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import Mathlib.Data.Real.Basic

namespace C03S03
section
variable (a b : ℝ)

def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop :=
  ∀ x, f x ≤ a

def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop :=
  ∀ x, a ≤ f x

def FnHasUb (f : ℝ → ℝ) :=
  ∃ a, FnUb f a

def FnHasLb (f : ℝ → ℝ) :=
  ∃ a, FnLb f a

variable (f : ℝ → ℝ)

example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := by
  rintro ⟨a, ha⟩
  rcases h a with ⟨x, hx⟩
  have := ha x
  linarith

example : ¬FnHasUb fun x ↦ x := by
  rintro ⟨a, ha⟩
  have : a + 1 ≤ a := ha (a + 1)
  linarith

example (h : Monotone f) (h' : f a < f b) : a < b := by
  apply lt_of_not_ge
  intro h''
  apply absurd h'
  apply not_lt_of_ge (h h'')

example (h : a ≤ b) (h' : f b < f a) : ¬Monotone f := by
  intro h''
  apply absurd h'
  apply not_lt_of_ge
  apply h'' h

example : ¬∀ {f : ℝ → ℝ}, Monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := by
  intro h
  let f := fun x : ℝ ↦ (0 : ℝ)
  have monof : Monotone f := by
    intro a b leab
    rfl
  have h' : f 1 ≤ f 0 := le_refl _
  have : (1 : ℝ) ≤ 0 := h monof h'
  linarith

example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by
  apply le_of_not_gt
  intro h'
  linarith [h _ h']

end

section
variable {α : Type _} (P : α → Prop) (Q : Prop)

example (h : ¬∃ x, P x) : ∀ x, ¬P x := by
  intro x Px
  apply h
  use x
  exact Px

example (h : ∀ x, ¬P x) : ¬∃ x, P x := by
  rintro ⟨x, Px⟩
  exact h x Px

example (h : ∃ x, ¬P x) : ¬∀ x, P x := by
  intro h'
  rcases h with ⟨x, nPx⟩
  apply nPx
  apply h'

example (h : ¬¬Q) : Q := by
  by_contra h'
  exact h h'

example (h : Q) : ¬¬Q := by
  intro h'
  exact h' h

end

section
variable (f : ℝ → ℝ)

example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by
  intro a
  by_contra h'
  apply h
  use a
  intro x
  apply le_of_not_gt
  intro h''
  apply h'
  use x
  exact h''

example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by
  rw [Monotone] at h
  push_neg  at h
  exact h

end