1 (* Title: HOL/NatBin.thy |
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2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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3 Copyright 1999 University of Cambridge |
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4 *) |
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5 |
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6 header {* Binary arithmetic for the natural numbers *} |
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7 |
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8 theory NatBin |
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9 imports IntDiv |
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10 uses ("Tools/nat_simprocs.ML") |
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11 begin |
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12 |
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13 text {* |
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14 Arithmetic for naturals is reduced to that for the non-negative integers. |
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15 *} |
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16 |
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17 instantiation nat :: number |
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18 begin |
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19 |
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20 definition |
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21 nat_number_of_def [code inline, code del]: "number_of v = nat (number_of v)" |
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22 |
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23 instance .. |
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24 |
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25 end |
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26 |
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27 lemma [code post]: |
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28 "nat (number_of v) = number_of v" |
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29 unfolding nat_number_of_def .. |
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30 |
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31 abbreviation (xsymbols) |
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32 power2 :: "'a::power => 'a" ("(_\<twosuperior>)" [1000] 999) where |
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33 "x\<twosuperior> == x^2" |
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34 |
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35 notation (latex output) |
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36 power2 ("(_\<twosuperior>)" [1000] 999) |
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37 |
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38 notation (HTML output) |
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39 power2 ("(_\<twosuperior>)" [1000] 999) |
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40 |
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41 |
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42 subsection {* Predicate for negative binary numbers *} |
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43 |
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44 definition neg :: "int \<Rightarrow> bool" where |
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45 "neg Z \<longleftrightarrow> Z < 0" |
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46 |
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47 lemma not_neg_int [simp]: "~ neg (of_nat n)" |
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48 by (simp add: neg_def) |
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49 |
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50 lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))" |
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51 by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc) |
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52 |
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53 lemmas neg_eq_less_0 = neg_def |
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54 |
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55 lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)" |
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56 by (simp add: neg_def linorder_not_less) |
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57 |
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58 text{*To simplify inequalities when Numeral1 can get simplified to 1*} |
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59 |
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60 lemma not_neg_0: "~ neg 0" |
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61 by (simp add: One_int_def neg_def) |
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62 |
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63 lemma not_neg_1: "~ neg 1" |
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64 by (simp add: neg_def linorder_not_less zero_le_one) |
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65 |
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66 lemma neg_nat: "neg z ==> nat z = 0" |
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67 by (simp add: neg_def order_less_imp_le) |
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68 |
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69 lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z" |
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70 by (simp add: linorder_not_less neg_def) |
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71 |
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72 text {* |
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73 If @{term Numeral0} is rewritten to 0 then this rule can't be applied: |
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74 @{term Numeral0} IS @{term "number_of Pls"} |
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75 *} |
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76 |
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77 lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)" |
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78 by (simp add: neg_def) |
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79 |
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80 lemma neg_number_of_Min: "neg (number_of Int.Min)" |
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81 by (simp add: neg_def) |
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82 |
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83 lemma neg_number_of_Bit0: |
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84 "neg (number_of (Int.Bit0 w)) = neg (number_of w)" |
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85 by (simp add: neg_def) |
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86 |
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87 lemma neg_number_of_Bit1: |
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88 "neg (number_of (Int.Bit1 w)) = neg (number_of w)" |
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89 by (simp add: neg_def) |
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90 |
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91 lemmas neg_simps [simp] = |
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92 not_neg_0 not_neg_1 |
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93 not_neg_number_of_Pls neg_number_of_Min |
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94 neg_number_of_Bit0 neg_number_of_Bit1 |
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95 |
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96 |
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97 subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*} |
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98 |
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99 declare nat_0 [simp] nat_1 [simp] |
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100 |
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101 lemma nat_number_of [simp]: "nat (number_of w) = number_of w" |
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102 by (simp add: nat_number_of_def) |
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103 |
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104 lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)" |
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105 by (simp add: nat_number_of_def) |
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106 |
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107 lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)" |
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108 by (simp add: nat_1 nat_number_of_def) |
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109 |
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110 lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0" |
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111 by (simp add: nat_numeral_1_eq_1) |
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112 |
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113 lemma numeral_2_eq_2: "2 = Suc (Suc 0)" |
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114 apply (unfold nat_number_of_def) |
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115 apply (rule nat_2) |
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116 done |
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117 |
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118 |
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119 subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*} |
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120 |
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121 lemma int_nat_number_of [simp]: |
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122 "int (number_of v) = |
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123 (if neg (number_of v :: int) then 0 |
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124 else (number_of v :: int))" |
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125 unfolding nat_number_of_def number_of_is_id neg_def |
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126 by simp |
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127 |
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128 |
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129 subsubsection{*Successor *} |
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130 |
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131 lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)" |
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132 apply (rule sym) |
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133 apply (simp add: nat_eq_iff int_Suc) |
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134 done |
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135 |
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136 lemma Suc_nat_number_of_add: |
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137 "Suc (number_of v + n) = |
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138 (if neg (number_of v :: int) then 1+n else number_of (Int.succ v) + n)" |
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139 unfolding nat_number_of_def number_of_is_id neg_def numeral_simps |
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140 by (simp add: Suc_nat_eq_nat_zadd1 add_ac) |
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141 |
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142 lemma Suc_nat_number_of [simp]: |
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143 "Suc (number_of v) = |
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144 (if neg (number_of v :: int) then 1 else number_of (Int.succ v))" |
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145 apply (cut_tac n = 0 in Suc_nat_number_of_add) |
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146 apply (simp cong del: if_weak_cong) |
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147 done |
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148 |
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149 |
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150 subsubsection{*Addition *} |
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151 |
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152 lemma add_nat_number_of [simp]: |
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153 "(number_of v :: nat) + number_of v' = |
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154 (if v < Int.Pls then number_of v' |
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155 else if v' < Int.Pls then number_of v |
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156 else number_of (v + v'))" |
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157 unfolding nat_number_of_def number_of_is_id numeral_simps |
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158 by (simp add: nat_add_distrib) |
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159 |
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160 lemma nat_number_of_add_1 [simp]: |
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161 "number_of v + (1::nat) = |
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162 (if v < Int.Pls then 1 else number_of (Int.succ v))" |
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163 unfolding nat_number_of_def number_of_is_id numeral_simps |
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164 by (simp add: nat_add_distrib) |
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165 |
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166 lemma nat_1_add_number_of [simp]: |
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167 "(1::nat) + number_of v = |
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168 (if v < Int.Pls then 1 else number_of (Int.succ v))" |
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169 unfolding nat_number_of_def number_of_is_id numeral_simps |
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170 by (simp add: nat_add_distrib) |
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171 |
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172 lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)" |
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173 by (rule int_int_eq [THEN iffD1]) simp |
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174 |
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175 |
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176 subsubsection{*Subtraction *} |
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177 |
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178 lemma diff_nat_eq_if: |
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179 "nat z - nat z' = |
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180 (if neg z' then nat z |
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181 else let d = z-z' in |
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182 if neg d then 0 else nat d)" |
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183 by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0) |
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184 |
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185 |
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186 lemma diff_nat_number_of [simp]: |
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187 "(number_of v :: nat) - number_of v' = |
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188 (if v' < Int.Pls then number_of v |
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189 else let d = number_of (v + uminus v') in |
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190 if neg d then 0 else nat d)" |
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191 unfolding nat_number_of_def number_of_is_id numeral_simps neg_def |
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192 by auto |
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193 |
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194 lemma nat_number_of_diff_1 [simp]: |
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195 "number_of v - (1::nat) = |
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196 (if v \<le> Int.Pls then 0 else number_of (Int.pred v))" |
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197 unfolding nat_number_of_def number_of_is_id numeral_simps |
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198 by auto |
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199 |
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200 |
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201 subsubsection{*Multiplication *} |
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202 |
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203 lemma mult_nat_number_of [simp]: |
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204 "(number_of v :: nat) * number_of v' = |
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205 (if v < Int.Pls then 0 else number_of (v * v'))" |
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206 unfolding nat_number_of_def number_of_is_id numeral_simps |
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207 by (simp add: nat_mult_distrib) |
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208 |
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209 |
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210 subsubsection{*Quotient *} |
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211 |
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212 lemma div_nat_number_of [simp]: |
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213 "(number_of v :: nat) div number_of v' = |
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214 (if neg (number_of v :: int) then 0 |
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215 else nat (number_of v div number_of v'))" |
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216 unfolding nat_number_of_def number_of_is_id neg_def |
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217 by (simp add: nat_div_distrib) |
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218 |
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219 lemma one_div_nat_number_of [simp]: |
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220 "Suc 0 div number_of v' = nat (1 div number_of v')" |
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221 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) |
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222 |
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223 |
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224 subsubsection{*Remainder *} |
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225 |
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226 lemma mod_nat_number_of [simp]: |
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227 "(number_of v :: nat) mod number_of v' = |
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228 (if neg (number_of v :: int) then 0 |
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229 else if neg (number_of v' :: int) then number_of v |
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230 else nat (number_of v mod number_of v'))" |
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231 unfolding nat_number_of_def number_of_is_id neg_def |
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232 by (simp add: nat_mod_distrib) |
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233 |
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234 lemma one_mod_nat_number_of [simp]: |
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235 "Suc 0 mod number_of v' = |
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236 (if neg (number_of v' :: int) then Suc 0 |
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237 else nat (1 mod number_of v'))" |
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238 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) |
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239 |
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240 |
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241 subsubsection{* Divisibility *} |
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242 |
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243 lemmas dvd_eq_mod_eq_0_number_of = |
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244 dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard] |
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245 |
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246 declare dvd_eq_mod_eq_0_number_of [simp] |
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247 |
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248 ML |
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249 {* |
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250 val nat_number_of_def = thm"nat_number_of_def"; |
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251 |
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252 val nat_number_of = thm"nat_number_of"; |
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253 val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0"; |
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254 val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1"; |
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255 val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0"; |
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256 val numeral_2_eq_2 = thm"numeral_2_eq_2"; |
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257 val nat_div_distrib = thm"nat_div_distrib"; |
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258 val nat_mod_distrib = thm"nat_mod_distrib"; |
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259 val int_nat_number_of = thm"int_nat_number_of"; |
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260 val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1"; |
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261 val Suc_nat_number_of_add = thm"Suc_nat_number_of_add"; |
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262 val Suc_nat_number_of = thm"Suc_nat_number_of"; |
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263 val add_nat_number_of = thm"add_nat_number_of"; |
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264 val diff_nat_eq_if = thm"diff_nat_eq_if"; |
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265 val diff_nat_number_of = thm"diff_nat_number_of"; |
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266 val mult_nat_number_of = thm"mult_nat_number_of"; |
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267 val div_nat_number_of = thm"div_nat_number_of"; |
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268 val mod_nat_number_of = thm"mod_nat_number_of"; |
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269 *} |
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270 |
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271 |
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272 subsection{*Comparisons*} |
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273 |
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274 subsubsection{*Equals (=) *} |
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275 |
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276 lemma eq_nat_nat_iff: |
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277 "[| (0::int) <= z; 0 <= z' |] ==> (nat z = nat z') = (z=z')" |
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278 by (auto elim!: nonneg_eq_int) |
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279 |
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280 lemma eq_nat_number_of [simp]: |
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281 "((number_of v :: nat) = number_of v') = |
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282 (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0 |
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283 else if neg (number_of v' :: int) then (number_of v :: int) = 0 |
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284 else v = v')" |
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285 unfolding nat_number_of_def number_of_is_id neg_def |
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286 by auto |
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287 |
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288 |
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289 subsubsection{*Less-than (<) *} |
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290 |
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291 lemma less_nat_number_of [simp]: |
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292 "(number_of v :: nat) < number_of v' \<longleftrightarrow> |
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293 (if v < v' then Int.Pls < v' else False)" |
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294 unfolding nat_number_of_def number_of_is_id numeral_simps |
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295 by auto |
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296 |
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297 |
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298 subsubsection{*Less-than-or-equal *} |
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299 |
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300 lemma le_nat_number_of [simp]: |
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301 "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow> |
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302 (if v \<le> v' then True else v \<le> Int.Pls)" |
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303 unfolding nat_number_of_def number_of_is_id numeral_simps |
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304 by auto |
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305 |
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306 (*Maps #n to n for n = 0, 1, 2*) |
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307 lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2 |
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308 |
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309 |
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310 subsection{*Powers with Numeric Exponents*} |
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311 |
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312 text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}. |
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313 We cannot prove general results about the numeral @{term "-1"}, so we have to |
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314 use @{term "- 1"} instead.*} |
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315 |
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316 lemma power2_eq_square: "(a::'a::recpower)\<twosuperior> = a * a" |
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317 by (simp add: numeral_2_eq_2 Power.power_Suc) |
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318 |
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319 lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0" |
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320 by (simp add: power2_eq_square) |
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321 |
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322 lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1" |
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323 by (simp add: power2_eq_square) |
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324 |
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325 lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x" |
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326 apply (subgoal_tac "3 = Suc (Suc (Suc 0))") |
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327 apply (erule ssubst) |
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328 apply (simp add: power_Suc mult_ac) |
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329 apply (unfold nat_number_of_def) |
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330 apply (subst nat_eq_iff) |
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331 apply simp |
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332 done |
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333 |
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334 text{*Squares of literal numerals will be evaluated.*} |
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335 lemmas power2_eq_square_number_of = |
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336 power2_eq_square [of "number_of w", standard] |
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337 declare power2_eq_square_number_of [simp] |
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338 |
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339 |
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340 lemma zero_le_power2[simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,recpower})" |
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341 by (simp add: power2_eq_square) |
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342 |
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343 lemma zero_less_power2[simp]: |
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344 "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,recpower}))" |
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345 by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff) |
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346 |
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347 lemma power2_less_0[simp]: |
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348 fixes a :: "'a::{ordered_idom,recpower}" |
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349 shows "~ (a\<twosuperior> < 0)" |
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350 by (force simp add: power2_eq_square mult_less_0_iff) |
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351 |
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352 lemma zero_eq_power2[simp]: |
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353 "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,recpower}))" |
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354 by (force simp add: power2_eq_square mult_eq_0_iff) |
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355 |
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356 lemma abs_power2[simp]: |
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357 "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,recpower})" |
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358 by (simp add: power2_eq_square abs_mult abs_mult_self) |
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359 |
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360 lemma power2_abs[simp]: |
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361 "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,recpower})" |
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362 by (simp add: power2_eq_square abs_mult_self) |
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363 |
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364 lemma power2_minus[simp]: |
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365 "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,recpower})" |
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366 by (simp add: power2_eq_square) |
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367 |
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368 lemma power2_le_imp_le: |
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369 fixes x y :: "'a::{ordered_semidom,recpower}" |
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370 shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y" |
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371 unfolding numeral_2_eq_2 by (rule power_le_imp_le_base) |
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372 |
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373 lemma power2_less_imp_less: |
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374 fixes x y :: "'a::{ordered_semidom,recpower}" |
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375 shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y" |
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376 by (rule power_less_imp_less_base) |
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377 |
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378 lemma power2_eq_imp_eq: |
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379 fixes x y :: "'a::{ordered_semidom,recpower}" |
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380 shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y" |
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381 unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp) |
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382 |
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383 lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})" |
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384 proof (induct n) |
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385 case 0 show ?case by simp |
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386 next |
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387 case (Suc n) then show ?case by (simp add: power_Suc power_add) |
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388 qed |
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389 |
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390 lemma power_minus1_odd: "(- 1) ^ Suc(2*n) = -(1::'a::{comm_ring_1,recpower})" |
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391 by (simp add: power_Suc) |
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392 |
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393 lemma power_even_eq: "(a::'a::recpower) ^ (2*n) = (a^n)^2" |
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394 by (subst mult_commute) (simp add: power_mult) |
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395 |
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396 lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2" |
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397 by (simp add: power_even_eq) |
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398 |
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399 lemma power_minus_even [simp]: |
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400 "(-a) ^ (2*n) = (a::'a::{comm_ring_1,recpower}) ^ (2*n)" |
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401 by (simp add: power_minus1_even power_minus [of a]) |
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402 |
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403 lemma zero_le_even_power'[simp]: |
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404 "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (2*n)" |
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405 proof (induct "n") |
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406 case 0 |
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407 show ?case by (simp add: zero_le_one) |
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408 next |
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409 case (Suc n) |
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410 have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" |
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411 by (simp add: mult_ac power_add power2_eq_square) |
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412 thus ?case |
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413 by (simp add: prems zero_le_mult_iff) |
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414 qed |
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415 |
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416 lemma odd_power_less_zero: |
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417 "(a::'a::{ordered_idom,recpower}) < 0 ==> a ^ Suc(2*n) < 0" |
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418 proof (induct "n") |
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419 case 0 |
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420 then show ?case by simp |
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421 next |
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422 case (Suc n) |
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423 have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)" |
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424 by (simp add: mult_ac power_add power2_eq_square) |
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425 thus ?case |
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426 by (simp del: power_Suc add: prems mult_less_0_iff mult_neg_neg) |
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427 qed |
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428 |
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429 lemma odd_0_le_power_imp_0_le: |
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430 "0 \<le> a ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,recpower})" |
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431 apply (insert odd_power_less_zero [of a n]) |
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432 apply (force simp add: linorder_not_less [symmetric]) |
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433 done |
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434 |
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435 text{*Simprules for comparisons where common factors can be cancelled.*} |
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436 lemmas zero_compare_simps = |
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437 add_strict_increasing add_strict_increasing2 add_increasing |
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438 zero_le_mult_iff zero_le_divide_iff |
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439 zero_less_mult_iff zero_less_divide_iff |
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440 mult_le_0_iff divide_le_0_iff |
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441 mult_less_0_iff divide_less_0_iff |
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442 zero_le_power2 power2_less_0 |
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443 |
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444 subsubsection{*Nat *} |
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445 |
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446 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)" |
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447 by (simp add: numerals) |
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448 |
|
449 (*Expresses a natural number constant as the Suc of another one. |
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450 NOT suitable for rewriting because n recurs in the condition.*) |
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451 lemmas expand_Suc = Suc_pred' [of "number_of v", standard] |
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452 |
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453 subsubsection{*Arith *} |
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454 |
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455 lemma Suc_eq_add_numeral_1: "Suc n = n + 1" |
|
456 by (simp add: numerals) |
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457 |
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458 lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n" |
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459 by (simp add: numerals) |
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460 |
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461 (* These two can be useful when m = number_of... *) |
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462 |
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463 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))" |
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464 unfolding One_nat_def by (cases m) simp_all |
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465 |
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466 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))" |
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467 unfolding One_nat_def by (cases m) simp_all |
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468 |
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469 lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))" |
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470 unfolding One_nat_def by (cases m) simp_all |
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471 |
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472 |
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473 subsection{*Comparisons involving (0::nat) *} |
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474 |
|
475 text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*} |
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476 |
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477 lemma eq_number_of_0 [simp]: |
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478 "number_of v = (0::nat) \<longleftrightarrow> v \<le> Int.Pls" |
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479 unfolding nat_number_of_def number_of_is_id numeral_simps |
|
480 by auto |
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481 |
|
482 lemma eq_0_number_of [simp]: |
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483 "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls" |
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484 by (rule trans [OF eq_sym_conv eq_number_of_0]) |
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485 |
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486 lemma less_0_number_of [simp]: |
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487 "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v" |
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488 unfolding nat_number_of_def number_of_is_id numeral_simps |
|
489 by simp |
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490 |
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491 lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)" |
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492 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric]) |
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493 |
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494 |
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495 |
|
496 subsection{*Comparisons involving @{term Suc} *} |
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497 |
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498 lemma eq_number_of_Suc [simp]: |
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499 "(number_of v = Suc n) = |
|
500 (let pv = number_of (Int.pred v) in |
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501 if neg pv then False else nat pv = n)" |
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502 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less |
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503 number_of_pred nat_number_of_def |
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504 split add: split_if) |
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505 apply (rule_tac x = "number_of v" in spec) |
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506 apply (auto simp add: nat_eq_iff) |
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507 done |
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508 |
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509 lemma Suc_eq_number_of [simp]: |
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510 "(Suc n = number_of v) = |
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511 (let pv = number_of (Int.pred v) in |
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512 if neg pv then False else nat pv = n)" |
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513 by (rule trans [OF eq_sym_conv eq_number_of_Suc]) |
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514 |
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515 lemma less_number_of_Suc [simp]: |
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516 "(number_of v < Suc n) = |
|
517 (let pv = number_of (Int.pred v) in |
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518 if neg pv then True else nat pv < n)" |
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519 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less |
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520 number_of_pred nat_number_of_def |
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521 split add: split_if) |
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522 apply (rule_tac x = "number_of v" in spec) |
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523 apply (auto simp add: nat_less_iff) |
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524 done |
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525 |
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526 lemma less_Suc_number_of [simp]: |
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527 "(Suc n < number_of v) = |
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528 (let pv = number_of (Int.pred v) in |
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529 if neg pv then False else n < nat pv)" |
|
530 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less |
|
531 number_of_pred nat_number_of_def |
|
532 split add: split_if) |
|
533 apply (rule_tac x = "number_of v" in spec) |
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534 apply (auto simp add: zless_nat_eq_int_zless) |
|
535 done |
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536 |
|
537 lemma le_number_of_Suc [simp]: |
|
538 "(number_of v <= Suc n) = |
|
539 (let pv = number_of (Int.pred v) in |
|
540 if neg pv then True else nat pv <= n)" |
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541 by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric]) |
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542 |
|
543 lemma le_Suc_number_of [simp]: |
|
544 "(Suc n <= number_of v) = |
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545 (let pv = number_of (Int.pred v) in |
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546 if neg pv then False else n <= nat pv)" |
|
547 by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric]) |
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548 |
|
549 |
|
550 lemma eq_number_of_Pls_Min: "(Numeral0 ::int) ~= number_of Int.Min" |
|
551 by auto |
|
552 |
|
553 |
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554 |
|
555 subsection{*Max and Min Combined with @{term Suc} *} |
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556 |
|
557 lemma max_number_of_Suc [simp]: |
|
558 "max (Suc n) (number_of v) = |
|
559 (let pv = number_of (Int.pred v) in |
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560 if neg pv then Suc n else Suc(max n (nat pv)))" |
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561 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def |
|
562 split add: split_if nat.split) |
|
563 apply (rule_tac x = "number_of v" in spec) |
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564 apply auto |
|
565 done |
|
566 |
|
567 lemma max_Suc_number_of [simp]: |
|
568 "max (number_of v) (Suc n) = |
|
569 (let pv = number_of (Int.pred v) in |
|
570 if neg pv then Suc n else Suc(max (nat pv) n))" |
|
571 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def |
|
572 split add: split_if nat.split) |
|
573 apply (rule_tac x = "number_of v" in spec) |
|
574 apply auto |
|
575 done |
|
576 |
|
577 lemma min_number_of_Suc [simp]: |
|
578 "min (Suc n) (number_of v) = |
|
579 (let pv = number_of (Int.pred v) in |
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580 if neg pv then 0 else Suc(min n (nat pv)))" |
|
581 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def |
|
582 split add: split_if nat.split) |
|
583 apply (rule_tac x = "number_of v" in spec) |
|
584 apply auto |
|
585 done |
|
586 |
|
587 lemma min_Suc_number_of [simp]: |
|
588 "min (number_of v) (Suc n) = |
|
589 (let pv = number_of (Int.pred v) in |
|
590 if neg pv then 0 else Suc(min (nat pv) n))" |
|
591 apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def |
|
592 split add: split_if nat.split) |
|
593 apply (rule_tac x = "number_of v" in spec) |
|
594 apply auto |
|
595 done |
|
596 |
|
597 subsection{*Literal arithmetic involving powers*} |
|
598 |
|
599 lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n" |
|
600 apply (induct "n") |
|
601 apply (simp_all (no_asm_simp) add: nat_mult_distrib) |
|
602 done |
|
603 |
|
604 lemma power_nat_number_of: |
|
605 "(number_of v :: nat) ^ n = |
|
606 (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))" |
|
607 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq |
|
608 split add: split_if cong: imp_cong) |
|
609 |
|
610 |
|
611 lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard] |
|
612 declare power_nat_number_of_number_of [simp] |
|
613 |
|
614 |
|
615 |
|
616 text{*For arbitrary rings*} |
|
617 |
|
618 lemma power_number_of_even: |
|
619 fixes z :: "'a::{number_ring,recpower}" |
|
620 shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)" |
|
621 unfolding Let_def nat_number_of_def number_of_Bit0 |
|
622 apply (rule_tac x = "number_of w" in spec, clarify) |
|
623 apply (case_tac " (0::int) <= x") |
|
624 apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square) |
|
625 done |
|
626 |
|
627 lemma power_number_of_odd: |
|
628 fixes z :: "'a::{number_ring,recpower}" |
|
629 shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w |
|
630 then (let w = z ^ (number_of w) in z * w * w) else 1)" |
|
631 unfolding Let_def nat_number_of_def number_of_Bit1 |
|
632 apply (rule_tac x = "number_of w" in spec, auto) |
|
633 apply (simp only: nat_add_distrib nat_mult_distrib) |
|
634 apply simp |
|
635 apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat power_Suc) |
|
636 done |
|
637 |
|
638 lemmas zpower_number_of_even = power_number_of_even [where 'a=int] |
|
639 lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int] |
|
640 |
|
641 lemmas power_number_of_even_number_of [simp] = |
|
642 power_number_of_even [of "number_of v", standard] |
|
643 |
|
644 lemmas power_number_of_odd_number_of [simp] = |
|
645 power_number_of_odd [of "number_of v", standard] |
|
646 |
|
647 |
|
648 |
|
649 ML |
|
650 {* |
|
651 val numeral_ss = @{simpset} addsimps @{thms numerals}; |
|
652 |
|
653 val nat_bin_arith_setup = |
|
654 Lin_Arith.map_data |
|
655 (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} => |
|
656 {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, |
|
657 inj_thms = inj_thms, |
|
658 lessD = lessD, neqE = neqE, |
|
659 simpset = simpset addsimps @{thms neg_simps} @ |
|
660 [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}]}) |
|
661 *} |
|
662 |
|
663 declaration {* K nat_bin_arith_setup *} |
|
664 |
|
665 (* Enable arith to deal with div/mod k where k is a numeral: *) |
|
666 declare split_div[of _ _ "number_of k", standard, arith_split] |
|
667 declare split_mod[of _ _ "number_of k", standard, arith_split] |
|
668 |
|
669 lemma nat_number_of_Pls: "Numeral0 = (0::nat)" |
|
670 by (simp add: number_of_Pls nat_number_of_def) |
|
671 |
|
672 lemma nat_number_of_Min: "number_of Int.Min = (0::nat)" |
|
673 apply (simp only: number_of_Min nat_number_of_def nat_zminus_int) |
|
674 done |
|
675 |
|
676 lemma nat_number_of_Bit0: |
|
677 "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)" |
|
678 unfolding nat_number_of_def number_of_is_id numeral_simps Let_def |
|
679 by auto |
|
680 |
|
681 lemma nat_number_of_Bit1: |
|
682 "number_of (Int.Bit1 w) = |
|
683 (if neg (number_of w :: int) then 0 |
|
684 else let n = number_of w in Suc (n + n))" |
|
685 unfolding nat_number_of_def number_of_is_id numeral_simps neg_def Let_def |
|
686 by auto |
|
687 |
|
688 lemmas nat_number = |
|
689 nat_number_of_Pls nat_number_of_Min |
|
690 nat_number_of_Bit0 nat_number_of_Bit1 |
|
691 |
|
692 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)" |
|
693 by (simp add: Let_def) |
|
694 |
|
695 lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,recpower})" |
|
696 by (simp add: power_mult power_Suc); |
|
697 |
|
698 lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,recpower})" |
|
699 by (simp add: power_mult power_Suc); |
|
700 |
|
701 |
|
702 subsection{*Literal arithmetic and @{term of_nat}*} |
|
703 |
|
704 lemma of_nat_double: |
|
705 "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)" |
|
706 by (simp only: mult_2 nat_add_distrib of_nat_add) |
|
707 |
|
708 lemma nat_numeral_m1_eq_0: "-1 = (0::nat)" |
|
709 by (simp only: nat_number_of_def) |
|
710 |
|
711 lemma of_nat_number_of_lemma: |
|
712 "of_nat (number_of v :: nat) = |
|
713 (if 0 \<le> (number_of v :: int) |
|
714 then (number_of v :: 'a :: number_ring) |
|
715 else 0)" |
|
716 by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat); |
|
717 |
|
718 lemma of_nat_number_of_eq [simp]: |
|
719 "of_nat (number_of v :: nat) = |
|
720 (if neg (number_of v :: int) then 0 |
|
721 else (number_of v :: 'a :: number_ring))" |
|
722 by (simp only: of_nat_number_of_lemma neg_def, simp) |
|
723 |
|
724 |
|
725 subsection {*Lemmas for the Combination and Cancellation Simprocs*} |
|
726 |
|
727 lemma nat_number_of_add_left: |
|
728 "number_of v + (number_of v' + (k::nat)) = |
|
729 (if neg (number_of v :: int) then number_of v' + k |
|
730 else if neg (number_of v' :: int) then number_of v + k |
|
731 else number_of (v + v') + k)" |
|
732 unfolding nat_number_of_def number_of_is_id neg_def |
|
733 by auto |
|
734 |
|
735 lemma nat_number_of_mult_left: |
|
736 "number_of v * (number_of v' * (k::nat)) = |
|
737 (if v < Int.Pls then 0 |
|
738 else number_of (v * v') * k)" |
|
739 by simp |
|
740 |
|
741 |
|
742 subsubsection{*For @{text combine_numerals}*} |
|
743 |
|
744 lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)" |
|
745 by (simp add: add_mult_distrib) |
|
746 |
|
747 |
|
748 subsubsection{*For @{text cancel_numerals}*} |
|
749 |
|
750 lemma nat_diff_add_eq1: |
|
751 "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)" |
|
752 by (simp split add: nat_diff_split add: add_mult_distrib) |
|
753 |
|
754 lemma nat_diff_add_eq2: |
|
755 "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))" |
|
756 by (simp split add: nat_diff_split add: add_mult_distrib) |
|
757 |
|
758 lemma nat_eq_add_iff1: |
|
759 "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)" |
|
760 by (auto split add: nat_diff_split simp add: add_mult_distrib) |
|
761 |
|
762 lemma nat_eq_add_iff2: |
|
763 "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)" |
|
764 by (auto split add: nat_diff_split simp add: add_mult_distrib) |
|
765 |
|
766 lemma nat_less_add_iff1: |
|
767 "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)" |
|
768 by (auto split add: nat_diff_split simp add: add_mult_distrib) |
|
769 |
|
770 lemma nat_less_add_iff2: |
|
771 "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)" |
|
772 by (auto split add: nat_diff_split simp add: add_mult_distrib) |
|
773 |
|
774 lemma nat_le_add_iff1: |
|
775 "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)" |
|
776 by (auto split add: nat_diff_split simp add: add_mult_distrib) |
|
777 |
|
778 lemma nat_le_add_iff2: |
|
779 "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)" |
|
780 by (auto split add: nat_diff_split simp add: add_mult_distrib) |
|
781 |
|
782 |
|
783 subsubsection{*For @{text cancel_numeral_factors} *} |
|
784 |
|
785 lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)" |
|
786 by auto |
|
787 |
|
788 lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)" |
|
789 by auto |
|
790 |
|
791 lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)" |
|
792 by auto |
|
793 |
|
794 lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)" |
|
795 by auto |
|
796 |
|
797 lemma nat_mult_dvd_cancel_disj[simp]: |
|
798 "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))" |
|
799 by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric]) |
|
800 |
|
801 lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)" |
|
802 by(auto) |
|
803 |
|
804 |
|
805 subsubsection{*For @{text cancel_factor} *} |
|
806 |
|
807 lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)" |
|
808 by auto |
|
809 |
|
810 lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)" |
|
811 by auto |
|
812 |
|
813 lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)" |
|
814 by auto |
|
815 |
|
816 lemma nat_mult_div_cancel_disj[simp]: |
|
817 "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)" |
|
818 by (simp add: nat_mult_div_cancel1) |
|
819 |
|
820 |
|
821 subsection {* Simprocs for the Naturals *} |
|
822 |
|
823 use "Tools/nat_simprocs.ML" |
|
824 declaration {* K nat_simprocs_setup *} |
|
825 |
|
826 subsubsection{*For simplifying @{term "Suc m - K"} and @{term "K - Suc m"}*} |
|
827 |
|
828 text{*Where K above is a literal*} |
|
829 |
|
830 lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)" |
|
831 by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split) |
|
832 |
|
833 text {*Now just instantiating @{text n} to @{text "number_of v"} does |
|
834 the right simplification, but with some redundant inequality |
|
835 tests.*} |
|
836 lemma neg_number_of_pred_iff_0: |
|
837 "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))" |
|
838 apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ") |
|
839 apply (simp only: less_Suc_eq_le le_0_eq) |
|
840 apply (subst less_number_of_Suc, simp) |
|
841 done |
|
842 |
|
843 text{*No longer required as a simprule because of the @{text inverse_fold} |
|
844 simproc*} |
|
845 lemma Suc_diff_number_of: |
|
846 "Int.Pls < v ==> |
|
847 Suc m - (number_of v) = m - (number_of (Int.pred v))" |
|
848 apply (subst Suc_diff_eq_diff_pred) |
|
849 apply simp |
|
850 apply (simp del: nat_numeral_1_eq_1) |
|
851 apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric] |
|
852 neg_number_of_pred_iff_0) |
|
853 done |
|
854 |
|
855 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n" |
|
856 by (simp add: numerals split add: nat_diff_split) |
|
857 |
|
858 |
|
859 subsubsection{*For @{term nat_case} and @{term nat_rec}*} |
|
860 |
|
861 lemma nat_case_number_of [simp]: |
|
862 "nat_case a f (number_of v) = |
|
863 (let pv = number_of (Int.pred v) in |
|
864 if neg pv then a else f (nat pv))" |
|
865 by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0) |
|
866 |
|
867 lemma nat_case_add_eq_if [simp]: |
|
868 "nat_case a f ((number_of v) + n) = |
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869 (let pv = number_of (Int.pred v) in |
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870 if neg pv then nat_case a f n else f (nat pv + n))" |
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871 apply (subst add_eq_if) |
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872 apply (simp split add: nat.split |
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873 del: nat_numeral_1_eq_1 |
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874 add: nat_numeral_1_eq_1 [symmetric] |
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875 numeral_1_eq_Suc_0 [symmetric] |
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876 neg_number_of_pred_iff_0) |
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877 done |
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878 |
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879 lemma nat_rec_number_of [simp]: |
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880 "nat_rec a f (number_of v) = |
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881 (let pv = number_of (Int.pred v) in |
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882 if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))" |
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883 apply (case_tac " (number_of v) ::nat") |
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884 apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0) |
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885 apply (simp split add: split_if_asm) |
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886 done |
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887 |
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888 lemma nat_rec_add_eq_if [simp]: |
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889 "nat_rec a f (number_of v + n) = |
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890 (let pv = number_of (Int.pred v) in |
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891 if neg pv then nat_rec a f n |
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892 else f (nat pv + n) (nat_rec a f (nat pv + n)))" |
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893 apply (subst add_eq_if) |
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894 apply (simp split add: nat.split |
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895 del: nat_numeral_1_eq_1 |
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896 add: nat_numeral_1_eq_1 [symmetric] |
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897 numeral_1_eq_Suc_0 [symmetric] |
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898 neg_number_of_pred_iff_0) |
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899 done |
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900 |
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901 |
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902 subsubsection{*Various Other Lemmas*} |
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903 |
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904 text {*Evens and Odds, for Mutilated Chess Board*} |
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905 |
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906 text{*Lemmas for specialist use, NOT as default simprules*} |
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907 lemma nat_mult_2: "2 * z = (z+z::nat)" |
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908 proof - |
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909 have "2*z = (1 + 1)*z" by simp |
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910 also have "... = z+z" by (simp add: left_distrib) |
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911 finally show ?thesis . |
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912 qed |
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913 |
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914 lemma nat_mult_2_right: "z * 2 = (z+z::nat)" |
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915 by (subst mult_commute, rule nat_mult_2) |
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916 |
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917 text{*Case analysis on @{term "n<2"}*} |
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918 lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0" |
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919 by arith |
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920 |
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921 lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)" |
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922 by arith |
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923 |
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924 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)" |
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925 by (simp add: nat_mult_2 [symmetric]) |
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926 |
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927 lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2" |
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928 apply (subgoal_tac "m mod 2 < 2") |
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929 apply (erule less_2_cases [THEN disjE]) |
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930 apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1) |
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931 done |
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932 |
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933 lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)" |
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934 apply (subgoal_tac "m mod 2 < 2") |
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935 apply (force simp del: mod_less_divisor, simp) |
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936 done |
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937 |
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938 text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*} |
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939 |
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940 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)" |
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941 by simp |
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942 |
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943 lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)" |
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944 by simp |
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945 |
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946 text{*Can be used to eliminate long strings of Sucs, but not by default*} |
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947 lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n" |
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948 by simp |
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949 |
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950 |
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951 text{*These lemmas collapse some needless occurrences of Suc: |
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952 at least three Sucs, since two and fewer are rewritten back to Suc again! |
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953 We already have some rules to simplify operands smaller than 3.*} |
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954 |
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955 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)" |
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956 by (simp add: Suc3_eq_add_3) |
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957 |
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958 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)" |
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959 by (simp add: Suc3_eq_add_3) |
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960 |
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961 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n" |
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962 by (simp add: Suc3_eq_add_3) |
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963 |
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964 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n" |
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965 by (simp add: Suc3_eq_add_3) |
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966 |
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967 lemmas Suc_div_eq_add3_div_number_of = |
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968 Suc_div_eq_add3_div [of _ "number_of v", standard] |
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969 declare Suc_div_eq_add3_div_number_of [simp] |
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970 |
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971 lemmas Suc_mod_eq_add3_mod_number_of = |
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972 Suc_mod_eq_add3_mod [of _ "number_of v", standard] |
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973 declare Suc_mod_eq_add3_mod_number_of [simp] |
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974 |
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975 end |
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