69 |
69 |
70 lemma alpha5_symp: |
70 lemma alpha5_symp: |
71 "(a \<approx>5 b \<longrightarrow> b \<approx>5 a) \<and> |
71 "(a \<approx>5 b \<longrightarrow> b \<approx>5 a) \<and> |
72 (x \<approx>l y \<longrightarrow> y \<approx>l x) \<and> |
72 (x \<approx>l y \<longrightarrow> y \<approx>l x) \<and> |
73 (alpha_rbv5 x y \<longrightarrow> alpha_rbv5 y x)" |
73 (alpha_rbv5 x y \<longrightarrow> alpha_rbv5 y x)" |
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74 apply (tactic {* symp_tac @{thm alpha_rtrm5_alpha_rlts_alpha_rbv5.induct} @{thms alpha5_inj} @{thms alpha5_eqvt} @{context} 1 *}) |
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75 done |
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76 |
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77 lemma alpha5_symp1: |
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78 "(a \<approx>5 b \<longrightarrow> b \<approx>5 a) \<and> |
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79 (x \<approx>l y \<longrightarrow> y \<approx>l x) \<and> |
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80 (alpha_rbv5 x y \<longrightarrow> alpha_rbv5 y x)" |
74 apply (rule alpha_rtrm5_alpha_rlts_alpha_rbv5.induct) |
81 apply (rule alpha_rtrm5_alpha_rlts_alpha_rbv5.induct) |
75 apply (simp only: alpha5_inj) |
82 apply (simp_all add: alpha5_inj) |
76 apply (simp add: alpha5_inj) |
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77 apply (erule exE) |
83 apply (erule exE) |
78 apply (simp only: alpha5_inj) |
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79 apply (rule_tac x="- pi" in exI) |
84 apply (rule_tac x="- pi" in exI) |
80 apply (erule alpha_gen_compose_sym2) |
85 apply (simp add: alpha_gen) |
81 apply (simp_all add: alpha5_inj eqvts alpha5_eqvt) |
86 apply(simp add: fresh_star_def fresh_minus_perm) |
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87 apply clarify |
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88 apply (rule conjI) |
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89 apply (rotate_tac 3) |
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90 apply (frule_tac p="- pi" in alpha5_eqvt(2)) |
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91 apply simp |
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92 apply (rule conjI) |
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93 apply (rotate_tac 5) |
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94 apply (frule_tac p="- pi" in alpha5_eqvt(1)) |
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95 apply simp |
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96 apply (rotate_tac 6) |
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97 apply simp |
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98 apply (drule_tac p1="- pi" in permute_eq_iff[symmetric,THEN iffD1]) |
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99 apply (simp) |
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100 done |
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101 |
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102 thm alpha_gen_sym[no_vars] |
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103 thm alpha_gen_compose_sym[no_vars] |
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104 |
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105 lemma tt: |
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106 "\<lbrakk>R (- p \<bullet> x) y \<Longrightarrow> R (p \<bullet> y) x; (bs, x) \<approx>gen R f (- p) (cs, y)\<rbrakk> \<Longrightarrow> (cs, y) \<approx>gen R f p (bs, x)" |
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107 unfolding alphas |
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108 unfolding fresh_star_def |
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109 by (auto simp add: fresh_minus_perm) |
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110 |
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111 lemma alpha5_symp2: |
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112 shows "a \<approx>5 b \<Longrightarrow> b \<approx>5 a" |
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113 and "x \<approx>l y \<Longrightarrow> y \<approx>l x" |
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114 and "alpha_rbv5 x y \<Longrightarrow> alpha_rbv5 y x" |
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115 apply(induct rule: alpha_rtrm5_alpha_rlts_alpha_rbv5.inducts) |
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116 (* non-binding case *) |
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117 apply(simp add: alpha5_inj) |
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118 (* non-binding case *) |
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119 apply(simp add: alpha5_inj) |
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120 (* binding case *) |
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121 apply(simp add: alpha5_inj) |
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122 apply(erule exE) |
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123 apply(rule_tac x="- pi" in exI) |
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124 apply(rule tt) |
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125 apply(simp add: alphas) |
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126 apply(erule conjE)+ |
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127 apply(drule_tac p="- pi" in alpha5_eqvt(2)) |
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128 apply(drule_tac p="- pi" in alpha5_eqvt(2)) |
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129 apply(drule_tac p="- pi" in alpha5_eqvt(1)) |
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130 apply(drule_tac p="- pi" in alpha5_eqvt(1)) |
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131 apply(simp) |
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132 apply(simp add: alphas) |
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133 apply(erule conjE)+ |
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134 apply metis |
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135 (* non-binding case *) |
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136 apply(simp add: alpha5_inj) |
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137 (* non-binding case *) |
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138 apply(simp add: alpha5_inj) |
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139 (* non-binding case *) |
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140 apply(simp add: alpha5_inj) |
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141 (* non-binding case *) |
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142 apply(simp add: alpha5_inj) |
82 done |
143 done |
83 |
144 |
84 lemma alpha5_transp: |
145 lemma alpha5_transp: |
85 "(a \<approx>5 b \<longrightarrow> (\<forall>c. b \<approx>5 c \<longrightarrow> a \<approx>5 c)) \<and> |
146 "(a \<approx>5 b \<longrightarrow> (\<forall>c. b \<approx>5 c \<longrightarrow> a \<approx>5 c)) \<and> |
86 (x \<approx>l y \<longrightarrow> (\<forall>z. y \<approx>l z \<longrightarrow> x \<approx>l z)) \<and> |
147 (x \<approx>l y \<longrightarrow> (\<forall>z. y \<approx>l z \<longrightarrow> x \<approx>l z)) \<and> |
87 (alpha_rbv5 k l \<longrightarrow> (\<forall>m. alpha_rbv5 l m \<longrightarrow> alpha_rbv5 k m))" |
148 (alpha_rbv5 k l \<longrightarrow> (\<forall>m. alpha_rbv5 l m \<longrightarrow> alpha_rbv5 k m))" |
88 apply (tactic {* transp_tac @{context} @{thm alpha_rtrm5_alpha_rlts_alpha_rbv5.induct} @{thms alpha5_inj} @{thms rtrm5.distinct rtrm5.inject rlts.distinct rlts.inject} [] @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{thms alpha5_eqvt} 1 *}) |
149 (*apply (tactic {* transp_tac @{context} @{thm alpha_rtrm5_alpha_rlts_alpha_rbv5.induct} @{thms alpha5_inj} @{thms rtrm5.distinct rtrm5.inject rlts.distinct rlts.inject} [] @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{thms alpha5_eqvt} 1 *})*) |
89 done |
150 apply (rule alpha_rtrm5_alpha_rlts_alpha_rbv5.induct) |
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151 apply (rule_tac [!] allI) |
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152 apply (tactic {* (imp_elim_tac @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{context}) 1 *}) |
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153 apply (simp_all add: alpha5_inj) |
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154 apply (tactic {* (imp_elim_tac @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{context}) 1 *}) |
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155 apply (simp_all add: alpha5_inj) |
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156 apply (tactic {* (imp_elim_tac @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{context}) 1 *}) |
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157 apply (simp_all add: alpha5_inj) |
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158 defer |
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159 apply (tactic {* (imp_elim_tac @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{context}) 1 *}) |
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160 apply (simp_all add: alpha5_inj) |
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161 apply (tactic {* (imp_elim_tac @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{context}) 1 *}) |
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162 apply (simp_all add: alpha5_inj) |
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163 apply (tactic {* eetac @{thm exi_sum} @{context} 1 *}) |
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164 (* HERE *) |
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165 (* |
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166 apply(rule alpha_gen_trans) |
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167 thm alpha_gen_trans |
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168 defer |
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169 apply (simp add: alpha_gen) |
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170 apply clarify |
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171 apply(simp add: fresh_star_plus) |
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172 apply (rule conjI) |
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173 apply (erule_tac x="- pi \<bullet> rltsaa" in allE) |
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174 apply (rotate_tac 5) |
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175 apply (drule_tac p="- pi" in alpha5_eqvt(2)) |
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176 apply simp |
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177 apply (drule_tac p="pi" in alpha5_eqvt(2)) |
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178 apply simp |
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179 apply (erule_tac x="- pi \<bullet> rtrm5aa" in allE) |
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180 apply (rotate_tac 7) |
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181 apply (drule_tac p="- pi" in alpha5_eqvt(1)) |
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182 apply simp |
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183 apply (rotate_tac 3) |
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184 apply (drule_tac p="pi" in alpha5_eqvt(1)) |
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185 apply simp |
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186 done |
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187 *) |
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188 sorry |
90 |
189 |
91 lemma alpha5_equivp: |
190 lemma alpha5_equivp: |
92 "equivp alpha_rtrm5" |
191 "equivp alpha_rtrm5" |
93 "equivp alpha_rlts" |
192 "equivp alpha_rlts" |
94 unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def |
193 unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def |