1 (* Title: HOL/MiniML/MiniML.ML |
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2 ID: $Id$ |
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3 Author: Dieter Nazareth and Tobias Nipkow |
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4 Copyright 1995 TU Muenchen |
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5 *) |
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6 |
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7 Addsimps has_type.intrs; |
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8 Addsimps [Un_upper1,Un_upper2]; |
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9 |
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10 Addsimps [is_bound_typ_instance_closed_subst]; |
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11 |
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12 |
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13 Goal "!!t::typ. $(%n. if n : (free_tv A) Un (free_tv t) then (S n) else (TVar n)) t = $S t"; |
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14 by (rtac typ_substitutions_only_on_free_variables 1); |
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15 by (asm_full_simp_tac (simpset() addsimps [Ball_def]) 1); |
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16 qed "s'_t_equals_s_t"; |
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17 |
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18 Addsimps [s'_t_equals_s_t]; |
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19 |
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20 Goal "!!A::type_scheme list. $(%n. if n : (free_tv A) Un (free_tv t) then (S n) else (TVar n)) A = $S A"; |
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21 by (rtac scheme_list_substitutions_only_on_free_variables 1); |
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22 by (asm_full_simp_tac (simpset() addsimps [Ball_def]) 1); |
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23 qed "s'_a_equals_s_a"; |
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24 |
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25 Addsimps [s'_a_equals_s_a]; |
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26 |
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27 Goal |
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28 "$(%n. if n : (free_tv A) Un (free_tv t) then S n else TVar n) A |- \ |
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29 \ e :: $(%n. if n : (free_tv A) Un (free_tv t) then S n else TVar n) t \ |
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30 \ ==> $S A |- e :: $S t"; |
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31 |
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32 by (res_inst_tac [("P","%A. A |- e :: $S t")] ssubst 1); |
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33 by (rtac (s'_a_equals_s_a RS sym) 1); |
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34 by (res_inst_tac [("P","%t. $ (%n. if n : free_tv A Un free_tv (?t1 S) then S n else TVar n) A |- e :: t")] ssubst 1); |
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35 by (rtac (s'_t_equals_s_t RS sym) 1); |
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36 by (Asm_full_simp_tac 1); |
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37 qed "replace_s_by_s'"; |
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38 |
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39 Goal "!!A::type_scheme list. $ (%x. TVar (if x : free_tv A then x else n + x)) A = $ id_subst A"; |
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40 by (rtac scheme_list_substitutions_only_on_free_variables 1); |
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41 by (asm_full_simp_tac (simpset() addsimps [id_subst_def]) 1); |
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42 qed "alpha_A'"; |
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43 |
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44 Goal "!!A::type_scheme list. $ (%x. TVar (if x : free_tv A then x else n + x)) A = A"; |
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45 by (rtac (alpha_A' RS ssubst) 1); |
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46 by (Asm_full_simp_tac 1); |
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47 qed "alpha_A"; |
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48 |
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49 Goal "$ (S o alpha) (t::typ) = $ S ($ (%x. TVar (alpha x)) t)"; |
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50 by (induct_tac "t" 1); |
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51 by (ALLGOALS (Asm_full_simp_tac)); |
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52 qed "S_o_alpha_typ"; |
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53 |
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54 Goal "$ (%u. (S o alpha) u) (t::typ) = $ S ($ (%x. TVar (alpha x)) t)"; |
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55 by (induct_tac "t" 1); |
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56 by (ALLGOALS (Asm_full_simp_tac)); |
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57 val S_o_alpha_typ' = result (); |
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58 |
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59 Goal "$ (S o alpha) (sch::type_scheme) = $ S ($ (%x. TVar (alpha x)) sch)"; |
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60 by (induct_tac "sch" 1); |
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61 by (ALLGOALS (Asm_full_simp_tac)); |
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62 qed "S_o_alpha_type_scheme"; |
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63 |
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64 Goal "$ (S o alpha) (A::type_scheme list) = $ S ($ (%x. TVar (alpha x)) A)"; |
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65 by (induct_tac "A" 1); |
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66 by (ALLGOALS (Asm_full_simp_tac)); |
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67 by (rtac (rewrite_rule [o_def] S_o_alpha_type_scheme) 1); |
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68 qed "S_o_alpha_type_scheme_list"; |
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69 |
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70 Goal "!!A::type_scheme list. \ |
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71 \ $ (%n. if n : free_tv A Un free_tv t then S n else TVar n) A = \ |
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72 \ $ ((%x. if x : free_tv A Un free_tv t then S x else TVar x) o \ |
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73 \ (%x. if x : free_tv A then x else n + x)) A"; |
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74 by (stac S_o_alpha_type_scheme_list 1); |
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75 by (stac alpha_A 1); |
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76 by (rtac refl 1); |
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77 qed "S'_A_eq_S'_alpha_A"; |
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78 |
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79 Goalw [free_tv_subst,dom_def] |
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80 "dom (%n. if n : free_tv A Un free_tv t then S n else TVar n) <= \ |
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81 \ free_tv A Un free_tv t"; |
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82 by (Simp_tac 1); |
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83 by (Fast_tac 1); |
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84 qed "dom_S'"; |
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85 |
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86 Goalw [free_tv_subst,cod_def,subset_def] |
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87 "!!(A::type_scheme list) (t::typ). \ |
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88 \ cod (%n. if n : free_tv A Un free_tv t then S n else TVar n) <= \ |
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89 \ free_tv ($ S A) Un free_tv ($ S t)"; |
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90 by (rtac ballI 1); |
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91 by (etac UN_E 1); |
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92 by (dtac (dom_S' RS subsetD) 1); |
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93 by (Asm_full_simp_tac 1); |
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94 by (fast_tac (claset() addDs [free_tv_of_substitutions_extend_to_scheme_lists] |
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95 addIs [free_tv_of_substitutions_extend_to_types RS subsetD]) 1); |
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96 qed "cod_S'"; |
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97 |
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98 Goalw [free_tv_subst] |
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99 "!!(A::type_scheme list) (t::typ). \ |
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100 \ free_tv (%n. if n : free_tv A Un free_tv t then S n else TVar n) <= \ |
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101 \ free_tv A Un free_tv ($ S A) Un free_tv t Un free_tv ($ S t)"; |
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102 by (fast_tac (claset() addDs [dom_S' RS subsetD,cod_S' RS subsetD]) 1); |
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103 qed "free_tv_S'"; |
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104 |
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105 Goal "!!t1::typ. \ |
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106 \ (free_tv ($ (%x. TVar (if x : free_tv A then x else n + x)) t1) - free_tv A) <= \ |
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107 \ {x. ? y. x = n + y}"; |
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108 by (induct_tac "t1" 1); |
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109 by (Simp_tac 1); |
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110 by (Fast_tac 1); |
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111 by (Simp_tac 1); |
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112 by (Fast_tac 1); |
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113 qed "free_tv_alpha"; |
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114 |
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115 Goal "!!(k::nat). n <= n + k"; |
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116 by (induct_thm_tac nat_induct "k" 1); |
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117 by (Simp_tac 1); |
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118 by (Asm_simp_tac 1); |
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119 val aux_plus = result(); |
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120 |
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121 Addsimps [aux_plus]; |
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122 |
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123 Goal "!!t::typ. new_tv n t ==> {x. ? y. x = n + y} Int free_tv t = {}"; |
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124 by Safe_tac; |
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125 by (cut_facts_tac [aux_plus] 1); |
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126 by (dtac new_tv_le 1); |
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127 by (assume_tac 1); |
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128 by (rotate_tac 1 1); |
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129 by (dtac new_tv_not_free_tv 1); |
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130 by (Fast_tac 1); |
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131 val new_tv_Int_free_tv_empty_type = result (); |
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132 |
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133 Goal "!!sch::type_scheme. new_tv n sch ==> {x. ? y. x = n + y} Int free_tv sch = {}"; |
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134 by Safe_tac; |
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135 by (cut_facts_tac [aux_plus] 1); |
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136 by (dtac new_tv_le 1); |
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137 by (assume_tac 1); |
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138 by (rotate_tac 1 1); |
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139 by (dtac new_tv_not_free_tv 1); |
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140 by (Fast_tac 1); |
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141 val new_tv_Int_free_tv_empty_scheme = result (); |
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142 |
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143 Goal "!A::type_scheme list. new_tv n A --> {x. ? y. x = n + y} Int free_tv A = {}"; |
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144 by (rtac allI 1); |
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145 by (induct_tac "A" 1); |
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146 by (Simp_tac 1); |
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147 by (Simp_tac 1); |
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148 by (fast_tac (claset() addDs [new_tv_Int_free_tv_empty_scheme ]) 1); |
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149 val new_tv_Int_free_tv_empty_scheme_list = result (); |
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150 |
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151 Goalw [le_type_scheme_def,is_bound_typ_instance] |
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152 "new_tv n A --> gen A t <= gen A ($ (%x. TVar (if x : free_tv A then x else n + x)) t)"; |
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153 by (strip_tac 1); |
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154 by (etac exE 1); |
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155 by (hyp_subst_tac 1); |
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156 by (res_inst_tac [("x","(%x. S (if n <= x then x - n else x))")] exI 1); |
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157 by (induct_tac "t" 1); |
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158 by (Simp_tac 1); |
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159 by (case_tac "nat : free_tv A" 1); |
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160 by (Asm_simp_tac 1); |
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161 by (subgoal_tac "n <= n + nat" 1); |
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162 by (dtac new_tv_le 1); |
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163 by (assume_tac 1); |
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164 by (dtac new_tv_not_free_tv 1); |
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165 by (dtac new_tv_not_free_tv 1); |
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166 by (asm_simp_tac (simpset() addsimps [diff_add_inverse ]) 1); |
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167 by (Simp_tac 1); |
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168 by (Asm_simp_tac 1); |
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169 qed_spec_mp "gen_t_le_gen_alpha_t"; |
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170 |
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171 AddSIs has_type.intrs; |
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172 |
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173 Goal "A |- e::t ==> !B. A <= B --> B |- e::t"; |
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174 by (etac has_type.induct 1); |
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175 by (simp_tac (simpset() addsimps [le_env_def]) 1); |
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176 by (fast_tac (claset() addEs [bound_typ_instance_trans] addss simpset()) 1); |
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177 by (Asm_full_simp_tac 1); |
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178 by (Fast_tac 1); |
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179 by (slow_tac (claset() addEs [le_env_free_tv RS free_tv_subset_gen_le]) 1); |
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180 qed_spec_mp "has_type_le_env"; |
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181 |
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182 (* has_type is closed w.r.t. substitution *) |
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183 Goal "A |- e :: t ==> !S. $S A |- e :: $S t"; |
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184 by (etac has_type.induct 1); |
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185 (* case VarI *) |
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186 by (rtac allI 1); |
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187 by (rtac has_type.VarI 1); |
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188 by (asm_full_simp_tac (simpset() addsimps [app_subst_list]) 1); |
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189 by (asm_simp_tac (simpset() addsimps [app_subst_list]) 1); |
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190 (* case AbsI *) |
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191 by (rtac allI 1); |
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192 by (Simp_tac 1); |
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193 by (rtac has_type.AbsI 1); |
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194 by (Asm_full_simp_tac 1); |
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195 (* case AppI *) |
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196 by (rtac allI 1); |
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197 by (rtac has_type.AppI 1); |
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198 by (Asm_full_simp_tac 1); |
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199 by (etac spec 1); |
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200 by (etac spec 1); |
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201 (* case LetI *) |
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202 by (rtac allI 1); |
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203 by (rtac replace_s_by_s' 1); |
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204 by (cut_inst_tac [("A","$ S A"), |
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205 ("A'","A"), |
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206 ("t","t"), |
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207 ("t'","$ S t")] |
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208 ex_fresh_variable 1); |
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209 by (etac exE 1); |
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210 by (REPEAT (etac conjE 1)); |
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211 by (res_inst_tac [("t1.0","$((%x. if x : free_tv A Un free_tv t then S x else TVar x) o \ |
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212 \ (%x. if x : free_tv A then x else n + x)) t1")] |
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213 has_type.LETI 1); |
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214 by (dres_inst_tac [("x","(%x. if x : free_tv A Un free_tv t then S x else TVar x) o \ |
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215 \ (%x. if x : free_tv A then x else n + x)")] spec 1); |
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216 by (stac (S'_A_eq_S'_alpha_A) 1); |
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217 by (assume_tac 1); |
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218 by (stac S_o_alpha_typ 1); |
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219 by (stac gen_subst_commutes 1); |
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220 by (rtac subset_antisym 1); |
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221 by (rtac subsetI 1); |
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222 by (etac IntE 1); |
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223 by (dtac (free_tv_S' RS subsetD) 1); |
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224 by (dtac (free_tv_alpha RS subsetD) 1); |
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225 by (asm_full_simp_tac (simpset() delsimps [full_SetCompr_eq]) 1); |
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226 by (etac exE 1); |
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227 by (hyp_subst_tac 1); |
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228 by (subgoal_tac "new_tv (n + y) ($ S A)" 1); |
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229 by (subgoal_tac "new_tv (n + y) ($ S t)" 1); |
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230 by (subgoal_tac "new_tv (n + y) A" 1); |
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231 by (subgoal_tac "new_tv (n + y) t" 1); |
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232 by (REPEAT (dtac new_tv_not_free_tv 1)); |
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233 by (Fast_tac 1); |
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234 by (REPEAT ((rtac new_tv_le 1) THEN (assume_tac 2) THEN (Simp_tac 1))); |
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235 by (Fast_tac 1); |
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236 by (rtac has_type_le_env 1); |
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237 by (dtac spec 1); |
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238 by (dtac spec 1); |
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239 by (assume_tac 1); |
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240 by (rtac (app_subst_Cons RS subst) 1); |
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241 by (rtac S_compatible_le_scheme_lists 1); |
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242 by (Asm_simp_tac 1); |
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243 qed "has_type_cl_sub"; |
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