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import Mathlib.Data.Real.Basic

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

theorem fnUb_add {f g : ℝ → 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)

section

variable {f g : ℝ → ℝ}

example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by
  cases' lbf with a lbfa
  cases' lbg with b lbgb
  use a + b
  intro x
  exact add_le_add (lbfa x) (lbgb x)

example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x := by
  cases' ubf with a lbfa
  use c * a
  intro x
  exact mul_le_mul_of_nonneg_left (lbfa x) h

end

section

variable {a b c : ℕ}

example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by
  rcases divab with ⟨d, rfl⟩
  rcases divbc with ⟨e, rfl⟩
  use d * e; ring

example (divab : a ∣ b) (divac : a ∣ c) : a ∣ b + c := by
  rcases divab with ⟨d, rfl⟩
  rcases divac with ⟨e, rfl⟩
  use d + e; ring

end

section

open Function

example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by
  intro x
  use x / c
  dsimp; rw [mul_div_cancel' _ h]

example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by
  intro x
  use x / c
  field_simp [h] ; ring

end

section

open Function

variable {α : Type _} {β : Type _} {γ : Type _}

variable {g : β → γ} {f : α → β}

example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f x) := by
  intro z
  rcases surjg z with ⟨y, rfl⟩
  rcases surjf y with ⟨x, rfl⟩
  use x

end