1 (* Title: HOL/Subst/Subst.ML |
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2 ID: $Id$ |
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3 Author: Martin Coen, Cambridge University Computer Laboratory |
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4 Copyright 1993 University of Cambridge |
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5 |
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6 Substitutions on uterms |
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7 *) |
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8 |
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9 open Subst; |
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10 |
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11 |
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12 (**** Substitutions ****) |
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13 |
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14 Goal "t <| [] = t"; |
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15 by (induct_tac "t" 1); |
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16 by (ALLGOALS Asm_simp_tac); |
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17 qed "subst_Nil"; |
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18 |
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19 Addsimps [subst_Nil]; |
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20 |
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21 Goal "t <: u --> t <| s <: u <| s"; |
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22 by (induct_tac "u" 1); |
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23 by (ALLGOALS Asm_simp_tac); |
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24 qed_spec_mp "subst_mono"; |
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25 |
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26 Goal "~ (Var(v) <: t) --> t <| (v,t <| s) # s = t <| s"; |
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27 by (case_tac "t = Var(v)" 1); |
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28 by (etac rev_mp 2); |
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29 by (res_inst_tac [("P", |
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30 "%x.~x=Var(v) --> ~(Var(v) <: x) --> x <| (v,t<|s)#s=x<|s")] |
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31 uterm.induct 2); |
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32 by (ALLGOALS Asm_simp_tac); |
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33 by (Blast_tac 1); |
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34 qed_spec_mp "Var_not_occs"; |
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35 |
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36 Goal "(t <|r = t <|s) = (! v. v : vars_of(t) --> Var(v) <|r = Var(v) <|s)"; |
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37 by (induct_tac "t" 1); |
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38 by (ALLGOALS Asm_full_simp_tac); |
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39 by (ALLGOALS Blast_tac); |
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40 qed "agreement"; |
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41 |
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42 Goal "~ v: vars_of(t) --> t <| (v,u)#s = t <| s"; |
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43 by (simp_tac (simpset() addsimps [agreement]) 1); |
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44 qed_spec_mp"repl_invariance"; |
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45 |
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46 val asms = goal Subst.thy |
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47 "v : vars_of(t) --> w : vars_of(t <| (v,Var(w))#s)"; |
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48 by (induct_tac "t" 1); |
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49 by (ALLGOALS Asm_simp_tac); |
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50 qed_spec_mp"Var_in_subst"; |
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51 |
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52 |
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53 (**** Equality between Substitutions ****) |
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54 |
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55 Goalw [subst_eq_def] "r =$= s = (! t. t <| r = t <| s)"; |
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56 by (Simp_tac 1); |
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57 qed "subst_eq_iff"; |
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58 |
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59 |
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60 local fun prove s = prove_goal Subst.thy s |
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61 (fn prems => [cut_facts_tac prems 1, |
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62 REPEAT (etac rev_mp 1), |
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63 simp_tac (simpset() addsimps [subst_eq_iff]) 1]) |
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64 in |
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65 val subst_refl = prove "r =$= r"; |
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66 val subst_sym = prove "r =$= s ==> s =$= r"; |
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67 val subst_trans = prove "[| q =$= r; r =$= s |] ==> q =$= s"; |
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68 end; |
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69 |
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70 |
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71 AddIffs [subst_refl]; |
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72 |
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73 |
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74 val eq::prems = goalw Subst.thy [subst_eq_def] |
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75 "[| r =$= s; P (t <| r) (u <| r) |] ==> P (t <| s) (u <| s)"; |
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76 by (resolve_tac [eq RS spec RS subst] 1); |
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77 by (resolve_tac (prems RL [eq RS spec RS subst]) 1); |
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78 qed "subst_subst2"; |
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79 |
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80 val ssubst_subst2 = subst_sym RS subst_subst2; |
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81 |
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82 (**** Composition of Substitutions ****) |
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83 |
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84 let fun prove s = |
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85 prove_goalw Subst.thy [comp_def,sdom_def] s (fn _ => [Simp_tac 1]) |
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86 in |
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87 Addsimps |
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88 ( |
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89 map prove |
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90 [ "[] <> bl = bl", |
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91 "((a,b)#al) <> bl = (a,b <| bl) # (al <> bl)", |
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92 "sdom([]) = {}", |
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93 "sdom((a,b)#al) = (if Var(a)=b then (sdom al) - {a} else sdom al Un {a})"] |
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94 ) |
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95 end; |
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96 |
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97 |
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98 Goal "s <> [] = s"; |
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99 by (alist_ind_tac "s" 1); |
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100 by (ALLGOALS Asm_simp_tac); |
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101 qed "comp_Nil"; |
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102 |
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103 Addsimps [comp_Nil]; |
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104 |
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105 Goal "s =$= s <> []"; |
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106 by (Simp_tac 1); |
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107 qed "subst_comp_Nil"; |
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108 |
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109 Goal "(t <| r <> s) = (t <| r <| s)"; |
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110 by (induct_tac "t" 1); |
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111 by (ALLGOALS Asm_simp_tac); |
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112 by (alist_ind_tac "r" 1); |
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113 by (ALLGOALS Asm_simp_tac); |
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114 qed "subst_comp"; |
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115 |
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116 Addsimps [subst_comp]; |
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117 |
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118 Goal "(q <> r) <> s =$= q <> (r <> s)"; |
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119 by (simp_tac (simpset() addsimps [subst_eq_iff]) 1); |
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120 qed "comp_assoc"; |
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121 |
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122 Goal "[| theta =$= theta1; sigma =$= sigma1|] ==> \ |
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123 \ (theta <> sigma) =$= (theta1 <> sigma1)"; |
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124 by (asm_full_simp_tac (simpset() addsimps [subst_eq_def]) 1); |
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125 qed "subst_cong"; |
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126 |
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127 |
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128 Goal "(w, Var(w) <| s) # s =$= s"; |
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129 by (simp_tac (simpset() addsimps [subst_eq_iff]) 1); |
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130 by (rtac allI 1); |
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131 by (induct_tac "t" 1); |
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132 by (ALLGOALS Asm_full_simp_tac); |
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133 qed "Cons_trivial"; |
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134 |
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135 |
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136 Goal "q <> r =$= s ==> t <| q <| r = t <| s"; |
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137 by (asm_full_simp_tac (simpset() addsimps [subst_eq_iff]) 1); |
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138 qed "comp_subst_subst"; |
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139 |
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140 |
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141 (**** Domain and range of Substitutions ****) |
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142 |
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143 Goal "(v : sdom(s)) = (Var(v) <| s ~= Var(v))"; |
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144 by (alist_ind_tac "s" 1); |
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145 by (ALLGOALS Asm_simp_tac); |
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146 by (Blast_tac 1); |
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147 qed "sdom_iff"; |
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148 |
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149 |
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150 Goalw [srange_def] |
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151 "v : srange(s) = (? w. w : sdom(s) & v : vars_of(Var(w) <| s))"; |
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152 by (Blast_tac 1); |
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153 qed "srange_iff"; |
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154 |
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155 Goalw [empty_def] "(A = {}) = (ALL a.~ a:A)"; |
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156 by (Blast_tac 1); |
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157 qed "empty_iff_all_not"; |
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158 |
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159 Goal "(t <| s = t) = (sdom(s) Int vars_of(t) = {})"; |
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160 by (induct_tac "t" 1); |
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161 by (ALLGOALS |
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162 (asm_full_simp_tac (simpset() addsimps [empty_iff_all_not, sdom_iff]))); |
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163 by (ALLGOALS Blast_tac); |
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164 qed "invariance"; |
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165 |
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166 Goal "v : sdom(s) --> v : vars_of(t <| s) --> v : srange(s)"; |
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167 by (induct_tac "t" 1); |
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168 by (case_tac "a : sdom(s)" 1); |
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169 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [sdom_iff, srange_iff]))); |
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170 by (ALLGOALS Blast_tac); |
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171 qed_spec_mp "Var_in_srange"; |
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172 |
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173 Goal "[| v : sdom(s); v ~: srange(s) |] ==> v ~: vars_of(t <| s)"; |
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174 by (blast_tac (claset() addIs [Var_in_srange]) 1); |
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175 qed "Var_elim"; |
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176 |
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177 Goal "v : vars_of(t <| s) --> v : srange(s) | v : vars_of(t)"; |
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178 by (induct_tac "t" 1); |
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179 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [sdom_iff,srange_iff]))); |
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180 by (Blast_tac 2); |
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181 by (safe_tac (claset() addSIs [exI, vars_var_iff RS iffD1 RS sym])); |
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182 by Auto_tac; |
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183 qed_spec_mp "Var_intro"; |
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184 |
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185 Goal "v : srange(s) --> (? w. w : sdom(s) & v : vars_of(Var(w) <| s))"; |
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186 by (simp_tac (simpset() addsimps [srange_iff]) 1); |
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187 qed_spec_mp "srangeD"; |
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188 |
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189 Goal "sdom(s) Int srange(s) = {} = (! t. sdom(s) Int vars_of(t <| s) = {})"; |
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190 by (simp_tac (simpset() addsimps [empty_iff_all_not]) 1); |
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191 by (fast_tac (claset() addIs [Var_in_srange] addDs [srangeD]) 1); |
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192 qed "dom_range_disjoint"; |
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193 |
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194 Goal "~ u <| s = u ==> (? x. x : sdom(s))"; |
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195 by (full_simp_tac (simpset() addsimps [empty_iff_all_not, invariance]) 1); |
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196 by (Blast_tac 1); |
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197 qed "subst_not_empty"; |
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198 |
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199 |
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200 Goal "(M <| [(x, Var x)]) = M"; |
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201 by (induct_tac "M" 1); |
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202 by (ALLGOALS Asm_simp_tac); |
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203 qed "id_subst_lemma"; |
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204 |
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205 Addsimps [id_subst_lemma]; |
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