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1 {-# OPTIONS --allow-unsolved-metas #-}
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2 module nat where
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3
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4 open import Data.Nat
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5 open import Data.Nat.Properties
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6 open import Data.Empty
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7 open import Relation.Nullary
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8 open import Relation.Binary.PropositionalEquality
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9 open import Relation.Binary.Core
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10 open import Relation.Binary.Definitions
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11 open import logic
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12 open import Level hiding ( zero ; suc )
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13
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14 nat-<> : { x y : ℕ } → x < y → y < x → ⊥
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15 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x
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16
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17 nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥
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18 nat-≤> (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x
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19
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20 nat-<≡ : { x : ℕ } → x < x → ⊥
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21 nat-<≡ (s≤s lt) = nat-<≡ lt
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22
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23 nat-≡< : { x y : ℕ } → x ≡ y → x < y → ⊥
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24 nat-≡< refl lt = nat-<≡ lt
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25
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26 ¬a≤a : {la : ℕ} → suc la ≤ la → ⊥
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27 ¬a≤a (s≤s lt) = ¬a≤a lt
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28
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29 a<sa : {la : ℕ} → la < suc la
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30 a<sa {zero} = s≤s z≤n
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31 a<sa {suc la} = s≤s a<sa
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32
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33 =→¬< : {x : ℕ } → ¬ ( x < x )
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34 =→¬< {zero} ()
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35 =→¬< {suc x} (s≤s lt) = =→¬< lt
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36
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37 >→¬< : {x y : ℕ } → (x < y ) → ¬ ( y < x )
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38 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x
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39
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40 <-∨ : { x y : ℕ } → x < suc y → ( (x ≡ y ) ∨ (x < y) )
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41 <-∨ {zero} {zero} (s≤s z≤n) = case1 refl
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42 <-∨ {zero} {suc y} (s≤s lt) = case2 (s≤s z≤n)
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43 <-∨ {suc x} {zero} (s≤s ())
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44 <-∨ {suc x} {suc y} (s≤s lt) with <-∨ {x} {y} lt
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45 <-∨ {suc x} {suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → suc k ) eq)
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46 <-∨ {suc x} {suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1)
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47
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48 ≤-∨ : { x y : ℕ } → x ≤ y → ( (x ≡ y ) ∨ (x < y) )
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49 ≤-∨ {zero} {zero} z≤n = case1 refl
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50 ≤-∨ {zero} {suc y} z≤n = case2 (s≤s z≤n)
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51 ≤-∨ {suc x} {zero} ()
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52 ≤-∨ {suc x} {suc y} (s≤s lt) with ≤-∨ {x} {y} lt
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53 ≤-∨ {suc x} {suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → suc k ) eq)
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54 ≤-∨ {suc x} {suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1)
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55
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56 max : (x y : ℕ) → ℕ
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57 max zero zero = zero
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58 max zero (suc x) = (suc x)
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59 max (suc x) zero = (suc x)
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60 max (suc x) (suc y) = suc ( max x y )
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61
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62 x≤max : (x y : ℕ) → x ≤ max x y
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63 x≤max zero zero = ≤-refl
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64 x≤max zero (suc x) = z≤n
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65 x≤max (suc x) zero = ≤-refl
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66 x≤max (suc x) (suc y) = s≤s( x≤max x y )
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67
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68 y≤max : (x y : ℕ) → y ≤ max x y
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69 y≤max zero zero = ≤-refl
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70 y≤max zero (suc x) = ≤-refl
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71 y≤max (suc x) zero = z≤n
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72 y≤max (suc x) (suc y) = s≤s( y≤max x y )
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73
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74 -- _*_ : ℕ → ℕ → ℕ
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75 -- _*_ zero _ = zero
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76 -- _*_ (suc n) m = m + ( n * m )
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77
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78 -- x ^ y
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79 exp : ℕ → ℕ → ℕ
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80 exp _ zero = 1
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81 exp n (suc m) = n * ( exp n m )
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82
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83 div2 : ℕ → (ℕ ∧ Bool )
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84 div2 zero = ⟪ 0 , false ⟫
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85 div2 (suc zero) = ⟪ 0 , true ⟫
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86 div2 (suc (suc n)) = ⟪ suc (proj1 (div2 n)) , proj2 (div2 n) ⟫ where
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87 open _∧_
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88
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89 div2-rev : (ℕ ∧ Bool ) → ℕ
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90 div2-rev ⟪ x , true ⟫ = suc (x + x)
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91 div2-rev ⟪ x , false ⟫ = x + x
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92
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93 div2-eq : (x : ℕ ) → div2-rev ( div2 x ) ≡ x
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94 div2-eq zero = refl
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95 div2-eq (suc zero) = refl
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96 div2-eq (suc (suc x)) with div2 x | inspect div2 x
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97 ... | ⟪ x1 , true ⟫ | record { eq = eq1 } = begin -- eq1 : div2 x ≡ ⟪ x1 , true ⟫
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98 div2-rev ⟪ suc x1 , true ⟫ ≡⟨⟩
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99 suc (suc (x1 + suc x1)) ≡⟨ cong (λ k → suc (suc k )) (+-comm x1 _ ) ⟩
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100 suc (suc (suc (x1 + x1))) ≡⟨⟩
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101 suc (suc (div2-rev ⟪ x1 , true ⟫)) ≡⟨ cong (λ k → suc (suc (div2-rev k ))) (sym eq1) ⟩
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102 suc (suc (div2-rev (div2 x))) ≡⟨ cong (λ k → suc (suc k)) (div2-eq x) ⟩
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103 suc (suc x) ∎ where open ≡-Reasoning
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104 ... | ⟪ x1 , false ⟫ | record { eq = eq1 } = begin
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105 div2-rev ⟪ suc x1 , false ⟫ ≡⟨⟩
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106 suc (x1 + suc x1) ≡⟨ cong (λ k → (suc k )) (+-comm x1 _ ) ⟩
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107 suc (suc (x1 + x1)) ≡⟨⟩
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108 suc (suc (div2-rev ⟪ x1 , false ⟫)) ≡⟨ cong (λ k → suc (suc (div2-rev k ))) (sym eq1) ⟩
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109 suc (suc (div2-rev (div2 x))) ≡⟨ cong (λ k → suc (suc k)) (div2-eq x) ⟩
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110 suc (suc x) ∎ where open ≡-Reasoning
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111
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112 sucprd : {i : ℕ } → 0 < i → suc (pred i) ≡ i
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113 sucprd {suc i} 0<i = refl
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114
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115 0<s : {x : ℕ } → zero < suc x
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116 0<s {_} = s≤s z≤n
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117
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118 px<py : {x y : ℕ } → pred x < pred y → x < y
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119 px<py {zero} {suc y} lt = 0<s
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120 px<py {suc zero} {suc (suc y)} (s≤s lt) = s≤s 0<s
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121 px<py {suc (suc x)} {suc (suc y)} (s≤s lt) = s≤s (px<py {suc x} {suc y} lt)
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122
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123 minus : (a b : ℕ ) → ℕ
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124 minus a zero = a
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125 minus zero (suc b) = zero
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126 minus (suc a) (suc b) = minus a b
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127
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128 _-_ = minus
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129
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130 m+= : {i j m : ℕ } → m + i ≡ m + j → i ≡ j
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131 m+= {i} {j} {zero} refl = refl
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132 m+= {i} {j} {suc m} eq = m+= {i} {j} {m} ( cong (λ k → pred k ) eq )
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133
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134 +m= : {i j m : ℕ } → i + m ≡ j + m → i ≡ j
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135 +m= {i} {j} {m} eq = m+= ( subst₂ (λ j k → j ≡ k ) (+-comm i _ ) (+-comm j _ ) eq )
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136
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137 less-1 : { n m : ℕ } → suc n < m → n < m
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138 less-1 {zero} {suc (suc _)} (s≤s (s≤s z≤n)) = s≤s z≤n
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139 less-1 {suc n} {suc m} (s≤s lt) = s≤s (less-1 {n} {m} lt)
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140
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141 sa=b→a<b : { n m : ℕ } → suc n ≡ m → n < m
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142 sa=b→a<b {0} {suc zero} refl = s≤s z≤n
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143 sa=b→a<b {suc n} {suc (suc n)} refl = s≤s (sa=b→a<b refl)
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144
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145 minus+n : {x y : ℕ } → suc x > y → minus x y + y ≡ x
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146 minus+n {x} {zero} _ = trans (sym (+-comm zero _ )) refl
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147 minus+n {zero} {suc y} (s≤s ())
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148 minus+n {suc x} {suc y} (s≤s lt) = begin
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149 minus (suc x) (suc y) + suc y
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150 ≡⟨ +-comm _ (suc y) ⟩
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151 suc y + minus x y
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152 ≡⟨ cong ( λ k → suc k ) (
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153 begin
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154 y + minus x y
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155 ≡⟨ +-comm y _ ⟩
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156 minus x y + y
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157 ≡⟨ minus+n {x} {y} lt ⟩
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158 x
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159 ∎
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160 ) ⟩
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161 suc x
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162 ∎ where open ≡-Reasoning
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163
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164 <-minus-0 : {x y z : ℕ } → z + x < z + y → x < y
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165 <-minus-0 {x} {suc _} {zero} lt = lt
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166 <-minus-0 {x} {y} {suc z} (s≤s lt) = <-minus-0 {x} {y} {z} lt
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167
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168 <-minus : {x y z : ℕ } → x + z < y + z → x < y
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169 <-minus {x} {y} {z} lt = <-minus-0 ( subst₂ ( λ j k → j < k ) (+-comm x _) (+-comm y _ ) lt )
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170
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171 x≤x+y : {z y : ℕ } → z ≤ z + y
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172 x≤x+y {zero} {y} = z≤n
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173 x≤x+y {suc z} {y} = s≤s (x≤x+y {z} {y})
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174
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175 x≤y+x : {z y : ℕ } → z ≤ y + z
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176 x≤y+x {z} {y} = subst (λ k → z ≤ k ) (+-comm _ y ) x≤x+y
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177
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178 x≤x+sy : {x y : ℕ} → x < x + suc y
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179 x≤x+sy {x} {y} = begin
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180 suc x ≤⟨ x≤x+y ⟩
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181 suc x + y ≡⟨ cong (λ k → k + y) (+-comm 1 x ) ⟩
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182 (x + 1) + y ≡⟨ (+-assoc x 1 _) ⟩
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183 x + suc y ∎ where open ≤-Reasoning
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184
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185 <-plus : {x y z : ℕ } → x < y → x + z < y + z
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186 <-plus {zero} {suc y} {z} (s≤s z≤n) = s≤s (subst (λ k → z ≤ k ) (+-comm z _ ) x≤x+y )
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187 <-plus {suc x} {suc y} {z} (s≤s lt) = s≤s (<-plus {x} {y} {z} lt)
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188
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189 <-plus-0 : {x y z : ℕ } → x < y → z + x < z + y
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190 <-plus-0 {x} {y} {z} lt = subst₂ (λ j k → j < k ) (+-comm _ z) (+-comm _ z) ( <-plus {x} {y} {z} lt )
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191
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192 ≤-plus : {x y z : ℕ } → x ≤ y → x + z ≤ y + z
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193 ≤-plus {0} {y} {zero} z≤n = z≤n
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194 ≤-plus {0} {y} {suc z} z≤n = subst (λ k → z < k ) (+-comm _ y ) x≤x+y
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195 ≤-plus {suc x} {suc y} {z} (s≤s lt) = s≤s ( ≤-plus {x} {y} {z} lt )
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196
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197 ≤-plus-0 : {x y z : ℕ } → x ≤ y → z + x ≤ z + y
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198 ≤-plus-0 {x} {y} {zero} lt = lt
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199 ≤-plus-0 {x} {y} {suc z} lt = s≤s ( ≤-plus-0 {x} {y} {z} lt )
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200
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201 x+y<z→x<z : {x y z : ℕ } → x + y < z → x < z
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202 x+y<z→x<z {zero} {y} {suc z} (s≤s lt1) = s≤s z≤n
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203 x+y<z→x<z {suc x} {y} {suc z} (s≤s lt1) = s≤s ( x+y<z→x<z {x} {y} {z} lt1 )
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204
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205 *≤ : {x y z : ℕ } → x ≤ y → x * z ≤ y * z
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206 *≤ lt = *-mono-≤ lt ≤-refl
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207
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208 *< : {x y z : ℕ } → x < y → x * suc z < y * suc z
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209 *< {zero} {suc y} lt = s≤s z≤n
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210 *< {suc x} {suc y} (s≤s lt) = <-plus-0 (*< lt)
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211
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212 <to<s : {x y : ℕ } → x < y → x < suc y
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213 <to<s {zero} {suc y} (s≤s lt) = s≤s z≤n
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214 <to<s {suc x} {suc y} (s≤s lt) = s≤s (<to<s {x} {y} lt)
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215
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216 <tos<s : {x y : ℕ } → x < y → suc x < suc y
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217 <tos<s {zero} {suc y} (s≤s z≤n) = s≤s (s≤s z≤n)
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218 <tos<s {suc x} {suc y} (s≤s lt) = s≤s (<tos<s {x} {y} lt)
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219
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220 <to≤ : {x y : ℕ } → x < y → x ≤ y
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221 <to≤ {zero} {suc y} (s≤s z≤n) = z≤n
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222 <to≤ {suc x} {suc y} (s≤s lt) = s≤s (<to≤ {x} {y} lt)
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223
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224 <∨≤ : ( x y : ℕ ) → (x < y ) ∨ (y ≤ x)
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225 <∨≤ x y with <-cmp x y
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226 ... | tri< a ¬b ¬c = case1 a
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227 ... | tri≈ ¬a refl ¬c = case2 ≤-refl
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228 ... | tri> ¬a ¬b c = case2 (<to≤ c)
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229
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230 refl-≤s : {x : ℕ } → x ≤ suc x
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231 refl-≤s {zero} = z≤n
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232 refl-≤s {suc x} = s≤s (refl-≤s {x})
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233
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234 refl-≤ : {x : ℕ } → x ≤ x
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235 refl-≤ {zero} = z≤n
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236 refl-≤ {suc x} = s≤s (refl-≤ {x})
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237
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238 x<y→≤ : {x y : ℕ } → x < y → x ≤ suc y
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239 x<y→≤ {zero} {.(suc _)} (s≤s z≤n) = z≤n
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240 x<y→≤ {suc x} {suc y} (s≤s lt) = s≤s (x<y→≤ {x} {y} lt)
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241
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242 ≤→= : {i j : ℕ} → i ≤ j → j ≤ i → i ≡ j
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243 ≤→= {0} {0} z≤n z≤n = refl
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244 ≤→= {suc i} {suc j} (s≤s i<j) (s≤s j<i) = cong suc ( ≤→= {i} {j} i<j j<i )
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245
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246 px≤x : {x : ℕ } → pred x ≤ x
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247 px≤x {zero} = refl-≤
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248 px≤x {suc x} = refl-≤s
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249
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250 px≤py : {x y : ℕ } → x ≤ y → pred x ≤ pred y
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251 px≤py {zero} {zero} lt = refl-≤
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252 px≤py {zero} {suc y} lt = z≤n
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253 px≤py {suc x} {suc y} (s≤s lt) = lt
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254
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255 sx≤y→x≤y : {x y : ℕ } → suc x ≤ y → x ≤ y
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256 sx≤y→x≤y {zero} {suc y} (s≤s le) = z≤n
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257 sx≤y→x≤y {suc x} {suc y} (s≤s le) = s≤s (sx≤y→x≤y {x} {y} le)
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258
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259 x<sy→x≤y : {x y : ℕ } → x < suc y → x ≤ y
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260 x<sy→x≤y {zero} {suc y} (s≤s le) = z≤n
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261 x<sy→x≤y {suc x} {suc y} (s≤s le) = s≤s (x<sy→x≤y {x} {y} le)
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262 x<sy→x≤y {zero} {zero} (s≤s z≤n) = ≤-refl
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263
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264
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265 open import Data.Product
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266
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267 i-j=0→i=j : {i j : ℕ } → j ≤ i → i - j ≡ 0 → i ≡ j
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268 i-j=0→i=j {zero} {zero} _ refl = refl
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269 i-j=0→i=j {zero} {suc j} () refl
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270 i-j=0→i=j {suc i} {zero} z≤n ()
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271 i-j=0→i=j {suc i} {suc j} (s≤s lt) eq = cong suc (i-j=0→i=j {i} {j} lt eq)
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272
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273 m*n=0⇒m=0∨n=0 : {i j : ℕ} → i * j ≡ 0 → (i ≡ 0) ∨ ( j ≡ 0 )
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274 m*n=0⇒m=0∨n=0 {zero} {j} refl = case1 refl
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275 m*n=0⇒m=0∨n=0 {suc i} {zero} eq = case2 refl
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276
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277
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278 minus+1 : {x y : ℕ } → y ≤ x → suc (minus x y) ≡ minus (suc x) y
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279 minus+1 {zero} {zero} y≤x = refl
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280 minus+1 {suc x} {zero} y≤x = refl
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281 minus+1 {suc x} {suc y} (s≤s y≤x) = minus+1 {x} {y} y≤x
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282
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283 minus+yz : {x y z : ℕ } → z ≤ y → x + minus y z ≡ minus (x + y) z
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284 minus+yz {zero} {y} {z} _ = refl
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285 minus+yz {suc x} {y} {z} z≤y = begin
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286 suc x + minus y z ≡⟨ cong suc ( minus+yz z≤y ) ⟩
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287 suc (minus (x + y) z) ≡⟨ minus+1 {x + y} {z} (≤-trans z≤y (subst (λ g → y ≤ g) (+-comm y x) x≤x+y) ) ⟩
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288 minus (suc x + y) z ∎ where open ≡-Reasoning
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289
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290 minus<=0 : {x y : ℕ } → x ≤ y → minus x y ≡ 0
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291 minus<=0 {0} {zero} z≤n = refl
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292 minus<=0 {0} {suc y} z≤n = refl
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293 minus<=0 {suc x} {suc y} (s≤s le) = minus<=0 {x} {y} le
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294
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295 minus>0 : {x y : ℕ } → x < y → 0 < minus y x
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296 minus>0 {zero} {suc _} (s≤s z≤n) = s≤s z≤n
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297 minus>0 {suc x} {suc y} (s≤s lt) = minus>0 {x} {y} lt
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298
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299 minus>0→x<y : {x y : ℕ } → 0 < minus y x → x < y
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300 minus>0→x<y {x} {y} lt with <-cmp x y
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301 ... | tri< a ¬b ¬c = a
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302 ... | tri≈ ¬a refl ¬c = ⊥-elim ( nat-≡< (sym (minus<=0 {x} ≤-refl)) lt )
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303 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≡< (sym (minus<=0 {y} (≤-trans refl-≤s c ))) lt )
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304
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305 minus+y-y : {x y : ℕ } → (x + y) - y ≡ x
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306 minus+y-y {zero} {y} = minus<=0 {zero + y} {y} ≤-refl
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307 minus+y-y {suc x} {y} = begin
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308 (suc x + y) - y ≡⟨ sym (minus+1 {_} {y} x≤y+x) ⟩
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309 suc ((x + y) - y) ≡⟨ cong suc (minus+y-y {x} {y}) ⟩
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310 suc x ∎ where open ≡-Reasoning
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311
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312 minus+yx-yz : {x y z : ℕ } → (y + x) - (y + z) ≡ x - z
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313 minus+yx-yz {x} {zero} {z} = refl
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314 minus+yx-yz {x} {suc y} {z} = minus+yx-yz {x} {y} {z}
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315
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316 minus+xy-zy : {x y z : ℕ } → (x + y) - (z + y) ≡ x - z
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317 minus+xy-zy {x} {y} {z} = subst₂ (λ j k → j - k ≡ x - z ) (+-comm y x) (+-comm y z) (minus+yx-yz {x} {y} {z})
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318
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319 +cancel<l : (x z : ℕ ) {y : ℕ} → y + x < y + z → x < z
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320 +cancel<l x z {zero} lt = lt
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321 +cancel<l x z {suc y} (s≤s lt) = +cancel<l x z {y} lt
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322
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323 +cancel<r : (x z : ℕ ) {y : ℕ} → x + y < z + y → x < z
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324 +cancel<r x z {y} lt = +cancel<l x z (subst₂ (λ j k → j < k ) (+-comm x _) (+-comm z _) lt )
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325
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326 y-x<y : {x y : ℕ } → 0 < x → 0 < y → y - x < y
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327 y-x<y {x} {y} 0<x 0<y with <-cmp x (suc y)
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328 ... | tri< a ¬b ¬c = +cancel<r (y - x) _ ( begin
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329 suc ((y - x) + x) ≡⟨ cong suc (minus+n {y} {x} a ) ⟩
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330 suc y ≡⟨ +-comm 1 _ ⟩
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331 y + suc 0 ≤⟨ +-mono-≤ ≤-refl 0<x ⟩
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332 y + x ∎ ) where open ≤-Reasoning
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333 ... | tri≈ ¬a refl ¬c = subst ( λ k → k < y ) (sym (minus<=0 {y} {x} refl-≤s )) 0<y
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334 ... | tri> ¬a ¬b c = subst ( λ k → k < y ) (sym (minus<=0 {y} {x} (≤-trans (≤-trans refl-≤s refl-≤s) c))) 0<y -- suc (suc y) ≤ x → y ≤ x
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335
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336 open import Relation.Binary.Definitions
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337
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338 distr-minus-* : {x y z : ℕ } → (minus x y) * z ≡ minus (x * z) (y * z)
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339 distr-minus-* {x} {zero} {z} = refl
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340 distr-minus-* {x} {suc y} {z} with <-cmp x y
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341 distr-minus-* {x} {suc y} {z} | tri< a ¬b ¬c = begin
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342 minus x (suc y) * z
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343 ≡⟨ cong (λ k → k * z ) (minus<=0 {x} {suc y} (x<y→≤ a)) ⟩
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344 0 * z
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345 ≡⟨ sym (minus<=0 {x * z} {z + y * z} le ) ⟩
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346 minus (x * z) (z + y * z)
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347 ∎ where
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348 open ≡-Reasoning
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349 le : x * z ≤ z + y * z
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350 le = ≤-trans lemma (subst (λ k → y * z ≤ k ) (+-comm _ z ) (x≤x+y {y * z} {z} ) ) where
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351 lemma : x * z ≤ y * z
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352 lemma = *≤ {x} {y} {z} (<to≤ a)
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353 distr-minus-* {x} {suc y} {z} | tri≈ ¬a refl ¬c = begin
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354 minus x (suc y) * z
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355 ≡⟨ cong (λ k → k * z ) (minus<=0 {x} {suc y} refl-≤s ) ⟩
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356 0 * z
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357 ≡⟨ sym (minus<=0 {x * z} {z + y * z} (lt {x} {z} )) ⟩
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358 minus (x * z) (z + y * z)
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359 ∎ where
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360 open ≡-Reasoning
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361 lt : {x z : ℕ } → x * z ≤ z + x * z
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362 lt {zero} {zero} = z≤n
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363 lt {suc x} {zero} = lt {x} {zero}
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364 lt {x} {suc z} = ≤-trans lemma refl-≤s where
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365 lemma : x * suc z ≤ z + x * suc z
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366 lemma = subst (λ k → x * suc z ≤ k ) (+-comm _ z) (x≤x+y {x * suc z} {z})
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367 distr-minus-* {x} {suc y} {z} | tri> ¬a ¬b c = +m= {_} {_} {suc y * z} ( begin
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368 minus x (suc y) * z + suc y * z
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369 ≡⟨ sym (proj₂ *-distrib-+ z (minus x (suc y) ) _) ⟩
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370 ( minus x (suc y) + suc y ) * z
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371 ≡⟨ cong (λ k → k * z) (minus+n {x} {suc y} (s≤s c)) ⟩
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372 x * z
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373 ≡⟨ sym (minus+n {x * z} {suc y * z} (s≤s (lt c))) ⟩
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374 minus (x * z) (suc y * z) + suc y * z
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375 ∎ ) where
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376 open ≡-Reasoning
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377 lt : {x y z : ℕ } → suc y ≤ x → z + y * z ≤ x * z
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378 lt {x} {y} {z} le = *≤ le
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379
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380 distr-minus-*' : {z x y : ℕ } → z * (minus x y) ≡ minus (z * x) (z * y)
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381 distr-minus-*' {z} {x} {y} = begin
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382 z * (minus x y) ≡⟨ *-comm _ (x - y) ⟩
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383 (minus x y) * z ≡⟨ distr-minus-* {x} {y} {z} ⟩
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384 minus (x * z) (y * z) ≡⟨ cong₂ (λ j k → j - k ) (*-comm x z ) (*-comm y z) ⟩
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385 minus (z * x) (z * y) ∎ where open ≡-Reasoning
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431
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386
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387 minus- : {x y z : ℕ } → suc x > z + y → minus (minus x y) z ≡ minus x (y + z)
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388 minus- {x} {y} {z} gt = +m= {_} {_} {z} ( begin
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389 minus (minus x y) z + z
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390 ≡⟨ minus+n {_} {z} lemma ⟩
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391 minus x y
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392 ≡⟨ +m= {_} {_} {y} ( begin
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393 minus x y + y
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394 ≡⟨ minus+n {_} {y} lemma1 ⟩
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395 x
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396 ≡⟨ sym ( minus+n {_} {z + y} gt ) ⟩
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397 minus x (z + y) + (z + y)
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398 ≡⟨ sym ( +-assoc (minus x (z + y)) _ _ ) ⟩
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399 minus x (z + y) + z + y
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400 ∎ ) ⟩
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401 minus x (z + y) + z
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402 ≡⟨ cong (λ k → minus x k + z ) (+-comm _ y ) ⟩
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403 minus x (y + z) + z
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404 ∎ ) where
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405 open ≡-Reasoning
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406 lemma1 : suc x > y
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407 lemma1 = x+y<z→x<z (subst (λ k → k < suc x ) (+-comm z _ ) gt )
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408 lemma : suc (minus x y) > z
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409 lemma = <-minus {_} {_} {y} ( subst ( λ x → z + y < suc x ) (sym (minus+n {x} {y} lemma1 )) gt )
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410
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411 minus-* : {M k n : ℕ } → n < k → minus k (suc n) * M ≡ minus (minus k n * M ) M
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412 minus-* {zero} {k} {n} lt = begin
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413 minus k (suc n) * zero
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414 ≡⟨ *-comm (minus k (suc n)) zero ⟩
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415 zero * minus k (suc n)
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416 ≡⟨⟩
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417 0 * minus k n
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418 ≡⟨ *-comm 0 (minus k n) ⟩
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419 minus (minus k n * 0 ) 0
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420 ∎ where
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421 open ≡-Reasoning
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422 minus-* {suc m} {k} {n} lt with <-cmp k 1
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423 minus-* {suc m} {.0} {zero} lt | tri< (s≤s z≤n) ¬b ¬c = refl
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424 minus-* {suc m} {.0} {suc n} lt | tri< (s≤s z≤n) ¬b ¬c = refl
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425 minus-* {suc zero} {.1} {zero} lt | tri≈ ¬a refl ¬c = refl
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426 minus-* {suc (suc m)} {.1} {zero} lt | tri≈ ¬a refl ¬c = minus-* {suc m} {1} {zero} lt
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427 minus-* {suc m} {.1} {suc n} (s≤s ()) | tri≈ ¬a refl ¬c
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428 minus-* {suc m} {k} {n} lt | tri> ¬a ¬b c = begin
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429 minus k (suc n) * M
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430 ≡⟨ distr-minus-* {k} {suc n} {M} ⟩
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431 minus (k * M ) ((suc n) * M)
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432 ≡⟨⟩
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433 minus (k * M ) (M + n * M )
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434 ≡⟨ cong (λ x → minus (k * M) x) (+-comm M _ ) ⟩
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435 minus (k * M ) ((n * M) + M )
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436 ≡⟨ sym ( minus- {k * M} {n * M} (lemma lt) ) ⟩
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437 minus (minus (k * M ) (n * M)) M
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438 ≡⟨ cong (λ x → minus x M ) ( sym ( distr-minus-* {k} {n} )) ⟩
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439 minus (minus k n * M ) M
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440 ∎ where
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441 M = suc m
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442 lemma : {n k m : ℕ } → n < k → suc (k * suc m) > suc m + n * suc m
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443 lemma {zero} {suc k} {m} (s≤s lt) = s≤s (s≤s (subst (λ x → x ≤ m + k * suc m) (+-comm 0 _ ) x≤x+y ))
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444 lemma {suc n} {suc k} {m} lt = begin
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445 suc (suc m + suc n * suc m)
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446 ≡⟨⟩
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447 suc ( suc (suc n) * suc m)
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448 ≤⟨ ≤-plus-0 {_} {_} {1} (*≤ lt ) ⟩
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449 suc (suc k * suc m)
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450 ∎ where open ≤-Reasoning
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451 open ≡-Reasoning
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452
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453 x=y+z→x-z=y : {x y z : ℕ } → x ≡ y + z → x - z ≡ y
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454 x=y+z→x-z=y {x} {zero} {.x} refl = minus<=0 {x} {x} refl-≤ -- x ≡ suc (y + z) → (x ≡ y + z → x - z ≡ y) → (x - z) ≡ suc y
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455 x=y+z→x-z=y {suc x} {suc y} {zero} eq = begin -- suc x ≡ suc (y + zero) → (suc x - zero) ≡ suc y
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456 suc x - zero ≡⟨ refl ⟩
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457 suc x ≡⟨ eq ⟩
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458 suc y + zero ≡⟨ +-comm _ zero ⟩
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459 suc y ∎ where open ≡-Reasoning
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460 x=y+z→x-z=y {suc x} {suc y} {suc z} eq = x=y+z→x-z=y {x} {suc y} {z} ( begin
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461 x ≡⟨ cong pred eq ⟩
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462 pred (suc y + suc z) ≡⟨ +-comm _ (suc z) ⟩
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463 suc z + y ≡⟨ cong suc ( +-comm _ y ) ⟩
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464 suc y + z ∎ ) where open ≡-Reasoning
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465
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466 m*1=m : {m : ℕ } → m * 1 ≡ m
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467 m*1=m {zero} = refl
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468 m*1=m {suc m} = cong suc m*1=m
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469
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470 +-cancel-1 : (x y z : ℕ ) → x + y ≡ x + z → y ≡ z
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471 +-cancel-1 zero y z eq = eq
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472 +-cancel-1 (suc x) y z eq = +-cancel-1 x y z (cong pred eq )
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473
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474 +-cancel-0 : (x y z : ℕ ) → y + x ≡ z + x → y ≡ z
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475 +-cancel-0 x y z eq = +-cancel-1 x y z (trans (+-comm x y) (trans eq (sym (+-comm x z)) ))
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476
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477 *-cancel-left : {x y z : ℕ } → x > 0 → x * y ≡ x * z → y ≡ z
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478 *-cancel-left {suc x} {zero} {zero} lt eq = refl
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479 *-cancel-left {suc x} {zero} {suc z} lt eq = ⊥-elim ( nat-≡< eq (s≤s (begin
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480 x * zero ≡⟨ *-comm x _ ⟩
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481 zero ≤⟨ z≤n ⟩
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482 z + x * suc z ∎ ))) where open ≤-Reasoning
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483 *-cancel-left {suc x} {suc y} {zero} lt eq = ⊥-elim ( nat-≡< (sym eq) (s≤s (begin
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484 x * zero ≡⟨ *-comm x _ ⟩
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485 zero ≤⟨ z≤n ⟩
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486 _ ∎ ))) where open ≤-Reasoning
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487 *-cancel-left {suc x} {suc y} {suc z} lt eq with cong pred eq
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488 ... | eq1 = cong suc (*-cancel-left {suc x} {y} {z} lt (+-cancel-0 x _ _ (begin
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489 y + x * y + x ≡⟨ +-assoc y _ _ ⟩
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490 y + (x * y + x) ≡⟨ cong (λ k → y + (k + x)) (*-comm x _) ⟩
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491 y + (y * x + x) ≡⟨ cong (_+_ y) (+-comm _ x) ⟩
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492 y + (x + y * x ) ≡⟨ refl ⟩
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493 y + suc y * x ≡⟨ cong (_+_ y) (*-comm (suc y) _) ⟩
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494 y + x * suc y ≡⟨ eq1 ⟩
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495 z + x * suc z ≡⟨ refl ⟩
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496 _ ≡⟨ sym ( cong (_+_ z) (*-comm (suc z) _) ) ⟩
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497 _ ≡⟨ sym ( cong (_+_ z) (+-comm _ x)) ⟩
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498 z + (z * x + x) ≡⟨ sym ( cong (λ k → z + (k + x)) (*-comm x _) ) ⟩
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499 z + (x * z + x) ≡⟨ sym ( +-assoc z _ _) ⟩
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500 z + x * z + x ∎ ))) where open ≡-Reasoning
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501
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502 record Finduction {n m : Level} (P : Set n ) (Q : P → Set m ) (f : P → ℕ) : Set (n Level.⊔ m) where
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503 field
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504 fzero : {p : P} → f p ≡ zero → Q p
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505 pnext : (p : P ) → P
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506 decline : {p : P} → 0 < f p → f (pnext p) < f p
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507 ind : {p : P} → Q (pnext p) → Q p
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508
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509 y<sx→y≤x : {x y : ℕ} → y < suc x → y ≤ x
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510 y<sx→y≤x (s≤s lt) = lt
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511
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512 fi0 : (x : ℕ) → x ≤ zero → x ≡ zero
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513 fi0 .0 z≤n = refl
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514
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515 f-induction : {n m : Level} {P : Set n } → {Q : P → Set m }
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516 → (f : P → ℕ)
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517 → Finduction P Q f
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518 → (p : P ) → Q p
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519 f-induction {n} {m} {P} {Q} f I p with <-cmp 0 (f p)
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520 ... | tri> ¬a ¬b ()
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521 ... | tri≈ ¬a b ¬c = Finduction.fzero I (sym b)
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522 ... | tri< lt _ _ = f-induction0 p (f p) (<to≤ (Finduction.decline I lt)) where
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523 f-induction0 : (p : P) → (x : ℕ) → (f (Finduction.pnext I p)) ≤ x → Q p
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524 f-induction0 p zero le = Finduction.ind I (Finduction.fzero I (fi0 _ le))
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525 f-induction0 p (suc x) le with <-cmp (f (Finduction.pnext I p)) (suc x)
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526 ... | tri< (s≤s a) ¬b ¬c = f-induction0 p x a
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527 ... | tri≈ ¬a b ¬c = Finduction.ind I (f-induction0 (Finduction.pnext I p) x (y<sx→y≤x f1)) where
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528 f1 : f (Finduction.pnext I (Finduction.pnext I p)) < suc x
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529 f1 = subst (λ k → f (Finduction.pnext I (Finduction.pnext I p)) < k ) b ( Finduction.decline I {Finduction.pnext I p}
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530 (subst (λ k → 0 < k ) (sym b) (s≤s z≤n ) ))
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531 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> le c )
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532
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533
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534 record Ninduction {n m : Level} (P : Set n ) (Q : P → Set m ) (f : P → ℕ) : Set (n Level.⊔ m) where
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535 field
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536 pnext : (p : P ) → P
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537 fzero : {p : P} → f (pnext p) ≡ zero → Q p
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538 decline : {p : P} → 0 < f p → f (pnext p) < f p
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539 ind : {p : P} → Q (pnext p) → Q p
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540
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541 s≤s→≤ : { i j : ℕ} → suc i ≤ suc j → i ≤ j
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542 s≤s→≤ (s≤s lt) = lt
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431
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543
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1266
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544 n-induction : {n m : Level} {P : Set n } → {Q : P → Set m }
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545 → (f : P → ℕ)
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546 → Ninduction P Q f
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547 → (p : P ) → Q p
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548 n-induction {n} {m} {P} {Q} f I p = f-induction0 p (f (Ninduction.pnext I p)) ≤-refl where
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549 f-induction0 : (p : P) → (x : ℕ) → (f (Ninduction.pnext I p)) ≤ x → Q p
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550 f-induction0 p zero lt = Ninduction.fzero I {p} (fi0 _ lt)
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551 f-induction0 p (suc x) le with <-cmp (f (Ninduction.pnext I p)) (suc x)
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552 ... | tri< (s≤s a) ¬b ¬c = f-induction0 p x a
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553 ... | tri≈ ¬a b ¬c = Ninduction.ind I (f-induction0 (Ninduction.pnext I p) x (s≤s→≤ nle) ) where
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554 f>0 : 0 < f (Ninduction.pnext I p)
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555 f>0 = subst (λ k → 0 < k ) (sym b) ( s≤s z≤n )
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556 nle : suc (f (Ninduction.pnext I (Ninduction.pnext I p))) ≤ suc x
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557 nle = subst (λ k → suc (f (Ninduction.pnext I (Ninduction.pnext I p))) ≤ k) b (Ninduction.decline I {Ninduction.pnext I p} f>0 )
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558 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> le c )
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559
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560
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561 record Factor (n m : ℕ ) : Set where
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562 field
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563 factor : ℕ
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564 remain : ℕ
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565 is-factor : factor * n + remain ≡ m
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566
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567 record Dividable (n m : ℕ ) : Set where
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568 field
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569 factor : ℕ
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570 is-factor : factor * n + 0 ≡ m
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571
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572 open Factor
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573
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574 DtoF : {n m : ℕ} → Dividable n m → Factor n m
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575 DtoF {n} {m} record { factor = f ; is-factor = fa } = record { factor = f ; remain = 0 ; is-factor = fa }
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576
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577 FtoD : {n m : ℕ} → (fc : Factor n m) → remain fc ≡ 0 → Dividable n m
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578 FtoD {n} {m} record { factor = f ; remain = r ; is-factor = fa } refl = record { factor = f ; is-factor = fa }
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579
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580 --divdable^2 : ( n k : ℕ ) → Dividable k ( n * n ) → Dividable k n
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581 --divdable^2 n k dn2 = {!!}
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582
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583 decf : { n k : ℕ } → ( x : Factor k (suc n) ) → Factor k n
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584 decf {n} {k} record { factor = f ; remain = r ; is-factor = fa } =
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585 decf1 {n} {k} f r fa where
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586 decf1 : { n k : ℕ } → (f r : ℕ) → (f * k + r ≡ suc n) → Factor k n
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587 decf1 {n} {k} f (suc r) fa = -- this case must be the first
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588 record { factor = f ; remain = r ; is-factor = ( begin -- fa : f * k + suc r ≡ suc n
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589 f * k + r ≡⟨ cong pred ( begin
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590 suc ( f * k + r ) ≡⟨ +-comm _ r ⟩
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591 r + suc (f * k) ≡⟨ sym (+-assoc r 1 _) ⟩
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592 (r + 1) + f * k ≡⟨ cong (λ t → t + f * k ) (+-comm r 1) ⟩
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593 (suc r ) + f * k ≡⟨ +-comm (suc r) _ ⟩
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594 f * k + suc r ≡⟨ fa ⟩
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595 suc n ∎ ) ⟩
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596 n ∎ ) } where open ≡-Reasoning
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597 decf1 {n} {zero} (suc f) zero fa = ⊥-elim ( nat-≡< fa (
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598 begin suc (suc f * zero + zero) ≡⟨ cong suc (+-comm _ zero) ⟩
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599 suc (f * 0) ≡⟨ cong suc (*-comm f zero) ⟩
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600 suc zero ≤⟨ s≤s z≤n ⟩
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601 suc n ∎ )) where open ≤-Reasoning
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602 decf1 {n} {suc k} (suc f) zero fa =
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603 record { factor = f ; remain = k ; is-factor = ( begin -- fa : suc (k + f * suc k + zero) ≡ suc n
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604 f * suc k + k ≡⟨ +-comm _ k ⟩
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605 k + f * suc k ≡⟨ +-comm zero _ ⟩
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606 (k + f * suc k) + zero ≡⟨ cong pred fa ⟩
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607 n ∎ ) } where open ≡-Reasoning
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608
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609 div0 : {k : ℕ} → Dividable k 0
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610 div0 {k} = record { factor = 0; is-factor = refl }
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611
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612 div= : {k : ℕ} → Dividable k k
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613 div= {k} = record { factor = 1; is-factor = ( begin
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614 k + 0 * k + 0 ≡⟨ trans ( +-comm _ 0) ( +-comm _ 0) ⟩
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615 k ∎ ) } where open ≡-Reasoning
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616
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617 div1 : { k : ℕ } → k > 1 → ¬ Dividable k 1
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618 div1 {k} k>1 record { factor = (suc f) ; is-factor = fa } = ⊥-elim ( nat-≡< (sym fa) ( begin
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619 2 ≤⟨ k>1 ⟩
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620 k ≡⟨ +-comm 0 _ ⟩
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621 k + 0 ≡⟨ refl ⟩
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622 1 * k ≤⟨ *-mono-≤ {1} {suc f} (s≤s z≤n ) ≤-refl ⟩
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623 suc f * k ≡⟨ +-comm 0 _ ⟩
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624 suc f * k + 0 ∎ )) where open ≤-Reasoning
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625
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626 div+div : { i j k : ℕ } → Dividable k i → Dividable k j → Dividable k (i + j) ∧ Dividable k (j + i)
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627 div+div {i} {j} {k} di dj = ⟪ div+div1 , subst (λ g → Dividable k g) (+-comm i j) div+div1 ⟫ where
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628 fki = Dividable.factor di
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629 fkj = Dividable.factor dj
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630 div+div1 : Dividable k (i + j)
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631 div+div1 = record { factor = fki + fkj ; is-factor = ( begin
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632 (fki + fkj) * k + 0 ≡⟨ +-comm _ 0 ⟩
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633 (fki + fkj) * k ≡⟨ *-distribʳ-+ k fki _ ⟩
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634 fki * k + fkj * k ≡⟨ cong₂ ( λ i j → i + j ) (+-comm 0 (fki * k)) (+-comm 0 (fkj * k)) ⟩
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635 (fki * k + 0) + (fkj * k + 0) ≡⟨ cong₂ ( λ i j → i + j ) (Dividable.is-factor di) (Dividable.is-factor dj) ⟩
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636 i + j ∎ ) } where
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637 open ≡-Reasoning
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638
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639 div-div : { i j k : ℕ } → k > 1 → Dividable k i → Dividable k j → Dividable k (i - j) ∧ Dividable k (j - i)
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640 div-div {i} {j} {k} k>1 di dj = ⟪ div-div1 di dj , div-div1 dj di ⟫ where
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641 div-div1 : {i j : ℕ } → Dividable k i → Dividable k j → Dividable k (i - j)
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642 div-div1 {i} {j} di dj = record { factor = fki - fkj ; is-factor = ( begin
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643 (fki - fkj) * k + 0 ≡⟨ +-comm _ 0 ⟩
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644 (fki - fkj) * k ≡⟨ distr-minus-* {fki} {fkj} ⟩
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645 (fki * k) - (fkj * k) ≡⟨ cong₂ ( λ i j → i - j ) (+-comm 0 (fki * k)) (+-comm 0 (fkj * k)) ⟩
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646 (fki * k + 0) - (fkj * k + 0) ≡⟨ cong₂ ( λ i j → i - j ) (Dividable.is-factor di) (Dividable.is-factor dj) ⟩
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647 i - j ∎ ) } where
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648 open ≡-Reasoning
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649 fki = Dividable.factor di
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650 fkj = Dividable.factor dj
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651
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652 open _∧_
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653
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654 div+1 : { i k : ℕ } → k > 1 → Dividable k i → ¬ Dividable k (suc i)
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655 div+1 {i} {k} k>1 d d1 = div1 k>1 div+11 where
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656 div+11 : Dividable k 1
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657 div+11 = subst (λ g → Dividable k g) (minus+y-y {1} {i} ) ( proj2 (div-div k>1 d d1 ) )
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658
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659 div<k : { m k : ℕ } → k > 1 → m > 0 → m < k → ¬ Dividable k m
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660 div<k {m} {k} k>1 m>0 m<k d = ⊥-elim ( nat-≤> (div<k1 (Dividable.factor d) (Dividable.is-factor d)) m<k ) where
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661 div<k1 : (f : ℕ ) → f * k + 0 ≡ m → k ≤ m
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662 div<k1 zero eq = ⊥-elim (nat-≡< eq m>0 )
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663 div<k1 (suc f) eq = begin
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664 k ≤⟨ x≤x+y ⟩
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665 k + (f * k + 0) ≡⟨ sym (+-assoc k _ _) ⟩
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666 k + f * k + 0 ≡⟨ eq ⟩
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667 m ∎ where open ≤-Reasoning
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668
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669 0<factor : { m k : ℕ } → k > 0 → m > 0 → (d : Dividable k m ) → Dividable.factor d > 0
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670 0<factor {m} {k} k>0 m>0 d with Dividable.factor d | inspect Dividable.factor d
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671 ... | zero | record { eq = eq1 } = ⊥-elim ( nat-≡< ff1 m>0 ) where
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672 ff1 : 0 ≡ m
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673 ff1 = begin
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674 0 ≡⟨⟩
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675 0 * k + 0 ≡⟨ cong (λ j → j * k + 0) (sym eq1) ⟩
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676 Dividable.factor d * k + 0 ≡⟨ Dividable.is-factor d ⟩
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677 m ∎ where open ≡-Reasoning
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678 ... | suc t | _ = s≤s z≤n
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679
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680 div→k≤m : { m k : ℕ } → k > 1 → m > 0 → Dividable k m → m ≥ k
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681 div→k≤m {m} {k} k>1 m>0 d with <-cmp m k
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682 ... | tri< a ¬b ¬c = ⊥-elim ( div<k k>1 m>0 a d )
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683 ... | tri≈ ¬a refl ¬c = ≤-refl
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684 ... | tri> ¬a ¬b c = <to≤ c
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685
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686 div1*k+0=k : {k : ℕ } → 1 * k + 0 ≡ k
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687 div1*k+0=k {k} = begin
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688 1 * k + 0 ≡⟨ cong (λ g → g + 0) (+-comm _ 0) ⟩
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689 k + 0 ≡⟨ +-comm _ 0 ⟩
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690 k ∎ where open ≡-Reasoning
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691
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692 decD : {k m : ℕ} → k > 1 → Dec (Dividable k m )
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693 decD {k} {m} k>1 = n-induction {_} {_} {ℕ} {λ m → Dec (Dividable k m ) } F I m where
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694 F : ℕ → ℕ
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695 F m = m
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696 F0 : ( m : ℕ ) → F (m - k) ≡ 0 → Dec (Dividable k m )
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697 F0 0 eq = yes record { factor = 0 ; is-factor = refl }
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698 F0 (suc m) eq with <-cmp k (suc m)
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699 ... | tri< a ¬b ¬c = yes record { factor = 1 ; is-factor =
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700 subst (λ g → 1 * k + 0 ≡ g ) (sym (i-j=0→i=j (<to≤ a) eq )) div1*k+0=k } -- (suc m - k) ≡ 0 → k ≡ suc m, k ≤ suc m
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701 ... | tri≈ ¬a refl ¬c = yes record { factor = 1 ; is-factor = div1*k+0=k }
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702 ... | tri> ¬a ¬b c = no ( λ d → ⊥-elim (div<k k>1 (s≤s z≤n ) c d) )
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703 decl : {m : ℕ } → 0 < m → m - k < m
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704 decl {m} 0<m = y-x<y (<-trans a<sa k>1 ) 0<m
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705 ind : (p : ℕ ) → Dec (Dividable k (p - k) ) → Dec (Dividable k p )
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706 ind p (yes y) with <-cmp p k
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707 ... | tri≈ ¬a refl ¬c = yes (subst (λ g → Dividable k g) (minus+n ≤-refl ) (proj1 ( div+div y div= )))
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708 ... | tri> ¬a ¬b k<p = yes (subst (λ g → Dividable k g) (minus+n (<-trans k<p a<sa)) (proj1 ( div+div y div= )))
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709 ... | tri< a ¬b ¬c with <-cmp p 0
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710 ... | tri≈ ¬a refl ¬c₁ = yes div0
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711 ... | tri> ¬a ¬b₁ c = no (λ d → not-div p (Dividable.factor d) a c (Dividable.is-factor d) ) where
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712 not-div : (p f : ℕ) → p < k → 0 < p → f * k + 0 ≡ p → ⊥
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713 not-div (suc p) (suc f) p<k 0<p eq = nat-≡< (sym eq) ( begin -- ≤-trans p<k {!!}) -- suc p ≤ k
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714 suc (suc p) ≤⟨ p<k ⟩
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715 k ≤⟨ x≤x+y ⟩
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716 k + (f * k + 0) ≡⟨ sym (+-assoc k _ _) ⟩
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717 suc f * k + 0 ∎ ) where open ≤-Reasoning
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718 ind p (no n) = no (λ d → n (proj1 (div-div k>1 d div=)) )
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719 I : Ninduction ℕ _ F
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720 I = record {
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721 pnext = λ p → p - k
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722 ; fzero = λ {m} eq → F0 m eq
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723 ; decline = λ {m} lt → decl lt
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724 ; ind = λ {p} prev → ind p prev
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725 }
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726
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