feat(library/theories/topology/limit.lean): add topological filters
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Jeremy Avigad
Leonardo de Moura
parent
b8c230a55d
commit
eae80118bf
1 changed files with 203 additions and 3 deletions
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@ -62,7 +62,7 @@ and iff rules. Again, see also specific versions for metric spaces, normed space
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Note that the filters at_infty and at_ninfty don't rely on topological notions at all, only the
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existence of a suitable order.
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-/
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import data.set data.nat algebra.interval
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import data.set data.nat algebra.interval .basic
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import theories.move -- TODO: remove after move to Lean 3
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open set function set.filter interval
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@ -263,6 +263,7 @@ section linorderX
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(assume H, obtain x Hx, from H, eventually_at_infty Hx)
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end linorderX
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/- at_ninfty -/
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section linorderX
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@ -315,7 +316,8 @@ section linorderX
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(assume H, obtain x Hx, from H, eventually_at_ninfty Hx)
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end linorderX
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/- approaches infty -/
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/- approaches_infty -/
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section approaches_infty
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variable [linear_strong_order_pair Y]
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@ -355,7 +357,8 @@ section approaches_infty
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eventually_mpr H (eventually_mpr this (eventually_of_forall _ (take x, le.trans))))
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end approaches_infty
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/- approaches ninfty -/
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/- approaches_ninfty -/
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section approaches_ninfty
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variable [linear_strong_order_pair Y]
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@ -395,4 +398,201 @@ section approaches_ninfty
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(take x H₁ H₂, le.trans H₂ H₁))))
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end approaches_ninfty
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/- the nhds filter -/
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open topology
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section nhds_filter
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variables [topology X] {P : X → Prop}
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definition nhds (x : X) : filter X := Inf (principal ' {s | Open s ∧ x ∈ s})
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proposition eventually_nhds {x : X} {s : set X} (Os : Open s) (xs : x ∈ s) (H : ∀₀ x ∈ s, P x) :
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eventually P (nhds x) :=
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eventually_Inf (mem_image_of_mem _ (and.intro Os xs)) (eventually_principal H)
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proposition eventually_mem_nhds {x : X} {s : set X} (Os : Open s) (xs : x ∈ s) :
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eventually (λ x, x ∈ s) (nhds x) :=
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eventually_nhds Os xs (λ x Hx, Hx)
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proposition exists_of_eventually_nhds {x : X} (H : eventually P (nhds x)) :
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∃ s, Open s ∧ x ∈ s ∧ ∀₀ x ∈ s, P x :=
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let princS := principal ' {s | Open s ∧ x ∈ s} in
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have principal univ ∈ princS,
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from mem_image_of_mem principal (and.intro Open_univ !mem_univ),
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have ∃₀ F ∈ princS, eventually P F, from
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exists_eventually_of_eventually_Inf this H
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(λ F₁ (F₁mem : F₁ ∈ princS) F₂ (F₂mem : F₂ ∈ princS),
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obtain s₁ [[Os₁ xs₁] (F₁eq : principal s₁ = F₁)], from F₁mem,
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obtain s₂ [[Os₂ xs₂] (F₂eq : principal s₂ = F₂)], from F₂mem,
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have F₁ ⊓ F₂ ∈ princS,
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begin
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rewrite [-F₁eq, -F₂eq, principal_inf_principal],
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exact mem_image_of_mem _ (and.intro (Open_inter Os₁ Os₂) (mem_inter xs₁ xs₂))
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end,
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show ∃₀ F ∈ princS, F ≤ F₁ ⊓ F₂, from exists.intro _ (and.intro this !le.refl)),
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obtain F [Fmem ePF], from this,
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obtain s [[Os xs] (Feq : principal s = F)], from Fmem,
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exists.intro s (and.intro Os (and.intro xs
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(subset_of_eventually_principal (by rewrite Feq; exact ePF))))
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proposition eventually_nhds_iff (x : X) :
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eventually P (nhds x) ↔ ∃ s, Open s ∧ x ∈ s ∧ ∀₀ x ∈ s, P x :=
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iff.intro exists_of_eventually_nhds
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(assume H, obtain s [Os [xs Hs]], from H, eventually_nhds Os xs Hs)
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end nhds_filter
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/- the at_within filter -/
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section at_within
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variables [topology X] {P : X → Prop}
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definition at_within (x : X) (s : set X) : filter X := inf (nhds x) (principal (s \ '{x}))
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notation [at x ` within ` s] := at_within x s
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proposition eventually_at_within {x : X} {s t : set X} (Os : Open s) (xs : x ∈ s)
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(Hs : ∀₀ y ∈ s, y ≠ x → y ∈ t → P y) :
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eventually P [at x within t] :=
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have H₁ : eventually (λ y, y ≠ x → y ∈ t → P y) (nhds x),
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from eventually_nhds Os xs Hs,
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eventually_inf H₁ _ (mem_principal _)
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(take y, assume Hy,
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have H : y ∈ t \ '{x}, from and.right Hy,
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have H' : y ≠ x, from suppose y = x,
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have y ∈ '{x}, from mem_singleton_of_eq this,
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show false, from and.right H this,
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show P y, from and.left Hy `y ≠ x` (and.left H))
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proposition exists_of_eventually_at_within {x : X} {t : set X}
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(H : eventually P [at x within t]) :
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∃ s, Open s ∧ x ∈ s ∧ ∀₀ y ∈ s, y ≠ x → y ∈ t → P y :=
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obtain P₁ [eP₁nhds [P₂ [eP₂princ (H' : P ⊇ P₁ ∩ P₂)]]], from exists_of_eventually_inf H,
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obtain s [Os [xs Hs]], from exists_of_eventually_nhds eP₁nhds,
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have Ht : ∀₀ y ∈ t \ '{x}, P₂ y, from subset_of_eventually_principal eP₂princ,
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exists.intro s (and.intro Os (and.intro xs
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(take y, assume ys ynex yt,
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have y ∉ '{x}, from assume ymem, ynex (eq_of_mem_singleton ymem),
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show P y, from H' (and.intro (Hs y ys) (Ht (mem_diff yt this))))))
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theorem eventually_at_within_iff (x : X) {s : set X} :
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eventually P [at x within s] ↔ ∃ t, Open t ∧ x ∈ t ∧ ∀₀ y ∈ t, y ≠ x → y ∈ s → P y :=
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iff.intro exists_of_eventually_at_within
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(assume H, obtain t [Ot [xt Ht]], from H, eventually_at_within Ot xt Ht)
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abbreviation at_elt (x : X) : filter X := at_within x univ
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notation [at x] := at_elt x
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proposition eventually_at {x : X} {s : set X} (Os : Open s) (xs : x ∈ s)
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(Hs : ∀₀ y ∈ s, y ≠ x → P y) :
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eventually P [at x] :=
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eventually_at_within Os xs (take y, assume ys ynex yuniv, Hs ys ynex)
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proposition exists_of_eventually_at {x : X} (H : eventually P [at x]) :
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∃ s, Open s ∧ x ∈ s ∧ ∀₀ y ∈ s, y ≠ x → P y :=
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obtain s [Os [xs Hs]], from exists_of_eventually_at_within H,
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exists.intro s (and.intro Os (and.intro xs (λ y ys ynex, Hs y ys ynex trivial)))
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proposition eventually_at_iff (x : X) :
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eventually P [at x] ↔ ∃ s, Open s ∧ x ∈ s ∧ ∀₀ y ∈ s, y ≠ x → P y :=
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iff.intro exists_of_eventually_at
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(assume H, obtain s [Os [xs Hs]], from H, eventually_at Os xs Hs)
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proposition at_within_eq_of_Open {x : X} {s : set X} (Os : Open s) (xs : x ∈ s) :
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[at x within s] = [at x] :=
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filter.eq (ext (take P, iff.intro
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(assume H,
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obtain t [Ot [xt Ht]], from exists_of_eventually_at_within H,
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eventually_at (Open_inter Os Ot) (mem_inter xs xt)
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(λ y Hy yne, Ht _ (and.right Hy) yne (and.left Hy)))
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(assume H,
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obtain t [Ot [xt Ht]], from exists_of_eventually_at H,
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eventually_at_within (Open_inter Os Ot) (mem_inter xs xt)
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(λ y Hy yne ys, Ht _ (and.right Hy) yne))))
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end at_within
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/- at_left and at_right -/
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section at_left_at_right
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variables [topology X] [linear_strong_order_pair X] {P : X → Prop}
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abbreviation at_left (x : X) : filter X := [at x within '(-∞, x)]
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abbreviation at_right (x : X) : filter X := [at x within '(x, ∞)]
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notation [at x`-]` := at_left x
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notation [at x`+]` := at_right x
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proposition eventually_at_left {x : X} {s : set X} (Os : Open s) (xs : x ∈ s)
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(H : ∀₀ y ∈ s, y < x → P y) :
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eventually P [at x-] :=
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eventually_at_within Os xs (take y, assume ys ynex yltx, H ys yltx)
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proposition exists_of_eventually_at_left {x : X} (H : eventually P [at x-]) :
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∃ s, Open s ∧ x ∈ s ∧ ∀₀ y ∈ s, y < x → P y :=
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obtain s [Os [xs Hs]], from exists_of_eventually_at_within H,
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exists.intro s (and.intro Os (and.intro xs
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(take y, assume ys yltx,
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show P y, from Hs y ys (ne_of_lt yltx) yltx)))
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variable (P)
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proposition eventually_at_left_iff (x : X) :
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eventually P [at x-] ↔ ∃ s, Open s ∧ x ∈ s ∧ ∀₀ y ∈ s, y < x → P y :=
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iff.intro exists_of_eventually_at_left
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(take H, obtain s [Os [xs Hs]], from H, eventually_at_left Os xs Hs)
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variable {P}
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proposition eventually_at_right {x : X} {s : set X} (Os : Open s) (xs : x ∈ s)
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(H : ∀₀ y ∈ s, y > x → P y) :
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eventually P [at x+] :=
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eventually_at_within Os xs (take y, assume ys ynex yltx, H ys yltx)
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proposition exists_of_eventually_at_right {x : X} (H : eventually P [at x+]) :
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∃ s, Open s ∧ x ∈ s ∧ ∀₀ y ∈ s, y > x → P y :=
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obtain s [Os [xs Hs]], from exists_of_eventually_at_within H,
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exists.intro s (and.intro Os (and.intro xs
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(take y, assume ys yltx,
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show P y, from Hs y ys (ne_of_gt yltx) yltx)))
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variable (P)
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proposition eventually_at_right_iff (x : X) :
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eventually P [at x+] ↔ ∃ s, Open s ∧ x ∈ s ∧ ∀₀ y ∈ s, y > x → P y :=
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iff.intro exists_of_eventually_at_right
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(take H, obtain s [Os [xs Hs]], from H, eventually_at_right Os xs Hs)
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end at_left_at_right
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/- approaches -/
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section approaches
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variables [topology Y]
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variables {F : filter X} {f : X → Y} {y : Y}
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abbreviation approaches (f : X → Y) (y : Y) : filter X → Prop := tendsto f (nhds y)
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notation f ⟶ y := approaches f y
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proposition approaches_intro (H : ∀ s, Open s → y ∈ s → eventually (λ x, f x ∈ s) F) :
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(f ⟶ y) F :=
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tendsto_of_forall_eventually
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(take P, assume eventuallyP,
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obtain s [Os [(ys : y ∈ s) (H' : ∀₀ y' ∈ s, P y')]],
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from exists_of_eventually_nhds eventuallyP,
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have eventually (λ x, P (f x)) F, from eventually_mono (H s Os ys) (λ x Hx, H' Hx),
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show eventually P (mapfilter f F), from eventually_mapfilter this)
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proposition approaches_elim (H : (f ⟶ y) F) {s : set Y} (Os : Open s) (ys : y ∈ s) :
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eventually (λ x, f x ∈ s) F :=
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have eventually (λ y', y' ∈ s) (mapfilter f F),
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from eventually_mapfilter_of_tendsto H (eventually_mem_nhds Os ys),
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eventually_of_eventually_mapfilter this
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variables (F f y)
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proposition approaches_iff : (f ⟶ y) F ↔ (∀ s, Open s → y ∈ s → eventually (λ x, f x ∈ s) F) :=
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iff.intro approaches_elim approaches_intro
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end approaches
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end topology
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