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import Mathlib.Data.Real.Basic

#check ∀ x : ℝ, 0 ≤ x → abs x = x

#check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε

theorem my_lemma : ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε :=
  sorry

section
variable (a b δ : ℝ)
variable (h₀ : 0 < δ) (h₁ : δ ≤ 1)
variable (ha : abs a < δ) (hb : abs b < δ)

#check my_lemma a b δ
#check my_lemma a b δ h₀ h₁
#check my_lemma a b δ h₀ h₁ ha hb

end

theorem my_lemma2 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε :=
  sorry

section
variable (a b δ : ℝ)
variable (h₀ : 0 < δ) (h₁ : δ ≤ 1)
variable (ha : abs a < δ) (hb : abs b < δ)

#check my_lemma2 h₀ h₁ ha hb

end

theorem my_lemma3 :
    ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by
  intro x y ε epos ele1 xlt ylt
  sorry

theorem my_lemma4 :
    ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by
  intro x y ε epos ele1 xlt ylt
  calc
    abs (x * y) = abs x * abs y := sorry
    _ ≤ abs x * ε := sorry
    _ < 1 * ε := sorry
    _ = ε := sorry

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

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

section
variable (f g : ℝ → ℝ) (a b : ℝ)

example (hfa : FnUb f a) (hgb : FnUb g b) : FnUb (fun x ↦ f x + g x) (a + b) := by
  intro x
  dsimp
  apply add_le_add
  apply hfa
  apply hgb

example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) :=
  sorry

example (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x => f x * g x) 0 :=
  sorry

example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) :
    FnUb (fun x => f x * g x) (a * b) :=
  sorry

end

section
variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R]

#check @add_le_add

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

theorem fn_ub_add {f g : α → R} {a b : R} (hfa : FnUb' f a) (hgb : FnUb' g b) :
    FnUb' (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x)

end

example (f : ℝ → ℝ) (h : Monotone f) : ∀ {a b}, a ≤ b → f a ≤ f b :=
  @h

section
variable (f g : ℝ → ℝ)

example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := by
  intro a b aleb
  apply add_le_add
  apply mf aleb
  apply mg aleb

example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x :=
  fun a b aleb => add_le_add (mf aleb) (mg aleb)

example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x :=
  sorry

example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) :=
  sorry

def FnEven (f : ℝ → ℝ) : Prop :=
  ∀ x, f x = f (-x)

def FnOdd (f : ℝ → ℝ) : Prop :=
  ∀ x, f x = -f (-x)

example (ef : FnEven f) (eg : FnEven g) : FnEven fun x => f x + g x := by
  intro x
  calc
    (fun x => f x + g x) x = f x + g x := rfl
    _ = f (-x) + g (-x) := by rw [ef, eg]


example (of : FnOdd f) (og : FnOdd g) : FnEven fun x => f x * g x := by
  sorry

example (ef : FnEven f) (og : FnOdd g) : FnOdd fun x => f x * g x := by
  sorry

example (ef : FnEven f) (og : FnOdd g) : FnEven fun x => f (g x) := by
  sorry

end

section

variable {α : Type _} (r s t : Set α)

example : s ⊆ s := by
  intro x xs
  exact xs

theorem Subset.refl : s ⊆ s := fun x xs => xs

theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := by
  sorry

end

section
variable {α : Type _} [PartialOrder α]
variable (s : Set α) (a b : α)

def SetUb (s : Set α) (a : α) :=
  ∀ x, x ∈ s → x ≤ a

example (h : SetUb s a) (h' : a ≤ b) : SetUb s b :=
  sorry

end

section

open Function

example (c : ℝ) : Injective fun x => x + c := by
  intro x₁ x₂ h'
  exact (add_left_inj c).mp h'

example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by
  sorry

variable {α : Type _} {β : Type _} {γ : Type _}
variable {g : β → γ} {f : α → β}

example (injg : Injective g) (injf : Injective f) : Injective fun x => g (f x) := by
  sorry

end