bookshelf/Bookshelf/Avigad/Chapter_5.lean

/-! # Avigad.Chapter5

Tactics
-/

/-! #### Exercise 1

Go back to the exercises in Chapter 3 and Chapter 4 and redo as many as you can
now with tactic proofs, using also `rw` and `simp` as appropriate.
-/

namespace Avigad.Chapter5

namespace ex1

/-! ##### Exercises 3.1 -/

section ex3_1

variable (p q r : Prop)

-- Commutativity of ∧ and ∨
theorem and_comm' : p ∧ q ↔ q ∧ p := by
  apply Iff.intro
  · intro ⟨hp, hq⟩
    exact ⟨hq, hp⟩
  · intro ⟨hq, hp⟩
    exact ⟨hp, hq⟩

theorem or_comm' : p ∨ q ↔ q ∨ p := by
  apply Iff.intro
  · intro
    | Or.inl hp => exact Or.inr hp
    | Or.inr hq => exact Or.inl hq
  · intro
    | Or.inl hq => exact Or.inr hq
    | Or.inr hp => exact Or.inl hp

-- Associativity of ∧ and ∨
theorem and_assoc : (p ∧ q) ∧ r ↔ p ∧ (q ∧ r) := by
  apply Iff.intro
  · intro ⟨⟨hp, hq⟩, hr⟩
    exact ⟨hp, hq, hr⟩
  · intro ⟨hp, hq, hr⟩
    exact ⟨⟨hp, hq⟩, hr⟩

theorem or_assoc' : (p ∨ q) ∨ r ↔ p ∨ (q ∨ r) := by
  apply Iff.intro
  · intro
    | Or.inl (Or.inl hp) => exact Or.inl hp
    | Or.inl (Or.inr hq) => exact Or.inr (Or.inl hq)
    | Or.inr         hr  => exact Or.inr (Or.inr hr)
  · intro
    | Or.inl         hp  => exact Or.inl (Or.inl hp)
    | Or.inr (Or.inl hq) => exact Or.inl (Or.inr hq)
    | Or.inr (Or.inr hr) => exact Or.inr hr

-- Distributivity
theorem and_or_left : p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r) := by
  apply Iff.intro
  · intro
    | ⟨hp, Or.inl hq⟩ => exact Or.inl ⟨hp, hq⟩
    | ⟨hp, Or.inr hr⟩ => exact Or.inr ⟨hp, hr⟩
  · intro
    | Or.inl ⟨hp, hq⟩ => exact ⟨hp, Or.inl hq⟩
    | Or.inr ⟨hp, hr⟩ => exact ⟨hp, Or.inr hr⟩

theorem or_and_left : p ∨ (q ∧ r) ↔ (p ∨ q) ∧ (p ∨ r) := by
  apply Iff.intro
  · intro
    | Or.inl      hp  => exact ⟨Or.inl hp, Or.inl hp⟩
    | Or.inr ⟨hq, hr⟩ => exact ⟨Or.inr hq, Or.inr hr⟩
  · intro
    | ⟨Or.inl hp,         _⟩ => exact Or.inl hp
    | ⟨Or.inr  _, Or.inl hp⟩ => exact Or.inl hp
    | ⟨Or.inr hq, Or.inr hr⟩ => exact Or.inr ⟨hq, hr⟩

-- Other properties
theorem imp_imp_iff_and_imp : (p → (q → r)) ↔ (p ∧ q → r) := by
  apply Iff.intro
  · intro h ⟨hp, hq⟩
    exact h hp hq
  · intro h hp hq
    exact h ⟨hp, hq⟩

theorem or_imp : ((p ∨ q) → r) ↔ (p → r) ∧ (q → r) := by
  apply Iff.intro
  · intro h
    apply And.intro
    · intro hp
      exact h (Or.inl hp)
    · intro hq
      exact h (Or.inr hq)
  · intro ⟨hpr, hqr⟩ h
    apply Or.elim h
    · intro hp
      exact hpr hp
    · intro hq
      exact hqr hq

theorem nor_or : ¬(p ∨ q) ↔ ¬p ∧ ¬q := by
  apply Iff.intro
  · intro h
    apply And.intro
    · intro hp
      exact h (Or.inl hp)
    · intro hq
      exact h (Or.inr hq)
  · intro ⟨np, nq⟩
    intro
    | Or.inl hp => exact absurd hp np
    | Or.inr hq => exact absurd hq nq

theorem not_and_or_mpr : ¬p ∨ ¬q → ¬(p ∧ q) := by
  intro
  | Or.inl np => intro h; exact absurd h.left np
  | Or.inr nq => intro h; exact absurd h.right nq

theorem and_not_self : ¬(p ∧ ¬p) := by
  intro ⟨hp, np⟩
  exact absurd hp np

theorem not_imp_o_and_not : p ∧ ¬q → ¬(p → q) := by
  intro ⟨hp, nq⟩ h
  exact absurd (h hp) nq

theorem false_elim_self : ¬p → (p → q) := by
  intro np hp
  exact absurd hp np

theorem not_or_imp_imp : (¬p ∨ q) → (p → q) := by
  intro
  | Or.inl np => intro hp; exact absurd hp np
  | Or.inr hq => exact fun _ => hq

theorem or_false_iff : p ∨ False ↔ p := by
  apply Iff.intro
  · intro
    | Or.inl hp => exact hp
    | Or.inr ff => exact False.elim ff
  · intro hp
    exact Or.inl hp

theorem and_false_iff : p ∧ False ↔ False := by
  apply Iff.intro
  · intro ⟨_, ff⟩
    exact ff
  · intro ff
    exact False.elim ff

theorem imp_imp_not_imp_not : (p → q) → (¬q → ¬p) := by
  intro hpq nq hp
  exact absurd (hpq hp) nq

end ex3_1

/-! ##### Exercises 3.2 -/

section ex3_2

open Classical

variable (p q r s : Prop)

theorem imp_or_mp (hp : p) : (p → r ∨ s) → ((p → r) ∨ (p → s)) := by
  intro h
  apply (h hp).elim
  · intro hr
    exact Or.inl (fun _ => hr)
  · intro hs
    exact Or.inr (fun _ => hs)

theorem not_and_iff_or_not : ¬(p ∧ q) → ¬p ∨ ¬q := by
  intro h
  apply (em p).elim
  · intro hp
    apply (em q).elim
    · intro hq
      exact False.elim (h ⟨hp, hq⟩)
    · intro nq
      exact Or.inr nq
  · intro np
    exact Or.inl np

theorem not_imp_mp : ¬(p → q) → p ∧ ¬q := by
  intro h
  apply And.intro
  · apply byContradiction
    intro np
    apply h
    intro hp
    exact absurd hp np
  · intro hq
    apply h
    intro _
    exact hq

theorem not_or_of_imp : (p → q) → (¬p ∨ q) := by
  intro hpq
  apply (em p).elim
  · intro hp
    exact Or.inr (hpq hp)
  · intro np
    exact Or.inl np

theorem not_imp_not_imp_imp : (¬q → ¬p) → (p → q) := by
  intro hqp hp
  apply byContradiction
  intro nq
  exact absurd hp (hqp nq)

theorem or_not : p ∨ ¬p := by apply em

theorem imp_imp_imp : (((p → q) → p) → p) := by
  intro h
  apply (em p).elim
  · intro hp
    exact hp
  · intro np
    apply h
    intro hp
    exact absurd hp np

end ex3_2

/-! ##### Exercises 3.3 -/

section ex3_3

variable (p : Prop)

theorem iff_not_self (hp : p) : ¬(p ↔ ¬p) := by
  intro h
  exact absurd hp (h.mp hp)

end ex3_3

/-! ##### Exercises 4.1 -/

section ex4_1

variable (α : Type _)
variable (p q : α → Prop)

theorem forall_and : (∀ x, p x ∧ q x) ↔ (∀ x, p x) ∧ (∀ x, q x) := by
  apply Iff.intro
  · intro h
    apply And.intro
    · intro hx; exact And.left (h hx)
    · intro hx; exact And.right (h hx)
  · intro h hx
    have lhs : ∀ (x : α), p x := And.left h
    have rhs : ∀ (x : α), q x := And.right h
    exact ⟨lhs hx, rhs hx⟩

theorem forall_imp_distrib : (∀ x, p x → q x) → (∀ x, p x) → (∀ x, q x) := by
  intro h₁ h₂ hx
  exact h₁ hx (h₂ hx)

theorem forall_or_distrib : (∀ x, p x) ∨ (∀ x, q x) → ∀ x, p x ∨ q x := by
  intro
  | Or.inl h => intro hx; exact Or.inl (h hx)
  | Or.inr h => intro hx; exact Or.inr (h hx)

end ex4_1

/-! ##### Exercises 4.2 -/

section ex4_2

variable (α : Type _)
variable (p q : α → Prop)
variable (r : Prop)

theorem self_imp_forall : α → ((∀ _ : α, r) ↔ r) := by
  intro ha
  apply Iff.intro
  · intro har
    apply har
    exact ha
  · intro hr _
    exact hr

section

open Classical

theorem forall_or_right : (∀ x, p x ∨ r) ↔ (∀ x, p x) ∨ r := by
  apply Iff.intro
  · intro h
    apply (em r).elim
    · intro hr
      exact Or.inr hr
    · intro nr
      apply Or.inl
      · intro hx
        apply (h hx).elim
        · exact id
        · intro hr
          exact absurd hr nr
  · intro h₁ hx
    apply h₁.elim
    · intro h₂
      exact Or.inl (h₂ hx)
    · intro hr
      exact Or.inr hr

end

theorem forall_swap : (∀ x, r → p x) ↔ (r → ∀ x, p x) := by
  apply Iff.intro
  · intro h hr hx
    exact h hx hr
  · intro h hx hr
    exact h hr hx

end ex4_2

/-! ##### Exercises 4.3 -/

section ex4_3

open Classical

variable (men : Type _)
variable (barber : men)
variable (shaves : men → men → Prop)

theorem barber_paradox (h : ∀ x : men, shaves barber x ↔ ¬ shaves x x)
  : False := by
  apply (em (shaves barber barber)).elim
  · intro hb
    exact absurd hb ((h barber).mp hb)
  · intro nb
    exact absurd ((h barber).mpr nb) nb

end ex4_3

/-! ##### Exercises 4.5 -/

section ex4_5

open Classical

variable (α : Type _)
variable (p q : α → Prop)
variable (r s : Prop)

theorem exists_imp : (∃ _ : α, r) → r := by
  intro ⟨_, hr⟩
  exact hr

theorem exists_intro (a : α) : r → (∃ _ : α, r) := by
  intro hr
  exact ⟨a, hr⟩

theorem exists_and_right : (∃ x, p x ∧ r) ↔ (∃ x, p x) ∧ r := by
  apply Iff.intro
  · intro ⟨hx, hp, hr⟩
    exact ⟨⟨hx, hp⟩, hr⟩
  · intro ⟨⟨hx, hp⟩, hr⟩
    exact ⟨hx, hp, hr⟩

theorem exists_or : (∃ x, p x ∨ q x) ↔ (∃ x, p x) ∨ (∃ x, q x) := by
  apply Iff.intro
  · intro
    | ⟨hx, Or.inl hp⟩ => exact Or.inl ⟨hx, hp⟩
    | ⟨hx, Or.inr hq⟩ => exact Or.inr ⟨hx, hq⟩
  · intro
    | Or.inl ⟨hx, hp⟩ => exact ⟨hx, Or.inl hp⟩
    | Or.inr ⟨hx, hq⟩ => exact ⟨hx, Or.inr hq⟩

theorem forall_iff_not_exists : (∀ x, p x) ↔ ¬(∃ x, ¬p x) := by
  apply Iff.intro
  · intro ha ⟨hx, np⟩
    exact absurd (ha hx) np
  · intro he hx
    apply byContradiction
    intro np
    exact he ⟨hx, np⟩

theorem exists_iff_not_forall : (∃ x, p x) ↔ ¬(∀ x, ¬p x) := by
  apply Iff.intro
  · intro ⟨hx, hp⟩ h
    exact absurd hp (h hx)
  · intro h₁
    apply byContradiction
    intro h₂
    apply h₁
    intro hx hp
    exact h₂ ⟨hx, hp⟩

theorem not_exists : (¬∃ x, p x) ↔ (∀ x, ¬p x) := by
  apply Iff.intro
  · intro h hx hp
    exact h ⟨hx, hp⟩
  · intro h ⟨hx, hp⟩
    exact absurd hp (h hx)

theorem forall_negation : (¬∀ x, p x) ↔ (∃ x, ¬p x) := by
  apply Iff.intro
  · intro h₁
    apply byContradiction
    intro h₂
    exact h₁ (fun (x : α) => by
      apply byContradiction
      intro np
      exact h₂ ⟨x, np⟩)
  · intro ⟨hx, np⟩ h
    exact absurd (h hx) np

theorem not_forall : (¬∀ x, p x) ↔ (∃ x, ¬p x) := forall_negation α p

theorem forall_iff_exists_imp : (∀ x, p x → r) ↔ (∃ x, p x) → r := by
  apply Iff.intro
  · intro h ⟨hx, hp⟩
    exact h hx hp
  · intro h hx hp
    exact h ⟨hx, hp⟩

theorem exists_iff_forall_imp (a : α) : (∃ x, p x → r) ↔ (∀ x, p x) → r := by
  apply Iff.intro
  · intro ⟨hx, hp⟩ h
    apply hp
    exact h hx
  · intro h₁
    apply (em (∀ x, p x)).elim
    · intro h₂
      exact ⟨a, fun _ => h₁ h₂⟩
    · intro h₂
      have ⟨hx, np⟩ : (∃ x, ¬p x) := (forall_negation α p).mp h₂
      exact ⟨hx, fun hp => absurd hp np⟩

theorem exists_self_iff_self_exists (a : α)
  : (∃ x, r → p x) ↔ (r → ∃ x, p x) := by
  apply Iff.intro
  · intro ⟨hx, h⟩ hr
    exact ⟨hx, h hr⟩
  · intro h
    apply (em r).elim
    · intro hr
      have ⟨hx, hp⟩ := h hr
      exact ⟨hx, fun _ => hp⟩
    · intro nr
      exact ⟨a, fun hr => absurd hr nr⟩

end ex4_5

end ex1

/-! #### Exercise 2

Use tactic combinators to obtain a one line proof of the following:
-/

namespace ex2

theorem or_and_or_and_or (p q r : Prop) (hp : p)
  : (p ∨ q ∨ r) ∧ (q ∨ p ∨ r) ∧ (q ∨ r ∨ p) := by
  simp [*]

end ex2

end Avigad.Chapter5