bookshelf/Bookshelf/Avigad/Chapter_3.lean

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/-! # Avigad.Chapter3
Propositions and Proofs
-/
/-! ### Exercise 1
Prove the following identities.
-/
namespace Avigad.Chapter3
namespace ex1
open or
variable (p q r : Prop)
-- Commutativity of ∧ and
theorem and_comm' : p ∧ q ↔ q ∧ p :=
Iff.intro
(fun ⟨hp, hq⟩ => show q ∧ p from ⟨hq, hp⟩)
(fun ⟨hq, hp⟩ => show p ∧ q from ⟨hp, hq⟩)
theorem or_comm' : p q ↔ q p :=
Iff.intro
(fun h => h.elim Or.inr Or.inl)
(fun h => h.elim Or.inr Or.inl)
-- Associativity of ∧ and
theorem and_assoc : (p ∧ q) ∧ r ↔ p ∧ (q ∧ r) :=
Iff.intro
(fun ⟨⟨hp, hq⟩, hr⟩ => ⟨hp, hq, hr⟩)
(fun ⟨hp, hq, hr⟩ => ⟨⟨hp, hq⟩, hr⟩)
theorem or_assoc' : (p q) r ↔ p (q r) :=
Iff.intro
(fun h₁ => h₁.elim
(fun h₂ => h₂.elim Or.inl (Or.inr ∘ Or.inl))
(Or.inr ∘ Or.inr))
(fun h₁ => h₁.elim
(Or.inl ∘ Or.inl)
(fun h₂ => h₂.elim (Or.inl ∘ Or.inr) Or.inr))
-- Distributivity
theorem and_or_left : p ∧ (q r) ↔ (p ∧ q) (p ∧ r) :=
Iff.intro
(fun ⟨hp, hqr⟩ => hqr.elim (Or.inl ⟨hp, ·⟩) (Or.inr ⟨hp, ·⟩))
(fun h₁ => h₁.elim
(fun ⟨hp, hq⟩ => ⟨hp, Or.inl hq⟩)
(fun ⟨hp, hr⟩ => ⟨hp, Or.inr hr⟩))
theorem or_and_left : p (q ∧ r) ↔ (p q) ∧ (p r) :=
Iff.intro
(fun h => h.elim
(fun hp => ⟨Or.inl hp, Or.inl hp⟩)
(fun ⟨hq, hr⟩ => ⟨Or.inr hq, Or.inr hr⟩))
(fun ⟨h₁, h₂⟩ => h₁.elim
Or.inl
(fun hq => h₂.elim Or.inl (fun hr => Or.inr ⟨hq, hr⟩)))
-- Other properties
theorem imp_imp_iff_and_imp : (p → (q → r)) ↔ (p ∧ q → r) :=
Iff.intro
(fun h ⟨hp, hq⟩ => h hp hq)
(fun h hp hq => h ⟨hp, hq⟩)
theorem or_imp : ((p q) → r) ↔ (p → r) ∧ (q → r) :=
Iff.intro
(fun h =>
have h₁ : p → r := h ∘ Or.inl
have h₂ : q → r := h ∘ Or.inr
show (p → r) ∧ (q → r) from ⟨h₁, h₂⟩)
(fun ⟨h₁, h₂⟩ h => h.elim h₁ h₂)
theorem nor_or : ¬(p q) ↔ ¬p ∧ ¬q :=
Iff.intro
(fun h => ⟨h ∘ Or.inl, h ∘ Or.inr⟩)
(fun h₁ h₂ => h₂.elim (absurd · h₁.left) (absurd · h₁.right))
theorem not_and_or_mpr : ¬p ¬q → ¬(p ∧ q) :=
fun h₁ h₂ => h₁.elim (absurd h₂.left ·) (absurd h₂.right ·)
theorem and_not_self : ¬(p ∧ ¬p) :=
fun h => absurd h.left h.right
theorem not_imp_o_and_not : p ∧ ¬q → ¬(p → q) :=
fun ⟨hp, nq⟩ hpq => absurd (hpq hp) nq
theorem false_elim_self : ¬p → (p → q) :=
fun np hp => absurd hp np
theorem not_or_imp_imp : (¬p q) → (p → q) :=
fun npq hp => npq.elim (absurd hp ·) id
theorem or_false_iff : p False ↔ p :=
Iff.intro (fun hpf => hpf.elim id False.elim) Or.inl
theorem and_false_iff : p ∧ False ↔ False :=
Iff.intro (fun ⟨_, hf⟩ => hf) False.elim
theorem imp_imp_not_imp_not : (p → q) → (¬q → ¬p) :=
fun hpq nq hp => absurd (hpq hp) nq
end ex1
/-! ### Exercise 2
Prove the following identities. These require classical reasoning.
-/
namespace ex2
open Classical
variable (p q r s : Prop)
theorem imp_or_mp (hp : p) : (p → r s) → ((p → r) (p → s)) :=
fun h => (h hp).elim
(fun hr => Or.inl (fun _ => hr))
(fun hs => Or.inr (fun _ => hs))
theorem not_and_iff_or_not : ¬(p ∧ q) → ¬p ¬q :=
fun npq => (em p).elim
(fun hp => (em q).elim
(fun hq => False.elim (npq ⟨hp, hq⟩))
Or.inr)
Or.inl
theorem not_imp_mp : ¬(p → q) → p ∧ ¬q :=
fun h =>
have lhs : p := byContradiction
fun np => h (fun (hp : p) => absurd hp np)
⟨lhs, fun hq => h (fun _ => hq)⟩
theorem not_or_of_imp : (p → q) → (¬p q) :=
fun hpq => (em p).elim (fun hp => Or.inr (hpq hp)) Or.inl
theorem not_imp_not_imp_imp : (¬q → ¬p) → (p → q) :=
fun h hp => byContradiction
fun nq => absurd hp (h nq)
theorem or_not : p ¬p := em p
theorem imp_imp_imp : (((p → q) → p) → p) :=
fun h => byContradiction
fun np =>
suffices hp : p from absurd hp np
h (fun (hp : p) => absurd hp np)
end ex2
/-! ### Exercise 3
Prove `¬(p ↔ ¬p)` without using classical logic.
-/
namespace ex3
variable (p : Prop)
theorem iff_not_self (hp : p) : ¬(p ↔ ¬p) :=
fun h => absurd hp (Iff.mp h hp)
end ex3
end Avigad.Chapter3