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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 14 Jan 2022 14:39:36 +0900 |
| parents | 8b437dd616dd |
| children | b065ba2f68c5 |
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{-# OPTIONS --allow-unsolved-metas #-} module gcd where open import Data.Nat open import Data.Nat.Properties open import Data.Empty open import Data.Unit using (⊤ ; tt) open import Relation.Nullary open import Relation.Binary.PropositionalEquality open import Relation.Binary.Definitions open import nat open import logic open Factor gcd1 : ( i i0 j j0 : ℕ ) → ℕ gcd1 zero i0 zero j0 with <-cmp i0 j0 ... | tri< a ¬b ¬c = i0 ... | tri≈ ¬a refl ¬c = i0 ... | tri> ¬a ¬b c = j0 gcd1 zero i0 (suc zero) j0 = 1 gcd1 zero zero (suc (suc j)) j0 = j0 gcd1 zero (suc i0) (suc (suc j)) j0 = gcd1 i0 (suc i0) (suc j) (suc (suc j)) gcd1 (suc zero) i0 zero j0 = 1 gcd1 (suc (suc i)) i0 zero zero = i0 gcd1 (suc (suc i)) i0 zero (suc j0) = gcd1 (suc i) (suc (suc i)) j0 (suc j0) gcd1 (suc i) i0 (suc j) j0 = gcd1 i i0 j j0 gcd : ( i j : ℕ ) → ℕ gcd i j = gcd1 i i j j gcd20 : (i : ℕ) → gcd i 0 ≡ i gcd20 zero = refl gcd20 (suc i) = gcd201 (suc i) where gcd201 : (i : ℕ ) → gcd1 i i zero zero ≡ i gcd201 zero = refl gcd201 (suc zero) = refl gcd201 (suc (suc i)) = refl gcd22 : ( i i0 o o0 : ℕ ) → gcd1 (suc i) i0 (suc o) o0 ≡ gcd1 i i0 o o0 gcd22 zero i0 zero o0 = refl gcd22 zero i0 (suc o) o0 = refl gcd22 (suc i) i0 zero o0 = refl gcd22 (suc i) i0 (suc o) o0 = refl gcdmm : (n m : ℕ) → gcd1 n m n m ≡ m gcdmm zero m with <-cmp m m ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a refl ¬c = refl ... | tri> ¬a ¬b c = refl gcdmm (suc n) m = subst (λ k → k ≡ m) (sym (gcd22 n m n m )) (gcdmm n m ) gcdsym2 : (i j : ℕ) → gcd1 zero i zero j ≡ gcd1 zero j zero i gcdsym2 i j with <-cmp i j | <-cmp j i ... | tri< a ¬b ¬c | tri< a₁ ¬b₁ ¬c₁ = ⊥-elim (nat-<> a a₁) ... | tri< a ¬b ¬c | tri≈ ¬a b ¬c₁ = ⊥-elim (nat-≡< (sym b) a) ... | tri< a ¬b ¬c | tri> ¬a ¬b₁ c = refl ... | tri≈ ¬a b ¬c | tri< a ¬b ¬c₁ = ⊥-elim (nat-≡< (sym b) a) ... | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = refl ... | tri≈ ¬a b ¬c | tri> ¬a₁ ¬b c = ⊥-elim (nat-≡< b c) ... | tri> ¬a ¬b c | tri< a ¬b₁ ¬c = refl ... | tri> ¬a ¬b c | tri≈ ¬a₁ b ¬c = ⊥-elim (nat-≡< b c) ... | tri> ¬a ¬b c | tri> ¬a₁ ¬b₁ c₁ = ⊥-elim (nat-<> c c₁) gcdsym1 : ( i i0 j j0 : ℕ ) → gcd1 i i0 j j0 ≡ gcd1 j j0 i i0 gcdsym1 zero zero zero zero = refl gcdsym1 zero zero zero (suc j0) = refl gcdsym1 zero (suc i0) zero zero = refl gcdsym1 zero (suc i0) zero (suc j0) = gcdsym2 (suc i0) (suc j0) gcdsym1 zero zero (suc zero) j0 = refl gcdsym1 zero zero (suc (suc j)) j0 = refl gcdsym1 zero (suc i0) (suc zero) j0 = refl gcdsym1 zero (suc i0) (suc (suc j)) j0 = gcdsym1 i0 (suc i0) (suc j) (suc (suc j)) gcdsym1 (suc zero) i0 zero j0 = refl gcdsym1 (suc (suc i)) i0 zero zero = refl gcdsym1 (suc (suc i)) i0 zero (suc j0) = gcdsym1 (suc i) (suc (suc i))j0 (suc j0) gcdsym1 (suc i) i0 (suc j) j0 = subst₂ (λ j k → j ≡ k ) (sym (gcd22 i _ _ _)) (sym (gcd22 j _ _ _)) (gcdsym1 i i0 j j0 ) gcdsym : { n m : ℕ} → gcd n m ≡ gcd m n gcdsym {n} {m} = gcdsym1 n n m m gcd11 : ( i : ℕ ) → gcd i i ≡ i gcd11 i = gcdmm i i gcd203 : (i : ℕ) → gcd1 (suc i) (suc i) i i ≡ 1 gcd203 zero = refl gcd203 (suc i) = gcd205 (suc i) where gcd205 : (j : ℕ) → gcd1 (suc j) (suc (suc i)) j (suc i) ≡ 1 gcd205 zero = refl gcd205 (suc j) = subst (λ k → k ≡ 1) (gcd22 (suc j) (suc (suc i)) j (suc i)) (gcd205 j) gcd204 : (i : ℕ) → gcd1 1 1 i i ≡ 1 gcd204 zero = refl gcd204 (suc zero) = refl gcd204 (suc (suc zero)) = refl gcd204 (suc (suc (suc i))) = gcd204 (suc (suc i)) gcd+j : ( i j : ℕ ) → gcd (i + j) j ≡ gcd i j gcd+j i j = gcd200 i i j j refl refl where gcd202 : (i j1 : ℕ) → (i + suc j1) ≡ suc (i + j1) gcd202 zero j1 = refl gcd202 (suc i) j1 = cong suc (gcd202 i j1) gcd201 : (i i0 j j0 j1 : ℕ) → gcd1 (i + j1) (i0 + suc j) j1 j0 ≡ gcd1 i (i0 + suc j) zero j0 gcd201 i i0 j j0 zero = subst (λ k → gcd1 k (i0 + suc j) zero j0 ≡ gcd1 i (i0 + suc j) zero j0 ) (+-comm zero i) refl gcd201 i i0 j j0 (suc j1) = begin gcd1 (i + suc j1) (i0 + suc j) (suc j1) j0 ≡⟨ cong (λ k → gcd1 k (i0 + suc j) (suc j1) j0 ) (gcd202 i j1) ⟩ gcd1 (suc (i + j1)) (i0 + suc j) (suc j1) j0 ≡⟨ gcd22 (i + j1) (i0 + suc j) j1 j0 ⟩ gcd1 (i + j1) (i0 + suc j) j1 j0 ≡⟨ gcd201 i i0 j j0 j1 ⟩ gcd1 i (i0 + suc j) zero j0 ∎ where open ≡-Reasoning gcd200 : (i i0 j j0 : ℕ) → i ≡ i0 → j ≡ j0 → gcd1 (i + j) (i0 + j) j j0 ≡ gcd1 i i j0 j0 gcd200 i .i zero .0 refl refl = subst (λ k → gcd1 k k zero zero ≡ gcd1 i i zero zero ) (+-comm zero i) refl gcd200 (suc (suc i)) i0 (suc j) (suc j0) i=i0 j=j0 = gcd201 (suc (suc i)) i0 j (suc j0) (suc j) gcd200 zero zero (suc zero) .1 i=i0 refl = refl gcd200 zero zero (suc (suc j)) .(suc (suc j)) i=i0 refl = begin gcd1 (zero + suc (suc j)) (zero + suc (suc j)) (suc (suc j)) (suc (suc j)) ≡⟨ gcdmm (suc (suc j)) (suc (suc j)) ⟩ suc (suc j) ≡⟨ sym (gcd20 (suc (suc j))) ⟩ gcd1 zero zero (suc (suc j)) (suc (suc j)) ∎ where open ≡-Reasoning gcd200 zero (suc i0) (suc j) .(suc j) () refl gcd200 (suc zero) .1 (suc j) .(suc j) refl refl = begin gcd1 (1 + suc j) (1 + suc j) (suc j) (suc j) ≡⟨ gcd203 (suc j) ⟩ 1 ≡⟨ sym ( gcd204 (suc j)) ⟩ gcd1 1 1 (suc j) (suc j) ∎ where open ≡-Reasoning gcd200 (suc (suc i)) i0 (suc j) zero i=i0 () open _∧_ gcd-gt : ( i i0 j j0 k : ℕ ) → k > 1 → (if : Factor k i) (i0f : Dividable k i0 ) (jf : Factor k j ) (j0f : Dividable k j0) → Dividable k (i - j) ∧ Dividable k (j - i) → Dividable k ( gcd1 i i0 j j0 ) gcd-gt zero i0 zero j0 k k>1 if i0f jf j0f i-j with <-cmp i0 j0 ... | tri< a ¬b ¬c = i0f ... | tri≈ ¬a refl ¬c = i0f ... | tri> ¬a ¬b c = j0f gcd-gt zero i0 (suc zero) j0 k k>1 if i0f jf j0f i-j = ⊥-elim (div1 k>1 (proj2 i-j)) -- can't happen gcd-gt zero zero (suc (suc j)) j0 k k>1 if i0f jf j0f i-j = j0f gcd-gt zero (suc i0) (suc (suc j)) j0 k k>1 if i0f jf j0f i-j = gcd-gt i0 (suc i0) (suc j) (suc (suc j)) k k>1 (decf (DtoF i0f)) i0f (decf jf) (proj2 i-j) (div-div k>1 i0f (proj2 i-j)) gcd-gt (suc zero) i0 zero j0 k k>1 if i0f jf j0f i-j = ⊥-elim (div1 k>1 (proj1 i-j)) -- can't happen gcd-gt (suc (suc i)) i0 zero zero k k>1 if i0f jf j0f i-j = i0f gcd-gt (suc (suc i)) i0 zero (suc j0) k k>1 if i0f jf j0f i-j = -- gcd-gt (suc i) (suc (suc i)) j0 (suc j0) k k>1 (decf if) (proj1 i-j) (decf (DtoF j0f)) j0f (div-div k>1 (proj1 i-j) j0f ) gcd-gt (suc zero) i0 (suc j) j0 k k>1 if i0f jf j0f i-j = gcd-gt zero i0 j j0 k k>1 (decf if) i0f (decf jf) j0f i-j gcd-gt (suc (suc i)) i0 (suc j) j0 k k>1 if i0f jf j0f i-j = gcd-gt (suc i) i0 j j0 k k>1 (decf if) i0f (decf jf) j0f i-j gcd-div : ( i j k : ℕ ) → k > 1 → (if : Dividable k i) (jf : Dividable k j ) → Dividable k ( gcd i j ) gcd-div i j k k>1 if jf = gcd-gt i i j j k k>1 (DtoF if) if (DtoF jf) jf (div-div k>1 if jf) di-next : {i i0 j j0 : ℕ} → Dividable i0 ((j0 + suc i) - suc j ) ∧ Dividable j0 ((i0 + suc j) - suc i) → Dividable i0 ((j0 + i) - j ) ∧ Dividable j0 ((i0 + j) - i) di-next {i} {i0} {j} {j0} x = ⟪ ( subst (λ k → Dividable i0 (k - suc j)) ( begin j0 + suc i ≡⟨ sym (+-assoc j0 1 i ) ⟩ (j0 + 1) + i ≡⟨ cong (λ k → k + i) (+-comm j0 _ ) ⟩ suc (j0 + i) ∎ ) (proj1 x) ) , ( subst (λ k → Dividable j0 (k - suc i)) ( begin i0 + suc j ≡⟨ sym (+-assoc i0 1 j ) ⟩ (i0 + 1) + j ≡⟨ cong (λ k → k + j) (+-comm i0 _ ) ⟩ suc (i0 + j) ∎ ) (proj2 x) ) ⟫ where open ≡-Reasoning di-next1 : {i0 j j0 : ℕ} → Dividable (suc i0) ((j0 + 0) - (suc (suc j))) ∧ Dividable j0 (suc (i0 + suc (suc j))) → Dividable (suc i0) ((suc (suc j) + i0) - suc j) ∧ Dividable (suc (suc j)) ((suc i0 + suc j) - i0) di-next1 {i0} {j} {j0} x = ⟪ record { factor = 1 ; is-factor = begin 1 * suc i0 + 0 ≡⟨ cong suc ( trans (+-comm _ 0) (+-comm _ 0) ) ⟩ suc i0 ≡⟨ sym (minus+y-y {suc i0} {j}) ⟩ (suc i0 + j) - j ≡⟨ cong (λ k → k - j ) (+-comm (suc i0) _ ) ⟩ (suc j + suc i0 ) - suc j ≡⟨ cong (λ k → k - suc j) (sym (+-assoc (suc j) 1 i0 )) ⟩ ((suc j + 1) + i0) - suc j ≡⟨ cong (λ k → (k + i0) - suc j) (+-comm _ 1) ⟩ (suc (suc j) + i0) - suc j ∎ } , subst (λ k → Dividable (suc (suc j)) k) ( begin suc (suc j) ≡⟨ sym ( minus+y-y {suc (suc j)}{i0} ) ⟩ (suc (suc j) + i0 ) - i0 ≡⟨ cong (λ k → (k + i0) - i0) (cong suc (+-comm 1 _ )) ⟩ ((suc j + 1) + i0 ) - i0 ≡⟨ cong (λ k → k - i0) (+-assoc (suc j) 1 _ ) ⟩ (suc j + suc i0 ) - i0 ≡⟨ cong (λ k → k - i0) (+-comm (suc j) _) ⟩ ((suc i0 + suc j) - i0) ∎ ) div= ⟫ where open ≡-Reasoning gcd>0 : ( i j : ℕ ) → 0 < i → 0 < j → 0 < gcd i j gcd>0 i j 0<i 0<j = gcd>01 i i j j 0<i 0<j where gcd>01 : ( i i0 j j0 : ℕ ) → 0 < i0 → 0 < j0 → gcd1 i i0 j j0 > 0 gcd>01 zero i0 zero j0 0<i 0<j with <-cmp i0 j0 ... | tri< a ¬b ¬c = 0<i ... | tri≈ ¬a refl ¬c = 0<i ... | tri> ¬a ¬b c = 0<j gcd>01 zero i0 (suc zero) j0 0<i 0<j = s≤s z≤n gcd>01 zero zero (suc (suc j)) j0 0<i 0<j = 0<j gcd>01 zero (suc i0) (suc (suc j)) j0 0<i 0<j = gcd>01 i0 (suc i0) (suc j) (suc (suc j)) 0<i (s≤s z≤n) -- 0 < suc (suc j) gcd>01 (suc zero) i0 zero j0 0<i 0<j = s≤s z≤n gcd>01 (suc (suc i)) i0 zero zero 0<i 0<j = 0<i gcd>01 (suc (suc i)) i0 zero (suc j0) 0<i 0<j = gcd>01 (suc i) (suc (suc i)) j0 (suc j0) (s≤s z≤n) 0<j gcd>01 (suc i) i0 (suc j) j0 0<i 0<j = subst (λ k → 0 < k ) (sym (gcd033 i i0 j j0 )) (gcd>01 i i0 j j0 0<i 0<j ) where gcd033 : (i i0 j j0 : ℕ) → gcd1 (suc i) i0 (suc j) j0 ≡ gcd1 i i0 j j0 gcd033 zero zero zero zero = refl gcd033 zero zero (suc j) zero = refl gcd033 (suc i) zero j zero = refl gcd033 zero zero zero (suc j0) = refl gcd033 (suc i) zero zero (suc j0) = refl gcd033 zero zero (suc j) (suc j0) = refl gcd033 (suc i) zero (suc j) (suc j0) = refl gcd033 zero (suc i0) j j0 = refl gcd033 (suc i) (suc i0) j j0 = refl -- gcd loop invariant -- record GCD ( i i0 j j0 : ℕ ) : Set where field i<i0 : i ≤ i0 j<j0 : j ≤ j0 div-i : Dividable i0 ((j0 + i) - j) div-j : Dividable j0 ((i0 + j) - i) div-11 : {i j : ℕ } → Dividable i ((j + i) - j) div-11 {i} {j} = record { factor = 1 ; is-factor = begin 1 * i + 0 ≡⟨ +-comm _ 0 ⟩ i + 0 ≡⟨ +-comm _ 0 ⟩ i ≡⟨ sym (minus+y-y {i} {j}) ⟩ (i + j ) - j ≡⟨ cong (λ k → k - j ) (+-comm i j ) ⟩ (j + i) - j ∎ } where open ≡-Reasoning div→gcd : {n k : ℕ } → k > 1 → Dividable k n → gcd n k ≡ k div→gcd {n} {k} k>1 = n-induction {_} {_} {ℕ} {λ m → Dividable k m → gcd m k ≡ k } (λ x → x) I n where decl : {m : ℕ } → 0 < m → m - k < m decl {m} 0<m = y-x<y (<-trans a<sa k>1 ) 0<m ind : (m : ℕ ) → (Dividable k (m - k) → gcd (m - k) k ≡ k) → Dividable k m → gcd m k ≡ k ind m prev d with <-cmp (suc m) k ... | tri≈ ¬a refl ¬c = ⊥-elim ( div+1 k>1 d div= ) ... | tri> ¬a ¬b c = subst (λ g → g ≡ k) ind1 ( prev (proj2 (div-div k>1 div= d))) where ind1 : gcd (m - k) k ≡ gcd m k ind1 = begin gcd (m - k) k ≡⟨ sym (gcd+j (m - k) _) ⟩ gcd (m - k + k) k ≡⟨ cong (λ g → gcd g k) (minus+n {m} {k} c) ⟩ gcd m k ∎ where open ≡-Reasoning ... | tri< a ¬b ¬c with <-cmp 0 m ... | tri< a₁ ¬b₁ ¬c₁ = ⊥-elim ( div<k k>1 a₁ (<-trans a<sa a) d ) ... | tri≈ ¬a refl ¬c₁ = subst (λ g → g ≡ k ) (gcdsym {k} {0} ) (gcd20 k) fzero : (m : ℕ) → (m - k) ≡ zero → Dividable k m → gcd m k ≡ k fzero 0 eq d = trans (gcdsym {0} {k} ) (gcd20 k) fzero (suc m) eq d with <-cmp (suc m) k ... | tri< a ¬b ¬c = ⊥-elim ( div<k k>1 (s≤s z≤n) a d ) ... | tri≈ ¬a refl ¬c = gcdmm k k ... | tri> ¬a ¬b c = ⊥-elim ( nat-≡< (sym eq) (minus>0 c) ) I : Ninduction ℕ _ (λ x → x) I = record { pnext = λ p → p - k ; fzero = λ {m} eq → fzero m eq ; decline = λ {m} lt → decl lt ; ind = λ {p} prev → ind p prev } GCDi : {i j : ℕ } → GCD i i j j GCDi {i} {j} = record { i<i0 = refl-≤ ; j<j0 = refl-≤ ; div-i = div-11 {i} {j} ; div-j = div-11 {j} {i} } GCD-sym : {i i0 j j0 : ℕ} → GCD i i0 j j0 → GCD j j0 i i0 GCD-sym g = record { i<i0 = GCD.j<j0 g ; j<j0 = GCD.i<i0 g ; div-i = GCD.div-j g ; div-j = GCD.div-i g } pred-≤ : {i i0 : ℕ } → suc i ≤ suc i0 → i ≤ suc i0 pred-≤ {i} {i0} (s≤s lt) = ≤-trans lt refl-≤s gcd-next : {i i0 j j0 : ℕ} → GCD (suc i) i0 (suc j) j0 → GCD i i0 j j0 gcd-next {i} {0} {j} {0} () gcd-next {i} {suc i0} {j} {suc j0} g = record { i<i0 = pred-≤ (GCD.i<i0 g) ; j<j0 = pred-≤ (GCD.j<j0 g) ; div-i = proj1 (di-next {i} {suc i0} {j} {suc j0} ⟪ GCD.div-i g , GCD.div-j g ⟫ ) ; div-j = proj2 (di-next {i} {suc i0} {j} {suc j0} ⟪ GCD.div-i g , GCD.div-j g ⟫ ) } gcd-next1 : {i0 j j0 : ℕ} → GCD 0 (suc i0) (suc (suc j)) j0 → GCD i0 (suc i0) (suc j) (suc (suc j)) gcd-next1 {i0} {j} {j0} g = record { i<i0 = refl-≤s ; j<j0 = refl-≤s ; div-i = proj1 (di-next1 ⟪ GCD.div-i g , GCD.div-j g ⟫ ) ; div-j = proj2 (di-next1 ⟪ GCD.div-i g , GCD.div-j g ⟫ ) } -- gcd-dividable1 : ( i i0 j j0 : ℕ ) -- → Dividable i0 (j0 + i - j ) ∨ Dividable j0 (i0 + j - i) -- → Dividable ( gcd1 i i0 j j0 ) i0 ∧ Dividable ( gcd1 i i0 j j0 ) j0 -- gcd-dividable1 zero i0 zero j0 with <-cmp i0 j0 -- ... | tri< a ¬b ¬c = ⟪ div= , {!!} ⟫ -- Dividable i0 (j0 + i - j ) ∧ Dividable j0 (i0 + j - i) -- ... | tri≈ ¬a refl ¬c = {!!} -- ... | tri> ¬a ¬b c = {!!} -- gcd-dividable1 zero i0 (suc zero) j0 = {!!} -- gcd-dividable1 i i0 j j0 = {!!} gcd-dividable : ( i j : ℕ ) → Dividable ( gcd i j ) i ∧ Dividable ( gcd i j ) j gcd-dividable i j = f-induction {_} {_} {ℕ ∧ ℕ} {λ p → Dividable ( gcd (proj1 p) (proj2 p) ) (proj1 p) ∧ Dividable ( gcd (proj1 p) (proj2 p) ) (proj2 p)} F I ⟪ i , j ⟫ where F : ℕ ∧ ℕ → ℕ F ⟪ 0 , 0 ⟫ = 0 F ⟪ 0 , suc j ⟫ = 0 F ⟪ suc i , 0 ⟫ = 0 F ⟪ suc i , suc j ⟫ with <-cmp i j ... | tri< a ¬b ¬c = suc j ... | tri≈ ¬a b ¬c = 0 ... | tri> ¬a ¬b c = suc i F0 : { i j : ℕ } → F ⟪ i , j ⟫ ≡ 0 → (i ≡ j) ∨ (i ≡ 0 ) ∨ (j ≡ 0) F0 {zero} {zero} p = case1 refl F0 {zero} {suc j} p = case2 (case1 refl) F0 {suc i} {zero} p = case2 (case2 refl) F0 {suc i} {suc j} p with <-cmp i j ... | tri< a ¬b ¬c = ⊥-elim ( nat-≡< (sym p) (s≤s z≤n )) ... | tri≈ ¬a refl ¬c = case1 refl ... | tri> ¬a ¬b c = ⊥-elim ( nat-≡< (sym p) (s≤s z≤n )) F00 : {p : ℕ ∧ ℕ} → F p ≡ zero → Dividable (gcd (proj1 p) (proj2 p)) (proj1 p) ∧ Dividable (gcd (proj1 p) (proj2 p)) (proj2 p) F00 {⟪ i , j ⟫} eq with F0 {i} {j} eq ... | case1 refl = ⟪ subst (λ k → Dividable k i) (sym (gcdmm i i)) div= , subst (λ k → Dividable k i) (sym (gcdmm i i)) div= ⟫ ... | case2 (case1 refl) = ⟪ subst (λ k → Dividable k i) (sym (trans (gcdsym {0} {j} ) (gcd20 j)))div0 , subst (λ k → Dividable k j) (sym (trans (gcdsym {0} {j}) (gcd20 j))) div= ⟫ ... | case2 (case2 refl) = ⟪ subst (λ k → Dividable k i) (sym (gcd20 i)) div= , subst (λ k → Dividable k j) (sym (gcd20 i)) div0 ⟫ Fsym : {i j : ℕ } → F ⟪ i , j ⟫ ≡ F ⟪ j , i ⟫ Fsym {0} {0} = refl Fsym {0} {suc j} = refl Fsym {suc i} {0} = refl Fsym {suc i} {suc j} with <-cmp i j | <-cmp j i ... | tri< a ¬b ¬c | tri< a₁ ¬b₁ ¬c₁ = ⊥-elim (nat-<> a a₁) ... | tri< a ¬b ¬c | tri≈ ¬a b ¬c₁ = ⊥-elim (¬b (sym b)) ... | tri< a ¬b ¬c | tri> ¬a ¬b₁ c = refl ... | tri≈ ¬a refl ¬c | tri< a ¬b ¬c₁ = ⊥-elim (¬b refl) ... | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = refl ... | tri≈ ¬a refl ¬c | tri> ¬a₁ ¬b c = ⊥-elim (¬b refl) ... | tri> ¬a ¬b c | tri< a ¬b₁ ¬c = refl ... | tri> ¬a ¬b c | tri≈ ¬a₁ b ¬c = ⊥-elim (¬b (sym b)) ... | tri> ¬a ¬b c | tri> ¬a₁ ¬b₁ c₁ = ⊥-elim (nat-<> c c₁) record Fdec ( i j : ℕ ) : Set where field ni : ℕ nj : ℕ fdec : 0 < F ⟪ i , j ⟫ → F ⟪ ni , nj ⟫ < F ⟪ i , j ⟫ fd1 : ( i j k : ℕ ) → i < j → k ≡ j - i → F ⟪ suc i , k ⟫ < F ⟪ suc i , suc j ⟫ fd1 i j 0 i<j eq = ⊥-elim ( nat-≡< eq (minus>0 {i} {j} i<j )) fd1 i j (suc k) i<j eq = fd2 i j k i<j eq where fd2 : ( i j k : ℕ ) → i < j → suc k ≡ j - i → F ⟪ suc i , suc k ⟫ < F ⟪ suc i , suc j ⟫ fd2 i j k i<j eq with <-cmp i k | <-cmp i j ... | tri< a ¬b ¬c | tri< a₁ ¬b₁ ¬c₁ = fd3 where fd3 : suc k < suc j -- suc j - suc i < suc j fd3 = subst (λ g → g < suc j) (sym eq) (y-x<y {suc i} {suc j} (s≤s z≤n) (s≤s z≤n)) ... | tri< a ¬b ¬c | tri≈ ¬a b ¬c₁ = ⊥-elim (⊥-elim (¬a i<j)) ... | tri< a ¬b ¬c | tri> ¬a ¬b₁ c = ⊥-elim (⊥-elim (¬a i<j)) ... | tri≈ ¬a b ¬c | tri< a ¬b ¬c₁ = s≤s z≤n ... | tri≈ ¬a b ¬c | tri≈ ¬a₁ b₁ ¬c₁ = ⊥-elim (¬a₁ i<j) ... | tri≈ ¬a b ¬c | tri> ¬a₁ ¬b c = s≤s z≤n -- i > j ... | tri> ¬a ¬b c | tri< a ¬b₁ ¬c = fd4 where fd4 : suc i < suc j fd4 = s≤s a ... | tri> ¬a ¬b c | tri≈ ¬a₁ b ¬c = ⊥-elim (¬a₁ i<j) ... | tri> ¬a ¬b c | tri> ¬a₁ ¬b₁ c₁ = ⊥-elim (¬a₁ i<j) fedc0 : (i j : ℕ ) → Fdec i j fedc0 0 0 = record { ni = 0 ; nj = 0 ; fdec = λ () } fedc0 (suc i) 0 = record { ni = suc i ; nj = 0 ; fdec = λ () } fedc0 0 (suc j) = record { ni = 0 ; nj = suc j ; fdec = λ () } fedc0 (suc i) (suc j) with <-cmp i j ... | tri< i<j ¬b ¬c = record { ni = suc i ; nj = j - i ; fdec = λ lt → fd1 i j (j - i) i<j refl } ... | tri≈ ¬a refl ¬c = record { ni = suc i ; nj = suc j ; fdec = λ lt → ⊥-elim (nat-≡< fd0 lt) } where fd0 : {i : ℕ } → 0 ≡ F ⟪ suc i , suc i ⟫ fd0 {i} with <-cmp i i ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = ⊥-elim ( ¬b refl ) ... | tri> ¬a ¬b c = record { ni = i - j ; nj = suc j ; fdec = λ lt → subst₂ (λ s t → s < t) (Fsym {suc j} {i - j}) (Fsym {suc j} {suc i}) (fd1 j i (i - j) c refl ) } ind3 : {i j : ℕ } → i < j → Dividable (gcd (suc i) (j - i)) (suc i) → Dividable (gcd (suc i) (suc j)) (suc i) ind3 {i} {j} a prev = subst (λ k → Dividable k (suc i)) ( begin gcd (suc i) (j - i) ≡⟨ gcdsym {suc i} {j - i} ⟩ gcd (j - i ) (suc i) ≡⟨ sym (gcd+j (j - i) (suc i)) ⟩ gcd ((j - i) + suc i) (suc i) ≡⟨ cong (λ k → gcd k (suc i)) ( begin (suc j - suc i) + suc i ≡⟨ minus+n {suc j} {suc i} (<-trans ( s≤s a) a<sa ) ⟩ -- i ≤ n → suc (suc i) ≤ suc (suc (suc n)) suc j ∎ ) ⟩ gcd (suc j) (suc i) ≡⟨ gcdsym {suc j} {suc i} ⟩ gcd (suc i) (suc j) ∎ ) prev where open ≡-Reasoning ind7 : {i j : ℕ } → (i < j ) → (j - i) + suc i ≡ suc j ind7 {i} {j} a = begin (suc j - suc i) + suc i ≡⟨ minus+n {suc j} {suc i} (<-trans (s≤s a) a<sa) ⟩ suc j ∎ where open ≡-Reasoning ind6 : {i j k : ℕ } → i < j → Dividable k (j - i) → Dividable k (suc i) → Dividable k (suc j) ind6 {i} {j} {k} i<j dj di = subst (λ g → Dividable k g ) (ind7 i<j) (proj1 (div+div dj di)) ind4 : {i j : ℕ } → i < j → Dividable (gcd (suc i) (j - i)) (j - i) → Dividable (gcd (suc i) (suc j)) (j - i) ind4 {i} {j} i<j prev = subst (λ k → k) ( begin Dividable (gcd (suc i) (j - i)) (j - i) ≡⟨ cong (λ k → Dividable k (j - i)) (gcdsym {suc i} ) ⟩ Dividable (gcd (j - i ) (suc i) ) (j - i) ≡⟨ cong (λ k → Dividable k (j - i)) ( sym (gcd+j (j - i) (suc i))) ⟩ Dividable (gcd ((j - i) + suc i) (suc i)) (j - i) ≡⟨ cong (λ k → Dividable (gcd k (suc i)) (j - i)) (ind7 i<j ) ⟩ Dividable (gcd (suc j) (suc i)) (j - i) ≡⟨ cong (λ k → Dividable k (j - i)) (gcdsym {suc j} ) ⟩ Dividable (gcd (suc i) (suc j)) (j - i) ∎ ) prev where open ≡-Reasoning ind : ( i j : ℕ ) → Dividable (gcd (Fdec.ni (fedc0 i j)) (Fdec.nj (fedc0 i j))) (Fdec.ni (fedc0 i j)) ∧ Dividable (gcd (Fdec.ni (fedc0 i j)) (Fdec.nj (fedc0 i j))) (Fdec.nj (fedc0 i j)) → Dividable (gcd i j) i ∧ Dividable (gcd i j) j ind zero zero prev = ind0 where ind0 : Dividable (gcd zero zero) zero ∧ Dividable (gcd zero zero) zero ind0 = ⟪ div0 , div0 ⟫ ind zero (suc j) prev = ind1 where ind1 : Dividable (gcd zero (suc j)) zero ∧ Dividable (gcd zero (suc j)) (suc j) ind1 = ⟪ div0 , subst (λ k → Dividable k (suc j)) (sym (trans (gcdsym {zero} {suc j}) (gcd20 (suc j)))) div= ⟫ ind (suc i) zero prev = ind2 where ind2 : Dividable (gcd (suc i) zero) (suc i) ∧ Dividable (gcd (suc i) zero) zero ind2 = ⟪ subst (λ k → Dividable k (suc i)) (sym (trans refl (gcd20 (suc i)))) div= , div0 ⟫ ind (suc i) (suc j) prev with <-cmp i j ... | tri< a ¬b ¬c = ⟪ ind3 a (proj1 prev) , ind6 a (ind4 a (proj2 prev)) (ind3 a (proj1 prev) ) ⟫ ... | tri≈ ¬a refl ¬c = ⟪ ind5 , ind5 ⟫ where ind5 : Dividable (gcd (suc i) (suc i)) (suc i) ind5 = subst (λ k → Dividable k (suc j)) (sym (gcdmm (suc i) (suc i))) div= ... | tri> ¬a ¬b c = ⟪ ind8 c (proj1 prev) (proj2 prev) , ind10 c (proj2 prev) ⟫ where ind9 : {i j : ℕ} → i < j → gcd (j - i) (suc i) ≡ gcd (suc j) (suc i) ind9 {i} {j} i<j = begin gcd (j - i ) (suc i) ≡⟨ sym (gcd+j (j - i ) (suc i) ) ⟩ gcd (j - i + suc i) (suc i) ≡⟨ cong (λ k → gcd k (suc i)) (ind7 i<j ) ⟩ gcd (suc j) (suc i) ∎ where open ≡-Reasoning ind8 : { i j : ℕ } → i < j → Dividable (gcd (j - i) (suc i)) (j - i) → Dividable (gcd (j - i) (suc i)) (suc i) → Dividable (gcd (suc j) (suc i)) (suc j) ind8 {i} {j} i<j dji di = ind6 i<j (subst (λ k → Dividable k (j - i)) (ind9 i<j) dji) (subst (λ k → Dividable k (suc i)) (ind9 i<j) di) ind10 : { i j : ℕ } → j < i → Dividable (gcd (i - j) (suc j)) (suc j) → Dividable (gcd (suc i) (suc j)) (suc j) ind10 {i} {j} j<i dji = subst (λ g → Dividable g (suc j) ) (begin gcd (i - j) (suc j) ≡⟨ sym (gcd+j (i - j) (suc j)) ⟩ gcd (i - j + suc j) (suc j) ≡⟨ cong (λ k → gcd k (suc j)) (ind7 j<i ) ⟩ gcd (suc i) (suc j) ∎ ) dji where open ≡-Reasoning I : Finduction (ℕ ∧ ℕ) _ F I = record { fzero = F00 ; pnext = λ p → ⟪ Fdec.ni (fedc0 (proj1 p) (proj2 p)) , Fdec.nj (fedc0 (proj1 p) (proj2 p)) ⟫ ; decline = λ {p} lt → Fdec.fdec (fedc0 (proj1 p) (proj2 p)) lt ; ind = λ {p} prev → ind (proj1 p ) ( proj2 p ) prev } f-div>0 : { k i : ℕ } → (d : Dividable k i ) → 0 < i → 0 < Dividable.factor d f-div>0 {k} {i} d 0<i with <-cmp 0 (Dividable.factor d) ... | tri< a ¬b ¬c = a ... | tri≈ ¬a b ¬c = ⊥-elim ( nat-≡< (begin 0 * k + 0 ≡⟨ cong (λ g → g * k + 0) b ⟩ Dividable.factor d * k + 0 ≡⟨ Dividable.is-factor d ⟩ i ∎ ) 0<i ) where open ≡-Reasoning gcd-≤i : ( i j : ℕ ) → 0 < i → i ≤ j → gcd i j ≤ i gcd-≤i zero _ () z≤n gcd-≤i (suc i) (suc j) _ (s≤s i<j) = begin gcd (suc i) (suc j) ≡⟨ sym m*1=m ⟩ gcd (suc i) (suc j) * 1 ≤⟨ *-monoʳ-≤ (gcd (suc i) (suc j)) (f-div>0 d (s≤s z≤n)) ⟩ gcd (suc i) (suc j) * f ≡⟨ +-comm 0 _ ⟩ gcd (suc i) (suc j) * f + 0 ≡⟨ cong (λ k → k + 0) (*-comm (gcd (suc i) (suc j)) _ ) ⟩ Dividable.factor (proj1 (gcd-dividable (suc i) (suc j))) * gcd (suc i) (suc j) + 0 ≡⟨ Dividable.is-factor (proj1 (gcd-dividable (suc i) (suc j))) ⟩ suc i ∎ where d = proj1 (gcd-dividable (suc i) (suc j)) f = Dividable.factor (proj1 (gcd-dividable (suc i) (suc j))) open ≤-Reasoning gcd-≤ : { i j : ℕ } → 0 < i → 0 < j → gcd i j ≤ i gcd-≤ {i} {j} 0<i 0<j with <-cmp i j ... | tri< a ¬b ¬c = gcd-≤i i j 0<i (<to≤ a) ... | tri≈ ¬a refl ¬c = gcd-≤i i j 0<i refl-≤ ... | tri> ¬a ¬b c = ≤-trans (subst (λ k → k ≤ j) (gcdsym {j} {i}) (gcd-≤i j i 0<j (<to≤ c))) (<to≤ c) record Euclid (i j gcd : ℕ ) : Set where field eqa : ℕ eqb : ℕ is-equ< : eqa * i > eqb * j → (eqa * i) - (eqb * j) ≡ gcd is-equ> : eqb * j > eqa * i → (eqb * j) - (eqa * i) ≡ gcd is-equ= : eqa * i ≡ eqb * j → 0 ≡ gcd ge3 : {a b c d : ℕ } → b > a → b - a ≡ d - c → d > c ge3 {a} {b} {c} {d} b>a eq = minus>0→x<y (subst (λ k → 0 < k ) eq (minus>0 b>a)) ge01 : ( i0 j j0 ea eb : ℕ ) → ( di : GCD 0 (suc i0) (suc (suc j)) j0 ) → (((ea + eb * (Dividable.factor (GCD.div-i di))) * suc i0) ≡ (ea * suc i0) + (eb * (Dividable.factor (GCD.div-i di)) ) * suc i0 ) ∧ ( (eb * j0) ≡ (eb * suc (suc j) + (eb * (Dividable.factor (GCD.div-i di)) ) * suc i0) ) ge01 i0 j j0 ea eb di = ⟪ ge011 , ge012 ⟫ where f = Dividable.factor (GCD.div-i di) ge4 : suc (j0 + 0) > suc (suc j) ge4 = subst (λ k → k > suc (suc j)) (+-comm 0 _ ) ( s≤s (GCD.j<j0 di )) ge011 : (ea + eb * f) * suc i0 ≡ ea * suc i0 + eb * f * suc i0 ge011 = begin (ea + eb * f) * suc i0 ≡⟨ *-distribʳ-+ (suc i0) ea _ ⟩ ea * suc i0 + eb * f * suc i0 ∎ where open ≡-Reasoning ge012 : eb * j0 ≡ eb * suc (suc j) + eb * f * suc i0 ge012 = begin eb * j0 ≡⟨ cong (λ k → eb * k) ( begin j0 ≡⟨ +-comm 0 _ ⟩ j0 + 0 ≡⟨ sym (minus+n {j0 + 0} {suc (suc j)} ge4) ⟩ ((j0 + 0) - (suc (suc j))) + suc (suc j) ≡⟨ +-comm _ (suc (suc j)) ⟩ suc (suc j) + ((j0 + 0) - suc (suc j)) ≡⟨ cong (λ k → suc (suc j) + k ) (sym (Dividable.is-factor (GCD.div-i di))) ⟩ suc (suc j) + (f * suc i0 + 0) ≡⟨ cong (λ k → suc (suc j) + k ) ( +-comm _ 0 ) ⟩ suc (suc j) + (f * suc i0 ) ∎ ) ⟩ eb * (suc (suc j) + (f * suc i0 ) ) ≡⟨ *-distribˡ-+ eb (suc (suc j)) (f * suc i0) ⟩ eb * suc (suc j) + eb * (f * suc i0) ≡⟨ cong (λ k → eb * suc (suc j) + k ) ((sym (*-assoc eb _ _ )) ) ⟩ eb * suc (suc j) + eb * f * suc i0 ∎ where open ≡-Reasoning ge20 : {i0 j0 : ℕ } → i0 ≡ 0 → 0 ≡ gcd1 zero i0 zero j0 ge20 {i0} {zero} refl = refl ge20 {i0} {suc j0} refl = refl gcd-euclid1 : ( i i0 j j0 : ℕ ) → GCD i i0 j j0 → Euclid i0 j0 (gcd1 i i0 j j0) gcd-euclid1 zero i0 zero j0 di with <-cmp i0 j0 ... | tri< a' ¬b ¬c = record { eqa = 1 ; eqb = 0 ; is-equ< = λ _ → +-comm _ 0 ; is-equ> = λ () ; is-equ= = ge21 } where ge21 : 1 * i0 ≡ 0 * j0 → 0 ≡ i0 ge21 eq = trans (sym eq) (+-comm i0 0) ... | tri≈ ¬a refl ¬c = record { eqa = 1 ; eqb = 0 ; is-equ< = λ _ → +-comm _ 0 ; is-equ> = λ () ; is-equ= = λ eq → trans (sym eq) (+-comm i0 0) } ... | tri> ¬a ¬b c = record { eqa = 0 ; eqb = 1 ; is-equ< = λ () ; is-equ> = λ _ → +-comm _ 0 ; is-equ= = ge22 } where ge22 : 0 * i0 ≡ 1 * j0 → 0 ≡ j0 ge22 eq = trans eq (+-comm j0 0) -- i<i0 : zero ≤ i0 -- j<j0 : 1 ≤ j0 -- div-i : Dividable i0 ((j0 + zero) - 1) -- fi * i0 ≡ (j0 + zero) - 1 -- div-j : Dividable j0 ((i0 + 1) - zero) -- fj * j0 ≡ (i0 + 1) - zero gcd-euclid1 zero i0 (suc zero) j0 di = record { eqa = 1 ; eqb = Dividable.factor (GCD.div-j di) ; is-equ< = λ lt → ⊥-elim ( ge7 lt) ; is-equ> = λ _ → ge6 ; is-equ= = λ eq → ⊥-elim (nat-≡< (sym (minus<=0 (subst (λ k → k ≤ 1 * i0) eq refl-≤ ))) (subst (λ k → 0 < k) (sym ge6) a<sa )) } where ge6 : (Dividable.factor (GCD.div-j di) * j0) - (1 * i0) ≡ gcd1 zero i0 1 j0 ge6 = begin (Dividable.factor (GCD.div-j di) * j0) - (1 * i0) ≡⟨ cong (λ k → k - (1 * i0)) (+-comm 0 _) ⟩ (Dividable.factor (GCD.div-j di) * j0 + 0) - (1 * i0) ≡⟨ cong (λ k → k - (1 * i0)) (Dividable.is-factor (GCD.div-j di) ) ⟩ ((i0 + 1) - zero) - (1 * i0) ≡⟨ refl ⟩ (i0 + 1) - (i0 + 0) ≡⟨ minus+yx-yz {_} {i0} {0} ⟩ 1 ∎ where open ≡-Reasoning ge7 : ¬ ( 1 * i0 > Dividable.factor (GCD.div-j di) * j0 ) ge7 lt = ⊥-elim ( nat-≡< (sym ( minus<=0 (<to≤ lt))) (subst (λ k → 0 < k) (sym ge6) (s≤s z≤n))) gcd-euclid1 zero zero (suc (suc j)) j0 di = record { eqa = 0 ; eqb = 1 ; is-equ< = λ () ; is-equ> = λ _ → +-comm _ 0 ; is-equ= = λ eq → subst (λ k → 0 ≡ k) (+-comm _ 0) eq } gcd-euclid1 zero (suc i0) (suc (suc j)) j0 di with gcd-euclid1 i0 (suc i0) (suc j) (suc (suc j)) ( gcd-next1 di ) ... | e = record { eqa = ea + eb * f ; eqb = eb ; is-equ= = λ eq → Euclid.is-equ= e (ge23 eq) ; is-equ< = λ lt → subst (λ k → ((ea + eb * f) * suc i0) - (eb * j0) ≡ k ) (Euclid.is-equ< e (ge3 lt (ge1 ))) (ge1 ) ; is-equ> = λ lt → subst (λ k → (eb * j0) - ((ea + eb * f) * suc i0) ≡ k ) (Euclid.is-equ> e (ge3 lt (ge2 ))) (ge2 ) } where ea = Euclid.eqa e eb = Euclid.eqb e f = Dividable.factor (GCD.div-i di) ge1 : ((ea + eb * f) * suc i0) - (eb * j0) ≡ (ea * suc i0) - (eb * suc (suc j)) ge1 = begin ((ea + eb * f) * suc i0) - (eb * j0) ≡⟨ cong₂ (λ j k → j - k ) (proj1 (ge01 i0 j j0 ea eb di)) (proj2 (ge01 i0 j j0 ea eb di)) ⟩ (ea * suc i0 + (eb * f ) * suc i0 ) - ( eb * suc (suc j) + ((eb * f) * (suc i0)) ) ≡⟨ minus+xy-zy {ea * suc i0} {(eb * f ) * suc i0} {eb * suc (suc j)} ⟩ (ea * suc i0) - (eb * suc (suc j)) ∎ where open ≡-Reasoning ge2 : (eb * j0) - ((ea + eb * f) * suc i0) ≡ (eb * suc (suc j)) - (ea * suc i0) ge2 = begin (eb * j0) - ((ea + eb * f) * suc i0) ≡⟨ cong₂ (λ j k → j - k ) (proj2 (ge01 i0 j j0 ea eb di)) (proj1 (ge01 i0 j j0 ea eb di)) ⟩ ( eb * suc (suc j) + ((eb * f) * (suc i0)) ) - (ea * suc i0 + (eb * f ) * suc i0 ) ≡⟨ minus+xy-zy {eb * suc (suc j)}{(eb * f ) * suc i0} {ea * suc i0} ⟩ (eb * suc (suc j)) - (ea * suc i0) ∎ where open ≡-Reasoning ge23 : (ea + eb * f) * suc i0 ≡ eb * j0 → ea * suc i0 ≡ eb * suc (suc j) ge23 eq = begin ea * suc i0 ≡⟨ sym (minus+y-y {_} {(eb * f ) * suc i0} ) ⟩ (ea * suc i0 + ((eb * f ) * suc i0 )) - ((eb * f ) * suc i0 ) ≡⟨ cong (λ k → k - ((eb * f ) * suc i0 )) (sym ( proj1 (ge01 i0 j j0 ea eb di))) ⟩ ((ea + eb * f) * suc i0) - ((eb * f ) * suc i0 ) ≡⟨ cong (λ k → k - ((eb * f ) * suc i0 )) eq ⟩ (eb * j0) - ((eb * f ) * suc i0 ) ≡⟨ cong (λ k → k - ((eb * f ) * suc i0 )) ( proj2 (ge01 i0 j j0 ea eb di)) ⟩ (eb * suc (suc j) + ((eb * f ) * suc i0 )) - ((eb * f ) * suc i0 ) ≡⟨ minus+y-y {_} {(eb * f ) * suc i0 } ⟩ eb * suc (suc j) ∎ where open ≡-Reasoning gcd-euclid1 (suc zero) i0 zero j0 di = record { eqb = 1 ; eqa = Dividable.factor (GCD.div-i di) ; is-equ> = λ lt → ⊥-elim ( ge7' lt) ; is-equ< = λ _ → ge6' ; is-equ= = λ eq → ⊥-elim (nat-≡< (sym (minus<=0 (subst (λ k → k ≤ 1 * j0) (sym eq) refl-≤ ))) (subst (λ k → 0 < k) (sym ge6') a<sa )) } where ge6' : (Dividable.factor (GCD.div-i di) * i0) - (1 * j0) ≡ gcd1 (suc zero) i0 zero j0 ge6' = begin (Dividable.factor (GCD.div-i di) * i0) - (1 * j0) ≡⟨ cong (λ k → k - (1 * j0)) (+-comm 0 _) ⟩ (Dividable.factor (GCD.div-i di) * i0 + 0) - (1 * j0) ≡⟨ cong (λ k → k - (1 * j0)) (Dividable.is-factor (GCD.div-i di) ) ⟩ ((j0 + 1) - zero) - (1 * j0) ≡⟨ refl ⟩ (j0 + 1) - (j0 + 0) ≡⟨ minus+yx-yz {_} {j0} {0} ⟩ 1 ∎ where open ≡-Reasoning ge7' : ¬ ( 1 * j0 > Dividable.factor (GCD.div-i di) * i0 ) ge7' lt = ⊥-elim ( nat-≡< (sym ( minus<=0 (<to≤ lt))) (subst (λ k → 0 < k) (sym ge6') (s≤s z≤n))) gcd-euclid1 (suc (suc i)) i0 zero zero di = record { eqb = 0 ; eqa = 1 ; is-equ> = λ () ; is-equ< = λ _ → +-comm _ 0 ; is-equ= = λ eq → subst (λ k → 0 ≡ k) (+-comm _ 0) (sym eq) } gcd-euclid1 (suc (suc i)) i0 zero (suc j0) di with gcd-euclid1 (suc i) (suc (suc i)) j0 (suc j0) (GCD-sym (gcd-next1 (GCD-sym di))) ... | e = record { eqa = ea ; eqb = eb + ea * f ; is-equ= = λ eq → Euclid.is-equ= e (ge24 eq) ; is-equ< = λ lt → subst (λ k → ((ea * i0) - ((eb + ea * f) * suc j0)) ≡ k ) (Euclid.is-equ< e (ge3 lt ge4)) ge4 ; is-equ> = λ lt → subst (λ k → (((eb + ea * f) * suc j0) - (ea * i0)) ≡ k ) (Euclid.is-equ> e (ge3 lt ge5)) ge5 } where ea = Euclid.eqa e eb = Euclid.eqb e f = Dividable.factor (GCD.div-j di) ge5 : (((eb + ea * f) * suc j0) - (ea * i0)) ≡ ((eb * suc j0) - (ea * suc (suc i))) ge5 = begin ((eb + ea * f) * suc j0) - (ea * i0) ≡⟨ cong₂ (λ j k → j - k ) (proj1 (ge01 j0 i i0 eb ea (GCD-sym di) )) (proj2 (ge01 j0 i i0 eb ea (GCD-sym di) )) ⟩ ( eb * suc j0 + (ea * f )* suc j0) - (ea * suc (suc i) + (ea * f )* suc j0) ≡⟨ minus+xy-zy {_} {(ea * f )* suc j0} {ea * suc (suc i)} ⟩ (eb * suc j0) - (ea * suc (suc i)) ∎ where open ≡-Reasoning ge4 : ((ea * i0) - ((eb + ea * f) * suc j0)) ≡ ((ea * suc (suc i)) - (eb * suc j0)) ge4 = begin (ea * i0) - ((eb + ea * f) * suc j0) ≡⟨ cong₂ (λ j k → j - k ) (proj2 (ge01 j0 i i0 eb ea (GCD-sym di) )) (proj1 (ge01 j0 i i0 eb ea (GCD-sym di) )) ⟩ (ea * suc (suc i) + (ea * f )* suc j0) - ( eb * suc j0 + (ea * f )* suc j0) ≡⟨ minus+xy-zy {ea * suc (suc i)} {(ea * f )* suc j0} { eb * suc j0} ⟩ (ea * suc (suc i)) - (eb * suc j0) ∎ where open ≡-Reasoning ge24 : ea * i0 ≡ (eb + ea * f) * suc j0 → ea * suc (suc i) ≡ eb * suc j0 ge24 eq = begin ea * suc (suc i) ≡⟨ sym ( minus+y-y {_} {(ea * f ) * suc j0 }) ⟩ (ea * suc (suc i) + (ea * f ) * suc j0 ) - ((ea * f ) * suc j0) ≡⟨ cong (λ k → k - ((ea * f ) * suc j0 )) (sym (proj2 (ge01 j0 i i0 eb ea (GCD-sym di) ))) ⟩ (ea * i0) - ((ea * f ) * suc j0) ≡⟨ cong (λ k → k - ((ea * f ) * suc j0 )) eq ⟩ ((eb + ea * f) * suc j0) - ((ea * f ) * suc j0) ≡⟨ cong (λ k → k - ((ea * f ) * suc j0 )) ((proj1 (ge01 j0 i i0 eb ea (GCD-sym di)))) ⟩ ( eb * suc j0 + (ea * f ) * suc j0 ) - ((ea * f ) * suc j0) ≡⟨ minus+y-y {_} {(ea * f ) * suc j0 } ⟩ eb * suc j0 ∎ where open ≡-Reasoning gcd-euclid1 (suc zero) i0 (suc j) j0 di = gcd-euclid1 zero i0 j j0 (gcd-next di) gcd-euclid1 (suc (suc i)) i0 (suc j) j0 di = gcd-euclid1 (suc i) i0 j j0 (gcd-next di) ge12 : (p x : ℕ) → 0 < x → 1 < p → ((i : ℕ ) → i < p → 0 < i → gcd p i ≡ 1) → ( gcd p x ≡ 1 ) ∨ ( Dividable p x ) ge12 p x 0<x 1<p prime with decD {p} {x} 1<p ... | yes y = case2 y ... | no nx with <-cmp (gcd p x ) 1 ... | tri< a ¬b ¬c = ⊥-elim ( nat-≤> a (s≤s (gcd>0 p x (<-trans a<sa 1<p) 0<x) ) ) ... | tri≈ ¬a b ¬c = case1 b ... | tri> ¬a ¬b c = ⊥-elim ( nat-≡< (sym (prime (gcd p x) ge13 (<to≤ c) )) ge18 ) where -- 1 < gcd p x ge13 : gcd p x < p -- gcd p x ≡ p → ¬ nx ge13 with <-cmp (gcd p x ) p ... | tri< a ¬b ¬c = a ... | tri≈ ¬a b ¬c = ⊥-elim ( nx (subst (λ k → Dividable k x) b (proj2 (gcd-dividable p x )))) ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> (gcd-≤ (<-trans a<sa 1<p) 0<x) c ) ge19 : Dividable (gcd p x) p ge19 = proj1 (gcd-dividable p x ) ge18 : 1 < gcd p (gcd p x) -- Dividable p (gcd p x) → gcd p (gcd p x) ≡ (gcd p x) > 1 ge18 = subst (λ k → 1 < k ) (sym (div→gcd {p} {gcd p x} c ge19 )) c gcd-euclid : ( p a b : ℕ ) → 1 < p → 0 < a → 0 < b → ((i : ℕ ) → i < p → 0 < i → gcd p i ≡ 1) → Dividable p (a * b) → Dividable p a ∨ Dividable p b gcd-euclid p a b 1<p 0<a 0<b prime div-ab with decD {p} {a} 1<p ... | yes y = case1 y ... | no np = case2 ge16 where f = Dividable.factor div-ab ge10 : gcd p a ≡ 1 ge10 with ge12 p a 0<a 1<p prime ... | case1 x = x ... | case2 x = ⊥-elim ( np x ) ge11 : Euclid p a (gcd p a) ge11 = gcd-euclid1 p p a a GCDi ea = Euclid.eqa ge11 eb = Euclid.eqb ge11 ge18 : (f * eb) * p ≡ b * (a * eb ) ge18 = begin (f * eb) * p ≡⟨ *-assoc (f) (eb) p ⟩ f * (eb * p) ≡⟨ cong (λ k → f * k) (*-comm _ p) ⟩ f * (p * eb ) ≡⟨ sym (*-assoc (f) p (eb) ) ⟩ (f * p ) * eb ≡⟨ cong (λ k → k * eb ) (+-comm 0 (f * p )) ⟩ (f * p + 0) * eb ≡⟨ cong (λ k → k * eb) (((Dividable.is-factor div-ab))) ⟩ (a * b) * eb ≡⟨ cong (λ k → k * eb) (*-comm a b) ⟩ (b * a) * eb ≡⟨ *-assoc b a (eb ) ⟩ b * (a * eb ) ∎ where open ≡-Reasoning ge19 : ( ea * p ) ≡ ( eb * a ) → ((b * ea) - (f * eb)) * p + 0 ≡ b ge19 eq = ⊥-elim ( nat-≡< (Euclid.is-equ= ge11 eq) (subst (λ k → 0 < k ) (sym ge10) a<sa ) ) ge14 : ( ea * p ) > ( eb * a ) → ((b * ea) - (f * eb)) * p + 0 ≡ b ge14 lt = begin (((b * ea) - (f * eb)) * p) + 0 ≡⟨ +-comm _ 0 ⟩ ((b * ea) - ((f * eb)) * p) ≡⟨ distr-minus-* {_} {f * eb} {p} ⟩ ((b * ea) * p) - (((f * eb) * p)) ≡⟨ cong (λ k → ((b * ea) * p) - k ) ge18 ⟩ ((b * ea) * p) - (b * (a * eb )) ≡⟨ cong (λ k → k - (b * (a * eb)) ) (*-assoc b _ p) ⟩ (b * (ea * p)) - (b * (a * eb )) ≡⟨ sym ( distr-minus-*' {b} {ea * p} {a * eb} ) ⟩ b * (( ea * p) - (a * eb) ) ≡⟨ cong (λ k → b * ( ( ea * p) - k)) (*-comm a (eb)) ⟩ (b * ( (ea * p)) - (eb * a) ) ≡⟨ cong (b *_) (Euclid.is-equ< ge11 lt )⟩ b * gcd p a ≡⟨ cong (b *_) ge10 ⟩ b * 1 ≡⟨ m*1=m ⟩ b ∎ where open ≡-Reasoning ge15 : ( ea * p ) < ( eb * a ) → ((f * eb) - (b * ea ) ) * p + 0 ≡ b ge15 lt = begin ((f * eb) - (b * ea) ) * p + 0 ≡⟨ +-comm _ 0 ⟩ ((f * eb) - (b * ea) ) * p ≡⟨ distr-minus-* {_} {b * ea} {p} ⟩ ((f * eb) * p) - ((b * ea) * p) ≡⟨ cong (λ k → k - ((b * ea) * p) ) ge18 ⟩ (b * (a * eb )) - ((b * ea) * p ) ≡⟨ cong (λ k → (b * (a * eb)) - k ) (*-assoc b _ p) ⟩ (b * (a * eb )) - (b * (ea * p) ) ≡⟨ sym ( distr-minus-*' {b} {a * eb} {ea * p} ) ⟩ b * ( (a * eb) - (ea * p) ) ≡⟨ cong (λ k → b * ( k - ( ea * p) )) (*-comm a (eb)) ⟩ b * ( (eb * a) - (ea * p) ) ≡⟨ cong (b *_) (Euclid.is-equ> ge11 lt) ⟩ b * gcd p a ≡⟨ cong (b *_) ge10 ⟩ b * 1 ≡⟨ m*1=m ⟩ b ∎ where open ≡-Reasoning ge17 : (x y : ℕ ) → x ≡ y → x ≤ y ge17 x x refl = refl-≤ ge16 : Dividable p b ge16 with <-cmp ( ea * p ) ( eb * a ) ... | tri< a ¬b ¬c = record { factor = (f * eb) - (b * ea) ; is-factor = ge15 a } ... | tri≈ ¬a eq ¬c = record { factor = (b * ea) - ( f * eb) ; is-factor = ge19 eq } ... | tri> ¬a ¬b c = record { factor = (b * ea) - (f * eb) ; is-factor = ge14 c } gcdmul+1 : ( m n : ℕ ) → gcd (m * n + 1) n ≡ 1 gcdmul+1 zero n = gcd204 n gcdmul+1 (suc m) n = begin gcd (suc m * n + 1) n ≡⟨⟩ gcd (n + m * n + 1) n ≡⟨ cong (λ k → gcd k n ) (begin n + m * n + 1 ≡⟨ cong (λ k → k + 1) (+-comm n _) ⟩ m * n + n + 1 ≡⟨ +-assoc (m * n) _ _ ⟩ m * n + (n + 1) ≡⟨ cong (λ k → m * n + k) (+-comm n _) ⟩ m * n + (1 + n) ≡⟨ sym ( +-assoc (m * n) _ _ ) ⟩ m * n + 1 + n ∎ ) ⟩ gcd (m * n + 1 + n) n ≡⟨ gcd+j (m * n + 1) n ⟩ gcd (m * n + 1) n ≡⟨ gcdmul+1 m n ⟩ 1 ∎ where open ≡-Reasoning m*n=m→n : {m n : ℕ } → 0 < m → m * n ≡ m * 1 → n ≡ 1 m*n=m→n {suc m} {n} (s≤s lt) eq = *-cancelˡ-≡ m eq gcd-is-gratest : { i j k : ℕ } → i > 0 → j > 0 → k > 1 → Dividable k i → Dividable k j → k ≤ gcd i j gcd-is-gratest {i} {j} {k} i>0 j>0 k>1 ki kj = div→k≤m k>1 (gcd>0 i j i>0 j>0 ) gcd001 where gcd001 : Dividable k ( gcd i j ) gcd001 = gcd-div _ _ _ k>1 ki kj