refactor(library/theories/analysis/metric_space): refactor some proofs
This commit is contained in:
Jeremy Avigad
Leonardo de Moura
parent
e17c5c4f08
commit
2bc67cf936
1 changed files with 71 additions and 182 deletions
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@ -57,7 +57,7 @@ eq_of_dist_eq_zero (eq_zero_of_nonneg_of_forall_le !dist_nonneg H)
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definition open_ball (x : M) (ε : ℝ) := {y | dist y x < ε}
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theorem open_ball_empty_of_nonpos (x : M) {ε : ℝ} (Hε : ε ≤ 0) : open_ball x ε = ∅ :=
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theorem open_ball_eq_empty_of_nonpos (x : M) {ε : ℝ} (Hε : ε ≤ 0) : open_ball x ε = ∅ :=
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begin
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apply eq_empty_of_forall_not_mem,
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intro y Hlt,
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@ -65,11 +65,11 @@ theorem open_ball_empty_of_nonpos (x : M) {ε : ℝ} (Hε : ε ≤ 0) : open_bal
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apply lt_of_lt_of_le Hlt Hε
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end
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theorem radius_pos_of_nonempty {x : M} {ε : ℝ} {u : M} (Hu : u ∈ open_ball x ε) : ε > 0 :=
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theorem pos_of_mem_open_ball {x : M} {ε : ℝ} {u : M} (Hu : u ∈ open_ball x ε) : ε > 0 :=
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begin
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apply lt_of_not_ge,
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intro Hge,
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note Hop := open_ball_empty_of_nonpos x Hge,
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note Hop := open_ball_eq_empty_of_nonpos x Hge,
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rewrite Hop at Hu,
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apply not_mem_empty _ Hu
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end
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@ -77,205 +77,94 @@ theorem radius_pos_of_nonempty {x : M} {ε : ℝ} {u : M} (Hu : u ∈ open_ball
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theorem mem_open_ball (x : M) {ε : ℝ} (H : ε > 0) : x ∈ open_ball x ε :=
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show dist x x < ε, by rewrite dist_self; assumption
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definition closed_ball (x : M) (ε : ℝ) := {y | dist x y ≤ ε}
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definition closed_ball (x : M) (ε : ℝ) := {y | dist y x ≤ ε}
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theorem closed_ball_eq_compl (x : M) (ε : ℝ) : closed_ball x ε = - {y | dist x y > ε} :=
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begin
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apply ext,
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intro y,
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apply iff.intro,
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intro Hle,
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apply mem_compl,
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intro Hgt,
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apply not_le_of_gt Hgt Hle,
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intro Hx,
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note Hx' := not_mem_of_mem_compl Hx,
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apply le_of_not_gt,
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intro Hgt,
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apply Hx',
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exact Hgt
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end
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theorem closed_ball_eq_compl (x : M) (ε : ℝ) : closed_ball x ε = - {y | dist y x > ε} :=
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ext (take y, iff.intro
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(suppose dist y x ≤ ε, not_lt_of_ge this)
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(suppose ¬ dist y x > ε, le_of_not_gt this))
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variable (M)
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definition open_sets_basis :=
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image (λ pair : M × ℝ, open_ball (pr1 pair) (pr2 pair)) univ
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definition open_sets_basis : set (set M) := { s | ∃ x, ∃ ε, s = open_ball x ε }
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definition metric_topology [instance] : topology M :=
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topology.generated_by (open_sets_basis M)
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definition metric_topology [instance] : topology M := topology.generated_by (open_sets_basis M)
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variable {M}
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theorem open_ball_mem_open_sets_basis (x : M) (ε : ℝ) : open_ball x ε ∈ open_sets_basis M :=
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mem_image !mem_univ rfl
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exists.intro x (exists.intro ε rfl)
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theorem open_ball_open (x : M) (ε : ℝ) : Open (open_ball x ε) :=
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theorem Open_open_ball (x : M) (ε : ℝ) : Open (open_ball x ε) :=
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by apply generators_mem_topology_generated_by; apply open_ball_mem_open_sets_basis
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theorem closed_ball_closed (x : M) {ε : ℝ} (H : ε > 0) : closed (closed_ball x ε) :=
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begin
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apply iff.mpr !closed_iff_Open_compl,
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rewrite closed_ball_eq_compl,
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rewrite compl_compl,
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apply Open_of_forall_exists_Open_nbhd,
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intro y Hxy,
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existsi open_ball y (dist x y - ε),
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split,
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apply open_ball_open,
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split,
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apply mem_open_ball,
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apply sub_pos_of_lt Hxy,
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intros y' Hxy'd,
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apply lt_of_not_ge,
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intro Hxy',
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apply not_lt_self (dist x y),
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exact calc
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dist x y ≤ dist x y' + dist y' y : dist_triangle
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... ≤ ε + dist y' y : add_le_add_right Hxy'
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... < ε + (dist x y - ε) : add_lt_add_left Hxy'd
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... = dist x y : by rewrite [add.comm, sub_add_cancel]
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end
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theorem closed_closed_ball (x : M) {ε : ℝ} (H : ε > 0) : closed (closed_ball x ε) :=
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Open_of_forall_exists_Open_nbhd
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(take y, suppose ¬ dist y x ≤ ε,
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have dist y x > ε, from lt_of_not_ge this,
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let B := open_ball y (dist y x - ε) in
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have y ∈ B, from mem_open_ball y (sub_pos_of_lt this),
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have B ⊆ - closed_ball x ε, from
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take y',
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assume Hy'y : dist y' y < dist y x - ε,
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assume Hy'x : dist y' x ≤ ε,
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show false, from not_lt_self (dist y x)
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(calc
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dist y x ≤ dist y y' + dist y' x : dist_triangle
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... < dist y x - ε + dist y' x : by rewrite dist_comm; apply add_lt_add_right Hy'y
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... ≤ dist y x - ε + ε : add_le_add_left Hy'x
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... = dist y x : by rewrite [sub_add_cancel]),
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exists.intro B (and.intro (Open_open_ball _ _) (and.intro `y ∈ B` this)))
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private theorem not_mem_open_basis_of_boundary_pt {s : set M} (a : s ∈ open_sets_basis M) {x : M}
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(Hbd : ∀ ε : ℝ, ε > 0 → ∃ v : M, v ∉ s ∧ dist x v < ε) : ¬ x ∈ s :=
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begin
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intro HxU,
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cases a with pr Hpr,
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cases pr with y r,
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cases Hpr with _ Hs,
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rewrite -Hs at HxU,
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have H : dist x y < r, from HxU,
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cases Hbd _ (sub_pos_of_lt H) with v Hv,
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cases Hv with Hv Hvdist,
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apply Hv,
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rewrite -Hs,
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apply lt_of_le_of_lt,
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apply dist_triangle,
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exact x,
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esimp,
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rewrite dist_comm,
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exact add_lt_of_lt_sub_right Hvdist
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end
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proposition open_ball_subset_open_ball_of_le (x : M) {r₁ r₂ : ℝ} (H : r₁ ≤ r₂) :
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open_ball x r₁ ⊆ open_ball x r₂ :=
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take y, assume ymem, lt_of_lt_of_le ymem H
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private theorem not_mem_intersect_of_boundary_pt {s t : set M} (a : Open s) (a_1 : Open t) {x : M}
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(v_0 : (x ∈ s → ¬ (∀ (ε : ℝ), ε > 0 → (∃ (v : M), v ∉ s ∧ dist x v < ε))))
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(v_1 : (x ∈ t → ¬ (∀ (ε : ℝ), ε > 0 → (∃ (v : M), v ∉ t ∧ dist x v < ε))))
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(Hbd : ∀ (ε : ℝ), ε > 0 → (∃ (v : M), v ∉ s ∩ t ∧ dist x v < ε)) : ¬ (x ∈ s ∩ t) :=
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begin
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intro HxU,
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have Hxs : x ∈ s, from mem_of_mem_inter_left HxU,
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have Hxt : x ∈ t, from mem_of_mem_inter_right HxU,
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note Hsih := exists_not_of_not_forall (v_0 Hxs),
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note Htih := exists_not_of_not_forall (v_1 Hxt),
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cases Hsih with ε1 Hε1,
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cases Htih with ε2 Hε2,
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note Hε1' := and_not_of_not_implies Hε1,
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note Hε2' := and_not_of_not_implies Hε2,
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cases Hε1' with Hε1p Hε1',
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cases Hε2' with Hε2p Hε2',
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note Hε1'' := forall_not_of_not_exists Hε1',
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note Hε2'' := forall_not_of_not_exists Hε2',
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have Hmin : min ε1 ε2 > 0, from lt_min Hε1p Hε2p,
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cases Hbd _ Hmin with v Hv,
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cases Hv with Hvint Hvdist,
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note Hε1v := Hε1'' v,
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note Hε2v := Hε2'' v,
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cases em (v ∉ s) with Hnm Hmem,
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apply Hε1v,
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split,
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exact Hnm,
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apply lt_of_lt_of_le Hvdist,
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apply min_le_left,
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apply Hε2v,
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have Hmem' : v ∈ s, from not_not_elim Hmem,
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note Hnm := not_mem_of_mem_of_not_mem_inter_left Hmem' Hvint,
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split,
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exact Hnm,
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apply lt_of_lt_of_le Hvdist,
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apply min_le_right
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end
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private theorem not_mem_sUnion_of_boundary_pt {S : set (set M)} (a : ∀₀ s ∈ S, Open s) {x : M}
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(v_0 : ∀ ⦃x_1 : set M⦄,
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x_1 ∈ S → x ∈ x_1 → ¬ (∀ (ε : ℝ), ε > 0 → (∃ (v : M), v ∉ x_1 ∧ dist x v < ε)))
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(Hbd : ∀ (ε : ℝ), ε > 0 → (∃ (v : M), v ∉ ⋃₀ S ∧ dist x v < ε)) : ¬ x ∈ ⋃₀ S :=
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begin
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intro HxU,
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have Hex : ∃₀ s ∈ S, x ∈ s, from HxU,
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cases Hex with s Hs,
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cases Hs with Hs Hxs,
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cases exists_not_of_not_forall (v_0 Hs Hxs) with ε Hε,
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cases and_not_of_not_implies Hε with Hεp Hv,
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cases Hbd _ Hεp with v Hv',
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cases Hv' with Hvnm Hdist,
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apply Hv,
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existsi v,
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split,
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apply not_mem_of_not_mem_sUnion Hvnm Hs,
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exact Hdist
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end
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/-
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this should be doable by showing that the open-ball boundary definition
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is equivalent to topology.on_boundary, and applying topology.not_open_of_on_boundary.
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But the induction hypotheses don't work out nicely.
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-/
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theorem not_open_of_ex_boundary_pt {U : set M} {x : M} (HxU : x ∈ U)
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(Hbd : ∀ ε : ℝ, ε > 0 → ∃ v : M, v ∉ U ∧ dist x v < ε) : ¬ Open U :=
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begin
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intro HUopen,
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induction HUopen,
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{apply not_mem_open_basis_of_boundary_pt a Hbd HxU},
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{cases Hbd 1 zero_lt_one with v Hv,
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cases Hv with Hv _,
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exact Hv !mem_univ},
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{apply not_mem_intersect_of_boundary_pt a a_1 v_0 v_1 Hbd HxU},
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{apply not_mem_sUnion_of_boundary_pt a v_0 Hbd HxU}
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end
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theorem exists_Open_ball_subset_of_Open_of_mem {U : set M} (HU : Open U) {x : M} (Hx : x ∈ U) :
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theorem exists_open_ball_subset_of_Open_of_mem {U : set M} (HU : Open U) {x : M} (Hx : x ∈ U) :
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∃ (r : ℝ), r > 0 ∧ open_ball x r ⊆ U :=
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begin
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let balloon := {r ∈ univ | r > 0 ∧ open_ball x r ⊆ U},
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cases em (balloon = ∅),
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have H : ∀ r : ℝ, r > 0 → ∃ v : M, v ∉ U ∧ dist x v < r, begin
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intro r Hr,
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note Hor := not_or_not_of_not_and (forall_not_of_sep_empty a (mem_univ r)),
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note Hor' := or.neg_resolve_left Hor Hr,
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apply exists_of_not_forall_not,
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intro Hall,
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apply Hor',
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intro y Hy,
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cases not_or_not_of_not_and (Hall y) with Hmem Hge,
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apply not_not_elim Hmem,
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rewrite dist_comm at Hge,
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apply absurd Hy Hge
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end,
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apply absurd HU,
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apply not_open_of_ex_boundary_pt Hx H,
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cases exists_mem_of_ne_empty a with r Hr,
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cases Hr with _ Hr,
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cases Hr with Hrpos HxrU,
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existsi r,
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split,
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repeat assumption
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end
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begin
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induction HU with s sbasis s t sbasis tbasis ihs iht S Sbasis ihS,
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{cases sbasis with x' aux, cases aux with ε seq,
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have x ∈ open_ball x' ε, by rewrite -seq; exact Hx,
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have εpos : ε > 0, from pos_of_mem_open_ball this,
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have ε - dist x x' > 0, from sub_pos_of_lt `x ∈ open_ball x' ε`,
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existsi (ε - dist x x'), split, exact this, rewrite seq,
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show open_ball x (ε - dist x x') ⊆ open_ball x' ε, from
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take y, suppose dist y x < ε - dist x x',
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calc
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dist y x' ≤ dist y x + dist x x' : dist_triangle
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... < ε - dist x x' + dist x x' : add_lt_add_right this
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... = ε : sub_add_cancel},
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{existsi 1, split, exact zero_lt_one, exact subset_univ _},
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{cases ihs (and.left Hx) with rs aux, cases aux with rspos ballrs_sub,
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cases iht (and.right Hx) with rt aux, cases aux with rtpos ballrt_sub,
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let rmin := min rs rt,
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existsi rmin, split, exact lt_min rspos rtpos,
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have open_ball x rmin ⊆ s,
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from subset.trans (open_ball_subset_open_ball_of_le x !min_le_left) ballrs_sub,
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have open_ball x rmin ⊆ t,
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from subset.trans (open_ball_subset_open_ball_of_le x !min_le_right) ballrt_sub,
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show open_ball x (min rs rt) ⊆ s ∩ t,
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by apply subset_inter; repeat assumption},
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cases Hx with s aux, cases aux with sS xs,
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cases (ihS sS xs) with r aux, cases aux with rpos ballr_sub,
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existsi r, split, exact rpos,
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show open_ball x r ⊆ ⋃₀ S, from subset.trans ballr_sub (subset_sUnion_of_mem sS)
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end
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/- limits in metric spaces -/
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proposition eventually_nhds_intro {P : M → Prop} {ε : ℝ} (εpos : ε > 0) {x : M}
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(H : ∀ x', dist x' x < ε → P x') :
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eventually P (nhds x) :=
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topology.eventually_nhds_intro (open_ball_open x ε) (mem_open_ball x εpos) H
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topology.eventually_nhds_intro (Open_open_ball x ε) (mem_open_ball x εpos) H
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proposition eventually_nhds_dest {P : M → Prop} {x : M} (H : eventually P (nhds x)) :
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∃ ε, ε > 0 ∧ ∀ x', dist x' x < ε → P x' :=
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obtain s [(Os : Open s) [(xs : x ∈ s) (Hs : ∀₀ x' ∈ s, P x')]],
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from topology.eventually_nhds_dest H,
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obtain ε [(εpos : ε > 0) (Hε : open_ball x ε ⊆ s)],
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from exists_Open_ball_subset_of_Open_of_mem Os xs,
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from exists_open_ball_subset_of_Open_of_mem Os xs,
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exists.intro ε (and.intro εpos
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(take x', suppose dist x' x < ε,
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have x' ∈ s, from Hε this,
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@ -293,7 +182,7 @@ eventually_nhds_intro εpos (λ x' H, H)
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proposition eventually_at_within_intro {P : M → Prop} {ε : ℝ} (εpos : ε > 0) {x : M} {s : set M}
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(H : ∀₀ x' ∈ s, dist x' x < ε → x' ≠ x → P x') :
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eventually P [at x within s] :=
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topology.eventually_at_within_intro (open_ball_open x ε) (mem_open_ball x εpos)
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topology.eventually_at_within_intro (Open_open_ball x ε) (mem_open_ball x εpos)
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(λ x' x'mem x'ne x's, H x's x'mem x'ne)
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proposition eventually_at_within_dest {P : M → Prop} {x : M} {s : set M}
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@ -302,7 +191,7 @@ proposition eventually_at_within_dest {P : M → Prop} {x : M} {s : set M}
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obtain t [(Ot : Open t) [(xt : x ∈ t) (Ht : ∀₀ x' ∈ t, x' ≠ x → x' ∈ s → P x')]],
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from topology.eventually_at_within_dest H,
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obtain ε [(εpos : ε > 0) (Hε : open_ball x ε ⊆ t)],
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from exists_Open_ball_subset_of_Open_of_mem Ot xt,
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from exists_open_ball_subset_of_Open_of_mem Ot xt,
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exists.intro ε (and.intro εpos
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(take x', assume x's distx'x x'nex,
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have x' ∈ t, from Hε distx'x,
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@ -316,7 +205,7 @@ iff.intro eventually_at_within_dest
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proposition eventually_at_intro {P : M → Prop} {ε : ℝ} (εpos : ε > 0) {x : M}
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(H : ∀ x', dist x' x < ε → x' ≠ x → P x') :
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eventually P [at x] :=
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topology.eventually_at_intro (open_ball_open x ε) (mem_open_ball x εpos)
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topology.eventually_at_intro (Open_open_ball x ε) (mem_open_ball x εpos)
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(λ x' x'mem x'ne, H x' x'mem x'ne)
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proposition eventually_at_dest {P : M → Prop} {x : M} (H : eventually P [at x]) :
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@ -662,13 +551,13 @@ theorem continuous_at_intro {f : M → N} {x : M}
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begin
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rewrite ↑continuous_at,
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intros U Uopen HfU,
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cases exists_Open_ball_subset_of_Open_of_mem Uopen HfU with r Hr,
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cases exists_open_ball_subset_of_Open_of_mem Uopen HfU with r Hr,
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cases Hr with Hr HUr,
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cases H Hr with δ Hδ,
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cases Hδ with Hδ Hx'δ,
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existsi open_ball x δ,
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split,
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apply open_ball_open,
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apply Open_open_ball,
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split,
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apply mem_open_ball,
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exact Hδ,
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@ -684,10 +573,10 @@ theorem continuous_at_elim {f : M → N} {x : M} (Hfx : continuous_at f x) :
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begin
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intros ε Hε,
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rewrite [↑continuous_at at Hfx],
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cases @Hfx (open_ball (f x) ε) !open_ball_open (mem_open_ball _ Hε) with V HV,
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cases @Hfx (open_ball (f x) ε) !Open_open_ball (mem_open_ball _ Hε) with V HV,
|
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cases HV with HV HVx,
|
||||
cases HVx with HVx HVf,
|
||||
cases exists_Open_ball_subset_of_Open_of_mem HV HVx with δ Hδ,
|
||||
cases exists_open_ball_subset_of_Open_of_mem HV HVx with δ Hδ,
|
||||
cases Hδ with Hδ Hδx,
|
||||
existsi δ,
|
||||
split,
|
||||
|
|
|
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Loading…
Reference in a new issue