exercises.lean

import tactic
open classical

/-
Cheat sheet:

to prove `p → q`, use `intro h`
to use `h : p → q`, use `apply h`

to prove `p ∧ q`, use `split`
to use `h : p ∧ q`, use `cases h with hp hq`

to prove `p ∨ q`, use `left` or `right`
to use `p ∨ q`, use `cases h with h1 h1`

to prove `¬ p`, use `intro h`
to use `h : ¬ p`, use `apply h` (when the goal is false)

to prove `p ↔ q`, use `split`
to use `h : p ↔ q`, use `cases h with h1 h2`

to prove a change a goal to `false`: `exfalso`

to prove a goal by contradiction: use `by_contradiction h`
-/

/- Some exercises -/

section

variables p q r s : Prop

example : (p → q) ∧ (q → r) → p → r :=
begin
  sorry
end

example : (p ∨ q) ∧ ¬ p → q :=
begin
  sorry
end

end

/- Exercises from Theorem Proving in Lean -/

section

variables p q r s : Prop

-- commutativity of ∧ and ∨
example : p ∧ q ↔ q ∧ p := sorry
example : p ∨ q ↔ q ∨ p := sorry

-- associativity of ∧ and ∨
example : (p ∧ q) ∧ r ↔ p ∧ (q ∧ r) := sorry
example : (p ∨ q) ∨ r ↔ p ∨ (q ∨ r) := sorry

-- distributivity
example : p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r) := sorry
example : p ∨ (q ∧ r) ↔ (p ∨ q) ∧ (p ∨ r) := sorry

-- other properties
example : (p → (q → r)) ↔ (p ∧ q → r) := sorry
example : ((p ∨ q) → r) ↔ (p → r) ∧ (q → r) := sorry
example : ¬(p ∨ q) ↔ ¬p ∧ ¬q := sorry
example : ¬p ∨ ¬q → ¬(p ∧ q) := sorry
example : ¬(p ∧ ¬p) := sorry
example : p ∧ ¬q → ¬(p → q) := sorry
example : ¬p → (p → q) := sorry
example : (¬p ∨ q) → (p → q) := sorry
example : p ∨ false ↔ p := sorry
example : p ∧ false ↔ false := sorry
example : (p → q) → (¬q → ¬p) := sorry

example : ¬ (p ↔ ¬ p) := sorry

-- these require classical logic

example : (p → r ∨ s) → ((p → r) ∨ (p → s)) := sorry
example : ¬(p ∧ q) → ¬p ∨ ¬q := sorry
example : ¬(p → q) → p ∧ ¬q := sorry
example : (p → q) → (¬p ∨ q) := sorry
example : (¬q → ¬p) → (p → q) := sorry
example : p ∨ ¬p := sorry
example : (((p → q) → p) → p) := sorry

end

/-
Quantifiers
-/

section

variables (α : Type) (p q : α → Prop)

example : (∀ x, p x ∧ q x) ↔ (∀ x, p x) ∧ (∀ x, q x) := sorry
example : (∀ x, p x → q x) → (∀ x, p x) → (∀ x, q x) := sorry
example : (∀ x, p x) ∨ (∀ x, q x) → ∀ x, p x ∨ q x := sorry

end

/-
Equality
-/

section

variables (real : Type) [ordered_ring real]
variables (log exp : real → real)
variable  log_exp_eq : ∀ x, log (exp x) = x
variable  exp_log_eq : ∀ {x}, x > 0 → exp (log x) = x
variable  exp_pos    : ∀ x, exp x > 0
variable  exp_add    : ∀ x y, exp (x + y) = exp x * exp y

-- this ensures the assumptions are available in tactic proofs
include log_exp_eq exp_log_eq exp_pos exp_add

example (x y z : real) :
  exp (x + y + z) = exp x * exp y * exp z :=
by rw [exp_add, exp_add]

example (y : real) (h : y > 0)  : exp (log y) = y :=
exp_log_eq h

theorem log_mul {x y : real} (hx : x > 0) (hy : y > 0) :
  log (x * y) = log x + log y :=
sorry

end