273 lines
9.5 KiB
Text
273 lines
9.5 KiB
Text
/-
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Copyright (c) 2015 Robert Y. Lewis. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Robert Y. Lewis
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The intermediate value theorem.
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-/
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import .real_limit
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open real analysis set classical
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noncomputable theory
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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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private theorem Hinh : ∃ x, x ∈ {x | x < b ∧ f x < 0} := exists.intro a (and.intro Hab Ha)
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private theorem Hmem : ∀ x, x ∈ {x | x < b ∧ f x < 0} → x ≤ b := λ x Hx, le_of_lt (and.left Hx)
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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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krewrite [abs_sub, 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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krewrite [abs_sub, 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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apply abs_lt_of_lt_of_neg_lt,
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apply sub_lt_left_of_lt_add,
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apply lt_of_le_of_lt,
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apply Hsupleb,
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apply lt_add_of_pos_right Hpos,
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rewrite neg_sub,
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apply sub_lt_left_of_lt_add,
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apply lt_of_le_of_lt,
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apply Hgeb,
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apply add_lt_add_left,
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apply div_two_lt_of_pos Hpos
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end
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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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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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(suppose f (inter_sup a b f) < 0,
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obtain δ Hδ, from delta_of_lt this,
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have inter_sup a b f + δ ∈ {x | x < b ∧ f x < 0},
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from and.intro (and.left (and.right Hδ)) (and.right (and.right Hδ)),
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have ¬ inter_sup a b f < inter_sup a b f + δ,
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from not_lt_of_ge (le_sup this Hmem),
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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 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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(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, 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 Habs : abs (x - inter_sup a b f) < δ, begin
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rewrite abs_sub,
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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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exact Hngt,
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apply sub_lt_sub_left,
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exact div_two_lt_of_pos (and.left Hδ),
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rewrite neg_sub,
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apply lt_of_le_of_lt,
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rotate 1,
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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 Hxset,
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exact Hmem
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end,
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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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(sub_lt_of_pos _ (div_pos_of_pos_of_pos (and.left Hδ) (two_pos))) Hle),
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le_of_not_gt this
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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_aux2 : ∃ δ : ℝ, δ > 0 ∧ a + δ < b ∧ f (a + δ) < 0 :=
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begin
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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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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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krewrite [abs_sub, 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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exact sub_pos_of_lt Hab,
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exact two_pos},
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split,
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{apply add_midpoint Hab},
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{apply and.right Hδ,
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krewrite [abs_sub, 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_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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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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apply le_sup,
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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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split,
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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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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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apply le_of_not_gt,
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intro Hxgt,
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have Hxgt' : b - x < δ, from sub_lt_of_sub_lt Hxgt,
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krewrite [-(abs_of_pos (sub_pos_of_lt (and.left Hx))) at Hxgt', abs_sub at Hxgt'],
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note 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 sup_fn_interval}
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end
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end
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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' : - 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 continuous_neg_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,
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exact and.left Hc,
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split,
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exact and.left (and.right Hc),
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apply eq_zero_of_neg_eq_zero,
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apply and.right (and.right Hc)
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end
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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 continuous_offset_of_continuous Hf _,
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have Hiv : ∃ c, a < c ∧ c < b ∧ f c - v = 0, from intermediate_value_incr_zero Hcon Hab Hav' Hbv',
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obtain c Hc, from Hiv,
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exists.intro c
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(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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theorem intermediate_value_decr {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_pos_of_lt Hav,
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have Hbv' : f b - v < 0, from sub_neg_of_lt Hbv,
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have Hcon : continuous (λ x, f x - v), from continuous_offset_of_continuous Hf _,
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have Hiv : ∃ c, a < c ∧ c < b ∧ f c - v = 0, from intermediate_value_decr_zero Hcon Hab Hav' Hbv',
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obtain c Hc, from Hiv,
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exists.intro c
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(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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theorem intermediate_value_incr_weak {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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begin
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cases lt_or_eq_of_le Hab with Hlt Heq,
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cases lt_or_eq_of_le Hav with Hlt' Heq',
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cases lt_or_eq_of_le Hbv with Hlt'' Heq'',
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cases intermediate_value_incr Hf Hlt Hlt' Hlt'' with c Hc,
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cases Hc with Hc1 Hc2,
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cases Hc2 with Hc2 Hc3,
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existsi c,
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repeat (split; apply le_of_lt; assumption),
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assumption,
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existsi b,
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split,
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apply le_of_lt,
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assumption,
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split,
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apply le.refl,
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rewrite Heq'',
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existsi a,
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split,
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apply le.refl,
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split,
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apply le_of_lt,
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repeat assumption,
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existsi a,
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split,
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apply le.refl,
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split,
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assumption,
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apply eq_of_le_of_ge,
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apply Hav,
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rewrite Heq,
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apply Hbv
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end
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