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1 theory LamEx |
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2 imports Nominal "../Quotient" "../Quotient_List" |
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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> name set" |
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14 where |
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15 rfv_var: "rfv (rVar a) = {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) - {a}" |
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18 |
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19 overloading |
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20 perm_rlam \<equiv> "perm :: 'x prm \<Rightarrow> rlam \<Rightarrow> rlam" (unchecked) |
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21 begin |
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22 |
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23 fun |
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24 perm_rlam |
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25 where |
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26 "perm_rlam pi (rVar a) = rVar (pi \<bullet> a)" |
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27 | "perm_rlam pi (rApp t1 t2) = rApp (perm_rlam pi t1) (perm_rlam pi t2)" |
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28 | "perm_rlam pi (rLam a t) = rLam (pi \<bullet> a) (perm_rlam pi t)" |
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29 |
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30 end |
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31 |
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32 declare perm_rlam.simps[eqvt] |
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33 |
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34 instance rlam::pt_name |
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35 apply(default) |
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36 apply(induct_tac [!] x rule: rlam.induct) |
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37 apply(simp_all add: pt_name2 pt_name3) |
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38 done |
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39 |
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40 instance rlam::fs_name |
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41 apply(default) |
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42 apply(induct_tac [!] x rule: rlam.induct) |
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43 apply(simp add: supp_def) |
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44 apply(fold supp_def) |
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45 apply(simp add: supp_atm) |
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46 apply(simp add: supp_def Collect_imp_eq Collect_neg_eq) |
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47 apply(simp add: supp_def) |
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48 apply(simp add: supp_def Collect_imp_eq Collect_neg_eq[symmetric]) |
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49 apply(fold supp_def) |
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50 apply(simp add: supp_atm) |
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51 done |
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52 |
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53 declare set_diff_eqvt[eqvt] |
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54 |
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55 lemma rfv_eqvt[eqvt]: |
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56 fixes pi::"name prm" |
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57 shows "(pi\<bullet>rfv t) = rfv (pi\<bullet>t)" |
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58 apply(induct t) |
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59 apply(simp_all) |
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60 apply(simp add: perm_set_eq) |
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61 apply(simp add: union_eqvt) |
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62 apply(simp add: set_diff_eqvt) |
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63 apply(simp add: perm_set_eq) |
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64 done |
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65 |
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66 inductive |
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67 alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100) |
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68 where |
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69 a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)" |
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70 | a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2" |
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71 | a3: "\<exists>pi::name prm. (rfv t - {a} = rfv s - {b} \<and> (rfv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s \<and> (pi \<bullet> a) = b) |
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72 \<Longrightarrow> rLam a t \<approx> rLam b s" |
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73 |
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74 |
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75 (* should be automatic with new version of eqvt-machinery *) |
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76 lemma alpha_eqvt: |
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77 fixes pi::"name prm" |
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78 shows "t \<approx> s \<Longrightarrow> (pi \<bullet> t) \<approx> (pi \<bullet> s)" |
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79 apply(induct rule: alpha.induct) |
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80 apply(simp add: a1) |
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81 apply(simp add: a2) |
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82 apply(simp) |
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83 apply(rule a3) |
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84 apply(erule conjE) |
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85 apply(erule exE) |
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86 apply(erule conjE) |
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87 apply(rule_tac x="pi \<bullet> pia" in exI) |
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88 apply(rule conjI) |
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89 apply(rule_tac pi1="rev pi" in perm_bij[THEN iffD1]) |
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90 apply(perm_simp add: eqvts) |
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91 apply(rule conjI) |
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92 apply(rule_tac pi1="rev pi" in pt_fresh_star_bij(1)[OF pt_name_inst at_name_inst, THEN iffD1]) |
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93 apply(perm_simp add: eqvts) |
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94 apply(rule conjI) |
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95 apply(subst perm_compose[symmetric]) |
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96 apply(simp) |
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97 apply(subst perm_compose[symmetric]) |
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98 apply(simp) |
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99 done |
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100 |
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101 lemma alpha_refl: |
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102 shows "t \<approx> t" |
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103 apply(induct t rule: rlam.induct) |
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104 apply(simp add: a1) |
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105 apply(simp add: a2) |
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106 apply(rule a3) |
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107 apply(rule_tac x="[]" in exI) |
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108 apply(simp_all add: fresh_star_def fresh_list_nil) |
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109 done |
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110 |
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111 lemma alpha_sym: |
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112 shows "t \<approx> s \<Longrightarrow> s \<approx> t" |
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113 apply(induct rule: alpha.induct) |
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114 apply(simp add: a1) |
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115 apply(simp add: a2) |
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116 apply(rule a3) |
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117 apply(erule exE) |
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118 apply(rule_tac x="rev pi" in exI) |
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119 apply(simp) |
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120 apply(simp add: fresh_star_def fresh_list_rev) |
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121 apply(rule conjI) |
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122 apply(erule conjE)+ |
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123 apply(rotate_tac 3) |
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124 apply(drule_tac pi="rev pi" in alpha_eqvt) |
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125 apply(perm_simp) |
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126 apply(rule pt_bij2[OF pt_name_inst at_name_inst]) |
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127 apply(simp) |
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128 done |
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129 |
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130 lemma alpha_trans: |
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131 shows "t1 \<approx> t2 \<Longrightarrow> t2 \<approx> t3 \<Longrightarrow> t1 \<approx> t3" |
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132 apply(induct arbitrary: t3 rule: alpha.induct) |
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133 apply(erule alpha.cases) |
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134 apply(simp_all) |
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135 apply(simp add: a1) |
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136 apply(rotate_tac 4) |
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137 apply(erule alpha.cases) |
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138 apply(simp_all) |
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139 apply(simp add: a2) |
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140 apply(rotate_tac 1) |
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141 apply(erule alpha.cases) |
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142 apply(simp_all) |
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143 apply(erule conjE)+ |
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144 apply(erule exE)+ |
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145 apply(erule conjE)+ |
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146 apply(rule a3) |
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147 apply(rule_tac x="pia @ pi" in exI) |
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148 apply(simp add: fresh_star_def fresh_list_append) |
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149 apply(simp add: pt_name2) |
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150 apply(drule_tac x="rev pia \<bullet> sa" in spec) |
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151 apply(drule mp) |
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152 apply(rotate_tac 8) |
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153 apply(drule_tac pi="rev pia" in alpha_eqvt) |
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154 apply(perm_simp) |
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155 apply(rotate_tac 11) |
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156 apply(drule_tac pi="pia" in alpha_eqvt) |
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157 apply(perm_simp) |
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158 done |
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159 |
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160 lemma alpha_equivp: |
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161 shows "equivp alpha" |
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162 apply(rule equivpI) |
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163 unfolding reflp_def symp_def transp_def |
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164 apply(auto intro: alpha_refl alpha_sym alpha_trans) |
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165 done |
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166 |
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167 lemma alpha_rfv: |
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168 shows "t \<approx> s \<Longrightarrow> rfv t = rfv s" |
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169 apply(induct rule: alpha.induct) |
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170 apply(simp) |
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171 apply(simp) |
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172 apply(simp) |
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173 done |
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174 |
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175 quotient_type lam = rlam / alpha |
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176 by (rule alpha_equivp) |
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177 |
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178 |
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179 quotient_definition |
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180 "Var :: name \<Rightarrow> lam" |
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181 is |
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182 "rVar" |
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183 |
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184 quotient_definition |
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185 "App :: lam \<Rightarrow> lam \<Rightarrow> lam" |
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186 is |
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187 "rApp" |
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188 |
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189 quotient_definition |
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190 "Lam :: name \<Rightarrow> lam \<Rightarrow> lam" |
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191 is |
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192 "rLam" |
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193 |
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194 quotient_definition |
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195 "fv :: lam \<Rightarrow> name set" |
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196 is |
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197 "rfv" |
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198 |
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199 (* definition of overloaded permutation function *) |
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200 (* for the lifted type lam *) |
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201 overloading |
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202 perm_lam \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam" (unchecked) |
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203 begin |
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204 |
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205 quotient_definition |
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206 "perm_lam :: 'x prm \<Rightarrow> lam \<Rightarrow> lam" |
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207 is |
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208 "perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam" |
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209 |
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210 end |
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211 |
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212 lemma perm_rsp[quot_respect]: |
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213 "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>" |
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214 apply(auto) |
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215 (* this is propably true if some type conditions are imposed ;o) *) |
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216 sorry |
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217 |
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218 lemma fresh_rsp: |
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219 "(op = ===> alpha ===> op =) fresh fresh" |
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220 apply(auto) |
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221 (* this is probably only true if some type conditions are imposed *) |
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222 sorry |
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223 |
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224 lemma rVar_rsp[quot_respect]: |
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225 "(op = ===> alpha) rVar rVar" |
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226 by (auto intro: a1) |
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227 |
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228 lemma rApp_rsp[quot_respect]: "(alpha ===> alpha ===> alpha) rApp rApp" |
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229 by (auto intro: a2) |
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230 |
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231 lemma rLam_rsp[quot_respect]: "(op = ===> alpha ===> alpha) rLam rLam" |
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232 apply(auto) |
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233 apply(rule a3) |
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234 apply(rule_tac x="[]" in exI) |
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235 unfolding fresh_star_def |
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236 apply(simp add: fresh_list_nil) |
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237 apply(simp add: alpha_rfv) |
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238 done |
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239 |
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240 lemma rfv_rsp[quot_respect]: |
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241 "(alpha ===> op =) rfv rfv" |
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242 apply(simp add: alpha_rfv) |
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243 done |
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244 |
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245 section {* lifted theorems *} |
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246 |
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247 lemma lam_induct: |
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248 "\<lbrakk>\<And>name. P (Var name); |
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249 \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2); |
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250 \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk> |
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251 \<Longrightarrow> P lam" |
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252 by (lifting rlam.induct) |
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253 |
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254 ML {* show_all_types := true *} |
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255 |
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256 lemma perm_lam [simp]: |
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257 fixes pi::"'a prm" |
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258 shows "pi \<bullet> Var a = Var (pi \<bullet> a)" |
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259 and "pi \<bullet> App t1 t2 = App (pi \<bullet> t1) (pi \<bullet> t2)" |
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260 and "pi \<bullet> Lam a t = Lam (pi \<bullet> a) (pi \<bullet> t)" |
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261 apply(lifting perm_rlam.simps) |
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262 ML_prf {* |
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263 List.last (map (symmetric o #def) (Quotient_Info.qconsts_dest @{context})); |
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264 List.last (map (Thm.varifyT o symmetric o #def) (Quotient_Info.qconsts_dest @{context})) |
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265 *} |
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266 done |
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267 |
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268 instance lam::pt_name |
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269 apply(default) |
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270 apply(induct_tac [!] x rule: lam_induct) |
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271 apply(simp_all add: pt_name2 pt_name3) |
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272 done |
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273 |
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274 lemma fv_lam [simp]: |
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275 shows "fv (Var a) = {a}" |
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276 and "fv (App t1 t2) = fv t1 \<union> fv t2" |
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277 and "fv (Lam a t) = fv t - {a}" |
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278 apply(lifting rfv_var rfv_app rfv_lam) |
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279 done |
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280 |
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281 |
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282 lemma a1: |
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283 "a = b \<Longrightarrow> Var a = Var b" |
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284 by (lifting a1) |
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285 |
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286 lemma a2: |
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287 "\<lbrakk>x = xa; xb = xc\<rbrakk> \<Longrightarrow> App x xb = App xa xc" |
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288 by (lifting a2) |
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289 |
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290 lemma a3: |
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291 "\<lbrakk>\<exists>pi::name prm. (fv t - {a} = fv s - {b} \<and> (fv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) = s \<and> (pi \<bullet> a) = b)\<rbrakk> |
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292 \<Longrightarrow> Lam a t = Lam b s" |
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293 by (lifting a3) |
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294 |
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295 lemma alpha_cases: |
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296 "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P; |
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297 \<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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298 \<And>t a s b. \<lbrakk>a1 = Lam a t; a2 = Lam b s; |
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299 \<exists>pi::name prm. fv t - {a} = fv s - {b} \<and> (fv t - {a}) \<sharp>* pi \<and> (pi \<bullet> t) = s \<and> pi \<bullet> a = b\<rbrakk> \<Longrightarrow> P\<rbrakk> |
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300 \<Longrightarrow> P" |
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301 by (lifting alpha.cases) |
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302 |
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303 lemma alpha_induct: |
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304 "\<lbrakk>qx = qxa; \<And>a b. a = b \<Longrightarrow> qxb (Var a) (Var b); |
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305 \<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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306 \<And>t a s b. |
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307 \<lbrakk>\<exists>pi::name prm. fv t - {a} = fv s - {b} \<and> |
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308 (fv t - {a}) \<sharp>* pi \<and> ((pi \<bullet> t) = s \<and> qxb (pi \<bullet> t) s) \<and> pi \<bullet> a = b\<rbrakk> \<Longrightarrow> qxb (Lam a t) (Lam b s)\<rbrakk> |
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309 \<Longrightarrow> qxb qx qxa" |
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310 by (lifting alpha.induct) |
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311 |
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312 lemma lam_inject [simp]: |
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313 shows "(Var a = Var b) = (a = b)" |
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314 and "(App t1 t2 = App s1 s2) = (t1 = s1 \<and> t2 = s2)" |
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315 apply(lifting rlam.inject(1) rlam.inject(2)) |
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316 apply(auto) |
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317 apply(drule alpha.cases) |
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318 apply(simp_all) |
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319 apply(simp add: alpha.a1) |
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320 apply(drule alpha.cases) |
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321 apply(simp_all) |
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322 apply(drule alpha.cases) |
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323 apply(simp_all) |
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324 apply(rule alpha.a2) |
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325 apply(simp_all) |
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326 done |
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327 |
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328 lemma rlam_distinct: |
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329 shows "\<not>(rVar nam \<approx> rApp rlam1' rlam2')" |
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330 and "\<not>(rApp rlam1' rlam2' \<approx> rVar nam)" |
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331 and "\<not>(rVar nam \<approx> rLam nam' rlam')" |
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332 and "\<not>(rLam nam' rlam' \<approx> rVar nam)" |
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333 and "\<not>(rApp rlam1 rlam2 \<approx> rLam nam' rlam')" |
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334 and "\<not>(rLam nam' rlam' \<approx> rApp rlam1 rlam2)" |
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335 apply auto |
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336 apply(erule alpha.cases) |
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337 apply simp_all |
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338 apply(erule alpha.cases) |
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339 apply simp_all |
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340 apply(erule alpha.cases) |
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341 apply simp_all |
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342 apply(erule alpha.cases) |
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343 apply simp_all |
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344 apply(erule alpha.cases) |
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345 apply simp_all |
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346 apply(erule alpha.cases) |
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347 apply simp_all |
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348 done |
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349 |
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350 lemma lam_distinct[simp]: |
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351 shows "Var nam \<noteq> App lam1' lam2'" |
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352 and "App lam1' lam2' \<noteq> Var nam" |
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353 and "Var nam \<noteq> Lam nam' lam'" |
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354 and "Lam nam' lam' \<noteq> Var nam" |
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355 and "App lam1 lam2 \<noteq> Lam nam' lam'" |
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356 and "Lam nam' lam' \<noteq> App lam1 lam2" |
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357 apply(lifting rlam_distinct(1) rlam_distinct(2) rlam_distinct(3) rlam_distinct(4) rlam_distinct(5) rlam_distinct(6)) |
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358 done |
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359 |
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360 lemma var_supp1: |
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361 shows "(supp (Var a)) = ((supp a)::name set)" |
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362 by (simp add: supp_def) |
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363 |
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364 lemma var_supp: |
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365 shows "(supp (Var a)) = {a::name}" |
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366 using var_supp1 by (simp add: supp_atm) |
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367 |
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368 lemma app_supp: |
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369 shows "supp (App t1 t2) = (supp t1) \<union> ((supp t2)::name set)" |
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370 apply(simp only: perm_lam supp_def lam_inject) |
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371 apply(simp add: Collect_imp_eq Collect_neg_eq) |
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372 done |
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373 |
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374 lemma lam_supp: |
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375 shows "supp (Lam x t) = ((supp ([x].t))::name set)" |
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376 apply(simp add: supp_def) |
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377 apply(simp add: abs_perm) |
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378 sorry |
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379 |
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380 |
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381 instance lam::fs_name |
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382 apply(default) |
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383 apply(induct_tac x rule: lam_induct) |
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384 apply(simp add: var_supp) |
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385 apply(simp add: app_supp) |
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386 apply(simp add: lam_supp abs_supp) |
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387 done |
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388 |
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389 lemma fresh_lam: |
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390 "(a \<sharp> Lam b t) \<longleftrightarrow> (a = b) \<or> (a \<noteq> b \<and> a \<sharp> t)" |
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391 apply(simp add: fresh_def) |
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392 apply(simp add: lam_supp abs_supp) |
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393 apply(auto) |
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394 done |
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395 |
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396 lemma lam_induct_strong: |
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397 fixes a::"'a::fs_name" |
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398 assumes a1: "\<And>name b. P b (Var name)" |
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399 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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400 and a3: "\<And>name lam b. \<lbrakk>\<And>c. P c lam; name \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lam name lam)" |
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401 shows "P a lam" |
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402 proof - |
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403 have "\<And>(pi::name prm) a. P a (pi \<bullet> lam)" |
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404 proof (induct lam rule: lam_induct) |
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405 case (1 name pi) |
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406 show "P a (pi \<bullet> Var name)" |
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407 apply (simp) |
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408 apply (rule a1) |
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409 done |
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410 next |
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411 case (2 lam1 lam2 pi) |
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412 have b1: "\<And>(pi::name prm) a. P a (pi \<bullet> lam1)" by fact |
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413 have b2: "\<And>(pi::name prm) a. P a (pi \<bullet> lam2)" by fact |
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414 show "P a (pi \<bullet> App lam1 lam2)" |
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415 apply (simp) |
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416 apply (rule a2) |
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417 apply (rule b1) |
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418 apply (rule b2) |
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419 done |
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420 next |
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421 case (3 name lam pi a) |
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422 have b: "\<And>(pi::name prm) a. P a (pi \<bullet> lam)" by fact |
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423 obtain c::name where fr: "c\<sharp>(a, pi\<bullet>name, pi\<bullet>lam)" |
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424 apply(rule exists_fresh[of "(a, pi\<bullet>name, pi\<bullet>lam)"]) |
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425 apply(simp_all add: fs_name1) |
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426 done |
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427 from b fr have p: "P a (Lam c (([(c, pi\<bullet>name)]@pi)\<bullet>lam))" |
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428 apply - |
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429 apply(rule a3) |
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430 apply(blast) |
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431 apply(simp) |
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432 done |
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433 have eq: "[(c, pi\<bullet>name)] \<bullet> Lam (pi \<bullet> name) (pi \<bullet> lam) = Lam (pi \<bullet> name) (pi \<bullet> lam)" |
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434 apply(rule perm_fresh_fresh) |
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435 using fr |
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436 apply(simp add: fresh_lam) |
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437 apply(simp add: fresh_lam) |
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438 done |
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439 show "P a (pi \<bullet> Lam name lam)" |
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440 apply (simp) |
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441 apply(subst eq[symmetric]) |
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442 using p |
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443 apply(simp only: perm_lam pt_name2 swap_simps) |
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444 done |
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445 qed |
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446 then have "P a (([]::name prm) \<bullet> lam)" by blast |
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447 then show "P a lam" by simp |
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448 qed |
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449 |
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450 |
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451 lemma var_fresh: |
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452 fixes a::"name" |
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453 shows "(a \<sharp> (Var b)) = (a \<sharp> b)" |
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454 apply(simp add: fresh_def) |
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455 apply(simp add: var_supp1) |
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456 done |
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457 |
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458 (* lemma hom_reg: *) |
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459 |
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460 lemma rlam_rec_eqvt: |
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461 fixes pi::"name prm" |
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462 and f1::"name \<Rightarrow> ('a::pt_name)" |
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463 shows "(pi\<bullet>rlam_rec f1 f2 f3 t) = rlam_rec (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3) (pi\<bullet>t)" |
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464 apply(induct t) |
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465 apply(simp_all) |
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466 apply(simp add: perm_fun_def) |
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467 apply(perm_simp) |
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468 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
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469 back |
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470 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
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471 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
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472 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
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473 apply(simp) |
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474 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
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475 back |
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476 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
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477 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
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478 apply(simp) |
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479 done |
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480 |
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481 |
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482 lemma rlam_rec_respects: |
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483 assumes f1: "f_var \<in> Respects (op= ===> op=)" |
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484 and f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)" |
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485 and f3: "f_lam \<in> Respects (op= ===> alpha ===> op= ===> op=)" |
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486 shows "rlam_rec f_var f_app f_lam \<in> Respects (alpha ===> op =)" |
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487 apply(simp add: mem_def) |
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488 apply(simp add: Respects_def) |
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489 apply(rule allI) |
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490 apply(rule allI) |
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491 apply(rule impI) |
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492 apply(erule alpha.induct) |
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493 apply(simp) |
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494 apply(simp) |
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495 using f2 |
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496 apply(simp add: mem_def) |
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497 apply(simp add: Respects_def) |
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498 using f3[simplified mem_def Respects_def] |
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499 apply(simp) |
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500 apply(case_tac "a=b") |
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501 apply(clarify) |
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502 apply(simp) |
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503 (* probably true *) |
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504 sorry |
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505 |
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506 function |
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507 term1_hom :: "(name \<Rightarrow> 'a) \<Rightarrow> |
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508 (rlam \<Rightarrow> rlam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> |
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509 ((name \<Rightarrow> rlam) \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a) \<Rightarrow> rlam \<Rightarrow> 'a" |
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510 where |
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511 "term1_hom var app abs' (rVar x) = (var x)" |
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512 | "term1_hom var app abs' (rApp t u) = |
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513 app t u (term1_hom var app abs' t) (term1_hom var app abs' u)" |
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514 | "term1_hom var app abs' (rLam x u) = |
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515 abs' (\<lambda>y. [(x, y)] \<bullet> u) (\<lambda>y. term1_hom var app abs' ([(x, y)] \<bullet> u))" |
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516 apply(pat_completeness) |
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517 apply(auto) |
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518 done |
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519 |
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520 lemma pi_size: |
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521 fixes pi::"name prm" |
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522 and t::"rlam" |
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523 shows "size (pi \<bullet> t) = size t" |
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524 apply(induct t) |
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525 apply(auto) |
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526 done |
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527 |
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528 termination term1_hom |
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529 apply(relation "measure (\<lambda>(f1, f2, f3, t). size t)") |
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530 apply(auto simp add: pi_size) |
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531 done |
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532 |
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533 lemma lam_exhaust: |
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534 "\<lbrakk>\<And>name. y = Var name \<Longrightarrow> P; \<And>rlam1 rlam2. y = App rlam1 rlam2 \<Longrightarrow> P; \<And>name rlam. y = Lam name rlam \<Longrightarrow> P\<rbrakk> |
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535 \<Longrightarrow> P" |
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536 apply(lifting rlam.exhaust) |
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537 done |
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538 |
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539 (* THIS IS NOT TRUE, but it lets prove the existence of the hom function *) |
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540 lemma lam_inject': |
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541 "(Lam a x = Lam b y) = ((\<lambda>c. [(a, c)] \<bullet> x) = (\<lambda>c. [(b, c)] \<bullet> y))" |
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542 sorry |
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543 |
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544 function |
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545 hom :: "(name \<Rightarrow> 'a) \<Rightarrow> |
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546 (lam \<Rightarrow> lam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> |
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547 ((name \<Rightarrow> lam) \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a) \<Rightarrow> lam \<Rightarrow> 'a" |
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548 where |
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549 "hom f_var f_app f_lam (Var x) = f_var x" |
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550 | "hom f_var f_app f_lam (App l r) = f_app l r (hom f_var f_app f_lam l) (hom f_var f_app f_lam r)" |
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551 | "hom f_var f_app f_lam (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom f_var f_app f_lam ([(a,b)] \<bullet> x))" |
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552 defer |
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553 apply(simp_all add: lam_inject') (* inject, distinct *) |
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554 apply(tactic {* Cong_Tac.cong_tac @{thm cong} 1 *}) |
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555 apply(rule refl) |
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556 apply(rule ext) |
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557 apply(tactic {* Cong_Tac.cong_tac @{thm cong} 1 *}) |
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558 apply simp_all |
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559 apply(erule conjE)+ |
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560 apply(rule_tac x="b" in cong) |
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561 apply simp_all |
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562 apply auto |
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563 apply(rule_tac y="b" in lam_exhaust) |
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564 apply simp_all |
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565 apply auto |
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566 apply meson |
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567 apply(simp_all add: lam_inject') |
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568 apply metis |
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569 done |
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570 |
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571 termination hom |
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572 apply - |
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573 (* |
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574 ML_prf {* Size.size_thms @{theory} "LamEx.lam" *} |
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575 *) |
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576 sorry |
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577 |
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578 thm hom.simps |
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579 |
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580 lemma term1_hom_rsp: |
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581 "\<lbrakk>(alpha ===> alpha ===> op =) f_app f_app; ((op = ===> alpha) ===> op =) f_lam f_lam\<rbrakk> |
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582 \<Longrightarrow> (alpha ===> op =) (term1_hom f_var f_app f_lam) (term1_hom f_var f_app f_lam)" |
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583 apply(simp) |
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584 apply(rule allI)+ |
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585 apply(rule impI) |
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586 apply(erule alpha.induct) |
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587 apply(auto)[1] |
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588 apply(auto)[1] |
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589 apply(simp) |
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590 apply(erule conjE)+ |
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591 apply(erule exE)+ |
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592 apply(erule conjE)+ |
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593 apply(clarify) |
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594 sorry |
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595 |
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596 lemma hom: " |
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597 \<forall>f_var. \<forall>f_app \<in> Respects(alpha ===> alpha ===> op =). |
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598 \<forall>f_lam \<in> Respects((op = ===> alpha) ===> op =). |
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599 \<exists>hom\<in>Respects (alpha ===> op =). |
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600 ((\<forall>x. hom (rVar x) = f_var x) \<and> |
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601 (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and> |
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602 (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))" |
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603 apply(rule allI) |
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604 apply(rule ballI)+ |
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605 apply(rule_tac x="term1_hom f_var f_app f_lam" in bexI) |
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606 apply(simp_all) |
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607 apply(simp only: in_respects) |
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608 apply(rule term1_hom_rsp) |
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609 apply(assumption)+ |
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610 done |
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611 |
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612 lemma hom': |
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613 "\<exists>hom. |
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614 ((\<forall>x. hom (Var x) = f_var x) \<and> |
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615 (\<forall>l r. hom (App l r) = f_app l r (hom l) (hom r)) \<and> |
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616 (\<forall>x a. hom (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))" |
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617 apply (lifting hom) |
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618 done |
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619 |
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620 (* test test |
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621 lemma raw_hom_correct: |
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622 assumes f1: "f_var \<in> Respects (op= ===> op=)" |
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623 and f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)" |
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624 and f3: "f_lam \<in> Respects ((op= ===> alpha) ===> (op= ===> op=) ===> op=)" |
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625 shows "\<exists>!hom\<in>Respects (alpha ===> op =). |
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626 ((\<forall>x. hom (rVar x) = f_var x) \<and> |
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627 (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and> |
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628 (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))" |
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629 unfolding Bex1_def |
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630 apply(rule ex1I) |
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631 sorry |
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632 *) |
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633 |
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634 |
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635 end |
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636 |