------------------------------------------------------------------------
-- The Agda standard library
--
-- Boring lemmas used in Data.Nat.GCD and Data.Nat.Coprimality
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --safe #-}
module Data.Nat.GCD.Lemmas where
open import Data.Nat.Base
using (ℕ; suc; zero; _+_; _*_; _∸_; _≤_; s≤s; z≤n; _≤′_)
open import Data.Nat.Properties
open import Function.Base using (_$_)
open import Relation.Binary.PropositionalEquality.Core
using (_≡_; refl; cong; cong₂; sym)
open import Relation.Binary.PropositionalEquality.Properties
using (module ≡-Reasoning)
open ≡-Reasoning
private
comm-factor : ∀ x k n → x * k + x * n ≡ x * (n + k)
comm-factor x k n = begin
x * k + x * n ≡⟨ +-comm (x * k) _ ⟩
x * n + x * k ≡⟨ *-distribˡ-+ x n k ⟨
x * (n + k) ∎
distrib-comm₂ : ∀ d x k n → d + x * (n + k) ≡ d + x * k + x * n
distrib-comm₂ d x k n = begin
d + x * (n + k) ≡⟨ cong (d +_) (comm-factor x k n) ⟨
d + (x * k + x * n) ≡⟨ +-assoc d _ _ ⟨
d + x * k + x * n ∎
-- Other properties
-- TODO: Can this proof be simplified? An automatic solver which can
-- handle ∸ would be nice...
lem₀ : ∀ i j m n → i * m ≡ j * m + n → (i ∸ j) * m ≡ n
lem₀ i j m n eq = begin
(i ∸ j) * m ≡⟨ *-distribʳ-∸ m i j ⟩
(i * m) ∸ (j * m) ≡⟨ cong (_∸ j * m) eq ⟩
(j * m + n) ∸ (j * m) ≡⟨ cong (_∸ j * m) (+-comm (j * m) n) ⟩
(n + j * m) ∸ (j * m) ≡⟨ m+n∸n≡m n (j * m) ⟩
n ∎
lem₁ : ∀ i j → 2 + i ≤′ 2 + j + i
lem₁ i j = ≤⇒≤′ $ s≤s $ s≤s $ m≤n+m i j
private
times2 : ∀ x → x + x ≡ 2 * x
times2 x = cong (x +_) (sym (+-identityʳ x))
times2′ : ∀ x y → x * y + x * y ≡ 2 * x * y
times2′ x y = begin
x * y + x * y ≡⟨ times2 (x * y) ⟩
2 * (x * y) ≡⟨ *-assoc 2 x y ⟨
2 * x * y ∎
lem₂ : ∀ d x {k n} →
d + x * k ≡ x * n → d + x * (n + k) ≡ 2 * x * n
lem₂ d x {k} {n} eq = begin
d + x * (n + k) ≡⟨ distrib-comm₂ d x k n ⟩
d + x * k + x * n ≡⟨ cong (_+ x * n) eq ⟩
x * n + x * n ≡⟨ times2′ x n ⟩
2 * x * n ∎
private
distrib₃ : ∀ a b c x → (a + b + c) * x ≡ a * x + b * x + c * x
distrib₃ a b c x = begin
(a + b + c) * x ≡⟨ *-distribʳ-+ x (a + b) c ⟩
(a + b) * x + c * x ≡⟨ cong (_+ c * x) (*-distribʳ-+ x a b) ⟩
a * x + b * x + c * x ∎
lem₃₁ : ∀ a b c → a + (b + c) ≡ b + a + c
lem₃₁ a b c = begin
a + (b + c) ≡⟨ +-assoc a b c ⟨
(a + b) + c ≡⟨ cong (_+ c) (+-comm a b) ⟩
b + a + c ∎
+-assoc-comm : ∀ a b c d → a + (b + c + d) ≡ (a + c) + (b + d)
+-assoc-comm a b c d = begin
a + (b + c + d) ≡⟨ cong (a +_) (cong (_+ d) (+-comm b c)) ⟩
a + (c + b + d) ≡⟨ cong (a +_) (+-assoc c b d) ⟩
a + (c + (b + d)) ≡⟨ +-assoc a c _ ⟨
(a + c) + (b + d) ∎
*-on-right : ∀ a b c {d} → b * c ≡ d → a * b * c ≡ a * d
*-on-right a b c {d} eq = begin
a * b * c ≡⟨ *-assoc a b c ⟩
a * (b * c) ≡⟨ cong (a *_) eq ⟩
a * d ∎
*-on-left : ∀ a b c {d} → a * b ≡ d → a * (b * c) ≡ d * c
*-on-left a b c {d} eq = begin
a * (b * c) ≡⟨ *-assoc a b c ⟨
a * b * c ≡⟨ cong (_* c) eq ⟩
d * c ∎
+-on-right : ∀ a b c {d} → b + c ≡ d → a + b + c ≡ a + d
+-on-right a b c {d} eq = begin
a + b + c ≡⟨ +-assoc a b c ⟩
a + (b + c) ≡⟨ cong (a +_) eq ⟩
a + d ∎
+-on-left : ∀ a b c d → a + b ≡ d → a + (b + c) ≡ d + c
+-on-left a b c d eq = begin
a + (b + c) ≡⟨ +-assoc a b c ⟨
a + b + c ≡⟨ cong (_+ c) eq ⟩
d + c ∎
+-focus-mid : ∀ a b c d → a + b + c + d ≡ a + (b + c) + d
+-focus-mid a b c d = begin
a + b + c + d ≡⟨ cong (_+ d) (+-assoc a b c) ⟩
a + (b + c) + d ∎
+-assoc-comm′ : ∀ a b c d → a + b + c + d ≡ a + (b + d) + c
+-assoc-comm′ a b c d = begin
a + b + c + d ≡⟨ +-on-left a (b + c) d (a + b + c) (sym $ +-assoc a b c) ⟨
a + (b + c + d) ≡⟨ cong (a +_) (+-on-right b c d (+-comm c d)) ⟩
a + (b + (d + c)) ≡⟨ cong (a +_) (+-assoc b d c) ⟨
a + (b + d + c) ≡⟨ +-assoc a _ c ⟨
a + (b + d) + c ∎
lem₃₂ : ∀ a b c n → a * n + (b * n + a * n + c * n) ≡ (a + a + (b + c)) * n
lem₃₂ a b c n = begin
a * n + (b * n + a * n + c * n) ≡⟨ cong (a * n +_) (distrib₃ b a c n) ⟨
a * n + (b + a + c) * n ≡⟨ *-distribʳ-+ n a _ ⟨
(a + (b + a + c)) * n ≡⟨ cong (_* n) (+-assoc-comm a b a c) ⟩
(a + a + (b + c)) * n ∎
mid-to-right : ∀ a b c → a + 2 * b + c ≡ (a + b + c) + b
mid-to-right a b c = begin
a + 2 * b + c ≡⟨ cong (λ x → a + x + c) (times2 b) ⟨
a + (b + b) + c ≡⟨ +-assoc-comm′ a b c b ⟨
a + b + c + b ∎
mid-to-left : ∀ a b c → a + 2 * b + c ≡ b + (a + b + c)
mid-to-left a b c = begin
a + 2 * b + c ≡⟨ cong (λ x → a + x + c) (times2 b) ⟨
a + (b + b) + c ≡⟨ cong (_+ c) (+-on-left a b b _ (+-comm a b)) ⟩
b + a + b + c ≡⟨ +-on-left b (a + b) c (b + a + b) (sym $ +-assoc b a b) ⟨
b + (a + b + c) ∎
lem₃ : ∀ d x {i k n} →
d + (1 + x + i) * k ≡ x * n →
d + (1 + x + i) * (n + k) ≡ (1 + 2 * x + i) * n
lem₃ d x {i} {k} {n} eq = begin
d + y * (n + k) ≡⟨ distrib-comm₂ d y k n ⟩
d + y * k + y * n ≡⟨ cong (_+ y * n) eq ⟩
x * n + y * n ≡⟨ cong (x * n +_) (distrib₃ 1 x i n) ⟩
x * n + (1 * n + x * n + i * n) ≡⟨ lem₃₂ x 1 i n ⟩
(x + x + (1 + i)) * n ≡⟨ cong (_* n) (cong (_+ (1 + i)) (times2 x)) ⟩
(2 * x + (1 + i)) * n ≡⟨ cong (_* n) (lem₃₁ (2 * x) 1 i) ⟩
(1 + 2 * x + i) * n ∎
where y = 1 + x + i
lem₄ : ∀ d y {k i} n →
d + y * k ≡ (1 + y + i) * n →
d + y * (n + k) ≡ (1 + 2 * y + i) * n
lem₄ d y {k} {i} n eq = begin
d + y * (n + k) ≡⟨ distrib-comm₂ d y k n ⟩
d + y * k + y * n ≡⟨ cong (_+ y * n) eq ⟩
(1 + y + i) * n + y * n ≡⟨ *-distribʳ-+ n (1 + y + i) y ⟨
(1 + y + i + y) * n ≡⟨ cong (_* n) (mid-to-right 1 y i) ⟨
(1 + 2 * y + i) * n ∎
lem₅ : ∀ d x {n k} →
d + x * n ≡ x * k →
d + 2 * x * n ≡ x * (n + k)
lem₅ d x {n} {k} eq = begin
d + 2 * x * n ≡⟨ cong (d +_) (times2′ x n) ⟨
d + (x * n + x * n) ≡⟨ +-assoc d (x * n) _ ⟨
d + x * n + x * n ≡⟨ cong (_+ x * n) eq ⟩
x * k + x * n ≡⟨ comm-factor x k n ⟩
x * (n + k) ∎
lem₆ : ∀ d x {n i k} →
d + x * n ≡ (1 + x + i) * k →
d + (1 + 2 * x + i) * n ≡ (1 + x + i) * (n + k)
lem₆ d x {n} {i} {k} eq = begin
d + (1 + 2 * x + i) * n ≡⟨ cong (λ z → d + z * n) (mid-to-left 1 x i) ⟩
d + (x + y) * n ≡⟨ cong (d +_) (*-distribʳ-+ n x y) ⟩
d + (x * n + y * n) ≡⟨ +-on-left d _ _ _ eq ⟩
y * k + y * n ≡⟨ comm-factor y k n ⟩
y * (n + k) ∎
where y = 1 + x + i
lem₇ : ∀ d y {i} n {k} →
d + (1 + y + i) * n ≡ y * k →
d + (1 + 2 * y + i) * n ≡ y * (n + k)
lem₇ d y {i} n {k} eq = begin
d + (1 + 2 * y + i) * n ≡⟨ cong (λ z → d + z * n) (mid-to-right 1 y i) ⟩
d + (1 + y + i + y) * n ≡⟨ cong (d +_) (*-distribʳ-+ n (1 + y + i) y) ⟩
d + ((1 + y + i) * n + y * n) ≡⟨ +-on-left d _ _ _ eq ⟩
y * k + y * n ≡⟨ comm-factor y k n ⟩
y * (n + k) ∎
lem₈ : ∀ {i j k q} x y →
1 + y * j ≡ x * i → j * k ≡ q * i →
k ≡ (x * k ∸ y * q) * i
lem₈ {i} {j} {k} {q} x y eq eq′ =
sym (lem₀ (x * k) (y * q) i k lemma)
where
lemma = begin
x * k * i ≡⟨ *-on-right x k i (*-comm k i) ⟩
x * (i * k) ≡⟨ *-on-left x i k (sym eq) ⟩
(1 + y * j) * k ≡⟨ +-comm k _ ⟩
(y * j) * k + k ≡⟨ cong (_+ k) (*-assoc y j k) ⟩
y * (j * k) + k ≡⟨ cong (λ n → y * n + k) eq′ ⟩
y * (q * i) + k ≡⟨ cong (_+ k) (*-assoc y q i) ⟨
y * q * i + k ∎
lem₉ : ∀ {i j k q} x y →
1 + x * i ≡ y * j → j * k ≡ q * i →
k ≡ (y * q ∸ x * k) * i
lem₉ {i} {j} {k} {q} x y eq eq′ =
sym (lem₀ (y * q) (x * k) i k lemma)
where
lem : ∀ a b c → a * b * c ≡ b * c * a
lem a b c = begin
a * b * c ≡⟨ *-assoc a b c ⟩
a * (b * c) ≡⟨ *-comm a _ ⟩
b * c * a ∎
lemma = begin
y * q * i ≡⟨ lem y q i ⟩
q * i * y ≡⟨ cong (λ n → n * y) eq′ ⟨
j * k * y ≡⟨ lem y j k ⟨
y * j * k ≡⟨ cong (λ n → n * k) eq ⟨
(1 + x * i) * k ≡⟨ +-comm k _ ⟩
x * i * k + k ≡⟨ cong (_+ k) (*-on-right x i k (*-comm i k)) ⟩
x * (k * i) + k ≡⟨ cong (_+ k) (*-assoc x k i) ⟨
x * k * i + k ∎
lem₁₀ : ∀ {a′} b c {d} e f → let a = suc a′ in
a + b * (c * d * a) ≡ e * (f * d * a) →
d ≡ 1
lem₁₀ {a′} b c {d} e f eq =
m*n≡1⇒n≡1 (e * f ∸ b * c) d
(lem₀ (e * f) (b * c) d 1
(*-cancelʳ-≡ (e * f * d) (b * c * d + 1) _ (begin
e * f * d * a ≡⟨ *-assoc₄₃ e f d a ⟩
e * (f * d * a) ≡⟨ eq ⟨
a + b * (c * d * a) ≡⟨ cong (a +_) (*-assoc₄₃ b c d a) ⟨
a + b * c * d * a ≡⟨ +-comm a _ ⟩
b * c * d * a + a ≡⟨ cong (b * c * d * a +_) (+-identityʳ a) ⟨
b * c * d * a + (a + 0) ≡⟨ *-distribʳ-+ a (b * c * d) 1 ⟨
(b * c * d + 1) * a ∎)))
where a = suc a′
*-assoc₄₃ : ∀ w x y z → w * x * y * z ≡ w * (x * y * z)
*-assoc₄₃ w x y z = begin
w * x * y * z ≡⟨ cong (_* z) (*-assoc w x y) ⟩
w * (x * y) * z ≡⟨ *-assoc w _ z ⟩
w * (x * y * z) ∎
lem₁₁ : ∀ {i j m n k d} x y →
1 + y * j ≡ x * i → i * k ≡ m * d → j * k ≡ n * d →
k ≡ (x * m ∸ y * n) * d
lem₁₁ {i} {j} {m} {n} {k} {d} x y eq eq₁ eq₂ =
sym (lem₀ (x * m) (y * n) d k (begin
x * m * d ≡⟨ *-on-right x m d (sym eq₁) ⟩
x * (i * k) ≡⟨ *-on-left x i k (sym eq) ⟩
(1 + y * j) * k ≡⟨ +-comm k _ ⟩
y * j * k + k ≡⟨ cong (_+ k) (*-assoc y j k) ⟩
y * (j * k) + k ≡⟨ cong (λ p → y * p + k) eq₂ ⟩
y * (n * d) + k ≡⟨ cong (_+ k) (*-assoc y n d) ⟨
y * n * d + k ∎))
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