import LoVe.LoVelib
namespace LoVe

FPV Lab 3: Forward Proofs

Question 1: Connectives and Quantifiers

1.1.

Supply structured proofs of the following theorems.

theorem I (a : Prop) :
  a → a :=
  sorry

theorem K (a b : Prop) :
  a → b → b :=
  sorry

theorem C (a b c : Prop) :
  (a → b → c) → b → a → c :=
  sorry

theorem proj_fst (a : Prop) :
  a → a → a :=
  sorry

Please give a different answer than for proj_fst.

theorem proj_snd (a : Prop) :
  a → a → a :=
  sorry

theorem some_nonsense (a b c : Prop) :
  (a → b → c) → a → (a → c) → b → c :=
  sorry

1.2.

\ Supply a structured proof of the contraposition rule.

theorem contrapositive (a b : Prop) :
  (a → b) → ¬ b → ¬ a :=
  sorry

1.3.

\ Supply a structured proof of the distributivity of ∀ over ∧.

theorem forall_and {α : Type} (p q : α → Prop) :
  (∀x, p x ∧ q x) ↔ (∀x, p x) ∧ (∀x, q x) :=
  sorry

1.4 (optional).

\ Supply a structured proof of the following property, which can be used to pull a \ ∀ quantifier past an ∃ quantifier.

theorem forall_exists_of_exists_forall {α : Type} (p : α → α → Prop) :
  (∃x, ∀y, p x y) → (∀y, ∃x, p x y) :=
  sorry

Question 2: Chain of Equalities

2.1. Write the following proof using `calc`.

  (a + b) * (a + b)
= a * (a + b) + b * (a + b)
= a * a + a * b + b * a + b * b
= a * a + a * b + a * b + b * b
= a * a + 2 * a * b + b * b

Hint: This is a difficult question. You might need the tactics rw, simp and ac_rfl and some of the theorems listed below:

#check mul_add
#check add_mul
#check add_comm
#check add_assoc
#check mul_comm
#check mul_assoc
#check Nat.two_mul

theorem binomial_square (a b : ℕ) :
  (a + b) * (a + b) = a * a + 2 * a * b + b * b :=
  sorry

2.2 (optional).

\ Prove the same argument again, this time as a structured proof, with have steps corresponding to the calc equations.

You can use tactical proofs (some copy/pasting might be in order!) to prove each of your have steps.) You'll also want to use a tactical proof to show your final goal.

theorem binomial_square₂ (a b : ℕ) :
  (a + b) * (a + b) = a * a + 2 * a * b + b * b :=
  sorry

Question 3: One-Point Rules

3.1. Prove that the following wrong formulation of the one-point rule for `∀` is

inconsistent, using a structured proof.

How should this rule be modified to produce the correct one-point rule?

axiom All.one_point_wrong {α : Type} (t : α) (P : α → Prop) :
  (∀x : α, x = t ∧ P x) ↔ P t

theorem All.proof_of_False :
  False :=
  sorry

3.2. Prove that the following wrong formulation of the one-point rule for `∃`

is inconsistent, using a structured proof.