44 lemma alpha_bn_inducts_raw[consumes 1]:

    45   "\<lbrakk>alpha_bn_raw a b; P3 ANil_raw ANil_raw;

    46  \<And>trm_raw trm_rawa assn_raw assn_rawa name namea.

    47     \<lbrakk>alpha_trm_raw trm_raw trm_rawa; alpha_bn_raw assn_raw assn_rawa;

    48      P3 assn_raw assn_rawa\<rbrakk>

    49     \<Longrightarrow> P3 (ACons_raw name trm_raw assn_raw)

    50         (ACons_raw namea trm_rawa assn_rawa)\<rbrakk> \<Longrightarrow> P3 a b"

    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

    52 

    53 lemmas alpha_bn_inducts[consumes 1] = alpha_bn_inducts_raw[quot_lifted]

    54 

    55 

    56 

    57 lemma alpha_bn_refl: "alpha_bn x x"

    58   by (induct x rule: trm_assn.inducts(2))

    59      (rule TrueI, auto simp add: trm_assn.eq_iff)

    60 lemma alpha_bn_sym: "alpha_bn x y \<Longrightarrow> alpha_bn y x"

    61   sorry

    62 lemma alpha_bn_trans: "alpha_bn x y \<Longrightarrow> alpha_bn y z \<Longrightarrow> alpha_bn x z"

    63   sorry

    64 

    65 lemma bn_inj[rule_format]:

    66   assumes a: "alpha_bn x y"

    67   shows "bn x = bn y \<longrightarrow> x = y"

    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

    80 

    81 

    82 function

    83   apply_assn :: "(trm \<Rightarrow> nat) \<Rightarrow> assn \<Rightarrow> nat"

    84 where

    85   "apply_assn f ANil = (0 :: nat)"

    86 | "apply_assn f (ACons x t as) = max (f t) (apply_assn f as)"

    87 apply(case_tac x)

    88 apply(case_tac b rule: trm_assn.exhaust(2))

    89 apply(simp_all)

    90 apply(blast)

    91 done

    92 

    93 termination by lexicographic_order

    94 

    95 lemma [eqvt]:

    96   "p \<bullet> (apply_assn f a) = apply_assn (p \<bullet> f) (p \<bullet> a)"

    97   apply(induct f a rule: apply_assn.induct)

    98   apply simp_all

    99   apply(perm_simp)

   100   apply rule

   101   apply(perm_simp)

   102   apply simp

   103   done

   104 

   105 lemma alpha_bn_apply_assn:

   106   assumes "alpha_bn as bs"

   107   shows "apply_assn f as = apply_assn f bs"

   108   using assms

   109   apply (induct rule: alpha_bn_inducts)

   110   apply simp_all

   111   done

   112 

   113 nominal_primrec

   114     height_trm :: "trm \<Rightarrow> nat"

   115 where

   116   "height_trm (Var x) = 1"

   117 | "height_trm (App l r) = max (height_trm l) (height_trm r)"

   118 | "height_trm (Lam v b) = 1 + (height_trm b)"

   119 | "height_trm (Let as b) = max (apply_assn height_trm as) (height_trm b)"

   120   apply (simp only: eqvt_def height_trm_graph_def)

   121   apply (rule, perm_simp, rule, rule TrueI)

   122   apply (case_tac x rule: trm_assn.exhaust(1))

   123   apply (auto)[4]

   124   apply (drule_tac x="assn" in meta_spec)

   125   apply (drule_tac x="trm" in meta_spec)

   126   apply (simp add: alpha_bn_refl)

   127   apply(simp_all)

   128   apply (erule_tac c="()" in Abs_lst1_fcb2)

   129   apply (simp_all add: pure_fresh fresh_star_def eqvt_at_def)[4]

   130   apply (erule conjE)

   131   apply (subst alpha_bn_apply_assn)

   132   apply assumption

   133   apply (rule arg_cong) back

   134   apply (erule_tac c="()" in Abs_lst_fcb2)

   135   apply (simp_all add: pure_fresh fresh_star_def)[3]

   136   apply (simp_all add: eqvt_at_def)[2]

   137   done

   138 

   139 definition "height_assn = apply_assn height_trm"

   140 

   141 function

   142   apply_assn2 :: "(trm \<Rightarrow> trm) \<Rightarrow> assn \<Rightarrow> assn"

   143 where

   144   "apply_assn2 f ANil = ANil"

   145 | "apply_assn2 f (ACons x t as) = ACons x (f t) (apply_assn2 f as)"

   146   apply(case_tac x)

   147   apply(case_tac b rule: trm_assn.exhaust(2))

   148   apply(simp_all)

   149   apply(blast)

   150   done

   151 

   152 termination by lexicographic_order

   153 

   154 lemma [eqvt]:

   155   "p \<bullet> (apply_assn2 f a) = apply_assn2 (p \<bullet> f) (p \<bullet> a)"

   156   apply(induct f a rule: apply_assn2.induct)

   157   apply simp_all

   158   apply(perm_simp)

   159   apply rule

   160   done

   161 

   162 lemma bn_apply_assn2: "bn (apply_assn2 f as) = bn as"

   163   apply (induct as rule: trm_assn.inducts(2))

   164   apply (rule TrueI)

   165   apply (simp_all add: trm_assn.bn_defs)

   166   done

   167 

   168 nominal_primrec

   169     subst  :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"

   170 where

   171   "subst s t (Var x) = (if (s = x) then t else (Var x))"

   172 | "subst s t (App l r) = App (subst s t l) (subst s t r)"

   173 | "atom v \<sharp> (s, t) \<Longrightarrow> subst s t (Lam v b) = Lam v (subst s t b)"

   174 | "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (apply_assn2 (subst s t) as) (subst s t b)"

   175   apply (simp only: eqvt_def subst_graph_def)

   176   apply (rule, perm_simp, rule)

   177   apply (rule TrueI)

   178   apply (case_tac x)

   179   apply (rule_tac y="c" and c="(a,b)" in trm_assn.strong_exhaust(1))

   180   apply (auto simp add: fresh_star_def)[3]

   181   apply (drule_tac x="assn" in meta_spec)

   182   apply (simp add: Abs1_eq_iff alpha_bn_refl)

   183   apply simp_all[7]

   184   prefer 2

   185   apply(simp)

   186   apply(simp)

   187   apply(erule conjE)+

   188   apply (erule_tac c="(sa, ta)" in Abs_lst1_fcb2)

   189   apply (simp add: Abs_fresh_iff)

   190   apply (simp add: fresh_star_def)

   191   apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def)[2]

   192   apply (simp add: bn_apply_assn2)

   193   apply(erule conjE)+

   194   apply(rule conjI)

   195   apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)

   196   apply (simp add: fresh_star_def Abs_fresh_iff)

   197   apply assumption+

   198   apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def trm_assn.fv_bn_eqvt)[2]

   199   apply (erule alpha_bn_inducts)

   200   apply simp_all

   201   done

   202 

   203 lemma lets_bla:

   204   "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))"

   205   by (simp add: trm_assn.eq_iff)

   206 

   207 lemma lets_ok:

   208   "(Let (ACons x (Var y) ANil) (Var x)) = (Let (ACons y (Var y) ANil) (Var y))"

   209   apply (simp add: trm_assn.eq_iff Abs_eq_iff )

   210   apply (rule_tac x="(x \<leftrightarrow> y)" in exI)

   211   apply (simp_all add: alphas atom_eqvt supp_at_base fresh_star_def trm_assn.bn_defs trm_assn.supp)

   212   done

   213 

   214 lemma lets_ok3:

   215   "x \<noteq> y \<Longrightarrow>

   216    (Let (ACons x (App (Var y) (Var x)) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>

   217    (Let (ACons y (App (Var x) (Var y)) (ACons x (Var x) ANil)) (App (Var x) (Var y)))"

   218   apply (simp add: trm_assn.eq_iff)

   219   done

   220 

   221 lemma lets_not_ok1:

   222   "x \<noteq> y \<Longrightarrow>

   223    (Let (ACons x (Var x) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>

   224    (Let (ACons y (Var x) (ACons x (Var y) ANil)) (App (Var x) (Var y)))"

   225   apply (simp add: alphas trm_assn.eq_iff trm_assn.supp fresh_star_def atom_eqvt Abs_eq_iff trm_assn.bn_defs)

   226   done

   227 

   228 lemma lets_nok:

   229   "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>

   230    (Let (ACons x (App (Var z) (Var z)) (ACons y (Var z) ANil)) (App (Var x) (Var y))) \<noteq>

   231    (Let (ACons y (Var z) (ACons x (App (Var z) (Var z)) ANil)) (App (Var x) (Var y)))"

   232   apply (simp add: alphas trm_assn.eq_iff fresh_star_def trm_assn.bn_defs Abs_eq_iff trm_assn.supp trm_assn.distinct)

   233   done

   234 

   235 lemma

   236   fixes a b c :: name

   237   assumes x: "a \<noteq> c" and y: "b \<noteq> c"

   238   shows "\<exists>p.([atom a], Var c) \<approx>lst (op =) supp p ([atom b], Var c)"

   239   apply (rule_tac x="(a \<leftrightarrow> b)" in exI)

   240   apply (simp add: alphas trm_assn.supp supp_at_base x y fresh_star_def atom_eqvt)

   241   by (metis Rep_name_inverse atom_name_def flip_fresh_fresh fresh_atom fresh_perm x y)