CSCI 2041

ADVANCED PROGRAMMING PRINCIPLES

Reasoning about programs:

Natural Numbers

Induction on nat

What about inputs that are not integers?

type nat = Zero | Succ of nat
let rec to_int (n:nat) = match n with
| Zero -> 0
| Succ m -> 1 + (to_int m)
let rec plus_nat m n = match m with
| Zero -> n
| Succ m’ -> Succ (plus_nat m’ n)
let rec mul_nat m n = match m with
| Zero -> Zero
| Succ m’ -> plus_nat n (mul_nat m’ n)

Generalizing Induction

Principle of induction for nat:

For all x : nat, P(x) if

  • P(Zero) ( Base Case ) and
  • ∀x, P(x) ⇒ P(Succ x) ( Inductive Case )

E.g. Let’s prove that for all n1 : nat, P(n1), where

P(n1) = ∀ n2 : nat, to_int (plus_nat n1 n2) ≡ (to_int n1) + (to_int n2)

Base Case: n1 = Zero.

E.g. Let’s prove that for all n1 : nat, P(n1), where

P(n1) = ∀ n2 : nat, to_int (plus_nat n1 n2) ≡ (to_int n1) + (to_int n2)

Base Case: n1 = Zero.

Inductive Case:

to_int (plus_nat (Succ n) n2) ≡
to_int (Succ (plus_nat n n2)) ≡
1 + (to_int (plus_nat n n2)) ≡
1 + (to_int n) + (to_int n2) ≡
(to_int (Succ n)) + (to_int n2)

mul_nat

∀ n1 : nat, ∀n2 : nat, to_int (mul_nat n1 n2) ≡ (to_int n1) * (to_int n2)

Base Case n1 = Zero:

∀n2, to_int (mul_nat Zero n2) ≡ to_int Zero ≡ 0 ≡ (to_int Zero) * (to_int n2)

Inductive Case:

to_int (mul_nat (Succ n) n2) ≡
to_int (plus_nat n2 (mul_nat n n2)) ≡
(to_int n2) + (to_int (mul_nat n n2)) ≡
(to_int n2) + (to_int n) * (to_int n2) ≡
(1 + (to_int n)) * (to_int n2) ≡
(to_int (Succ n)) * (to_int n2)

Generalized Induction

For any inductive type of the form:

type t = C₀ of b (* b is some other type not referring to t *)
| C₁ of b₁*t
| C₂ of b₂ * t * t

The principle of induction for type t is:
For all x : t, P(x) if:

  • ∀ v : b, P(C₀ v), and
  • ∀ x : t, v : b₁, P(x) ⇒ P(C₁(v,x)), and
  • ∀ x1, x2 : t, v : b₂, P(x1) and P(x2) ⇒ P(C₂ (v,x1,x2))

nat: ∀ n : nat, P(n) if P(Zero) and ∀ m, P(m) ⇒ P(Succ m)

'a list: ∀ ℓ : 'a list', P(ℓ) if P([]) and ∀ x ∈ 'a, ∀ ℓ : 'a list, P(ℓ) => P(x::ℓ)

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