theory LamEx
imports Nominal "../QuotList"
begin

atom_decl name

thm abs_fresh(1)

nominal_datatype rlam =
  rVar "name"
| rApp "rlam" "rlam"
| rLam "name" "rlam"

print_theorems

function
  rfv :: "rlam \<Rightarrow> name set"
where
  rfv_var: "rfv (rVar a) = {a}"
| rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"
| rfv_lam: "rfv (rLam a t) = (rfv t) - {a}"
sorry

termination rfv sorry

inductive
  alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)
where
  a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"
| a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"
| a3: "\<lbrakk>t \<approx> ([(a,b)]\<bullet>s); a \<notin> rfv (rLam b t)\<rbrakk> \<Longrightarrow> rLam a t \<approx> rLam b s"

print_theorems

lemma alpha_refl:
  fixes t::"rlam"
  shows "t \<approx> t"
  apply(induct t rule: rlam.induct)
  apply(simp add: a1)
  apply(simp add: a2)
  apply(rule a3)
  apply(subst pt_swap_bij'')
  apply(rule pt_name_inst)
  apply(rule at_name_inst)
  apply(simp)
  apply(simp)
  done

lemma alpha_equivp:
  shows "equivp alpha"
sorry

quotient lam = rlam / alpha
  apply(rule alpha_equivp)
  done

print_quotients

quotient_def
   "Var :: name \<Rightarrow> lam"
as
  "rVar"

quotient_def
   "App :: lam \<Rightarrow> lam \<Rightarrow> lam"
as
  "rApp"

quotient_def
   "Lam :: name \<Rightarrow> lam \<Rightarrow> lam"
as
  "rLam"

thm Var_def
thm App_def
thm Lam_def

quotient_def
   "fv :: lam \<Rightarrow> name set"
as
  "rfv"

thm fv_def

(* definition of overloaded permutation function *)
(* for the lifted type lam                       *)
overloading
  perm_lam \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"   (unchecked)
begin

quotient_def
   "perm_lam :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"
as
  "perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam"

end

thm perm_lam_def

(* lemmas that need to lift *)
lemma pi_var_com:
  fixes pi::"'x prm"
  shows "(pi\<bullet>rVar a) \<approx> rVar (pi\<bullet>a)"
  sorry

lemma pi_app_com:
  fixes pi::"'x prm"
  shows "(pi\<bullet>rApp t1 t2) \<approx> rApp (pi\<bullet>t1) (pi\<bullet>t2)"
  sorry

lemma pi_lam_com:
  fixes pi::"'x prm"
  shows "(pi\<bullet>rLam a t) \<approx> rLam (pi\<bullet>a) (pi\<bullet>t)"
  sorry



lemma real_alpha:
  assumes a: "t = [(a,b)]\<bullet>s" "a\<sharp>[b].s"
  shows "Lam a t = Lam b s"
using a
unfolding fresh_def supp_def
sorry

lemma perm_rsp[quot_respect]:
  "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"
  apply(auto)
  (* this is propably true if some type conditions are imposed ;o) *)
  sorry

lemma fresh_rsp:
  "(op = ===> alpha ===> op =) fresh fresh"
  apply(auto)
  (* this is probably only true if some type conditions are imposed *)
  sorry

lemma rVar_rsp[quot_respect]:
  "(op = ===> alpha) rVar rVar"
by (auto intro:a1)

lemma rApp_rsp[quot_respect]: "(alpha ===> alpha ===> alpha) rApp rApp"
by (auto intro:a2)

lemma rLam_rsp[quot_respect]: "(op = ===> alpha ===> alpha) rLam rLam"
  apply(auto)
  apply(rule a3)
  apply(rule_tac t="[(x,x)]\<bullet>y" and s="y" in subst)
  apply(rule sym)
  apply(rule trans)
  apply(rule pt_name3)
  apply(rule at_ds1[OF at_name_inst])
  apply(simp add: pt_name1)
  apply(assumption)
  apply(simp add: abs_fresh)
  done

lemma rfv_rsp[quot_respect]: "(alpha ===> op =) rfv rfv"
  sorry

lemma rvar_inject: "rVar a \<approx> rVar b = (a = b)"
apply (auto)
apply (erule alpha.cases)
apply (simp_all add: rlam.inject alpha_refl)
done


lemma pi_var: "(pi\<Colon>('x \<times> 'x) list) \<bullet> Var a = Var (pi \<bullet> a)"
apply (tactic {* lift_tac @{context} @{thm pi_var_com} 1 *})
done

lemma pi_app: "(pi\<Colon>('x \<times> 'x) list) \<bullet> App (x\<Colon>lam) (xa\<Colon>lam) = App (pi \<bullet> x) (pi \<bullet> xa)"
apply (tactic {* lift_tac @{context} @{thm pi_app_com} 1 *})
done

lemma pi_lam: "(pi\<Colon>('x \<times> 'x) list) \<bullet> Lam (a\<Colon>name) (x\<Colon>lam) = Lam (pi \<bullet> a) (pi \<bullet> x)"
apply (tactic {* lift_tac @{context} @{thm pi_lam_com} 1 *})
done

lemma fv_var: "fv (Var (a\<Colon>name)) = {a}"
apply (tactic {* lift_tac @{context} @{thm rfv_var} 1 *})
done

lemma fv_app: "fv (App (x\<Colon>lam) (xa\<Colon>lam)) = fv x \<union> fv xa"
apply (tactic {* lift_tac @{context} @{thm rfv_app} 1 *})
done

lemma fv_lam: "fv (Lam (a\<Colon>name) (x\<Colon>lam)) = fv x - {a}"
apply (tactic {* lift_tac @{context} @{thm rfv_lam} 1 *})
done

lemma a1: "(a\<Colon>name) = (b\<Colon>name) \<Longrightarrow> Var a = Var b"
apply (tactic {* lift_tac @{context} @{thm a1} 1 *})
done

lemma a2: "\<lbrakk>(x\<Colon>lam) = (xa\<Colon>lam); (xb\<Colon>lam) = (xc\<Colon>lam)\<rbrakk> \<Longrightarrow> App x xb = App xa xc"
apply (tactic {* lift_tac @{context} @{thm a2} 1 *})
done

lemma a3: "\<lbrakk>(x\<Colon>lam) = [(a\<Colon>name, b\<Colon>name)] \<bullet> (xa\<Colon>lam); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> Lam a x = Lam b xa"
apply (tactic {* lift_tac @{context} @{thm a3} 1 *})
done

lemma alpha_cases: "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;
     \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P;
     \<And>x a b xa. \<lbrakk>a1 = Lam a x; a2 = Lam b xa; x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> P\<rbrakk>
    \<Longrightarrow> P"
apply (tactic {* lift_tac @{context} @{thm alpha.cases} 1 *})
done

lemma alpha_induct: "\<lbrakk>(qx\<Colon>lam) = (qxa\<Colon>lam); \<And>(a\<Colon>name) b\<Colon>name. a = b \<Longrightarrow> (qxb\<Colon>lam \<Rightarrow> lam \<Rightarrow> bool) (Var a) (Var b);
     \<And>(x\<Colon>lam) (xa\<Colon>lam) (xb\<Colon>lam) xc\<Colon>lam. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc);
     \<And>(x\<Colon>lam) (a\<Colon>name) (b\<Colon>name) xa\<Colon>lam.
        \<lbrakk>x = [(a, b)] \<bullet> xa; qxb x ([(a, b)] \<bullet> xa); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> qxb (Lam a x) (Lam b xa)\<rbrakk>
    \<Longrightarrow> qxb qx qxa"
apply (tactic {* lift_tac @{context} @{thm alpha.induct} 1 *})
done

lemma var_inject: "(Var a = Var b) = (a = b)"
apply (tactic {* lift_tac @{context} @{thm rvar_inject} 1 *})
done

lemma lam_induct:" \<lbrakk>\<And>name. P (Var name); \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2);
              \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk> \<Longrightarrow> P lam"
apply (tactic {* lift_tac @{context} @{thm rlam.induct} 1 *})
done

lemma var_supp:
  shows "supp (Var a) = ((supp a)::name set)"
  apply(simp add: supp_def)
  apply(simp add: pi_var)
  apply(simp add: var_inject)
  done

lemma var_fresh:
  fixes a::"name"
  shows "(a\<sharp>(Var b)) = (a\<sharp>b)"
  apply(simp add: fresh_def)
  apply(simp add: var_supp)
  done

end
