ++ [l, a :: l]) r #[]
(by simp)
simpa using this
theorem sublistsAux_eq_bind :
sublistsAux = fun (a : α) (r : List (List α)) => r.bind fun l => [l, a :: l] :=
funext fun a => funext fun r =>
List.reverseRecOn r
(by simp [sublistsAux])
(fun r l ih => by
rw [append_bind, ← ih, bind_singleton, sublistsAux, foldl_append]
simp [sublistsAux])
@[csimp] theorem sublists_eq_sublistsFast : @sublists = @sublistsFast := by
ext α l : 2
trans l.foldr sublistsAux [[]]
· rw [sublistsAux_eq_bind, sublists]
· simp only [sublistsFast, sublistsAux_eq_array_foldl, Array.foldr_eq_foldr_data]
rw [← foldr_hom Array.toList]
· rfl
· intros _ _; congr <;> simp
#noalign list.sublists_aux₁_eq_sublists_aux
#noalign list.sublists_aux_cons_eq_sublists_aux₁
#noalign list.sublists_aux_eq_foldr.aux
#noalign list.sublists_aux_eq_foldr
#noalign list.sublists_aux_cons_cons
#noalign list.sublists_aux₁_append
#noalign list.sublists_aux₁_concat
#noalign list.sublists_aux₁_bind
#noalign list.sublists_aux_cons_append
theorem sublists_append (l₁ l₂ : List α) :
sublists (l₁ ++ l₂) = (sublists l₂) >>= (fun x => (sublists l₁).map (· ++ x)) := by
simp only [sublists, foldr_append]
induction l₁ with
| nil => simp
| cons a l₁ ih =>
rw [foldr_cons, ih]
simp [List.bind, join_join, Function.comp]
#align list.sublists_append List.sublists_append
-- Porting note (#10756): new theorem
theorem sublists_cons (a : α) (l : List α) :
sublists (a :: l) = sublists l >>= (fun x => [x, a :: x]) :=
show sublists ([a] ++ l) = _ by
rw [sublists_append]
simp only [sublists_singleton, map_cons, bind_eq_bind, nil_append, cons_append, map_nil]
@[simp]
theorem sublists_concat (l : List α) (a : α) :
sublists (l ++ [a]) = sublists l ++ map (fun x => x ++ [a]) (sublists l) := by
rw [sublists_append, sublists_singleton, bind_eq_bind, cons_bind, cons_bind, nil_bind,
map_id'' append_nil, append_nil]
#align list.sublists_concat List.sublists_concat
theorem sublists_reverse (l : List α) : sublists (reverse l) = map reverse (sublists' l) := by
induction' l with hd tl ih <;> [rfl;
simp only [reverse_cons, sublists_append, sublists'_cons, map_append, ih, sublists_singleton,
map_eq_map, bind_eq_bind, map_map, cons_bind, append_nil, nil_bind, (· ∘ ·)]]
#align list.sublists_reverse List.sublists_reverse
theorem sublists_eq_sublists' (l : List α) : sublists l = map reverse (sublists' (reverse l)) := by
rw [← sublists_reverse, reverse_reverse]
#align list.sublists_eq_sublists' List.sublists_eq_sublists'
theorem sublists'_reverse (l : List α) : sublists' (reverse l) = map reverse (sublists l) := by
simp only [sublists_eq_sublists', map_map, map_id'' reverse_reverse, Function.comp]
#align list.sublists'_reverse List.sublists'_reverse
theorem sublists'_eq_sublists (l : List α) : sublists' l = map reverse (sublists (reverse l)) := by
rw [← sublists'_reverse, reverse_reverse]
#align list.sublists'_eq_sublists List.sublists'_eq_sublists
#noalign list.sublists_aux_ne_nil
@[simp]
theorem mem_sublists {s t : List α} : s ∈ sublists t ↔ s <+ t := by
rw [← reverse_sublist, ← mem_sublists', sublists'_reverse,
mem_map_of_injective reverse_injective]
#align list.mem_sublists List.mem_sublists
@[simp]
| Mathlib/Data/List/Sublists.lean | 210 | 211 | theorem length_sublists (l : List α) : length (sublists l) = 2 ^ length l := by |
simp only [sublists_eq_sublists', length_map, length_sublists', length_reverse]
|
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Analysis.InnerProductSpace.Projection
#align_import analysis.convex.cone.dual from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
open scoped Classical
open Pointwise
variable {𝕜 E F G : Type*}
section Dual
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s t : Set H)
open RealInnerProductSpace
def Set.innerDualCone (s : Set H) : ConvexCone ℝ H where
carrier := { y | ∀ x ∈ s, 0 ≤ ⟪x, y⟫ }
smul_mem' c hc y hy x hx := by
rw [real_inner_smul_right]
exact mul_nonneg hc.le (hy x hx)
add_mem' u hu v hv x hx := by
rw [inner_add_right]
exact add_nonneg (hu x hx) (hv x hx)
#align set.inner_dual_cone Set.innerDualCone
@[simp]
theorem mem_innerDualCone (y : H) (s : Set H) : y ∈ s.innerDualCone ↔ ∀ x ∈ s, 0 ≤ ⟪x, y⟫ :=
Iff.rfl
#align mem_inner_dual_cone mem_innerDualCone
@[simp]
theorem innerDualCone_empty : (∅ : Set H).innerDualCone = ⊤ :=
eq_top_iff.mpr fun _ _ _ => False.elim
#align inner_dual_cone_empty innerDualCone_empty
@[simp]
theorem innerDualCone_zero : (0 : Set H).innerDualCone = ⊤ :=
eq_top_iff.mpr fun _ _ y (hy : y = 0) => hy.symm ▸ (inner_zero_left _).ge
#align inner_dual_cone_zero innerDualCone_zero
@[simp]
theorem innerDualCone_univ : (univ : Set H).innerDualCone = 0 := by
suffices ∀ x : H, x ∈ (univ : Set H).innerDualCone → x = 0 by
apply SetLike.coe_injective
exact eq_singleton_iff_unique_mem.mpr ⟨fun x _ => (inner_zero_right _).ge, this⟩
exact fun x hx => by simpa [← real_inner_self_nonpos] using hx (-x) (mem_univ _)
#align inner_dual_cone_univ innerDualCone_univ
theorem innerDualCone_le_innerDualCone (h : t ⊆ s) : s.innerDualCone ≤ t.innerDualCone :=
fun _ hy x hx => hy x (h hx)
#align inner_dual_cone_le_inner_dual_cone innerDualCone_le_innerDualCone
theorem pointed_innerDualCone : s.innerDualCone.Pointed := fun x _ => by rw [inner_zero_right]
#align pointed_inner_dual_cone pointed_innerDualCone
theorem innerDualCone_singleton (x : H) :
({x} : Set H).innerDualCone = (ConvexCone.positive ℝ ℝ).comap (innerₛₗ ℝ x) :=
ConvexCone.ext fun _ => forall_eq
#align inner_dual_cone_singleton innerDualCone_singleton
theorem innerDualCone_union (s t : Set H) :
(s ∪ t).innerDualCone = s.innerDualCone ⊓ t.innerDualCone :=
le_antisymm (le_inf (fun _ hx _ hy => hx _ <| Or.inl hy) fun _ hx _ hy => hx _ <| Or.inr hy)
fun _ hx _ => Or.rec (hx.1 _) (hx.2 _)
#align inner_dual_cone_union innerDualCone_union
| Mathlib/Analysis/Convex/Cone/InnerDual.lean | 105 | 107 | theorem innerDualCone_insert (x : H) (s : Set H) :
(insert x s).innerDualCone = Set.innerDualCone {x} ⊓ s.innerDualCone := by |
rw [insert_eq, innerDualCone_union]
|
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 : Type*} [NormedAddCommGroup E]
theorem RCLike.norm_coe_norm {z : E} : ‖(‖z‖ : 𝕜)‖ = ‖z‖ := by simp
#align is_R_or_C.norm_coe_norm RCLike.norm_coe_norm
variable [NormedSpace 𝕜 E]
@[simp]
theorem norm_smul_inv_norm {x : E} (hx : x ≠ 0) : ‖(‖x‖⁻¹ : 𝕜) • x‖ = 1 := by
have : ‖x‖ ≠ 0 := by simp [hx]
field_simp [norm_smul]
#align norm_smul_inv_norm norm_smul_inv_norm
| Mathlib/Analysis/NormedSpace/RCLike.lean | 49 | 52 | theorem norm_smul_inv_norm' {r : ℝ} (r_nonneg : 0 ≤ r) {x : E} (hx : x ≠ 0) :
‖((r : 𝕜) * (‖x‖ : 𝕜)⁻¹) • x‖ = r := by |
have : ‖x‖ ≠ 0 := by simp [hx]
field_simp [norm_smul, r_nonneg, rclike_simps]
|
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic
#align_import category_theory.monoidal.of_chosen_finite_products.symmetric from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C] {X Y : C}
open CategoryTheory.Limits
variable (𝒯 : LimitCone (Functor.empty.{0} C))
variable (ℬ : ∀ X Y : C, LimitCone (pair X Y))
open MonoidalOfChosenFiniteProducts
namespace MonoidalOfChosenFiniteProducts
open MonoidalCategory
theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
tensorHom ℬ f g ≫ (Limits.BinaryFan.braiding (ℬ Y Y').isLimit (ℬ Y' Y).isLimit).hom =
(Limits.BinaryFan.braiding (ℬ X X').isLimit (ℬ X' X).isLimit).hom ≫ tensorHom ℬ g f := by
dsimp [tensorHom, Limits.BinaryFan.braiding]
apply (ℬ _ _).isLimit.hom_ext
rintro ⟨⟨⟩⟩ <;> · dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
#align category_theory.monoidal_of_chosen_finite_products.braiding_naturality CategoryTheory.MonoidalOfChosenFiniteProducts.braiding_naturality
theorem hexagon_forward (X Y Z : C) :
(BinaryFan.associatorOfLimitCone ℬ X Y Z).hom ≫
(Limits.BinaryFan.braiding (ℬ X (tensorObj ℬ Y Z)).isLimit
(ℬ (tensorObj ℬ Y Z) X).isLimit).hom ≫
(BinaryFan.associatorOfLimitCone ℬ Y Z X).hom =
tensorHom ℬ (Limits.BinaryFan.braiding (ℬ X Y).isLimit (ℬ Y X).isLimit).hom (𝟙 Z) ≫
(BinaryFan.associatorOfLimitCone ℬ Y X Z).hom ≫
tensorHom ℬ (𝟙 Y) (Limits.BinaryFan.braiding (ℬ X Z).isLimit (ℬ Z X).isLimit).hom := by
dsimp [tensorHom, Limits.BinaryFan.braiding]
apply (ℬ _ _).isLimit.hom_ext; rintro ⟨⟨⟩⟩
· dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
· apply (ℬ _ _).isLimit.hom_ext
rintro ⟨⟨⟩⟩ <;> · dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
#align category_theory.monoidal_of_chosen_finite_products.hexagon_forward CategoryTheory.MonoidalOfChosenFiniteProducts.hexagon_forward
| Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean | 57 | 74 | theorem hexagon_reverse (X Y Z : C) :
(BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫
(Limits.BinaryFan.braiding (ℬ (tensorObj ℬ X Y) Z).isLimit
(ℬ Z (tensorObj ℬ X Y)).isLimit).hom ≫
(BinaryFan.associatorOfLimitCone ℬ Z X Y).inv =
tensorHom ℬ (𝟙 X) (Limits.BinaryFan.braiding (ℬ Y Z).isLimit (ℬ Z Y).isLimit).hom ≫
(BinaryFan.associatorOfLimitCone ℬ X Z Y).inv ≫
tensorHom ℬ (Limits.BinaryFan.braiding (ℬ X Z).isLimit (ℬ Z X).isLimit).hom (𝟙 Y) := by |
dsimp [tensorHom, Limits.BinaryFan.braiding]
apply (ℬ _ _).isLimit.hom_ext; rintro ⟨⟨⟩⟩
· apply (ℬ _ _).isLimit.hom_ext
rintro ⟨⟨⟩⟩ <;>
· dsimp [BinaryFan.associatorOfLimitCone, BinaryFan.associator,
Limits.IsLimit.conePointUniqueUpToIso]
simp
· dsimp [BinaryFan.associatorOfLimitCone, BinaryFan.associator,
Limits.IsLimit.conePointUniqueUpToIso]
simp
|
import Mathlib.Control.Traversable.Equiv
import Mathlib.Control.Traversable.Instances
import Batteries.Data.LazyList
import Mathlib.Lean.Thunk
#align_import data.lazy_list.basic from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
universe u
namespace LazyList
open Function
def listEquivLazyList (α : Type*) : List α ≃ LazyList α where
toFun := LazyList.ofList
invFun := LazyList.toList
right_inv := by
intro xs
induction xs using toList.induct
· simp [toList, ofList]
· simp [toList, ofList, *]; rfl
left_inv := by
intro xs
induction xs
· simp [toList, ofList]
· simpa [ofList, toList]
#align lazy_list.list_equiv_lazy_list LazyList.listEquivLazyList
-- Porting note: Added a name to make the recursion work.
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
| nil, nil => isTrue rfl
| cons x xs, cons y ys =>
if h : x = y then
match decidableEq xs.get ys.get with
| isFalse h2 => by
apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
| isTrue h2 => by apply isTrue; congr; ext; exact h2
else by apply isFalse; simp only [cons.injEq, not_and]; intro; contradiction
| nil, cons _ _ => by apply isFalse; simp
| cons _ _, nil => by apply isFalse; simp
protected def traverse {m : Type u → Type u} [Applicative m] {α β : Type u} (f : α → m β) :
LazyList α → m (LazyList β)
| LazyList.nil => pure LazyList.nil
| LazyList.cons x xs => LazyList.cons <$> f x <*> Thunk.pure <$> xs.get.traverse f
#align lazy_list.traverse LazyList.traverse
instance : Traversable LazyList where
map := @LazyList.traverse Id _
traverse := @LazyList.traverse
instance : LawfulTraversable LazyList := by
apply Equiv.isLawfulTraversable' listEquivLazyList <;> intros <;> ext <;> rename_i f xs
· induction' xs using LazyList.rec with _ _ _ _ ih
· simp only [Functor.map, LazyList.traverse, pure, Equiv.map, listEquivLazyList,
Equiv.coe_fn_symm_mk, toList, Equiv.coe_fn_mk, ofList]
· simpa only [Equiv.map, Functor.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk,
LazyList.traverse, Seq.seq, toList, ofList, cons.injEq, true_and]
· ext; apply ih
· simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
Functor.mapConst]
induction' xs using LazyList.rec with _ _ _ _ ih
· simp only [LazyList.traverse, pure, Functor.map, toList, ofList]
· simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]
· congr; apply ih
· simp only [traverse, Equiv.traverse, listEquivLazyList, Equiv.coe_fn_mk, Equiv.coe_fn_symm_mk]
induction' xs using LazyList.rec with _ tl ih _ ih
· simp only [LazyList.traverse, toList, List.traverse, map_pure, ofList]
· replace ih : tl.get.traverse f = ofList <$> tl.get.toList.traverse f := ih
simp [traverse.eq_2, ih, Functor.map_map, seq_map_assoc, toList, List.traverse, map_seq,
Function.comp, Thunk.pure, ofList]
· apply ih
def init {α} : LazyList α → LazyList α
| LazyList.nil => LazyList.nil
| LazyList.cons x xs =>
let xs' := xs.get
match xs' with
| LazyList.nil => LazyList.nil
| LazyList.cons _ _ => LazyList.cons x (init xs')
#align lazy_list.init LazyList.init
def find {α} (p : α → Prop) [DecidablePred p] : LazyList α → Option α
| nil => none
| cons h t => if p h then some h else t.get.find p
#align lazy_list.find LazyList.find
def interleave {α} : LazyList α → LazyList α → LazyList α
| LazyList.nil, xs => xs
| a@(LazyList.cons _ _), LazyList.nil => a
| LazyList.cons x xs, LazyList.cons y ys =>
LazyList.cons x (LazyList.cons y (interleave xs.get ys.get))
#align lazy_list.interleave LazyList.interleave
def interleaveAll {α} : List (LazyList α) → LazyList α
| [] => LazyList.nil
| x :: xs => interleave x (interleaveAll xs)
#align lazy_list.interleave_all LazyList.interleaveAll
protected def bind {α β} : LazyList α → (α → LazyList β) → LazyList β
| LazyList.nil, _ => LazyList.nil
| LazyList.cons x xs, f => (f x).append (xs.get.bind f)
#align lazy_list.bind LazyList.bind
def reverse {α} (xs : LazyList α) : LazyList α :=
ofList xs.toList.reverse
#align lazy_list.reverse LazyList.reverse
instance : Monad LazyList where
pure := @LazyList.singleton
bind := @LazyList.bind
-- Porting note: Added `Thunk.pure` to definition.
theorem append_nil {α} (xs : LazyList α) : xs.append (Thunk.pure LazyList.nil) = xs := by
induction' xs using LazyList.rec with _ _ _ _ ih
· simp only [Thunk.pure, append, Thunk.get]
· simpa only [append, cons.injEq, true_and]
· ext; apply ih
#align lazy_list.append_nil LazyList.append_nil
| Mathlib/Data/LazyList/Basic.lean | 150 | 155 | theorem append_assoc {α} (xs ys zs : LazyList α) :
(xs.append ys).append zs = xs.append (ys.append zs) := by |
induction' xs using LazyList.rec with _ _ _ _ ih
· simp only [append, Thunk.get]
· simpa only [append, cons.injEq, true_and]
· ext; apply ih
|
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
protected def product (s : Finset α) (t : Finset β) : Finset (α × β) :=
⟨_, s.nodup.product t.nodup⟩
#align finset.product Finset.product
instance instSProd : SProd (Finset α) (Finset β) (Finset (α × β)) where
sprod := Finset.product
@[simp]
theorem product_val : (s ×ˢ t).1 = s.1 ×ˢ t.1 :=
rfl
#align finset.product_val Finset.product_val
@[simp]
theorem mem_product {p : α × β} : p ∈ s ×ˢ t ↔ p.1 ∈ s ∧ p.2 ∈ t :=
Multiset.mem_product
#align finset.mem_product Finset.mem_product
theorem mk_mem_product (ha : a ∈ s) (hb : b ∈ t) : (a, b) ∈ s ×ˢ t :=
mem_product.2 ⟨ha, hb⟩
#align finset.mk_mem_product Finset.mk_mem_product
@[simp, norm_cast]
theorem coe_product (s : Finset α) (t : Finset β) :
(↑(s ×ˢ t) : Set (α × β)) = (s : Set α) ×ˢ t :=
Set.ext fun _ => Finset.mem_product
#align finset.coe_product Finset.coe_product
theorem subset_product_image_fst [DecidableEq α] : (s ×ˢ t).image Prod.fst ⊆ s := fun i => by
simp (config := { contextual := true }) [mem_image]
#align finset.subset_product_image_fst Finset.subset_product_image_fst
theorem subset_product_image_snd [DecidableEq β] : (s ×ˢ t).image Prod.snd ⊆ t := fun i => by
simp (config := { contextual := true }) [mem_image]
#align finset.subset_product_image_snd Finset.subset_product_image_snd
theorem product_image_fst [DecidableEq α] (ht : t.Nonempty) : (s ×ˢ t).image Prod.fst = s := by
ext i
simp [mem_image, ht.exists_mem]
#align finset.product_image_fst Finset.product_image_fst
theorem product_image_snd [DecidableEq β] (ht : s.Nonempty) : (s ×ˢ t).image Prod.snd = t := by
ext i
simp [mem_image, ht.exists_mem]
#align finset.product_image_snd Finset.product_image_snd
theorem subset_product [DecidableEq α] [DecidableEq β] {s : Finset (α × β)} :
s ⊆ s.image Prod.fst ×ˢ s.image Prod.snd := fun _ hp =>
mem_product.2 ⟨mem_image_of_mem _ hp, mem_image_of_mem _ hp⟩
#align finset.subset_product Finset.subset_product
@[gcongr]
theorem product_subset_product (hs : s ⊆ s') (ht : t ⊆ t') : s ×ˢ t ⊆ s' ×ˢ t' := fun ⟨_, _⟩ h =>
mem_product.2 ⟨hs (mem_product.1 h).1, ht (mem_product.1 h).2⟩
#align finset.product_subset_product Finset.product_subset_product
@[gcongr]
theorem product_subset_product_left (hs : s ⊆ s') : s ×ˢ t ⊆ s' ×ˢ t :=
product_subset_product hs (Subset.refl _)
#align finset.product_subset_product_left Finset.product_subset_product_left
@[gcongr]
theorem product_subset_product_right (ht : t ⊆ t') : s ×ˢ t ⊆ s ×ˢ t' :=
product_subset_product (Subset.refl _) ht
#align finset.product_subset_product_right Finset.product_subset_product_right
theorem map_swap_product (s : Finset α) (t : Finset β) :
(t ×ˢ s).map ⟨Prod.swap, Prod.swap_injective⟩ = s ×ˢ t :=
coe_injective <| by
push_cast
exact Set.image_swap_prod _ _
#align finset.map_swap_product Finset.map_swap_product
@[simp]
theorem image_swap_product [DecidableEq (α × β)] (s : Finset α) (t : Finset β) :
(t ×ˢ s).image Prod.swap = s ×ˢ t :=
coe_injective <| by
push_cast
exact Set.image_swap_prod _ _
#align finset.image_swap_product Finset.image_swap_product
theorem product_eq_biUnion [DecidableEq (α × β)] (s : Finset α) (t : Finset β) :
s ×ˢ t = s.biUnion fun a => t.image fun b => (a, b) :=
ext fun ⟨x, y⟩ => by
simp only [mem_product, mem_biUnion, mem_image, exists_prop, Prod.mk.inj_iff, and_left_comm,
exists_and_left, exists_eq_right, exists_eq_left]
#align finset.product_eq_bUnion Finset.product_eq_biUnion
theorem product_eq_biUnion_right [DecidableEq (α × β)] (s : Finset α) (t : Finset β) :
s ×ˢ t = t.biUnion fun b => s.image fun a => (a, b) :=
ext fun ⟨x, y⟩ => by
simp only [mem_product, mem_biUnion, mem_image, exists_prop, Prod.mk.inj_iff, and_left_comm,
exists_and_left, exists_eq_right, exists_eq_left]
#align finset.product_eq_bUnion_right Finset.product_eq_biUnion_right
@[simp]
theorem product_biUnion [DecidableEq γ] (s : Finset α) (t : Finset β) (f : α × β → Finset γ) :
(s ×ˢ t).biUnion f = s.biUnion fun a => t.biUnion fun b => f (a, b) := by
classical simp_rw [product_eq_biUnion, biUnion_biUnion, image_biUnion]
#align finset.product_bUnion Finset.product_biUnion
@[simp]
theorem card_product (s : Finset α) (t : Finset β) : card (s ×ˢ t) = card s * card t :=
Multiset.card_product _ _
#align finset.card_product Finset.card_product
lemma nontrivial_prod_iff : (s ×ˢ t).Nontrivial ↔
s.Nonempty ∧ t.Nonempty ∧ (s.Nontrivial ∨ t.Nontrivial) := by
simp_rw [← card_pos, ← one_lt_card_iff_nontrivial, card_product]; apply Nat.one_lt_mul_iff
theorem filter_product (p : α → Prop) (q : β → Prop) [DecidablePred p] [DecidablePred q] :
((s ×ˢ t).filter fun x : α × β => p x.1 ∧ q x.2) = s.filter p ×ˢ t.filter q := by
ext ⟨a, b⟩
simp [mem_filter, mem_product, decide_eq_true_eq, and_comm, and_left_comm, and_assoc]
#align finset.filter_product Finset.filter_product
| Mathlib/Data/Finset/Prod.lean | 159 | 161 | theorem filter_product_left (p : α → Prop) [DecidablePred p] :
((s ×ˢ t).filter fun x : α × β => p x.1) = s.filter p ×ˢ t := by |
simpa using filter_product p fun _ => true
|
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open scoped ComplexConjugate
abbrev GaussianInt : Type :=
Zsqrtd (-1)
#align gaussian_int GaussianInt
local notation "ℤ[i]" => GaussianInt
namespace GaussianInt
instance : Repr ℤ[i] :=
⟨fun x _ => "⟨" ++ repr x.re ++ ", " ++ repr x.im ++ "⟩"⟩
instance instCommRing : CommRing ℤ[i] :=
Zsqrtd.commRing
#align gaussian_int.comm_ring GaussianInt.instCommRing
section
attribute [-instance] Complex.instField -- Avoid making things noncomputable unnecessarily.
def toComplex : ℤ[i] →+* ℂ :=
Zsqrtd.lift ⟨I, by simp⟩
#align gaussian_int.to_complex GaussianInt.toComplex
end
instance : Coe ℤ[i] ℂ :=
⟨toComplex⟩
theorem toComplex_def (x : ℤ[i]) : (x : ℂ) = x.re + x.im * I :=
rfl
#align gaussian_int.to_complex_def GaussianInt.toComplex_def
theorem toComplex_def' (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ) = x + y * I := by simp [toComplex_def]
#align gaussian_int.to_complex_def' GaussianInt.toComplex_def'
theorem toComplex_def₂ (x : ℤ[i]) : (x : ℂ) = ⟨x.re, x.im⟩ := by
apply Complex.ext <;> simp [toComplex_def]
#align gaussian_int.to_complex_def₂ GaussianInt.toComplex_def₂
@[simp]
theorem to_real_re (x : ℤ[i]) : ((x.re : ℤ) : ℝ) = (x : ℂ).re := by simp [toComplex_def]
#align gaussian_int.to_real_re GaussianInt.to_real_re
@[simp]
| Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 93 | 93 | theorem to_real_im (x : ℤ[i]) : ((x.im : ℤ) : ℝ) = (x : ℂ).im := by | simp [toComplex_def]
|
import Batteries.Data.RBMap.Basic
import Batteries.Tactic.SeqFocus
namespace Batteries
namespace RBNode
open RBColor
attribute [simp] All
theorem All.trivial (H : ∀ {x : α}, p x) : ∀ {t : RBNode α}, t.All p
| nil => _root_.trivial
| node .. => ⟨H, All.trivial H, All.trivial H⟩
theorem All_and {t : RBNode α} : t.All (fun a => p a ∧ q a) ↔ t.All p ∧ t.All q := by
induction t <;> simp [*, and_assoc, and_left_comm]
protected theorem cmpLT.flip (h₁ : cmpLT cmp x y) : cmpLT (flip cmp) y x :=
⟨have : TransCmp cmp := inferInstanceAs (TransCmp (flip (flip cmp))); h₁.1⟩
theorem cmpLT.trans (h₁ : cmpLT cmp x y) (h₂ : cmpLT cmp y z) : cmpLT cmp x z :=
⟨TransCmp.lt_trans h₁.1 h₂.1⟩
theorem cmpLT.trans_l {cmp x y} (H : cmpLT cmp x y) {t : RBNode α}
(h : t.All (cmpLT cmp y ·)) : t.All (cmpLT cmp x ·) := h.imp fun h => H.trans h
theorem cmpLT.trans_r {cmp x y} (H : cmpLT cmp x y) {a : RBNode α}
(h : a.All (cmpLT cmp · x)) : a.All (cmpLT cmp · y) := h.imp fun h => h.trans H
theorem cmpEq.lt_congr_left (H : cmpEq cmp x y) : cmpLT cmp x z ↔ cmpLT cmp y z :=
⟨fun ⟨h⟩ => ⟨TransCmp.cmp_congr_left H.1 ▸ h⟩, fun ⟨h⟩ => ⟨TransCmp.cmp_congr_left H.1 ▸ h⟩⟩
theorem cmpEq.lt_congr_right (H : cmpEq cmp y z) : cmpLT cmp x y ↔ cmpLT cmp x z :=
⟨fun ⟨h⟩ => ⟨TransCmp.cmp_congr_right H.1 ▸ h⟩, fun ⟨h⟩ => ⟨TransCmp.cmp_congr_right H.1 ▸ h⟩⟩
@[simp] theorem reverse_reverse (t : RBNode α) : t.reverse.reverse = t := by
induction t <;> simp [*]
theorem reverse_eq_iff {t t' : RBNode α} : t.reverse = t' ↔ t = t'.reverse := by
constructor <;> rintro rfl <;> simp
@[simp] theorem reverse_balance1 (l : RBNode α) (v : α) (r : RBNode α) :
(balance1 l v r).reverse = balance2 r.reverse v l.reverse := by
unfold balance1 balance2; split <;> simp
· rw [balance2.match_1.eq_2]; simp [reverse_eq_iff]; intros; solve_by_elim
· rw [balance2.match_1.eq_3] <;> (simp [reverse_eq_iff]; intros; solve_by_elim)
@[simp] theorem reverse_balance2 (l : RBNode α) (v : α) (r : RBNode α) :
(balance2 l v r).reverse = balance1 r.reverse v l.reverse := by
refine Eq.trans ?_ (reverse_reverse _); rw [reverse_balance1]; simp
@[simp] theorem All.reverse {t : RBNode α} : t.reverse.All p ↔ t.All p := by
induction t <;> simp [*, and_comm]
protected theorem Ordered.reverse : ∀ {t : RBNode α}, t.Ordered cmp → t.reverse.Ordered (flip cmp)
| .nil, _ => ⟨⟩
| .node .., ⟨lv, vr, hl, hr⟩ =>
⟨(All.reverse.2 vr).imp cmpLT.flip, (All.reverse.2 lv).imp cmpLT.flip, hr.reverse, hl.reverse⟩
protected theorem Balanced.reverse {t : RBNode α} : t.Balanced c n → t.reverse.Balanced c n
| .nil => .nil
| .black hl hr => .black hr.reverse hl.reverse
| .red hl hr => .red hr.reverse hl.reverse
protected theorem Ordered.balance1 {l : RBNode α} {v : α} {r : RBNode α}
(lv : l.All (cmpLT cmp · v)) (vr : r.All (cmpLT cmp v ·))
(hl : l.Ordered cmp) (hr : r.Ordered cmp) : (balance1 l v r).Ordered cmp := by
unfold balance1; split
· next a x b y c =>
have ⟨yv, _, cv⟩ := lv; have ⟨xy, yc, hx, hc⟩ := hl
exact ⟨xy, ⟨yv, yc, yv.trans_l vr⟩, hx, cv, vr, hc, hr⟩
· next a x b y c _ =>
have ⟨_, _, yv, _, cv⟩ := lv; have ⟨ax, ⟨xy, xb, _⟩, ha, by_, yc, hb, hc⟩ := hl
exact ⟨⟨xy, xy.trans_r ax, by_⟩, ⟨yv, yc, yv.trans_l vr⟩, ⟨ax, xb, ha, hb⟩, cv, vr, hc, hr⟩
· exact ⟨lv, vr, hl, hr⟩
@[simp] theorem balance1_All {l : RBNode α} {v : α} {r : RBNode α} :
(balance1 l v r).All p ↔ p v ∧ l.All p ∧ r.All p := by
unfold balance1; split <;> simp [and_assoc, and_left_comm]
protected theorem Ordered.balance2 {l : RBNode α} {v : α} {r : RBNode α}
(lv : l.All (cmpLT cmp · v)) (vr : r.All (cmpLT cmp v ·))
(hl : l.Ordered cmp) (hr : r.Ordered cmp) : (balance2 l v r).Ordered cmp := by
rw [← reverse_reverse (balance2 ..), reverse_balance2]
exact .reverse <| hr.reverse.balance1
((All.reverse.2 vr).imp cmpLT.flip) ((All.reverse.2 lv).imp cmpLT.flip) hl.reverse
@[simp] theorem balance2_All {l : RBNode α} {v : α} {r : RBNode α} :
(balance2 l v r).All p ↔ p v ∧ l.All p ∧ r.All p := by
unfold balance2; split <;> simp [and_assoc, and_left_comm]
@[simp] theorem reverse_setBlack {t : RBNode α} : (setBlack t).reverse = setBlack t.reverse := by
unfold setBlack; split <;> simp
protected theorem Ordered.setBlack {t : RBNode α} : (setBlack t).Ordered cmp ↔ t.Ordered cmp := by
unfold setBlack; split <;> simp [Ordered]
protected theorem Balanced.setBlack : t.Balanced c n → ∃ n', (setBlack t).Balanced black n'
| .nil => ⟨_, .nil⟩
| .black hl hr | .red hl hr => ⟨_, hl.black hr⟩
theorem setBlack_idem {t : RBNode α} : t.setBlack.setBlack = t.setBlack := by cases t <;> rfl
@[simp] theorem reverse_ins [inst : @OrientedCmp α cmp] {t : RBNode α} :
(ins cmp x t).reverse = ins (flip cmp) x t.reverse := by
induction t <;> [skip; (rename_i c a y b iha ihb; cases c)] <;> simp [ins, flip]
<;> rw [← inst.symm x y] <;> split <;> simp [*, Ordering.swap, iha, ihb]
protected theorem All.ins {x : α} {t : RBNode α}
(h₁ : p x) (h₂ : t.All p) : (ins cmp x t).All p := by
induction t <;> unfold ins <;> try simp [*]
split <;> cases ‹_=_› <;> split <;> simp at h₂ <;> simp [*]
protected theorem Ordered.ins : ∀ {t : RBNode α}, t.Ordered cmp → (ins cmp x t).Ordered cmp
| nil, _ => ⟨⟨⟩, ⟨⟩, ⟨⟩, ⟨⟩⟩
| node red a y b, ⟨ay, yb, ha, hb⟩ => by
unfold ins; split
· next h => exact ⟨ay.ins ⟨h⟩, yb, ha.ins, hb⟩
· next h => exact ⟨ay, yb.ins ⟨OrientedCmp.cmp_eq_gt.1 h⟩, ha, hb.ins⟩
· next h => exact (⟨
ay.imp fun ⟨h'⟩ => ⟨(TransCmp.cmp_congr_right h).trans h'⟩,
yb.imp fun ⟨h'⟩ => ⟨(TransCmp.cmp_congr_left h).trans h'⟩, ha, hb⟩)
| node black a y b, ⟨ay, yb, ha, hb⟩ => by
unfold ins; split
· next h => exact ha.ins.balance1 (ay.ins ⟨h⟩) yb hb
· next h => exact ha.balance2 ay (yb.ins ⟨OrientedCmp.cmp_eq_gt.1 h⟩) hb.ins
· next h => exact (⟨
ay.imp fun ⟨h'⟩ => ⟨(TransCmp.cmp_congr_right h).trans h'⟩,
yb.imp fun ⟨h'⟩ => ⟨(TransCmp.cmp_congr_left h).trans h'⟩, ha, hb⟩)
@[simp] theorem isRed_reverse {t : RBNode α} : t.reverse.isRed = t.isRed := by
cases t <;> simp [isRed]
@[simp] theorem reverse_insert [inst : @OrientedCmp α cmp] {t : RBNode α} :
(insert cmp t x).reverse = insert (flip cmp) t.reverse x := by
simp [insert] <;> split <;> simp
theorem insert_setBlack {t : RBNode α} :
(t.insert cmp v).setBlack = (t.ins cmp v).setBlack := by
unfold insert; split <;> simp [setBlack_idem]
protected theorem Ordered.insert (h : t.Ordered cmp) : (insert cmp t v).Ordered cmp := by
unfold RBNode.insert; split <;> simp [Ordered.setBlack, h.ins (x := v)]
inductive RedRed (p : Prop) : RBNode α → Nat → Prop where
| balanced : Balanced t c n → RedRed p t n
| redred : p → Balanced a c₁ n → Balanced b c₂ n → RedRed p (node red a x b) n
protected theorem RedRed.of_false (h : ¬p) : RedRed p t n → ∃ c, Balanced t c n
| .balanced h => ⟨_, h⟩
| .redred hp .. => nomatch h hp
protected theorem RedRed.of_red : RedRed p (node red a x b) n →
∃ c₁ c₂, Balanced a c₁ n ∧ Balanced b c₂ n
| .balanced (.red ha hb) | .redred _ ha hb => ⟨_, _, ha, hb⟩
protected theorem RedRed.imp (h : p → q) : RedRed p t n → RedRed q t n
| .balanced h => .balanced h
| .redred hp ha hb => .redred (h hp) ha hb
protected theorem RedRed.reverse : RedRed p t n → RedRed p t.reverse n
| .balanced h => .balanced h.reverse
| .redred hp ha hb => .redred hp hb.reverse ha.reverse
protected theorem RedRed.setBlack : t.RedRed p n → ∃ n', (setBlack t).Balanced black n'
| .balanced h => h.setBlack
| .redred _ hl hr => ⟨_, hl.black hr⟩
protected theorem RedRed.balance1 {l : RBNode α} {v : α} {r : RBNode α}
(hl : l.RedRed p n) (hr : r.Balanced c n) : ∃ c, (balance1 l v r).Balanced c (n + 1) := by
unfold balance1; split
· have .redred _ (.red ha hb) hc := hl; exact ⟨_, .red (.black ha hb) (.black hc hr)⟩
· have .redred _ ha (.red hb hc) := hl; exact ⟨_, .red (.black ha hb) (.black hc hr)⟩
· next H1 H2 => match hl with
| .balanced hl => exact ⟨_, .black hl hr⟩
| .redred _ (c₁ := black) (c₂ := black) ha hb => exact ⟨_, .black (.red ha hb) hr⟩
| .redred _ (c₁ := red) (.red ..) _ => cases H1 _ _ _ _ _ rfl
| .redred _ (c₂ := red) _ (.red ..) => cases H2 _ _ _ _ _ rfl
protected theorem RedRed.balance2 {l : RBNode α} {v : α} {r : RBNode α}
(hl : l.Balanced c n) (hr : r.RedRed p n) : ∃ c, (balance2 l v r).Balanced c (n + 1) :=
(hr.reverse.balance1 hl.reverse (v := v)).imp fun _ h => by simpa using h.reverse
theorem balance1_eq {l : RBNode α} {v : α} {r : RBNode α}
(hl : l.Balanced c n) : balance1 l v r = node black l v r := by
unfold balance1; split <;> first | rfl | nomatch hl
theorem balance2_eq {l : RBNode α} {v : α} {r : RBNode α}
(hr : r.Balanced c n) : balance2 l v r = node black l v r :=
(reverse_reverse _).symm.trans <| by simp [balance1_eq hr.reverse]
protected theorem Balanced.ins (cmp v) {t : RBNode α}
(h : t.Balanced c n) : (ins cmp v t).RedRed (t.isRed = red) n := by
induction h with
| nil => exact .balanced (.red .nil .nil)
| @red a n b x hl hr ihl ihr =>
unfold ins; split
· match ins cmp v a, ihl with
| _, .balanced .nil => exact .balanced (.red .nil hr)
| _, .balanced (.red ha hb) => exact .redred rfl (.red ha hb) hr
| _, .balanced (.black ha hb) => exact .balanced (.red (.black ha hb) hr)
| _, .redred h .. => cases hl <;> cases h
· match ins cmp v b, ihr with
| _, .balanced .nil => exact .balanced (.red hl .nil)
| _, .balanced (.red ha hb) => exact .redred rfl hl (.red ha hb)
| _, .balanced (.black ha hb) => exact .balanced (.red hl (.black ha hb))
| _, .redred h .. => cases hr <;> cases h
· exact .balanced (.red hl hr)
| @black a ca n b cb x hl hr ihl ihr =>
unfold ins; split
· exact have ⟨c, h⟩ := ihl.balance1 hr; .balanced h
· exact have ⟨c, h⟩ := ihr.balance2 hl; .balanced h
· exact .balanced (.black hl hr)
theorem Balanced.insert {t : RBNode α} (h : t.Balanced c n) :
∃ c' n', (insert cmp t v).Balanced c' n' := by
unfold insert; match ins cmp v t, h.ins cmp v with
| _, .balanced h => split <;> [exact ⟨_, h.setBlack⟩; exact ⟨_, _, h⟩]
| _, .redred _ ha hb => have .node red .. := t; exact ⟨_, _, .black ha hb⟩
@[simp] theorem reverse_setRed {t : RBNode α} : (setRed t).reverse = setRed t.reverse := by
unfold setRed; split <;> simp
protected theorem All.setRed {t : RBNode α} (h : t.All p) : (setRed t).All p := by
unfold setRed; split <;> simp_all
protected theorem Ordered.setRed {t : RBNode α} : (setRed t).Ordered cmp ↔ t.Ordered cmp := by
unfold setRed; split <;> simp [Ordered]
@[simp] theorem reverse_balLeft (l : RBNode α) (v : α) (r : RBNode α) :
(balLeft l v r).reverse = balRight r.reverse v l.reverse := by
unfold balLeft balRight; split
· simp
· rw [balLeft.match_2.eq_2 _ _ _ _ (by simp [reverse_eq_iff]; intros; solve_by_elim)]
split <;> simp
rw [balRight.match_1.eq_3] <;> (simp [reverse_eq_iff]; intros; solve_by_elim)
@[simp] theorem reverse_balRight (l : RBNode α) (v : α) (r : RBNode α) :
(balRight l v r).reverse = balLeft r.reverse v l.reverse := by
rw [← reverse_reverse (balLeft ..)]; simp
protected theorem All.balLeft
(hl : l.All p) (hv : p v) (hr : r.All p) : (balLeft l v r).All p := by
unfold balLeft; split <;> (try simp_all); split <;> simp_all [All.setRed]
protected theorem Ordered.balLeft {l : RBNode α} {v : α} {r : RBNode α}
(lv : l.All (cmpLT cmp · v)) (vr : r.All (cmpLT cmp v ·))
(hl : l.Ordered cmp) (hr : r.Ordered cmp) : (balLeft l v r).Ordered cmp := by
unfold balLeft; split
· exact ⟨lv, vr, hl, hr⟩
split
· exact hl.balance2 lv vr hr
· have ⟨vy, va, _⟩ := vr.2.1; have ⟨⟨yz, _, bz⟩, zc, ⟨ay, yb, ha, hb⟩, hc⟩ := hr
exact ⟨⟨vy, vy.trans_r lv, ay⟩, balance2_All.2 ⟨yz, yb, (yz.trans_l zc).setRed⟩,
⟨lv, va, hl, ha⟩, hb.balance2 bz zc.setRed (Ordered.setRed.2 hc)⟩
· exact ⟨lv, vr, hl, hr⟩
protected theorem Balanced.balLeft (hl : l.RedRed True n) (hr : r.Balanced cr (n + 1)) :
(balLeft l v r).RedRed (cr = red) (n + 1) := by
unfold balLeft; split
· next a x b => exact
let ⟨ca, cb, ha, hb⟩ := hl.of_red
match cr with
| red => .redred rfl (.black ha hb) hr
| black => .balanced (.red (.black ha hb) hr)
· next H => exact match hl with
| .redred .. => nomatch H _ _ _ rfl
| .balanced hl => match hr with
| .black ha hb =>
let ⟨c, h⟩ := RedRed.balance2 hl (.redred trivial ha hb); .balanced h
| .red (.black ha hb) (.black hc hd) =>
let ⟨c, h⟩ := RedRed.balance2 hb (.redred trivial hc hd); .redred rfl (.black hl ha) h
protected theorem All.balRight
(hl : l.All p) (hv : p v) (hr : r.All p) : (balRight l v r).All p :=
All.reverse.1 <| reverse_balRight .. ▸ (All.reverse.2 hr).balLeft hv (All.reverse.2 hl)
protected theorem Ordered.balRight {l : RBNode α} {v : α} {r : RBNode α}
(lv : l.All (cmpLT cmp · v)) (vr : r.All (cmpLT cmp v ·))
(hl : l.Ordered cmp) (hr : r.Ordered cmp) : (balRight l v r).Ordered cmp := by
rw [← reverse_reverse (balRight ..), reverse_balRight]
exact .reverse <| hr.reverse.balLeft
((All.reverse.2 vr).imp cmpLT.flip) ((All.reverse.2 lv).imp cmpLT.flip) hl.reverse
protected theorem Balanced.balRight (hl : l.Balanced cl (n + 1)) (hr : r.RedRed True n) :
(balRight l v r).RedRed (cl = red) (n + 1) := by
rw [← reverse_reverse (balRight ..), reverse_balRight]
exact .reverse <| hl.reverse.balLeft hr.reverse
-- note: reverse_append is false!
protected theorem All.append (hl : l.All p) (hr : r.All p) : (append l r).All p := by
unfold append; split <;> try simp [*]
· have ⟨hx, ha, hb⟩ := hl; have ⟨hy, hc, hd⟩ := hr
have := hb.append hc; split <;> simp_all
· have ⟨hx, ha, hb⟩ := hl; have ⟨hy, hc, hd⟩ := hr
have := hb.append hc; split <;> simp_all [All.balLeft]
· simp_all [hl.append hr.2.1]
· simp_all [hl.2.2.append hr]
termination_by l.size + r.size
protected theorem Ordered.append {l : RBNode α} {v : α} {r : RBNode α}
(lv : l.All (cmpLT cmp · v)) (vr : r.All (cmpLT cmp v ·))
(hl : l.Ordered cmp) (hr : r.Ordered cmp) : (append l r).Ordered cmp := by
unfold append; split
· exact hr
· exact hl
· have ⟨xv, _, bv⟩ := lv; have ⟨ax, xb, ha, hb⟩ := hl
have ⟨vy, vc, _⟩ := vr; have ⟨cy, yd, hc, hd⟩ := hr
have : _ ∧ _ ∧ _ := ⟨hb.append bv vc hc, xb.append (xv.trans_l vc), (vy.trans_r bv).append cy⟩
split
· next H =>
have ⟨⟨b'z, c'z, hb', hc'⟩, ⟨xz, xb', _⟩, zy, _, c'y⟩ := H ▸ this
have az := xz.trans_r ax; have zd := zy.trans_l yd
exact ⟨⟨xz, az, b'z⟩, ⟨zy, c'z, zd⟩, ⟨ax, xb', ha, hb'⟩, c'y, yd, hc', hd⟩
· have ⟨hbc, xbc, bcy⟩ := this; have xy := xv.trans vy
exact ⟨ax, ⟨xy, xbc, xy.trans_l yd⟩, ha, bcy, yd, hbc, hd⟩
· have ⟨xv, _, bv⟩ := lv; have ⟨ax, xb, ha, hb⟩ := hl
have ⟨vy, vc, _⟩ := vr; have ⟨cy, yd, hc, hd⟩ := hr
have : _ ∧ _ ∧ _ := ⟨hb.append bv vc hc, xb.append (xv.trans_l vc), (vy.trans_r bv).append cy⟩
split
· next H =>
have ⟨⟨b'z, c'z, hb', hc'⟩, ⟨xz, xb', _⟩, zy, _, c'y⟩ := H ▸ this
have az := xz.trans_r ax; have zd := zy.trans_l yd
exact ⟨⟨xz, az, b'z⟩, ⟨zy, c'z, zd⟩, ⟨ax, xb', ha, hb'⟩, c'y, yd, hc', hd⟩
· have ⟨hbc, xbc, bcy⟩ := this; have xy := xv.trans vy
exact ha.balLeft ax ⟨xy, xbc, xy.trans_l yd⟩ ⟨bcy, yd, hbc, hd⟩
· have ⟨vx, vb, _⟩ := vr; have ⟨bx, yc, hb, hc⟩ := hr
exact ⟨(vx.trans_r lv).append bx, yc, hl.append lv vb hb, hc⟩
· have ⟨xv, _, bv⟩ := lv; have ⟨ax, xb, ha, hb⟩ := hl
exact ⟨ax, xb.append (xv.trans_l vr), ha, hb.append bv vr hr⟩
termination_by l.size + r.size
protected theorem Balanced.append {l r : RBNode α}
(hl : l.Balanced c₁ n) (hr : r.Balanced c₂ n) :
(l.append r).RedRed (c₁ = black → c₂ ≠ black) n := by
unfold append; split
· exact .balanced hr
· exact .balanced hl
· next b c _ _ =>
have .red ha hb := hl; have .red hc hd := hr
have ⟨_, IH⟩ := (hb.append hc).of_false (· rfl rfl); split
· next e =>
have .red hb' hc' := e ▸ IH
exact .redred nofun (.red ha hb') (.red hc' hd)
· next bcc _ H =>
match bcc, append b c, IH, H with
| black, _, IH, _ => exact .redred nofun ha (.red IH hd)
| red, _, .red .., H => cases H _ _ _ rfl
· next b c _ _ =>
have .black ha hb := hl; have .black hc hd := hr
have IH := hb.append hc; split
· next e => match e ▸ IH with
| .balanced (.red hb' hc') | .redred _ hb' hc' =>
exact .balanced (.red (.black ha hb') (.black hc' hd))
· next H =>
match append b c, IH, H with
| bc, .balanced hbc, _ =>
unfold balLeft; split
· have .red ha' hb' := ha
exact .balanced (.red (.black ha' hb') (.black hbc hd))
· exact have ⟨c, h⟩ := RedRed.balance2 ha (.redred trivial hbc hd); .balanced h
| _, .redred .., H => cases H _ _ _ rfl
· have .red hc hd := hr; have IH := hl.append hc
have .black ha hb := hl; have ⟨c, IH⟩ := IH.of_false (· rfl rfl)
exact .redred nofun IH hd
· have .red ha hb := hl; have IH := hb.append hr
have .black hc hd := hr; have ⟨c, IH⟩ := IH.of_false (· rfl rfl)
exact .redred nofun ha IH
termination_by l.size + r.size
def DelProp (p : RBColor) (t : RBNode α) (n : Nat) : Prop :=
match p with
| black => ∃ n', n = n' + 1 ∧ RedRed True t n'
| red => ∃ c, Balanced t c n
| .lake/packages/batteries/Batteries/Data/RBMap/WF.lean | 441 | 445 | theorem DelProp.redred (h : DelProp c t n) : ∃ n', RedRed (c = black) t n' := by |
unfold DelProp at h
exact match c, h with
| red, ⟨_, h⟩ => ⟨_, .balanced h⟩
| black, ⟨_, _, h⟩ => ⟨_, h.imp fun _ => rfl⟩
|
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open scoped Classical
open Set
variable {ι : Sort*} {α : Type*} (s : Set α)
namespace Set.Iic
variable [CompleteLattice α] {a : α}
instance instCompleteLattice : CompleteLattice (Iic a) where
sSup S := ⟨sSup ((↑) '' S), by simpa using fun b hb _ ↦ hb⟩
sInf S := ⟨a ⊓ sInf ((↑) '' S), by simp⟩
le_sSup S b hb := le_sSup <| mem_image_of_mem Subtype.val hb
sSup_le S b hb := sSup_le <| fun c' ⟨c, hc, hc'⟩ ↦ hc' ▸ hb c hc
sInf_le S b hb := inf_le_of_right_le <| sInf_le <| mem_image_of_mem Subtype.val hb
le_sInf S b hb := le_inf_iff.mpr ⟨b.property, le_sInf fun d' ⟨d, hd, hd'⟩ ↦ hd' ▸ hb d hd⟩
le_top := by simp
bot_le := by simp
variable (S : Set <| Iic a) (f : ι → Iic a) (p : ι → Prop)
@[simp] theorem coe_sSup : (↑(sSup S) : α) = sSup ((↑) '' S) := rfl
@[simp] theorem coe_iSup : (↑(⨆ i, f i) : α) = ⨆ i, (f i : α) := by
rw [iSup, coe_sSup]; congr; ext; simp
| Mathlib/Order/CompleteLatticeIntervals.lean | 265 | 265 | theorem coe_biSup : (↑(⨆ i, ⨆ (_ : p i), f i) : α) = ⨆ i, ⨆ (_ : p i), (f i : α) := by | simp
|
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import algebra.order.to_interval_mod from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829ce42efec"
noncomputable section
section LinearOrderedAddCommGroup
variable {α : Type*} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p)
{a b c : α} {n : ℤ}
def toIcoDiv (a b : α) : ℤ :=
(existsUnique_sub_zsmul_mem_Ico hp b a).choose
#align to_Ico_div toIcoDiv
theorem sub_toIcoDiv_zsmul_mem_Ico (a b : α) : b - toIcoDiv hp a b • p ∈ Set.Ico a (a + p) :=
(existsUnique_sub_zsmul_mem_Ico hp b a).choose_spec.1
#align sub_to_Ico_div_zsmul_mem_Ico sub_toIcoDiv_zsmul_mem_Ico
theorem toIcoDiv_eq_of_sub_zsmul_mem_Ico (h : b - n • p ∈ Set.Ico a (a + p)) :
toIcoDiv hp a b = n :=
((existsUnique_sub_zsmul_mem_Ico hp b a).choose_spec.2 _ h).symm
#align to_Ico_div_eq_of_sub_zsmul_mem_Ico toIcoDiv_eq_of_sub_zsmul_mem_Ico
def toIocDiv (a b : α) : ℤ :=
(existsUnique_sub_zsmul_mem_Ioc hp b a).choose
#align to_Ioc_div toIocDiv
theorem sub_toIocDiv_zsmul_mem_Ioc (a b : α) : b - toIocDiv hp a b • p ∈ Set.Ioc a (a + p) :=
(existsUnique_sub_zsmul_mem_Ioc hp b a).choose_spec.1
#align sub_to_Ioc_div_zsmul_mem_Ioc sub_toIocDiv_zsmul_mem_Ioc
theorem toIocDiv_eq_of_sub_zsmul_mem_Ioc (h : b - n • p ∈ Set.Ioc a (a + p)) :
toIocDiv hp a b = n :=
((existsUnique_sub_zsmul_mem_Ioc hp b a).choose_spec.2 _ h).symm
#align to_Ioc_div_eq_of_sub_zsmul_mem_Ioc toIocDiv_eq_of_sub_zsmul_mem_Ioc
def toIcoMod (a b : α) : α :=
b - toIcoDiv hp a b • p
#align to_Ico_mod toIcoMod
def toIocMod (a b : α) : α :=
b - toIocDiv hp a b • p
#align to_Ioc_mod toIocMod
theorem toIcoMod_mem_Ico (a b : α) : toIcoMod hp a b ∈ Set.Ico a (a + p) :=
sub_toIcoDiv_zsmul_mem_Ico hp a b
#align to_Ico_mod_mem_Ico toIcoMod_mem_Ico
theorem toIcoMod_mem_Ico' (b : α) : toIcoMod hp 0 b ∈ Set.Ico 0 p := by
convert toIcoMod_mem_Ico hp 0 b
exact (zero_add p).symm
#align to_Ico_mod_mem_Ico' toIcoMod_mem_Ico'
theorem toIocMod_mem_Ioc (a b : α) : toIocMod hp a b ∈ Set.Ioc a (a + p) :=
sub_toIocDiv_zsmul_mem_Ioc hp a b
#align to_Ioc_mod_mem_Ioc toIocMod_mem_Ioc
theorem left_le_toIcoMod (a b : α) : a ≤ toIcoMod hp a b :=
(Set.mem_Ico.1 (toIcoMod_mem_Ico hp a b)).1
#align left_le_to_Ico_mod left_le_toIcoMod
theorem left_lt_toIocMod (a b : α) : a < toIocMod hp a b :=
(Set.mem_Ioc.1 (toIocMod_mem_Ioc hp a b)).1
#align left_lt_to_Ioc_mod left_lt_toIocMod
theorem toIcoMod_lt_right (a b : α) : toIcoMod hp a b < a + p :=
(Set.mem_Ico.1 (toIcoMod_mem_Ico hp a b)).2
#align to_Ico_mod_lt_right toIcoMod_lt_right
theorem toIocMod_le_right (a b : α) : toIocMod hp a b ≤ a + p :=
(Set.mem_Ioc.1 (toIocMod_mem_Ioc hp a b)).2
#align to_Ioc_mod_le_right toIocMod_le_right
@[simp]
theorem self_sub_toIcoDiv_zsmul (a b : α) : b - toIcoDiv hp a b • p = toIcoMod hp a b :=
rfl
#align self_sub_to_Ico_div_zsmul self_sub_toIcoDiv_zsmul
@[simp]
theorem self_sub_toIocDiv_zsmul (a b : α) : b - toIocDiv hp a b • p = toIocMod hp a b :=
rfl
#align self_sub_to_Ioc_div_zsmul self_sub_toIocDiv_zsmul
@[simp]
theorem toIcoDiv_zsmul_sub_self (a b : α) : toIcoDiv hp a b • p - b = -toIcoMod hp a b := by
rw [toIcoMod, neg_sub]
#align to_Ico_div_zsmul_sub_self toIcoDiv_zsmul_sub_self
@[simp]
theorem toIocDiv_zsmul_sub_self (a b : α) : toIocDiv hp a b • p - b = -toIocMod hp a b := by
rw [toIocMod, neg_sub]
#align to_Ioc_div_zsmul_sub_self toIocDiv_zsmul_sub_self
@[simp]
theorem toIcoMod_sub_self (a b : α) : toIcoMod hp a b - b = -toIcoDiv hp a b • p := by
rw [toIcoMod, sub_sub_cancel_left, neg_smul]
#align to_Ico_mod_sub_self toIcoMod_sub_self
@[simp]
theorem toIocMod_sub_self (a b : α) : toIocMod hp a b - b = -toIocDiv hp a b • p := by
rw [toIocMod, sub_sub_cancel_left, neg_smul]
#align to_Ioc_mod_sub_self toIocMod_sub_self
@[simp]
theorem self_sub_toIcoMod (a b : α) : b - toIcoMod hp a b = toIcoDiv hp a b • p := by
rw [toIcoMod, sub_sub_cancel]
#align self_sub_to_Ico_mod self_sub_toIcoMod
@[simp]
theorem self_sub_toIocMod (a b : α) : b - toIocMod hp a b = toIocDiv hp a b • p := by
rw [toIocMod, sub_sub_cancel]
#align self_sub_to_Ioc_mod self_sub_toIocMod
@[simp]
theorem toIcoMod_add_toIcoDiv_zsmul (a b : α) : toIcoMod hp a b + toIcoDiv hp a b • p = b := by
rw [toIcoMod, sub_add_cancel]
#align to_Ico_mod_add_to_Ico_div_zsmul toIcoMod_add_toIcoDiv_zsmul
@[simp]
theorem toIocMod_add_toIocDiv_zsmul (a b : α) : toIocMod hp a b + toIocDiv hp a b • p = b := by
rw [toIocMod, sub_add_cancel]
#align to_Ioc_mod_add_to_Ioc_div_zsmul toIocMod_add_toIocDiv_zsmul
@[simp]
theorem toIcoDiv_zsmul_sub_toIcoMod (a b : α) : toIcoDiv hp a b • p + toIcoMod hp a b = b := by
rw [add_comm, toIcoMod_add_toIcoDiv_zsmul]
#align to_Ico_div_zsmul_sub_to_Ico_mod toIcoDiv_zsmul_sub_toIcoMod
@[simp]
theorem toIocDiv_zsmul_sub_toIocMod (a b : α) : toIocDiv hp a b • p + toIocMod hp a b = b := by
rw [add_comm, toIocMod_add_toIocDiv_zsmul]
#align to_Ioc_div_zsmul_sub_to_Ioc_mod toIocDiv_zsmul_sub_toIocMod
theorem toIcoMod_eq_iff : toIcoMod hp a b = c ↔ c ∈ Set.Ico a (a + p) ∧ ∃ z : ℤ, b = c + z • p := by
refine
⟨fun h =>
⟨h ▸ toIcoMod_mem_Ico hp a b, toIcoDiv hp a b, h ▸ (toIcoMod_add_toIcoDiv_zsmul _ _ _).symm⟩,
?_⟩
simp_rw [← @sub_eq_iff_eq_add]
rintro ⟨hc, n, rfl⟩
rw [← toIcoDiv_eq_of_sub_zsmul_mem_Ico hp hc, toIcoMod]
#align to_Ico_mod_eq_iff toIcoMod_eq_iff
theorem toIocMod_eq_iff : toIocMod hp a b = c ↔ c ∈ Set.Ioc a (a + p) ∧ ∃ z : ℤ, b = c + z • p := by
refine
⟨fun h =>
⟨h ▸ toIocMod_mem_Ioc hp a b, toIocDiv hp a b, h ▸ (toIocMod_add_toIocDiv_zsmul hp _ _).symm⟩,
?_⟩
simp_rw [← @sub_eq_iff_eq_add]
rintro ⟨hc, n, rfl⟩
rw [← toIocDiv_eq_of_sub_zsmul_mem_Ioc hp hc, toIocMod]
#align to_Ioc_mod_eq_iff toIocMod_eq_iff
@[simp]
theorem toIcoDiv_apply_left (a : α) : toIcoDiv hp a a = 0 :=
toIcoDiv_eq_of_sub_zsmul_mem_Ico hp <| by simp [hp]
#align to_Ico_div_apply_left toIcoDiv_apply_left
@[simp]
theorem toIocDiv_apply_left (a : α) : toIocDiv hp a a = -1 :=
toIocDiv_eq_of_sub_zsmul_mem_Ioc hp <| by simp [hp]
#align to_Ioc_div_apply_left toIocDiv_apply_left
@[simp]
theorem toIcoMod_apply_left (a : α) : toIcoMod hp a a = a := by
rw [toIcoMod_eq_iff hp, Set.left_mem_Ico]
exact ⟨lt_add_of_pos_right _ hp, 0, by simp⟩
#align to_Ico_mod_apply_left toIcoMod_apply_left
@[simp]
theorem toIocMod_apply_left (a : α) : toIocMod hp a a = a + p := by
rw [toIocMod_eq_iff hp, Set.right_mem_Ioc]
exact ⟨lt_add_of_pos_right _ hp, -1, by simp⟩
#align to_Ioc_mod_apply_left toIocMod_apply_left
theorem toIcoDiv_apply_right (a : α) : toIcoDiv hp a (a + p) = 1 :=
toIcoDiv_eq_of_sub_zsmul_mem_Ico hp <| by simp [hp]
#align to_Ico_div_apply_right toIcoDiv_apply_right
theorem toIocDiv_apply_right (a : α) : toIocDiv hp a (a + p) = 0 :=
toIocDiv_eq_of_sub_zsmul_mem_Ioc hp <| by simp [hp]
#align to_Ioc_div_apply_right toIocDiv_apply_right
theorem toIcoMod_apply_right (a : α) : toIcoMod hp a (a + p) = a := by
rw [toIcoMod_eq_iff hp, Set.left_mem_Ico]
exact ⟨lt_add_of_pos_right _ hp, 1, by simp⟩
#align to_Ico_mod_apply_right toIcoMod_apply_right
theorem toIocMod_apply_right (a : α) : toIocMod hp a (a + p) = a + p := by
rw [toIocMod_eq_iff hp, Set.right_mem_Ioc]
exact ⟨lt_add_of_pos_right _ hp, 0, by simp⟩
#align to_Ioc_mod_apply_right toIocMod_apply_right
@[simp]
theorem toIcoDiv_add_zsmul (a b : α) (m : ℤ) : toIcoDiv hp a (b + m • p) = toIcoDiv hp a b + m :=
toIcoDiv_eq_of_sub_zsmul_mem_Ico hp <| by
simpa only [add_smul, add_sub_add_right_eq_sub] using sub_toIcoDiv_zsmul_mem_Ico hp a b
#align to_Ico_div_add_zsmul toIcoDiv_add_zsmul
@[simp]
theorem toIcoDiv_add_zsmul' (a b : α) (m : ℤ) :
toIcoDiv hp (a + m • p) b = toIcoDiv hp a b - m := by
refine toIcoDiv_eq_of_sub_zsmul_mem_Ico _ ?_
rw [sub_smul, ← sub_add, add_right_comm]
simpa using sub_toIcoDiv_zsmul_mem_Ico hp a b
#align to_Ico_div_add_zsmul' toIcoDiv_add_zsmul'
@[simp]
theorem toIocDiv_add_zsmul (a b : α) (m : ℤ) : toIocDiv hp a (b + m • p) = toIocDiv hp a b + m :=
toIocDiv_eq_of_sub_zsmul_mem_Ioc hp <| by
simpa only [add_smul, add_sub_add_right_eq_sub] using sub_toIocDiv_zsmul_mem_Ioc hp a b
#align to_Ioc_div_add_zsmul toIocDiv_add_zsmul
@[simp]
theorem toIocDiv_add_zsmul' (a b : α) (m : ℤ) :
toIocDiv hp (a + m • p) b = toIocDiv hp a b - m := by
refine toIocDiv_eq_of_sub_zsmul_mem_Ioc _ ?_
rw [sub_smul, ← sub_add, add_right_comm]
simpa using sub_toIocDiv_zsmul_mem_Ioc hp a b
#align to_Ioc_div_add_zsmul' toIocDiv_add_zsmul'
@[simp]
theorem toIcoDiv_zsmul_add (a b : α) (m : ℤ) : toIcoDiv hp a (m • p + b) = m + toIcoDiv hp a b := by
rw [add_comm, toIcoDiv_add_zsmul, add_comm]
#align to_Ico_div_zsmul_add toIcoDiv_zsmul_add
@[simp]
theorem toIocDiv_zsmul_add (a b : α) (m : ℤ) : toIocDiv hp a (m • p + b) = m + toIocDiv hp a b := by
rw [add_comm, toIocDiv_add_zsmul, add_comm]
#align to_Ioc_div_zsmul_add toIocDiv_zsmul_add
@[simp]
theorem toIcoDiv_sub_zsmul (a b : α) (m : ℤ) : toIcoDiv hp a (b - m • p) = toIcoDiv hp a b - m := by
rw [sub_eq_add_neg, ← neg_smul, toIcoDiv_add_zsmul, sub_eq_add_neg]
#align to_Ico_div_sub_zsmul toIcoDiv_sub_zsmul
@[simp]
theorem toIcoDiv_sub_zsmul' (a b : α) (m : ℤ) :
toIcoDiv hp (a - m • p) b = toIcoDiv hp a b + m := by
rw [sub_eq_add_neg, ← neg_smul, toIcoDiv_add_zsmul', sub_neg_eq_add]
#align to_Ico_div_sub_zsmul' toIcoDiv_sub_zsmul'
@[simp]
theorem toIocDiv_sub_zsmul (a b : α) (m : ℤ) : toIocDiv hp a (b - m • p) = toIocDiv hp a b - m := by
rw [sub_eq_add_neg, ← neg_smul, toIocDiv_add_zsmul, sub_eq_add_neg]
#align to_Ioc_div_sub_zsmul toIocDiv_sub_zsmul
@[simp]
theorem toIocDiv_sub_zsmul' (a b : α) (m : ℤ) :
toIocDiv hp (a - m • p) b = toIocDiv hp a b + m := by
rw [sub_eq_add_neg, ← neg_smul, toIocDiv_add_zsmul', sub_neg_eq_add]
#align to_Ioc_div_sub_zsmul' toIocDiv_sub_zsmul'
@[simp]
theorem toIcoDiv_add_right (a b : α) : toIcoDiv hp a (b + p) = toIcoDiv hp a b + 1 := by
simpa only [one_zsmul] using toIcoDiv_add_zsmul hp a b 1
#align to_Ico_div_add_right toIcoDiv_add_right
@[simp]
theorem toIcoDiv_add_right' (a b : α) : toIcoDiv hp (a + p) b = toIcoDiv hp a b - 1 := by
simpa only [one_zsmul] using toIcoDiv_add_zsmul' hp a b 1
#align to_Ico_div_add_right' toIcoDiv_add_right'
@[simp]
theorem toIocDiv_add_right (a b : α) : toIocDiv hp a (b + p) = toIocDiv hp a b + 1 := by
simpa only [one_zsmul] using toIocDiv_add_zsmul hp a b 1
#align to_Ioc_div_add_right toIocDiv_add_right
@[simp]
theorem toIocDiv_add_right' (a b : α) : toIocDiv hp (a + p) b = toIocDiv hp a b - 1 := by
simpa only [one_zsmul] using toIocDiv_add_zsmul' hp a b 1
#align to_Ioc_div_add_right' toIocDiv_add_right'
@[simp]
theorem toIcoDiv_add_left (a b : α) : toIcoDiv hp a (p + b) = toIcoDiv hp a b + 1 := by
rw [add_comm, toIcoDiv_add_right]
#align to_Ico_div_add_left toIcoDiv_add_left
@[simp]
theorem toIcoDiv_add_left' (a b : α) : toIcoDiv hp (p + a) b = toIcoDiv hp a b - 1 := by
rw [add_comm, toIcoDiv_add_right']
#align to_Ico_div_add_left' toIcoDiv_add_left'
@[simp]
theorem toIocDiv_add_left (a b : α) : toIocDiv hp a (p + b) = toIocDiv hp a b + 1 := by
rw [add_comm, toIocDiv_add_right]
#align to_Ioc_div_add_left toIocDiv_add_left
@[simp]
theorem toIocDiv_add_left' (a b : α) : toIocDiv hp (p + a) b = toIocDiv hp a b - 1 := by
rw [add_comm, toIocDiv_add_right']
#align to_Ioc_div_add_left' toIocDiv_add_left'
@[simp]
theorem toIcoDiv_sub (a b : α) : toIcoDiv hp a (b - p) = toIcoDiv hp a b - 1 := by
simpa only [one_zsmul] using toIcoDiv_sub_zsmul hp a b 1
#align to_Ico_div_sub toIcoDiv_sub
@[simp]
theorem toIcoDiv_sub' (a b : α) : toIcoDiv hp (a - p) b = toIcoDiv hp a b + 1 := by
simpa only [one_zsmul] using toIcoDiv_sub_zsmul' hp a b 1
#align to_Ico_div_sub' toIcoDiv_sub'
@[simp]
theorem toIocDiv_sub (a b : α) : toIocDiv hp a (b - p) = toIocDiv hp a b - 1 := by
simpa only [one_zsmul] using toIocDiv_sub_zsmul hp a b 1
#align to_Ioc_div_sub toIocDiv_sub
@[simp]
theorem toIocDiv_sub' (a b : α) : toIocDiv hp (a - p) b = toIocDiv hp a b + 1 := by
simpa only [one_zsmul] using toIocDiv_sub_zsmul' hp a b 1
#align to_Ioc_div_sub' toIocDiv_sub'
theorem toIcoDiv_sub_eq_toIcoDiv_add (a b c : α) :
toIcoDiv hp a (b - c) = toIcoDiv hp (a + c) b := by
apply toIcoDiv_eq_of_sub_zsmul_mem_Ico
rw [← sub_right_comm, Set.sub_mem_Ico_iff_left, add_right_comm]
exact sub_toIcoDiv_zsmul_mem_Ico hp (a + c) b
#align to_Ico_div_sub_eq_to_Ico_div_add toIcoDiv_sub_eq_toIcoDiv_add
theorem toIocDiv_sub_eq_toIocDiv_add (a b c : α) :
toIocDiv hp a (b - c) = toIocDiv hp (a + c) b := by
apply toIocDiv_eq_of_sub_zsmul_mem_Ioc
rw [← sub_right_comm, Set.sub_mem_Ioc_iff_left, add_right_comm]
exact sub_toIocDiv_zsmul_mem_Ioc hp (a + c) b
#align to_Ioc_div_sub_eq_to_Ioc_div_add toIocDiv_sub_eq_toIocDiv_add
theorem toIcoDiv_sub_eq_toIcoDiv_add' (a b c : α) :
toIcoDiv hp (a - c) b = toIcoDiv hp a (b + c) := by
rw [← sub_neg_eq_add, toIcoDiv_sub_eq_toIcoDiv_add, sub_eq_add_neg]
#align to_Ico_div_sub_eq_to_Ico_div_add' toIcoDiv_sub_eq_toIcoDiv_add'
theorem toIocDiv_sub_eq_toIocDiv_add' (a b c : α) :
toIocDiv hp (a - c) b = toIocDiv hp a (b + c) := by
rw [← sub_neg_eq_add, toIocDiv_sub_eq_toIocDiv_add, sub_eq_add_neg]
#align to_Ioc_div_sub_eq_to_Ioc_div_add' toIocDiv_sub_eq_toIocDiv_add'
theorem toIcoDiv_neg (a b : α) : toIcoDiv hp a (-b) = -(toIocDiv hp (-a) b + 1) := by
suffices toIcoDiv hp a (-b) = -toIocDiv hp (-(a + p)) b by
rwa [neg_add, ← sub_eq_add_neg, toIocDiv_sub_eq_toIocDiv_add', toIocDiv_add_right] at this
rw [← neg_eq_iff_eq_neg, eq_comm]
apply toIocDiv_eq_of_sub_zsmul_mem_Ioc
obtain ⟨hc, ho⟩ := sub_toIcoDiv_zsmul_mem_Ico hp a (-b)
rw [← neg_lt_neg_iff, neg_sub' (-b), neg_neg, ← neg_smul] at ho
rw [← neg_le_neg_iff, neg_sub' (-b), neg_neg, ← neg_smul] at hc
refine ⟨ho, hc.trans_eq ?_⟩
rw [neg_add, neg_add_cancel_right]
#align to_Ico_div_neg toIcoDiv_neg
theorem toIcoDiv_neg' (a b : α) : toIcoDiv hp (-a) b = -(toIocDiv hp a (-b) + 1) := by
simpa only [neg_neg] using toIcoDiv_neg hp (-a) (-b)
#align to_Ico_div_neg' toIcoDiv_neg'
theorem toIocDiv_neg (a b : α) : toIocDiv hp a (-b) = -(toIcoDiv hp (-a) b + 1) := by
rw [← neg_neg b, toIcoDiv_neg, neg_neg, neg_neg, neg_add', neg_neg, add_sub_cancel_right]
#align to_Ioc_div_neg toIocDiv_neg
theorem toIocDiv_neg' (a b : α) : toIocDiv hp (-a) b = -(toIcoDiv hp a (-b) + 1) := by
simpa only [neg_neg] using toIocDiv_neg hp (-a) (-b)
#align to_Ioc_div_neg' toIocDiv_neg'
@[simp]
theorem toIcoMod_add_zsmul (a b : α) (m : ℤ) : toIcoMod hp a (b + m • p) = toIcoMod hp a b := by
rw [toIcoMod, toIcoDiv_add_zsmul, toIcoMod, add_smul]
abel
#align to_Ico_mod_add_zsmul toIcoMod_add_zsmul
@[simp]
theorem toIcoMod_add_zsmul' (a b : α) (m : ℤ) :
toIcoMod hp (a + m • p) b = toIcoMod hp a b + m • p := by
simp only [toIcoMod, toIcoDiv_add_zsmul', sub_smul, sub_add]
#align to_Ico_mod_add_zsmul' toIcoMod_add_zsmul'
@[simp]
theorem toIocMod_add_zsmul (a b : α) (m : ℤ) : toIocMod hp a (b + m • p) = toIocMod hp a b := by
rw [toIocMod, toIocDiv_add_zsmul, toIocMod, add_smul]
abel
#align to_Ioc_mod_add_zsmul toIocMod_add_zsmul
@[simp]
theorem toIocMod_add_zsmul' (a b : α) (m : ℤ) :
toIocMod hp (a + m • p) b = toIocMod hp a b + m • p := by
simp only [toIocMod, toIocDiv_add_zsmul', sub_smul, sub_add]
#align to_Ioc_mod_add_zsmul' toIocMod_add_zsmul'
@[simp]
theorem toIcoMod_zsmul_add (a b : α) (m : ℤ) : toIcoMod hp a (m • p + b) = toIcoMod hp a b := by
rw [add_comm, toIcoMod_add_zsmul]
#align to_Ico_mod_zsmul_add toIcoMod_zsmul_add
@[simp]
theorem toIcoMod_zsmul_add' (a b : α) (m : ℤ) :
toIcoMod hp (m • p + a) b = m • p + toIcoMod hp a b := by
rw [add_comm, toIcoMod_add_zsmul', add_comm]
#align to_Ico_mod_zsmul_add' toIcoMod_zsmul_add'
@[simp]
theorem toIocMod_zsmul_add (a b : α) (m : ℤ) : toIocMod hp a (m • p + b) = toIocMod hp a b := by
rw [add_comm, toIocMod_add_zsmul]
#align to_Ioc_mod_zsmul_add toIocMod_zsmul_add
@[simp]
theorem toIocMod_zsmul_add' (a b : α) (m : ℤ) :
toIocMod hp (m • p + a) b = m • p + toIocMod hp a b := by
rw [add_comm, toIocMod_add_zsmul', add_comm]
#align to_Ioc_mod_zsmul_add' toIocMod_zsmul_add'
@[simp]
theorem toIcoMod_sub_zsmul (a b : α) (m : ℤ) : toIcoMod hp a (b - m • p) = toIcoMod hp a b := by
rw [sub_eq_add_neg, ← neg_smul, toIcoMod_add_zsmul]
#align to_Ico_mod_sub_zsmul toIcoMod_sub_zsmul
@[simp]
theorem toIcoMod_sub_zsmul' (a b : α) (m : ℤ) :
toIcoMod hp (a - m • p) b = toIcoMod hp a b - m • p := by
simp_rw [sub_eq_add_neg, ← neg_smul, toIcoMod_add_zsmul']
#align to_Ico_mod_sub_zsmul' toIcoMod_sub_zsmul'
@[simp]
theorem toIocMod_sub_zsmul (a b : α) (m : ℤ) : toIocMod hp a (b - m • p) = toIocMod hp a b := by
rw [sub_eq_add_neg, ← neg_smul, toIocMod_add_zsmul]
#align to_Ioc_mod_sub_zsmul toIocMod_sub_zsmul
@[simp]
theorem toIocMod_sub_zsmul' (a b : α) (m : ℤ) :
toIocMod hp (a - m • p) b = toIocMod hp a b - m • p := by
simp_rw [sub_eq_add_neg, ← neg_smul, toIocMod_add_zsmul']
#align to_Ioc_mod_sub_zsmul' toIocMod_sub_zsmul'
@[simp]
theorem toIcoMod_add_right (a b : α) : toIcoMod hp a (b + p) = toIcoMod hp a b := by
simpa only [one_zsmul] using toIcoMod_add_zsmul hp a b 1
#align to_Ico_mod_add_right toIcoMod_add_right
@[simp]
theorem toIcoMod_add_right' (a b : α) : toIcoMod hp (a + p) b = toIcoMod hp a b + p := by
simpa only [one_zsmul] using toIcoMod_add_zsmul' hp a b 1
#align to_Ico_mod_add_right' toIcoMod_add_right'
@[simp]
theorem toIocMod_add_right (a b : α) : toIocMod hp a (b + p) = toIocMod hp a b := by
simpa only [one_zsmul] using toIocMod_add_zsmul hp a b 1
#align to_Ioc_mod_add_right toIocMod_add_right
@[simp]
theorem toIocMod_add_right' (a b : α) : toIocMod hp (a + p) b = toIocMod hp a b + p := by
simpa only [one_zsmul] using toIocMod_add_zsmul' hp a b 1
#align to_Ioc_mod_add_right' toIocMod_add_right'
@[simp]
theorem toIcoMod_add_left (a b : α) : toIcoMod hp a (p + b) = toIcoMod hp a b := by
rw [add_comm, toIcoMod_add_right]
#align to_Ico_mod_add_left toIcoMod_add_left
@[simp]
theorem toIcoMod_add_left' (a b : α) : toIcoMod hp (p + a) b = p + toIcoMod hp a b := by
rw [add_comm, toIcoMod_add_right', add_comm]
#align to_Ico_mod_add_left' toIcoMod_add_left'
@[simp]
theorem toIocMod_add_left (a b : α) : toIocMod hp a (p + b) = toIocMod hp a b := by
rw [add_comm, toIocMod_add_right]
#align to_Ioc_mod_add_left toIocMod_add_left
@[simp]
theorem toIocMod_add_left' (a b : α) : toIocMod hp (p + a) b = p + toIocMod hp a b := by
rw [add_comm, toIocMod_add_right', add_comm]
#align to_Ioc_mod_add_left' toIocMod_add_left'
@[simp]
theorem toIcoMod_sub (a b : α) : toIcoMod hp a (b - p) = toIcoMod hp a b := by
simpa only [one_zsmul] using toIcoMod_sub_zsmul hp a b 1
#align to_Ico_mod_sub toIcoMod_sub
@[simp]
theorem toIcoMod_sub' (a b : α) : toIcoMod hp (a - p) b = toIcoMod hp a b - p := by
simpa only [one_zsmul] using toIcoMod_sub_zsmul' hp a b 1
#align to_Ico_mod_sub' toIcoMod_sub'
@[simp]
theorem toIocMod_sub (a b : α) : toIocMod hp a (b - p) = toIocMod hp a b := by
simpa only [one_zsmul] using toIocMod_sub_zsmul hp a b 1
#align to_Ioc_mod_sub toIocMod_sub
@[simp]
theorem toIocMod_sub' (a b : α) : toIocMod hp (a - p) b = toIocMod hp a b - p := by
simpa only [one_zsmul] using toIocMod_sub_zsmul' hp a b 1
#align to_Ioc_mod_sub' toIocMod_sub'
theorem toIcoMod_sub_eq_sub (a b c : α) : toIcoMod hp a (b - c) = toIcoMod hp (a + c) b - c := by
simp_rw [toIcoMod, toIcoDiv_sub_eq_toIcoDiv_add, sub_right_comm]
#align to_Ico_mod_sub_eq_sub toIcoMod_sub_eq_sub
theorem toIocMod_sub_eq_sub (a b c : α) : toIocMod hp a (b - c) = toIocMod hp (a + c) b - c := by
simp_rw [toIocMod, toIocDiv_sub_eq_toIocDiv_add, sub_right_comm]
#align to_Ioc_mod_sub_eq_sub toIocMod_sub_eq_sub
theorem toIcoMod_add_right_eq_add (a b c : α) :
toIcoMod hp a (b + c) = toIcoMod hp (a - c) b + c := by
simp_rw [toIcoMod, toIcoDiv_sub_eq_toIcoDiv_add', sub_add_eq_add_sub]
#align to_Ico_mod_add_right_eq_add toIcoMod_add_right_eq_add
theorem toIocMod_add_right_eq_add (a b c : α) :
toIocMod hp a (b + c) = toIocMod hp (a - c) b + c := by
simp_rw [toIocMod, toIocDiv_sub_eq_toIocDiv_add', sub_add_eq_add_sub]
#align to_Ioc_mod_add_right_eq_add toIocMod_add_right_eq_add
theorem toIcoMod_neg (a b : α) : toIcoMod hp a (-b) = p - toIocMod hp (-a) b := by
simp_rw [toIcoMod, toIocMod, toIcoDiv_neg, neg_smul, add_smul]
abel
#align to_Ico_mod_neg toIcoMod_neg
theorem toIcoMod_neg' (a b : α) : toIcoMod hp (-a) b = p - toIocMod hp a (-b) := by
simpa only [neg_neg] using toIcoMod_neg hp (-a) (-b)
#align to_Ico_mod_neg' toIcoMod_neg'
theorem toIocMod_neg (a b : α) : toIocMod hp a (-b) = p - toIcoMod hp (-a) b := by
simp_rw [toIocMod, toIcoMod, toIocDiv_neg, neg_smul, add_smul]
abel
#align to_Ioc_mod_neg toIocMod_neg
theorem toIocMod_neg' (a b : α) : toIocMod hp (-a) b = p - toIcoMod hp a (-b) := by
simpa only [neg_neg] using toIocMod_neg hp (-a) (-b)
#align to_Ioc_mod_neg' toIocMod_neg'
theorem toIcoMod_eq_toIcoMod : toIcoMod hp a b = toIcoMod hp a c ↔ ∃ n : ℤ, c - b = n • p := by
refine ⟨fun h => ⟨toIcoDiv hp a c - toIcoDiv hp a b, ?_⟩, fun h => ?_⟩
· conv_lhs => rw [← toIcoMod_add_toIcoDiv_zsmul hp a b, ← toIcoMod_add_toIcoDiv_zsmul hp a c]
rw [h, sub_smul]
abel
· rcases h with ⟨z, hz⟩
rw [sub_eq_iff_eq_add] at hz
rw [hz, toIcoMod_zsmul_add]
#align to_Ico_mod_eq_to_Ico_mod toIcoMod_eq_toIcoMod
theorem toIocMod_eq_toIocMod : toIocMod hp a b = toIocMod hp a c ↔ ∃ n : ℤ, c - b = n • p := by
refine ⟨fun h => ⟨toIocDiv hp a c - toIocDiv hp a b, ?_⟩, fun h => ?_⟩
· conv_lhs => rw [← toIocMod_add_toIocDiv_zsmul hp a b, ← toIocMod_add_toIocDiv_zsmul hp a c]
rw [h, sub_smul]
abel
· rcases h with ⟨z, hz⟩
rw [sub_eq_iff_eq_add] at hz
rw [hz, toIocMod_zsmul_add]
#align to_Ioc_mod_eq_to_Ioc_mod toIocMod_eq_toIocMod
section IcoIoc
open AddCommGroup
theorem toIcoMod_eq_self : toIcoMod hp a b = b ↔ b ∈ Set.Ico a (a + p) := by
rw [toIcoMod_eq_iff, and_iff_left]
exact ⟨0, by simp⟩
#align to_Ico_mod_eq_self toIcoMod_eq_self
theorem toIocMod_eq_self : toIocMod hp a b = b ↔ b ∈ Set.Ioc a (a + p) := by
rw [toIocMod_eq_iff, and_iff_left]
exact ⟨0, by simp⟩
#align to_Ioc_mod_eq_self toIocMod_eq_self
@[simp]
theorem toIcoMod_toIcoMod (a₁ a₂ b : α) : toIcoMod hp a₁ (toIcoMod hp a₂ b) = toIcoMod hp a₁ b :=
(toIcoMod_eq_toIcoMod _).2 ⟨toIcoDiv hp a₂ b, self_sub_toIcoMod hp a₂ b⟩
#align to_Ico_mod_to_Ico_mod toIcoMod_toIcoMod
@[simp]
theorem toIcoMod_toIocMod (a₁ a₂ b : α) : toIcoMod hp a₁ (toIocMod hp a₂ b) = toIcoMod hp a₁ b :=
(toIcoMod_eq_toIcoMod _).2 ⟨toIocDiv hp a₂ b, self_sub_toIocMod hp a₂ b⟩
#align to_Ico_mod_to_Ioc_mod toIcoMod_toIocMod
@[simp]
theorem toIocMod_toIocMod (a₁ a₂ b : α) : toIocMod hp a₁ (toIocMod hp a₂ b) = toIocMod hp a₁ b :=
(toIocMod_eq_toIocMod _).2 ⟨toIocDiv hp a₂ b, self_sub_toIocMod hp a₂ b⟩
#align to_Ioc_mod_to_Ioc_mod toIocMod_toIocMod
@[simp]
theorem toIocMod_toIcoMod (a₁ a₂ b : α) : toIocMod hp a₁ (toIcoMod hp a₂ b) = toIocMod hp a₁ b :=
(toIocMod_eq_toIocMod _).2 ⟨toIcoDiv hp a₂ b, self_sub_toIcoMod hp a₂ b⟩
#align to_Ioc_mod_to_Ico_mod toIocMod_toIcoMod
theorem toIcoMod_periodic (a : α) : Function.Periodic (toIcoMod hp a) p :=
toIcoMod_add_right hp a
#align to_Ico_mod_periodic toIcoMod_periodic
theorem toIocMod_periodic (a : α) : Function.Periodic (toIocMod hp a) p :=
toIocMod_add_right hp a
#align to_Ioc_mod_periodic toIocMod_periodic
-- helper lemmas for when `a = 0`
section Zero
theorem toIcoMod_zero_sub_comm (a b : α) : toIcoMod hp 0 (a - b) = p - toIocMod hp 0 (b - a) := by
rw [← neg_sub, toIcoMod_neg, neg_zero]
#align to_Ico_mod_zero_sub_comm toIcoMod_zero_sub_comm
theorem toIocMod_zero_sub_comm (a b : α) : toIocMod hp 0 (a - b) = p - toIcoMod hp 0 (b - a) := by
rw [← neg_sub, toIocMod_neg, neg_zero]
#align to_Ioc_mod_zero_sub_comm toIocMod_zero_sub_comm
theorem toIcoDiv_eq_sub (a b : α) : toIcoDiv hp a b = toIcoDiv hp 0 (b - a) := by
rw [toIcoDiv_sub_eq_toIcoDiv_add, zero_add]
#align to_Ico_div_eq_sub toIcoDiv_eq_sub
theorem toIocDiv_eq_sub (a b : α) : toIocDiv hp a b = toIocDiv hp 0 (b - a) := by
rw [toIocDiv_sub_eq_toIocDiv_add, zero_add]
#align to_Ioc_div_eq_sub toIocDiv_eq_sub
theorem toIcoMod_eq_sub (a b : α) : toIcoMod hp a b = toIcoMod hp 0 (b - a) + a := by
rw [toIcoMod_sub_eq_sub, zero_add, sub_add_cancel]
#align to_Ico_mod_eq_sub toIcoMod_eq_sub
| Mathlib/Algebra/Order/ToIntervalMod.lean | 784 | 785 | theorem toIocMod_eq_sub (a b : α) : toIocMod hp a b = toIocMod hp 0 (b - a) + a := by |
rw [toIocMod_sub_eq_sub, zero_add, sub_add_cancel]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Filter.Bases
#align_import order.filter.at_top_bot from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
set_option autoImplicit true
variable {ι ι' α β γ : Type*}
open Set
namespace Filter
def atTop [Preorder α] : Filter α :=
⨅ a, 𝓟 (Ici a)
#align filter.at_top Filter.atTop
def atBot [Preorder α] : Filter α :=
⨅ a, 𝓟 (Iic a)
#align filter.at_bot Filter.atBot
theorem mem_atTop [Preorder α] (a : α) : { b : α | a ≤ b } ∈ @atTop α _ :=
mem_iInf_of_mem a <| Subset.refl _
#align filter.mem_at_top Filter.mem_atTop
theorem Ici_mem_atTop [Preorder α] (a : α) : Ici a ∈ (atTop : Filter α) :=
mem_atTop a
#align filter.Ici_mem_at_top Filter.Ici_mem_atTop
theorem Ioi_mem_atTop [Preorder α] [NoMaxOrder α] (x : α) : Ioi x ∈ (atTop : Filter α) :=
let ⟨z, hz⟩ := exists_gt x
mem_of_superset (mem_atTop z) fun _ h => lt_of_lt_of_le hz h
#align filter.Ioi_mem_at_top Filter.Ioi_mem_atTop
theorem mem_atBot [Preorder α] (a : α) : { b : α | b ≤ a } ∈ @atBot α _ :=
mem_iInf_of_mem a <| Subset.refl _
#align filter.mem_at_bot Filter.mem_atBot
theorem Iic_mem_atBot [Preorder α] (a : α) : Iic a ∈ (atBot : Filter α) :=
mem_atBot a
#align filter.Iic_mem_at_bot Filter.Iic_mem_atBot
theorem Iio_mem_atBot [Preorder α] [NoMinOrder α] (x : α) : Iio x ∈ (atBot : Filter α) :=
let ⟨z, hz⟩ := exists_lt x
mem_of_superset (mem_atBot z) fun _ h => lt_of_le_of_lt h hz
#align filter.Iio_mem_at_bot Filter.Iio_mem_atBot
theorem disjoint_atBot_principal_Ioi [Preorder α] (x : α) : Disjoint atBot (𝓟 (Ioi x)) :=
disjoint_of_disjoint_of_mem (Iic_disjoint_Ioi le_rfl) (Iic_mem_atBot x) (mem_principal_self _)
#align filter.disjoint_at_bot_principal_Ioi Filter.disjoint_atBot_principal_Ioi
theorem disjoint_atTop_principal_Iio [Preorder α] (x : α) : Disjoint atTop (𝓟 (Iio x)) :=
@disjoint_atBot_principal_Ioi αᵒᵈ _ _
#align filter.disjoint_at_top_principal_Iio Filter.disjoint_atTop_principal_Iio
theorem disjoint_atTop_principal_Iic [Preorder α] [NoMaxOrder α] (x : α) :
Disjoint atTop (𝓟 (Iic x)) :=
disjoint_of_disjoint_of_mem (Iic_disjoint_Ioi le_rfl).symm (Ioi_mem_atTop x)
(mem_principal_self _)
#align filter.disjoint_at_top_principal_Iic Filter.disjoint_atTop_principal_Iic
theorem disjoint_atBot_principal_Ici [Preorder α] [NoMinOrder α] (x : α) :
Disjoint atBot (𝓟 (Ici x)) :=
@disjoint_atTop_principal_Iic αᵒᵈ _ _ _
#align filter.disjoint_at_bot_principal_Ici Filter.disjoint_atBot_principal_Ici
theorem disjoint_pure_atTop [Preorder α] [NoMaxOrder α] (x : α) : Disjoint (pure x) atTop :=
Disjoint.symm <| (disjoint_atTop_principal_Iic x).mono_right <| le_principal_iff.2 <|
mem_pure.2 right_mem_Iic
#align filter.disjoint_pure_at_top Filter.disjoint_pure_atTop
theorem disjoint_pure_atBot [Preorder α] [NoMinOrder α] (x : α) : Disjoint (pure x) atBot :=
@disjoint_pure_atTop αᵒᵈ _ _ _
#align filter.disjoint_pure_at_bot Filter.disjoint_pure_atBot
theorem not_tendsto_const_atTop [Preorder α] [NoMaxOrder α] (x : α) (l : Filter β) [l.NeBot] :
¬Tendsto (fun _ => x) l atTop :=
tendsto_const_pure.not_tendsto (disjoint_pure_atTop x)
#align filter.not_tendsto_const_at_top Filter.not_tendsto_const_atTop
theorem not_tendsto_const_atBot [Preorder α] [NoMinOrder α] (x : α) (l : Filter β) [l.NeBot] :
¬Tendsto (fun _ => x) l atBot :=
tendsto_const_pure.not_tendsto (disjoint_pure_atBot x)
#align filter.not_tendsto_const_at_bot Filter.not_tendsto_const_atBot
theorem disjoint_atBot_atTop [PartialOrder α] [Nontrivial α] :
Disjoint (atBot : Filter α) atTop := by
rcases exists_pair_ne α with ⟨x, y, hne⟩
by_cases hle : x ≤ y
· refine disjoint_of_disjoint_of_mem ?_ (Iic_mem_atBot x) (Ici_mem_atTop y)
exact Iic_disjoint_Ici.2 (hle.lt_of_ne hne).not_le
· refine disjoint_of_disjoint_of_mem ?_ (Iic_mem_atBot y) (Ici_mem_atTop x)
exact Iic_disjoint_Ici.2 hle
#align filter.disjoint_at_bot_at_top Filter.disjoint_atBot_atTop
theorem disjoint_atTop_atBot [PartialOrder α] [Nontrivial α] : Disjoint (atTop : Filter α) atBot :=
disjoint_atBot_atTop.symm
#align filter.disjoint_at_top_at_bot Filter.disjoint_atTop_atBot
theorem hasAntitoneBasis_atTop [Nonempty α] [Preorder α] [IsDirected α (· ≤ ·)] :
(@atTop α _).HasAntitoneBasis Ici :=
.iInf_principal fun _ _ ↦ Ici_subset_Ici.2
theorem atTop_basis [Nonempty α] [SemilatticeSup α] : (@atTop α _).HasBasis (fun _ => True) Ici :=
hasAntitoneBasis_atTop.1
#align filter.at_top_basis Filter.atTop_basis
theorem atTop_eq_generate_Ici [SemilatticeSup α] : atTop = generate (range (Ici (α := α))) := by
rcases isEmpty_or_nonempty α with hα|hα
· simp only [eq_iff_true_of_subsingleton]
· simp [(atTop_basis (α := α)).eq_generate, range]
theorem atTop_basis' [SemilatticeSup α] (a : α) : (@atTop α _).HasBasis (fun x => a ≤ x) Ici :=
⟨fun _ =>
(@atTop_basis α ⟨a⟩ _).mem_iff.trans
⟨fun ⟨x, _, hx⟩ => ⟨x ⊔ a, le_sup_right, fun _y hy => hx (le_trans le_sup_left hy)⟩,
fun ⟨x, _, hx⟩ => ⟨x, trivial, hx⟩⟩⟩
#align filter.at_top_basis' Filter.atTop_basis'
theorem atBot_basis [Nonempty α] [SemilatticeInf α] : (@atBot α _).HasBasis (fun _ => True) Iic :=
@atTop_basis αᵒᵈ _ _
#align filter.at_bot_basis Filter.atBot_basis
theorem atBot_basis' [SemilatticeInf α] (a : α) : (@atBot α _).HasBasis (fun x => x ≤ a) Iic :=
@atTop_basis' αᵒᵈ _ _
#align filter.at_bot_basis' Filter.atBot_basis'
@[instance]
theorem atTop_neBot [Nonempty α] [SemilatticeSup α] : NeBot (atTop : Filter α) :=
atTop_basis.neBot_iff.2 fun _ => nonempty_Ici
#align filter.at_top_ne_bot Filter.atTop_neBot
@[instance]
theorem atBot_neBot [Nonempty α] [SemilatticeInf α] : NeBot (atBot : Filter α) :=
@atTop_neBot αᵒᵈ _ _
#align filter.at_bot_ne_bot Filter.atBot_neBot
@[simp]
theorem mem_atTop_sets [Nonempty α] [SemilatticeSup α] {s : Set α} :
s ∈ (atTop : Filter α) ↔ ∃ a : α, ∀ b ≥ a, b ∈ s :=
atTop_basis.mem_iff.trans <| exists_congr fun _ => true_and_iff _
#align filter.mem_at_top_sets Filter.mem_atTop_sets
@[simp]
theorem mem_atBot_sets [Nonempty α] [SemilatticeInf α] {s : Set α} :
s ∈ (atBot : Filter α) ↔ ∃ a : α, ∀ b ≤ a, b ∈ s :=
@mem_atTop_sets αᵒᵈ _ _ _
#align filter.mem_at_bot_sets Filter.mem_atBot_sets
@[simp]
theorem eventually_atTop [SemilatticeSup α] [Nonempty α] {p : α → Prop} :
(∀ᶠ x in atTop, p x) ↔ ∃ a, ∀ b ≥ a, p b :=
mem_atTop_sets
#align filter.eventually_at_top Filter.eventually_atTop
@[simp]
theorem eventually_atBot [SemilatticeInf α] [Nonempty α] {p : α → Prop} :
(∀ᶠ x in atBot, p x) ↔ ∃ a, ∀ b ≤ a, p b :=
mem_atBot_sets
#align filter.eventually_at_bot Filter.eventually_atBot
theorem eventually_ge_atTop [Preorder α] (a : α) : ∀ᶠ x in atTop, a ≤ x :=
mem_atTop a
#align filter.eventually_ge_at_top Filter.eventually_ge_atTop
theorem eventually_le_atBot [Preorder α] (a : α) : ∀ᶠ x in atBot, x ≤ a :=
mem_atBot a
#align filter.eventually_le_at_bot Filter.eventually_le_atBot
theorem eventually_gt_atTop [Preorder α] [NoMaxOrder α] (a : α) : ∀ᶠ x in atTop, a < x :=
Ioi_mem_atTop a
#align filter.eventually_gt_at_top Filter.eventually_gt_atTop
theorem eventually_ne_atTop [Preorder α] [NoMaxOrder α] (a : α) : ∀ᶠ x in atTop, x ≠ a :=
(eventually_gt_atTop a).mono fun _ => ne_of_gt
#align filter.eventually_ne_at_top Filter.eventually_ne_atTop
protected theorem Tendsto.eventually_gt_atTop [Preorder β] [NoMaxOrder β] {f : α → β} {l : Filter α}
(hf : Tendsto f l atTop) (c : β) : ∀ᶠ x in l, c < f x :=
hf.eventually (eventually_gt_atTop c)
#align filter.tendsto.eventually_gt_at_top Filter.Tendsto.eventually_gt_atTop
protected theorem Tendsto.eventually_ge_atTop [Preorder β] {f : α → β} {l : Filter α}
(hf : Tendsto f l atTop) (c : β) : ∀ᶠ x in l, c ≤ f x :=
hf.eventually (eventually_ge_atTop c)
#align filter.tendsto.eventually_ge_at_top Filter.Tendsto.eventually_ge_atTop
protected theorem Tendsto.eventually_ne_atTop [Preorder β] [NoMaxOrder β] {f : α → β} {l : Filter α}
(hf : Tendsto f l atTop) (c : β) : ∀ᶠ x in l, f x ≠ c :=
hf.eventually (eventually_ne_atTop c)
#align filter.tendsto.eventually_ne_at_top Filter.Tendsto.eventually_ne_atTop
protected theorem Tendsto.eventually_ne_atTop' [Preorder β] [NoMaxOrder β] {f : α → β}
{l : Filter α} (hf : Tendsto f l atTop) (c : α) : ∀ᶠ x in l, x ≠ c :=
(hf.eventually_ne_atTop (f c)).mono fun _ => ne_of_apply_ne f
#align filter.tendsto.eventually_ne_at_top' Filter.Tendsto.eventually_ne_atTop'
theorem eventually_lt_atBot [Preorder α] [NoMinOrder α] (a : α) : ∀ᶠ x in atBot, x < a :=
Iio_mem_atBot a
#align filter.eventually_lt_at_bot Filter.eventually_lt_atBot
theorem eventually_ne_atBot [Preorder α] [NoMinOrder α] (a : α) : ∀ᶠ x in atBot, x ≠ a :=
(eventually_lt_atBot a).mono fun _ => ne_of_lt
#align filter.eventually_ne_at_bot Filter.eventually_ne_atBot
protected theorem Tendsto.eventually_lt_atBot [Preorder β] [NoMinOrder β] {f : α → β} {l : Filter α}
(hf : Tendsto f l atBot) (c : β) : ∀ᶠ x in l, f x < c :=
hf.eventually (eventually_lt_atBot c)
#align filter.tendsto.eventually_lt_at_bot Filter.Tendsto.eventually_lt_atBot
protected theorem Tendsto.eventually_le_atBot [Preorder β] {f : α → β} {l : Filter α}
(hf : Tendsto f l atBot) (c : β) : ∀ᶠ x in l, f x ≤ c :=
hf.eventually (eventually_le_atBot c)
#align filter.tendsto.eventually_le_at_bot Filter.Tendsto.eventually_le_atBot
protected theorem Tendsto.eventually_ne_atBot [Preorder β] [NoMinOrder β] {f : α → β} {l : Filter α}
(hf : Tendsto f l atBot) (c : β) : ∀ᶠ x in l, f x ≠ c :=
hf.eventually (eventually_ne_atBot c)
#align filter.tendsto.eventually_ne_at_bot Filter.Tendsto.eventually_ne_atBot
| Mathlib/Order/Filter/AtTopBot.lean | 253 | 259 | theorem eventually_forall_ge_atTop [Preorder α] {p : α → Prop} :
(∀ᶠ x in atTop, ∀ y, x ≤ y → p y) ↔ ∀ᶠ x in atTop, p x := by |
refine ⟨fun h ↦ h.mono fun x hx ↦ hx x le_rfl, fun h ↦ ?_⟩
rcases (hasBasis_iInf_principal_finite _).eventually_iff.1 h with ⟨S, hSf, hS⟩
refine mem_iInf_of_iInter hSf (V := fun x ↦ Ici x.1) (fun _ ↦ Subset.rfl) fun x hx y hy ↦ ?_
simp only [mem_iInter] at hS hx
exact hS fun z hz ↦ le_trans (hx ⟨z, hz⟩) hy
|
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scoped Pointwise
universe u v w x
namespace QuotientGroup
variable {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [Group H]
{M : Type x} [Monoid M]
@[to_additive "The additive congruence relation generated by a normal additive subgroup."]
protected def con : Con G where
toSetoid := leftRel N
mul' := @fun a b c d hab hcd => by
rw [leftRel_eq] at hab hcd ⊢
dsimp only
calc
(a * c)⁻¹ * (b * d) = c⁻¹ * (a⁻¹ * b) * c⁻¹⁻¹ * (c⁻¹ * d) := by
simp only [mul_inv_rev, mul_assoc, inv_mul_cancel_left]
_ ∈ N := N.mul_mem (nN.conj_mem _ hab _) hcd
#align quotient_group.con QuotientGroup.con
#align quotient_add_group.con QuotientAddGroup.con
@[to_additive]
instance Quotient.group : Group (G ⧸ N) :=
(QuotientGroup.con N).group
#align quotient_group.quotient.group QuotientGroup.Quotient.group
#align quotient_add_group.quotient.add_group QuotientAddGroup.Quotient.addGroup
@[to_additive "The additive group homomorphism from `G` to `G/N`."]
def mk' : G →* G ⧸ N :=
MonoidHom.mk' QuotientGroup.mk fun _ _ => rfl
#align quotient_group.mk' QuotientGroup.mk'
#align quotient_add_group.mk' QuotientAddGroup.mk'
@[to_additive (attr := simp)]
theorem coe_mk' : (mk' N : G → G ⧸ N) = mk :=
rfl
#align quotient_group.coe_mk' QuotientGroup.coe_mk'
#align quotient_add_group.coe_mk' QuotientAddGroup.coe_mk'
@[to_additive (attr := simp)]
theorem mk'_apply (x : G) : mk' N x = x :=
rfl
#align quotient_group.mk'_apply QuotientGroup.mk'_apply
#align quotient_add_group.mk'_apply QuotientAddGroup.mk'_apply
@[to_additive]
theorem mk'_surjective : Surjective <| mk' N :=
@mk_surjective _ _ N
#align quotient_group.mk'_surjective QuotientGroup.mk'_surjective
#align quotient_add_group.mk'_surjective QuotientAddGroup.mk'_surjective
@[to_additive]
theorem mk'_eq_mk' {x y : G} : mk' N x = mk' N y ↔ ∃ z ∈ N, x * z = y :=
QuotientGroup.eq'.trans <| by
simp only [← _root_.eq_inv_mul_iff_mul_eq, exists_prop, exists_eq_right]
#align quotient_group.mk'_eq_mk' QuotientGroup.mk'_eq_mk'
#align quotient_add_group.mk'_eq_mk' QuotientAddGroup.mk'_eq_mk'
open scoped Pointwise in
@[to_additive]
theorem sound (U : Set (G ⧸ N)) (g : N.op) :
g • (mk' N) ⁻¹' U = (mk' N) ⁻¹' U := by
ext x
simp only [Set.mem_preimage, Set.mem_smul_set_iff_inv_smul_mem]
congr! 1
exact Quotient.sound ⟨g⁻¹, rfl⟩
@[to_additive (attr := ext 1100) "Two `AddMonoidHom`s from an additive quotient group are equal if
their compositions with `AddQuotientGroup.mk'` are equal.
See note [partially-applied ext lemmas]. "]
theorem monoidHom_ext ⦃f g : G ⧸ N →* M⦄ (h : f.comp (mk' N) = g.comp (mk' N)) : f = g :=
MonoidHom.ext fun x => QuotientGroup.induction_on x <| (DFunLike.congr_fun h : _)
#align quotient_group.monoid_hom_ext QuotientGroup.monoidHom_ext
#align quotient_add_group.add_monoid_hom_ext QuotientAddGroup.addMonoidHom_ext
@[to_additive (attr := simp)]
theorem eq_one_iff {N : Subgroup G} [nN : N.Normal] (x : G) : (x : G ⧸ N) = 1 ↔ x ∈ N := by
refine QuotientGroup.eq.trans ?_
rw [mul_one, Subgroup.inv_mem_iff]
#align quotient_group.eq_one_iff QuotientGroup.eq_one_iff
#align quotient_add_group.eq_zero_iff QuotientAddGroup.eq_zero_iff
@[to_additive]
theorem ker_le_range_iff {I : Type w} [Group I] (f : G →* H) [f.range.Normal] (g : H →* I) :
g.ker ≤ f.range ↔ (mk' f.range).comp g.ker.subtype = 1 :=
⟨fun h => MonoidHom.ext fun ⟨_, hx⟩ => (eq_one_iff _).mpr <| h hx,
fun h x hx => (eq_one_iff _).mp <| by exact DFunLike.congr_fun h ⟨x, hx⟩⟩
@[to_additive (attr := simp)]
theorem ker_mk' : MonoidHom.ker (QuotientGroup.mk' N : G →* G ⧸ N) = N :=
Subgroup.ext eq_one_iff
#align quotient_group.ker_mk QuotientGroup.ker_mk'
#align quotient_add_group.ker_mk QuotientAddGroup.ker_mk'
-- Porting note: I think this is misnamed without the prime
@[to_additive]
theorem eq_iff_div_mem {N : Subgroup G} [nN : N.Normal] {x y : G} :
(x : G ⧸ N) = y ↔ x / y ∈ N := by
refine eq_comm.trans (QuotientGroup.eq.trans ?_)
rw [nN.mem_comm_iff, div_eq_mul_inv]
#align quotient_group.eq_iff_div_mem QuotientGroup.eq_iff_div_mem
#align quotient_add_group.eq_iff_sub_mem QuotientAddGroup.eq_iff_sub_mem
-- for commutative groups we don't need normality assumption
@[to_additive]
instance Quotient.commGroup {G : Type*} [CommGroup G] (N : Subgroup G) : CommGroup (G ⧸ N) :=
{ toGroup := @QuotientGroup.Quotient.group _ _ N N.normal_of_comm
mul_comm := fun a b => Quotient.inductionOn₂' a b fun a b => congr_arg mk (mul_comm a b) }
#align quotient_group.quotient.comm_group QuotientGroup.Quotient.commGroup
#align quotient_add_group.quotient.add_comm_group QuotientAddGroup.Quotient.addCommGroup
local notation " Q " => G ⧸ N
@[to_additive (attr := simp)]
theorem mk_one : ((1 : G) : Q) = 1 :=
rfl
#align quotient_group.coe_one QuotientGroup.mk_one
#align quotient_add_group.coe_zero QuotientAddGroup.mk_zero
@[to_additive (attr := simp)]
theorem mk_mul (a b : G) : ((a * b : G) : Q) = a * b :=
rfl
#align quotient_group.coe_mul QuotientGroup.mk_mul
#align quotient_add_group.coe_add QuotientAddGroup.mk_add
@[to_additive (attr := simp)]
theorem mk_inv (a : G) : ((a⁻¹ : G) : Q) = (a : Q)⁻¹ :=
rfl
#align quotient_group.coe_inv QuotientGroup.mk_inv
#align quotient_add_group.coe_neg QuotientAddGroup.mk_neg
@[to_additive (attr := simp)]
theorem mk_div (a b : G) : ((a / b : G) : Q) = a / b :=
rfl
#align quotient_group.coe_div QuotientGroup.mk_div
#align quotient_add_group.coe_sub QuotientAddGroup.mk_sub
@[to_additive (attr := simp)]
theorem mk_pow (a : G) (n : ℕ) : ((a ^ n : G) : Q) = (a : Q) ^ n :=
rfl
#align quotient_group.coe_pow QuotientGroup.mk_pow
#align quotient_add_group.coe_nsmul QuotientAddGroup.mk_nsmul
@[to_additive (attr := simp)]
theorem mk_zpow (a : G) (n : ℤ) : ((a ^ n : G) : Q) = (a : Q) ^ n :=
rfl
#align quotient_group.coe_zpow QuotientGroup.mk_zpow
#align quotient_add_group.coe_zsmul QuotientAddGroup.mk_zsmul
@[to_additive (attr := simp)]
theorem mk_prod {G ι : Type*} [CommGroup G] (N : Subgroup G) (s : Finset ι) {f : ι → G} :
((Finset.prod s f : G) : G ⧸ N) = Finset.prod s (fun i => (f i : G ⧸ N)) :=
map_prod (QuotientGroup.mk' N) _ _
@[to_additive (attr := simp)] lemma map_mk'_self : N.map (mk' N) = ⊥ := by aesop
@[to_additive "An `AddGroup` homomorphism `φ : G →+ M` with `N ⊆ ker(φ)` descends (i.e. `lift`s)
to a group homomorphism `G/N →* M`."]
def lift (φ : G →* M) (HN : N ≤ φ.ker) : Q →* M :=
(QuotientGroup.con N).lift φ fun x y h => by
simp only [QuotientGroup.con, leftRel_apply, Con.rel_mk] at h
rw [Con.ker_rel]
calc
φ x = φ (y * (x⁻¹ * y)⁻¹) := by rw [mul_inv_rev, inv_inv, mul_inv_cancel_left]
_ = φ y := by rw [φ.map_mul, HN (N.inv_mem h), mul_one]
#align quotient_group.lift QuotientGroup.lift
#align quotient_add_group.lift QuotientAddGroup.lift
@[to_additive (attr := simp)]
theorem lift_mk {φ : G →* M} (HN : N ≤ φ.ker) (g : G) : lift N φ HN (g : Q) = φ g :=
rfl
#align quotient_group.lift_mk QuotientGroup.lift_mk
#align quotient_add_group.lift_mk QuotientAddGroup.lift_mk
@[to_additive (attr := simp)]
theorem lift_mk' {φ : G →* M} (HN : N ≤ φ.ker) (g : G) : lift N φ HN (mk g : Q) = φ g :=
rfl
-- TODO: replace `mk` with `mk'`)
#align quotient_group.lift_mk' QuotientGroup.lift_mk'
#align quotient_add_group.lift_mk' QuotientAddGroup.lift_mk'
@[to_additive (attr := simp)]
theorem lift_quot_mk {φ : G →* M} (HN : N ≤ φ.ker) (g : G) :
lift N φ HN (Quot.mk _ g : Q) = φ g :=
rfl
#align quotient_group.lift_quot_mk QuotientGroup.lift_quot_mk
#align quotient_add_group.lift_quot_mk QuotientAddGroup.lift_quot_mk
@[to_additive
"An `AddGroup` homomorphism `f : G →+ H` induces a map `G/N →+ H/M` if `N ⊆ f⁻¹(M)`."]
def map (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ M.comap f) : G ⧸ N →* H ⧸ M := by
refine QuotientGroup.lift N ((mk' M).comp f) ?_
intro x hx
refine QuotientGroup.eq.2 ?_
rw [mul_one, Subgroup.inv_mem_iff]
exact h hx
#align quotient_group.map QuotientGroup.map
#align quotient_add_group.map QuotientAddGroup.map
@[to_additive (attr := simp)]
theorem map_mk (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ M.comap f) (x : G) :
map N M f h ↑x = ↑(f x) :=
rfl
#align quotient_group.map_coe QuotientGroup.map_mk
#align quotient_add_group.map_coe QuotientAddGroup.map_mk
@[to_additive]
theorem map_mk' (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ M.comap f) (x : G) :
map N M f h (mk' _ x) = ↑(f x) :=
rfl
#align quotient_group.map_mk' QuotientGroup.map_mk'
#align quotient_add_group.map_mk' QuotientAddGroup.map_mk'
@[to_additive]
theorem map_id_apply (h : N ≤ Subgroup.comap (MonoidHom.id _) N := (Subgroup.comap_id N).le) (x) :
map N N (MonoidHom.id _) h x = x :=
induction_on' x fun _x => rfl
#align quotient_group.map_id_apply QuotientGroup.map_id_apply
#align quotient_add_group.map_id_apply QuotientAddGroup.map_id_apply
@[to_additive (attr := simp)]
theorem map_id (h : N ≤ Subgroup.comap (MonoidHom.id _) N := (Subgroup.comap_id N).le) :
map N N (MonoidHom.id _) h = MonoidHom.id _ :=
MonoidHom.ext (map_id_apply N h)
#align quotient_group.map_id QuotientGroup.map_id
#align quotient_add_group.map_id QuotientAddGroup.map_id
@[to_additive (attr := simp)]
| Mathlib/GroupTheory/QuotientGroup.lean | 285 | 291 | theorem map_map {I : Type*} [Group I] (M : Subgroup H) (O : Subgroup I) [M.Normal] [O.Normal]
(f : G →* H) (g : H →* I) (hf : N ≤ Subgroup.comap f M) (hg : M ≤ Subgroup.comap g O)
(hgf : N ≤ Subgroup.comap (g.comp f) O :=
hf.trans ((Subgroup.comap_mono hg).trans_eq (Subgroup.comap_comap _ _ _)))
(x : G ⧸ N) : map M O g hg (map N M f hf x) = map N O (g.comp f) hgf x := by |
refine induction_on' x fun x => ?_
simp only [map_mk, MonoidHom.comp_apply]
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Regular
import Mathlib.Tactic.Common
#align_import algebra.gcd_monoid.basic from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
variable {α : Type*}
-- Porting note: mathlib3 had a `@[protect_proj]` here, but adding `protected` to all the fields
-- adds unnecessary clutter to later code
class NormalizationMonoid (α : Type*) [CancelCommMonoidWithZero α] where
normUnit : α → αˣ
normUnit_zero : normUnit 0 = 1
normUnit_mul : ∀ {a b}, a ≠ 0 → b ≠ 0 → normUnit (a * b) = normUnit a * normUnit b
normUnit_coe_units : ∀ u : αˣ, normUnit u = u⁻¹
#align normalization_monoid NormalizationMonoid
export NormalizationMonoid (normUnit normUnit_zero normUnit_mul normUnit_coe_units)
attribute [simp] normUnit_coe_units normUnit_zero normUnit_mul
section NormalizationMonoid
variable [CancelCommMonoidWithZero α] [NormalizationMonoid α]
@[simp]
theorem normUnit_one : normUnit (1 : α) = 1 :=
normUnit_coe_units 1
#align norm_unit_one normUnit_one
-- Porting note (#11083): quite slow. Improve performance?
def normalize : α →*₀ α where
toFun x := x * normUnit x
map_zero' := by
simp only [normUnit_zero]
exact mul_one (0:α)
map_one' := by dsimp only; rw [normUnit_one, one_mul]; rfl
map_mul' x y :=
(by_cases fun hx : x = 0 => by dsimp only; rw [hx, zero_mul, zero_mul, zero_mul]) fun hx =>
(by_cases fun hy : y = 0 => by dsimp only; rw [hy, mul_zero, zero_mul, mul_zero]) fun hy => by
simp only [normUnit_mul hx hy, Units.val_mul]; simp only [mul_assoc, mul_left_comm y]
#align normalize normalize
theorem associated_normalize (x : α) : Associated x (normalize x) :=
⟨_, rfl⟩
#align associated_normalize associated_normalize
theorem normalize_associated (x : α) : Associated (normalize x) x :=
(associated_normalize _).symm
#align normalize_associated normalize_associated
theorem associated_normalize_iff {x y : α} : Associated x (normalize y) ↔ Associated x y :=
⟨fun h => h.trans (normalize_associated y), fun h => h.trans (associated_normalize y)⟩
#align associated_normalize_iff associated_normalize_iff
theorem normalize_associated_iff {x y : α} : Associated (normalize x) y ↔ Associated x y :=
⟨fun h => (associated_normalize _).trans h, fun h => (normalize_associated _).trans h⟩
#align normalize_associated_iff normalize_associated_iff
theorem Associates.mk_normalize (x : α) : Associates.mk (normalize x) = Associates.mk x :=
Associates.mk_eq_mk_iff_associated.2 (normalize_associated _)
#align associates.mk_normalize Associates.mk_normalize
@[simp]
theorem normalize_apply (x : α) : normalize x = x * normUnit x :=
rfl
#align normalize_apply normalize_apply
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem normalize_zero : normalize (0 : α) = 0 :=
normalize.map_zero
#align normalize_zero normalize_zero
-- Porting note (#10618): `simp` can prove this
-- @[simp]
theorem normalize_one : normalize (1 : α) = 1 :=
normalize.map_one
#align normalize_one normalize_one
| Mathlib/Algebra/GCDMonoid/Basic.lean | 148 | 148 | theorem normalize_coe_units (u : αˣ) : normalize (u : α) = 1 := by | simp
|
import Mathlib.MeasureTheory.Integral.Bochner
open MeasureTheory Filter
open scoped ENNReal NNReal BoundedContinuousFunction Topology
namespace BoundedContinuousFunction
section NNRealValued
lemma apply_le_nndist_zero {X : Type*} [TopologicalSpace X] (f : X →ᵇ ℝ≥0) (x : X) :
f x ≤ nndist 0 f := by
convert nndist_coe_le_nndist x
simp only [coe_zero, Pi.zero_apply, NNReal.nndist_zero_eq_val]
variable {X : Type*} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X]
lemma lintegral_le_edist_mul (f : X →ᵇ ℝ≥0) (μ : Measure X) :
(∫⁻ x, f x ∂μ) ≤ edist 0 f * (μ Set.univ) :=
le_trans (lintegral_mono (fun x ↦ ENNReal.coe_le_coe.mpr (f.apply_le_nndist_zero x))) (by simp)
theorem measurable_coe_ennreal_comp (f : X →ᵇ ℝ≥0) :
Measurable fun x ↦ (f x : ℝ≥0∞) :=
measurable_coe_nnreal_ennreal.comp f.continuous.measurable
#align bounded_continuous_function.nnreal.to_ennreal_comp_measurable BoundedContinuousFunction.measurable_coe_ennreal_comp
variable (μ : Measure X) [IsFiniteMeasure μ]
theorem lintegral_lt_top_of_nnreal (f : X →ᵇ ℝ≥0) : ∫⁻ x, f x ∂μ < ∞ := by
apply IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal
refine ⟨nndist f 0, fun x ↦ ?_⟩
have key := BoundedContinuousFunction.NNReal.upper_bound f x
rwa [ENNReal.coe_le_coe]
#align measure_theory.lintegral_lt_top_of_bounded_continuous_to_nnreal BoundedContinuousFunction.lintegral_lt_top_of_nnreal
| Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean | 49 | 52 | theorem integrable_of_nnreal (f : X →ᵇ ℝ≥0) : Integrable (((↑) : ℝ≥0 → ℝ) ∘ ⇑f) μ := by |
refine ⟨(NNReal.continuous_coe.comp f.continuous).measurable.aestronglyMeasurable, ?_⟩
simp only [HasFiniteIntegral, Function.comp_apply, NNReal.nnnorm_eq]
exact lintegral_lt_top_of_nnreal _ f
|
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
variable {C : Type u} [Category.{v} C] {X Y Z : C}
namespace CategoryTheory
namespace Limits
section Image
variable (f : X ⟶ Y) [HasImage f]
abbrev imageSubobject : Subobject Y :=
Subobject.mk (image.ι f)
#align category_theory.limits.image_subobject CategoryTheory.Limits.imageSubobject
def imageSubobjectIso : (imageSubobject f : C) ≅ image f :=
Subobject.underlyingIso (image.ι f)
#align category_theory.limits.image_subobject_iso CategoryTheory.Limits.imageSubobjectIso
@[reassoc (attr := simp)]
theorem imageSubobject_arrow :
(imageSubobjectIso f).hom ≫ image.ι f = (imageSubobject f).arrow := by simp [imageSubobjectIso]
#align category_theory.limits.image_subobject_arrow CategoryTheory.Limits.imageSubobject_arrow
@[reassoc (attr := simp)]
theorem imageSubobject_arrow' :
(imageSubobjectIso f).inv ≫ (imageSubobject f).arrow = image.ι f := by simp [imageSubobjectIso]
#align category_theory.limits.image_subobject_arrow' CategoryTheory.Limits.imageSubobject_arrow'
def factorThruImageSubobject : X ⟶ imageSubobject f :=
factorThruImage f ≫ (imageSubobjectIso f).inv
#align category_theory.limits.factor_thru_image_subobject CategoryTheory.Limits.factorThruImageSubobject
instance [HasEqualizers C] : Epi (factorThruImageSubobject f) := by
dsimp [factorThruImageSubobject]
apply epi_comp
@[reassoc (attr := simp), elementwise (attr := simp)]
theorem imageSubobject_arrow_comp : factorThruImageSubobject f ≫ (imageSubobject f).arrow = f := by
simp [factorThruImageSubobject, imageSubobject_arrow]
#align category_theory.limits.image_subobject_arrow_comp CategoryTheory.Limits.imageSubobject_arrow_comp
theorem imageSubobject_arrow_comp_eq_zero [HasZeroMorphisms C] {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z}
[HasImage f] [Epi (factorThruImageSubobject f)] (h : f ≫ g = 0) :
(imageSubobject f).arrow ≫ g = 0 :=
zero_of_epi_comp (factorThruImageSubobject f) <| by simp [h]
#align category_theory.limits.image_subobject_arrow_comp_eq_zero CategoryTheory.Limits.imageSubobject_arrow_comp_eq_zero
theorem imageSubobject_factors_comp_self {W : C} (k : W ⟶ X) : (imageSubobject f).Factors (k ≫ f) :=
⟨k ≫ factorThruImage f, by simp⟩
#align category_theory.limits.image_subobject_factors_comp_self CategoryTheory.Limits.imageSubobject_factors_comp_self
@[simp]
| Mathlib/CategoryTheory/Subobject/Limits.lean | 343 | 346 | theorem factorThruImageSubobject_comp_self {W : C} (k : W ⟶ X) (h) :
(imageSubobject f).factorThru (k ≫ f) h = k ≫ factorThruImageSubobject f := by |
ext
simp
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.specific_limits.floor_pow from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Filter Finset
open Topology
theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (l : ℝ)
(hmono : Monotone u)
(hlim : ∀ a : ℝ, 1 < a → ∃ c : ℕ → ℕ, (∀ᶠ n in atTop, (c (n + 1) : ℝ) ≤ a * c n) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / c n) atTop (𝓝 l)) :
Tendsto (fun n => u n / n) atTop (𝓝 l) := by
have lnonneg : 0 ≤ l := by
rcases hlim 2 one_lt_two with ⟨c, _, ctop, clim⟩
have : Tendsto (fun n => u 0 / c n) atTop (𝓝 0) :=
tendsto_const_nhds.div_atTop (tendsto_natCast_atTop_iff.2 ctop)
apply le_of_tendsto_of_tendsto' this clim fun n => ?_
gcongr
exact hmono (zero_le _)
have A : ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop, u n - n * l ≤ ε * (1 + ε + l) * n := by
intro ε εpos
rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
have L : ∀ᶠ n in atTop, u (c n) - c n * l ≤ ε * c n := by
rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
Asymptotics.isLittleO_iff] at clim
filter_upwards [clim εpos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'
have cnpos : 0 < c n := cnpos'
calc
u (c n) - c n * l = (u (c n) / c n - l) * c n := by
simp only [cnpos.ne', Ne, Nat.cast_eq_zero, not_false_iff, field_simps]
_ ≤ ε * c n := by
gcongr
refine (le_abs_self _).trans ?_
simpa using hn
obtain ⟨a, ha⟩ :
∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ u (c b) - c b * l ≤ ε * c b :=
eventually_atTop.1 (cgrowth.and L)
let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
filter_upwards [Ici_mem_atTop M] with n hn
have exN : ∃ N, n < c N := by
rcases (tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
exact ⟨N, by linarith only [hN]⟩
let N := Nat.find exN
have ncN : n < c N := Nat.find_spec exN
have aN : a + 1 ≤ N := by
by_contra! h
have cNM : c N ≤ M := by
apply le_max'
apply mem_image_of_mem
exact mem_range.2 h
exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)
have Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN
have cNn : c (N - 1) ≤ n := by
have : N - 1 < N := Nat.pred_lt Npos.ne'
simpa only [not_lt] using Nat.find_min exN this
have IcN : (c N : ℝ) ≤ (1 + ε) * c (N - 1) := by
have A : a ≤ N - 1 := by
apply @Nat.le_of_add_le_add_right a 1 (N - 1)
rw [Nat.sub_add_cancel Npos]
exact aN
have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
have := (ha _ A).1
rwa [B] at this
calc
u n - n * l ≤ u (c N) - c (N - 1) * l := by gcongr; exact hmono ncN.le
_ = u (c N) - c N * l + (c N - c (N - 1)) * l := by ring
_ ≤ ε * c N + ε * c (N - 1) * l := by
gcongr
· exact (ha N (a.le_succ.trans aN)).2
· linarith only [IcN]
_ ≤ ε * ((1 + ε) * c (N - 1)) + ε * c (N - 1) * l := by gcongr
_ = ε * (1 + ε + l) * c (N - 1) := by ring
_ ≤ ε * (1 + ε + l) * n := by gcongr
have B : ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, (n : ℝ) * l - u n ≤ ε * (1 + l) * n := by
intro ε εpos
rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
have L : ∀ᶠ n : ℕ in atTop, (c n : ℝ) * l - u (c n) ≤ ε * c n := by
rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
Asymptotics.isLittleO_iff] at clim
filter_upwards [clim εpos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'
have cnpos : 0 < c n := cnpos'
calc
(c n : ℝ) * l - u (c n) = -(u (c n) / c n - l) * c n := by
simp only [cnpos.ne', Ne, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
_ ≤ ε * c n := by
gcongr
refine le_trans (neg_le_abs _) ?_
simpa using hn
obtain ⟨a, ha⟩ :
∃ a : ℕ,
∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ (c b : ℝ) * l - u (c b) ≤ ε * c b :=
eventually_atTop.1 (cgrowth.and L)
let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
filter_upwards [Ici_mem_atTop M] with n hn
have exN : ∃ N, n < c N := by
rcases (tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
exact ⟨N, by linarith only [hN]⟩
let N := Nat.find exN
have ncN : n < c N := Nat.find_spec exN
have aN : a + 1 ≤ N := by
by_contra! h
have cNM : c N ≤ M := by
apply le_max'
apply mem_image_of_mem
exact mem_range.2 h
exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)
have Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN
have aN' : a ≤ N - 1 := by
apply @Nat.le_of_add_le_add_right a 1 (N - 1)
rw [Nat.sub_add_cancel Npos]
exact aN
have cNn : c (N - 1) ≤ n := by
have : N - 1 < N := Nat.pred_lt Npos.ne'
simpa only [not_lt] using Nat.find_min exN this
calc
(n : ℝ) * l - u n ≤ c N * l - u (c (N - 1)) := by
gcongr
exact hmono cNn
_ ≤ (1 + ε) * c (N - 1) * l - u (c (N - 1)) := by
gcongr
have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
simpa [B] using (ha _ aN').1
_ = c (N - 1) * l - u (c (N - 1)) + ε * c (N - 1) * l := by ring
_ ≤ ε * c (N - 1) + ε * c (N - 1) * l := add_le_add (ha _ aN').2 le_rfl
_ = ε * (1 + l) * c (N - 1) := by ring
_ ≤ ε * (1 + l) * n := by gcongr
refine tendsto_order.2 ⟨fun d hd => ?_, fun d hd => ?_⟩
· obtain ⟨ε, hε, εpos⟩ : ∃ ε : ℝ, d + ε * (1 + l) < l ∧ 0 < ε := by
have L : Tendsto (fun ε => d + ε * (1 + l)) (𝓝[>] 0) (𝓝 (d + 0 * (1 + l))) := by
apply Tendsto.mono_left _ nhdsWithin_le_nhds
exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)
simp only [zero_mul, add_zero] at L
exact (((tendsto_order.1 L).2 l hd).and self_mem_nhdsWithin).exists
filter_upwards [B ε εpos, Ioi_mem_atTop 0] with n hn npos
simp_rw [div_eq_inv_mul]
calc
d < (n : ℝ)⁻¹ * n * (l - ε * (1 + l)) := by
rw [inv_mul_cancel, one_mul]
· linarith only [hε]
· exact Nat.cast_ne_zero.2 (ne_of_gt npos)
_ = (n : ℝ)⁻¹ * (n * l - ε * (1 + l) * n) := by ring
_ ≤ (n : ℝ)⁻¹ * u n := by gcongr; linarith only [hn]
· obtain ⟨ε, hε, εpos⟩ : ∃ ε : ℝ, l + ε * (1 + ε + l) < d ∧ 0 < ε := by
have L : Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 (l + 0 * (1 + 0 + l))) := by
apply Tendsto.mono_left _ nhdsWithin_le_nhds
exact
tendsto_const_nhds.add
(tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))
simp only [zero_mul, add_zero] at L
exact (((tendsto_order.1 L).2 d hd).and self_mem_nhdsWithin).exists
filter_upwards [A ε εpos, Ioi_mem_atTop 0] with n hn (npos : 0 < n)
calc
u n / n ≤ (n * l + ε * (1 + ε + l) * n) / n := by gcongr; linarith only [hn]
_ = (l + ε * (1 + ε + l)) := by field_simp; ring
_ < d := hε
#align tendsto_div_of_monotone_of_exists_subseq_tendsto_div tendsto_div_of_monotone_of_exists_subseq_tendsto_div
| Mathlib/Analysis/SpecificLimits/FloorPow.lean | 188 | 218 | theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
(c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
(hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
Tendsto (fun n => u n / n) atTop (𝓝 l) := by |
apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
intro a ha
obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
refine
⟨fun n => ⌊c k ^ n⌋₊, ?_,
(tendsto_nat_floor_atTop (α := ℝ)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
have H : ∀ n : ℕ, (0 : ℝ) < ⌊c k ^ n⌋₊ := by
intro n
refine zero_lt_one.trans_le ?_
simp only [Real.rpow_natCast, Nat.one_le_cast, Nat.one_le_floor_iff,
one_le_pow_of_one_le (cone k).le n]
have A :
Tendsto (fun n : ℕ => (⌊c k ^ (n + 1)⌋₊ : ℝ) / c k ^ (n + 1) * c k / (⌊c k ^ n⌋₊ / c k ^ n))
atTop (𝓝 (1 * c k / 1)) := by
refine Tendsto.div (Tendsto.mul ?_ tendsto_const_nhds) ?_ one_ne_zero
· refine tendsto_nat_floor_div_atTop.comp ?_
exact (tendsto_pow_atTop_atTop_of_one_lt (cone k)).comp (tendsto_add_atTop_nat 1)
· refine tendsto_nat_floor_div_atTop.comp ?_
exact tendsto_pow_atTop_atTop_of_one_lt (cone k)
have B : Tendsto (fun n : ℕ => (⌊c k ^ (n + 1)⌋₊ : ℝ) / ⌊c k ^ n⌋₊) atTop (𝓝 (c k)) := by
simp only [one_mul, div_one] at A
convert A using 1
ext1 n
field_simp [(zero_lt_one.trans (cone k)).ne', (H n).ne']
ring
filter_upwards [(tendsto_order.1 B).2 a hk] with n hn
exact (div_le_iff (H n)).1 hn.le
|
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.Order.Basic
#align_import topology.fiber_bundle.trivialization from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open TopologicalSpace Filter Set Bundle Function
open scoped Topology Classical Bundle
variable {ι : Type*} {B : Type*} {F : Type*} {E : B → Type*}
variable (F) {Z : Type*} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B}
structure Pretrivialization (proj : Z → B) extends PartialEquiv Z (B × F) where
open_target : IsOpen target
baseSet : Set B
open_baseSet : IsOpen baseSet
source_eq : source = proj ⁻¹' baseSet
target_eq : target = baseSet ×ˢ univ
proj_toFun : ∀ p ∈ source, (toFun p).1 = proj p
#align pretrivialization Pretrivialization
namespace Pretrivialization
variable {F}
variable (e : Pretrivialization F proj) {x : Z}
@[coe] def toFun' : Z → (B × F) := e.toFun
instance : CoeFun (Pretrivialization F proj) fun _ => Z → B × F := ⟨toFun'⟩
@[ext]
lemma ext' (e e' : Pretrivialization F proj) (h₁ : e.toPartialEquiv = e'.toPartialEquiv)
(h₂ : e.baseSet = e'.baseSet) : e = e' := by
cases e; cases e'; congr
#align pretrivialization.ext Pretrivialization.ext'
-- Porting note (#11215): TODO: move `ext` here?
lemma ext {e e' : Pretrivialization F proj} (h₁ : ∀ x, e x = e' x)
(h₂ : ∀ x, e.toPartialEquiv.symm x = e'.toPartialEquiv.symm x) (h₃ : e.baseSet = e'.baseSet) :
e = e' := by
ext1 <;> [ext1; exact h₃]
· apply h₁
· apply h₂
· rw [e.source_eq, e'.source_eq, h₃]
lemma toPartialEquiv_injective [Nonempty F] :
Injective (toPartialEquiv : Pretrivialization F proj → PartialEquiv Z (B × F)) := by
refine fun e e' h ↦ ext' _ _ h ?_
simpa only [fst_image_prod, univ_nonempty, target_eq]
using congr_arg (Prod.fst '' PartialEquiv.target ·) h
@[simp, mfld_simps]
theorem coe_coe : ⇑e.toPartialEquiv = e :=
rfl
#align pretrivialization.coe_coe Pretrivialization.coe_coe
@[simp, mfld_simps]
theorem coe_fst (ex : x ∈ e.source) : (e x).1 = proj x :=
e.proj_toFun x ex
#align pretrivialization.coe_fst Pretrivialization.coe_fst
theorem mem_source : x ∈ e.source ↔ proj x ∈ e.baseSet := by rw [e.source_eq, mem_preimage]
#align pretrivialization.mem_source Pretrivialization.mem_source
theorem coe_fst' (ex : proj x ∈ e.baseSet) : (e x).1 = proj x :=
e.coe_fst (e.mem_source.2 ex)
#align pretrivialization.coe_fst' Pretrivialization.coe_fst'
protected theorem eqOn : EqOn (Prod.fst ∘ e) proj e.source := fun _ hx => e.coe_fst hx
#align pretrivialization.eq_on Pretrivialization.eqOn
theorem mk_proj_snd (ex : x ∈ e.source) : (proj x, (e x).2) = e x :=
Prod.ext (e.coe_fst ex).symm rfl
#align pretrivialization.mk_proj_snd Pretrivialization.mk_proj_snd
theorem mk_proj_snd' (ex : proj x ∈ e.baseSet) : (proj x, (e x).2) = e x :=
Prod.ext (e.coe_fst' ex).symm rfl
#align pretrivialization.mk_proj_snd' Pretrivialization.mk_proj_snd'
def setSymm : e.target → Z :=
e.target.restrict e.toPartialEquiv.symm
#align pretrivialization.set_symm Pretrivialization.setSymm
theorem mem_target {x : B × F} : x ∈ e.target ↔ x.1 ∈ e.baseSet := by
rw [e.target_eq, prod_univ, mem_preimage]
#align pretrivialization.mem_target Pretrivialization.mem_target
theorem proj_symm_apply {x : B × F} (hx : x ∈ e.target) : proj (e.toPartialEquiv.symm x) = x.1 := by
have := (e.coe_fst (e.map_target hx)).symm
rwa [← e.coe_coe, e.right_inv hx] at this
#align pretrivialization.proj_symm_apply Pretrivialization.proj_symm_apply
theorem proj_symm_apply' {b : B} {x : F} (hx : b ∈ e.baseSet) :
proj (e.toPartialEquiv.symm (b, x)) = b :=
e.proj_symm_apply (e.mem_target.2 hx)
#align pretrivialization.proj_symm_apply' Pretrivialization.proj_symm_apply'
theorem proj_surjOn_baseSet [Nonempty F] : Set.SurjOn proj e.source e.baseSet := fun b hb =>
let ⟨y⟩ := ‹Nonempty F›
⟨e.toPartialEquiv.symm (b, y), e.toPartialEquiv.map_target <| e.mem_target.2 hb,
e.proj_symm_apply' hb⟩
#align pretrivialization.proj_surj_on_base_set Pretrivialization.proj_surjOn_baseSet
theorem apply_symm_apply {x : B × F} (hx : x ∈ e.target) : e (e.toPartialEquiv.symm x) = x :=
e.toPartialEquiv.right_inv hx
#align pretrivialization.apply_symm_apply Pretrivialization.apply_symm_apply
theorem apply_symm_apply' {b : B} {x : F} (hx : b ∈ e.baseSet) :
e (e.toPartialEquiv.symm (b, x)) = (b, x) :=
e.apply_symm_apply (e.mem_target.2 hx)
#align pretrivialization.apply_symm_apply' Pretrivialization.apply_symm_apply'
theorem symm_apply_apply {x : Z} (hx : x ∈ e.source) : e.toPartialEquiv.symm (e x) = x :=
e.toPartialEquiv.left_inv hx
#align pretrivialization.symm_apply_apply Pretrivialization.symm_apply_apply
@[simp, mfld_simps]
theorem symm_apply_mk_proj {x : Z} (ex : x ∈ e.source) :
e.toPartialEquiv.symm (proj x, (e x).2) = x := by
rw [← e.coe_fst ex, ← e.coe_coe, e.left_inv ex]
#align pretrivialization.symm_apply_mk_proj Pretrivialization.symm_apply_mk_proj
@[simp, mfld_simps]
theorem preimage_symm_proj_baseSet :
e.toPartialEquiv.symm ⁻¹' (proj ⁻¹' e.baseSet) ∩ e.target = e.target := by
refine inter_eq_right.mpr fun x hx => ?_
simp only [mem_preimage, PartialEquiv.invFun_as_coe, e.proj_symm_apply hx]
exact e.mem_target.mp hx
#align pretrivialization.preimage_symm_proj_base_set Pretrivialization.preimage_symm_proj_baseSet
@[simp, mfld_simps]
theorem preimage_symm_proj_inter (s : Set B) :
e.toPartialEquiv.symm ⁻¹' (proj ⁻¹' s) ∩ e.baseSet ×ˢ univ = (s ∩ e.baseSet) ×ˢ univ := by
ext ⟨x, y⟩
suffices x ∈ e.baseSet → (proj (e.toPartialEquiv.symm (x, y)) ∈ s ↔ x ∈ s) by
simpa only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true_iff, mem_univ, and_congr_left_iff]
intro h
rw [e.proj_symm_apply' h]
#align pretrivialization.preimage_symm_proj_inter Pretrivialization.preimage_symm_proj_inter
theorem target_inter_preimage_symm_source_eq (e f : Pretrivialization F proj) :
f.target ∩ f.toPartialEquiv.symm ⁻¹' e.source = (e.baseSet ∩ f.baseSet) ×ˢ univ := by
rw [inter_comm, f.target_eq, e.source_eq, f.preimage_symm_proj_inter]
#align pretrivialization.target_inter_preimage_symm_source_eq Pretrivialization.target_inter_preimage_symm_source_eq
theorem trans_source (e f : Pretrivialization F proj) :
(f.toPartialEquiv.symm.trans e.toPartialEquiv).source = (e.baseSet ∩ f.baseSet) ×ˢ univ := by
rw [PartialEquiv.trans_source, PartialEquiv.symm_source, e.target_inter_preimage_symm_source_eq]
#align pretrivialization.trans_source Pretrivialization.trans_source
theorem symm_trans_symm (e e' : Pretrivialization F proj) :
(e.toPartialEquiv.symm.trans e'.toPartialEquiv).symm
= e'.toPartialEquiv.symm.trans e.toPartialEquiv := by
rw [PartialEquiv.trans_symm_eq_symm_trans_symm, PartialEquiv.symm_symm]
#align pretrivialization.symm_trans_symm Pretrivialization.symm_trans_symm
theorem symm_trans_source_eq (e e' : Pretrivialization F proj) :
(e.toPartialEquiv.symm.trans e'.toPartialEquiv).source = (e.baseSet ∩ e'.baseSet) ×ˢ univ := by
rw [PartialEquiv.trans_source, e'.source_eq, PartialEquiv.symm_source, e.target_eq, inter_comm,
e.preimage_symm_proj_inter, inter_comm]
#align pretrivialization.symm_trans_source_eq Pretrivialization.symm_trans_source_eq
theorem symm_trans_target_eq (e e' : Pretrivialization F proj) :
(e.toPartialEquiv.symm.trans e'.toPartialEquiv).target = (e.baseSet ∩ e'.baseSet) ×ˢ univ := by
rw [← PartialEquiv.symm_source, symm_trans_symm, symm_trans_source_eq, inter_comm]
#align pretrivialization.symm_trans_target_eq Pretrivialization.symm_trans_target_eq
variable (e' : Pretrivialization F (π F E)) {x' : TotalSpace F E} {b : B} {y : E b}
@[simp]
theorem coe_mem_source : ↑y ∈ e'.source ↔ b ∈ e'.baseSet :=
e'.mem_source
#align pretrivialization.coe_mem_source Pretrivialization.coe_mem_source
@[simp, mfld_simps]
theorem coe_coe_fst (hb : b ∈ e'.baseSet) : (e' y).1 = b :=
e'.coe_fst (e'.mem_source.2 hb)
#align pretrivialization.coe_coe_fst Pretrivialization.coe_coe_fst
theorem mk_mem_target {x : B} {y : F} : (x, y) ∈ e'.target ↔ x ∈ e'.baseSet :=
e'.mem_target
#align pretrivialization.mk_mem_target Pretrivialization.mk_mem_target
theorem symm_coe_proj {x : B} {y : F} (e' : Pretrivialization F (π F E)) (h : x ∈ e'.baseSet) :
(e'.toPartialEquiv.symm (x, y)).1 = x :=
e'.proj_symm_apply' h
#align pretrivialization.symm_coe_proj Pretrivialization.symm_coe_proj
section Zero
variable [∀ x, Zero (E x)]
protected noncomputable def symm (e : Pretrivialization F (π F E)) (b : B) (y : F) : E b :=
if hb : b ∈ e.baseSet then
cast (congr_arg E (e.proj_symm_apply' hb)) (e.toPartialEquiv.symm (b, y)).2
else 0
#align pretrivialization.symm Pretrivialization.symm
theorem symm_apply (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
e.symm b y = cast (congr_arg E (e.symm_coe_proj hb)) (e.toPartialEquiv.symm (b, y)).2 :=
dif_pos hb
#align pretrivialization.symm_apply Pretrivialization.symm_apply
theorem symm_apply_of_not_mem (e : Pretrivialization F (π F E)) {b : B} (hb : b ∉ e.baseSet)
(y : F) : e.symm b y = 0 :=
dif_neg hb
#align pretrivialization.symm_apply_of_not_mem Pretrivialization.symm_apply_of_not_mem
theorem coe_symm_of_not_mem (e : Pretrivialization F (π F E)) {b : B} (hb : b ∉ e.baseSet) :
(e.symm b : F → E b) = 0 :=
funext fun _ => dif_neg hb
#align pretrivialization.coe_symm_of_not_mem Pretrivialization.coe_symm_of_not_mem
theorem mk_symm (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
TotalSpace.mk b (e.symm b y) = e.toPartialEquiv.symm (b, y) := by
simp only [e.symm_apply hb, TotalSpace.mk_cast (e.proj_symm_apply' hb), TotalSpace.eta]
#align pretrivialization.mk_symm Pretrivialization.mk_symm
theorem symm_proj_apply (e : Pretrivialization F (π F E)) (z : TotalSpace F E)
(hz : z.proj ∈ e.baseSet) : e.symm z.proj (e z).2 = z.2 := by
rw [e.symm_apply hz, cast_eq_iff_heq, e.mk_proj_snd' hz, e.symm_apply_apply (e.mem_source.mpr hz)]
#align pretrivialization.symm_proj_apply Pretrivialization.symm_proj_apply
theorem symm_apply_apply_mk (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet)
(y : E b) : e.symm b (e ⟨b, y⟩).2 = y :=
e.symm_proj_apply ⟨b, y⟩ hb
#align pretrivialization.symm_apply_apply_mk Pretrivialization.symm_apply_apply_mk
| Mathlib/Topology/FiberBundle/Trivialization.lean | 288 | 290 | theorem apply_mk_symm (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
e ⟨b, e.symm b y⟩ = (b, y) := by |
rw [e.mk_symm hb, e.apply_symm_apply (e.mk_mem_target.mpr hb)]
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y : Type v} {Z W ε ζ : Type*}
section Constructions
instance instTopologicalSpaceSubtype {p : X → Prop} [t : TopologicalSpace X] :
TopologicalSpace (Subtype p) :=
induced (↑) t
instance {r : X → X → Prop} [t : TopologicalSpace X] : TopologicalSpace (Quot r) :=
coinduced (Quot.mk r) t
instance instTopologicalSpaceQuotient {s : Setoid X} [t : TopologicalSpace X] :
TopologicalSpace (Quotient s) :=
coinduced Quotient.mk' t
instance instTopologicalSpaceProd [t₁ : TopologicalSpace X] [t₂ : TopologicalSpace Y] :
TopologicalSpace (X × Y) :=
induced Prod.fst t₁ ⊓ induced Prod.snd t₂
instance instTopologicalSpaceSum [t₁ : TopologicalSpace X] [t₂ : TopologicalSpace Y] :
TopologicalSpace (X ⊕ Y) :=
coinduced Sum.inl t₁ ⊔ coinduced Sum.inr t₂
instance instTopologicalSpaceSigma {ι : Type*} {X : ι → Type v} [t₂ : ∀ i, TopologicalSpace (X i)] :
TopologicalSpace (Sigma X) :=
⨆ i, coinduced (Sigma.mk i) (t₂ i)
instance Pi.topologicalSpace {ι : Type*} {Y : ι → Type v} [t₂ : (i : ι) → TopologicalSpace (Y i)] :
TopologicalSpace ((i : ι) → Y i) :=
⨅ i, induced (fun f => f i) (t₂ i)
#align Pi.topological_space Pi.topologicalSpace
instance ULift.topologicalSpace [t : TopologicalSpace X] : TopologicalSpace (ULift.{v, u} X) :=
t.induced ULift.down
#align ulift.topological_space ULift.topologicalSpace
section
variable [TopologicalSpace X]
open Additive Multiplicative
instance : TopologicalSpace (Additive X) := ‹TopologicalSpace X›
instance : TopologicalSpace (Multiplicative X) := ‹TopologicalSpace X›
instance [DiscreteTopology X] : DiscreteTopology (Additive X) := ‹DiscreteTopology X›
instance [DiscreteTopology X] : DiscreteTopology (Multiplicative X) := ‹DiscreteTopology X›
theorem continuous_ofMul : Continuous (ofMul : X → Additive X) := continuous_id
#align continuous_of_mul continuous_ofMul
theorem continuous_toMul : Continuous (toMul : Additive X → X) := continuous_id
#align continuous_to_mul continuous_toMul
theorem continuous_ofAdd : Continuous (ofAdd : X → Multiplicative X) := continuous_id
#align continuous_of_add continuous_ofAdd
theorem continuous_toAdd : Continuous (toAdd : Multiplicative X → X) := continuous_id
#align continuous_to_add continuous_toAdd
theorem isOpenMap_ofMul : IsOpenMap (ofMul : X → Additive X) := IsOpenMap.id
#align is_open_map_of_mul isOpenMap_ofMul
theorem isOpenMap_toMul : IsOpenMap (toMul : Additive X → X) := IsOpenMap.id
#align is_open_map_to_mul isOpenMap_toMul
theorem isOpenMap_ofAdd : IsOpenMap (ofAdd : X → Multiplicative X) := IsOpenMap.id
#align is_open_map_of_add isOpenMap_ofAdd
theorem isOpenMap_toAdd : IsOpenMap (toAdd : Multiplicative X → X) := IsOpenMap.id
#align is_open_map_to_add isOpenMap_toAdd
theorem isClosedMap_ofMul : IsClosedMap (ofMul : X → Additive X) := IsClosedMap.id
#align is_closed_map_of_mul isClosedMap_ofMul
theorem isClosedMap_toMul : IsClosedMap (toMul : Additive X → X) := IsClosedMap.id
#align is_closed_map_to_mul isClosedMap_toMul
theorem isClosedMap_ofAdd : IsClosedMap (ofAdd : X → Multiplicative X) := IsClosedMap.id
#align is_closed_map_of_add isClosedMap_ofAdd
theorem isClosedMap_toAdd : IsClosedMap (toAdd : Multiplicative X → X) := IsClosedMap.id
#align is_closed_map_to_add isClosedMap_toAdd
theorem nhds_ofMul (x : X) : 𝓝 (ofMul x) = map ofMul (𝓝 x) := rfl
#align nhds_of_mul nhds_ofMul
theorem nhds_ofAdd (x : X) : 𝓝 (ofAdd x) = map ofAdd (𝓝 x) := rfl
#align nhds_of_add nhds_ofAdd
theorem nhds_toMul (x : Additive X) : 𝓝 (toMul x) = map toMul (𝓝 x) := rfl
#align nhds_to_mul nhds_toMul
theorem nhds_toAdd (x : Multiplicative X) : 𝓝 (toAdd x) = map toAdd (𝓝 x) := rfl
#align nhds_to_add nhds_toAdd
end
section
variable [TopologicalSpace X]
open OrderDual
instance : TopologicalSpace Xᵒᵈ := ‹TopologicalSpace X›
instance [DiscreteTopology X] : DiscreteTopology Xᵒᵈ := ‹DiscreteTopology X›
theorem continuous_toDual : Continuous (toDual : X → Xᵒᵈ) := continuous_id
#align continuous_to_dual continuous_toDual
theorem continuous_ofDual : Continuous (ofDual : Xᵒᵈ → X) := continuous_id
#align continuous_of_dual continuous_ofDual
theorem isOpenMap_toDual : IsOpenMap (toDual : X → Xᵒᵈ) := IsOpenMap.id
#align is_open_map_to_dual isOpenMap_toDual
theorem isOpenMap_ofDual : IsOpenMap (ofDual : Xᵒᵈ → X) := IsOpenMap.id
#align is_open_map_of_dual isOpenMap_ofDual
theorem isClosedMap_toDual : IsClosedMap (toDual : X → Xᵒᵈ) := IsClosedMap.id
#align is_closed_map_to_dual isClosedMap_toDual
theorem isClosedMap_ofDual : IsClosedMap (ofDual : Xᵒᵈ → X) := IsClosedMap.id
#align is_closed_map_of_dual isClosedMap_ofDual
theorem nhds_toDual (x : X) : 𝓝 (toDual x) = map toDual (𝓝 x) := rfl
#align nhds_to_dual nhds_toDual
theorem nhds_ofDual (x : X) : 𝓝 (ofDual x) = map ofDual (𝓝 x) := rfl
#align nhds_of_dual nhds_ofDual
end
theorem Quotient.preimage_mem_nhds [TopologicalSpace X] [s : Setoid X] {V : Set <| Quotient s}
{x : X} (hs : V ∈ 𝓝 (Quotient.mk' x)) : Quotient.mk' ⁻¹' V ∈ 𝓝 x :=
preimage_nhds_coinduced hs
#align quotient.preimage_mem_nhds Quotient.preimage_mem_nhds
theorem Dense.quotient [Setoid X] [TopologicalSpace X] {s : Set X} (H : Dense s) :
Dense (Quotient.mk' '' s) :=
Quotient.surjective_Quotient_mk''.denseRange.dense_image continuous_coinduced_rng H
#align dense.quotient Dense.quotient
theorem DenseRange.quotient [Setoid X] [TopologicalSpace X] {f : Y → X} (hf : DenseRange f) :
DenseRange (Quotient.mk' ∘ f) :=
Quotient.surjective_Quotient_mk''.denseRange.comp hf continuous_coinduced_rng
#align dense_range.quotient DenseRange.quotient
theorem continuous_map_of_le {α : Type*} [TopologicalSpace α]
{s t : Setoid α} (h : s ≤ t) : Continuous (Setoid.map_of_le h) :=
continuous_coinduced_rng
theorem continuous_map_sInf {α : Type*} [TopologicalSpace α]
{S : Set (Setoid α)} {s : Setoid α} (h : s ∈ S) : Continuous (Setoid.map_sInf h) :=
continuous_coinduced_rng
instance {p : X → Prop} [TopologicalSpace X] [DiscreteTopology X] : DiscreteTopology (Subtype p) :=
⟨bot_unique fun s _ => ⟨(↑) '' s, isOpen_discrete _, preimage_image_eq _ Subtype.val_injective⟩⟩
instance Sum.discreteTopology [TopologicalSpace X] [TopologicalSpace Y] [h : DiscreteTopology X]
[hY : DiscreteTopology Y] : DiscreteTopology (X ⊕ Y) :=
⟨sup_eq_bot_iff.2 <| by simp [h.eq_bot, hY.eq_bot]⟩
#align sum.discrete_topology Sum.discreteTopology
instance Sigma.discreteTopology {ι : Type*} {Y : ι → Type v} [∀ i, TopologicalSpace (Y i)]
[h : ∀ i, DiscreteTopology (Y i)] : DiscreteTopology (Sigma Y) :=
⟨iSup_eq_bot.2 fun _ => by simp only [(h _).eq_bot, coinduced_bot]⟩
#align sigma.discrete_topology Sigma.discreteTopology
section Top
variable [TopologicalSpace X]
theorem mem_nhds_subtype (s : Set X) (x : { x // x ∈ s }) (t : Set { x // x ∈ s }) :
t ∈ 𝓝 x ↔ ∃ u ∈ 𝓝 (x : X), Subtype.val ⁻¹' u ⊆ t :=
mem_nhds_induced _ x t
#align mem_nhds_subtype mem_nhds_subtype
theorem nhds_subtype (s : Set X) (x : { x // x ∈ s }) : 𝓝 x = comap (↑) (𝓝 (x : X)) :=
nhds_induced _ x
#align nhds_subtype nhds_subtype
theorem nhdsWithin_subtype_eq_bot_iff {s t : Set X} {x : s} :
𝓝[((↑) : s → X) ⁻¹' t] x = ⊥ ↔ 𝓝[t] (x : X) ⊓ 𝓟 s = ⊥ := by
rw [inf_principal_eq_bot_iff_comap, nhdsWithin, nhdsWithin, comap_inf, comap_principal,
nhds_induced]
#align nhds_within_subtype_eq_bot_iff nhdsWithin_subtype_eq_bot_iff
theorem nhds_ne_subtype_eq_bot_iff {S : Set X} {x : S} :
𝓝[≠] x = ⊥ ↔ 𝓝[≠] (x : X) ⊓ 𝓟 S = ⊥ := by
rw [← nhdsWithin_subtype_eq_bot_iff, preimage_compl, ← image_singleton,
Subtype.coe_injective.preimage_image]
#align nhds_ne_subtype_eq_bot_iff nhds_ne_subtype_eq_bot_iff
theorem nhds_ne_subtype_neBot_iff {S : Set X} {x : S} :
(𝓝[≠] x).NeBot ↔ (𝓝[≠] (x : X) ⊓ 𝓟 S).NeBot := by
rw [neBot_iff, neBot_iff, not_iff_not, nhds_ne_subtype_eq_bot_iff]
#align nhds_ne_subtype_ne_bot_iff nhds_ne_subtype_neBot_iff
| Mathlib/Topology/Constructions.lean | 260 | 262 | theorem discreteTopology_subtype_iff {S : Set X} :
DiscreteTopology S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥ := by |
simp_rw [discreteTopology_iff_nhds_ne, SetCoe.forall', nhds_ne_subtype_eq_bot_iff]
|
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace AffineSubspace
section StrictOrderedCommRing
variable [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P]
variable [AddCommGroup V'] [Module R V'] [AddTorsor V' P']
def WSameSide (s : AffineSubspace R P) (x y : P) : Prop :=
∃ᵉ (p₁ ∈ s) (p₂ ∈ s), SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
#align affine_subspace.w_same_side AffineSubspace.WSameSide
def SSameSide (s : AffineSubspace R P) (x y : P) : Prop :=
s.WSameSide x y ∧ x ∉ s ∧ y ∉ s
#align affine_subspace.s_same_side AffineSubspace.SSameSide
def WOppSide (s : AffineSubspace R P) (x y : P) : Prop :=
∃ᵉ (p₁ ∈ s) (p₂ ∈ s), SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
#align affine_subspace.w_opp_side AffineSubspace.WOppSide
def SOppSide (s : AffineSubspace R P) (x y : P) : Prop :=
s.WOppSide x y ∧ x ∉ s ∧ y ∉ s
#align affine_subspace.s_opp_side AffineSubspace.SOppSide
theorem WSameSide.map {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y) (f : P →ᵃ[R] P') :
(s.map f).WSameSide (f x) (f y) := by
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
simp_rw [← linearMap_vsub]
exact h.map f.linear
#align affine_subspace.w_same_side.map AffineSubspace.WSameSide.map
theorem _root_.Function.Injective.wSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).WSameSide (f x) (f y) ↔ s.WSameSide x y := by
refine ⟨fun h => ?_, fun h => h.map _⟩
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩
rw [mem_map] at hfp₁ hfp₂
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩
refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩
simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h
exact h
#align function.injective.w_same_side_map_iff Function.Injective.wSameSide_map_iff
theorem _root_.Function.Injective.sSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).SSameSide (f x) (f y) ↔ s.SSameSide x y := by
simp_rw [SSameSide, hf.wSameSide_map_iff, mem_map_iff_mem_of_injective hf]
#align function.injective.s_same_side_map_iff Function.Injective.sSameSide_map_iff
@[simp]
theorem _root_.AffineEquiv.wSameSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P') :
(s.map ↑f).WSameSide (f x) (f y) ↔ s.WSameSide x y :=
(show Function.Injective f.toAffineMap from f.injective).wSameSide_map_iff
#align affine_equiv.w_same_side_map_iff AffineEquiv.wSameSide_map_iff
@[simp]
theorem _root_.AffineEquiv.sSameSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P') :
(s.map ↑f).SSameSide (f x) (f y) ↔ s.SSameSide x y :=
(show Function.Injective f.toAffineMap from f.injective).sSameSide_map_iff
#align affine_equiv.s_same_side_map_iff AffineEquiv.sSameSide_map_iff
theorem WOppSide.map {s : AffineSubspace R P} {x y : P} (h : s.WOppSide x y) (f : P →ᵃ[R] P') :
(s.map f).WOppSide (f x) (f y) := by
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
simp_rw [← linearMap_vsub]
exact h.map f.linear
#align affine_subspace.w_opp_side.map AffineSubspace.WOppSide.map
theorem _root_.Function.Injective.wOppSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).WOppSide (f x) (f y) ↔ s.WOppSide x y := by
refine ⟨fun h => ?_, fun h => h.map _⟩
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩
rw [mem_map] at hfp₁ hfp₂
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩
refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩
simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h
exact h
#align function.injective.w_opp_side_map_iff Function.Injective.wOppSide_map_iff
theorem _root_.Function.Injective.sOppSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).SOppSide (f x) (f y) ↔ s.SOppSide x y := by
simp_rw [SOppSide, hf.wOppSide_map_iff, mem_map_iff_mem_of_injective hf]
#align function.injective.s_opp_side_map_iff Function.Injective.sOppSide_map_iff
@[simp]
theorem _root_.AffineEquiv.wOppSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P') :
(s.map ↑f).WOppSide (f x) (f y) ↔ s.WOppSide x y :=
(show Function.Injective f.toAffineMap from f.injective).wOppSide_map_iff
#align affine_equiv.w_opp_side_map_iff AffineEquiv.wOppSide_map_iff
@[simp]
theorem _root_.AffineEquiv.sOppSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P') :
(s.map ↑f).SOppSide (f x) (f y) ↔ s.SOppSide x y :=
(show Function.Injective f.toAffineMap from f.injective).sOppSide_map_iff
#align affine_equiv.s_opp_side_map_iff AffineEquiv.sOppSide_map_iff
theorem WSameSide.nonempty {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y) :
(s : Set P).Nonempty :=
⟨h.choose, h.choose_spec.left⟩
#align affine_subspace.w_same_side.nonempty AffineSubspace.WSameSide.nonempty
theorem SSameSide.nonempty {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y) :
(s : Set P).Nonempty :=
⟨h.1.choose, h.1.choose_spec.left⟩
#align affine_subspace.s_same_side.nonempty AffineSubspace.SSameSide.nonempty
theorem WOppSide.nonempty {s : AffineSubspace R P} {x y : P} (h : s.WOppSide x y) :
(s : Set P).Nonempty :=
⟨h.choose, h.choose_spec.left⟩
#align affine_subspace.w_opp_side.nonempty AffineSubspace.WOppSide.nonempty
theorem SOppSide.nonempty {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) :
(s : Set P).Nonempty :=
⟨h.1.choose, h.1.choose_spec.left⟩
#align affine_subspace.s_opp_side.nonempty AffineSubspace.SOppSide.nonempty
theorem SSameSide.wSameSide {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y) :
s.WSameSide x y :=
h.1
#align affine_subspace.s_same_side.w_same_side AffineSubspace.SSameSide.wSameSide
theorem SSameSide.left_not_mem {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y) : x ∉ s :=
h.2.1
#align affine_subspace.s_same_side.left_not_mem AffineSubspace.SSameSide.left_not_mem
theorem SSameSide.right_not_mem {s : AffineSubspace R P} {x y : P} (h : s.SSameSide x y) : y ∉ s :=
h.2.2
#align affine_subspace.s_same_side.right_not_mem AffineSubspace.SSameSide.right_not_mem
theorem SOppSide.wOppSide {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) :
s.WOppSide x y :=
h.1
#align affine_subspace.s_opp_side.w_opp_side AffineSubspace.SOppSide.wOppSide
theorem SOppSide.left_not_mem {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) : x ∉ s :=
h.2.1
#align affine_subspace.s_opp_side.left_not_mem AffineSubspace.SOppSide.left_not_mem
theorem SOppSide.right_not_mem {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) : y ∉ s :=
h.2.2
#align affine_subspace.s_opp_side.right_not_mem AffineSubspace.SOppSide.right_not_mem
theorem wSameSide_comm {s : AffineSubspace R P} {x y : P} : s.WSameSide x y ↔ s.WSameSide y x :=
⟨fun ⟨p₁, hp₁, p₂, hp₂, h⟩ => ⟨p₂, hp₂, p₁, hp₁, h.symm⟩,
fun ⟨p₁, hp₁, p₂, hp₂, h⟩ => ⟨p₂, hp₂, p₁, hp₁, h.symm⟩⟩
#align affine_subspace.w_same_side_comm AffineSubspace.wSameSide_comm
alias ⟨WSameSide.symm, _⟩ := wSameSide_comm
#align affine_subspace.w_same_side.symm AffineSubspace.WSameSide.symm
| Mathlib/Analysis/Convex/Side.lean | 194 | 195 | theorem sSameSide_comm {s : AffineSubspace R P} {x y : P} : s.SSameSide x y ↔ s.SSameSide y x := by |
rw [SSameSide, SSameSide, wSameSide_comm, and_comm (b := x ∉ s)]
|
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f fun a n => n • {a}
-- Porting note: times out if h is not specified
map_add' _f _g := sum_add_index' (h := fun a n => n • ({a} : Multiset α))
(fun _ ↦ zero_nsmul _) (fun _ ↦ add_nsmul _)
map_zero' := sum_zero_index
theorem toMultiset_zero : toMultiset (0 : α →₀ ℕ) = 0 :=
rfl
#align finsupp.to_multiset_zero Finsupp.toMultiset_zero
theorem toMultiset_add (m n : α →₀ ℕ) : toMultiset (m + n) = toMultiset m + toMultiset n :=
toMultiset.map_add m n
#align finsupp.to_multiset_add Finsupp.toMultiset_add
theorem toMultiset_apply (f : α →₀ ℕ) : toMultiset f = f.sum fun a n => n • {a} :=
rfl
#align finsupp.to_multiset_apply Finsupp.toMultiset_apply
@[simp]
theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by
rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
#align finsupp.to_multiset_single Finsupp.toMultiset_single
theorem toMultiset_sum {f : ι → α →₀ ℕ} (s : Finset ι) :
Finsupp.toMultiset (∑ i ∈ s, f i) = ∑ i ∈ s, Finsupp.toMultiset (f i) :=
map_sum Finsupp.toMultiset _ _
#align finsupp.to_multiset_sum Finsupp.toMultiset_sum
theorem toMultiset_sum_single (s : Finset ι) (n : ℕ) :
Finsupp.toMultiset (∑ i ∈ s, single i n) = n • s.val := by
simp_rw [toMultiset_sum, Finsupp.toMultiset_single, sum_nsmul, sum_multiset_singleton]
#align finsupp.to_multiset_sum_single Finsupp.toMultiset_sum_single
@[simp]
theorem card_toMultiset (f : α →₀ ℕ) : Multiset.card (toMultiset f) = f.sum fun _ => id := by
simp [toMultiset_apply, map_finsupp_sum, Function.id_def]
#align finsupp.card_to_multiset Finsupp.card_toMultiset
theorem toMultiset_map (f : α →₀ ℕ) (g : α → β) :
f.toMultiset.map g = toMultiset (f.mapDomain g) := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAddMonoidHom g).map_nsmul]
rfl
#align finsupp.to_multiset_map Finsupp.toMultiset_map
@[to_additive (attr := simp)]
| Mathlib/Data/Finsupp/Multiset.lean | 83 | 90 | theorem prod_toMultiset [CommMonoid α] (f : α →₀ ℕ) :
f.toMultiset.prod = f.prod fun a n => a ^ n := by |
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.prod_zero, Finsupp.prod_zero_index]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.prod_add, ih, toMultiset_single, Multiset.prod_nsmul,
Finsupp.prod_add_index' pow_zero pow_add, Finsupp.prod_single_index, Multiset.prod_singleton]
exact pow_zero a
|
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section BagInter
@[simp]
| Mathlib/Data/List/Lattice.lean | 195 | 195 | theorem nil_bagInter (l : List α) : [].bagInter l = [] := by | cases l <;> rfl
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
section RealDerivOfComplex
open Complex
variable {e : ℂ → ℂ} {e' : ℂ} {z : ℝ}
theorem HasStrictDerivAt.real_of_complex (h : HasStrictDerivAt e e' z) :
HasStrictDerivAt (fun x : ℝ => (e x).re) e'.re z := by
have A : HasStrictFDerivAt ((↑) : ℝ → ℂ) ofRealCLM z := ofRealCLM.hasStrictFDerivAt
have B :
HasStrictFDerivAt e ((ContinuousLinearMap.smulRight 1 e' : ℂ →L[ℂ] ℂ).restrictScalars ℝ)
(ofRealCLM z) :=
h.hasStrictFDerivAt.restrictScalars ℝ
have C : HasStrictFDerivAt re reCLM (e (ofRealCLM z)) := reCLM.hasStrictFDerivAt
-- Porting note: this should be by:
-- simpa using (C.comp z (B.comp z A)).hasStrictDerivAt
-- but for some reason simp can not use `ContinuousLinearMap.comp_apply`
convert (C.comp z (B.comp z A)).hasStrictDerivAt
rw [ContinuousLinearMap.comp_apply, ContinuousLinearMap.comp_apply]
simp
#align has_strict_deriv_at.real_of_complex HasStrictDerivAt.real_of_complex
theorem HasDerivAt.real_of_complex (h : HasDerivAt e e' z) :
HasDerivAt (fun x : ℝ => (e x).re) e'.re z := by
have A : HasFDerivAt ((↑) : ℝ → ℂ) ofRealCLM z := ofRealCLM.hasFDerivAt
have B :
HasFDerivAt e ((ContinuousLinearMap.smulRight 1 e' : ℂ →L[ℂ] ℂ).restrictScalars ℝ)
(ofRealCLM z) :=
h.hasFDerivAt.restrictScalars ℝ
have C : HasFDerivAt re reCLM (e (ofRealCLM z)) := reCLM.hasFDerivAt
-- Porting note: this should be by:
-- simpa using (C.comp z (B.comp z A)).hasStrictDerivAt
-- but for some reason simp can not use `ContinuousLinearMap.comp_apply`
convert (C.comp z (B.comp z A)).hasDerivAt
rw [ContinuousLinearMap.comp_apply, ContinuousLinearMap.comp_apply]
simp
#align has_deriv_at.real_of_complex HasDerivAt.real_of_complex
theorem ContDiffAt.real_of_complex {n : ℕ∞} (h : ContDiffAt ℂ n e z) :
ContDiffAt ℝ n (fun x : ℝ => (e x).re) z := by
have A : ContDiffAt ℝ n ((↑) : ℝ → ℂ) z := ofRealCLM.contDiff.contDiffAt
have B : ContDiffAt ℝ n e z := h.restrict_scalars ℝ
have C : ContDiffAt ℝ n re (e z) := reCLM.contDiff.contDiffAt
exact C.comp z (B.comp z A)
#align cont_diff_at.real_of_complex ContDiffAt.real_of_complex
theorem ContDiff.real_of_complex {n : ℕ∞} (h : ContDiff ℂ n e) :
ContDiff ℝ n fun x : ℝ => (e x).re :=
contDiff_iff_contDiffAt.2 fun _ => h.contDiffAt.real_of_complex
#align cont_diff.real_of_complex ContDiff.real_of_complex
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
theorem HasStrictDerivAt.complexToReal_fderiv' {f : ℂ → E} {x : ℂ} {f' : E}
(h : HasStrictDerivAt f f' x) :
HasStrictFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x := by
simpa only [Complex.restrictScalars_one_smulRight'] using
h.hasStrictFDerivAt.restrictScalars ℝ
#align has_strict_deriv_at.complex_to_real_fderiv' HasStrictDerivAt.complexToReal_fderiv'
theorem HasDerivAt.complexToReal_fderiv' {f : ℂ → E} {x : ℂ} {f' : E} (h : HasDerivAt f f' x) :
HasFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x := by
simpa only [Complex.restrictScalars_one_smulRight'] using h.hasFDerivAt.restrictScalars ℝ
#align has_deriv_at.complex_to_real_fderiv' HasDerivAt.complexToReal_fderiv'
theorem HasDerivWithinAt.complexToReal_fderiv' {f : ℂ → E} {s : Set ℂ} {x : ℂ} {f' : E}
(h : HasDerivWithinAt f f' s x) :
HasFDerivWithinAt f (reCLM.smulRight f' + I • imCLM.smulRight f') s x := by
simpa only [Complex.restrictScalars_one_smulRight'] using
h.hasFDerivWithinAt.restrictScalars ℝ
#align has_deriv_within_at.complex_to_real_fderiv' HasDerivWithinAt.complexToReal_fderiv'
theorem HasStrictDerivAt.complexToReal_fderiv {f : ℂ → ℂ} {f' x : ℂ} (h : HasStrictDerivAt f f' x) :
HasStrictFDerivAt f (f' • (1 : ℂ →L[ℝ] ℂ)) x := by
simpa only [Complex.restrictScalars_one_smulRight] using h.hasStrictFDerivAt.restrictScalars ℝ
#align has_strict_deriv_at.complex_to_real_fderiv HasStrictDerivAt.complexToReal_fderiv
theorem HasDerivAt.complexToReal_fderiv {f : ℂ → ℂ} {f' x : ℂ} (h : HasDerivAt f f' x) :
HasFDerivAt f (f' • (1 : ℂ →L[ℝ] ℂ)) x := by
simpa only [Complex.restrictScalars_one_smulRight] using h.hasFDerivAt.restrictScalars ℝ
#align has_deriv_at.complex_to_real_fderiv HasDerivAt.complexToReal_fderiv
theorem HasDerivWithinAt.complexToReal_fderiv {f : ℂ → ℂ} {s : Set ℂ} {f' x : ℂ}
(h : HasDerivWithinAt f f' s x) : HasFDerivWithinAt f (f' • (1 : ℂ →L[ℝ] ℂ)) s x := by
simpa only [Complex.restrictScalars_one_smulRight] using h.hasFDerivWithinAt.restrictScalars ℝ
#align has_deriv_within_at.complex_to_real_fderiv HasDerivWithinAt.complexToReal_fderiv
theorem HasDerivAt.comp_ofReal (hf : HasDerivAt e e' ↑z) : HasDerivAt (fun y : ℝ => e ↑y) e' z := by
simpa only [ofRealCLM_apply, ofReal_one, mul_one] using hf.comp z ofRealCLM.hasDerivAt
#align has_deriv_at.comp_of_real HasDerivAt.comp_ofReal
| Mathlib/Analysis/Complex/RealDeriv.lean | 141 | 144 | theorem HasDerivAt.ofReal_comp {f : ℝ → ℝ} {u : ℝ} (hf : HasDerivAt f u z) :
HasDerivAt (fun y : ℝ => ↑(f y) : ℝ → ℂ) u z := by |
simpa only [ofRealCLM_apply, ofReal_one, real_smul, mul_one] using
ofRealCLM.hasDerivAt.scomp z hf
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Chebyshev
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.polynomial.dickson from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial
variable {R S : Type*} [CommRing R] [CommRing S] (k : ℕ) (a : R)
noncomputable def dickson : ℕ → R[X]
| 0 => 3 - k
| 1 => X
| n + 2 => X * dickson (n + 1) - C a * dickson n
#align polynomial.dickson Polynomial.dickson
@[simp]
theorem dickson_zero : dickson k a 0 = 3 - k :=
rfl
#align polynomial.dickson_zero Polynomial.dickson_zero
@[simp]
theorem dickson_one : dickson k a 1 = X :=
rfl
#align polynomial.dickson_one Polynomial.dickson_one
theorem dickson_two : dickson k a 2 = X ^ 2 - C a * (3 - k : R[X]) := by
simp only [dickson, sq]
#align polynomial.dickson_two Polynomial.dickson_two
@[simp]
theorem dickson_add_two (n : ℕ) :
dickson k a (n + 2) = X * dickson k a (n + 1) - C a * dickson k a n := by rw [dickson]
#align polynomial.dickson_add_two Polynomial.dickson_add_two
| Mathlib/RingTheory/Polynomial/Dickson.lean | 86 | 90 | theorem dickson_of_two_le {n : ℕ} (h : 2 ≤ n) :
dickson k a n = X * dickson k a (n - 1) - C a * dickson k a (n - 2) := by |
obtain ⟨n, rfl⟩ := Nat.exists_eq_add_of_le h
rw [add_comm]
exact dickson_add_two k a n
|
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
namespace List
open Nat
@[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : ∀ l : List α, drop 1 l = tail l
| [] | _ :: _ => rfl
theorem zipWith_distrib_tail : (zipWith f l l').tail = zipWith f l.tail l'.tail := by
rw [← drop_one]; simp [zipWith_distrib_drop]
theorem subset_def {l₁ l₂ : List α} : l₁ ⊆ l₂ ↔ ∀ {a : α}, a ∈ l₁ → a ∈ l₂ := .rfl
@[simp] theorem nil_subset (l : List α) : [] ⊆ l := nofun
@[simp] theorem Subset.refl (l : List α) : l ⊆ l := fun _ i => i
theorem Subset.trans {l₁ l₂ l₃ : List α} (h₁ : l₁ ⊆ l₂) (h₂ : l₂ ⊆ l₃) : l₁ ⊆ l₃ :=
fun _ i => h₂ (h₁ i)
instance : Trans (Membership.mem : α → List α → Prop) Subset Membership.mem :=
⟨fun h₁ h₂ => h₂ h₁⟩
instance : Trans (Subset : List α → List α → Prop) Subset Subset :=
⟨Subset.trans⟩
@[simp] theorem subset_cons (a : α) (l : List α) : l ⊆ a :: l := fun _ => Mem.tail _
theorem subset_of_cons_subset {a : α} {l₁ l₂ : List α} : a :: l₁ ⊆ l₂ → l₁ ⊆ l₂ :=
fun s _ i => s (mem_cons_of_mem _ i)
theorem subset_cons_of_subset (a : α) {l₁ l₂ : List α} : l₁ ⊆ l₂ → l₁ ⊆ a :: l₂ :=
fun s _ i => .tail _ (s i)
theorem cons_subset_cons {l₁ l₂ : List α} (a : α) (s : l₁ ⊆ l₂) : a :: l₁ ⊆ a :: l₂ :=
fun _ => by simp only [mem_cons]; exact Or.imp_right (@s _)
@[simp] theorem subset_append_left (l₁ l₂ : List α) : l₁ ⊆ l₁ ++ l₂ := fun _ => mem_append_left _
@[simp] theorem subset_append_right (l₁ l₂ : List α) : l₂ ⊆ l₁ ++ l₂ := fun _ => mem_append_right _
theorem subset_append_of_subset_left (l₂ : List α) : l ⊆ l₁ → l ⊆ l₁ ++ l₂ :=
fun s => Subset.trans s <| subset_append_left _ _
theorem subset_append_of_subset_right (l₁ : List α) : l ⊆ l₂ → l ⊆ l₁ ++ l₂ :=
fun s => Subset.trans s <| subset_append_right _ _
@[simp] theorem cons_subset : a :: l ⊆ m ↔ a ∈ m ∧ l ⊆ m := by
simp only [subset_def, mem_cons, or_imp, forall_and, forall_eq]
@[simp] theorem append_subset {l₁ l₂ l : List α} :
l₁ ++ l₂ ⊆ l ↔ l₁ ⊆ l ∧ l₂ ⊆ l := by simp [subset_def, or_imp, forall_and]
theorem subset_nil {l : List α} : l ⊆ [] ↔ l = [] :=
⟨fun h => match l with | [] => rfl | _::_ => (nomatch h (.head ..)), fun | rfl => Subset.refl _⟩
theorem map_subset {l₁ l₂ : List α} (f : α → β) (H : l₁ ⊆ l₂) : map f l₁ ⊆ map f l₂ :=
fun x => by simp only [mem_map]; exact .imp fun a => .imp_left (@H _)
@[simp] theorem nil_sublist : ∀ l : List α, [] <+ l
| [] => .slnil
| a :: l => (nil_sublist l).cons a
@[simp] theorem Sublist.refl : ∀ l : List α, l <+ l
| [] => .slnil
| a :: l => (Sublist.refl l).cons₂ a
theorem Sublist.trans {l₁ l₂ l₃ : List α} (h₁ : l₁ <+ l₂) (h₂ : l₂ <+ l₃) : l₁ <+ l₃ := by
induction h₂ generalizing l₁ with
| slnil => exact h₁
| cons _ _ IH => exact (IH h₁).cons _
| @cons₂ l₂ _ a _ IH =>
generalize e : a :: l₂ = l₂'
match e ▸ h₁ with
| .slnil => apply nil_sublist
| .cons a' h₁' => cases e; apply (IH h₁').cons
| .cons₂ a' h₁' => cases e; apply (IH h₁').cons₂
instance : Trans (@Sublist α) Sublist Sublist := ⟨Sublist.trans⟩
@[simp] theorem sublist_cons (a : α) (l : List α) : l <+ a :: l := (Sublist.refl l).cons _
theorem sublist_of_cons_sublist : a :: l₁ <+ l₂ → l₁ <+ l₂ :=
(sublist_cons a l₁).trans
@[simp] theorem sublist_append_left : ∀ l₁ l₂ : List α, l₁ <+ l₁ ++ l₂
| [], _ => nil_sublist _
| _ :: l₁, l₂ => (sublist_append_left l₁ l₂).cons₂ _
@[simp] theorem sublist_append_right : ∀ l₁ l₂ : List α, l₂ <+ l₁ ++ l₂
| [], _ => Sublist.refl _
| _ :: l₁, l₂ => (sublist_append_right l₁ l₂).cons _
theorem sublist_append_of_sublist_left (s : l <+ l₁) : l <+ l₁ ++ l₂ :=
s.trans <| sublist_append_left ..
theorem sublist_append_of_sublist_right (s : l <+ l₂) : l <+ l₁ ++ l₂ :=
s.trans <| sublist_append_right ..
@[simp]
theorem cons_sublist_cons : a :: l₁ <+ a :: l₂ ↔ l₁ <+ l₂ :=
⟨fun | .cons _ s => sublist_of_cons_sublist s | .cons₂ _ s => s, .cons₂ _⟩
@[simp] theorem append_sublist_append_left : ∀ l, l ++ l₁ <+ l ++ l₂ ↔ l₁ <+ l₂
| [] => Iff.rfl
| _ :: l => cons_sublist_cons.trans (append_sublist_append_left l)
theorem Sublist.append_left : l₁ <+ l₂ → ∀ l, l ++ l₁ <+ l ++ l₂ :=
fun h l => (append_sublist_append_left l).mpr h
theorem Sublist.append_right : l₁ <+ l₂ → ∀ l, l₁ ++ l <+ l₂ ++ l
| .slnil, _ => Sublist.refl _
| .cons _ h, _ => (h.append_right _).cons _
| .cons₂ _ h, _ => (h.append_right _).cons₂ _
theorem sublist_or_mem_of_sublist (h : l <+ l₁ ++ a :: l₂) : l <+ l₁ ++ l₂ ∨ a ∈ l := by
induction l₁ generalizing l with
| nil => match h with
| .cons _ h => exact .inl h
| .cons₂ _ h => exact .inr (.head ..)
| cons b l₁ IH =>
match h with
| .cons _ h => exact (IH h).imp_left (Sublist.cons _)
| .cons₂ _ h => exact (IH h).imp (Sublist.cons₂ _) (.tail _)
theorem Sublist.reverse : l₁ <+ l₂ → l₁.reverse <+ l₂.reverse
| .slnil => Sublist.refl _
| .cons _ h => by rw [reverse_cons]; exact sublist_append_of_sublist_left h.reverse
| .cons₂ _ h => by rw [reverse_cons, reverse_cons]; exact h.reverse.append_right _
@[simp] theorem reverse_sublist : l₁.reverse <+ l₂.reverse ↔ l₁ <+ l₂ :=
⟨fun h => l₁.reverse_reverse ▸ l₂.reverse_reverse ▸ h.reverse, Sublist.reverse⟩
@[simp] theorem append_sublist_append_right (l) : l₁ ++ l <+ l₂ ++ l ↔ l₁ <+ l₂ :=
⟨fun h => by
have := h.reverse
simp only [reverse_append, append_sublist_append_left, reverse_sublist] at this
exact this,
fun h => h.append_right l⟩
theorem Sublist.append (hl : l₁ <+ l₂) (hr : r₁ <+ r₂) : l₁ ++ r₁ <+ l₂ ++ r₂ :=
(hl.append_right _).trans ((append_sublist_append_left _).2 hr)
theorem Sublist.subset : l₁ <+ l₂ → l₁ ⊆ l₂
| .slnil, _, h => h
| .cons _ s, _, h => .tail _ (s.subset h)
| .cons₂ .., _, .head .. => .head ..
| .cons₂ _ s, _, .tail _ h => .tail _ (s.subset h)
instance : Trans (@Sublist α) Subset Subset :=
⟨fun h₁ h₂ => trans h₁.subset h₂⟩
instance : Trans Subset (@Sublist α) Subset :=
⟨fun h₁ h₂ => trans h₁ h₂.subset⟩
instance : Trans (Membership.mem : α → List α → Prop) Sublist Membership.mem :=
⟨fun h₁ h₂ => h₂.subset h₁⟩
theorem Sublist.length_le : l₁ <+ l₂ → length l₁ ≤ length l₂
| .slnil => Nat.le_refl 0
| .cons _l s => le_succ_of_le (length_le s)
| .cons₂ _ s => succ_le_succ (length_le s)
@[simp] theorem sublist_nil {l : List α} : l <+ [] ↔ l = [] :=
⟨fun s => subset_nil.1 s.subset, fun H => H ▸ Sublist.refl _⟩
theorem Sublist.eq_of_length : l₁ <+ l₂ → length l₁ = length l₂ → l₁ = l₂
| .slnil, _ => rfl
| .cons a s, h => nomatch Nat.not_lt.2 s.length_le (h ▸ lt_succ_self _)
| .cons₂ a s, h => by rw [s.eq_of_length (succ.inj h)]
theorem Sublist.eq_of_length_le (s : l₁ <+ l₂) (h : length l₂ ≤ length l₁) : l₁ = l₂ :=
s.eq_of_length <| Nat.le_antisymm s.length_le h
@[simp] theorem singleton_sublist {a : α} {l} : [a] <+ l ↔ a ∈ l := by
refine ⟨fun h => h.subset (mem_singleton_self _), fun h => ?_⟩
obtain ⟨_, _, rfl⟩ := append_of_mem h
exact ((nil_sublist _).cons₂ _).trans (sublist_append_right ..)
@[simp] theorem replicate_sublist_replicate {m n} (a : α) :
replicate m a <+ replicate n a ↔ m ≤ n := by
refine ⟨fun h => ?_, fun h => ?_⟩
· have := h.length_le; simp only [length_replicate] at this ⊢; exact this
· induction h with
| refl => apply Sublist.refl
| step => simp [*, replicate, Sublist.cons]
theorem isSublist_iff_sublist [BEq α] [LawfulBEq α] {l₁ l₂ : List α} :
l₁.isSublist l₂ ↔ l₁ <+ l₂ := by
cases l₁ <;> cases l₂ <;> simp [isSublist]
case cons.cons hd₁ tl₁ hd₂ tl₂ =>
if h_eq : hd₁ = hd₂ then
simp [h_eq, cons_sublist_cons, isSublist_iff_sublist]
else
simp only [beq_iff_eq, h_eq]
constructor
· intro h_sub
apply Sublist.cons
exact isSublist_iff_sublist.mp h_sub
· intro h_sub
cases h_sub
case cons h_sub =>
exact isSublist_iff_sublist.mpr h_sub
case cons₂ =>
contradiction
instance [DecidableEq α] (l₁ l₂ : List α) : Decidable (l₁ <+ l₂) :=
decidable_of_iff (l₁.isSublist l₂) isSublist_iff_sublist
theorem tail_eq_tailD (l) : @tail α l = tailD l [] := by cases l <;> rfl
theorem tail_eq_tail? (l) : @tail α l = (tail? l).getD [] := by simp [tail_eq_tailD]
@[simp] theorem next?_nil : @next? α [] = none := rfl
@[simp] theorem next?_cons (a l) : @next? α (a :: l) = some (a, l) := rfl
theorem get_eq_iff : List.get l n = x ↔ l.get? n.1 = some x := by simp [get?_eq_some]
theorem get?_inj
(h₀ : i < xs.length) (h₁ : Nodup xs) (h₂ : xs.get? i = xs.get? j) : i = j := by
induction xs generalizing i j with
| nil => cases h₀
| cons x xs ih =>
match i, j with
| 0, 0 => rfl
| i+1, j+1 => simp; cases h₁ with
| cons ha h₁ => exact ih (Nat.lt_of_succ_lt_succ h₀) h₁ h₂
| i+1, 0 => ?_ | 0, j+1 => ?_
all_goals
simp at h₂
cases h₁; rename_i h' h
have := h x ?_ rfl; cases this
rw [mem_iff_get?]
exact ⟨_, h₂⟩; exact ⟨_ , h₂.symm⟩
theorem tail_drop (l : List α) (n : Nat) : (l.drop n).tail = l.drop (n + 1) := by
induction l generalizing n with
| nil => simp
| cons hd tl hl =>
cases n
· simp
· simp [hl]
@[simp] theorem modifyNth_nil (f : α → α) (n) : [].modifyNth f n = [] := by cases n <;> rfl
@[simp] theorem modifyNth_zero_cons (f : α → α) (a : α) (l : List α) :
(a :: l).modifyNth f 0 = f a :: l := rfl
@[simp] theorem modifyNth_succ_cons (f : α → α) (a : α) (l : List α) (n) :
(a :: l).modifyNth f (n + 1) = a :: l.modifyNth f n := by rfl
theorem modifyNthTail_id : ∀ n (l : List α), l.modifyNthTail id n = l
| 0, _ => rfl
| _+1, [] => rfl
| n+1, a :: l => congrArg (cons a) (modifyNthTail_id n l)
theorem eraseIdx_eq_modifyNthTail : ∀ n (l : List α), eraseIdx l n = modifyNthTail tail n l
| 0, l => by cases l <;> rfl
| n+1, [] => rfl
| n+1, a :: l => congrArg (cons _) (eraseIdx_eq_modifyNthTail _ _)
@[deprecated] alias removeNth_eq_nth_tail := eraseIdx_eq_modifyNthTail
theorem get?_modifyNth (f : α → α) :
∀ n (l : List α) m, (modifyNth f n l).get? m = (fun a => if n = m then f a else a) <$> l.get? m
| n, l, 0 => by cases l <;> cases n <;> rfl
| n, [], _+1 => by cases n <;> rfl
| 0, _ :: l, m+1 => by cases h : l.get? m <;> simp [h, modifyNth, m.succ_ne_zero.symm]
| n+1, a :: l, m+1 =>
(get?_modifyNth f n l m).trans <| by
cases h' : l.get? m <;> by_cases h : n = m <;>
simp [h, if_pos, if_neg, Option.map, mt Nat.succ.inj, not_false_iff, h']
theorem modifyNthTail_length (f : List α → List α) (H : ∀ l, length (f l) = length l) :
∀ n l, length (modifyNthTail f n l) = length l
| 0, _ => H _
| _+1, [] => rfl
| _+1, _ :: _ => congrArg (·+1) (modifyNthTail_length _ H _ _)
theorem modifyNthTail_add (f : List α → List α) (n) (l₁ l₂ : List α) :
modifyNthTail f (l₁.length + n) (l₁ ++ l₂) = l₁ ++ modifyNthTail f n l₂ := by
induction l₁ <;> simp [*, Nat.succ_add]
theorem exists_of_modifyNthTail (f : List α → List α) {n} {l : List α} (h : n ≤ l.length) :
∃ l₁ l₂, l = l₁ ++ l₂ ∧ l₁.length = n ∧ modifyNthTail f n l = l₁ ++ f l₂ :=
have ⟨_, _, eq, hl⟩ : ∃ l₁ l₂, l = l₁ ++ l₂ ∧ l₁.length = n :=
⟨_, _, (take_append_drop n l).symm, length_take_of_le h⟩
⟨_, _, eq, hl, hl ▸ eq ▸ modifyNthTail_add (n := 0) ..⟩
@[simp] theorem modify_get?_length (f : α → α) : ∀ n l, length (modifyNth f n l) = length l :=
modifyNthTail_length _ fun l => by cases l <;> rfl
@[simp] theorem get?_modifyNth_eq (f : α → α) (n) (l : List α) :
(modifyNth f n l).get? n = f <$> l.get? n := by
simp only [get?_modifyNth, if_pos]
@[simp] theorem get?_modifyNth_ne (f : α → α) {m n} (l : List α) (h : m ≠ n) :
(modifyNth f m l).get? n = l.get? n := by
simp only [get?_modifyNth, if_neg h, id_map']
theorem exists_of_modifyNth (f : α → α) {n} {l : List α} (h : n < l.length) :
∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ l₁.length = n ∧ modifyNth f n l = l₁ ++ f a :: l₂ :=
match exists_of_modifyNthTail _ (Nat.le_of_lt h) with
| ⟨_, _::_, eq, hl, H⟩ => ⟨_, _, _, eq, hl, H⟩
| ⟨_, [], eq, hl, _⟩ => nomatch Nat.ne_of_gt h (eq ▸ append_nil _ ▸ hl)
theorem modifyNthTail_eq_take_drop (f : List α → List α) (H : f [] = []) :
∀ n l, modifyNthTail f n l = take n l ++ f (drop n l)
| 0, _ => rfl
| _ + 1, [] => H.symm
| n + 1, b :: l => congrArg (cons b) (modifyNthTail_eq_take_drop f H n l)
theorem modifyNth_eq_take_drop (f : α → α) :
∀ n l, modifyNth f n l = take n l ++ modifyHead f (drop n l) :=
modifyNthTail_eq_take_drop _ rfl
theorem modifyNth_eq_take_cons_drop (f : α → α) {n l} (h) :
modifyNth f n l = take n l ++ f (get l ⟨n, h⟩) :: drop (n + 1) l := by
rw [modifyNth_eq_take_drop, drop_eq_get_cons h]; rfl
theorem set_eq_modifyNth (a : α) : ∀ n (l : List α), set l n a = modifyNth (fun _ => a) n l
| 0, l => by cases l <;> rfl
| n+1, [] => rfl
| n+1, b :: l => congrArg (cons _) (set_eq_modifyNth _ _ _)
theorem set_eq_take_cons_drop (a : α) {n l} (h : n < length l) :
set l n a = take n l ++ a :: drop (n + 1) l := by
rw [set_eq_modifyNth, modifyNth_eq_take_cons_drop _ h]
theorem modifyNth_eq_set_get? (f : α → α) :
∀ n (l : List α), l.modifyNth f n = ((fun a => l.set n (f a)) <$> l.get? n).getD l
| 0, l => by cases l <;> rfl
| n+1, [] => rfl
| n+1, b :: l =>
(congrArg (cons _) (modifyNth_eq_set_get? ..)).trans <| by cases h : l.get? n <;> simp [h]
theorem modifyNth_eq_set_get (f : α → α) {n} {l : List α} (h) :
l.modifyNth f n = l.set n (f (l.get ⟨n, h⟩)) := by
rw [modifyNth_eq_set_get?, get?_eq_get h]; rfl
theorem exists_of_set {l : List α} (h : n < l.length) :
∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ l₁.length = n ∧ l.set n a' = l₁ ++ a' :: l₂ := by
rw [set_eq_modifyNth]; exact exists_of_modifyNth _ h
theorem exists_of_set' {l : List α} (h : n < l.length) :
∃ l₁ l₂, l = l₁ ++ l.get ⟨n, h⟩ :: l₂ ∧ l₁.length = n ∧ l.set n a' = l₁ ++ a' :: l₂ :=
have ⟨_, _, _, h₁, h₂, h₃⟩ := exists_of_set h; ⟨_, _, get_of_append h₁ h₂ ▸ h₁, h₂, h₃⟩
@[simp]
theorem get?_set_eq (a : α) (n) (l : List α) : (set l n a).get? n = (fun _ => a) <$> l.get? n := by
simp only [set_eq_modifyNth, get?_modifyNth_eq]
theorem get?_set_eq_of_lt (a : α) {n} {l : List α} (h : n < length l) :
(set l n a).get? n = some a := by rw [get?_set_eq, get?_eq_get h]; rfl
@[simp]
theorem get?_set_ne (a : α) {m n} (l : List α) (h : m ≠ n) : (set l m a).get? n = l.get? n := by
simp only [set_eq_modifyNth, get?_modifyNth_ne _ _ h]
theorem get?_set (a : α) {m n} (l : List α) :
(set l m a).get? n = if m = n then (fun _ => a) <$> l.get? n else l.get? n := by
by_cases m = n <;> simp [*, get?_set_eq, get?_set_ne]
theorem get?_set_of_lt (a : α) {m n} (l : List α) (h : n < length l) :
(set l m a).get? n = if m = n then some a else l.get? n := by
simp [get?_set, get?_eq_get h]
theorem get?_set_of_lt' (a : α) {m n} (l : List α) (h : m < length l) :
(set l m a).get? n = if m = n then some a else l.get? n := by
simp [get?_set]; split <;> subst_vars <;> simp [*, get?_eq_get h]
theorem drop_set_of_lt (a : α) {n m : Nat} (l : List α) (h : n < m) :
(l.set n a).drop m = l.drop m :=
List.ext fun i => by rw [get?_drop, get?_drop, get?_set_ne _ _ (by omega)]
theorem take_set_of_lt (a : α) {n m : Nat} (l : List α) (h : m < n) :
(l.set n a).take m = l.take m :=
List.ext fun i => by
rw [get?_take_eq_if, get?_take_eq_if]
split
· next h' => rw [get?_set_ne _ _ (by omega)]
· rfl
theorem length_eraseIdx : ∀ {l i}, i < length l → length (@eraseIdx α l i) = length l - 1
| [], _, _ => rfl
| _::_, 0, _ => by simp [eraseIdx]
| x::xs, i+1, h => by
have : i < length xs := Nat.lt_of_succ_lt_succ h
simp [eraseIdx, ← Nat.add_one]
rw [length_eraseIdx this, Nat.sub_add_cancel (Nat.lt_of_le_of_lt (Nat.zero_le _) this)]
@[deprecated] alias length_removeNth := length_eraseIdx
@[simp] theorem length_tail (l : List α) : length (tail l) = length l - 1 := by cases l <;> rfl
@[simp] theorem eraseP_nil : [].eraseP p = [] := rfl
theorem eraseP_cons (a : α) (l : List α) :
(a :: l).eraseP p = bif p a then l else a :: l.eraseP p := rfl
@[simp] theorem eraseP_cons_of_pos {l : List α} (p) (h : p a) : (a :: l).eraseP p = l := by
simp [eraseP_cons, h]
@[simp] theorem eraseP_cons_of_neg {l : List α} (p) (h : ¬p a) :
(a :: l).eraseP p = a :: l.eraseP p := by simp [eraseP_cons, h]
theorem eraseP_of_forall_not {l : List α} (h : ∀ a, a ∈ l → ¬p a) : l.eraseP p = l := by
induction l with
| nil => rfl
| cons _ _ ih => simp [h _ (.head ..), ih (forall_mem_cons.1 h).2]
theorem exists_of_eraseP : ∀ {l : List α} {a} (al : a ∈ l) (pa : p a),
∃ a l₁ l₂, (∀ b ∈ l₁, ¬p b) ∧ p a ∧ l = l₁ ++ a :: l₂ ∧ l.eraseP p = l₁ ++ l₂
| b :: l, a, al, pa =>
if pb : p b then
⟨b, [], l, forall_mem_nil _, pb, by simp [pb]⟩
else
match al with
| .head .. => nomatch pb pa
| .tail _ al =>
let ⟨c, l₁, l₂, h₁, h₂, h₃, h₄⟩ := exists_of_eraseP al pa
⟨c, b::l₁, l₂, (forall_mem_cons ..).2 ⟨pb, h₁⟩,
h₂, by rw [h₃, cons_append], by simp [pb, h₄]⟩
theorem exists_or_eq_self_of_eraseP (p) (l : List α) :
l.eraseP p = l ∨
∃ a l₁ l₂, (∀ b ∈ l₁, ¬p b) ∧ p a ∧ l = l₁ ++ a :: l₂ ∧ l.eraseP p = l₁ ++ l₂ :=
if h : ∃ a ∈ l, p a then
let ⟨_, ha, pa⟩ := h
.inr (exists_of_eraseP ha pa)
else
.inl (eraseP_of_forall_not (h ⟨·, ·, ·⟩))
@[simp] theorem length_eraseP_of_mem (al : a ∈ l) (pa : p a) :
length (l.eraseP p) = Nat.pred (length l) := by
let ⟨_, l₁, l₂, _, _, e₁, e₂⟩ := exists_of_eraseP al pa
rw [e₂]; simp [length_append, e₁]; rfl
theorem eraseP_append_left {a : α} (pa : p a) :
∀ {l₁ : List α} l₂, a ∈ l₁ → (l₁++l₂).eraseP p = l₁.eraseP p ++ l₂
| x :: xs, l₂, h => by
by_cases h' : p x <;> simp [h']
rw [eraseP_append_left pa l₂ ((mem_cons.1 h).resolve_left (mt _ h'))]
intro | rfl => exact pa
theorem eraseP_append_right :
∀ {l₁ : List α} l₂, (∀ b ∈ l₁, ¬p b) → eraseP p (l₁++l₂) = l₁ ++ l₂.eraseP p
| [], l₂, _ => rfl
| x :: xs, l₂, h => by
simp [(forall_mem_cons.1 h).1, eraseP_append_right _ (forall_mem_cons.1 h).2]
theorem eraseP_sublist (l : List α) : l.eraseP p <+ l := by
match exists_or_eq_self_of_eraseP p l with
| .inl h => rw [h]; apply Sublist.refl
| .inr ⟨c, l₁, l₂, _, _, h₃, h₄⟩ => rw [h₄, h₃]; simp
theorem eraseP_subset (l : List α) : l.eraseP p ⊆ l := (eraseP_sublist l).subset
protected theorem Sublist.eraseP : l₁ <+ l₂ → l₁.eraseP p <+ l₂.eraseP p
| .slnil => Sublist.refl _
| .cons a s => by
by_cases h : p a <;> simp [h]
exacts [s.eraseP.trans (eraseP_sublist _), s.eraseP.cons _]
| .cons₂ a s => by
by_cases h : p a <;> simp [h]
exacts [s, s.eraseP]
theorem mem_of_mem_eraseP {l : List α} : a ∈ l.eraseP p → a ∈ l := (eraseP_subset _ ·)
@[simp] theorem mem_eraseP_of_neg {l : List α} (pa : ¬p a) : a ∈ l.eraseP p ↔ a ∈ l := by
refine ⟨mem_of_mem_eraseP, fun al => ?_⟩
match exists_or_eq_self_of_eraseP p l with
| .inl h => rw [h]; assumption
| .inr ⟨c, l₁, l₂, h₁, h₂, h₃, h₄⟩ =>
rw [h₄]; rw [h₃] at al
have : a ≠ c := fun h => (h ▸ pa).elim h₂
simp [this] at al; simp [al]
theorem eraseP_map (f : β → α) : ∀ (l : List β), (map f l).eraseP p = map f (l.eraseP (p ∘ f))
| [] => rfl
| b::l => by by_cases h : p (f b) <;> simp [h, eraseP_map f l, eraseP_cons_of_pos]
@[simp] theorem extractP_eq_find?_eraseP
(l : List α) : extractP p l = (find? p l, eraseP p l) := by
let rec go (acc) : ∀ xs, l = acc.data ++ xs →
extractP.go p l xs acc = (xs.find? p, acc.data ++ xs.eraseP p)
| [] => fun h => by simp [extractP.go, find?, eraseP, h]
| x::xs => by
simp [extractP.go, find?, eraseP]; cases p x <;> simp
· intro h; rw [go _ xs]; {simp}; simp [h]
exact go #[] _ rfl
section erase
variable [BEq α]
theorem erase_eq_eraseP' (a : α) (l : List α) : l.erase a = l.eraseP (· == a) := by
induction l
· simp
· next b t ih =>
rw [erase_cons, eraseP_cons, ih]
if h : b == a then simp [h] else simp [h]
theorem erase_eq_eraseP [LawfulBEq α] (a : α) : ∀ l : List α, l.erase a = l.eraseP (a == ·)
| [] => rfl
| b :: l => by
if h : a = b then simp [h] else simp [h, Ne.symm h, erase_eq_eraseP a l]
theorem exists_erase_eq [LawfulBEq α] {a : α} {l : List α} (h : a ∈ l) :
∃ l₁ l₂, a ∉ l₁ ∧ l = l₁ ++ a :: l₂ ∧ l.erase a = l₁ ++ l₂ := by
let ⟨_, l₁, l₂, h₁, e, h₂, h₃⟩ := exists_of_eraseP h (beq_self_eq_true _)
rw [erase_eq_eraseP]; exact ⟨l₁, l₂, fun h => h₁ _ h (beq_self_eq_true _), eq_of_beq e ▸ h₂, h₃⟩
@[simp] theorem length_erase_of_mem [LawfulBEq α] {a : α} {l : List α} (h : a ∈ l) :
length (l.erase a) = Nat.pred (length l) := by
rw [erase_eq_eraseP]; exact length_eraseP_of_mem h (beq_self_eq_true a)
theorem erase_append_left [LawfulBEq α] {l₁ : List α} (l₂) (h : a ∈ l₁) :
(l₁ ++ l₂).erase a = l₁.erase a ++ l₂ := by
simp [erase_eq_eraseP]; exact eraseP_append_left (beq_self_eq_true a) l₂ h
| .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 564 | 567 | theorem erase_append_right [LawfulBEq α] {a : α} {l₁ : List α} (l₂ : List α) (h : a ∉ l₁) :
(l₁ ++ l₂).erase a = (l₁ ++ l₂.erase a) := by |
rw [erase_eq_eraseP, erase_eq_eraseP, eraseP_append_right]
intros b h' h''; rw [eq_of_beq h''] at h; exact h h'
|
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