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1 theory LamEx |
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2 imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" |
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3 begin |
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4 |
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5 atom_decl name |
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6 |
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7 datatype rlam = |
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8 rVar "name" |
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9 | rApp "rlam" "rlam" |
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10 | rLam "name" "rlam" |
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11 |
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12 fun |
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13 rfv :: "rlam \<Rightarrow> atom set" |
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14 where |
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15 rfv_var: "rfv (rVar a) = {atom a}" |
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16 | rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)" |
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17 | rfv_lam: "rfv (rLam a t) = (rfv t) - {atom a}" |
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18 |
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19 instantiation rlam :: pt |
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20 begin |
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21 |
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22 primrec |
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23 permute_rlam |
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24 where |
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25 "permute_rlam pi (rVar a) = rVar (pi \<bullet> a)" |
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26 | "permute_rlam pi (rApp t1 t2) = rApp (permute_rlam pi t1) (permute_rlam pi t2)" |
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27 | "permute_rlam pi (rLam a t) = rLam (pi \<bullet> a) (permute_rlam pi t)" |
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28 |
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29 instance |
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30 apply default |
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31 apply(induct_tac [!] x) |
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32 apply(simp_all) |
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33 done |
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34 |
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35 end |
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36 |
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37 instantiation rlam :: fs |
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38 begin |
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39 |
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40 lemma neg_conj: |
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41 "\<not>(P \<and> Q) \<longleftrightarrow> (\<not>P) \<or> (\<not>Q)" |
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42 by simp |
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43 |
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44 lemma infinite_Un: |
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45 "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite T" |
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46 by simp |
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47 |
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48 instance |
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49 apply default |
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50 apply(induct_tac x) |
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51 (* var case *) |
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52 apply(simp add: supp_def) |
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53 apply(fold supp_def)[1] |
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54 apply(simp add: supp_at_base) |
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55 (* app case *) |
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56 apply(simp only: supp_def) |
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57 apply(simp only: permute_rlam.simps) |
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58 apply(simp only: rlam.inject) |
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59 apply(simp only: neg_conj) |
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60 apply(simp only: Collect_disj_eq) |
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61 apply(simp only: infinite_Un) |
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62 apply(simp only: Collect_disj_eq) |
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63 apply(simp) |
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64 (* lam case *) |
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65 apply(simp only: supp_def) |
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66 apply(simp only: permute_rlam.simps) |
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67 apply(simp only: rlam.inject) |
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68 apply(simp only: neg_conj) |
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69 apply(simp only: Collect_disj_eq) |
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70 apply(simp only: infinite_Un) |
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71 apply(simp only: Collect_disj_eq) |
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72 apply(simp) |
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73 apply(fold supp_def)[1] |
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74 apply(simp add: supp_at_base) |
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75 done |
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76 |
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77 end |
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78 |
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79 |
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80 (* for the eqvt proof of the alpha-equivalence *) |
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81 declare permute_rlam.simps[eqvt] |
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82 |
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83 lemma rfv_eqvt[eqvt]: |
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84 shows "(pi\<bullet>rfv t) = rfv (pi\<bullet>t)" |
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85 apply(induct t) |
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86 apply(simp_all) |
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87 apply(simp add: permute_set_eq atom_eqvt) |
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88 apply(simp add: union_eqvt) |
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89 apply(simp add: Diff_eqvt) |
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90 apply(simp add: permute_set_eq atom_eqvt) |
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91 done |
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92 |
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93 inductive |
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94 alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100) |
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95 where |
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96 a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)" |
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97 | a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2" |
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98 | a3: "\<exists>pi. (rfv t - {atom a} = rfv s - {atom b} \<and> (rfv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s) |
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99 \<Longrightarrow> rLam a t \<approx> rLam b s" |
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100 |
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101 lemma a3_inverse: |
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102 assumes "rLam a t \<approx> rLam b s" |
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103 shows "\<exists>pi. (rfv t - {atom a} = rfv s - {atom b} \<and> (rfv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s)" |
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104 using assms |
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105 apply(erule_tac alpha.cases) |
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106 apply(auto) |
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107 done |
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108 |
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109 text {* should be automatic with new version of eqvt-machinery *} |
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110 lemma alpha_eqvt: |
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111 shows "t \<approx> s \<Longrightarrow> (pi \<bullet> t) \<approx> (pi \<bullet> s)" |
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112 apply(induct rule: alpha.induct) |
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113 apply(simp add: a1) |
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114 apply(simp add: a2) |
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115 apply(simp) |
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116 apply(rule a3) |
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117 apply(erule conjE) |
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118 apply(erule exE) |
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119 apply(erule conjE) |
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120 apply(rule_tac x="pi \<bullet> pia" in exI) |
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121 apply(rule conjI) |
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122 apply(rule_tac ?p1="- pi" in permute_eq_iff[THEN iffD1]) |
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123 apply(simp only: Diff_eqvt rfv_eqvt insert_eqvt atom_eqvt empty_eqvt) |
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124 apply(simp) |
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125 apply(rule conjI) |
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126 apply(rule_tac ?p1="- pi" in fresh_star_permute_iff[THEN iffD1]) |
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127 apply(simp add: Diff_eqvt rfv_eqvt atom_eqvt insert_eqvt empty_eqvt) |
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128 apply(subst permute_eqvt[symmetric]) |
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129 apply(simp) |
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130 done |
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131 |
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132 lemma alpha_refl: |
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133 shows "t \<approx> t" |
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134 apply(induct t rule: rlam.induct) |
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135 apply(simp add: a1) |
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136 apply(simp add: a2) |
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137 apply(rule a3) |
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138 apply(rule_tac x="0" in exI) |
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139 apply(simp_all add: fresh_star_def fresh_zero_perm) |
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140 done |
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141 |
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142 lemma alpha_sym: |
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143 shows "t \<approx> s \<Longrightarrow> s \<approx> t" |
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144 apply(induct rule: alpha.induct) |
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145 apply(simp add: a1) |
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146 apply(simp add: a2) |
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147 apply(rule a3) |
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148 apply(erule exE) |
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149 apply(rule_tac x="- pi" in exI) |
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150 apply(simp) |
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151 apply(simp add: fresh_star_def fresh_minus_perm) |
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152 apply(erule conjE)+ |
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153 apply(rotate_tac 3) |
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154 apply(drule_tac pi="- pi" in alpha_eqvt) |
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155 apply(simp) |
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156 done |
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157 |
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158 lemma alpha_trans: |
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159 shows "t1 \<approx> t2 \<Longrightarrow> t2 \<approx> t3 \<Longrightarrow> t1 \<approx> t3" |
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160 apply(induct arbitrary: t3 rule: alpha.induct) |
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161 apply(erule alpha.cases) |
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162 apply(simp_all) |
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163 apply(simp add: a1) |
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164 apply(rotate_tac 4) |
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165 apply(erule alpha.cases) |
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166 apply(simp_all) |
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167 apply(simp add: a2) |
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168 apply(rotate_tac 1) |
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169 apply(erule alpha.cases) |
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170 apply(simp_all) |
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171 apply(erule conjE)+ |
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172 apply(erule exE)+ |
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173 apply(erule conjE)+ |
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174 apply(rule a3) |
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175 apply(rule_tac x="pia + pi" in exI) |
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176 apply(simp add: fresh_star_plus) |
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177 apply(drule_tac x="- pia \<bullet> sa" in spec) |
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178 apply(drule mp) |
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179 apply(rotate_tac 7) |
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180 apply(drule_tac pi="- pia" in alpha_eqvt) |
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181 apply(simp) |
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182 apply(rotate_tac 9) |
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183 apply(drule_tac pi="pia" in alpha_eqvt) |
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184 apply(simp) |
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185 done |
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186 |
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187 lemma alpha_equivp: |
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188 shows "equivp alpha" |
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189 apply(rule equivpI) |
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190 unfolding reflp_def symp_def transp_def |
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191 apply(auto intro: alpha_refl alpha_sym alpha_trans) |
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192 done |
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193 |
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194 lemma alpha_rfv: |
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195 shows "t \<approx> s \<Longrightarrow> rfv t = rfv s" |
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196 apply(induct rule: alpha.induct) |
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197 apply(simp_all) |
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198 done |
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199 |
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200 inductive |
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201 alpha2 :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx>2 _" [100, 100] 100) |
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202 where |
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203 a21: "a = b \<Longrightarrow> (rVar a) \<approx>2 (rVar b)" |
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204 | a22: "\<lbrakk>t1 \<approx>2 t2; s1 \<approx>2 s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx>2 rApp t2 s2" |
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205 | a23: "(a = b \<and> t \<approx>2 s) \<or> (a \<noteq> b \<and> ((a \<leftrightarrow> b) \<bullet> t) \<approx>2 s \<and> atom b \<notin> rfv t)\<Longrightarrow> rLam a t \<approx>2 rLam b s" |
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206 |
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207 lemma fv_vars: |
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208 fixes a::name |
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209 assumes a1: "\<forall>x \<in> rfv t - {atom a}. pi \<bullet> x = x" |
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210 shows "(pi \<bullet> t) \<approx>2 ((a \<leftrightarrow> (pi \<bullet> a)) \<bullet> t)" |
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211 using a1 |
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212 apply(induct t) |
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213 apply(auto) |
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214 apply(rule a21) |
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215 apply(case_tac "name = a") |
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216 apply(simp) |
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217 apply(simp) |
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218 defer |
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219 apply(rule a22) |
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220 apply(simp) |
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221 apply(simp) |
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222 apply(rule a23) |
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223 apply(case_tac "a = name") |
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224 apply(simp) |
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225 oops |
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226 |
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227 |
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228 lemma |
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229 assumes a1: "t \<approx>2 s" |
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230 shows "t \<approx> s" |
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231 using a1 |
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232 apply(induct) |
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233 apply(rule alpha.intros) |
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234 apply(simp) |
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235 apply(rule alpha.intros) |
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236 apply(simp) |
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237 apply(simp) |
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238 apply(rule alpha.intros) |
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239 apply(erule disjE) |
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240 apply(rule_tac x="0" in exI) |
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241 apply(simp add: fresh_star_def fresh_zero_perm) |
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242 apply(erule conjE)+ |
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243 apply(drule alpha_rfv) |
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244 apply(simp) |
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245 apply(rule_tac x="(a \<leftrightarrow> b)" in exI) |
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246 apply(simp) |
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247 apply(erule conjE)+ |
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248 apply(rule conjI) |
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249 apply(drule alpha_rfv) |
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250 apply(drule sym) |
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251 apply(simp) |
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252 apply(simp add: rfv_eqvt[symmetric]) |
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253 defer |
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254 apply(subgoal_tac "atom a \<sharp> (rfv t - {atom a})") |
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255 apply(subgoal_tac "atom b \<sharp> (rfv t - {atom a})") |
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256 |
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257 defer |
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258 sorry |
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259 |
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260 lemma |
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261 assumes a1: "t \<approx> s" |
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262 shows "t \<approx>2 s" |
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263 using a1 |
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264 apply(induct) |
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265 apply(rule alpha2.intros) |
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266 apply(simp) |
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267 apply(rule alpha2.intros) |
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268 apply(simp) |
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269 apply(simp) |
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270 apply(clarify) |
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271 apply(rule alpha2.intros) |
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272 apply(frule alpha_rfv) |
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273 apply(rotate_tac 4) |
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274 apply(drule sym) |
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275 apply(simp) |
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276 apply(drule sym) |
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277 apply(simp) |
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278 oops |
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279 |
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280 quotient_type lam = rlam / alpha |
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281 by (rule alpha_equivp) |
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282 |
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283 quotient_definition |
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284 "Var :: name \<Rightarrow> lam" |
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285 is |
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286 "rVar" |
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287 |
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288 quotient_definition |
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289 "App :: lam \<Rightarrow> lam \<Rightarrow> lam" |
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290 is |
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291 "rApp" |
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292 |
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293 quotient_definition |
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294 "Lam :: name \<Rightarrow> lam \<Rightarrow> lam" |
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295 is |
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296 "rLam" |
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297 |
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298 quotient_definition |
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299 "fv :: lam \<Rightarrow> atom set" |
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300 is |
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301 "rfv" |
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302 |
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303 lemma perm_rsp[quot_respect]: |
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304 "(op = ===> alpha ===> alpha) permute permute" |
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305 apply(auto) |
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306 apply(rule alpha_eqvt) |
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307 apply(simp) |
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308 done |
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309 |
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310 lemma rVar_rsp[quot_respect]: |
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311 "(op = ===> alpha) rVar rVar" |
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312 by (auto intro: a1) |
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313 |
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314 lemma rApp_rsp[quot_respect]: |
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315 "(alpha ===> alpha ===> alpha) rApp rApp" |
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316 by (auto intro: a2) |
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317 |
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318 lemma rLam_rsp[quot_respect]: |
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319 "(op = ===> alpha ===> alpha) rLam rLam" |
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320 apply(auto) |
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321 apply(rule a3) |
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322 apply(rule_tac x="0" in exI) |
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323 unfolding fresh_star_def |
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324 apply(simp add: fresh_star_def fresh_zero_perm) |
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325 apply(simp add: alpha_rfv) |
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326 done |
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327 |
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328 lemma rfv_rsp[quot_respect]: |
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329 "(alpha ===> op =) rfv rfv" |
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330 apply(simp add: alpha_rfv) |
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331 done |
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332 |
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333 |
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334 section {* lifted theorems *} |
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335 |
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336 lemma lam_induct: |
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337 "\<lbrakk>\<And>name. P (Var name); |
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338 \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2); |
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339 \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk> |
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340 \<Longrightarrow> P lam" |
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341 apply (lifting rlam.induct) |
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342 done |
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343 |
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344 instantiation lam :: pt |
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345 begin |
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346 |
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347 quotient_definition |
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348 "permute_lam :: perm \<Rightarrow> lam \<Rightarrow> lam" |
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349 is |
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350 "permute :: perm \<Rightarrow> rlam \<Rightarrow> rlam" |
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351 |
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352 lemma permute_lam [simp]: |
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353 shows "pi \<bullet> Var a = Var (pi \<bullet> a)" |
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354 and "pi \<bullet> App t1 t2 = App (pi \<bullet> t1) (pi \<bullet> t2)" |
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355 and "pi \<bullet> Lam a t = Lam (pi \<bullet> a) (pi \<bullet> t)" |
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356 apply(lifting permute_rlam.simps) |
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357 done |
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358 |
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359 instance |
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360 apply default |
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361 apply(induct_tac [!] x rule: lam_induct) |
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362 apply(simp_all) |
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363 done |
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364 |
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365 end |
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366 |
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367 lemma fv_lam [simp]: |
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368 shows "fv (Var a) = {atom a}" |
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369 and "fv (App t1 t2) = fv t1 \<union> fv t2" |
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370 and "fv (Lam a t) = fv t - {atom a}" |
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371 apply(lifting rfv_var rfv_app rfv_lam) |
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372 done |
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373 |
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374 lemma fv_eqvt: |
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375 shows "(p \<bullet> fv t) = fv (p \<bullet> t)" |
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376 apply(lifting rfv_eqvt) |
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377 done |
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378 |
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379 lemma a1: |
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380 "a = b \<Longrightarrow> Var a = Var b" |
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381 by (lifting a1) |
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382 |
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383 lemma a2: |
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384 "\<lbrakk>x = xa; xb = xc\<rbrakk> \<Longrightarrow> App x xb = App xa xc" |
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385 by (lifting a2) |
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386 |
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387 lemma a3: |
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388 "\<lbrakk>\<exists>pi. (fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) = s)\<rbrakk> |
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389 \<Longrightarrow> Lam a t = Lam b s" |
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390 apply (lifting a3) |
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391 done |
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392 |
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393 lemma a3_inv: |
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394 assumes "Lam a t = Lam b s" |
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395 shows "\<exists>pi. (fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) = s)" |
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396 using assms |
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397 apply(lifting a3_inverse) |
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398 done |
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399 |
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400 lemma alpha_cases: |
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401 "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P; |
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402 \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P; |
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403 \<And>t a s b. \<lbrakk>a1 = Lam a t; a2 = Lam b s; |
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404 \<exists>pi. fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a}) \<sharp>* pi \<and> (pi \<bullet> t) = s\<rbrakk> |
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405 \<Longrightarrow> P\<rbrakk> |
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406 \<Longrightarrow> P" |
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407 by (lifting alpha.cases) |
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408 |
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409 (* not sure whether needed *) |
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410 lemma alpha_induct: |
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411 "\<lbrakk>qx = qxa; \<And>a b. a = b \<Longrightarrow> qxb (Var a) (Var b); |
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412 \<And>x xa xb xc. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc); |
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413 \<And>t a s b. |
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414 \<lbrakk>\<exists>pi. fv t - {atom a} = fv s - {atom b} \<and> |
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415 (fv t - {atom a}) \<sharp>* pi \<and> ((pi \<bullet> t) = s \<and> qxb (pi \<bullet> t) s)\<rbrakk> |
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416 \<Longrightarrow> qxb (Lam a t) (Lam b s)\<rbrakk> |
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417 \<Longrightarrow> qxb qx qxa" |
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418 by (lifting alpha.induct) |
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419 |
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420 (* should they lift automatically *) |
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421 lemma lam_inject [simp]: |
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422 shows "(Var a = Var b) = (a = b)" |
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423 and "(App t1 t2 = App s1 s2) = (t1 = s1 \<and> t2 = s2)" |
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424 apply(lifting rlam.inject(1) rlam.inject(2)) |
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425 apply(regularize) |
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426 prefer 2 |
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427 apply(regularize) |
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428 prefer 2 |
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429 apply(auto) |
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430 apply(drule alpha.cases) |
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431 apply(simp_all) |
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432 apply(simp add: alpha.a1) |
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433 apply(drule alpha.cases) |
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434 apply(simp_all) |
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435 apply(drule alpha.cases) |
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436 apply(simp_all) |
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437 apply(rule alpha.a2) |
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438 apply(simp_all) |
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439 done |
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440 |
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441 lemma Lam_pseudo_inject: |
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442 shows "(Lam a t = Lam b s) = |
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443 (\<exists>pi. (fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) = s))" |
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444 apply(rule iffI) |
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445 apply(rule a3_inv) |
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446 apply(assumption) |
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447 apply(rule a3) |
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448 apply(assumption) |
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449 done |
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450 |
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451 lemma rlam_distinct: |
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452 shows "\<not>(rVar nam \<approx> rApp rlam1' rlam2')" |
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453 and "\<not>(rApp rlam1' rlam2' \<approx> rVar nam)" |
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454 and "\<not>(rVar nam \<approx> rLam nam' rlam')" |
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455 and "\<not>(rLam nam' rlam' \<approx> rVar nam)" |
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456 and "\<not>(rApp rlam1 rlam2 \<approx> rLam nam' rlam')" |
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457 and "\<not>(rLam nam' rlam' \<approx> rApp rlam1 rlam2)" |
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458 apply auto |
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459 apply (erule alpha.cases) |
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460 apply (simp_all only: rlam.distinct) |
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461 apply (erule alpha.cases) |
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462 apply (simp_all only: rlam.distinct) |
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463 apply (erule alpha.cases) |
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464 apply (simp_all only: rlam.distinct) |
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465 apply (erule alpha.cases) |
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466 apply (simp_all only: rlam.distinct) |
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467 apply (erule alpha.cases) |
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468 apply (simp_all only: rlam.distinct) |
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469 apply (erule alpha.cases) |
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470 apply (simp_all only: rlam.distinct) |
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471 done |
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472 |
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473 lemma lam_distinct[simp]: |
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474 shows "Var nam \<noteq> App lam1' lam2'" |
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475 and "App lam1' lam2' \<noteq> Var nam" |
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476 and "Var nam \<noteq> Lam nam' lam'" |
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477 and "Lam nam' lam' \<noteq> Var nam" |
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478 and "App lam1 lam2 \<noteq> Lam nam' lam'" |
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479 and "Lam nam' lam' \<noteq> App lam1 lam2" |
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480 apply(lifting rlam_distinct(1) rlam_distinct(2) rlam_distinct(3) rlam_distinct(4) rlam_distinct(5) rlam_distinct(6)) |
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481 done |
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482 |
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483 lemma var_supp1: |
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484 shows "(supp (Var a)) = (supp a)" |
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485 apply (simp add: supp_def) |
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486 done |
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487 |
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488 lemma var_supp: |
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489 shows "(supp (Var a)) = {a:::name}" |
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490 using var_supp1 by (simp add: supp_at_base) |
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491 |
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492 lemma app_supp: |
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493 shows "supp (App t1 t2) = (supp t1) \<union> (supp t2)" |
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494 apply(simp only: supp_def lam_inject) |
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495 apply(simp add: Collect_imp_eq Collect_neg_eq) |
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496 done |
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497 |
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498 (* supp for lam *) |
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499 lemma lam_supp1: |
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500 shows "(supp (atom x, t)) supports (Lam x t) " |
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501 apply(simp add: supports_def) |
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502 apply(fold fresh_def) |
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503 apply(simp add: fresh_Pair swap_fresh_fresh) |
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504 apply(clarify) |
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505 apply(subst swap_at_base_simps(3)) |
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506 apply(simp_all add: fresh_atom) |
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507 done |
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508 |
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509 lemma lam_fsupp1: |
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510 assumes a: "finite (supp t)" |
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511 shows "finite (supp (Lam x t))" |
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512 apply(rule supports_finite) |
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513 apply(rule lam_supp1) |
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514 apply(simp add: a supp_Pair supp_atom) |
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515 done |
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516 |
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517 instance lam :: fs |
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518 apply(default) |
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519 apply(induct_tac x rule: lam_induct) |
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520 apply(simp add: var_supp) |
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521 apply(simp add: app_supp) |
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522 apply(simp add: lam_fsupp1) |
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523 done |
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524 |
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525 lemma supp_fv: |
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526 shows "supp t = fv t" |
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527 apply(induct t rule: lam_induct) |
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528 apply(simp add: var_supp) |
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529 apply(simp add: app_supp) |
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530 apply(subgoal_tac "supp (Lam name lam) = supp (Abs {atom name} lam)") |
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531 apply(simp add: supp_Abs) |
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532 apply(simp (no_asm) add: supp_def permute_set_eq atom_eqvt) |
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533 apply(simp add: Lam_pseudo_inject) |
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534 apply(simp add: Abs_eq_iff alpha_gen) |
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535 apply(simp add: supp_eqvt[symmetric] fv_eqvt[symmetric]) |
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536 done |
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537 |
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538 lemma lam_supp2: |
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539 shows "supp (Lam x t) = supp (Abs {atom x} t)" |
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540 apply(simp add: supp_def permute_set_eq atom_eqvt) |
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541 apply(simp add: Lam_pseudo_inject) |
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542 apply(simp add: Abs_eq_iff supp_fv alpha_gen) |
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543 done |
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544 |
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545 lemma lam_supp: |
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546 shows "supp (Lam x t) = ((supp t) - {atom x})" |
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547 apply(simp add: lam_supp2) |
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548 apply(simp add: supp_Abs) |
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549 done |
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550 |
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551 lemma fresh_lam: |
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552 "(atom a \<sharp> Lam b t) \<longleftrightarrow> (a = b) \<or> (a \<noteq> b \<and> atom a \<sharp> t)" |
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553 apply(simp add: fresh_def) |
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554 apply(simp add: lam_supp) |
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555 apply(auto) |
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556 done |
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557 |
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558 lemma lam_induct_strong: |
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559 fixes a::"'a::fs" |
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560 assumes a1: "\<And>name b. P b (Var name)" |
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561 and a2: "\<And>lam1 lam2 b. \<lbrakk>\<And>c. P c lam1; \<And>c. P c lam2\<rbrakk> \<Longrightarrow> P b (App lam1 lam2)" |
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562 and a3: "\<And>name lam b. \<lbrakk>\<And>c. P c lam; (atom name) \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lam name lam)" |
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563 shows "P a lam" |
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564 proof - |
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565 have "\<And>pi a. P a (pi \<bullet> lam)" |
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566 proof (induct lam rule: lam_induct) |
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567 case (1 name pi) |
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568 show "P a (pi \<bullet> Var name)" |
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569 apply (simp) |
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570 apply (rule a1) |
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571 done |
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572 next |
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573 case (2 lam1 lam2 pi) |
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574 have b1: "\<And>pi a. P a (pi \<bullet> lam1)" by fact |
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575 have b2: "\<And>pi a. P a (pi \<bullet> lam2)" by fact |
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576 show "P a (pi \<bullet> App lam1 lam2)" |
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577 apply (simp) |
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578 apply (rule a2) |
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579 apply (rule b1) |
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580 apply (rule b2) |
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581 done |
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582 next |
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583 case (3 name lam pi a) |
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584 have b: "\<And>pi a. P a (pi \<bullet> lam)" by fact |
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585 obtain c::name where fr: "atom c\<sharp>(a, pi\<bullet>name, pi\<bullet>lam)" |
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586 apply(rule obtain_atom) |
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587 apply(auto) |
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588 sorry |
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589 from b fr have p: "P a (Lam c (((c \<leftrightarrow> (pi \<bullet> name)) + pi)\<bullet>lam))" |
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590 apply - |
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591 apply(rule a3) |
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592 apply(blast) |
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593 apply(simp add: fresh_Pair) |
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594 done |
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595 have eq: "(atom c \<rightleftharpoons> atom (pi\<bullet>name)) \<bullet> Lam (pi \<bullet> name) (pi \<bullet> lam) = Lam (pi \<bullet> name) (pi \<bullet> lam)" |
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596 apply(rule swap_fresh_fresh) |
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597 using fr |
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598 apply(simp add: fresh_lam fresh_Pair) |
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599 apply(simp add: fresh_lam fresh_Pair) |
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600 done |
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601 show "P a (pi \<bullet> Lam name lam)" |
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602 apply (simp) |
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603 apply(subst eq[symmetric]) |
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604 using p |
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605 apply(simp only: permute_lam) |
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606 apply(simp add: flip_def) |
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607 done |
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608 qed |
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609 then have "P a (0 \<bullet> lam)" by blast |
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610 then show "P a lam" by simp |
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611 qed |
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612 |
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613 |
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614 lemma var_fresh: |
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615 fixes a::"name" |
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616 shows "(atom a \<sharp> (Var b)) = (atom a \<sharp> b)" |
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617 apply(simp add: fresh_def) |
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618 apply(simp add: var_supp1) |
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619 done |
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620 |
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621 |
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622 |
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623 end |
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624 |