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Nominal/Ex/Let.thy
changeset 2931 aaef9dec5e1d
parent 2926 37c0d7953cba
child 2932 e8ab80062061
equal deleted inserted replaced
2930:1d9e50934bc5 2931:aaef9dec5e1d 
    37 thm trm_assn.distinct
 
    37 thm trm_assn.distinct
 
    38 thm trm_assn.supp
 
    38 thm trm_assn.supp
 
    39 thm trm_assn.fresh
 
    39 thm trm_assn.fresh
 
    40 thm trm_assn.exhaust
 
    40 thm trm_assn.exhaust
 
    41 thm trm_assn.strong_exhaust
 
    41 thm trm_assn.strong_exhaust
 
    42 
    42 thm trm_assn.perm_bn_simps
 
    43 lemma alpha_bn_inducts_raw:
 
    43 
       
    44 lemma alpha_bn_inducts_raw[consumes 1]:
 
    44   "\<lbrakk>alpha_bn_raw a b; P3 ANil_raw ANil_raw;
 
    45   "\<lbrakk>alpha_bn_raw a b; P3 ANil_raw ANil_raw;
 
    45  \<And>trm_raw trm_rawa assn_raw assn_rawa name namea.
 
    46  \<And>trm_raw trm_rawa assn_raw assn_rawa name namea.
 
    46     \<lbrakk>alpha_trm_raw trm_raw trm_rawa; alpha_bn_raw assn_raw assn_rawa;
 
    47     \<lbrakk>alpha_trm_raw trm_raw trm_rawa; alpha_bn_raw assn_raw assn_rawa;
 
    47      P3 assn_raw assn_rawa\<rbrakk>
 
    48      P3 assn_raw assn_rawa\<rbrakk>
 
    48     \<Longrightarrow> P3 (ACons_raw name trm_raw assn_raw)
 
    49     \<Longrightarrow> P3 (ACons_raw name trm_raw assn_raw)
 
    49         (ACons_raw namea trm_rawa assn_rawa)\<rbrakk> \<Longrightarrow> P3 a b"
 
    50         (ACons_raw namea trm_rawa assn_rawa)\<rbrakk> \<Longrightarrow> P3 a b"
 
    50   by (erule alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.inducts(3)[of _ _ "\<lambda>x y. True" _ "\<lambda>x y. True", simplified]) auto
 
    51   by (erule alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.inducts(3)[of _ _ "\<lambda>x y. True" _ "\<lambda>x y. True", simplified]) auto
 
    51 
    52 
    52 lemmas alpha_bn_inducts = alpha_bn_inducts_raw[quot_lifted]
 
    53 lemmas alpha_bn_inducts[consumes 1] = alpha_bn_inducts_raw[quot_lifted]
 
    53 
    54 
    54 
    55 
    55 
    56 
    56 lemma alpha_bn_refl: "alpha_bn x x"
 
    57 lemma alpha_bn_refl: "alpha_bn x x"
 
    57   by (induct x rule: trm_assn.inducts(2))
 
    58   by (induct x rule: trm_assn.inducts(2))
 
    63 
    64 
    64 lemma bn_inj[rule_format]:
 
    65 lemma bn_inj[rule_format]:
 
    65   assumes a: "alpha_bn x y"
 
    66   assumes a: "alpha_bn x y"
 
    66   shows "bn x = bn y \<longrightarrow> x = y"
 
    67   shows "bn x = bn y \<longrightarrow> x = y"
 
    67   by (rule alpha_bn_inducts[OF a]) (simp_all add: trm_assn.bn_defs)
 
    68   by (rule alpha_bn_inducts[OF a]) (simp_all add: trm_assn.bn_defs)
 
       
    69 
       
    70 lemma bn_inj2:
 
       
    71   assumes a: "alpha_bn x y"
 
       
    72   shows "\<And>q r. (q \<bullet> bn x) = (r \<bullet> bn y) \<Longrightarrow> permute_bn q x = permute_bn r y"
 
       
    73 using a
 
       
    74 apply(induct rule: alpha_bn_inducts)
 
       
    75 apply(simp add: trm_assn.perm_bn_simps)
 
       
    76 apply(simp add: trm_assn.perm_bn_simps)
 
       
    77 apply(simp add: trm_assn.bn_defs)
 
       
    78 apply(simp add: atom_eqvt)
 
       
    79 done
 
    68 
    80 
    69 (*lemma alpha_bn_permute:
 
    81 (*lemma alpha_bn_permute:
 
    70   assumes a: "alpha_bn x y"
 
    82   assumes a: "alpha_bn x y"
 
    71       and b: "q \<bullet> bn x = r \<bullet> bn y"
 
    83       and b: "q \<bullet> bn x = r \<bullet> bn y"
 
    72     shows "alpha_bn (q \<bullet> x) (r \<bullet> y)"
 
    84     shows "alpha_bn (q \<bullet> x) (r \<bullet> y)"
 
    83     using b trm_assn.permute_bn by simp
 
    95     using b trm_assn.permute_bn by simp
 
    84   ultimately have "permute_bn q x = permute_bn r y"
 
    96   ultimately have "permute_bn q x = permute_bn r y"
 
    85     using bn_inj by simp
 
    97     using bn_inj by simp
 
    86 *)
 
    98 *)
 
    87 
    99 
    88 lemma lets_bla:
 
       
    89   "x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Let (ACons x (Var y) ANil) (Var x)) \<noteq> (Let (ACons x (Var z) ANil) (Var x))"
 
       
    90   by (simp add: trm_assn.eq_iff)
 
       
    91 
       
    92 
       
    93 lemma lets_ok:
 
       
    94   "(Let (ACons x (Var y) ANil) (Var x)) = (Let (ACons y (Var y) ANil) (Var y))"
 
       
    95   apply (simp add: trm_assn.eq_iff Abs_eq_iff )
 
       
    96   apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
 
       
    97   apply (simp_all add: alphas atom_eqvt supp_at_base fresh_star_def trm_assn.bn_defs trm_assn.supp)
 
       
    98   done
 
       
    99 
       
   100 lemma lets_ok3:
 
       
   101   "x \<noteq> y \<Longrightarrow>
 
       
   102    (Let (ACons x (App (Var y) (Var x)) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>
 
       
   103    (Let (ACons y (App (Var x) (Var y)) (ACons x (Var x) ANil)) (App (Var x) (Var y)))"
 
       
   104   apply (simp add: trm_assn.eq_iff)
 
       
   105   done
 
       
   106 
       
   107 lemma lets_not_ok1:
 
       
   108   "x \<noteq> y \<Longrightarrow>
 
       
   109    (Let (ACons x (Var x) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>
 
       
   110    (Let (ACons y (Var x) (ACons x (Var y) ANil)) (App (Var x) (Var y)))"
 
       
   111   apply (simp add: alphas trm_assn.eq_iff trm_assn.supp fresh_star_def atom_eqvt Abs_eq_iff trm_assn.bn_defs)
 
       
   112   done
 
       
   113 
       
   114 lemma lets_nok:
 
       
   115   "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>
 
       
   116    (Let (ACons x (App (Var z) (Var z)) (ACons y (Var z) ANil)) (App (Var x) (Var y))) \<noteq>
 
       
   117    (Let (ACons y (Var z) (ACons x (App (Var z) (Var z)) ANil)) (App (Var x) (Var y)))"
 
       
   118   apply (simp add: alphas trm_assn.eq_iff fresh_star_def trm_assn.bn_defs Abs_eq_iff trm_assn.supp trm_assn.distinct)
 
       
   119   done
 
       
   120 
       
   121 lemma
 
       
   122   fixes a b c :: name
 
       
   123   assumes x: "a \<noteq> c" and y: "b \<noteq> c"
 
       
   124   shows "\<exists>p.([atom a], Var c) \<approx>lst (op =) supp p ([atom b], Var c)"
 
       
   125   apply (rule_tac x="(a \<leftrightarrow> b)" in exI)
 
       
   126   apply (simp add: alphas trm_assn.supp supp_at_base x y fresh_star_def atom_eqvt)
 
       
   127   by (metis Rep_name_inverse atom_name_def flip_fresh_fresh fresh_atom fresh_perm x y)
 
       
   128 
       
   129 
   100 
   130 lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
 
   101 lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
 
   131   by (simp add: permute_pure)
 
   102   by (simp add: permute_pure)
 
   132 
   103 
   133 
   104 
   134 lemma Abs_lst_fcb2:
 
   105 lemma Abs_lst_fcb2:
 
   135   fixes as bs :: "'a :: fs"
 
   106   fixes as bs :: "'a :: fs"
 
   136     and x y :: "'b :: fs"
 
   107     and x y :: "'b :: fs"
 
   137     and c::"'c::fs"
 
       
   138   assumes eq: "[ba as]lst. x = [ba bs]lst. y"
 
   108   assumes eq: "[ba as]lst. x = [ba bs]lst. y"
 
       
   109   and ctxt: "finite (supp c)"
 
   139   and fcb1: "set (ba as) \<sharp>* f as x c"
 
   110   and fcb1: "set (ba as) \<sharp>* f as x c"
 
   140   and fresh1: "set (ba as) \<sharp>* c"
 
   111   and fresh1: "set (ba as) \<sharp>* c"
 
   141   and fresh2: "set (ba bs) \<sharp>* c"
 
   112   and fresh2: "set (ba bs) \<sharp>* c"
 
   142   and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
 
   113   and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
 
   143   and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
 
   114   and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
 
   144 (* What we would like in this proof, and lets this proof finish *)
 
   115 (* What we would like in this proof, and lets this proof finish *)
 
   145   and ba_inj: "\<And>q r. q \<bullet> ba as = r \<bullet> ba bs \<Longrightarrow> q \<bullet> as = r \<bullet> bs"
 
   116   and ba_inj: "\<And>q r. q \<bullet> ba as = r \<bullet> ba bs \<Longrightarrow> pn q as = pn r bs"
 
   146 (* What the user can supply with the help of alpha_bn *)
 
       
   147 (*  and bainj: "ba as = ba bs \<Longrightarrow> as = bs"*)
 
       
   148   shows "f as x c = f bs y c"
 
   117   shows "f as x c = f bs y c"
 
   149 proof -
 
   118 proof -
 
   150   have "supp (as, x, c) supports (f as x c)"
 
   119   have "supp (as, x, c) supports (f as x c)"
 
   151     unfolding  supports_def fresh_def[symmetric]
 
   120     unfolding  supports_def fresh_def[symmetric]
 
   152     by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
 
   121     apply (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
 
       
   122     sorry
 
   153   then have fin1: "finite (supp (f as x c))"
 
   123   then have fin1: "finite (supp (f as x c))"
 
   154     by (auto intro: supports_finite simp add: finite_supp)
 
   124     by (auto intro: supports_finite simp add: finite_supp supp_Pair ctxt)
 
   155   have "supp (bs, y, c) supports (f bs y c)"
 
   125   have "supp (bs, y, c) supports (f bs y c)"
 
   156     unfolding  supports_def fresh_def[symmetric]
 
   126     unfolding  supports_def fresh_def[symmetric]
 
   157     by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
 
   127     apply (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
 
       
   128     sorry
 
   158   then have fin2: "finite (supp (f bs y c))"
 
   129   then have fin2: "finite (supp (f bs y c))"
 
   159     by (auto intro: supports_finite simp add: finite_supp)
 
   130     by (auto intro: supports_finite simp add: finite_supp supp_Pair ctxt)
 
   160   obtain q::"perm" where 
   131   obtain q::"perm" where 
   161     fr1: "(q \<bullet> (set (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and 
   132     fr1: "(q \<bullet> (set (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and 
   162     fr2: "supp q \<sharp>* ([ba as]lst. x)" and 
   133     fr2: "supp q \<sharp>* ([ba as]lst. x)" and 
   163     inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))"
 
   134     inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))"
 
   164     using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)" 
   135     using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)" 
   165       and x="[ba as]lst. x"]  fin1 fin2
 
   136       and x="[ba as]lst. x"]  fin1 fin2
 
   166     by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
 
   137     by (auto simp add: supp_Pair finite_supp ctxt Abs_fresh_star dest: fresh_star_supp_conv)
 
   167   have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp
 
   138   have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp
 
   168   also have "\<dots> = [ba as]lst. x"
 
   139   also have "\<dots> = [ba as]lst. x"
 
   169     by (simp only: fr2 perm_supp_eq)
 
   140     by (simp only: fr2 perm_supp_eq)
 
   170   finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. y" using eq by simp
 
   141   finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. y" using eq by simp
 
   171   then obtain r::perm where 
   142   then obtain r::perm where 
   177     apply(erule exE)
 
   148     apply(erule exE)
 
   178     apply(erule conjE)+
 
   149     apply(erule conjE)+
 
   179     apply(drule_tac x="p" in meta_spec)
 
   150     apply(drule_tac x="p" in meta_spec)
 
   180     apply(simp add: set_eqvt)
 
   151     apply(simp add: set_eqvt)
 
   181     apply(blast)
 
   152     apply(blast)
 
   182     done
 
   153     done 
   183   have "(set (ba as)) \<sharp>* f as x c" by (rule fcb1)
 
   154   have "(set (ba as)) \<sharp>* f as x c" by (rule fcb1)
 
   184   then have "q \<bullet> ((set (ba as)) \<sharp>* f as x c)"
 
   155   then have "q \<bullet> ((set (ba as)) \<sharp>* f as x c)"
 
   185     by (simp add: permute_bool_def)
 
   156     by (simp add: permute_bool_def)
 
   186   then have "set (q \<bullet> (ba as)) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
 
   157   then have "set (q \<bullet> (ba as)) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
 
   187     apply(simp add: fresh_star_eqvt set_eqvt)
 
   158     apply(simp add: fresh_star_eqvt set_eqvt)
 
   188     apply(subst (asm) perm1)
 
   159     apply(subst (asm) perm1)
 
   189     using inc fresh1 fr1
 
   160     using inc fresh1 fr1
 
   190     apply(auto simp add: fresh_star_def fresh_Pair)
 
   161     apply(auto simp add: fresh_star_def fresh_Pair)
 
   191     done
 
   162     done
 
   192   then have "set (r \<bullet> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj 
   163   then have "set (r \<bullet> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj
 
   193     by simp
 
   164     apply simp
 
       
   165     sorry
 
   194   then have "r \<bullet> ((set (ba bs)) \<sharp>* f bs y c)"
 
   166   then have "r \<bullet> ((set (ba bs)) \<sharp>* f bs y c)"
 
   195     apply(simp add: fresh_star_eqvt set_eqvt)
 
   167     apply(simp add: fresh_star_eqvt set_eqvt)
 
   196     apply(subst (asm) perm2[symmetric])
 
   168     apply(subst (asm) perm2[symmetric])
 
   197     using qq3 fresh2 fr1
 
   169     using qq3 fresh2 fr1
 
   198     apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
 
   170     apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
 
   202     apply(rule perm_supp_eq[symmetric])
 
   174     apply(rule perm_supp_eq[symmetric])
 
   203     using inc fcb1 fr1 by (auto simp add: fresh_star_def)
 
   175     using inc fcb1 fr1 by (auto simp add: fresh_star_def)
 
   204   also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c"
 
   176   also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c"
 
   205     apply(rule perm1)
 
   177     apply(rule perm1)
 
   206     using inc fresh1 fr1 by (auto simp add: fresh_star_def)
 
   178     using inc fresh1 fr1 by (auto simp add: fresh_star_def)
 
   207   also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj by simp
 
   179   also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj
 
       
   180     apply(simp)
 
       
   181     sorry
 
   208   also have "\<dots> = r \<bullet> (f bs y c)"
 
   182   also have "\<dots> = r \<bullet> (f bs y c)"
 
   209     apply(rule perm2[symmetric])
 
   183     apply(rule perm2[symmetric])
 
   210     using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
 
   184     using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
 
   211   also have "... = f bs y c"
 
   185   also have "... = f bs y c"
 
   212     apply(rule perm_supp_eq)
 
   186     apply(rule perm_supp_eq)
 
   232   apply (drule_tac x="assn" in meta_spec)
 
   206   apply (drule_tac x="assn" in meta_spec)
 
   233   apply (drule_tac x="trm" in meta_spec)
 
   207   apply (drule_tac x="trm" in meta_spec)
 
   234   apply (simp add: alpha_bn_refl)
 
   208   apply (simp add: alpha_bn_refl)
 
   235   apply (case_tac b rule: trm_assn.exhaust(2))
 
   209   apply (case_tac b rule: trm_assn.exhaust(2))
 
   236   apply (auto)[2]
 
   210   apply (auto)[2]
 
   237   apply(simp_all)
 
   211   apply(simp_all del: trm_assn.eq_iff)
 
   238   apply (erule_tac c="()" in Abs_lst_fcb2)
 
   212   apply(simp)
 
       
   213   prefer 3
 
       
   214   apply(simp)
 
       
   215   apply(simp)
 
       
   216   apply (erule_tac c="()" and pn="permute" in Abs_lst_fcb2)
 
       
   217   apply(simp add: finite_supp)
 
   239   apply (simp_all add: pure_fresh fresh_star_def)[3]
 
   218   apply (simp_all add: pure_fresh fresh_star_def)[3]
 
   240   apply (simp add: eqvt_at_def)
 
   219   apply (simp add: eqvt_at_def)
 
   241   apply (simp add: eqvt_at_def)
 
   220   apply (simp add: eqvt_at_def)
 
   242   apply assumption
 
   221   apply(auto)[1]
 
   243   apply(erule conjE)
 
   222   --"other case"
 
   244   apply (simp add: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])
 
   223   apply (simp only: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])
 
   245   apply (simp add: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff])
 
   224   apply (simp only: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff])
 
   246   apply (subgoal_tac "eqvt_at height_assn as")
 
   225   apply (subgoal_tac "eqvt_at height_assn as")
 
   247   apply (subgoal_tac "eqvt_at height_assn asa")
 
   226   apply (subgoal_tac "eqvt_at height_assn asa")
 
   248   apply (subgoal_tac "eqvt_at height_trm b")
 
   227   apply (subgoal_tac "eqvt_at height_trm b")
 
   249   apply (subgoal_tac "eqvt_at height_trm ba")
 
   228   apply (subgoal_tac "eqvt_at height_trm ba")
 
   250   apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)")
 
   229   apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)")
 
   257   apply (simp add: eqvt_at_def height_assn_def)
 
   236   apply (simp add: eqvt_at_def height_assn_def)
 
   258   apply (simp add: eqvt_at_def height_assn_def)
 
   237   apply (simp add: eqvt_at_def height_assn_def)
 
   259   apply (subgoal_tac "height_assn as = height_assn asa")
 
   238   apply (subgoal_tac "height_assn as = height_assn asa")
 
   260   apply (subgoal_tac "height_trm b = height_trm ba")
 
   239   apply (subgoal_tac "height_trm b = height_trm ba")
 
   261   apply simp
 
   240   apply simp
 
   262   apply (erule_tac c="()" in Abs_lst_fcb2)
 
   241   apply(simp)
 
       
   242   apply(erule conjE)
 
       
   243   apply (erule_tac c="()" and pn="permute_bn" in Abs_lst_fcb2)
 
       
   244   apply(simp add: finite_supp)
 
   263   apply (simp_all add: pure_fresh fresh_star_def)[3]
 
   245   apply (simp_all add: pure_fresh fresh_star_def)[3]
 
   264   apply (simp_all add: eqvt_at_def)[2]
 
   246   apply (simp_all add: eqvt_at_def)[2]
 
   265   apply assumption
 
   247   apply(simp add: bn_inj2)
 
   266   apply (erule_tac c="()" and ba="bn" in Abs_lst_fcb2)
 
   248   apply(simp)
 
       
   249   apply(erule conjE)
 
       
   250   thm trm_assn.fv_defs
 
       
   251   (*apply(simp add: Abs_eq_iff alphas)*)
 
       
   252   apply (erule_tac c="()" and pn="permute_bn" and ba="bn" in Abs_lst_fcb2)
 
       
   253   defer
 
   267   apply (simp_all add: pure_fresh fresh_star_def)[3]
 
   254   apply (simp_all add: pure_fresh fresh_star_def)[3]
 
       
   255   defer
 
       
   256   defer
 
   268   apply (simp_all add: eqvt_at_def)[2]
 
   257   apply (simp_all add: eqvt_at_def)[2]
 
   269   apply (rule bn_inj)
 
   258   apply (rule bn_inj)
 
   270   prefer 2
 
   259   prefer 2
 
   271   apply (simp add: eqvts)
 
   260   apply (simp add: eqvts)
 
   272   oops
 
   261   oops
 
   305   prefer 6
 
   294   prefer 6
 
   306   apply (erule alpha_bn_inducts)
 
   295   apply (erule alpha_bn_inducts)
 
   307  oops
 
   296  oops
 
   308 
   297 
   309 
   298 
       
   299 lemma lets_bla:
 
       
   300   "x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Let (ACons x (Var y) ANil) (Var x)) \<noteq> (Let (ACons x (Var z) ANil) (Var x))"
 
       
   301   by (simp add: trm_assn.eq_iff)
 
       
   302 
       
   303 
       
   304 lemma lets_ok:
 
       
   305   "(Let (ACons x (Var y) ANil) (Var x)) = (Let (ACons y (Var y) ANil) (Var y))"
 
       
   306   apply (simp add: trm_assn.eq_iff Abs_eq_iff )
 
       
   307   apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
 
       
   308   apply (simp_all add: alphas atom_eqvt supp_at_base fresh_star_def trm_assn.bn_defs trm_assn.supp)
 
       
   309   done
 
       
   310 
       
   311 lemma lets_ok3:
 
       
   312   "x \<noteq> y \<Longrightarrow>
 
       
   313    (Let (ACons x (App (Var y) (Var x)) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>
 
       
   314    (Let (ACons y (App (Var x) (Var y)) (ACons x (Var x) ANil)) (App (Var x) (Var y)))"
 
       
   315   apply (simp add: trm_assn.eq_iff)
 
       
   316   done
 
       
   317 
       
   318 lemma lets_not_ok1:
 
       
   319   "x \<noteq> y \<Longrightarrow>
 
       
   320    (Let (ACons x (Var x) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>
 
       
   321    (Let (ACons y (Var x) (ACons x (Var y) ANil)) (App (Var x) (Var y)))"
 
       
   322   apply (simp add: alphas trm_assn.eq_iff trm_assn.supp fresh_star_def atom_eqvt Abs_eq_iff trm_assn.bn_defs)
 
       
   323   done
 
       
   324 
       
   325 lemma lets_nok:
 
       
   326   "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>
 
       
   327    (Let (ACons x (App (Var z) (Var z)) (ACons y (Var z) ANil)) (App (Var x) (Var y))) \<noteq>
 
       
   328    (Let (ACons y (Var z) (ACons x (App (Var z) (Var z)) ANil)) (App (Var x) (Var y)))"
 
       
   329   apply (simp add: alphas trm_assn.eq_iff fresh_star_def trm_assn.bn_defs Abs_eq_iff trm_assn.supp trm_assn.distinct)
 
       
   330   done
 
       
   331 
       
   332 lemma
 
       
   333   fixes a b c :: name
 
       
   334   assumes x: "a \<noteq> c" and y: "b \<noteq> c"
 
       
   335   shows "\<exists>p.([atom a], Var c) \<approx>lst (op =) supp p ([atom b], Var c)"
 
       
   336   apply (rule_tac x="(a \<leftrightarrow> b)" in exI)
 
       
   337   apply (simp add: alphas trm_assn.supp supp_at_base x y fresh_star_def atom_eqvt)
 
       
   338   by (metis Rep_name_inverse atom_name_def flip_fresh_fresh fresh_atom fresh_perm x y)
 
       
   339 
   310 end
 
   340 end
 
   311 
       
   312 
       
   313
nominal2