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1 theory Let |
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2 imports "../Nominal2" |
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3 begin |
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4 |
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5 |
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6 lemma Abs_lst_fcb2: |
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7 fixes as bs :: "atom list" |
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8 and x y :: "'b :: fs" |
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9 and c::"'c::fs" |
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10 assumes eq: "[as]lst. x = [bs]lst. y" |
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11 and fcb1: "(set as) \<sharp>* c \<Longrightarrow> (set as) \<sharp>* f as x c" |
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12 and fresh1: "set as \<sharp>* c" |
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13 and fresh2: "set bs \<sharp>* c" |
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14 and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c" |
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15 and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c" |
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16 shows "f as x c = f bs y c" |
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17 proof - |
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18 have "supp (as, x, c) supports (f as x c)" |
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19 unfolding supports_def fresh_def[symmetric] |
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20 by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh) |
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21 then have fin1: "finite (supp (f as x c))" |
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22 by (auto intro: supports_finite simp add: finite_supp) |
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23 have "supp (bs, y, c) supports (f bs y c)" |
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24 unfolding supports_def fresh_def[symmetric] |
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25 by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh) |
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26 then have fin2: "finite (supp (f bs y c))" |
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27 by (auto intro: supports_finite simp add: finite_supp) |
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28 obtain q::"perm" where |
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29 fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and |
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30 fr2: "supp q \<sharp>* Abs_lst as x" and |
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31 inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)" |
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32 using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"] |
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33 fin1 fin2 |
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34 by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv) |
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35 have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp |
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36 also have "\<dots> = Abs_lst as x" |
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37 by (simp only: fr2 perm_supp_eq) |
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38 finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp |
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39 then obtain r::perm where |
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40 qq1: "q \<bullet> x = r \<bullet> y" and |
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41 qq2: "q \<bullet> as = r \<bullet> bs" and |
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42 qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs" |
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43 apply(drule_tac sym) |
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44 apply(simp only: Abs_eq_iff2 alphas) |
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45 apply(erule exE) |
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46 apply(erule conjE)+ |
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47 apply(drule_tac x="p" in meta_spec) |
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48 apply(simp add: set_eqvt) |
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49 apply(blast) |
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50 done |
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51 have "(set as) \<sharp>* f as x c" |
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52 apply(rule fcb1) |
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53 apply(rule fresh1) |
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54 done |
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55 then have "q \<bullet> ((set as) \<sharp>* f as x c)" |
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56 by (simp add: permute_bool_def) |
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57 then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c" |
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58 apply(simp add: fresh_star_eqvt set_eqvt) |
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59 apply(subst (asm) perm1) |
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60 using inc fresh1 fr1 |
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61 apply(auto simp add: fresh_star_def fresh_Pair) |
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62 done |
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63 then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp |
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64 then have "r \<bullet> ((set bs) \<sharp>* f bs y c)" |
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65 apply(simp add: fresh_star_eqvt set_eqvt) |
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66 apply(subst (asm) perm2[symmetric]) |
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67 using qq3 fresh2 fr1 |
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68 apply(auto simp add: set_eqvt fresh_star_def fresh_Pair) |
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69 done |
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70 then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def) |
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71 have "f as x c = q \<bullet> (f as x c)" |
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72 apply(rule perm_supp_eq[symmetric]) |
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73 using inc fcb1[OF fresh1] fr1 by (auto simp add: fresh_star_def) |
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74 also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" |
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75 apply(rule perm1) |
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76 using inc fresh1 fr1 by (auto simp add: fresh_star_def) |
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77 also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp |
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78 also have "\<dots> = r \<bullet> (f bs y c)" |
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79 apply(rule perm2[symmetric]) |
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80 using qq3 fresh2 fr1 by (auto simp add: fresh_star_def) |
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81 also have "... = f bs y c" |
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82 apply(rule perm_supp_eq) |
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83 using qq3 fr1 fcb2 by (auto simp add: fresh_star_def) |
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84 finally show ?thesis by simp |
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85 qed |
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86 |
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87 lemma Abs_lst1_fcb2: |
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88 fixes a b :: "atom" |
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89 and x y :: "'b :: fs" |
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90 and c::"'c :: fs" |
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91 assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)" |
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92 and fcb1: "a \<sharp> c \<Longrightarrow> a \<sharp> f a x c" |
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93 and fresh: "{a, b} \<sharp>* c" |
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94 and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c" |
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95 and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c" |
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96 shows "f a x c = f b y c" |
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97 using e |
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98 apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"]) |
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99 apply(simp_all) |
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100 using fcb1 fresh perm1 perm2 |
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101 apply(simp_all add: fresh_star_def) |
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102 done |
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103 |
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104 atom_decl name |
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105 |
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106 nominal_datatype trm = |
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107 Var "name" |
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108 | App "trm" "trm" |
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109 | Let as::"assn" t::"trm" bind "bn as" in t |
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110 and assn = |
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111 Assn "name" "trm" |
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112 binder |
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113 bn |
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114 where |
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115 "bn (Assn x t) = [atom x]" |
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116 |
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117 print_theorems |
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118 |
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119 |
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120 |
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121 thm bn_raw.simps |
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122 thm permute_bn_raw.simps |
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123 thm trm_assn.perm_bn_alpha |
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124 thm trm_assn.permute_bn |
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125 |
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126 thm trm_assn.fv_defs |
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127 thm trm_assn.eq_iff |
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128 thm trm_assn.bn_defs |
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129 thm trm_assn.bn_inducts |
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130 thm trm_assn.perm_simps |
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131 thm trm_assn.induct |
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132 thm trm_assn.inducts |
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133 thm trm_assn.distinct |
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134 thm trm_assn.supp |
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135 thm trm_assn.fresh |
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136 thm trm_assn.exhaust |
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137 thm trm_assn.strong_exhaust |
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138 thm trm_assn.perm_bn_simps |
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139 |
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140 thm alpha_bn_raw.cases |
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141 |
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142 |
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143 lemmas alpha_bn_cases[consumes 1] = alpha_bn_raw.cases[quot_lifted] |
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144 |
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145 lemma alpha_bn_refl: "alpha_bn x x" |
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146 by (induct x rule: trm_assn.inducts(2)) |
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147 (rule TrueI, auto simp add: trm_assn.eq_iff) |
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148 lemma alpha_bn_sym: "alpha_bn x y \<Longrightarrow> alpha_bn y x" |
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149 apply(erule alpha_bn_cases) |
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150 apply(auto) |
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151 done |
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152 |
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153 lemma alpha_bn_trans: "alpha_bn x y \<Longrightarrow> alpha_bn y z \<Longrightarrow> alpha_bn x z" |
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154 sorry |
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155 |
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156 lemma k: "A \<Longrightarrow> A \<and> A" by blast |
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157 |
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158 lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)" |
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159 by (simp add: permute_pure) |
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160 |
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161 |
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162 section {* definition with helper functions *} |
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163 |
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164 function |
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165 apply_assn |
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166 where |
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167 "apply_assn f (Assn x t) = (f t)" |
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168 apply(case_tac x) |
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169 apply(simp) |
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170 apply(case_tac b rule: trm_assn.exhaust(2)) |
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171 apply(blast) |
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172 apply(simp) |
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173 done |
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174 |
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175 termination |
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176 by lexicographic_order |
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177 |
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178 function |
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179 apply_assn2 |
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180 where |
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181 "apply_assn2 f (Assn x t) = Assn x (f t)" |
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182 apply(case_tac x) |
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183 apply(simp) |
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184 apply(case_tac b rule: trm_assn.exhaust(2)) |
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185 apply(blast) |
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186 apply(simp) |
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187 done |
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188 |
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189 termination |
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190 by lexicographic_order |
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191 |
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192 lemma [eqvt]: |
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193 shows "p \<bullet> (apply_assn f as) = apply_assn (p \<bullet> f) (p \<bullet> as)" |
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194 apply(induct f as rule: apply_assn.induct) |
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195 apply(simp) |
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196 apply(perm_simp) |
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197 apply(rule) |
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198 done |
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199 |
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200 lemma [eqvt]: |
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201 shows "p \<bullet> (apply_assn2 f as) = apply_assn2 (p \<bullet> f) (p \<bullet> as)" |
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202 apply(induct f as rule: apply_assn.induct) |
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203 apply(simp) |
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204 apply(perm_simp) |
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205 apply(rule) |
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206 done |
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207 |
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208 |
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209 nominal_primrec |
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210 height_trm :: "trm \<Rightarrow> nat" |
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211 where |
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212 "height_trm (Var x) = 1" |
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213 | "height_trm (App l r) = max (height_trm l) (height_trm r)" |
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214 | "height_trm (Let as b) = max (apply_assn height_trm as) (height_trm b)" |
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215 apply (simp only: eqvt_def height_trm_graph_def) |
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216 apply (rule, perm_simp) |
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217 apply(rule) |
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218 apply(rule TrueI) |
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219 apply (case_tac x rule: trm_assn.exhaust(1)) |
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220 apply (auto simp add: alpha_bn_refl)[3] |
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221 apply (drule_tac x="assn" in meta_spec) |
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222 apply (drule_tac x="trm" in meta_spec) |
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223 apply(simp add: alpha_bn_refl) |
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224 apply(simp_all)[5] |
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225 apply(simp) |
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226 apply(erule conjE)+ |
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227 thm alpha_bn_cases |
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228 apply(erule alpha_bn_cases) |
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229 apply(simp) |
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230 apply (subgoal_tac "height_trm_sumC b = height_trm_sumC ba") |
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231 apply simp |
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232 apply(simp add: trm_assn.bn_defs) |
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233 apply(erule_tac c="()" in Abs_lst_fcb2) |
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234 apply(simp_all add: pure_fresh fresh_star_def)[3] |
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235 apply(simp_all add: eqvt_at_def) |
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236 done |
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237 |
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238 |
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239 nominal_primrec |
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240 subst_trm :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm" ("_ [_ ::= _]" [90, 90, 90] 90) |
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241 where |
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242 "(Var x)[y ::= s] = (if x = y then s else (Var x))" |
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243 | "(App t1 t2)[y ::= s] = App (t1[y ::= s]) (t2[y ::= s])" |
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244 | "(set (bn as)) \<sharp>* (y, s) \<Longrightarrow> |
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245 (Let as t)[y ::= s] = Let (apply_assn2 (\<lambda>t. t[y ::=s]) as) (t[y ::= s])" |
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246 apply (simp only: eqvt_def subst_trm_graph_def) |
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247 apply (rule, perm_simp) |
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248 apply(rule) |
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249 apply(rule TrueI) |
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250 apply(case_tac x) |
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251 apply(simp) |
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252 apply (rule_tac y="a" and c="(b,c)" in trm_assn.strong_exhaust(1)) |
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253 apply (auto simp add: alpha_bn_refl)[3] |
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254 apply(simp_all)[5] |
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255 apply(simp) |
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256 apply(erule conjE)+ |
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257 thm alpha_bn_cases |
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258 apply(erule alpha_bn_cases) |
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259 apply(simp) |
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260 apply(simp add: trm_assn.bn_defs) |
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261 apply(erule_tac c="(ya,sa)" in Abs_lst1_fcb2) |
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262 apply(simp add: Abs_fresh_iff fresh_star_def) |
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263 apply(simp add: fresh_star_def) |
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264 apply(simp_all add: eqvt_at_def perm_supp_eq fresh_star_Pair)[2] |
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265 done |
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266 |
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267 |
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268 section {* direct definitions --- problems *} |
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269 |
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270 |
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271 nominal_primrec |
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272 height_trm :: "trm \<Rightarrow> nat" |
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273 and height_assn :: "assn \<Rightarrow> nat" |
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274 where |
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275 "height_trm (Var x) = 1" |
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276 | "height_trm (App l r) = max (height_trm l) (height_trm r)" |
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277 | "height_trm (Let as b) = max (height_assn as) (height_trm b)" |
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278 | "height_assn (Assn x t) = (height_trm t)" |
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279 apply (simp only: eqvt_def height_trm_height_assn_graph_def) |
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280 apply (rule, perm_simp, rule, rule TrueI) |
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281 apply (case_tac x) |
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282 apply(simp) |
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283 apply (case_tac a rule: trm_assn.exhaust(1)) |
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284 apply (auto simp add: alpha_bn_refl)[3] |
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285 apply (drule_tac x="assn" in meta_spec) |
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286 apply (drule_tac x="trm" in meta_spec) |
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287 apply(simp add: alpha_bn_refl) |
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288 apply(simp) |
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289 apply (case_tac b rule: trm_assn.exhaust(2)) |
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290 apply (auto)[1] |
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291 apply(simp_all)[7] |
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292 prefer 2 |
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293 apply(simp) |
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294 --"let case" |
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295 apply (simp only: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff]) |
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296 apply (simp only: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff]) |
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297 apply (subgoal_tac "eqvt_at height_assn as") |
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298 apply (subgoal_tac "eqvt_at height_assn asa") |
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299 apply (subgoal_tac "eqvt_at height_trm b") |
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300 apply (subgoal_tac "eqvt_at height_trm ba") |
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301 apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)") |
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302 apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr asa)") |
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303 apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl b)") |
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304 apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl ba)") |
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305 defer |
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306 apply (simp add: eqvt_at_def height_trm_def) |
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307 apply (simp add: eqvt_at_def height_trm_def) |
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308 apply (simp add: eqvt_at_def height_assn_def) |
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309 apply (simp add: eqvt_at_def height_assn_def) |
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310 prefer 2 |
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311 apply (subgoal_tac "height_assn as = height_assn asa") |
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312 apply (subgoal_tac "height_trm b = height_trm ba") |
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313 apply simp |
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314 apply(simp) |
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315 apply(erule conjE)+ |
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316 apply(erule alpha_bn_cases) |
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317 apply(simp) |
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318 apply(simp add: trm_assn.bn_defs) |
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319 thm Abs_lst_fcb2 |
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320 apply(erule_tac c="()" in Abs_lst_fcb2) |
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321 apply(simp_all add: fresh_star_def pure_fresh)[3] |
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322 apply(simp add: eqvt_at_def) |
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323 apply(simp add: eqvt_at_def) |
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324 defer |
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325 apply(simp) |
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326 apply(frule Inl_inject) |
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327 apply(subst (asm) trm_assn.eq_iff) |
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328 apply(drule Inl_inject) |
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329 apply(clarify) |
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330 apply(erule alpha_bn_cases) |
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331 apply(simp del: trm_assn.eq_iff) |
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332 apply(rename_tac as s as' s' t' t x x') |
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333 apply(simp only: trm_assn.bn_defs) |
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334 (* HERE *) |
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335 oops |
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336 |
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337 |
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338 lemma ww1: |
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339 shows "finite (fv_trm t)" |
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340 and "finite (fv_bn as)" |
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341 apply(induct t and as rule: trm_assn.inducts) |
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342 apply(simp_all add: trm_assn.fv_defs supp_at_base) |
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343 done |
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344 |
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345 text {* works, but only because no recursion in as *} |
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346 |
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347 nominal_primrec (invariant "\<lambda>x (y::atom set). finite y") |
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348 frees_set :: "trm \<Rightarrow> atom set" |
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349 where |
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350 "frees_set (Var x) = {atom x}" |
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351 | "frees_set (App t1 t2) = frees_set t1 \<union> frees_set t2" |
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352 | "frees_set (Let as t) = (frees_set t) - (set (bn as)) \<union> (fv_bn as)" |
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353 apply(simp add: eqvt_def frees_set_graph_def) |
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354 apply(rule, perm_simp, rule) |
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355 apply(erule frees_set_graph.induct) |
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356 apply(auto simp add: ww1)[3] |
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357 apply(rule_tac y="x" in trm_assn.exhaust(1)) |
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358 apply(auto simp add: alpha_bn_refl)[3] |
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359 apply(drule_tac x="assn" in meta_spec) |
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360 apply(drule_tac x="trm" in meta_spec) |
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361 apply(simp add: alpha_bn_refl) |
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362 apply(simp_all)[5] |
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363 apply(simp) |
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364 apply(erule conjE) |
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365 apply(erule alpha_bn_cases) |
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366 apply(simp add: trm_assn.bn_defs) |
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367 apply(simp add: trm_assn.fv_defs) |
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368 (* apply(erule_tac c="(trm_rawa)" in Abs_lst1_fcb2) *) |
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369 apply(subgoal_tac " frees_set_sumC t - {atom name} = frees_set_sumC ta - {atom namea}") |
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370 apply(simp) |
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371 apply(erule_tac c="()" in Abs_lst1_fcb2) |
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372 apply(simp add: fresh_minus_atom_set) |
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373 apply(simp add: fresh_star_def fresh_Unit) |
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374 apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl) |
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375 apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl) |
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376 done |
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377 |
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378 termination |
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379 by lexicographic_order |
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380 |
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381 lemma test: |
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382 assumes a: "\<exists>y. f x = Inl y" |
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383 shows "(p \<bullet> (Sum_Type.Projl (f x))) = Sum_Type.Projl ((p \<bullet> f) (p \<bullet> x))" |
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384 using a |
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385 apply clarify |
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386 apply(frule_tac p="p" in permute_boolI) |
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387 apply(simp (no_asm_use) only: eqvts) |
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388 apply(subst (asm) permute_fun_app_eq) |
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389 back |
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390 apply(simp) |
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391 done |
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392 |
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393 |
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394 nominal_primrec (default "sum_case (\<lambda>x. Inl undefined) (\<lambda>x. Inr undefined)") |
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395 subst_trm :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm" ("_ [_ ::trm= _]" [90, 90, 90] 90) and |
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396 subst_assn :: "assn \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> assn" ("_ [_ ::assn= _]" [90, 90, 90] 90) |
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397 where |
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398 "(Var x)[y ::trm= s] = (if x = y then s else (Var x))" |
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399 | "(App t1 t2)[y ::trm= s] = App (t1[y ::trm= s]) (t2[y ::trm= s])" |
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400 | "(set (bn as)) \<sharp>* (y, s) \<Longrightarrow> (Let as t)[y ::trm= s] = Let (ast[y ::assn= s]) (t[y ::trm= s])" |
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401 | "(Assn x t)[y ::assn= s] = Assn x (t[y ::trm= s])" |
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402 apply(subgoal_tac "\<And>p x r. subst_trm_subst_assn_graph x r \<Longrightarrow> subst_trm_subst_assn_graph (p \<bullet> x) (p \<bullet> r)") |
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403 apply(simp add: eqvt_def) |
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404 apply(rule allI) |
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405 apply(simp add: permute_fun_def permute_bool_def) |
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406 apply(rule ext) |
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407 apply(rule ext) |
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408 apply(rule iffI) |
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409 apply(drule_tac x="p" in meta_spec) |
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410 apply(drule_tac x="- p \<bullet> x" in meta_spec) |
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411 apply(drule_tac x="- p \<bullet> xa" in meta_spec) |
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412 apply(simp) |
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413 apply(drule_tac x="-p" in meta_spec) |
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414 apply(drule_tac x="x" in meta_spec) |
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415 apply(drule_tac x="xa" in meta_spec) |
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416 apply(simp) |
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417 --"Eqvt One way" |
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418 defer |
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419 apply(rule TrueI) |
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420 apply(case_tac x) |
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421 apply(simp) |
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422 apply(case_tac a) |
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423 apply(simp) |
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424 apply(rule_tac y="aa" and c="(b, c)" in trm_assn.strong_exhaust(1)) |
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425 apply(blast)+ |
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426 apply(simp) |
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427 apply(case_tac b) |
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428 apply(simp) |
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429 apply(rule_tac y="a" in trm_assn.exhaust(2)) |
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430 apply(simp) |
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431 apply(blast) |
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432 apply(simp_all)[7] |
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433 prefer 2 |
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434 apply(simp) |
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435 prefer 2 |
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436 apply(simp) |
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437 apply(simp) |
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438 apply (simp only: meta_eq_to_obj_eq[OF subst_trm_def, symmetric, unfolded fun_eq_iff]) |
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439 apply (simp only: meta_eq_to_obj_eq[OF subst_assn_def, symmetric, unfolded fun_eq_iff]) |
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440 apply (subgoal_tac "eqvt_at (\<lambda>ast. subst_assn ast ya sa) ast") |
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441 apply (subgoal_tac "eqvt_at (\<lambda>asta. subst_assn asta ya sa) asta") |
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442 apply (subgoal_tac "eqvt_at (\<lambda>t. subst_trm t ya sa) t") |
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443 apply (subgoal_tac "eqvt_at (\<lambda>ta. subst_trm ta ya sa) ta") |
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444 apply (thin_tac "eqvt_at subst_trm_subst_assn_sumC (Inr (ast, ya, sa))") |
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445 apply (thin_tac "eqvt_at subst_trm_subst_assn_sumC (Inr (asta, ya, sa))") |
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446 apply (thin_tac "eqvt_at subst_trm_subst_assn_sumC (Inl (t, ya, sa))") |
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447 apply (thin_tac "eqvt_at subst_trm_subst_assn_sumC (Inl (ta, ya, sa))") |
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448 defer |
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449 defer |
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450 defer |
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451 defer |
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452 defer |
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453 defer |
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454 apply(rule conjI) |
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455 apply (subgoal_tac "subst_assn ast ya sa= subst_assn asta ya sa") |
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456 apply (subgoal_tac "subst_trm t ya sa = subst_trm ta ya sa") |
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457 apply(simp) |
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458 apply(erule_tac conjE)+ |
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459 apply(erule alpha_bn_cases) |
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460 apply(simp add: trm_assn.bn_defs) |
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461 apply(rotate_tac 7) |
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462 apply(drule k) |
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463 apply(erule conjE) |
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464 apply(subst (asm) Abs1_eq_iff) |
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465 apply(rule sort_of_atom_eq) |
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466 apply(rule sort_of_atom_eq) |
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467 apply(erule disjE) |
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468 apply(simp) |
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469 apply(rotate_tac 12) |
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470 apply(drule sym) |
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471 apply(rule sym) |
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472 apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2) |
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473 apply(erule fresh_eqvt_at) |
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474 |
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475 thm fresh_eqvt_at |
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476 apply(simp add: Abs_fresh_iff) |
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477 apply(simp add: fresh_star_def fresh_Pair) |
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478 apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq) |
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479 apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq) |
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480 |
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481 |
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482 |
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483 apply(simp_all add: fresh_star_def fresh_Pair_elim)[1] |
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484 apply(blast) |
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485 apply(simp_all)[5] |
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486 apply(simp (no_asm_use)) |
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487 apply(simp) |
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488 apply(erule conjE)+ |
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489 apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2) |
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490 apply(simp add: Abs_fresh_iff) |
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491 apply(simp add: fresh_star_def fresh_Pair) |
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492 apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq) |
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493 apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq) |
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494 done |
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495 |
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496 |
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497 end |