1 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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2 | (* Nat.ma: Natural numbers and common arithmetical functions. *) |
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3 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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4 | include "Cartesian.ma". |
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5 | include "Bool.ma". |
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6 | include "Connectives.ma". |
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7 | |
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8 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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9 | (* The datatype. *) |
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10 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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11 | ninductive Nat: Type[0] ≝ |
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12 | Z: Nat |
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13 | | S: Nat → Nat. |
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14 | |
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15 | ntheorem S_inj: ∀n,m:Nat. S n = S m → n=m. |
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16 | #n m H; |
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17 | nchange with (n = match S m with [ Z ⇒ Z | S x ⇒ x]); |
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18 | napply (match H return λx.λ_. n = match x with [Z ⇒ Z | S x ⇒ x] with [ refl ⇒ ? ]); |
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19 | nnormalize; /3/. |
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20 | nqed. |
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21 | |
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22 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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23 | (* Orderings. *) |
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24 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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25 | |
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26 | ninductive less_than_or_equal_p (n: Nat): Nat → Prop ≝ |
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27 | ltoe_refl: less_than_or_equal_p n n |
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28 | | ltoe_step: ∀m: Nat. less_than_or_equal_p n m → less_than_or_equal_p n (S m). |
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29 | |
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30 | nlet rec less_than_or_equal_b (m: Nat) (n: Nat) on m: Bool ≝ |
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31 | match m with |
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32 | [ Z ⇒ true |
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33 | | S o ⇒ |
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34 | match n with |
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35 | [ Z ⇒ false |
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36 | | S p ⇒ less_than_or_equal_b o p |
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37 | ] |
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38 | ]. |
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39 | |
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40 | notation "hvbox(n break ≤ m)" |
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41 | non associative with precedence 47 |
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42 | for @{ 'less_than_or_equal $n $m }. |
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43 | |
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44 | interpretation "Nat less than or equal prop" 'less_than_or_equal n m = (less_than_or_equal_p n m). |
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45 | |
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46 | notation "hvbox(n break ≲ m)" |
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47 | non associative with precedence 47 |
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48 | for @{ 'less_than_or_equalb $n $m }. |
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49 | |
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50 | interpretation "Nat less than or equal bool" 'less_than_or_equalb n m = (less_than_or_equal_b n m). |
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51 | |
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52 | |
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53 | ndefinition greater_than_or_equal_p: Nat → Nat → Prop ≝ |
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54 | λm, n: Nat. |
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55 | n ≤ m. |
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56 | |
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57 | ndefinition greater_than_or_equal_b: ∀m, n: Nat. Bool ≝ |
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58 | λm, n: Nat. |
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59 | less_than_or_equal_b n m. |
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60 | |
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61 | notation "hvbox(n break ≥ m)" |
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62 | non associative with precedence 47 |
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63 | for @{ 'greater_than_or_equal $n $m }. |
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64 | |
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65 | |
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66 | interpretation "Nat greater than or equal prop" 'greater_than_or_equal n m = (greater_than_or_equal_p n m). |
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67 | |
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68 | notation "hvbox(n break ≳ m)" |
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69 | non associative with precedence 47 |
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70 | for @{ 'greater_than_or_equalb $n $m }. |
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71 | |
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72 | interpretation "Nat greater than or equal bool" 'greater_than_or_equalb n m = (greater_than_or_equal_b n m). |
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73 | |
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74 | |
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75 | (* Add Boolean versions. *) |
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76 | ndefinition less_than_p ≝ |
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77 | λm: Nat. |
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78 | λn: Nat. |
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79 | less_than_or_equal_p (S m) n. |
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80 | |
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81 | notation "hvbox(n break < m)" |
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82 | non associative with precedence 47 |
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83 | for @{ 'less_than $n $m }. |
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84 | |
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85 | interpretation "Nat less than prop" 'less_than n m = (less_than_p n m). |
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86 | |
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87 | ndefinition greater_than_p ≝ |
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88 | λm, n: Nat. |
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89 | n < m. |
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90 | |
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91 | notation "hvbox(n break > m)" |
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92 | non associative with precedence 47 |
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93 | for @{ 'greater_than $n $m }. |
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94 | |
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95 | interpretation "Nat greater than prop" 'greater_than n m = (greater_than_p n m). |
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96 | |
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97 | nlet rec less_than_b (m: Nat) (n: Nat) on m ≝ |
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98 | match m with |
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99 | [ Z ⇒ |
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100 | match n with |
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101 | [ Z ⇒ false |
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102 | | S o ⇒ true |
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103 | ] |
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104 | | S o ⇒ |
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105 | match n with |
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106 | [ Z ⇒ false |
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107 | | S p ⇒ less_than_b o p |
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108 | ] |
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109 | ]. |
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110 | |
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111 | (* interpretation "Nat less than bool" 'less_than n m = (less_than_b n m). *) |
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112 | |
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113 | ndefinition greater_than_b ≝ |
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114 | λm, n: Nat. |
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115 | less_than_b n m. |
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116 | |
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117 | (* interpretation "Nat greater than bool" 'greater_than n m = (greater_than_b n m). *) |
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118 | |
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119 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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120 | (* Addition and subtraction. *) |
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121 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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122 | nlet rec plus (n: Nat) (o: Nat) on n ≝ |
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123 | match n with |
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124 | [ Z ⇒ o |
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125 | | S p ⇒ S (plus p o) |
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126 | ]. |
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127 | |
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128 | notation "hvbox(n break + m)" |
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129 | right associative with precedence 52 |
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130 | for @{ 'plus $n $m }. |
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131 | |
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132 | interpretation "Nat plus" 'plus n m = (plus n m). |
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133 | |
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134 | nlet rec minus (n: Nat) (o: Nat) on n ≝ |
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135 | match n with |
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136 | [ Z ⇒ Z |
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137 | | S p ⇒ |
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138 | match o with |
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139 | [ Z ⇒ S p |
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140 | | S q ⇒ minus p q |
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141 | ] |
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142 | ]. |
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143 | |
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144 | notation "hvbox(n break - m)" |
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145 | right associative with precedence 47 |
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146 | for @{ 'minus $n $m }. |
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147 | |
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148 | interpretation "Nat minus" 'minus n m = (minus n m). |
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149 | |
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150 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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151 | (* Multiplication, modulus and division. *) |
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152 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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153 | |
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154 | nlet rec multiplication (n: Nat) (o: Nat) on n ≝ |
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155 | match n with |
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156 | [ Z ⇒ Z |
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157 | | S p ⇒ o + (multiplication p o) |
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158 | ]. |
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159 | |
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160 | notation "hvbox(n break * m)" |
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161 | right associative with precedence 47 |
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162 | for @{ 'multiplication $n $m }. |
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163 | |
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164 | interpretation "Nat multiplication" 'times n m = (multiplication n m). |
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165 | |
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166 | nlet rec division_aux (m: Nat) (n : Nat) (p: Nat) ≝ |
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167 | match less_than_or_equal_b n p with |
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168 | [ true ⇒ Z |
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169 | | false ⇒ |
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170 | match m with |
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171 | [ Z ⇒ Z |
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172 | | (S q) ⇒ S (division_aux q (n - (S p)) p) |
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173 | ] |
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174 | ]. |
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175 | |
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176 | ndefinition division ≝ |
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177 | λm, n: Nat. |
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178 | match n with |
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179 | [ Z ⇒ S m |
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180 | | S o ⇒ division_aux m m o |
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181 | ]. |
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182 | |
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183 | notation "hvbox(n break ÷ m)" |
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184 | right associative with precedence 47 |
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185 | for @{ 'division $n $m }. |
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186 | |
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187 | interpretation "Nat division" 'division n m = (division n m). |
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188 | |
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189 | nlet rec modulus_aux (m: Nat) (n: Nat) (p: Nat) ≝ |
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190 | match less_than_or_equal_b n p with |
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191 | [ true ⇒ n |
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192 | | false ⇒ |
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193 | match m with |
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194 | [ Z ⇒ n |
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195 | | S o ⇒ modulus_aux o (n - (S p)) p |
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196 | ] |
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197 | ]. |
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198 | |
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199 | ndefinition modulus ≝ |
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200 | λm, n: Nat. |
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201 | match n with |
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202 | [ Z ⇒ m |
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203 | | S o ⇒ modulus_aux m m o |
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204 | ]. |
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205 | |
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206 | notation "hvbox(n break 'mod' m)" |
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207 | right associative with precedence 47 |
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208 | for @{ 'modulus $n $m }. |
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209 | |
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210 | interpretation "Nat modulus" 'modulus m n = (modulus m n). |
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211 | |
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212 | ndefinition divide_with_remainder ≝ |
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213 | λm, n: Nat. |
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214 | mk_Cartesian Nat Nat (m ÷ n) (modulus m n). |
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215 | |
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216 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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217 | (* Exponentials, and square roots. *) |
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218 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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219 | |
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220 | nlet rec exponential (m: Nat) (n: Nat) on n ≝ |
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221 | match n with |
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222 | [ Z ⇒ S (Z) |
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223 | | S o ⇒ m * exponential m o |
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224 | ]. |
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225 | |
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226 | notation "hvbox(n ^ m)" |
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227 | left associative with precedence 52 |
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228 | for @{ 'exponential $n $m }. |
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229 | |
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230 | interpretation "Nat exponential" 'exponential n m = (exponential n m). |
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231 | |
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232 | nlet rec eq_n (m: Nat) (n: Nat) on m: Bool ≝ |
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233 | match m with |
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234 | [ Z ⇒ |
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235 | match n with |
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236 | [ Z ⇒ true |
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237 | | _ ⇒ false |
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238 | ] |
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239 | | S o ⇒ |
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240 | match n with |
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241 | [ S p ⇒ eq_n o p |
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242 | | _ ⇒ false |
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243 | ] |
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244 | ]. |
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245 | |
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246 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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247 | (* Greatest common divisor and least common multiple. *) |
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248 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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249 | |
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250 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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251 | (* Axioms. *) |
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252 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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253 | |
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254 | (*naxiom plus_minus_inverse_left: |
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255 | ∀m, n: Nat. |
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256 | (m + n) - n = m. |
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257 | *) |
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258 | |
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259 | ntheorem less_than_or_equal_monotone: |
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260 | ∀m, n: Nat. |
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261 | m ≤ n → (S m) ≤ (S n). |
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262 | #m n H; nelim H; /2/. |
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263 | nqed. |
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264 | |
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265 | nlemma trans_le: ∀n,m,l. n ≤ m → m ≤ l → n ≤ l. |
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266 | #n m l H H1; nelim H1; /2/. |
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267 | nqed. |
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268 | |
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269 | ntheorem less_than_or_equal_injective: |
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270 | ∀m, n: Nat. |
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271 | (S m) ≤ (S n) → m ≤ n. |
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272 | #m n H; |
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273 | nchange with (match S m with [ Z ⇒ Z | S x ⇒ x] ≤ match S n with [ Z ⇒ Z | S x ⇒ x]); |
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274 | napply (match H return λx.λ_. m ≤ match x with [Z ⇒ Z | S x ⇒ x] with [ ltoe_refl ⇒ ? | ltoe_step y H ⇒ ? ]); |
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275 | nnormalize; /3/. |
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276 | nqed. |
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277 | |
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278 | ntheorem succ_less_than_injective: |
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279 | ∀m, n: Nat. |
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280 | less_than_p (S m) (S n) → m < n. |
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281 | /2/. |
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282 | nqed. |
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283 | |
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284 | nlemma not_less_than_S_Z: |
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285 | ∀m,n: Nat. S m ≤ n → ¬ (n = Z). |
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286 | #m n H; nelim H [ @; #K; ndestruct | #y H1 H2; @; #K; ndestruct ] |
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287 | nqed. |
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288 | |
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289 | ntheorem nothing_less_than_Z: |
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290 | ∀m: Nat. |
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291 | ¬(m < Z). |
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292 | #m; @; #H; nlapply (not_less_than_S_Z m Z H); /2/; |
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293 | nqed. |
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294 | |
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295 | |
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296 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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297 | (* Lemmas. *) |
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298 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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299 | |
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300 | nlemma less_than_or_equal_zero: |
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301 | ∀n: Nat. |
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302 | Z ≤ n. |
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303 | #n. |
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304 | nelim n. |
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305 | //. |
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306 | #N. |
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307 | napply ltoe_step. |
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308 | nqed. |
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309 | |
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310 | (* |
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311 | nlemma less_than_or_equal_zero_equal_zero: |
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312 | ∀m: Nat. |
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313 | m ≤ Z → m = Z. |
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314 | #m. |
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315 | nelim m. |
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316 | //. |
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317 | #N H H2. |
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318 | nnormalize. |
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319 | *) |
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320 | |
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321 | nlemma less_than_or_equal_reflexive: |
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322 | ∀n: Nat. |
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323 | n ≤ n. |
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324 | #n. |
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325 | nelim n. |
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326 | nnormalize. |
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327 | @. |
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328 | #N H. |
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329 | nnormalize. |
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330 | //. |
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331 | nqed. |
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332 | |
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333 | (* |
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334 | nlemma less_than_or_equal_succ: |
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335 | ∀m, n: Nat. |
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336 | S m ≤ n → m ≤ n. |
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337 | #m n. |
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338 | nelim m. |
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339 | #H. |
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340 | napplyS less_than_or_equal_zero. |
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341 | #N H H2. |
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342 | nrewrite > H. |
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343 | nnormalize. |
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344 | |
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345 | nlemma less_than_or_equal_transitive: |
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346 | ∀m, n, o: Nat. |
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347 | m ≤ n ∧ n ≤ o → m ≤ o. |
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348 | #m n o. |
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349 | nelim m. |
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350 | #H. |
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351 | napply less_than_or_equal_zero. |
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352 | #N H H2. |
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353 | nnormalize. |
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354 | *) |
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355 | |
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356 | nlemma plus_zero: |
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357 | ∀n: Nat. |
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358 | n + Z = n. |
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359 | #n. |
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360 | nelim n. |
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361 | nnormalize. |
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362 | @. |
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363 | #N H. |
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364 | nnormalize. |
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365 | nrewrite > H. |
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366 | @. |
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367 | nqed. |
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368 | |
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369 | nlemma plus_associative: |
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370 | ∀m, n, o: Nat. |
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371 | (m + n) + o = m + (n + o). |
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372 | #m n o. |
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373 | nelim m. |
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374 | nnormalize. |
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375 | @. |
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376 | #N H. |
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377 | nnormalize. |
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378 | nrewrite > H. |
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379 | @. |
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380 | nqed. |
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381 | |
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382 | nlemma succ_plus: |
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383 | ∀m, n: Nat. |
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384 | S(m + n) = m + S(n). |
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385 | #m n. |
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386 | nelim m. |
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387 | nnormalize. |
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388 | @. |
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389 | #N H. |
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390 | nnormalize. |
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391 | nrewrite > H. |
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392 | @. |
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393 | nqed. |
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394 | |
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395 | nlemma succ_plus_succ_zero: |
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396 | ∀n: Nat. |
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397 | S n = plus n (S Z). |
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398 | #n. |
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399 | nelim n. |
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400 | //. |
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401 | #N H. |
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402 | nnormalize. |
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403 | nrewrite < H. |
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404 | @. |
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405 | nqed. |
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406 | |
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407 | nlemma plus_symmetrical: |
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408 | ∀m, n: Nat. |
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409 | m + n = n + m. |
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410 | #m n. |
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411 | nelim m. |
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412 | nnormalize. |
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413 | nelim n. |
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414 | nnormalize. |
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415 | @. |
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416 | #N H. |
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417 | nnormalize. |
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418 | nrewrite < H. |
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419 | @. |
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420 | #N H. |
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421 | nnormalize. |
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422 | nrewrite > H. |
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423 | napply succ_plus. |
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424 | nqed. |
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425 | |
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426 | nlemma multiplication_zero_right_neutral: |
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427 | ∀m: Nat. |
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428 | m * Z = Z. |
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429 | #m. |
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430 | nelim m. |
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431 | nnormalize. |
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432 | @. |
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433 | #N H. |
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434 | nnormalize. |
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435 | nrewrite > H. |
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436 | @. |
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437 | nqed. |
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438 | |
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439 | nlemma multiplication_succ: |
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440 | ∀m, n: Nat. |
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441 | m * S(n) = m + (m * n). |
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442 | #m n. |
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443 | nelim m. |
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444 | nnormalize. |
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445 | @. |
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446 | #N H. |
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447 | nnormalize. |
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448 | nrewrite > H. |
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449 | nrewrite < (plus_associative n N ?). |
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450 | nrewrite > (plus_symmetrical n N). |
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451 | nrewrite > (plus_associative N n ?). |
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452 | @. |
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453 | nqed. |
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454 | |
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455 | nlemma multiplication_symmetrical: |
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456 | ∀m, n: Nat. |
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457 | m * n = n * m. |
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458 | #m n. |
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459 | nelim m. |
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460 | nnormalize. |
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461 | nelim n. |
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462 | nnormalize. |
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463 | @. |
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464 | #N H. |
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465 | nnormalize. |
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466 | nrewrite < H. |
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467 | @. |
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468 | #N H. |
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469 | nnormalize. |
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470 | nrewrite > H. |
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471 | nrewrite > (multiplication_succ ? ?). |
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472 | @. |
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473 | nqed. |
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474 | |
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475 | nlemma multiplication_succ_zero_left_neutral: |
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476 | ∀m: Nat. |
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477 | (S Z) * m = m. |
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478 | #m. |
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479 | nelim m. |
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480 | nnormalize. |
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481 | @. |
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482 | #N H. |
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483 | nnormalize. |
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484 | nrewrite > (succ_plus ? ?). |
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485 | nrewrite < (succ_plus_succ_zero ?). |
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486 | @. |
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487 | nqed. |
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488 | |
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489 | nlemma multiplication_succ_zero_right_neutral: |
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490 | ∀m: Nat. |
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491 | m * (S Z) = m. |
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492 | #m. |
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493 | nelim m. |
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494 | nnormalize. |
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495 | @. |
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496 | #N H. |
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497 | nnormalize. |
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498 | nrewrite > H. |
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499 | @. |
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500 | nqed. |
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501 | |
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502 | nlemma multiplication_distributes_right_plus: |
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503 | ∀m, n, o: Nat. |
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504 | (m + n) * o = m * o + n * o. |
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505 | #m n o. |
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506 | nelim m. |
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507 | nnormalize. |
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508 | @. |
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509 | #N H. |
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510 | nnormalize. |
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511 | nrewrite > H. |
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512 | nrewrite < (plus_associative ? ? ?). |
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513 | @. |
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514 | nqed. |
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515 | |
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516 | nlemma multiplication_distributes_left_plus: |
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517 | ∀m, n, o: Nat. |
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518 | o * (m + n) = o * m + o * n. |
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519 | #m n o. |
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520 | nelim o. |
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521 | //. |
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522 | #N H. |
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523 | nnormalize. |
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524 | nrewrite > H. |
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525 | nrewrite < (plus_associative ? n (N * n)). |
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526 | nrewrite > (plus_symmetrical (m + N * m) n). |
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527 | nrewrite < (plus_associative n m (N * m)). |
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528 | nrewrite < (plus_symmetrical n m). |
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529 | nrewrite > (plus_associative (n + m) (N * m) (N * n)). |
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530 | @. |
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531 | nqed. |
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532 | |
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533 | nlemma mutliplication_associative: |
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534 | ∀m, n, o: Nat. |
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535 | m * (n * o) = (m * n) * o. |
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536 | #m n o. |
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537 | nelim m. |
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538 | nnormalize. |
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539 | @. |
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540 | #N H. |
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541 | nnormalize. |
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542 | nrewrite > H. |
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543 | nrewrite > (multiplication_distributes_right_plus ? ? ?). |
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544 | @. |
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545 | nqed. |
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546 | |
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547 | nlemma minus_minus: |
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548 | ∀n: Nat. |
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549 | n - n = Z. |
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550 | #n. |
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551 | nelim n. |
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552 | nnormalize. |
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553 | @. |
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554 | #N H. |
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555 | nnormalize. |
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556 | nrewrite > H. |
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557 | @. |
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558 | nqed. |
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559 | |
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560 | |
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561 | nlemma succ_injective: |
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562 | ∀m, n: Nat. |
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563 | S m = S n → m = n. |
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564 | #m n H; |
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565 | napply (match H return λy.λ_.m = match y with [Z ⇒ Z | S z ⇒ z] with |
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566 | [refl ⇒ ? ]); |
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567 | @; |
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568 | nqed. |
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569 | |
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570 | ntheorem eq_f: ∀A,B:Type[0].∀f:A→B.∀a,a'. a=a' → f a = f a'. |
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571 | //; |
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572 | nqed. |
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573 | |
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574 | ntheorem plus_minus_inverse_right: |
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575 | ∀m, n: Nat. |
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576 | n ≤ m → (m - n) + n = m. |
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577 | #m; nelim m |
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578 | [ #n; nelim n; //; #H1 H2 H3; ncases (nothing_less_than_Z H1); |
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579 | #K; ncases (K H3) |
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580 | | #y H1 x H2; nnormalize; ngeneralize in match H2; ncases x; nnormalize; /2/; |
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581 | #w; nrewrite < succ_plus; /4/. ##] |
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582 | nqed. |
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583 | |
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584 | (* |
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585 | |
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586 | nlemma plus_minus_associate: |
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587 | ∀m, n, o: Nat. |
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588 | (m + n) - o = m + (n - o). |
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589 | #m n o. |
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590 | nelim m. |
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591 | nnormalize. |
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592 | @. |
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593 | #N H. |
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594 | |
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595 | |
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596 | nlemma plus_minus_inverses: |
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597 | ∀m, n: Nat. |
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598 | (m + n) - n = m. |
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599 | #m n. |
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600 | nelim m. |
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601 | nnormalize. |
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602 | napply minus_minus. |
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603 | #N H. |
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604 | *) |
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605 | |
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606 | ntheorem less_than_or_equal_b_correct: ∀m,n. less_than_or_equal_b m n = true → m ≤ n. |
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607 | #m; nelim m; //; #y H1 z; ncases z; nnormalize |
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608 | [ #H; ndestruct | /3/ ] |
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609 | nqed. |
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610 | |
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611 | ntheorem less_than_or_equal_b_complete: ∀m,n. less_than_or_equal_b m n = false → ¬(m ≤ n). |
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612 | #m; nelim m; nnormalize |
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613 | [ #n H; ndestruct | #y H1 z; ncases z; nnormalize |
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614 | [ #H; /2/ | /3/ ] |
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615 | nqed. |
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616 | |
---|
617 | ndefinition less_than_or_equal_b_elim: |
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618 | ∀m,n. ∀P: Bool → Type[0]. (m ≤ n → P true) → (¬(m ≤ n) → P false) → |
---|
619 | P (less_than_or_equal_b m n). |
---|
620 | #m n P H1 H2; nlapply (less_than_or_equal_b_correct m n); |
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621 | nlapply (less_than_or_equal_b_complete m n); |
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622 | ncases (less_than_or_equal_b m n); /3/. |
---|
623 | nqed. |
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624 | |
---|
625 | ndefinition greater_than_or_equal_b_elim: |
---|
626 | ∀m,n. ∀P: Bool → Type[0]. (m ≥ n → P true) → (¬(m ≥ n) → P false) → |
---|
627 | P (greater_than_or_equal_b m n). |
---|
628 | #m n; napply less_than_or_equal_b_elim; |
---|
629 | nqed. |
---|
630 | |
---|
631 | ntheorem less_than_b_correct: ∀m,n. less_than_b m n = true → m < n. |
---|
632 | #m; nelim m |
---|
633 | [ #n; ncases n [ #H; nnormalize in H; ndestruct | #z H; /2/ ] |
---|
634 | ##| #y H z; ncases z [ nnormalize; #K; ndestruct | #o K; |
---|
635 | napply less_than_or_equal_monotone; napply H; napply K ] |
---|
636 | nqed. |
---|
637 | |
---|
638 | ntheorem less_than_b_complete: ∀m,n. less_than_b m n = false → ¬(m < n). |
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639 | #m; nelim m |
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640 | [ #n; ncases n; //; #y H1; nnormalize in H1; ndestruct |
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641 | | #y H z; ncases z; //; #o H2; nlapply (H … H2); /3/. |
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642 | nqed. |
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643 | |
---|
644 | ndefinition less_than_b_elim: |
---|
645 | ∀m,n. ∀P: Bool → Type[0]. (m < n → P true) → (¬(m < n) → P false) → |
---|
646 | P (less_than_b m n). |
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647 | #m n; nlapply (less_than_b_correct m n); nlapply (less_than_b_complete m n); |
---|
648 | ncases (less_than_b m n); /3/. |
---|
649 | nqed. |
---|
650 | |
---|
651 | ndefinition greater_than_b_elim: |
---|
652 | ∀m,n. ∀P: Bool → Type[0]. (m > n → P true) → (¬(m > n) → P false) → |
---|
653 | P (greater_than_b m n). |
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654 | #m n; napply less_than_b_elim. |
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655 | nqed. |
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656 | |
---|
657 | nlemma less_than_or_equal_plus: |
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658 | ∀n,m: Nat. |
---|
659 | n ≤ n + m. |
---|
660 | #n m. |
---|
661 | nelim n. |
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662 | //. |
---|
663 | #N H. |
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664 | napply (less_than_or_equal_monotone). |
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665 | /2/. |
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666 | nqed. |
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667 | |
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668 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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669 | (* Numbers. *) |
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670 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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671 | |
---|
672 | ndefinition one ≝ S Z. |
---|
673 | ndefinition two ≝ (S(S(Z))). |
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674 | ndefinition three ≝ two + one. |
---|
675 | ndefinition four ≝ two + two. |
---|
676 | ndefinition five ≝ three + two. |
---|
677 | ndefinition six ≝ three + three. |
---|
678 | ndefinition seven ≝ three + four. |
---|
679 | ndefinition eight ≝ four + four. |
---|
680 | ndefinition nine ≝ five + four. |
---|
681 | ndefinition ten ≝ five + five. |
---|
682 | ndefinition eleven ≝ six + five. |
---|
683 | ndefinition twelve ≝ six + six. |
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684 | ndefinition thirteen ≝ seven + six. |
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685 | ndefinition fourteen ≝ seven + seven. |
---|
686 | ndefinition fifteen ≝ eight + seven. |
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687 | ndefinition sixteen ≝ eight + eight. |
---|
688 | ndefinition seventeen ≝ nine + eight. |
---|
689 | ndefinition eighteen ≝ nine + nine. |
---|
690 | ndefinition nineteen ≝ ten + nine. |
---|
691 | ndefinition twenty_four ≝ sixteen + eight. |
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692 | ndefinition thirty_two ≝ sixteen + sixteen. |
---|
693 | ndefinition one_hundred ≝ ten * ten. |
---|
694 | ndefinition one_hundred_and_twenty_eight ≝ sixteen * eight. |
---|
695 | ndefinition one_hundred_and_twenty_nine ≝ one_hundred_and_twenty_eight + one. |
---|
696 | ndefinition one_hundred_and_thirty ≝ one_hundred_and_twenty_nine + one. |
---|
697 | ndefinition one_hundred_and_thirty_one ≝ one_hundred_and_thirty + one. |
---|
698 | ndefinition one_hundred_and_thirty_five ≝ one_hundred_and_twenty_eight + seven. |
---|
699 | ndefinition one_hundred_and_thirty_six ≝ one_hundred_and_thirty_five + one. |
---|
700 | ndefinition one_hundred_and_thirty_seven ≝ one_hundred_and_thirty_six + one. |
---|
701 | ndefinition one_hundred_and_thirty_eight ≝ one_hundred_and_twenty_eight + ten. |
---|
702 | ndefinition one_hundred_and_thirty_nine ≝ one_hundred_and_thirty_eight + one. |
---|
703 | ndefinition one_hundred_and_forty ≝ one_hundred_and_thirty_nine + one. |
---|
704 | ndefinition one_hundred_and_forty_one ≝ one_hundred_and_forty + one. |
---|
705 | ndefinition one_hundred_and_forty_four ≝ one_hundred_and_twenty_eight + sixteen. |
---|
706 | ndefinition one_hundred_and_fifty_two ≝ one_hundred_and_forty_four + eight. |
---|
707 | ndefinition one_hundred_and_fifty_three ≝ one_hundred_and_forty_four + nine. |
---|
708 | ndefinition one_hundred_and_sixty ≝ one_hundred_and_forty_four + sixteen. |
---|
709 | ndefinition one_hundred_and_sixty_eight ≝ one_hundred_and_sixty + eight. |
---|
710 | ndefinition one_hundred_and_seventy_six ≝ one_hundred_and_sixty + sixteen. |
---|
711 | ndefinition one_hundred_and_eighty_four ≝ one_hundred_and_seventy_six + eight. |
---|
712 | ndefinition two_hundred ≝ one_hundred + one_hundred. |
---|
713 | ndefinition two_hundred_and_two ≝ two_hundred + two. |
---|
714 | ndefinition two_hundred_and_three ≝ two_hundred_and_two + one. |
---|
715 | ndefinition two_hundred_and_four ≝ two_hundred_and_three + one. |
---|
716 | ndefinition two_hundred_and_five ≝ two_hundred_and_four + one. |
---|
717 | ndefinition two_hundred_and_eight ≝ two_hundred_and_five + three. |
---|
718 | ndefinition two_hundred_and_twenty_four ≝ two_hundred_and_eight + sixteen. |
---|
719 | ndefinition two_hundred_and_forty ≝ two_hundred_and_twenty_four + sixteen. |
---|
720 | ndefinition two_hundred_and_fifty_six ≝ |
---|
721 | one_hundred_and_twenty_eight + one_hundred_and_twenty_eight. |
---|
722 | ndefinition sixty_five_thousand_five_hundred_and_thirty_six ≝ |
---|
723 | two_hundred_and_fifty_six * two_hundred_and_fifty_six. |
---|