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1 module root2 where
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2
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141
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3 open import Data.Nat
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4 open import Data.Nat.Properties
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5 open import Data.Empty
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141
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6 open import Data.Unit using (⊤ ; tt)
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7 open import Relation.Nullary
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8 open import Relation.Binary.PropositionalEquality
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141
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9 open import Relation.Binary.Definitions
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10
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11 even : (n : ℕ ) → Set
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141
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12 even zero = ⊤
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13 even (suc zero) = ⊥
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14 even (suc (suc n)) = even n
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15
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16 even? : (n : ℕ ) → Dec ( even n )
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17 even? zero = yes tt
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18 even? (suc zero) = no (λ ())
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19 even? (suc (suc n)) = even? n
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20
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21 n+even : {n m : ℕ } → even n → even m → even ( n + m )
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22 n+even {zero} {zero} tt tt = tt
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23 n+even {zero} {suc m} tt em = em
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24 n+even {suc (suc n)} {m} en em = n+even {n} {m} en em
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25
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26 n*even : {m n : ℕ } → even n → even ( m * n )
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27 n*even {zero} {n} en = tt
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28 n*even {suc m} {n} en = n+even {n} {m * n} en (n*even {m} {n} en)
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29
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30 even*n : {n m : ℕ } → even n → even ( n * m )
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31 even*n {n} {m} en = subst even (*-comm m n) (n*even {m} {n} en)
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32
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33 gcd1 : ( i i0 j j0 : ℕ ) → ℕ
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34 gcd1 zero i0 zero j0 = i0
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35 gcd1 zero i0 (suc zero) j0 = 1
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36 gcd1 zero zero (suc (suc j)) j0 = j0
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37 gcd1 zero (suc i0) (suc (suc j)) j0 = gcd1 i0 (suc i0) (suc j) (suc (suc j))
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38 gcd1 (suc zero) i0 zero j0 = 1
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39 gcd1 (suc (suc i)) i0 zero zero = i0
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40 gcd1 (suc (suc i)) i0 zero (suc j0) = gcd1 (suc i) (suc (suc i)) j0 (suc j0)
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41 gcd1 (suc i) i0 (suc j) j0 = gcd1 i i0 j j0
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42
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43 gcd : ( i j : ℕ ) → ℕ
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44 gcd i j = gcd1 i i j j
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45
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46 even→gcd=2 : {n : ℕ} → even n → n > 0 → gcd n 2 ≡ 2
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47 even→gcd=2 {suc (suc zero)} en (s≤s z≤n) = refl
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48 even→gcd=2 {suc (suc (suc (suc n)))} en (s≤s z≤n) = begin
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49 gcd (suc (suc (suc (suc n)))) 2
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50 ≡⟨⟩
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51 gcd (suc (suc n)) 2
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52 ≡⟨ even→gcd=2 {suc (suc n)} en (s≤s z≤n) ⟩
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53 2
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54 ∎ where open ≡-Reasoning
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55
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56 open import nat
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57
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145
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58 -- gcd26 : { n m : ℕ} → n > 1 → m > 1 → n - m > 0 → gcd n m ≡ gcd (n - m) m
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59 -- gcd25 : { n m : ℕ} → n > 1 → m > 1 → n - m > 0 → gcd n m ≡ gcd m n
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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 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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63 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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64 gcd22 : ( i i0 o o0 : ℕ ) → gcd1 (suc i) i0 (suc o) o0 ≡ gcd1 i i0 o o0
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65 gcd22 zero i0 zero o0 = refl
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66 gcd22 zero i0 (suc o) o0 = refl
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67 gcd22 (suc i) i0 zero o0 = refl
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68 gcd22 (suc i) i0 (suc o) o0 = refl
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69 gcd23 : { j j0 : ℕ } → 1 < j → 1 < j0 → ¬ (gcd1 zero zero j j0 ≡ 1)
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70 gcd23 {zero} {suc j0} 1<j 1<j0 ()
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71 gcd23 {suc zero} {suc j0} 1<j 1<j0 gn = nat-<≡ 1<j
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72 gcd23 {suc (suc j)} {suc .0} 1<j 1<j0 refl = nat-<≡ 1<j0
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73 1<ss : {j : ℕ} → 1 < suc (suc j)
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74 1<ss = s≤s (s≤s z≤n)
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75 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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76 gcd21 zero i0 zero j0 o o0 1<i 1<j 1<o refl gm gnm = nat-≡< (sym gnm) 1<i
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77 --- gcd1 zero i0 o o0 ≡ k o = suc zero → k ≡ 1
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78 --- i0 = zero , o = suc (suc o) → o0 = k -> gcd1 zero zero (suc j) j0 ≡ 1
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79 -- i0 = suc i0, o = suc (suc o) → gn = gcd1 i0 (suc i0) (suc o) (suc (suc o))
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80 gcd21 zero i0 (suc j) j0 o o0 1<i 1<j 1<o gn gm gnm = {!!}
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81 gcd21 (suc i) i0 zero j0 o o0 1<i 1<j 1<o gn gm gnm = {!!}
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82 -- ?
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83 gcd21 (suc i) i0 (suc j) j0 zero o0 1<i 1<j 1<o gn gm gnm = {!!}
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84 gcd21 (suc i) i0 (suc j) j0 (suc o) o0 1<i 1<j 1<o gn gm gnm =
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85 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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86 (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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87
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88 record Even (i : ℕ) : Set where
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89 field
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90 j : ℕ
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91 is-twice : i ≡ 2 * j
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92
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93 e2 : (i : ℕ) → even i → Even i
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94 e2 zero en = record { j = 0 ; is-twice = refl }
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95 e2 (suc (suc i)) en = record { j = suc (Even.j (e2 i en )) ; is-twice = e21 } where
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96 e21 : suc (suc i) ≡ 2 * suc (Even.j (e2 i en))
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97 e21 = begin
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98 suc (suc i) ≡⟨ cong (λ k → suc (suc k)) (Even.is-twice (e2 i en)) ⟩
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99 suc (suc (2 * Even.j (e2 i en))) ≡⟨ sym (*-distribˡ-+ 2 1 _) ⟩
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100 2 * suc (Even.j (e2 i en)) ∎ where open ≡-Reasoning
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101
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102 record Odd (i : ℕ) : Set where
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103 field
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104 j : ℕ
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105 is-twice : i ≡ suc (2 * j )
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106
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107 odd2 : (i : ℕ) → ¬ even i → even (suc i)
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108 odd2 zero ne = ⊥-elim ( ne tt )
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109 odd2 (suc zero) ne = tt
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110 odd2 (suc (suc i)) ne = odd2 i ne
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111
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112 odd3 : (i : ℕ) → ¬ even i → Odd i
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113 odd3 zero ne = ⊥-elim ( ne tt )
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114 odd3 (suc zero) ne = record { j = 0 ; is-twice = refl }
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115 odd3 (suc (suc i)) ne = record { j = Even.j (e2 (suc i) (odd2 i ne)) ; is-twice = odd31 } where
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116 odd31 : suc (suc i) ≡ suc (2 * Even.j (e2 (suc i) (odd2 i ne)))
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117 odd31 = begin
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118 suc (suc i) ≡⟨ cong suc (Even.is-twice (e2 (suc i) (odd2 i ne))) ⟩
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119 suc (2 * (Even.j (e2 (suc i) (odd2 i ne)))) ∎ where open ≡-Reasoning
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120
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121 odd4 : (i : ℕ) → even i → ¬ even ( suc i )
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122 odd4 (suc (suc i)) en en1 = odd4 i en en1
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123
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124 even^2 : {n : ℕ} → even ( n * n ) → even n
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125 even^2 {n} en with even? n
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126 ... | yes y = y
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127 ... | 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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128 m : ℕ
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129 m = Odd.j ( odd3 n ne )
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130 ee3 : even (2 * m)
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131 ee3 = subst (λ k → even k ) (*-comm m 2) (n*even {m} {2} tt )
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132 ee4 : even ((2 * m) * suc (2 * m))
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133 ee4 = even*n {(2 * m)} {suc (2 * m)} (even*n {2} {m} tt )
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134 ee2 : n * n ≡ suc (2 * m) + ((2 * m) * (suc (2 * m) ))
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135 ee2 = begin n * n ≡⟨ cong ( λ k → k * k) (Odd.is-twice (odd3 n ne)) ⟩
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136 suc (2 * m) * suc (2 * m) ≡⟨ *-distribʳ-+ (suc (2 * m)) 1 ((2 * m) ) ⟩
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137 (1 * suc (2 * m)) + 2 * m * suc (2 * m) ≡⟨ cong (λ k → k + 2 * m * suc (2 * m)) (begin
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138 suc m + 1 * m + 0 * (suc m + 1 * m ) ≡⟨ +-comm (suc m + 1 * m) 0 ⟩
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139 suc m + 1 * m ≡⟨⟩
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140 suc (2 * m)
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141 ∎) ⟩ suc (2 * m) + 2 * m * suc (2 * m) ∎ where open ≡-Reasoning
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142
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143 open import nat
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144
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145 e3 : {i j : ℕ } → 2 * i ≡ 2 * j → i ≡ j
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146 e3 {zero} {zero} refl = refl
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147 e3 {suc x} {suc y} eq with <-cmp x y
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148 ... | tri< a ¬b ¬c = ⊥-elim ( nat-≡< eq (s≤s (<-trans (<-plus a) (<-plus-0 (s≤s (<-plus a ))))))
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149 ... | tri≈ ¬a b ¬c = cong suc b
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150 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≡< (sym eq) (s≤s (<-trans (<-plus c) (<-plus-0 (s≤s (<-plus c ))))))
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151
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152 record Rational : Set where
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153 field
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154 i j : ℕ
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155 coprime : gcd i j ≡ 1
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156
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157 root2-irrational : ( n m : ℕ ) → n > 1 → m > 1 → 2 * n * n ≡ m * m → ¬ (gcd n m ≡ 1)
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158 root2-irrational n m n>1 m>1 2nm = gcd24 {n} {m} n>1 m>1 (s≤s (s≤s z≤n)) (even→gcd=2 rot7 (rot5 n>1)) (even→gcd=2 ( even^2 {m} ( rot1)) (rot5 m>1))where
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159 rot5 : {n : ℕ} → n > 1 → n > 0
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160 rot5 {n} lt = <-trans a<sa lt
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161 rot1 : even ( m * m )
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162 rot1 = subst (λ k → even k ) rot4 (n*even {n * n} {2} tt ) where
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163 rot4 : (n * n) * 2 ≡ m * m
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164 rot4 = begin
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165 (n * n) * 2 ≡⟨ *-comm (n * n) 2 ⟩
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166 2 * ( n * n ) ≡⟨ sym (*-assoc 2 n n) ⟩
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167 2 * n * n ≡⟨ 2nm ⟩
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168 m * m ∎ where open ≡-Reasoning
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169 E : Even m
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170 E = e2 m ( even^2 {m} ( rot1 ))
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171 rot2 : 2 * n * n ≡ 2 * Even.j E * m
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172 rot2 = subst (λ k → 2 * n * n ≡ k * m ) (Even.is-twice E) 2nm
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173 rot3 : n * n ≡ Even.j E * m
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174 rot3 = e3 ( begin
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175 2 * (n * n) ≡⟨ sym (*-assoc 2 n _) ⟩
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176 2 * n * n ≡⟨ rot2 ⟩
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177 2 * Even.j E * m ≡⟨ *-assoc 2 (Even.j E) m ⟩
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178 2 * (Even.j E * m) ∎ ) where open ≡-Reasoning
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179 rot7 : even n
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180 rot7 = even^2 {n} (subst (λ k → even k) (sym rot3) ((n*even {Even.j E} {m} ( even^2 {m} ( rot1 )))))
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