module Data.Nat.Properties where
Natural numbers - properties🔗
This module contains proofs of arithmetic identities for the natural numbers. Since they’re mostly simple inductive arguments written in equational reasoning style, they are very lightly commented:
Addition🔗
+-associative : (x y z : Nat) → x + (y + z) ≡ (x + y) + z +-associative zero y z = refl +-associative (suc x) y z = suc (x + (y + z)) ≡⟨ ap suc (+-associative x y z) ⟩≡ suc ((x + y) + z) ∎ +-zeror : (x : Nat) → x + 0 ≡ x +-zeror zero = refl +-zeror (suc x) = suc (x + 0) ≡⟨ ap suc (+-zeror x) ⟩≡ suc x ∎ +-sucr : (x y : Nat) → x + suc y ≡ suc (x + y) +-sucr zero y = refl +-sucr (suc x) y = ap suc (+-sucr x y) +-commutative : (x y : Nat) → x + y ≡ y + x +-commutative zero y = sym (+-zeror y) +-commutative (suc x) y = suc (x + y) ≡⟨ ap suc (+-commutative x y) ⟩≡ suc (y + x) ≡⟨ sym (+-sucr y x) ⟩≡ y + suc x ∎ +-inj : ∀ k x y → k + x ≡ k + y → x ≡ y +-inj zero x y p = p +-inj (suc k) x y p = +-inj k x y (suc-inj p)
Multiplication🔗
*-distrib-+r : (x y z : Nat) → (x + y) * z ≡ x * z + y * z *-distrib-+r zero y z = refl *-distrib-+r (suc x) y z = z + (x + y) * z ≡⟨ ap₂ _+_ refl (*-distrib-+r x y z) ⟩≡ z + (x * z + y * z) ≡⟨ +-associative z (x * z) (y * z) ⟩≡ z + x * z + y * z ∎ *-sucr : (m n : Nat) → m * suc n ≡ m + m * n *-sucr zero n = refl *-sucr (suc m) n = suc m * suc n ≡⟨⟩ suc n + m * suc n ≡⟨ ap₂ _+_ refl (*-sucr m n) ⟩≡ suc n + (m + m * n) ≡⟨⟩ suc (n + (m + m * n)) ≡⟨ ap suc (+-associative n m (m * n)) ⟩≡ suc (n + m + m * n) ≡⟨ ap (λ x → suc (x + m * n)) (+-commutative n m) ⟩≡ suc (m + n + m * n) ≡˘⟨ ap suc (+-associative m n (m * n)) ⟩≡˘ suc (m + (n + m * n)) ≡⟨⟩ suc m + suc m * n ∎ *-onel : (x : Nat) → 1 * x ≡ x *-onel x = +-zeror x *-oner : (x : Nat) → x * 1 ≡ x *-oner zero = refl *-oner (suc x) = suc (x * 1) ≡⟨ ap suc (*-oner x) ⟩≡ suc x ∎ *-zeror : (x : Nat) → x * 0 ≡ 0 *-zeror zero = refl *-zeror (suc x) = *-zeror x *-commutative : (x y : Nat) → x * y ≡ y * x *-commutative zero y = sym (*-zeror y) *-commutative (suc x) y = y + x * y ≡⟨ ap₂ _+_ refl (*-commutative x y) ⟩≡ y + y * x ≡⟨ sym (*-sucr y x) ⟩≡ y * suc x ∎ *-distrib-+l : (x y z : Nat) → z * (x + y) ≡ z * x + z * y *-distrib-+l x y z = z * (x + y) ≡⟨ *-commutative z (x + y) ⟩≡ (x + y) * z ≡⟨ *-distrib-+r x y z ⟩≡ x * z + y * z ≡⟨ ap₂ _+_ (*-commutative x z) (*-commutative y z) ⟩≡ z * x + z * y ∎ *-associative : (x y z : Nat) → (x * y) * z ≡ x * (y * z) *-associative zero y z = refl *-associative (suc x) y z = (y + x * y) * z ≡⟨ *-distrib-+r y (x * y) z ⟩≡ y * z + (x * y) * z ≡⟨ ap₂ _+_ refl (*-associative x y z) ⟩≡ y * z + x * (y * z) ∎ *-suc-inj : ∀ k x y → x * suc k ≡ y * suc k → x ≡ y *-suc-inj k zero zero p = refl *-suc-inj k zero (suc y) p = absurd (zero≠suc p) *-suc-inj k (suc x) zero p = absurd (suc≠zero p) *-suc-inj k (suc x) (suc y) p = ap suc (*-suc-inj k x y (+-inj _ _ _ p)) *-suc-inj' : ∀ k x y → suc k * x ≡ suc k * y → x ≡ y *-suc-inj' k x y p = *-suc-inj k x y (*-commutative x (suc k) ∙∙ p ∙∙ *-commutative (suc k) y) *-injr : ∀ k x y .⦃ _ : Positive k ⦄ → x * k ≡ y * k → x ≡ y *-injr (suc k) x y p = *-suc-inj k x y p *-injl : ∀ k x y .⦃ _ : Positive k ⦄ → k * x ≡ k * y → x ≡ y *-injl (suc k) x y p = *-suc-inj' k x y p *-is-onel : ∀ x n → x * n ≡ 1 → x ≡ 1 *-is-onel zero n p = p *-is-onel (suc zero) zero p = refl *-is-onel (suc (suc x)) zero p = absurd (zero≠suc (sym (*-zeror x) ∙ p)) *-is-onel (suc x) (suc zero) p = ap suc (sym (*-oner x)) ∙ p *-is-onel (suc x) (suc (suc n)) p = absurd (zero≠suc (sym (suc-inj p))) *-is-oner : ∀ x n → x * n ≡ 1 → n ≡ 1 *-is-oner x n p = *-is-onel n x (*-commutative n x ∙ p) *-is-zero : ∀ x y → x * y ≡ 0 → (x ≡ 0) ⊎ (y ≡ 0) *-is-zero zero y p = inl refl *-is-zero (suc x) zero p = inr refl *-is-zero (suc x) (suc y) p = absurd (suc≠zero p) *-is-zerol : ∀ x y ⦃ _ : Positive y ⦄ → x * y ≡ 0 → x ≡ 0 *-is-zerol x (suc y) p with *-is-zero x (suc y) p ... | inl p = p ... | inr q = absurd (suc≠zero q) *-is-zeror : ∀ x y ⦃ _ : Positive x ⦄ → x * y ≡ 0 → y ≡ 0 *-is-zeror (suc x) y p with *-is-zero (suc x) y p ... | inl p = absurd (suc≠zero p) ... | inr q = q
Exponentiation🔗
^-oner : (x : Nat) → x ^ 1 ≡ x ^-oner x = *-oner x ^-onel : (x : Nat) → 1 ^ x ≡ 1 ^-onel zero = refl ^-onel (suc x) = (1 ^ x) + 0 ≡⟨ +-zeror (1 ^ x) ⟩≡ (1 ^ x) ≡⟨ ^-onel x ⟩≡ 1 ∎ ^-+-hom-*r : (x y z : Nat) → x ^ (y + z) ≡ (x ^ y) * (x ^ z) ^-+-hom-*r x zero z = sym (+-zeror (x ^ z)) ^-+-hom-*r x (suc y) z = x * x ^ (y + z) ≡⟨ ap (x *_) (^-+-hom-*r x y z) ⟩≡ x * (x ^ y * x ^ z) ≡⟨ sym (*-associative x (x ^ y) (x ^ z)) ⟩≡ x * x ^ y * x ^ z ∎ ^-distrib-*r : (x y z : Nat) → (x * y) ^ z ≡ x ^ z * y ^ z ^-distrib-*r x y zero = refl ^-distrib-*r x y (suc z) = x * y * (x * y) ^ z ≡⟨ ap (λ a → x * y * a) (^-distrib-*r x y z) ⟩≡ x * y * (x ^ z * y ^ z) ≡⟨ sym (*-associative (x * y) (x ^ z) (y ^ z)) ⟩≡ x * y * x ^ z * y ^ z ≡⟨ ap (_* y ^ z) (*-associative x y (x ^ z)) ⟩≡ x * (y * x ^ z) * y ^ z ≡⟨ ap (λ a → x * a * y ^ z) (*-commutative y (x ^ z)) ⟩≡ x * (x ^ z * y) * y ^ z ≡⟨ ap (_* y ^ z) (sym (*-associative x (x ^ z) y)) ⟩≡ x * x ^ z * y * y ^ z ≡⟨ *-associative (x * x ^ z) y (y ^ z) ⟩≡ x * x ^ z * (y * y ^ z) ∎ ^-*-adjunct : (x y z : Nat) → (x ^ y) ^ z ≡ x ^ (y * z) ^-*-adjunct x zero z = ^-onel z ^-*-adjunct x (suc y) zero = ^-*-adjunct x y zero ^-*-adjunct x (suc y) (suc z) = x * x ^ y * (x * x ^ y) ^ z ≡⟨ ap (λ a → x * x ^ y * a) (^-distrib-*r x (x ^ y) z) ⟩≡ x * x ^ y * (x ^ z * (x ^ y) ^ z) ≡⟨ ap (λ a → x * x ^ y * (x ^ z * a)) (^-*-adjunct x y z) ⟩≡ x * x ^ y * (x ^ z * x ^ (y * z)) ≡⟨ ap (λ a → x * x ^ y * a) (sym (^-+-hom-*r x z (y * z))) ⟩≡ x * x ^ y * (x ^ (z + (y * z))) ≡⟨ *-associative x (x ^ y) (x ^ (z + y * z)) ⟩≡ x * (x ^ y * (x ^ (z + (y * z)))) ≡⟨ ap (x *_) (sym (^-+-hom-*r x y (z + y * z))) ⟩≡ x * x ^ (y + (z + y * z)) ≡⟨ ap (λ a → x * x ^ a) (+-associative y z (y * z)) ⟩≡ x * x ^ (y + z + y * z) ≡⟨ ap (λ a → x * x ^ (a + y * z)) (+-commutative y z) ⟩≡ x * x ^ (z + y + y * z) ≡˘⟨ ap (λ a → x * x ^ a) (+-associative z y (y * z)) ⟩≡˘ x * x ^ (z + (y + y * z)) ≡⟨ ap (λ a → x * x ^ (z + a)) (sym (*-sucr y z)) ⟩≡ x * x ^ (z + y * suc z) ∎
Compatibility🔗
The order relation on the natural numbers is also compatible with the arithmetic operators:
+-≤l : (x y : Nat) → x ≤ (x + y) +-≤l zero y = 0≤x +-≤l (suc x) y = s≤s (+-≤l x y) +-≤r : (x y : Nat) → y ≤ (x + y) +-≤r x zero = 0≤x +-≤r x (suc y) = subst (λ p → suc y ≤ p) (sym (+-sucr x y)) (s≤s (+-≤r x y)) monus-≤ : (x y : Nat) → x - y ≤ x monus-≤ x zero = x≤x monus-≤ zero (suc y) = 0≤x monus-≤ (suc x) (suc y) = ≤-sucr (monus-≤ x y) +-preserves-≤l : (x y z : Nat) → x ≤ y → (z + x) ≤ (z + y) +-preserves-≤l .0 y zero 0≤x = 0≤x +-preserves-≤l .0 y (suc z) 0≤x = s≤s (+-preserves-≤l zero y z 0≤x) +-preserves-≤l .(suc _) .(suc _) zero (s≤s p) = s≤s p +-preserves-≤l .(suc _) .(suc _) (suc z) (s≤s p) = s≤s (+-preserves-≤l (suc _) (suc _) z (s≤s p)) +-preserves-≤r : (x y z : Nat) → x ≤ y → (x + z) ≤ (y + z) +-preserves-≤r x y z prf = subst (λ a → a ≤ (y + z)) (+-commutative z x) (subst (λ a → (z + x) ≤ a) (+-commutative z y) (+-preserves-≤l x y z prf)) +-preserves-≤ : (x y x' y' : Nat) → x ≤ y → x' ≤ y' → (x + x') ≤ (y + y') +-preserves-≤ x y x' y' prf prf' = ≤-trans (+-preserves-≤r x y x' prf) (+-preserves-≤l x' y' y prf') +-preserves-<l : (x y z : Nat) → x < y → (z + x) < (z + y) +-preserves-<l x (suc y) z (s≤s p) = ≤-trans (s≤s (+-preserves-≤l x y z p)) (≤-refl' (sym (+-sucr z y))) +-preserves-<r : (x y z : Nat) → x < y → (x + z) < (y + z) +-preserves-<r x y z p = subst₂ _<_ (+-commutative z x) (+-commutative z y) (+-preserves-<l x y z p) +-preserves-< : ∀ x y x' y' → x < y → x' < y' → (x + x') < (y + y') +-preserves-< x y x' y' p q = <-trans _ _ _ (+-preserves-<r x y x' p) (+-preserves-<l x' y' y q) *-preserves-≤l : (x y z : Nat) → x ≤ y → (z * x) ≤ (z * y) *-preserves-≤l x y zero prf = 0≤x *-preserves-≤l x y (suc z) prf = +-preserves-≤ x y (z * x) (z * y) prf (*-preserves-≤l x y z prf) *-preserves-≤r : (x y z : Nat) → x ≤ y → (x * z) ≤ (y * z) *-preserves-≤r x y z prf = subst (λ a → a ≤ (y * z)) (*-commutative z x) (subst (λ a → (z * x) ≤ a) (*-commutative z y) (*-preserves-≤l x y z prf)) *-preserves-≤ : (x y x' y' : Nat) → x ≤ y → x' ≤ y' → (x * x') ≤ (y * y') *-preserves-≤ x y x' y' prf prf' = ≤-trans (*-preserves-≤r x y x' prf) (*-preserves-≤l x' y' y prf') +-reflects-≤l : (x y z : Nat) → (z + x) ≤ (z + y) → x ≤ y +-reflects-≤l x y zero prf = prf +-reflects-≤l x y (suc z) (s≤s prf) = +-reflects-≤l x y z prf +-reflects-≤r : (x y z : Nat) → (x + z) ≤ (y + z) → x ≤ y +-reflects-≤r x y z prf = +-reflects-≤l x y z (subst (_≤ (z + y)) (+-commutative x z) (subst ((x + z) ≤_) (+-commutative y z) prf)) difference→≤ : ∀ {x z} y → x + y ≡ z → x ≤ z difference→≤ {x} {z} zero p = subst (x ≤_) (sym (+-zeror x) ∙ p) ≤-refl difference→≤ {zero} {z} (suc y) p = 0≤x difference→≤ {suc x} {zero} (suc y) p = absurd (suc≠zero p) difference→≤ {suc x} {suc z} (suc y) p = s≤s (difference→≤ (suc y) (suc-inj p)) nonzero→positive : ∀ {x} → x ≠0 → 0 < x nonzero→positive {zero} p = absurd (p refl) nonzero→positive {suc x} p = s≤s 0≤x *-cancel-≤r : ∀ x {y z} .⦃ _ : Positive x ⦄ → (y * x) ≤ (z * x) → y ≤ z *-cancel-≤r (suc x) {zero} {z} p = 0≤x *-cancel-≤r (suc x) {suc y} {suc z} (s≤s p) = s≤s (*-cancel-≤r (suc x) {y} {z} (+-reflects-≤l (y * suc x) (z * suc x) x p))
Monus🔗
monus-zero : ∀ a → 0 - a ≡ 0 monus-zero zero = refl monus-zero (suc a) = refl monus-cancell : ∀ k m n → (k + m) - (k + n) ≡ m - n monus-cancell zero = λ _ _ → refl monus-cancell (suc k) = monus-cancell k monus-inversel : ∀ n m → m ≤ n → m + (n - m) ≡ n monus-inversel n m 0≤x = refl monus-inversel (suc n) (suc m) (s≤s p) = ap suc (monus-inversel n m p) monus-distribr : ∀ m n k → (m - n) * k ≡ m * k - n * k monus-distribr m zero k = refl monus-distribr zero (suc n) k = sym (monus-zero (k + n * k)) monus-distribr (suc m) (suc n) k = monus-distribr m n k ∙ sym (monus-cancell k (m * k) (n * k)) monus-distribl : ∀ k m n → k * (m - n) ≡ k * m - k * n monus-distribl k m n = *-commutative k (m - n) ∙ monus-distribr m n k ∙ ap₂ _-_ (*-commutative m k) (*-commutative n k) monus-cancelr : ∀ m n k → (m + k) - (n + k) ≡ m - n monus-cancelr m n k = (λ i → +-commutative m k i - +-commutative n k i) ∙ monus-cancell k m n monus-addl : ∀ m n k → m - (n + k) ≡ m - n - k monus-addl zero zero k = refl monus-addl zero (suc n) k = sym (monus-zero k) monus-addl (suc m) zero k = refl monus-addl (suc m) (suc n) k = monus-addl m n k monus-commute : ∀ m n k → m - n - k ≡ m - k - n monus-commute m n k = m - n - k ≡˘⟨ monus-addl m n k ⟩≡˘ m - (n + k) ≡⟨ ap (m -_) (+-commutative n k) ⟩≡ m - (k + n) ≡⟨ monus-addl m k n ⟩≡ m - k - n ∎ monus-swapl : ∀ x y z → x + y ≡ z → y ≡ z - x monus-swapl x y z p = sym (monus-cancell x y 0) ∙ ap (x + y -_) (+-zeror x) ∙ ap (_- x) p monus-inverser : ∀ n m → (m + n) - m ≡ n monus-inverser n m = sym (monus-swapl m n (m + n) refl) monus-swapr : ∀ x y z → x + y ≡ z → x ≡ z - y monus-swapr x y z p = sym (monus-cancelr x 0 y) ∙ ap (_- y) p
Maximum🔗
max-assoc : (x y z : Nat) → max x (max y z) ≡ max (max x y) z max-assoc zero zero zero = refl max-assoc zero zero (suc z) = refl max-assoc zero (suc y) zero = refl max-assoc zero (suc y) (suc z) = refl max-assoc (suc x) zero zero = refl max-assoc (suc x) zero (suc z) = refl max-assoc (suc x) (suc y) zero = refl max-assoc (suc x) (suc y) (suc z) = ap suc (max-assoc x y z) max-≤l : (x y : Nat) → x ≤ max x y max-≤l zero zero = 0≤x max-≤l zero (suc y) = 0≤x max-≤l (suc x) zero = ≤-refl max-≤l (suc x) (suc y) = s≤s (max-≤l x y) max-≤r : (x y : Nat) → y ≤ max x y max-≤r zero zero = 0≤x max-≤r zero (suc y) = ≤-refl max-≤r (suc x) zero = 0≤x max-≤r (suc x) (suc y) = s≤s (max-≤r x y) max-univ : (x y z : Nat) → x ≤ z → y ≤ z → max x y ≤ z max-univ zero zero z 0≤x 0≤x = 0≤x max-univ zero (suc y) (suc z) 0≤x (s≤s q) = s≤s q max-univ (suc x) zero (suc z) (s≤s p) 0≤x = s≤s p max-univ (suc x) (suc y) (suc z) (s≤s p) (s≤s q) = s≤s (max-univ x y z p q) max-zerol : (x : Nat) → max 0 x ≡ x max-zerol zero = refl max-zerol (suc x) = refl max-zeror : (x : Nat) → max x 0 ≡ x max-zeror zero = refl max-zeror (suc x) = refl
Minimum🔗
min-assoc : (x y z : Nat) → min x (min y z) ≡ min (min x y) z min-assoc zero zero zero = refl min-assoc zero zero (suc z) = refl min-assoc zero (suc y) zero = refl min-assoc zero (suc y) (suc z) = refl min-assoc (suc x) zero zero = refl min-assoc (suc x) zero (suc z) = refl min-assoc (suc x) (suc y) zero = refl min-assoc (suc x) (suc y) (suc z) = ap suc (min-assoc x y z) min-≤l : (x y : Nat) → min x y ≤ x min-≤l zero zero = 0≤x min-≤l zero (suc y) = 0≤x min-≤l (suc x) zero = 0≤x min-≤l (suc x) (suc y) = s≤s (min-≤l x y) min-≤r : (x y : Nat) → min x y ≤ y min-≤r zero zero = 0≤x min-≤r zero (suc y) = 0≤x min-≤r (suc x) zero = 0≤x min-≤r (suc x) (suc y) = s≤s (min-≤r x y) min-univ : (x y z : Nat) → z ≤ x → z ≤ y → z ≤ min x y min-univ x y zero 0≤x 0≤x = 0≤x min-univ (suc x) (suc y) (suc z) (s≤s p) (s≤s q) = s≤s (min-univ x y z p q) min-zerol : (x : Nat) → min 0 x ≡ 0 min-zerol zero = refl min-zerol (suc x) = refl min-zeror : (x : Nat) → min x 0 ≡ 0 min-zeror zero = refl min-zeror (suc x) = refl
The factorial function🔗
factorial-positive : ∀ n → Positive (factorial n) factorial-positive zero = s≤s 0≤x factorial-positive (suc n) = *-preserves-≤ 1 (suc n) 1 (factorial n) (s≤s 0≤x) (factorial-positive n) ≤-factorial : ∀ n → n ≤ factorial n ≤-factorial zero = 0≤x ≤-factorial (suc n) = subst (_≤ factorial (suc n)) (*-oner (suc n)) (*-preserves-≤ (suc n) (suc n) 1 (factorial n) ≤-refl (factorial-positive n))