theory Let

imports "../Nominal2" 

begin

atom_decl name

nominal_datatype trm =

  Var "name"

| App "trm" "trm"

| Lam x::"name" t::"trm"  bind  x in t

| Let as::"assn" t::"trm"   bind "bn as" in t

and assn =

  ANil

| ACons "name" "trm" "assn"

binder

  bn

where

  "bn ANil = []"

| "bn (ACons x t as) = (atom x) # (bn as)"

thm trm_assn.fv_defs

thm trm_assn.bn_defs

thm trm_assn.bn_inducts

thm trm_assn.perm_simps

thm trm_assn.induct

thm trm_assn.inducts

thm trm_assn.distinct

thm trm_assn.supp

thm trm_assn.fresh

thm trm_assn.exhaust

thm trm_assn.strong_exhaust

lemma alpha_bn_inducts_raw:

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

 \<And>trm_raw trm_rawa assn_raw assn_rawa name namea.

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

     P3 assn_raw assn_rawa\<rbrakk>

    \<Longrightarrow> P3 (ACons_raw name trm_raw assn_raw)

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

  by (erule alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.inducts(3)[of _ _ "\<lambda>x y. True" _ "\<lambda>x y. True", simplified]) auto

lemmas alpha_bn_inducts = alpha_bn_inducts_raw[quot_lifted]

lemma alpha_bn_refl: "alpha_bn x x"

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

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

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

  sorry

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

  sorry

lemma bn_inj[rule_format]:

  assumes a: "alpha_bn x y"

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

  by (rule alpha_bn_inducts[OF a]) (simp_all add: trm_assn.bn_defs)

(*lemma alpha_bn_permute:

  assumes a: "alpha_bn x y"

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

    shows "alpha_bn (q \<bullet> x) (r \<bullet> y)"

proof -

  have "alpha_bn x (permute_bn r y)"

    by (rule alpha_bn_trans[OF a]) (rule trm_assn.perm_bn_alpha)

  then have "alpha_bn (permute_bn r y) x"

    by (rule alpha_bn_sym)

  then have "alpha_bn (permute_bn r y) (permute_bn q x)"

    by (rule alpha_bn_trans) (rule trm_assn.perm_bn_alpha)

  then have "alpha_bn (permute_bn q x) (permute_bn r y)"

    by (rule alpha_bn_sym)

  moreover have "bn (permute_bn q x) = bn (permute_bn r y)"

    using b trm_assn.permute_bn by simp

  ultimately have "permute_bn q x = permute_bn r y"

    using bn_inj by simp

*)

lemma lets_bla:

  "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))"

  by (simp add: trm_assn.eq_iff)

lemma lets_ok:

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

  apply (simp add: trm_assn.eq_iff Abs_eq_iff )

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

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

  done

lemma lets_ok3:

  "x \<noteq> y \<Longrightarrow>

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

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

  apply (simp add: trm_assn.eq_iff)

  done

lemma lets_not_ok1:

  "x \<noteq> y \<Longrightarrow>

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

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

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

  done

lemma lets_nok:

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

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

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

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

  done

lemma

  fixes a b c :: name

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

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

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

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

  by (metis Rep_name_inverse atom_name_def flip_fresh_fresh fresh_atom fresh_perm x y)

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"

    and c::"'c::fs"

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

  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 *)

(* What the user can supply with the help of alpha_bn *)

(*  and bainj: "ba as = ba bs \<Longrightarrow> as = 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]

    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)

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

    by (auto intro: supports_finite simp add: finite_supp)

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

    unfolding  supports_def fresh_def[symmetric]

    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)

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

    by (auto intro: supports_finite simp add: finite_supp)

  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 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

    by simp

  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> = 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)

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

  apply (simp_all add: pure_fresh fresh_star_def)[3]

  apply (simp add: eqvt_at_def)

  apply (simp add: eqvt_at_def)

  apply assumption

  apply(erule conjE)

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

  apply (simp add: 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 (erule_tac c="()" in Abs_lst_fcb2)

  apply (simp_all add: pure_fresh fresh_star_def)[3]

  apply (simp_all add: eqvt_at_def)[2]

  apply assumption

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

  apply (simp_all add: pure_fresh fresh_star_def)[3]

  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