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1 {-# OPTIONS --allow-unsolved-metas #-}
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2 module gcd where
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3
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141
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4 open import Data.Nat
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5 open import Data.Nat.Properties
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57
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6 open import Data.Empty
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141
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7 open import Data.Unit using (⊤ ; tt)
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57
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8 open import Relation.Nullary
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9 open import Relation.Binary.PropositionalEquality
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141
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10 open import Relation.Binary.Definitions
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149
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11 open import nat
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151
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12 open import logic
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13
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14 even : (n : ℕ ) → Set
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141
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15 even zero = ⊤
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16 even (suc zero) = ⊥
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17 even (suc (suc n)) = even n
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18
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19 even? : (n : ℕ ) → Dec ( even n )
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20 even? zero = yes tt
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21 even? (suc zero) = no (λ ())
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22 even? (suc (suc n)) = even? n
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23
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24 n+even : {n m : ℕ } → even n → even m → even ( n + m )
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25 n+even {zero} {zero} tt tt = tt
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26 n+even {zero} {suc m} tt em = em
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27 n+even {suc (suc n)} {m} en em = n+even {n} {m} en em
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28
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29 n*even : {m n : ℕ } → even n → even ( m * n )
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30 n*even {zero} {n} en = tt
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31 n*even {suc m} {n} en = n+even {n} {m * n} en (n*even {m} {n} en)
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32
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33 even*n : {n m : ℕ } → even n → even ( n * m )
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34 even*n {n} {m} en = subst even (*-comm m n) (n*even {m} {n} en)
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35
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36 gcd1 : ( i i0 j j0 : ℕ ) → ℕ
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37 gcd1 zero i0 zero j0 with <-cmp i0 j0
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38 ... | tri< a ¬b ¬c = j0
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39 ... | tri≈ ¬a refl ¬c = i0
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40 ... | tri> ¬a ¬b c = i0
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41 gcd1 zero i0 (suc zero) j0 = 1
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42 gcd1 zero zero (suc (suc j)) j0 = j0
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43 gcd1 zero (suc i0) (suc (suc j)) j0 = gcd1 i0 (suc i0) (suc j) (suc (suc j))
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44 gcd1 (suc zero) i0 zero j0 = 1
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45 gcd1 (suc (suc i)) i0 zero zero = i0
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46 gcd1 (suc (suc i)) i0 zero (suc j0) = gcd1 (suc i) (suc (suc i)) j0 (suc j0)
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47 gcd1 (suc i) i0 (suc j) j0 = gcd1 i i0 j j0
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48
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49 gcd : ( i j : ℕ ) → ℕ
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50 gcd i j = gcd1 i i j j
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51
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52 even→gcd=2 : {n : ℕ} → even n → n > 0 → gcd n 2 ≡ 2
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53 even→gcd=2 {suc (suc zero)} en (s≤s z≤n) = refl
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54 even→gcd=2 {suc (suc (suc (suc n)))} en (s≤s z≤n) = begin
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55 gcd (suc (suc (suc (suc n)))) 2 ≡⟨⟩
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56 gcd (suc (suc n)) 2 ≡⟨ even→gcd=2 {suc (suc n)} en (s≤s z≤n) ⟩
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57 2 ∎ where open ≡-Reasoning
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58
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59 -- gcd26 : { n m : ℕ} → n > 1 → m > 1 → n - m > 0 → gcd n m ≡ gcd (n - m) m
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60 -- gcd27 : { n m : ℕ} → n > 1 → m > 1 → n - m > 0 → gcd n k ≡ k → k ≤ n
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143
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61
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62 gcd22 : ( i i0 o o0 : ℕ ) → gcd1 (suc i) i0 (suc o) o0 ≡ gcd1 i i0 o o0
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63 gcd22 zero i0 zero o0 = refl
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64 gcd22 zero i0 (suc o) o0 = refl
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65 gcd22 (suc i) i0 zero o0 = refl
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66 gcd22 (suc i) i0 (suc o) o0 = refl
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67
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68 gcd20 : (i : ℕ) → gcd i 0 ≡ i
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69 gcd20 zero = refl
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70 gcd20 (suc i) = gcd201 (suc i) where
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71 gcd201 : (i : ℕ ) → gcd1 i i zero zero ≡ i
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72 gcd201 zero = refl
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73 gcd201 (suc zero) = refl
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74 gcd201 (suc (suc i)) = refl
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75
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76 gcdmm : (n m : ℕ) → gcd1 n m n m ≡ m
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77 gcdmm zero m with <-cmp m m
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78 ... | tri< a ¬b ¬c = refl
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79 ... | tri≈ ¬a refl ¬c = refl
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80 ... | tri> ¬a ¬b c = refl
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81 gcdmm (suc n) m = subst (λ k → k ≡ m) (sym (gcd22 n m n m )) (gcdmm n m )
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82
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83 record Comp ( m n : ℕ ) : Set where
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84 field
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85 non-1 : 1 < m
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86 comp : ℕ
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87 is-comp : n * comp ≡ m
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88
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89 gcd27 : (n m i : ℕ) → 1 < m → gcd n i ≡ m → Comp m n ∨ Comp m i
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90 gcd27 n m i 1<m gn = gcd271 n n i i gn where
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91 gcd271 : (n n0 i i0 : ℕ ) → gcd1 n n0 i i0 ≡ m → Comp m n0 ∨ Comp m i0
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92 gcd271 zero n0 zero i0 eq with <-cmp n0 i0
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93 ... | tri< a ¬b ¬c = case2 (record { non-1 = 1<m ; comp = 1 ; is-comp = subst (λ k → k ≡ m) {!!} eq })
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94 ... | tri≈ ¬a refl ¬c = case1 (record { non-1 = 1<m ; comp = 1 ; is-comp = subst (λ k → k ≡ m) {!!} eq })
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95 ... | tri> ¬a ¬b c = case1 (record { non-1 = 1<m ; comp = 1 ; is-comp = subst (λ k → k ≡ m) {!!} eq })
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96 gcd271 zero n0 (suc zero) i0 eq = ⊥-elim ( nat-≡< eq 1<m )
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97 gcd271 zero zero (suc (suc i)) i0 eq = case2 (record { non-1 = 1<m ; comp = 1 ; is-comp = subst (λ k → k ≡ m) {!!} eq })
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98 gcd271 zero (suc n0) (suc (suc i)) i0 eq with gcd271 n0 (suc n0) (suc i) (suc (suc i)) eq
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99 ... | case1 t = case1 t
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100 -- t : Comp m (suc (suc i)), (suc n0) + (suc (suc i)) ≡ i0
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101 ... | case2 t = case1 (gcd272 t) where
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102 gcd272 : Comp m (suc (suc i)) → Comp m (suc n0)
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103 gcd272 = {!!}
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104 gcd271 (suc zero) n0 zero i0 eq = ⊥-elim ( nat-≡< eq 1<m )
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105 gcd271 (suc (suc n)) n0 zero zero eq = case2 (record { non-1 = 1<m ; comp = n0 ; is-comp = subst (λ k → k ≡ m) {!!} eq })
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106 gcd271 (suc (suc n)) n0 zero (suc i0) eq = {!!}
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107 gcd271 (suc n) n0 (suc i) i0 eq = gcd271 n n0 i i0 (trans (sym (gcd22 n n0 i i0)) eq )
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108
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109 gcd26 : (n m i : ℕ) → 1 < n → 1 < m → gcd n i ≡ m → ¬ ( gcd n m ≡ 1 )
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110 gcd26 n m i 1<n 1<m gi g1 = gcd261 n n m m i i 1<n 1<m gi g1 where
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111 gcd261 : (n n0 m m0 i i0 : ℕ) → 1 < n → 1 < m0 → gcd1 n n0 i i0 ≡ m0 → ¬ ( gcd1 n n0 m m0 ≡ 1 )
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112 gcd261 zero n0 m m0 i i0 () 1<m gi g1
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113 -- gi : gcd1 (suc n) n0 zero i0 ≡ m0
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114 -- g1 : gcd1 (suc n) n0 m m0 ≡ 1
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115 gcd261 (suc zero) n0 m m0 zero i0 1<n 1<m gi g1 = ⊥-elim ( nat-≡< gi 1<m )
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116 gcd261 (suc (suc n)) n0 zero m0 zero zero 1<n 1<m gi g1 = {!!}
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117 gcd261 (suc (suc n)) n0 zero m0 zero (suc i0) 1<n 1<m gi g1 = {!!}
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118 gcd261 (suc (suc n)) n0 (suc zero) m0 zero i0 1<n 1<m gi g1 = {!!}
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119 gcd261 (suc (suc n)) n0 (suc (suc m)) m0 zero zero 1<n 1<m gi g1 = {!!}
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120 gcd261 (suc (suc n)) n0 (suc (suc m)) m0 zero (suc i0) 1<n 1<m gi g1 = {!!}
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121 -- gi : gcd1 (suc n) n0 (suc i) i0 ≡ m0
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122 -- g1 : gcd1 (suc n) n0 zero m0 ≡ 1
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123 gcd261 (suc n) n0 zero m0 (suc i) i0 1<n 1<m gi g1 = {!!}
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124 gcd261 (suc zero) n0 (suc m) m0 (suc i) i0 1<n 1<m gi g1 = ⊥-elim ( nat-<≡ 1<n )
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125 gcd261 (suc (suc zero)) n0 (suc m) m0 (suc zero) i0 1<n 1<m gi g1 = ⊥-elim ( nat-≡< gi 1<m )
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126 -- gi : gcd1 0 n0 i i0 ≡ m0
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127 gcd261 (suc (suc zero)) n0 (suc zero) m0 (suc (suc i)) i0 1<n 1<m gi refl = {!!}
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128 -- gi : gcd1 0 n0 i i0 ≡ m0
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129 -- g1 : gcd1 0 n0 m m0 ≡ 1
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130 gcd261 (suc (suc zero)) n0 (suc (suc m)) m0 (suc (suc i)) i0 1<n 1<m gi g1 = {!!}
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131 gcd261 (suc (suc (suc n))) n0 (suc m) m0 (suc i) i0 1<n 1<m gi g1 =
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132 gcd261 (suc (suc n)) n0 m m0 i i0 (s≤s (s≤s z≤n)) 1<m gi g1
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133
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134 gcdsym2 : (i j : ℕ) → gcd1 zero i zero j ≡ gcd1 zero j zero i
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135 gcdsym2 i j with <-cmp i j | <-cmp j i
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136 ... | tri< a ¬b ¬c | tri< a₁ ¬b₁ ¬c₁ = ⊥-elim (nat-<> a a₁)
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137 ... | tri< a ¬b ¬c | tri≈ ¬a b ¬c₁ = ⊥-elim (nat-≡< (sym b) a)
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138 ... | tri< a ¬b ¬c | tri> ¬a ¬b₁ c = refl
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139 ... | tri≈ ¬a b ¬c | tri< a ¬b ¬c₁ = ⊥-elim (nat-≡< (sym b) a)
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140 ... | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = refl
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141 ... | tri≈ ¬a b ¬c | tri> ¬a₁ ¬b c = ⊥-elim (nat-≡< b c)
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142 ... | tri> ¬a ¬b c | tri< a ¬b₁ ¬c = refl
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143 ... | tri> ¬a ¬b c | tri≈ ¬a₁ b ¬c = ⊥-elim (nat-≡< b c)
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144 ... | tri> ¬a ¬b c | tri> ¬a₁ ¬b₁ c₁ = ⊥-elim (nat-<> c c₁)
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145 gcdsym1 : ( i i0 j j0 : ℕ ) → gcd1 i i0 j j0 ≡ gcd1 j j0 i i0
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146 gcdsym1 zero zero zero zero = refl
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147 gcdsym1 zero zero zero (suc j0) = refl
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148 gcdsym1 zero (suc i0) zero zero = refl
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149 gcdsym1 zero (suc i0) zero (suc j0) = gcdsym2 (suc i0) (suc j0)
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150 gcdsym1 zero zero (suc zero) j0 = refl
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151 gcdsym1 zero zero (suc (suc j)) j0 = refl
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152 gcdsym1 zero (suc i0) (suc zero) j0 = refl
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153 gcdsym1 zero (suc i0) (suc (suc j)) j0 = gcdsym1 i0 (suc i0) (suc j) (suc (suc j))
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154 gcdsym1 (suc zero) i0 zero j0 = refl
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155 gcdsym1 (suc (suc i)) i0 zero zero = refl
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156 gcdsym1 (suc (suc i)) i0 zero (suc j0) = gcdsym1 (suc i) (suc (suc i))j0 (suc j0)
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157 gcdsym1 (suc i) i0 (suc j) j0 = subst₂ (λ j k → j ≡ k ) (sym (gcd22 i _ _ _)) (sym (gcd22 j _ _ _)) (gcdsym1 i i0 j j0 )
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158
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159 gcdsym : { n m : ℕ} → gcd n m ≡ gcd m n
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160 gcdsym {n} {m} = gcdsym1 n n m m
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161
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162 gcd11 : ( i : ℕ ) → gcd i i ≡ i
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163 gcd11 i = gcdmm i i
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164
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165 gcd2 : ( i j : ℕ ) → gcd (i + j) j ≡ gcd i j
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166 gcd2 zero zero = refl
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167 gcd2 zero (suc j) = begin
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168 gcd (zero + suc j) (suc j) ≡⟨⟩
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169 gcd (suc j) (suc j) ≡⟨ gcd11 (suc j) ⟩
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170 suc j ≡⟨ sym (gcd20 (suc j)) ⟩
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171 gcd (suc j) zero ≡⟨ gcdsym {suc j} ⟩
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172 gcd zero (suc j) ∎ where open ≡-Reasoning
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173 gcd2 (suc i) zero = begin
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174 gcd (suc i + zero) zero ≡⟨ cong (λ k → gcd k zero) (+-comm (suc i) _ ) ⟩
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175 gcd (suc i) zero ∎ where open ≡-Reasoning
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176 gcd2 (suc i) (suc j) = gcd200 (suc i) (suc i) (suc j) (suc j) where
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177 gcd200 : (i i0 j j0 : ℕ) → gcd1 (i + j0) (i0 + j0) j j0 ≡ gcd1 i i0 j j0
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178 gcd200 i i0 j j0 = {!!}
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179
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180 gcd24 : { n m k : ℕ} → n > 1 → m > 1 → k > 1 → gcd n k ≡ k → gcd m k ≡ k → ¬ ( gcd n m ≡ 1 )
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181 gcd24 {n} {m} {k} 1<n 1<m 1<k gn gm gnm = gcd21 n n m m k k 1<n 1<m 1<k gn gm gnm where
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182 gcd23 : {i0 j0 : ℕ } → 1 < i0 → 1 < j0 → 1 < gcd1 zero i0 zero j0
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183 gcd23 {i0} {j0} 1<i 1<j with <-cmp i0 j0
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184 ... | tri< a ¬b ¬c = 1<j
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185 ... | tri≈ ¬a refl ¬c = 1<i
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186 ... | tri> ¬a ¬b c = 1<i
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187 1<ss : {j : ℕ} → 1 < suc (suc j)
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188 1<ss = s≤s (s≤s z≤n)
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142
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189 gcd21 : ( i i0 j j0 o o0 : ℕ ) → 1 < i0 → 1 < j0 → 1 < o0 → gcd1 i i0 o o0 ≡ k → gcd1 j j0 o o0 ≡ k → gcd1 i i0 j j0 ≡ 1 → ⊥
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190 gcd21 zero i0 zero j0 o o0 1<i 1<j 1<o refl gm gnm = nat-≡< (sym gnm) (gcd23 1<i 1<j)
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191 gcd21 zero i0 (suc j) j0 zero o0 1<i 1<j 1<o refl gm gnm = gcd25 i0 o0 j j0 1<o 1<i 1<k gm (subst (λ k → k ≡ 1) (gcdsym1 zero _ (suc j) _) gnm) where
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148
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192 -- gcd1 (suc j) (suc (suc j)) (suc o0) (suc (suc o0)) ≡ suc (suc i0) , gcd1 (suc j) (suc (suc j)) (suc i0) (suc (suc i0)) ≡ 1
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193 gcd25 : (i0 o0 j j0 : ℕ) → 1 < o0 → 1 < i0
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194 → 1 < gcd1 zero i0 zero o0
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195 → ( gm : gcd1 (suc j) j0 zero o0 ≡ gcd1 zero i0 zero o0 ) → (gnm : gcd1 (suc j) j0 zero i0 ≡ 1) → ⊥
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196 gcd25 i0 o0 zero j0 1<o 1<i 1<k gm refl with <-cmp i0 o0
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197 ... | tri< a ¬b ¬c = ⊥-elim ( nat-≡< gm 1<o )
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198 ... | tri≈ ¬a refl ¬c = ⊥-elim ( nat-≡< gm 1<o )
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199 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≡< gm 1<i )
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200 -- gm : gcd1 (suc j) (suc (suc j)) (suc o0) (suc (suc o0)) ≡ (gcd1 zero (suc i0) zero (suc (suc o0))
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201 -- gcd1 j (suc (suc j)) o0 (suc (suc o0))
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202 -- gnm : gcd1 (suc j) (suc (suc j)) i0 (suc i0) ≡ 1
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203 gcd25 i0 (suc zero) (suc j) j0 1<o 1<i 1<k gm gnm = {!!} -- ⊥-elim ( nat-≤> (subst (λ k → k ≤ 1 ) gm (gcd27 (suc j) (suc (suc j)))) 1<k )
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204 gcd25 (suc zero) (suc (suc o0)) (suc j) j0 1<o 1<i 1<k gm gnm = {!!}
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148
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205 -- (suc (suc i0)) > (suc (suc o0)) → gm = gnm → (suc (suc i0)) ≡ 1
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206 -- (suc (suc i0)) < (suc (suc o0)) → ? gcd1 (suc j) (suc (suc j)) (suc o0) (suc (suc o0)) ≡ (suc (suc o0))
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207 -- gcd1 (suc j) (suc (suc j)) (suc i0) (suc (suc i0)) ≡ 1
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208 gcd25 (suc (suc i0)) (suc (suc o0)) (suc j) j0 1<o 1<i 1<k gm gnm with <-cmp (suc (suc i0)) (suc (suc o0))
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209 ... | tri< a ¬b ¬c = {!!}
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210 ... | tri≈ ¬a b ¬c = {!!}
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211 ... | tri> ¬a ¬b c = gcd26 {!!} {!!} {!!} {!!} {!!} {!!} {!!}
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146
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212 gcd21 zero i0 (suc j) j0 (suc zero) o0 1<i 1<j 1<o refl gm gnm = nat-<≡ 1<k
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213 gcd21 zero (suc i0) (suc j) j0 (suc (suc o)) o0 1<i 1<j 1<o gn gm gnm =
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147
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214 gcd21 i0 {!!} (suc j) j0 (suc o) (suc (suc o)) 1<i 1<j {!!} gn {!!} {!!}
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145
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215 gcd21 (suc i) i0 zero j0 o o0 1<i 1<j 1<o gn gm gnm = {!!}
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142
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216 gcd21 (suc i) i0 (suc j) j0 zero o0 1<i 1<j 1<o gn gm gnm = {!!}
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217 gcd21 (suc i) i0 (suc j) j0 (suc o) o0 1<i 1<j 1<o gn gm gnm =
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218 gcd21 i i0 j j0 o o0 1<i 1<j 1<o (subst (λ m → m ≡ k) (gcd22 i i0 _ _ ) gn)
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219 (subst (λ m → m ≡ k) (gcd22 j j0 _ _ ) gm) (subst (λ k → k ≡ 1) (gcd22 i i0 _ _ ) gnm)
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141
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220
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221 record Even (i : ℕ) : Set where
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222 field
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223 j : ℕ
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224 is-twice : i ≡ 2 * j
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225
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226 e2 : (i : ℕ) → even i → Even i
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227 e2 zero en = record { j = 0 ; is-twice = refl }
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228 e2 (suc (suc i)) en = record { j = suc (Even.j (e2 i en )) ; is-twice = e21 } where
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229 e21 : suc (suc i) ≡ 2 * suc (Even.j (e2 i en))
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230 e21 = begin
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231 suc (suc i) ≡⟨ cong (λ k → suc (suc k)) (Even.is-twice (e2 i en)) ⟩
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232 suc (suc (2 * Even.j (e2 i en))) ≡⟨ sym (*-distribˡ-+ 2 1 _) ⟩
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233 2 * suc (Even.j (e2 i en)) ∎ where open ≡-Reasoning
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234
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235 record Odd (i : ℕ) : Set where
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236 field
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237 j : ℕ
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238 is-twice : i ≡ suc (2 * j )
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239
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240 odd2 : (i : ℕ) → ¬ even i → even (suc i)
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241 odd2 zero ne = ⊥-elim ( ne tt )
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242 odd2 (suc zero) ne = tt
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243 odd2 (suc (suc i)) ne = odd2 i ne
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244
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245 odd3 : (i : ℕ) → ¬ even i → Odd i
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246 odd3 zero ne = ⊥-elim ( ne tt )
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247 odd3 (suc zero) ne = record { j = 0 ; is-twice = refl }
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248 odd3 (suc (suc i)) ne = record { j = Even.j (e2 (suc i) (odd2 i ne)) ; is-twice = odd31 } where
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249 odd31 : suc (suc i) ≡ suc (2 * Even.j (e2 (suc i) (odd2 i ne)))
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250 odd31 = begin
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251 suc (suc i) ≡⟨ cong suc (Even.is-twice (e2 (suc i) (odd2 i ne))) ⟩
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252 suc (2 * (Even.j (e2 (suc i) (odd2 i ne)))) ∎ where open ≡-Reasoning
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253
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254 odd4 : (i : ℕ) → even i → ¬ even ( suc i )
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255 odd4 (suc (suc i)) en en1 = odd4 i en en1
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256
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257 even^2 : {n : ℕ} → even ( n * n ) → even n
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258 even^2 {n} en with even? n
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259 ... | yes y = y
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260 ... | no ne = ⊥-elim ( odd4 ((2 * m) + 2 * m * suc (2 * m)) (n+even {2 * m} {2 * m * suc (2 * m)} ee3 ee4) (subst (λ k → even k) ee2 en )) where
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261 m : ℕ
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262 m = Odd.j ( odd3 n ne )
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263 ee3 : even (2 * m)
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264 ee3 = subst (λ k → even k ) (*-comm m 2) (n*even {m} {2} tt )
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265 ee4 : even ((2 * m) * suc (2 * m))
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266 ee4 = even*n {(2 * m)} {suc (2 * m)} (even*n {2} {m} tt )
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267 ee2 : n * n ≡ suc (2 * m) + ((2 * m) * (suc (2 * m) ))
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268 ee2 = begin n * n ≡⟨ cong ( λ k → k * k) (Odd.is-twice (odd3 n ne)) ⟩
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269 suc (2 * m) * suc (2 * m) ≡⟨ *-distribʳ-+ (suc (2 * m)) 1 ((2 * m) ) ⟩
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270 (1 * suc (2 * m)) + 2 * m * suc (2 * m) ≡⟨ cong (λ k → k + 2 * m * suc (2 * m)) (begin
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271 suc m + 1 * m + 0 * (suc m + 1 * m ) ≡⟨ +-comm (suc m + 1 * m) 0 ⟩
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272 suc m + 1 * m ≡⟨⟩
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273 suc (2 * m)
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274 ∎) ⟩ suc (2 * m) + 2 * m * suc (2 * m) ∎ where open ≡-Reasoning
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275
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276 open import nat
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277
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278 e3 : {i j : ℕ } → 2 * i ≡ 2 * j → i ≡ j
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279 e3 {zero} {zero} refl = refl
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280 e3 {suc x} {suc y} eq with <-cmp x y
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281 ... | tri< a ¬b ¬c = ⊥-elim ( nat-≡< eq (s≤s (<-trans (<-plus a) (<-plus-0 (s≤s (<-plus a ))))))
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282 ... | tri≈ ¬a b ¬c = cong suc b
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283 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≡< (sym eq) (s≤s (<-trans (<-plus c) (<-plus-0 (s≤s (<-plus c ))))))
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284
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