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1 (* Title: Integ.ML |
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2 ID: $Id$ |
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3 Authors: Riccardo Mattolini, Dip. Sistemi e Informatica |
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4 Lawrence C Paulson, Cambridge University Computer Laboratory |
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5 Copyright 1994 Universita' di Firenze |
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6 Copyright 1993 University of Cambridge |
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7 |
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8 The integers as equivalence classes over nat*nat. |
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9 |
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10 Could also prove... |
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11 "znegative(z) ==> $# zmagnitude(z) = $~ z" |
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12 "~ znegative(z) ==> $# zmagnitude(z) = z" |
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13 < is a linear ordering |
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14 + and * are monotonic wrt < |
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15 *) |
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16 |
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17 open Integ; |
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18 |
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19 |
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20 (*** Proving that intrel is an equivalence relation ***) |
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21 |
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22 val eqa::eqb::prems = goal Arith.thy |
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23 "[| (x1::nat) + y2 = x2 + y1; x2 + y3 = x3 + y2 |] ==> \ |
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24 \ x1 + y3 = x3 + y1"; |
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25 by (res_inst_tac [("k2","x2")] (add_left_cancel RS iffD1) 1); |
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26 by (rtac (add_left_commute RS trans) 1); |
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27 by (rtac (eqb RS ssubst) 1); |
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28 by (rtac (add_left_commute RS trans) 1); |
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29 by (rtac (eqa RS ssubst) 1); |
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30 by (rtac (add_left_commute) 1); |
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31 qed "integ_trans_lemma"; |
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32 |
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33 (** Natural deduction for intrel **) |
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34 |
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35 val prems = goalw Integ.thy [intrel_def] |
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36 "[| x1+y2 = x2+y1|] ==> \ |
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37 \ <<x1,y1>,<x2,y2>>: intrel"; |
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38 by (fast_tac (rel_cs addIs prems) 1); |
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39 qed "intrelI"; |
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40 |
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41 (*intrelE is hard to derive because fast_tac tries hyp_subst_tac so soon*) |
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42 goalw Integ.thy [intrel_def] |
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43 "p: intrel --> (EX x1 y1 x2 y2. \ |
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44 \ p = <<x1,y1>,<x2,y2>> & x1+y2 = x2+y1)"; |
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45 by (fast_tac rel_cs 1); |
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46 qed "intrelE_lemma"; |
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47 |
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48 val [major,minor] = goal Integ.thy |
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49 "[| p: intrel; \ |
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50 \ !!x1 y1 x2 y2. [| p = <<x1,y1>,<x2,y2>>; x1+y2 = x2+y1|] ==> Q |] \ |
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51 \ ==> Q"; |
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52 by (cut_facts_tac [major RS (intrelE_lemma RS mp)] 1); |
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53 by (REPEAT (eresolve_tac [asm_rl,exE,conjE,minor] 1)); |
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54 qed "intrelE"; |
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55 |
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56 val intrel_cs = rel_cs addSIs [intrelI] addSEs [intrelE]; |
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57 |
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58 goal Integ.thy "<<x1,y1>,<x2,y2>>: intrel = (x1+y2 = x2+y1)"; |
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59 by (fast_tac intrel_cs 1); |
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60 qed "intrel_iff"; |
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61 |
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62 goal Integ.thy "<x,x>: intrel"; |
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63 by (rtac (surjective_pairing RS ssubst) 1 THEN rtac (refl RS intrelI) 1); |
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64 qed "intrel_refl"; |
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65 |
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66 goalw Integ.thy [equiv_def, refl_def, sym_def, trans_def] |
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67 "equiv {x::(nat*nat).True} intrel"; |
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68 by (fast_tac (intrel_cs addSIs [intrel_refl] |
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69 addSEs [sym, integ_trans_lemma]) 1); |
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70 qed "equiv_intrel"; |
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71 |
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72 val equiv_intrel_iff = |
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73 [TrueI, TrueI] MRS |
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74 ([CollectI, CollectI] MRS |
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75 (equiv_intrel RS eq_equiv_class_iff)); |
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76 |
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77 goalw Integ.thy [Integ_def,intrel_def,quotient_def] "intrel^^{<x,y>}:Integ"; |
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78 by (fast_tac set_cs 1); |
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79 qed "intrel_in_integ"; |
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80 |
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81 goal Integ.thy "inj_onto Abs_Integ Integ"; |
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82 by (rtac inj_onto_inverseI 1); |
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83 by (etac Abs_Integ_inverse 1); |
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84 qed "inj_onto_Abs_Integ"; |
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85 |
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86 val intrel_ss = |
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87 arith_ss addsimps [equiv_intrel_iff, inj_onto_Abs_Integ RS inj_onto_iff, |
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88 intrel_iff, intrel_in_integ, Abs_Integ_inverse]; |
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89 |
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90 goal Integ.thy "inj(Rep_Integ)"; |
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91 by (rtac inj_inverseI 1); |
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92 by (rtac Rep_Integ_inverse 1); |
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93 qed "inj_Rep_Integ"; |
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94 |
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95 |
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96 |
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97 |
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98 (** znat: the injection from nat to Integ **) |
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99 |
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100 goal Integ.thy "inj(znat)"; |
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101 by (rtac injI 1); |
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102 by (rewtac znat_def); |
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103 by (dtac (inj_onto_Abs_Integ RS inj_ontoD) 1); |
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104 by (REPEAT (rtac intrel_in_integ 1)); |
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105 by (dtac eq_equiv_class 1); |
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106 by (rtac equiv_intrel 1); |
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107 by (fast_tac set_cs 1); |
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108 by (safe_tac intrel_cs); |
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109 by (asm_full_simp_tac arith_ss 1); |
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110 qed "inj_znat"; |
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111 |
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112 |
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113 (**** zminus: unary negation on Integ ****) |
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114 |
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115 goalw Integ.thy [congruent_def] |
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116 "congruent intrel (%p. split (%x y. intrel^^{<y,x>}) p)"; |
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117 by (safe_tac intrel_cs); |
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118 by (asm_simp_tac (intrel_ss addsimps add_ac) 1); |
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119 qed "zminus_congruent"; |
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120 |
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121 |
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122 (*Resolve th against the corresponding facts for zminus*) |
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123 val zminus_ize = RSLIST [equiv_intrel, zminus_congruent]; |
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124 |
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125 goalw Integ.thy [zminus_def] |
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126 "$~ Abs_Integ(intrel^^{<x,y>}) = Abs_Integ(intrel ^^ {<y,x>})"; |
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127 by (res_inst_tac [("f","Abs_Integ")] arg_cong 1); |
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128 by (simp_tac (set_ss addsimps |
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129 [intrel_in_integ RS Abs_Integ_inverse,zminus_ize UN_equiv_class]) 1); |
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130 by (rewtac split_def); |
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131 by (simp_tac prod_ss 1); |
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132 qed "zminus"; |
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133 |
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134 (*by lcp*) |
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135 val [prem] = goal Integ.thy |
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136 "(!!x y. z = Abs_Integ(intrel^^{<x,y>}) ==> P) ==> P"; |
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137 by (res_inst_tac [("x1","z")] |
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138 (rewrite_rule [Integ_def] Rep_Integ RS quotientE) 1); |
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139 by (dres_inst_tac [("f","Abs_Integ")] arg_cong 1); |
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140 by (res_inst_tac [("p","x")] PairE 1); |
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141 by (rtac prem 1); |
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142 by (asm_full_simp_tac (HOL_ss addsimps [Rep_Integ_inverse]) 1); |
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143 qed "eq_Abs_Integ"; |
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144 |
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145 goal Integ.thy "$~ ($~ z) = z"; |
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146 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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147 by (asm_simp_tac (HOL_ss addsimps [zminus]) 1); |
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148 qed "zminus_zminus"; |
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149 |
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150 goal Integ.thy "inj(zminus)"; |
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151 by (rtac injI 1); |
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152 by (dres_inst_tac [("f","zminus")] arg_cong 1); |
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153 by (asm_full_simp_tac (HOL_ss addsimps [zminus_zminus]) 1); |
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154 qed "inj_zminus"; |
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155 |
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156 goalw Integ.thy [znat_def] "$~ ($#0) = $#0"; |
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157 by (simp_tac (arith_ss addsimps [zminus]) 1); |
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158 qed "zminus_0"; |
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159 |
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160 |
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161 (**** znegative: the test for negative integers ****) |
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162 |
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163 goal Arith.thy "!!m x n::nat. n+m=x ==> m<=x"; |
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164 by (dtac (disjI2 RS less_or_eq_imp_le) 1); |
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165 by (asm_full_simp_tac (arith_ss addsimps add_ac) 1); |
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166 by (dtac add_leD1 1); |
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167 by (assume_tac 1); |
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168 qed "not_znegative_znat_lemma"; |
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169 |
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170 |
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171 goalw Integ.thy [znegative_def, znat_def] |
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172 "~ znegative($# n)"; |
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173 by (simp_tac intrel_ss 1); |
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174 by (safe_tac intrel_cs); |
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175 by (rtac ccontr 1); |
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176 by (etac notE 1); |
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177 by (asm_full_simp_tac arith_ss 1); |
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178 by (dtac not_znegative_znat_lemma 1); |
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179 by (fast_tac (HOL_cs addDs [leD]) 1); |
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180 qed "not_znegative_znat"; |
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181 |
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182 goalw Integ.thy [znegative_def, znat_def] "znegative($~ $# Suc(n))"; |
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183 by (simp_tac (intrel_ss addsimps [zminus]) 1); |
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184 by (REPEAT (ares_tac [exI, conjI] 1)); |
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185 by (rtac (intrelI RS ImageI) 2); |
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186 by (rtac singletonI 3); |
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187 by (simp_tac arith_ss 2); |
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188 by (rtac less_add_Suc1 1); |
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189 qed "znegative_zminus_znat"; |
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190 |
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191 |
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192 (**** zmagnitude: magnitide of an integer, as a natural number ****) |
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193 |
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194 goal Arith.thy "!!n::nat. n - Suc(n+m)=0"; |
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195 by (nat_ind_tac "n" 1); |
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196 by (ALLGOALS(asm_simp_tac arith_ss)); |
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197 qed "diff_Suc_add_0"; |
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198 |
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199 goal Arith.thy "Suc((n::nat)+m)-n=Suc(m)"; |
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200 by (nat_ind_tac "n" 1); |
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201 by (ALLGOALS(asm_simp_tac arith_ss)); |
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202 qed "diff_Suc_add_inverse"; |
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203 |
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204 goalw Integ.thy [congruent_def] |
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205 "congruent intrel (split (%x y. intrel^^{<(y-x) + (x-(y::nat)),0>}))"; |
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206 by (safe_tac intrel_cs); |
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207 by (asm_simp_tac intrel_ss 1); |
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208 by (etac rev_mp 1); |
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209 by (res_inst_tac [("m","x1"),("n","y1")] diff_induct 1); |
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210 by (asm_simp_tac (arith_ss addsimps [inj_Suc RS inj_eq]) 3); |
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211 by (asm_simp_tac (arith_ss addsimps [diff_add_inverse,diff_add_0]) 2); |
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212 by (asm_simp_tac arith_ss 1); |
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213 by (rtac impI 1); |
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214 by (etac subst 1); |
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215 by (res_inst_tac [("m1","x")] (add_commute RS ssubst) 1); |
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216 by (asm_simp_tac (arith_ss addsimps [diff_add_inverse,diff_add_0]) 1); |
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217 by (rtac impI 1); |
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218 by (asm_simp_tac (arith_ss addsimps |
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219 [diff_add_inverse, diff_add_0, diff_Suc_add_0, |
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220 diff_Suc_add_inverse]) 1); |
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221 qed "zmagnitude_congruent"; |
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222 |
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223 (*Resolve th against the corresponding facts for zmagnitude*) |
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224 val zmagnitude_ize = RSLIST [equiv_intrel, zmagnitude_congruent]; |
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225 |
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226 |
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227 goalw Integ.thy [zmagnitude_def] |
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228 "zmagnitude (Abs_Integ(intrel^^{<x,y>})) = \ |
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229 \ Abs_Integ(intrel^^{<(y - x) + (x - y),0>})"; |
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230 by (res_inst_tac [("f","Abs_Integ")] arg_cong 1); |
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231 by (asm_simp_tac (intrel_ss addsimps [zmagnitude_ize UN_equiv_class]) 1); |
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232 qed "zmagnitude"; |
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233 |
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234 goalw Integ.thy [znat_def] "zmagnitude($# n) = $#n"; |
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235 by (asm_simp_tac (intrel_ss addsimps [zmagnitude]) 1); |
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236 qed "zmagnitude_znat"; |
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237 |
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238 goalw Integ.thy [znat_def] "zmagnitude($~ $# n) = $#n"; |
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239 by (asm_simp_tac (intrel_ss addsimps [zmagnitude, zminus]) 1); |
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240 qed "zmagnitude_zminus_znat"; |
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241 |
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242 |
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243 (**** zadd: addition on Integ ****) |
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244 |
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245 (** Congruence property for addition **) |
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246 |
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247 goalw Integ.thy [congruent2_def] |
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248 "congruent2 intrel (%p1 p2. \ |
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249 \ split (%x1 y1. split (%x2 y2. intrel^^{<x1+x2, y1+y2>}) p2) p1)"; |
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250 (*Proof via congruent2_commuteI seems longer*) |
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251 by (safe_tac intrel_cs); |
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252 by (asm_simp_tac (intrel_ss addsimps [add_assoc]) 1); |
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253 (*The rest should be trivial, but rearranging terms is hard*) |
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254 by (res_inst_tac [("x1","x1a")] (add_left_commute RS ssubst) 1); |
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255 by (asm_simp_tac (arith_ss addsimps [add_assoc RS sym]) 1); |
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256 by (asm_simp_tac (arith_ss addsimps add_ac) 1); |
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257 qed "zadd_congruent2"; |
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258 |
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259 (*Resolve th against the corresponding facts for zadd*) |
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260 val zadd_ize = RSLIST [equiv_intrel, zadd_congruent2]; |
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261 |
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262 goalw Integ.thy [zadd_def] |
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263 "Abs_Integ(intrel^^{<x1,y1>}) + Abs_Integ(intrel^^{<x2,y2>}) = \ |
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264 \ Abs_Integ(intrel^^{<x1+x2, y1+y2>})"; |
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265 by (asm_simp_tac |
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266 (intrel_ss addsimps [zadd_ize UN_equiv_class2]) 1); |
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267 qed "zadd"; |
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268 |
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269 goalw Integ.thy [znat_def] "$#0 + z = z"; |
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270 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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271 by (asm_simp_tac (arith_ss addsimps [zadd]) 1); |
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272 qed "zadd_0"; |
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273 |
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274 goal Integ.thy "$~ (z + w) = $~ z + $~ w"; |
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275 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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276 by (res_inst_tac [("z","w")] eq_Abs_Integ 1); |
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277 by (asm_simp_tac (arith_ss addsimps [zminus,zadd]) 1); |
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278 qed "zminus_zadd_distrib"; |
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279 |
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280 goal Integ.thy "(z::int) + w = w + z"; |
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281 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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282 by (res_inst_tac [("z","w")] eq_Abs_Integ 1); |
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283 by (asm_simp_tac (intrel_ss addsimps (add_ac @ [zadd])) 1); |
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284 qed "zadd_commute"; |
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285 |
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286 goal Integ.thy "((z1::int) + z2) + z3 = z1 + (z2 + z3)"; |
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287 by (res_inst_tac [("z","z1")] eq_Abs_Integ 1); |
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288 by (res_inst_tac [("z","z2")] eq_Abs_Integ 1); |
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289 by (res_inst_tac [("z","z3")] eq_Abs_Integ 1); |
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290 by (asm_simp_tac (arith_ss addsimps [zadd, add_assoc]) 1); |
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291 qed "zadd_assoc"; |
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292 |
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293 (*For AC rewriting*) |
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294 goal Integ.thy "(x::int)+(y+z)=y+(x+z)"; |
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295 by (rtac (zadd_commute RS trans) 1); |
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296 by (rtac (zadd_assoc RS trans) 1); |
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297 by (rtac (zadd_commute RS arg_cong) 1); |
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298 qed "zadd_left_commute"; |
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299 |
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300 (*Integer addition is an AC operator*) |
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301 val zadd_ac = [zadd_assoc,zadd_commute,zadd_left_commute]; |
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302 |
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303 goalw Integ.thy [znat_def] "$# (m + n) = ($#m) + ($#n)"; |
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304 by (asm_simp_tac (arith_ss addsimps [zadd]) 1); |
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305 qed "znat_add"; |
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306 |
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307 goalw Integ.thy [znat_def] "z + ($~ z) = $#0"; |
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308 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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309 by (asm_simp_tac (intrel_ss addsimps [zminus, zadd, add_commute]) 1); |
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310 qed "zadd_zminus_inverse"; |
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311 |
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312 goal Integ.thy "($~ z) + z = $#0"; |
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313 by (rtac (zadd_commute RS trans) 1); |
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314 by (rtac zadd_zminus_inverse 1); |
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315 qed "zadd_zminus_inverse2"; |
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316 |
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317 goal Integ.thy "z + $#0 = z"; |
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318 by (rtac (zadd_commute RS trans) 1); |
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319 by (rtac zadd_0 1); |
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320 qed "zadd_0_right"; |
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321 |
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322 |
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323 (*Need properties of subtraction? Or use $- just as an abbreviation!*) |
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324 |
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325 (**** zmult: multiplication on Integ ****) |
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326 |
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327 (** Congruence property for multiplication **) |
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328 |
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329 goal Integ.thy "((k::nat) + l) + (m + n) = (k + m) + (n + l)"; |
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330 by (simp_tac (arith_ss addsimps add_ac) 1); |
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331 qed "zmult_congruent_lemma"; |
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332 |
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333 goal Integ.thy |
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334 "congruent2 intrel (%p1 p2. \ |
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335 \ split (%x1 y1. split (%x2 y2. \ |
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336 \ intrel^^{<x1*x2 + y1*y2, x1*y2 + y1*x2>}) p2) p1)"; |
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337 by (rtac (equiv_intrel RS congruent2_commuteI) 1); |
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338 by (safe_tac intrel_cs); |
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339 by (rewtac split_def); |
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340 by (simp_tac (arith_ss addsimps add_ac@mult_ac) 1); |
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341 by (asm_simp_tac (arith_ss addsimps add_ac@mult_ac) 1); |
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342 by (rtac (intrelI RS(equiv_intrel RS equiv_class_eq)) 1); |
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343 by (rtac (zmult_congruent_lemma RS trans) 1); |
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344 by (rtac (zmult_congruent_lemma RS trans RS sym) 1); |
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345 by (rtac (zmult_congruent_lemma RS trans RS sym) 1); |
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346 by (rtac (zmult_congruent_lemma RS trans RS sym) 1); |
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347 by (asm_simp_tac (HOL_ss addsimps [add_mult_distrib RS sym]) 1); |
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348 by (asm_simp_tac (arith_ss addsimps add_ac@mult_ac) 1); |
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349 qed "zmult_congruent2"; |
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350 |
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351 (*Resolve th against the corresponding facts for zmult*) |
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352 val zmult_ize = RSLIST [equiv_intrel, zmult_congruent2]; |
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353 |
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354 goalw Integ.thy [zmult_def] |
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355 "Abs_Integ((intrel^^{<x1,y1>})) * Abs_Integ((intrel^^{<x2,y2>})) = \ |
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356 \ Abs_Integ(intrel ^^ {<x1*x2 + y1*y2, x1*y2 + y1*x2>})"; |
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357 by (simp_tac (intrel_ss addsimps [zmult_ize UN_equiv_class2]) 1); |
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358 qed "zmult"; |
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359 |
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360 goalw Integ.thy [znat_def] "$#0 * z = $#0"; |
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361 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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362 by (asm_simp_tac (arith_ss addsimps [zmult]) 1); |
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363 qed "zmult_0"; |
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364 |
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365 goalw Integ.thy [znat_def] "$#Suc(0) * z = z"; |
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366 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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367 by (asm_simp_tac (arith_ss addsimps [zmult, add_0_right]) 1); |
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368 qed "zmult_1"; |
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369 |
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370 goal Integ.thy "($~ z) * w = $~ (z * w)"; |
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371 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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372 by (res_inst_tac [("z","w")] eq_Abs_Integ 1); |
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373 by (asm_simp_tac (intrel_ss addsimps ([zminus, zmult] @ add_ac)) 1); |
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374 qed "zmult_zminus"; |
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375 |
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376 |
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377 goal Integ.thy "($~ z) * ($~ w) = (z * w)"; |
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378 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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379 by (res_inst_tac [("z","w")] eq_Abs_Integ 1); |
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380 by (asm_simp_tac (intrel_ss addsimps ([zminus, zmult] @ add_ac)) 1); |
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381 qed "zmult_zminus_zminus"; |
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382 |
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383 goal Integ.thy "(z::int) * w = w * z"; |
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384 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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385 by (res_inst_tac [("z","w")] eq_Abs_Integ 1); |
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386 by (asm_simp_tac (intrel_ss addsimps ([zmult] @ add_ac @ mult_ac)) 1); |
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387 qed "zmult_commute"; |
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388 |
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389 goal Integ.thy "z * $# 0 = $#0"; |
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390 by (rtac ([zmult_commute, zmult_0] MRS trans) 1); |
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391 qed "zmult_0_right"; |
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392 |
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393 goal Integ.thy "z * $#Suc(0) = z"; |
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394 by (rtac ([zmult_commute, zmult_1] MRS trans) 1); |
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395 qed "zmult_1_right"; |
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396 |
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397 goal Integ.thy "((z1::int) * z2) * z3 = z1 * (z2 * z3)"; |
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398 by (res_inst_tac [("z","z1")] eq_Abs_Integ 1); |
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399 by (res_inst_tac [("z","z2")] eq_Abs_Integ 1); |
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400 by (res_inst_tac [("z","z3")] eq_Abs_Integ 1); |
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401 by (asm_simp_tac (intrel_ss addsimps ([zmult] @ add_ac @ mult_ac)) 1); |
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402 qed "zmult_assoc"; |
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403 |
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404 (*For AC rewriting*) |
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405 qed_goal "zmult_left_commute" Integ.thy |
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406 "(z1::int)*(z2*z3) = z2*(z1*z3)" |
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407 (fn _ => [rtac (zmult_commute RS trans) 1, rtac (zmult_assoc RS trans) 1, |
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408 rtac (zmult_commute RS arg_cong) 1]); |
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409 |
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410 (*Integer multiplication is an AC operator*) |
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411 val zmult_ac = [zmult_assoc, zmult_commute, zmult_left_commute]; |
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412 |
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413 goal Integ.thy "((z1::int) + z2) * w = (z1 * w) + (z2 * w)"; |
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414 by (res_inst_tac [("z","z1")] eq_Abs_Integ 1); |
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415 by (res_inst_tac [("z","z2")] eq_Abs_Integ 1); |
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416 by (res_inst_tac [("z","w")] eq_Abs_Integ 1); |
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417 by (asm_simp_tac |
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418 (intrel_ss addsimps ([zadd, zmult, add_mult_distrib] @ |
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419 add_ac @ mult_ac)) 1); |
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420 qed "zadd_zmult_distrib"; |
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421 |
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422 val zmult_commute'= read_instantiate [("z","w")] zmult_commute; |
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423 |
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424 goal Integ.thy "w * ($~ z) = $~ (w * z)"; |
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425 by (simp_tac (HOL_ss addsimps [zmult_commute', zmult_zminus]) 1); |
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426 qed "zmult_zminus_right"; |
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427 |
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428 goal Integ.thy "(w::int) * (z1 + z2) = (w * z1) + (w * z2)"; |
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429 by (simp_tac (HOL_ss addsimps [zmult_commute',zadd_zmult_distrib]) 1); |
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430 qed "zadd_zmult_distrib2"; |
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431 |
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432 val zadd_simps = |
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433 [zadd_0, zadd_0_right, zadd_zminus_inverse, zadd_zminus_inverse2]; |
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434 |
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435 val zminus_simps = [zminus_zminus, zminus_0, zminus_zadd_distrib]; |
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436 |
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437 val zmult_simps = [zmult_0, zmult_1, zmult_0_right, zmult_1_right, |
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438 zmult_zminus, zmult_zminus_right]; |
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439 |
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440 val integ_ss = |
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441 arith_ss addsimps (zadd_simps @ zminus_simps @ zmult_simps @ |
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442 [zmagnitude_znat, zmagnitude_zminus_znat]); |
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443 |
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444 |
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445 (**** Additional Theorems (by Mattolini; proofs mainly by lcp) ****) |
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446 |
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447 (* Some Theorems about zsuc and zpred *) |
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448 goalw Integ.thy [zsuc_def] "$#(Suc(n)) = zsuc($# n)"; |
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449 by (simp_tac (arith_ss addsimps [znat_add RS sym]) 1); |
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450 qed "znat_Suc"; |
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451 |
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452 goalw Integ.thy [zpred_def,zsuc_def,zdiff_def] "$~ zsuc(z) = zpred($~ z)"; |
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453 by (simp_tac integ_ss 1); |
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454 qed "zminus_zsuc"; |
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455 |
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456 goalw Integ.thy [zpred_def,zsuc_def,zdiff_def] "$~ zpred(z) = zsuc($~ z)"; |
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457 by (simp_tac integ_ss 1); |
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458 qed "zminus_zpred"; |
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459 |
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460 goalw Integ.thy [zsuc_def,zpred_def,zdiff_def] |
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461 "zpred(zsuc(z)) = z"; |
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462 by (simp_tac (integ_ss addsimps [zadd_assoc]) 1); |
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463 qed "zpred_zsuc"; |
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464 |
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465 goalw Integ.thy [zsuc_def,zpred_def,zdiff_def] |
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466 "zsuc(zpred(z)) = z"; |
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467 by (simp_tac (integ_ss addsimps [zadd_assoc]) 1); |
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468 qed "zsuc_zpred"; |
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469 |
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470 goal Integ.thy "(zpred(z)=w) = (z=zsuc(w))"; |
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471 by (safe_tac HOL_cs); |
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472 by (rtac (zsuc_zpred RS sym) 1); |
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473 by (rtac zpred_zsuc 1); |
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474 qed "zpred_to_zsuc"; |
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475 |
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476 goal Integ.thy "(zsuc(z)=w)=(z=zpred(w))"; |
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477 by (safe_tac HOL_cs); |
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478 by (rtac (zpred_zsuc RS sym) 1); |
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479 by (rtac zsuc_zpred 1); |
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480 qed "zsuc_to_zpred"; |
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481 |
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482 goal Integ.thy "($~ z = w) = (z = $~ w)"; |
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483 by (safe_tac HOL_cs); |
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484 by (rtac (zminus_zminus RS sym) 1); |
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485 by (rtac zminus_zminus 1); |
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486 qed "zminus_exchange"; |
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487 |
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488 goal Integ.thy"(zsuc(z)=zsuc(w)) = (z=w)"; |
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489 by (safe_tac intrel_cs); |
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490 by (dres_inst_tac [("f","zpred")] arg_cong 1); |
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491 by (asm_full_simp_tac (HOL_ss addsimps [zpred_zsuc]) 1); |
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492 qed "bijective_zsuc"; |
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493 |
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494 goal Integ.thy"(zpred(z)=zpred(w)) = (z=w)"; |
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495 by (safe_tac intrel_cs); |
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496 by (dres_inst_tac [("f","zsuc")] arg_cong 1); |
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497 by (asm_full_simp_tac (HOL_ss addsimps [zsuc_zpred]) 1); |
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498 qed "bijective_zpred"; |
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499 |
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500 (* Additional Theorems about zadd *) |
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501 |
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502 goalw Integ.thy [zsuc_def] "zsuc(z) + w = zsuc(z+w)"; |
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503 by (simp_tac (arith_ss addsimps zadd_ac) 1); |
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504 qed "zadd_zsuc"; |
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505 |
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506 goalw Integ.thy [zsuc_def] "w + zsuc(z) = zsuc(w+z)"; |
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507 by (simp_tac (arith_ss addsimps zadd_ac) 1); |
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508 qed "zadd_zsuc_right"; |
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509 |
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510 goalw Integ.thy [zpred_def,zdiff_def] "zpred(z) + w = zpred(z+w)"; |
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511 by (simp_tac (arith_ss addsimps zadd_ac) 1); |
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512 qed "zadd_zpred"; |
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513 |
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514 goalw Integ.thy [zpred_def,zdiff_def] "w + zpred(z) = zpred(w+z)"; |
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515 by (simp_tac (arith_ss addsimps zadd_ac) 1); |
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516 qed "zadd_zpred_right"; |
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517 |
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518 |
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519 (* Additional Theorems about zmult *) |
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520 |
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521 goalw Integ.thy [zsuc_def] "zsuc(w) * z = z + w * z"; |
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522 by (simp_tac (integ_ss addsimps [zadd_zmult_distrib, zadd_commute]) 1); |
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523 qed "zmult_zsuc"; |
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524 |
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525 goalw Integ.thy [zsuc_def] "z * zsuc(w) = z + w * z"; |
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526 by (simp_tac |
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527 (integ_ss addsimps [zadd_zmult_distrib2, zadd_commute, zmult_commute]) 1); |
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528 qed "zmult_zsuc_right"; |
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529 |
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530 goalw Integ.thy [zpred_def, zdiff_def] "zpred(w) * z = w * z - z"; |
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531 by (simp_tac (integ_ss addsimps [zadd_zmult_distrib]) 1); |
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532 qed "zmult_zpred"; |
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533 |
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534 goalw Integ.thy [zpred_def, zdiff_def] "z * zpred(w) = w * z - z"; |
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535 by (simp_tac (integ_ss addsimps [zadd_zmult_distrib2, zmult_commute]) 1); |
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536 qed "zmult_zpred_right"; |
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537 |
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538 (* Further Theorems about zsuc and zpred *) |
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539 goal Integ.thy "$#Suc(m) ~= $#0"; |
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540 by (simp_tac (integ_ss addsimps [inj_znat RS inj_eq]) 1); |
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541 qed "znat_Suc_not_znat_Zero"; |
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542 |
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543 bind_thm ("znat_Zero_not_znat_Suc", (znat_Suc_not_znat_Zero RS not_sym)); |
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544 |
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545 |
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546 goalw Integ.thy [zsuc_def,znat_def] "w ~= zsuc(w)"; |
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547 by (res_inst_tac [("z","w")] eq_Abs_Integ 1); |
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548 by (asm_full_simp_tac (intrel_ss addsimps [zadd]) 1); |
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549 qed "n_not_zsuc_n"; |
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550 |
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551 val zsuc_n_not_n = n_not_zsuc_n RS not_sym; |
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552 |
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553 goal Integ.thy "w ~= zpred(w)"; |
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554 by (safe_tac HOL_cs); |
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555 by (dres_inst_tac [("x","w"),("f","zsuc")] arg_cong 1); |
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556 by (asm_full_simp_tac (HOL_ss addsimps [zsuc_zpred,zsuc_n_not_n]) 1); |
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557 qed "n_not_zpred_n"; |
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558 |
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559 val zpred_n_not_n = n_not_zpred_n RS not_sym; |
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560 |
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561 |
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562 (* Theorems about less and less_equal *) |
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563 |
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564 goalw Integ.thy [zless_def, znegative_def, zdiff_def, znat_def] |
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565 "!!w. w<z ==> ? n. z = w + $#(Suc(n))"; |
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566 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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567 by (res_inst_tac [("z","w")] eq_Abs_Integ 1); |
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568 by (safe_tac intrel_cs); |
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569 by (asm_full_simp_tac (intrel_ss addsimps [zadd, zminus]) 1); |
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570 by (safe_tac (intrel_cs addSDs [less_eq_Suc_add])); |
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571 by (res_inst_tac [("x","k")] exI 1); |
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572 by (asm_full_simp_tac (HOL_ss addsimps ([add_Suc RS sym] @ add_ac)) 1); |
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573 (*To cancel x2, rename it to be first!*) |
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574 by (rename_tac "a b c" 1); |
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575 by (asm_full_simp_tac (HOL_ss addsimps (add_left_cancel::add_ac)) 1); |
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576 qed "zless_eq_zadd_Suc"; |
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577 |
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578 goalw Integ.thy [zless_def, znegative_def, zdiff_def, znat_def] |
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579 "z < z + $#(Suc(n))"; |
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580 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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581 by (safe_tac intrel_cs); |
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582 by (simp_tac (intrel_ss addsimps [zadd, zminus]) 1); |
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583 by (REPEAT_SOME (ares_tac [refl, exI, singletonI, ImageI, conjI, intrelI])); |
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584 by (rtac le_less_trans 1); |
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585 by (rtac lessI 2); |
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586 by (asm_simp_tac (arith_ss addsimps ([le_add1,add_left_cancel_le]@add_ac)) 1); |
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587 qed "zless_zadd_Suc"; |
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588 |
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589 goal Integ.thy "!!z1 z2 z3. [| z1<z2; z2<z3 |] ==> z1 < (z3::int)"; |
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590 by (safe_tac (HOL_cs addSDs [zless_eq_zadd_Suc])); |
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591 by (simp_tac |
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592 (arith_ss addsimps [zadd_assoc, zless_zadd_Suc, znat_add RS sym]) 1); |
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593 qed "zless_trans"; |
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594 |
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595 goalw Integ.thy [zsuc_def] "z<zsuc(z)"; |
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596 by (rtac zless_zadd_Suc 1); |
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597 qed "zlessI"; |
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598 |
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599 val zless_zsucI = zlessI RSN (2,zless_trans); |
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600 |
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601 goal Integ.thy "!!z w::int. z<w ==> ~w<z"; |
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602 by (safe_tac (HOL_cs addSDs [zless_eq_zadd_Suc])); |
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603 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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604 by (safe_tac intrel_cs); |
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605 by (asm_full_simp_tac (intrel_ss addsimps ([znat_def, zadd])) 1); |
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606 by (asm_full_simp_tac |
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607 (HOL_ss addsimps [add_left_cancel, add_assoc, add_Suc_right RS sym]) 1); |
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608 by (resolve_tac [less_not_refl2 RS notE] 1); |
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609 by (etac sym 2); |
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610 by (REPEAT (resolve_tac [lessI, trans_less_add2, less_SucI] 1)); |
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611 qed "zless_not_sym"; |
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612 |
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613 (* [| n<m; m<n |] ==> R *) |
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614 bind_thm ("zless_asym", (zless_not_sym RS notE)); |
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615 |
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616 goal Integ.thy "!!z::int. ~ z<z"; |
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617 by (resolve_tac [zless_asym RS notI] 1); |
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618 by (REPEAT (assume_tac 1)); |
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619 qed "zless_not_refl"; |
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620 |
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621 (* z<z ==> R *) |
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622 bind_thm ("zless_anti_refl", (zless_not_refl RS notE)); |
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623 |
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624 goal Integ.thy "!!w. z<w ==> w ~= (z::int)"; |
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625 by(fast_tac (HOL_cs addEs [zless_anti_refl]) 1); |
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626 qed "zless_not_refl2"; |
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627 |
|
628 |
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629 (*"Less than" is a linear ordering*) |
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630 goalw Integ.thy [zless_def, znegative_def, zdiff_def] |
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631 "z<w | z=w | w<(z::int)"; |
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632 by (res_inst_tac [("z","z")] eq_Abs_Integ 1); |
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633 by (res_inst_tac [("z","w")] eq_Abs_Integ 1); |
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634 by (safe_tac intrel_cs); |
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635 by (asm_full_simp_tac |
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636 (intrel_ss addsimps [zadd, zminus, Image_iff, Bex_def]) 1); |
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637 by (res_inst_tac [("m1", "x+ya"), ("n1", "xa+y")] (less_linear RS disjE) 1); |
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638 by (etac disjE 2); |
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639 by (assume_tac 2); |
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640 by (DEPTH_SOLVE |
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641 (swap_res_tac [exI] 1 THEN |
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642 swap_res_tac [exI] 1 THEN |
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643 etac conjI 1 THEN |
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644 simp_tac (arith_ss addsimps add_ac) 1)); |
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645 qed "zless_linear"; |
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646 |
|
647 |
|
648 (*** Properties of <= ***) |
|
649 |
|
650 goalw Integ.thy [zless_def, znegative_def, zdiff_def, znat_def] |
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651 "($#m < $#n) = (m<n)"; |
|
652 by (simp_tac |
|
653 (intrel_ss addsimps [zadd, zminus, Image_iff, Bex_def]) 1); |
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654 by (fast_tac (HOL_cs addIs [add_commute] addSEs [less_add_eq_less]) 1); |
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655 qed "zless_eq_less"; |
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656 |
|
657 goalw Integ.thy [zle_def, le_def] "($#m <= $#n) = (m<=n)"; |
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658 by (simp_tac (HOL_ss addsimps [zless_eq_less]) 1); |
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659 qed "zle_eq_le"; |
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660 |
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661 goalw Integ.thy [zle_def] "!!w. ~(w<z) ==> z<=(w::int)"; |
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662 by (assume_tac 1); |
|
663 qed "zleI"; |
|
664 |
|
665 goalw Integ.thy [zle_def] "!!w. z<=w ==> ~(w<(z::int))"; |
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666 by (assume_tac 1); |
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667 qed "zleD"; |
|
668 |
|
669 val zleE = make_elim zleD; |
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670 |
|
671 goalw Integ.thy [zle_def] "!!z. ~ z <= w ==> w<(z::int)"; |
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672 by (fast_tac HOL_cs 1); |
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673 qed "not_zleE"; |
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674 |
|
675 goalw Integ.thy [zle_def] "!!z. z < w ==> z <= (w::int)"; |
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676 by (fast_tac (HOL_cs addEs [zless_asym]) 1); |
|
677 qed "zless_imp_zle"; |
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678 |
|
679 goalw Integ.thy [zle_def] "!!z. z <= w ==> z < w | z=(w::int)"; |
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680 by (cut_facts_tac [zless_linear] 1); |
|
681 by (fast_tac (HOL_cs addEs [zless_anti_refl,zless_asym]) 1); |
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682 qed "zle_imp_zless_or_eq"; |
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683 |
|
684 goalw Integ.thy [zle_def] "!!z. z<w | z=w ==> z <=(w::int)"; |
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685 by (cut_facts_tac [zless_linear] 1); |
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686 by (fast_tac (HOL_cs addEs [zless_anti_refl,zless_asym]) 1); |
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687 qed "zless_or_eq_imp_zle"; |
|
688 |
|
689 goal Integ.thy "(x <= (y::int)) = (x < y | x=y)"; |
|
690 by (REPEAT(ares_tac [iffI, zless_or_eq_imp_zle, zle_imp_zless_or_eq] 1)); |
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691 qed "zle_eq_zless_or_eq"; |
|
692 |
|
693 goal Integ.thy "w <= (w::int)"; |
|
694 by (simp_tac (HOL_ss addsimps [zle_eq_zless_or_eq]) 1); |
|
695 qed "zle_refl"; |
|
696 |
|
697 val prems = goal Integ.thy "!!i. [| i <= j; j < k |] ==> i < (k::int)"; |
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698 by (dtac zle_imp_zless_or_eq 1); |
|
699 by (fast_tac (HOL_cs addIs [zless_trans]) 1); |
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700 qed "zle_zless_trans"; |
|
701 |
|
702 goal Integ.thy "!!i. [| i <= j; j <= k |] ==> i <= (k::int)"; |
|
703 by (EVERY1 [dtac zle_imp_zless_or_eq, dtac zle_imp_zless_or_eq, |
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704 rtac zless_or_eq_imp_zle, fast_tac (HOL_cs addIs [zless_trans])]); |
|
705 qed "zle_trans"; |
|
706 |
|
707 goal Integ.thy "!!z. [| z <= w; w <= z |] ==> z = (w::int)"; |
|
708 by (EVERY1 [dtac zle_imp_zless_or_eq, dtac zle_imp_zless_or_eq, |
|
709 fast_tac (HOL_cs addEs [zless_anti_refl,zless_asym])]); |
|
710 qed "zle_anti_sym"; |
|
711 |
|
712 |
|
713 goal Integ.thy "!!w w' z::int. z + w' = z + w ==> w' = w"; |
|
714 by (dres_inst_tac [("f", "%x. x + $~z")] arg_cong 1); |
|
715 by (asm_full_simp_tac (integ_ss addsimps zadd_ac) 1); |
|
716 qed "zadd_left_cancel"; |
|
717 |
|
718 |
|
719 (*** Monotonicity results ***) |
|
720 |
|
721 goal Integ.thy "!!v w z::int. v < w ==> v + z < w + z"; |
|
722 by (safe_tac (HOL_cs addSDs [zless_eq_zadd_Suc])); |
|
723 by (simp_tac (HOL_ss addsimps zadd_ac) 1); |
|
724 by (simp_tac (HOL_ss addsimps [zadd_assoc RS sym, zless_zadd_Suc]) 1); |
|
725 qed "zadd_zless_mono1"; |
|
726 |
|
727 goal Integ.thy "!!v w z::int. (v+z < w+z) = (v < w)"; |
|
728 by (safe_tac (HOL_cs addSEs [zadd_zless_mono1])); |
|
729 by (dres_inst_tac [("z", "$~z")] zadd_zless_mono1 1); |
|
730 by (asm_full_simp_tac (integ_ss addsimps [zadd_assoc]) 1); |
|
731 qed "zadd_left_cancel_zless"; |
|
732 |
|
733 goal Integ.thy "!!v w z::int. (v+z <= w+z) = (v <= w)"; |
|
734 by (asm_full_simp_tac |
|
735 (integ_ss addsimps [zle_def, zadd_left_cancel_zless]) 1); |
|
736 qed "zadd_left_cancel_zle"; |
|
737 |
|
738 (*"v<=w ==> v+z <= w+z"*) |
|
739 bind_thm ("zadd_zle_mono1", zadd_left_cancel_zle RS iffD2); |
|
740 |
|
741 |
|
742 goal Integ.thy "!!z' z::int. [| w'<=w; z'<=z |] ==> w' + z' <= w + z"; |
|
743 by (etac (zadd_zle_mono1 RS zle_trans) 1); |
|
744 by (simp_tac (HOL_ss addsimps [zadd_commute]) 1); |
|
745 (*w moves to the end because it is free while z', z are bound*) |
|
746 by (etac zadd_zle_mono1 1); |
|
747 qed "zadd_zle_mono"; |
|
748 |
|
749 goal Integ.thy "!!w z::int. z<=$#0 ==> w+z <= w"; |
|
750 by (dres_inst_tac [("z", "w")] zadd_zle_mono1 1); |
|
751 by (asm_full_simp_tac (integ_ss addsimps [zadd_commute]) 1); |
|
752 qed "zadd_zle_self"; |