feat(library/theories/analysis): refactor IVT proof, add more general version of IVT
This commit is contained in:
Rob Lewis
parent
856a09d70e
commit
d6be32e4ef
2 changed files with 167 additions and 133 deletions
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@ -150,13 +150,11 @@ exists.intro δ (and.intro
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if Heq : x = x' then
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by rewrite [Heq, dist_self]; assumption
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else
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(suffices dist x x' < δ, from and.right Hδ _ (and.intro Heq this),
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(suffices dist x x' < δ, from and.right Hδ x' (and.intro Heq this),
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this)))
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definition image_seq (X : ℕ → M) (f : M → N) : ℕ → N := λ n, f (X n)
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theorem image_seq_converges_of_converges [instance] (X : ℕ → M) [HX : converges_seq X] {f : M → N} (Hf : continuous f) :
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converges_seq (image_seq X f) :=
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converges_seq (λ n, f (X n)) :=
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begin
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cases HX with xlim Hxlim,
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existsi f xlim,
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@ -168,9 +166,8 @@ theorem image_seq_converges_of_converges [instance] (X : ℕ → M) [HX : conver
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existsi B,
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intro n Hn,
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cases em (xlim = X n),
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rewrite [↑image_seq, a, dist_self],
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rewrite [a, dist_self],
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assumption,
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rewrite ↑image_seq,
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apply and.right Hδ,
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split,
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exact a,
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@ -190,23 +190,78 @@ end
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end nat
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section inter_val
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open set
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section continuous
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-- this definition should be inherited from metric_space once a migrate is done.
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definition continuous (f : ℝ → ℝ) :=
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∀ x : ℝ, ∀ ⦃ε : ℝ⦄, ε > 0 → ∃ δ : ℝ, δ > 0 ∧ ∀ x' : ℝ, abs (x - x') < δ → abs (f x - f x') < ε
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private definition inter_sup (a b : ℝ) (f : ℝ → ℝ) := sup {x | x < b ∧ f x < 0}
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private theorem add_midpoint {a b : ℝ} (H : a < b) : a + (b - a) / 2 < b :=
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theorem pos_on_nbhd_of_cts_of_pos {f : ℝ → ℝ} (Hf : continuous f) {b : ℝ} (Hb : f b > 0) :
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∃ δ : ℝ, δ > 0 ∧ ∀ y, abs (b - y) < δ → f y > 0 :=
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begin
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rewrite [-div_sub_div_same, sub_eq_add_neg, {b / 2 + _}add.comm, -add.assoc, -sub_eq_add_neg],
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apply add_lt_of_lt_sub_right,
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krewrite *sub_self_div_two,
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apply div_lt_div_of_lt_of_pos H two_pos
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let Hcont := Hf b Hb,
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cases Hcont with δ Hδ,
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existsi δ,
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split,
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exact and.left Hδ,
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intro y Hy,
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let Hy' := and.right Hδ y Hy,
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let Hlt := sub_lt_of_abs_sub_lt_right Hy',
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rewrite sub_self at Hlt,
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assumption
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end
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theorem neg_on_nbhd_of_cts_of_neg {f : ℝ → ℝ} (Hf : continuous f) {b : ℝ} (Hb : f b < 0) :
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∃ δ : ℝ, δ > 0 ∧ ∀ y, abs (b - y) < δ → f y < 0 :=
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begin
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let Hcont := Hf b (neg_pos_of_neg Hb),
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cases Hcont with δ Hδ,
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existsi δ,
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split,
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exact and.left Hδ,
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intro y Hy,
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let Hy' := and.right Hδ y Hy,
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let Hlt := sub_lt_of_abs_sub_lt_left Hy',
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let Hlt' := lt_add_of_sub_lt_right _ _ _ Hlt,
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rewrite [-sub_eq_add_neg at Hlt', sub_self at Hlt'],
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assumption
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end
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theorem neg_continuous_of_continuous {f : ℝ → ℝ} (Hcon : continuous f) : continuous (λ x, - f x) :=
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begin
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intros x ε Hε,
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cases Hcon x Hε with δ Hδ,
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cases Hδ with Hδ₁ Hδ₂,
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existsi δ,
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split,
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assumption,
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intros x' Hx',
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let HD := Hδ₂ x' Hx',
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rewrite [-abs_neg, neg_neg_sub_neg],
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assumption
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end
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theorem translate_continuous_of_continuous {f : ℝ → ℝ} (Hcon : continuous f) (a : ℝ) :
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continuous (λ x, (f x) + a) :=
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begin
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intros x ε Hε,
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cases Hcon x Hε with δ Hδ,
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cases Hδ with Hδ₁ Hδ₂,
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existsi δ,
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split,
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assumption,
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intros x' Hx',
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rewrite [add_sub_comm, sub_self, add_zero],
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apply Hδ₂,
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assumption
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end
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end continuous
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section inter_val
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open set
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private definition inter_sup (a b : ℝ) (f : ℝ → ℝ) := sup {x | x < b ∧ f x < 0}
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section
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parameters {f : ℝ → ℝ} (Hf : continuous f) {a b : ℝ} (Hab : a < b) (Ha : f a < 0) (Hb : f b > 0)
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include Hf Ha Hb Hab
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@ -219,6 +274,39 @@ private theorem Hsupleb : inter_sup a b f ≤ b := sup_le (Hinh) Hmem
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local notation 2 := of_num 1 + of_num 1
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private theorem ex_delta_lt {x : ℝ} (Hx : f x < 0) (Hxb : x < b) : ∃ δ : ℝ, δ > 0 ∧ x + δ < b ∧ f (x + δ) < 0 :=
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begin
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let Hcont := neg_on_nbhd_of_cts_of_neg Hf Hx,
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cases Hcont with δ Hδ,
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{cases em (x + δ < b) with Haδ Haδ,
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existsi δ / 2,
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split,
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{exact div_pos_of_pos_of_pos (and.left Hδ) two_pos},
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split,
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{apply lt.trans,
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apply add_lt_add_left,
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exact div_two_lt_of_pos (and.left Hδ),
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exact Haδ},
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{apply and.right Hδ,
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rewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
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abs_of_pos (div_pos_of_pos_of_pos (and.left Hδ) two_pos)],
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exact div_two_lt_of_pos (and.left Hδ)},
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existsi (b - x) / 2,
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split,
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{apply div_pos_of_pos_of_pos,
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exact sub_pos_of_lt _ _ Hxb,
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exact two_pos},
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split,
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{apply add_midpoint Hxb},
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{apply and.right Hδ,
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rewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
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abs_of_pos (div_pos_of_pos_of_pos (sub_pos_of_lt _ _ Hxb) two_pos)],
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apply lt_of_lt_of_le,
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apply div_two_lt_of_pos (sub_pos_of_lt _ _ Hxb),
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apply sub_left_le_of_le_add,
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apply le_of_not_gt Haδ}}
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end
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private lemma sup_near_b {δ : ℝ} (Hpos : 0 < δ)
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(Hgeb : inter_sup a b f + δ / 2 ≥ b) : abs (inter_sup a b f - b) < δ :=
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begin
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@ -237,29 +325,15 @@ private lemma sup_near_b {δ : ℝ} (Hpos : 0 < δ)
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private lemma delta_of_lt (Hflt : f (inter_sup a b f) < 0) :
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∃ δ : ℝ, δ > 0 ∧ inter_sup a b f + δ < b ∧ f (inter_sup a b f + δ) < 0 :=
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begin
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let Hcont := Hf (inter_sup a b f) (neg_pos_of_neg Hflt),
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cases Hcont with δ Hδ,
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cases em (inter_sup a b f + δ / 2 < b),
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existsi δ / 2,
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split,
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apply div_pos_of_pos_of_pos (and.left Hδ) two_pos,
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split,
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exact a_1,
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have Habs : abs (inter_sup a b f - (inter_sup a b f + δ / 2)) < δ, begin
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rewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
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abs_of_pos (div_pos_of_pos_of_pos (and.left Hδ) two_pos)],
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apply div_two_lt_of_pos (and.left Hδ)
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end,
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let Hlt := and.right Hδ _ Habs,
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let Hlt' := lt_add_of_sub_lt_right _ _ _ (sub_lt_of_abs_sub_lt_left Hlt),
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rewrite [-sub_eq_add_neg at Hlt', sub_self at Hlt'],
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exact Hlt',
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let Hlt := and.right Hδ _ (sup_near_b (and.left Hδ) (le_of_not_gt a_1)),
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let Hlt' := lt_add_of_sub_lt_right _ _ _ (sub_lt_of_abs_sub_lt_left Hlt),
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rewrite [-sub_eq_add_neg at Hlt', sub_self at Hlt'],
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apply absurd Hb (not_lt_of_ge (le_of_lt Hlt'))
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end
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if Hlt : inter_sup a b f < b then ex_delta_lt Hflt Hlt else
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begin
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let Heq := eq_of_le_of_ge Hsupleb (le_of_not_gt Hlt),
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apply absurd Hflt,
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apply not_lt_of_ge,
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apply le_of_lt,
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rewrite Heq,
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exact Hb
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end
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private theorem sup_fn_interval_aux1 : f (inter_sup a b f) ≥ 0 :=
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have ¬ f (inter_sup a b f) < 0, from
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@ -272,31 +346,15 @@ private theorem sup_fn_interval_aux1 : f (inter_sup a b f) ≥ 0 :=
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show false, from this (lt_add_of_pos_right (and.left Hδ))),
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le_of_not_gt this
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private lemma delta_of_gt (Hfpos : f (inter_sup a b f) > 0) :
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∃ δ : ℝ, δ > 0 ∧ ∀ x' : ℝ, abs (inter_sup a b f - x') < δ → f x' > 0 :=
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begin
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let Hcont := Hf (inter_sup a b f) Hfpos,
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cases Hcont with δ Hδ,
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existsi δ,
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split,
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exact and.left Hδ,
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intro x Hx,
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let Hlt := and.right Hδ _ Hx,
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let Hlt' := sub_lt_of_abs_sub_lt_right Hlt,
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rewrite sub_self at Hlt',
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exact Hlt'
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end
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private theorem sup_fn_interval_aux2 : f (inter_sup a b f) ≤ 0 :=
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have ¬ f (inter_sup a b f) > 0, from
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(suppose f (inter_sup a b f) > 0,
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obtain δ Hδ, from delta_of_gt this,
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(assume Hfsup : f (inter_sup a b f) > 0,
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obtain δ Hδ, from pos_on_nbhd_of_cts_of_pos Hf Hfsup,
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have ∀ x, x ∈ {x | x < b ∧ f x < 0} → x ≤ inter_sup a b f - δ / 2, from
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(take x, suppose x ∈ {x | x < b ∧ f x < 0},
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have x ≤ inter_sup a b f, from le_sup this Hmem,
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(take x, assume Hxset : x ∈ {x | x < b ∧ f x < 0},
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have ¬ x > inter_sup a b f - δ / 2, from
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(assume Hngt,
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have abs (inter_sup a b f - x) < δ, begin
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have Habs : abs (inter_sup a b f - x) < δ, begin
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apply abs_lt_of_lt_of_neg_lt,
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apply sub_lt_of_sub_lt,
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apply gt.trans,
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@ -309,11 +367,11 @@ private theorem sup_fn_interval_aux2 : f (inter_sup a b f) ≤ 0 :=
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apply and.left Hδ,
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apply sub_nonpos_of_le,
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apply le_sup,
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exact this,
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exact Hxset,
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exact Hmem
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end,
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have f x > 0, from and.right Hδ _ this,
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show false, from (not_lt_of_gt this) (and.right `x ∈ {x | x < b ∧ f x < 0}`)),
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have f x > 0, from and.right Hδ x Habs,
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show false, from (not_lt_of_gt this) (and.right Hxset)),
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le_of_not_gt this),
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have Hle : inter_sup a b f ≤ inter_sup a b f - δ / 2, from sup_le Hinh this,
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show false, from not_le_of_gt
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@ -323,71 +381,45 @@ private theorem sup_fn_interval_aux2 : f (inter_sup a b f) ≤ 0 :=
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private theorem sup_fn_interval : f (inter_sup a b f) = 0 :=
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eq_of_le_of_ge sup_fn_interval_aux2 sup_fn_interval_aux1
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private theorem intermediate_value_incr_aux1 : ∃ δ : ℝ, δ > 0 ∧ ∀ y, abs (b - y) < δ → f y > 0 :=
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begin
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let Hcont := Hf b Hb,
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cases Hcont with δ Hδ,
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existsi δ,
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split,
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exact and.left Hδ,
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intro y Hy,
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let Hy' := and.right Hδ _ Hy,
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let Hlt := sub_lt_of_abs_sub_lt_right Hy',
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rewrite sub_self at Hlt,
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assumption
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end
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private theorem intermediate_value_incr_aux2 : ∃ δ : ℝ, δ > 0 ∧ a + δ < b ∧ f (a + δ) < 0 :=
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begin
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have Hnfa : - (f a) > 0, from neg_pos_of_neg Ha,
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let Hcont := Hf a Hnfa,
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let Hcont := neg_on_nbhd_of_cts_of_neg Hf Ha,
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cases Hcont with δ Hδ,
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cases em (a + δ < b) with Haδ Haδ,
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{cases em (a + δ < b) with Haδ Haδ,
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existsi δ / 2,
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split,
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exact div_pos_of_pos_of_pos (and.left Hδ) two_pos,
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{exact div_pos_of_pos_of_pos (and.left Hδ) two_pos},
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split,
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apply lt.trans,
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{apply lt.trans,
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apply add_lt_add_left,
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exact div_two_lt_of_pos (and.left Hδ),
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assumption,
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have Habs : abs (a - (a + δ / 2)) < δ, begin
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rewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
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abs_of_pos (div_pos_of_pos_of_pos (and.left Hδ) two_pos)],
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exact div_two_lt_of_pos (and.left Hδ)
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end,
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let Hlt := and.right Hδ _ Habs,
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let Hlt' := sub_lt_of_abs_sub_lt_left Hlt,
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let Hlt'' := lt_add_of_sub_lt_right _ _ _ Hlt',
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rewrite [-sub_eq_add_neg at Hlt'', sub_self at Hlt''],
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assumption,
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exact Haδ},
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{apply and.right Hδ,
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rewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
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abs_of_pos (div_pos_of_pos_of_pos (and.left Hδ) two_pos)],
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exact div_two_lt_of_pos (and.left Hδ)},
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existsi (b - a) / 2,
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split,
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apply div_pos_of_pos_of_pos,
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{apply div_pos_of_pos_of_pos,
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exact sub_pos_of_lt _ _ Hab,
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exact two_pos,
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exact two_pos},
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split,
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apply add_midpoint Hab,
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have Habs : abs (a - (a + (b - a) / 2)) < δ, begin
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krewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
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abs_of_pos (div_pos_of_pos_of_pos (sub_pos_of_lt _ _ Hab) two_pos)],
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apply lt_of_lt_of_le,
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apply div_two_lt_of_pos (sub_pos_of_lt _ _ Hab),
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apply sub_left_le_of_le_add,
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apply le_of_not_gt Haδ
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end,
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let Hlt := and.right Hδ _ Habs,
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let Hlt' := sub_lt_of_abs_sub_lt_left Hlt,
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let Hlt'' := lt_add_of_sub_lt_right _ _ _ Hlt',
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rewrite [-sub_eq_add_neg at Hlt'', sub_self at Hlt''],
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assumption
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{apply add_midpoint Hab},
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{apply and.right Hδ,
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rewrite [sub_add_eq_sub_sub, sub_self, zero_sub, abs_neg,
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abs_of_pos (div_pos_of_pos_of_pos (sub_pos_of_lt _ _ Hab) two_pos)],
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apply lt_of_lt_of_le,
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apply div_two_lt_of_pos (sub_pos_of_lt _ _ Hab),
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apply sub_left_le_of_le_add,
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apply le_of_not_gt Haδ}}
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end
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theorem intermediate_value_incr : ∃ c, a < c ∧ c < b ∧ f c = 0 :=
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theorem intermediate_value_incr_zero : ∃ c, a < c ∧ c < b ∧ f c = 0 :=
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begin
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existsi inter_sup a b f,
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split,
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cases intermediate_value_incr_aux2 with δ Hδ,
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{cases intermediate_value_incr_aux2 with δ Hδ,
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apply lt_of_lt_of_le,
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apply lt_add_of_pos_right,
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exact and.left Hδ,
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@ -395,14 +427,14 @@ theorem intermediate_value_incr : ∃ c, a < c ∧ c < b ∧ f c = 0 :=
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exact and.right Hδ,
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intro x Hx,
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apply le_of_lt,
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exact and.left Hx,
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exact and.left Hx},
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split,
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cases intermediate_value_incr_aux1 with δ Hδ,
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{cases pos_on_nbhd_of_cts_of_pos Hf Hb with δ Hδ,
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apply lt_of_le_of_lt,
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rotate 1,
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apply sub_lt_of_pos,
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exact and.left Hδ,
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exact sup_fn_interval,
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rotate_right 1,
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apply sup_le,
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exact exists.intro a (and.intro Hab Ha),
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intro x Hx,
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@ -411,34 +443,19 @@ theorem intermediate_value_incr : ∃ c, a < c ∧ c < b ∧ f c = 0 :=
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have Hxgt' : b - x < δ, from sub_lt_of_sub_lt _ _ _ Hxgt,
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rewrite -(abs_of_pos (sub_pos_of_lt _ _ (and.left Hx))) at Hxgt',
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let Hxgt'' := and.right Hδ _ Hxgt',
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exact not_lt_of_ge (le_of_lt Hxgt'') (and.right Hx)
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exact not_lt_of_ge (le_of_lt Hxgt'') (and.right Hx)},
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{exact sup_fn_interval}
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end
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end
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private definition neg_f (f : ℝ → ℝ) := λ x, - f x
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private theorem neg_continuous_of_continuous {f : ℝ → ℝ} (Hcon : continuous f) : continuous (neg_f f) :=
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begin
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intros x ε Hε,
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cases Hcon x Hε with δ Hδ,
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cases Hδ with Hδ₁ Hδ₂,
|
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existsi δ,
|
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split,
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assumption,
|
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intros x' Hx',
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let HD := Hδ₂ x' Hx',
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rewrite [-abs_neg, ↑neg_f, neg_neg_sub_neg],
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assumption
|
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end
|
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|
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theorem intermediate_value_decr {f : ℝ → ℝ} (Hf : continuous f) {a b : ℝ} (Hab : a < b)
|
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theorem intermediate_value_decr_zero {f : ℝ → ℝ} (Hf : continuous f) {a b : ℝ} (Hab : a < b)
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(Ha : f a > 0) (Hb : f b < 0) : ∃ c, a < c ∧ c < b ∧ f c = 0 :=
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begin
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have Ha' : (neg_f f) a < 0, from neg_neg_of_pos Ha,
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have Hb' : (neg_f f) b > 0, from neg_pos_of_neg Hb,
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have Hcon : continuous (neg_f f), from neg_continuous_of_continuous Hf,
|
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let Hiv := intermediate_value_incr Hcon Hab Ha' Hb',
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have Ha' : - f a < 0, from neg_neg_of_pos Ha,
|
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have Hb' : - f b > 0, from neg_pos_of_neg Hb,
|
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have Hcon : continuous (λ x, - f x), from neg_continuous_of_continuous Hf,
|
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let Hiv := intermediate_value_incr_zero Hcon Hab Ha' Hb',
|
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cases Hiv with c Hc,
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existsi c,
|
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split,
|
||||
|
|
@ -449,6 +466,26 @@ theorem intermediate_value_decr {f : ℝ → ℝ} (Hf : continuous f) {a b : ℝ
|
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apply and.right (and.right Hc)
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end
|
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|
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theorem intermediate_value_incr {f : ℝ → ℝ} (Hf : continuous f) {a b : ℝ} (Hab : a < b) {v : ℝ}
|
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(Hav : f a < v) (Hbv : f b > v) : ∃ c, a < c ∧ c < b ∧ f c = v :=
|
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have Hav' : f a - v < 0, from sub_neg_of_lt _ _ Hav,
|
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have Hbv' : f b - v > 0, from sub_pos_of_lt _ _ Hbv,
|
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have Hcon : continuous (λ x, f x - v), from translate_continuous_of_continuous Hf _,
|
||||
have Hiv : ∃ c, a < c ∧ c < b ∧ f c - v = 0, from intermediate_value_incr_zero Hcon Hab Hav' Hbv',
|
||||
obtain c Hc, from Hiv,
|
||||
exists.intro c
|
||||
(and.intro (and.left Hc) (and.intro (and.left (and.right Hc)) (eq_of_sub_eq_zero (and.right (and.right Hc)))))
|
||||
|
||||
theorem intermediate_value_decr {f : ℝ → ℝ} (Hf : continuous f) {a b : ℝ} (Hab : a < b) {v : ℝ}
|
||||
(Hav : f a > v) (Hbv : f b < v) : ∃ c, a < c ∧ c < b ∧ f c = v :=
|
||||
have Hav' : f a - v > 0, from sub_pos_of_lt _ _ Hav,
|
||||
have Hbv' : f b - v < 0, from sub_neg_of_lt _ _ Hbv,
|
||||
have Hcon : continuous (λ x, f x - v), from translate_continuous_of_continuous Hf _,
|
||||
have Hiv : ∃ c, a < c ∧ c < b ∧ f c - v = 0, from intermediate_value_decr_zero Hcon Hab Hav' Hbv',
|
||||
obtain c Hc, from Hiv,
|
||||
exists.intro c
|
||||
(and.intro (and.left Hc) (and.intro (and.left (and.right Hc)) (eq_of_sub_eq_zero (and.right (and.right Hc)))))
|
||||
|
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end inter_val
|
||||
|
||||
/-
|
||||
|
|
|
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Reference in a new issue