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Lemmas.agda
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Boring lemmas used in Data.Nat.GCD and Data.Nat.Coprimality
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Nat.GCD.Lemmas where
open import Data.Nat.Base
open import Data.Nat.Properties
open import Data.Nat.Solver
open import Function
open import Relation.Binary.PropositionalEquality
open +-*-Solver
open ≡-Reasoning
private
distrib-comm : ∀ x k n → x * k + x * n ≡ x * (n + k)
distrib-comm =
solve 3 (λ x k n → x :* k :+ x :* n := x :* (n :+ k)) refl
distrib-comm₂ : ∀ d x k n → d + x * (n + k) ≡ d + x * k + x * n
distrib-comm₂ =
solve 4 (λ d x k n → d :+ x :* (n :+ k) := d :+ x :* k :+ x :* n) refl
-- 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
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 ≡⟨ cong₂ _+_ eq refl ⟩
x * n + x * n ≡⟨ solve 3 (λ x n k → x :* n :+ x :* n
:= con 2 :* x :* n)
refl x n k ⟩
2 * x * n ∎
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 ≡⟨ cong₂ _+_ eq refl ⟩
x * n + y * n ≡⟨ solve 3 (λ x n i → x :* n :+ (con 1 :+ x :+ i) :* n
:= (con 1 :+ con 2 :* x :+ i) :* n)
refl x n 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 ≡⟨ cong₂ _+_ eq refl ⟩
(1 + y + i) * n + y * n ≡⟨ solve 3 (λ y i n → (con 1 :+ y :+ i) :* n :+ y :* n
:= (con 1 :+ con 2 :* y :+ i) :* n)
refl y i n ⟩
(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 ≡⟨ solve 3 (λ d x n → d :+ con 2 :* x :* n
:= d :+ x :* n :+ x :* n)
refl d x n ⟩
d + x * n + x * n ≡⟨ cong₂ _+_ eq refl ⟩
x * k + x * n ≡⟨ distrib-comm 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 ≡⟨ solve 4 (λ d x i n → d :+ (con 1 :+ con 2 :* x :+ i) :* n
:= d :+ x :* n :+ (con 1 :+ x :+ i) :* n)
refl d x i n ⟩
d + x * n + y * n ≡⟨ cong₂ _+_ eq refl ⟩
y * k + y * n ≡⟨ distrib-comm 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 ≡⟨ solve 4 (λ d y i n → d :+ (con 1 :+ con 2 :* y :+ i) :* n
:= d :+ (con 1 :+ y :+ i) :* n :+ y :* n)
refl d y i n ⟩
d + (1 + y + i) * n + y * n ≡⟨ cong₂ _+_ eq refl ⟩
y * k + y * n ≡⟨ distrib-comm 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 ≡⟨ solve 3 (λ x k i → x :* k :* i
:= x :* i :* k)
refl x k i ⟩
x * i * k ≡⟨ cong (_* k) (sym eq) ⟩
(1 + y * j) * k ≡⟨ solve 3 (λ y j k → (con 1 :+ y :* j) :* k
:= y :* (j :* k) :+ k)
refl y j k ⟩
y * (j * k) + k ≡⟨ cong (λ n → y * n + k) eq′ ⟩
y * (q * i) + k ≡⟨ solve 4 (λ y q i k → y :* (q :* i) :+ k
:= y :* q :* i :+ k)
refl y q i k ⟩
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 = solve 3 (λ a b c → a :* b :* c := b :* c :* a) refl
lemma = begin
y * q * i ≡⟨ lem y q i ⟩
q * i * y ≡⟨ cong (λ n → n * y) (sym eq′) ⟩
j * k * y ≡⟨ sym (lem y j k) ⟩
y * j * k ≡⟨ cong (λ n → n * k) (sym eq) ⟩
(1 + x * i) * k ≡⟨ solve 3 (λ x i k → (con 1 :+ x :* i) :* k
:= x :* k :* i :+ k)
refl x i k ⟩
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 ≡⟨ solve 4 (λ e f d a → e :* f :* d :* a
:= e :* (f :* d :* a))
refl e f d a ⟩
e * (f * d * a) ≡⟨ sym eq ⟩
a + b * (c * d * a) ≡⟨ solve 4 (λ a b c d → a :+ b :* (c :* d :* a)
:= (b :* c :* d :+ con 1) :* a)
refl a b c d ⟩
(b * c * d + 1) * a ∎)))
where a = suc a′
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 ≡⟨ *-assoc x m d ⟩
x * (m * d) ≡⟨ cong (x *_) (sym eq₁) ⟩
x * (i * k) ≡⟨ sym (*-assoc x i k) ⟩
x * i * k ≡⟨ cong₂ _*_ (sym eq) refl ⟩
(1 + y * j) * k ≡⟨ solve 3 (λ y j k → (con 1 :+ y :* j) :* k
:= y :* (j :* k) :+ k)
refl y j k ⟩
y * (j * k) + k ≡⟨ cong (λ p → y * p + k) eq₂ ⟩
y * (n * d) + k ≡⟨ cong₂ _+_ (sym $ *-assoc y n d) refl ⟩
y * n * d + k ∎))