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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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74
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10 open import Relation.Binary.Definitions
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72
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11 open import logic
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12
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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 refl-≤s : {x : ℕ } → x ≤ suc x
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34 refl-≤s {zero} = z≤n
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35 refl-≤s {suc x} = s≤s (refl-≤s {x})
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36
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37 =→¬< : {x : ℕ } → ¬ ( x < x )
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38 =→¬< {zero} ()
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39 =→¬< {suc x} (s≤s lt) = =→¬< lt
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40
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41 >→¬< : {x y : ℕ } → (x < y ) → ¬ ( y < x )
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42 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x
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43
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44 <-∨ : { x y : ℕ } → x < suc y → ( (x ≡ y ) ∨ (x < y) )
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45 <-∨ {zero} {zero} (s≤s z≤n) = case1 refl
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46 <-∨ {zero} {suc y} (s≤s lt) = case2 (s≤s z≤n)
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47 <-∨ {suc x} {zero} (s≤s ())
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48 <-∨ {suc x} {suc y} (s≤s lt) with <-∨ {x} {y} lt
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49 <-∨ {suc x} {suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → suc k ) eq)
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50 <-∨ {suc x} {suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1)
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51
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52 max : (x y : ℕ) → ℕ
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53 max zero zero = zero
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54 max zero (suc x) = (suc x)
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55 max (suc x) zero = (suc x)
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56 max (suc x) (suc y) = suc ( max x y )
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57
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58 -- _*_ : ℕ → ℕ → ℕ
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59 -- _*_ zero _ = zero
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60 -- _*_ (suc n) m = m + ( n * m )
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61
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62 exp : ℕ → ℕ → ℕ
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63 exp _ zero = 1
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64 exp n (suc m) = n * ( exp n m )
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65
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66 minus : (a b : ℕ ) → ℕ
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67 minus a zero = a
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68 minus zero (suc b) = zero
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69 minus (suc a) (suc b) = minus a b
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70
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71 _-_ = minus
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72
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73 m+= : {i j m : ℕ } → m + i ≡ m + j → i ≡ j
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74 m+= {i} {j} {zero} refl = refl
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75 m+= {i} {j} {suc m} eq = m+= {i} {j} {m} ( cong (λ k → pred k ) eq )
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76
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77 +m= : {i j m : ℕ } → i + m ≡ j + m → i ≡ j
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78 +m= {i} {j} {m} eq = m+= ( subst₂ (λ j k → j ≡ k ) (+-comm i _ ) (+-comm j _ ) eq )
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79
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80 less-1 : { n m : ℕ } → suc n < m → n < m
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81 less-1 {zero} {suc (suc _)} (s≤s (s≤s z≤n)) = s≤s z≤n
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82 less-1 {suc n} {suc m} (s≤s lt) = s≤s (less-1 {n} {m} lt)
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83
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84 sa=b→a<b : { n m : ℕ } → suc n ≡ m → n < m
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85 sa=b→a<b {0} {suc zero} refl = s≤s z≤n
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86 sa=b→a<b {suc n} {suc (suc n)} refl = s≤s (sa=b→a<b refl)
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87
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88 minus+n : {x y : ℕ } → suc x > y → minus x y + y ≡ x
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89 minus+n {x} {zero} _ = trans (sym (+-comm zero _ )) refl
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90 minus+n {zero} {suc y} (s≤s ())
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91 minus+n {suc x} {suc y} (s≤s lt) = begin
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92 minus (suc x) (suc y) + suc y
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93 ≡⟨ +-comm _ (suc y) ⟩
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94 suc y + minus x y
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95 ≡⟨ cong ( λ k → suc k ) (
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96 begin
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97 y + minus x y
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98 ≡⟨ +-comm y _ ⟩
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99 minus x y + y
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100 ≡⟨ minus+n {x} {y} lt ⟩
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101 x
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102 ∎
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103 ) ⟩
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104 suc x
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105 ∎ where open ≡-Reasoning
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106
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107 sn-m=sn-m : {m n : ℕ } → m ≤ n → suc n - m ≡ suc ( n - m )
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108 sn-m=sn-m {0} {n} z≤n = refl
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109 sn-m=sn-m {suc m} {suc n} (s≤s m<n) = sn-m=sn-m m<n
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110
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111 si-sn=i-n : {i n : ℕ } → n < i → suc (i - suc n) ≡ (i - n)
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112 si-sn=i-n {i} {n} n<i = begin
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113 suc (i - suc n) ≡⟨ sym (sn-m=sn-m n<i ) ⟩
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114 suc i - suc n ≡⟨⟩
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115 i - n
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116 ∎ where
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117 open ≡-Reasoning
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118
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119 n-m<n : (n m : ℕ ) → n - m ≤ n
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120 n-m<n zero zero = z≤n
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121 n-m<n (suc n) zero = s≤s (n-m<n n zero)
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122 n-m<n zero (suc m) = z≤n
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123 n-m<n (suc n) (suc m) = ≤-trans (n-m<n n m ) refl-≤s
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124
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125 n-n-m=m : {m n : ℕ } → m ≤ n → m ≡ (n - (n - m))
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126 n-n-m=m {0} {zero} z≤n = refl
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127 n-n-m=m {0} {suc n} z≤n = n-n-m=m {0} {n} z≤n
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128 n-n-m=m {suc m} {suc n} (s≤s m≤n) = sym ( begin
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129 suc n - ( n - m ) ≡⟨ sn-m=sn-m (n-m<n n m) ⟩
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130 suc (n - ( n - m )) ≡⟨ cong (λ k → suc k ) (sym (n-n-m=m m≤n)) ⟩
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131 suc m
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132 ∎ ) where
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133 open ≡-Reasoning
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134
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135 0<s : {x : ℕ } → zero < suc x
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136 0<s {_} = s≤s z≤n
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137
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138 <-minus-0 : {x y z : ℕ } → z + x < z + y → x < y
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139 <-minus-0 {x} {suc _} {zero} lt = lt
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140 <-minus-0 {x} {y} {suc z} (s≤s lt) = <-minus-0 {x} {y} {z} lt
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141
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142 <-minus : {x y z : ℕ } → x + z < y + z → x < y
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143 <-minus {x} {y} {z} lt = <-minus-0 ( subst₂ ( λ j k → j < k ) (+-comm x _) (+-comm y _ ) lt )
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144
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145 x≤x+y : {z y : ℕ } → z ≤ z + y
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146 x≤x+y {zero} {y} = z≤n
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147 x≤x+y {suc z} {y} = s≤s (x≤x+y {z} {y})
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148
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149 <-plus : {x y z : ℕ } → x < y → x + z < y + z
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150 <-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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151 <-plus {suc x} {suc y} {z} (s≤s lt) = s≤s (<-plus {x} {y} {z} lt)
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152
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153 <-plus-0 : {x y z : ℕ } → x < y → z + x < z + y
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154 <-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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155
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156 ≤-plus : {x y z : ℕ } → x ≤ y → x + z ≤ y + z
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157 ≤-plus {0} {y} {zero} z≤n = z≤n
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158 ≤-plus {0} {y} {suc z} z≤n = subst (λ k → z < k ) (+-comm _ y ) x≤x+y
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159 ≤-plus {suc x} {suc y} {z} (s≤s lt) = s≤s ( ≤-plus {x} {y} {z} lt )
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160
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161 ≤-plus-0 : {x y z : ℕ } → x ≤ y → z + x ≤ z + y
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162 ≤-plus-0 {x} {y} {zero} lt = lt
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163 ≤-plus-0 {x} {y} {suc z} lt = s≤s ( ≤-plus-0 {x} {y} {z} lt )
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164
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165 x+y<z→x<z : {x y z : ℕ } → x + y < z → x < z
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166 x+y<z→x<z {zero} {y} {suc z} (s≤s lt1) = s≤s z≤n
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167 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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168
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169 *≤ : {x y z : ℕ } → x ≤ y → x * z ≤ y * z
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170 *≤ lt = *-mono-≤ lt ≤-refl
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171
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172 *< : {x y z : ℕ } → x < y → x * suc z < y * suc z
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173 *< {zero} {suc y} lt = s≤s z≤n
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174 *< {suc x} {suc y} (s≤s lt) = <-plus-0 (*< lt)
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175
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176 <to<s : {x y : ℕ } → x < y → x < suc y
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177 <to<s {zero} {suc y} (s≤s lt) = s≤s z≤n
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178 <to<s {suc x} {suc y} (s≤s lt) = s≤s (<to<s {x} {y} lt)
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179
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180 <tos<s : {x y : ℕ } → x < y → suc x < suc y
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181 <tos<s {zero} {suc y} (s≤s z≤n) = s≤s (s≤s z≤n)
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182 <tos<s {suc x} {suc y} (s≤s lt) = s≤s (<tos<s {x} {y} lt)
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183
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184 ≤to< : {x y : ℕ } → x < y → x ≤ y
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185 ≤to< {zero} {suc y} (s≤s z≤n) = z≤n
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186 ≤to< {suc x} {suc y} (s≤s lt) = s≤s (≤to< {x} {y} lt)
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187
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188 x<y→≤ : {x y : ℕ } → x < y → x ≤ suc y
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189 x<y→≤ {zero} {.(suc _)} (s≤s z≤n) = z≤n
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190 x<y→≤ {suc x} {suc y} (s≤s lt) = s≤s (x<y→≤ {x} {y} lt)
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191
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192 open import Data.Product
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193
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194 minus<=0 : {x y : ℕ } → x ≤ y → minus x y ≡ 0
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195 minus<=0 {0} {zero} z≤n = refl
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196 minus<=0 {0} {suc y} z≤n = refl
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197 minus<=0 {suc x} {suc y} (s≤s le) = minus<=0 {x} {y} le
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198
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199 minus>0 : {x y : ℕ } → x < y → 0 < minus y x
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200 minus>0 {zero} {suc _} (s≤s z≤n) = s≤s z≤n
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201 minus>0 {suc x} {suc y} (s≤s lt) = minus>0 {x} {y} lt
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202
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203 distr-minus-* : {x y z : ℕ } → (minus x y) * z ≡ minus (x * z) (y * z)
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204 distr-minus-* {x} {zero} {z} = refl
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205 distr-minus-* {x} {suc y} {z} with <-cmp x y
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206 distr-minus-* {x} {suc y} {z} | tri< a ¬b ¬c = begin
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207 minus x (suc y) * z
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208 ≡⟨ cong (λ k → k * z ) (minus<=0 {x} {suc y} (x<y→≤ a)) ⟩
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209 0 * z
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210 ≡⟨ sym (minus<=0 {x * z} {z + y * z} le ) ⟩
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211 minus (x * z) (z + y * z)
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212 ∎ where
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213 open ≡-Reasoning
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214 le : x * z ≤ z + y * z
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215 le = ≤-trans lemma (subst (λ k → y * z ≤ k ) (+-comm _ z ) (x≤x+y {y * z} {z} ) ) where
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216 lemma : x * z ≤ y * z
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217 lemma = *≤ {x} {y} {z} (≤to< a)
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218 distr-minus-* {x} {suc y} {z} | tri≈ ¬a refl ¬c = begin
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219 minus x (suc y) * z
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220 ≡⟨ cong (λ k → k * z ) (minus<=0 {x} {suc y} refl-≤s ) ⟩
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221 0 * z
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222 ≡⟨ sym (minus<=0 {x * z} {z + y * z} (lt {x} {z} )) ⟩
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223 minus (x * z) (z + y * z)
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224 ∎ where
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225 open ≡-Reasoning
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226 lt : {x z : ℕ } → x * z ≤ z + x * z
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227 lt {zero} {zero} = z≤n
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228 lt {suc x} {zero} = lt {x} {zero}
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229 lt {x} {suc z} = ≤-trans lemma refl-≤s where
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230 lemma : x * suc z ≤ z + x * suc z
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231 lemma = subst (λ k → x * suc z ≤ k ) (+-comm _ z) (x≤x+y {x * suc z} {z})
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232 distr-minus-* {x} {suc y} {z} | tri> ¬a ¬b c = +m= {_} {_} {suc y * z} ( begin
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233 minus x (suc y) * z + suc y * z
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234 ≡⟨ sym (proj₂ *-distrib-+ z (minus x (suc y) ) _) ⟩
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235 ( minus x (suc y) + suc y ) * z
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236 ≡⟨ cong (λ k → k * z) (minus+n {x} {suc y} (s≤s c)) ⟩
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237 x * z
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238 ≡⟨ sym (minus+n {x * z} {suc y * z} (s≤s (lt c))) ⟩
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239 minus (x * z) (suc y * z) + suc y * z
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240 ∎ ) where
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241 open ≡-Reasoning
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242 lt : {x y z : ℕ } → suc y ≤ x → z + y * z ≤ x * z
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243 lt {x} {y} {z} le = *≤ le
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244
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245 minus- : {x y z : ℕ } → suc x > z + y → minus (minus x y) z ≡ minus x (y + z)
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246 minus- {x} {y} {z} gt = +m= {_} {_} {z} ( begin
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247 minus (minus x y) z + z
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248 ≡⟨ minus+n {_} {z} lemma ⟩
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249 minus x y
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250 ≡⟨ +m= {_} {_} {y} ( begin
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251 minus x y + y
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252 ≡⟨ minus+n {_} {y} lemma1 ⟩
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253 x
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254 ≡⟨ sym ( minus+n {_} {z + y} gt ) ⟩
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255 minus x (z + y) + (z + y)
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256 ≡⟨ sym ( +-assoc (minus x (z + y)) _ _ ) ⟩
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257 minus x (z + y) + z + y
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258 ∎ ) ⟩
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259 minus x (z + y) + z
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260 ≡⟨ cong (λ k → minus x k + z ) (+-comm _ y ) ⟩
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261 minus x (y + z) + z
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262 ∎ ) where
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263 open ≡-Reasoning
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264 lemma1 : suc x > y
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265 lemma1 = x+y<z→x<z (subst (λ k → k < suc x ) (+-comm z _ ) gt )
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266 lemma : suc (minus x y) > z
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267 lemma = <-minus {_} {_} {y} ( subst ( λ x → z + y < suc x ) (sym (minus+n {x} {y} lemma1 )) gt )
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268
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269 minus-* : {M k n : ℕ } → n < k → minus k (suc n) * M ≡ minus (minus k n * M ) M
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270 minus-* {zero} {k} {n} lt = begin
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271 minus k (suc n) * zero
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272 ≡⟨ *-comm (minus k (suc n)) zero ⟩
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273 zero * minus k (suc n)
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274 ≡⟨⟩
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275 0 * minus k n
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276 ≡⟨ *-comm 0 (minus k n) ⟩
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277 minus (minus k n * 0 ) 0
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278 ∎ where
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279 open ≡-Reasoning
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280 minus-* {suc m} {k} {n} lt with <-cmp k 1
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281 minus-* {suc m} {.0} {zero} lt | tri< (s≤s z≤n) ¬b ¬c = refl
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282 minus-* {suc m} {.0} {suc n} lt | tri< (s≤s z≤n) ¬b ¬c = refl
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283 minus-* {suc zero} {.1} {zero} lt | tri≈ ¬a refl ¬c = refl
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284 minus-* {suc (suc m)} {.1} {zero} lt | tri≈ ¬a refl ¬c = minus-* {suc m} {1} {zero} lt
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285 minus-* {suc m} {.1} {suc n} (s≤s ()) | tri≈ ¬a refl ¬c
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286 minus-* {suc m} {k} {n} lt | tri> ¬a ¬b c = begin
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287 minus k (suc n) * M
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288 ≡⟨ distr-minus-* {k} {suc n} {M} ⟩
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289 minus (k * M ) ((suc n) * M)
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290 ≡⟨⟩
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291 minus (k * M ) (M + n * M )
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292 ≡⟨ cong (λ x → minus (k * M) x) (+-comm M _ ) ⟩
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293 minus (k * M ) ((n * M) + M )
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294 ≡⟨ sym ( minus- {k * M} {n * M} (lemma lt) ) ⟩
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295 minus (minus (k * M ) (n * M)) M
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296 ≡⟨ cong (λ x → minus x M ) ( sym ( distr-minus-* {k} {n} )) ⟩
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297 minus (minus k n * M ) M
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298 ∎ where
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299 M = suc m
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300 lemma : {n k m : ℕ } → n < k → suc (k * suc m) > suc m + n * suc m
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301 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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302 lemma {suc n} {suc k} {m} lt = begin
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303 suc (suc m + suc n * suc m)
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304 ≡⟨⟩
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305 suc ( suc (suc n) * suc m)
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306 ≤⟨ ≤-plus-0 {_} {_} {1} (*≤ lt ) ⟩
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307 suc (suc k * suc m)
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308 ∎ where open ≤-Reasoning
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309 open ≡-Reasoning
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