Examples of Proofs

We see our next proofs, most of which involve the ≤ relation on natural numbers.

We will see that the natural numbers are "defined" in terms of the zero and succ constructors.

• Nat.zero : ℕ
• Nat.succ : ℕ → ℕ

Analogous to this, (modulo renaming) the ≤ relation is defined in terms of the le_refl and le_step constructors.

• Nat.le : ℕ → ℕ → Prop
• Nat.le_refl : ∀ n : ℕ, n ≤ n
• Nat.le_step : ∀ {n m : ℕ}, n ≤ m → n ≤ Nat.succ m

The first proof we see is of 3 ≤ 3. This is a direct application of Nat.le_refl. This is analogous to applying Nat.le_refl as a function to the argument 3.

theorem three_le_three : 3 ≤ 3 :=
  Nat.le_refl 3

Our second result is similar and has a similar proof. However in the proof we did not specify the argument 4 and instead used the placeholder _. Lean deduced that the unique way to get types correct is to fill in 4.

/-- The result `4 ≤ 4`. -/
def four_le_four : 4 ≤ 4 :=
  Nat.le_refl _

theorem three_le_three : 3 ≤ 3

The result 3 ≤ 3.

def four_le_four : 4 ≤ 4

The result 4 ≤ 4.

Some more complex proofs. In the first case the proof is given fully while in the second we allow a parameter to be inferred

theorem three_le_five : 3 ≤ 5 :=
  Nat.le_step (Nat.le_step (Nat.le_refl 3))

theorem three_le_six : 3 ≤ 6 :=
  Nat.le_step (
    Nat.le_step 
     (Nat.le_step (Nat.le_refl _))) 

theorem three_le_five : 3 ≤ 5

theorem three_le_six : 3 ≤ 6

In the next proof we use tactics, specifically the apply tactic. We also see a case where it fails.

theorem four_le_seven : 4 ≤ 7 := by
  apply Nat.le_step
  -- apply Nat.le_refl 
  /-tactic 'apply' failed, failed to unify
  ?n ≤ ?n
    with
  4 ≤ 6-/ 
  apply Nat.le_step
  apply Nat.le_step
  apply Nat.le_refl

theorem four_le_seven : 4 ≤ 7

Note that the proofs produced by tactics are proofs in the usual sense.

#print four_le_seven /- theorem four_le_seven : 4 ≤ 7 :=
 Nat.le_step (Nat.le_step (Nat.le_step (Nat.le_refl 4))) -/

Lean has many powerful tactics. The decide tactic can prove propositions that (are true and) can be decided by an algorithm corresponding to the Decidable typeclass (which we see later).

theorem four_le_ten : 4 ≤ 10 :=
  by decide

The proof produced by the decide tacic is similar to the above proofs.

def four_le_ten' : 4 ≤ 10 :=
  by decide


#reduce four_le_ten' /- Nat.le.step (Nat.le.step (Nat.le.step (Nat.le.step (Nat.le.step (Nat.le.step Nat.le.refl)))))
-/

theorem four_le_ten : 4 ≤ 10

def four_le_ten' : 4 ≤ 10

We can combine tactics. repeat applies a tactic as long as it is valid; first applies the first applicable tactic.

example : 4 ≤ 10 :=
  by
    -- repeat (apply Nat.le_step) -- goal: 4 ≤ 0
    repeat (first | 
          apply Nat.le_refl | 
          apply Nat.le_step)
    done

We will write some basic tactics

def tacticNat_le : Lean.ParserDescr

A tactic for proving ≤ for natural numbers

def tacticFinish_with_Steps_ : Lean.ParserDescr

A tactic where we try repeatedly to finish with a theorem or take a step with another.

We can use our more general tactic in different ways.

example : 4 ≤ 44 := by
  finish_with Nat.le_refl steps Nat.le_step

#check Nat.succ_le_succ -- ∀ {n m : ℕ}, n ≤ m → Nat.succ n ≤ Nat.succ m
#check Nat.zero_le -- ∀ (n : ℕ), 0 ≤ n

example : 4 ≤ 44 := by
  finish_with Nat.zero_le steps Nat.succ_le_succ