thm trm_assn.perm_bn_simps

lemma alpha_bn_inducts_raw[consumes 1]:

lemmas alpha_bn_inducts[consumes 1] = alpha_bn_inducts_raw[quot_lifted]

lemma bn_inj2:

  assumes a: "alpha_bn x y"

  shows "\<And>q r. (q \<bullet> bn x) = (r \<bullet> bn y) \<Longrightarrow> permute_bn q x = permute_bn r y"

using a

apply(induct rule: alpha_bn_inducts)

apply(simp add: trm_assn.perm_bn_simps)

apply(simp add: trm_assn.perm_bn_simps)

apply(simp add: trm_assn.bn_defs)

apply(simp add: atom_eqvt)

done

lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"

  by (simp add: permute_pure)

lemma Abs_lst_fcb2:

  fixes as bs :: "'a :: fs"

    and x y :: "'b :: fs"

  assumes eq: "[ba as]lst. x = [ba bs]lst. y"

  and ctxt: "finite (supp c)"

  and fcb1: "set (ba as) \<sharp>* f as x c"

  and fresh1: "set (ba as) \<sharp>* c"

  and fresh2: "set (ba bs) \<sharp>* c"

  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"

  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"

(* What we would like in this proof, and lets this proof finish *)

  and ba_inj: "\<And>q r. q \<bullet> ba as = r \<bullet> ba bs \<Longrightarrow> pn q as = pn r bs"

  shows "f as x c = f bs y c"

proof -

  have "supp (as, x, c) supports (f as x c)"

    unfolding  supports_def fresh_def[symmetric]

    apply (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)

    sorry

  then have fin1: "finite (supp (f as x c))"

    by (auto intro: supports_finite simp add: finite_supp supp_Pair ctxt)

  have "supp (bs, y, c) supports (f bs y c)"

    unfolding  supports_def fresh_def[symmetric]

    apply (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)

    sorry

  then have fin2: "finite (supp (f bs y c))"

    by (auto intro: supports_finite simp add: finite_supp supp_Pair ctxt)

  obtain q::"perm" where 

    fr1: "(q \<bullet> (set (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and 

    fr2: "supp q \<sharp>* ([ba as]lst. x)" and 

    inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))"

    using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)" 

      and x="[ba as]lst. x"]  fin1 fin2

    by (auto simp add: supp_Pair finite_supp ctxt Abs_fresh_star dest: fresh_star_supp_conv)

  have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp

  also have "\<dots> = [ba as]lst. x"

    by (simp only: fr2 perm_supp_eq)

  finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. y" using eq by simp

  then obtain r::perm where 

    qq1: "q \<bullet> x = r \<bullet> y" and 

    qq2: "q \<bullet> (ba as) = r \<bullet> (ba bs)" and 

    qq3: "supp r \<subseteq> (q \<bullet> (set (ba as))) \<union> set (ba bs)"

    apply(drule_tac sym)

    apply(simp only: Abs_eq_iff2 alphas)

    apply(erule exE)

    apply(erule conjE)+

    apply(drule_tac x="p" in meta_spec)

    apply(simp add: set_eqvt)

    apply(blast)

    done 

  have "(set (ba as)) \<sharp>* f as x c" by (rule fcb1)

  then have "q \<bullet> ((set (ba as)) \<sharp>* f as x c)"

    by (simp add: permute_bool_def)

  then have "set (q \<bullet> (ba as)) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"

    apply(simp add: fresh_star_eqvt set_eqvt)

    apply(subst (asm) perm1)

    using inc fresh1 fr1

    apply(auto simp add: fresh_star_def fresh_Pair)

    done

  then have "set (r \<bullet> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj

    apply simp

    sorry

  then have "r \<bullet> ((set (ba bs)) \<sharp>* f bs y c)"

    apply(simp add: fresh_star_eqvt set_eqvt)

    apply(subst (asm) perm2[symmetric])

    using qq3 fresh2 fr1

    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)

    done

  then have fcb2: "(set (ba bs)) \<sharp>* f bs y c" by (simp add: permute_bool_def)

  have "f as x c = q \<bullet> (f as x c)"

    apply(rule perm_supp_eq[symmetric])

    using inc fcb1 fr1 by (auto simp add: fresh_star_def)

  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c"

    apply(rule perm1)

    using inc fresh1 fr1 by (auto simp add: fresh_star_def)

  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj

    apply(simp)

    sorry

  also have "\<dots> = r \<bullet> (f bs y c)"

    apply(rule perm2[symmetric])

    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)

  also have "... = f bs y c"

    apply(rule perm_supp_eq)

    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)

  finally show ?thesis by simp

qed

nominal_primrec

    height_trm :: "trm \<Rightarrow> nat"

and height_assn :: "assn \<Rightarrow> nat"

where

  "height_trm (Var x) = 1"

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

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

| "height_trm (Let as b) = max (height_assn as) (height_trm b)"

| "height_assn ANil = 0"

| "height_assn (ACons v t as) = max (height_trm t) (height_assn as)"

  apply (simp only: eqvt_def height_trm_height_assn_graph_def)

  apply (rule, perm_simp, rule, rule TrueI)

  apply (case_tac x)

  apply (case_tac a rule: trm_assn.exhaust(1))

  apply (auto)[4]

  apply (drule_tac x="assn" in meta_spec)

  apply (drule_tac x="trm" in meta_spec)

  apply (simp add: alpha_bn_refl)

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

  apply (auto)[2]

  apply(simp_all del: trm_assn.eq_iff)

  apply(simp)

  prefer 3

  apply(simp)

  apply(simp)

  apply (erule_tac c="()" and pn="permute" in Abs_lst_fcb2)

  apply(simp add: finite_supp)

  apply (simp_all add: pure_fresh fresh_star_def)[3]

  apply (simp add: eqvt_at_def)

  apply (simp add: eqvt_at_def)

  apply(auto)[1]

  --"other case"

  apply (simp only: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])

  apply (simp only: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff])

  apply (subgoal_tac "eqvt_at height_assn as")

  apply (subgoal_tac "eqvt_at height_assn asa")

  apply (subgoal_tac "eqvt_at height_trm b")

  apply (subgoal_tac "eqvt_at height_trm ba")

  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)")

  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr asa)")

  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl b)")

  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl ba)")

  defer

  apply (simp add: eqvt_at_def height_trm_def)

  apply (simp add: eqvt_at_def height_trm_def)

  apply (simp add: eqvt_at_def height_assn_def)

  apply (simp add: eqvt_at_def height_assn_def)

  apply (subgoal_tac "height_assn as = height_assn asa")

  apply (subgoal_tac "height_trm b = height_trm ba")

  apply simp

  apply(simp)

  apply(erule conjE)

  apply (erule_tac c="()" and pn="permute_bn" in Abs_lst_fcb2)

  apply(simp add: finite_supp)

  apply (simp_all add: pure_fresh fresh_star_def)[3]

  apply (simp_all add: eqvt_at_def)[2]

  apply(simp add: bn_inj2)

  apply(simp)

  apply(erule conjE)

  thm trm_assn.fv_defs

  (*apply(simp add: Abs_eq_iff alphas)*)

  apply (erule_tac c="()" and pn="permute_bn" and ba="bn" in Abs_lst_fcb2)

  defer

  apply (simp_all add: pure_fresh fresh_star_def)[3]

  defer

  defer

  apply (simp_all add: eqvt_at_def)[2]

  apply (rule bn_inj)

  prefer 2

  apply (simp add: eqvts)

  oops

nominal_primrec

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

and substa :: "name \<Rightarrow> trm \<Rightarrow> assn \<Rightarrow> assn"

where

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

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

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

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

| "substa s t ANil = ANil"

| "substa s t (ACons v t' as) = ACons v (subst v t t') as"

(*unfolding eqvt_def subst_substa_graph_def

  apply (rule, perm_simp)*)

  defer

  apply (rule TrueI)

  apply (case_tac x)

  apply (case_tac a)

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

  apply (auto simp add: fresh_star_def)[3]

  apply (drule_tac x="assn" in meta_spec)

  apply (simp add: Abs1_eq_iff alpha_bn_refl)

  apply (case_tac b)

  apply (case_tac c rule: trm_assn.exhaust(2))

  apply (auto)[2]

  apply blast

  apply blast

  apply auto

  apply (simp_all add: meta_eq_to_obj_eq[OF subst_def, symmetric, unfolded fun_eq_iff])

  apply (simp_all add: meta_eq_to_obj_eq[OF substa_def, symmetric, unfolded fun_eq_iff])

  prefer 2

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

  apply (simp_all add: fresh_star_Pair)

  prefer 6

  apply (erule alpha_bn_inducts)

 oops

end