theory Lambda

imports "../Nominal2" Quotient_Option

begin

atom_decl name

declare [[STEPS = 100]]

nominal_datatype lam =

  Var "name"

| App "lam" "lam"

| Lam x::"name" l::"lam"  bind x in l

thm lam.distinct

thm lam.induct

thm lam.exhaust

thm lam.fv_defs

thm lam.bn_defs

thm lam.perm_simps

thm lam.eq_iff

thm lam.fv_bn_eqvt

thm lam.size_eqvt

thm lam.size

thm lam_raw.size

thm lam.fresh

primrec match_Var_raw where

  "match_Var_raw (Var_raw x) = Some x"

| "match_Var_raw (App_raw x y) = None"

| "match_Var_raw (Lam_raw n t) = None"

quotient_definition

  "match_Var :: lam \<Rightarrow> name option"

is match_Var_raw

lemma [quot_respect]: "(alpha_lam_raw ===> op =) match_Var_raw match_Var_raw"

  by (rule, induct_tac a b rule: alpha_lam_raw.induct, simp_all)

lemmas match_Var_simps = match_Var_raw.simps[quot_lifted]

primrec match_App_raw where

  "match_App_raw (Var_raw x) = None"

| "match_App_raw (App_raw x y) = Some (x, y)"

| "match_App_raw (Lam_raw n t) = None"

quotient_definition

  "match_App :: lam \<Rightarrow> (lam \<times> lam) option"

is match_App_raw

lemma [quot_respect]:

  "(alpha_lam_raw ===> option_rel (prod_rel alpha_lam_raw alpha_lam_raw)) match_App_raw match_App_raw"

  by (intro fun_relI, induct_tac a b rule: alpha_lam_raw.induct, simp_all)

lemmas match_App_simps = match_App_raw.simps[quot_lifted]

definition next_name where

  "next_name (s :: 'a :: fs) = (THE x. \<forall>a \<in> supp s. atom x \<noteq> a)"

lemma lam_eq: "Lam a t = Lam b s \<longleftrightarrow> (a \<leftrightarrow> b) \<bullet> t = s"

  apply (simp add: lam.eq_iff Abs_eq_iff alphas)

  sorry

lemma lam_half_inj: "(Lam z s = Lam z sa) = (s = sa)"

  by (auto simp add: lam_eq)

definition

  "match_Lam (S :: 'a :: fs) t = (if (\<exists>n s. (t = Lam n s)) then

    (let z = next_name (S, t) in Some (z, THE s. t = Lam z s)) else None)"

lemma match_Lam_simps:

  "match_Lam S (Var n) = None"

  "match_Lam S (App l r) = None"

  "match_Lam S (Lam z s) = (let n = next_name (S, (Lam z s)) in Some (n, (z \<leftrightarrow> n) \<bullet> s))"

  apply (simp_all add: match_Lam_def lam.distinct)

  apply (auto simp add: lam_eq)

  done

lemma app_some: "match_App x = Some (a, b) \<Longrightarrow> x = App a b"

by (induct x rule: lam.induct) (simp_all add: match_App_simps)

lemma lam_some: "match_Lam S x = Some (n, t) \<Longrightarrow> x = Lam n t"

  apply (induct x rule: lam.induct)

  apply (simp add: match_Lam_simps)

  apply (simp add: match_Lam_simps)

  apply (simp add: match_Lam_simps Let_def lam_eq)

  apply clarify

  done

function subst where

"subst v s t = (

  case match_Var t of Some n \<Rightarrow> if n = v then s else Var n | None \<Rightarrow>

  case match_App t of Some (l, r) \<Rightarrow> App (subst v s l) (subst v s r) | None \<Rightarrow>

  case match_Lam (v, s) t of Some (n, s') \<Rightarrow> Lam n (subst v s s') | None \<Rightarrow> undefined)"

print_theorems

thm subst_rel.intros[no_vars]

by pat_completeness auto

termination apply (relation "measure (\<lambda>(_, _, t). size t)")

  apply auto[1]

  apply (case_tac a) apply simp

  apply (frule lam_some) apply (simp add: lam.size)

  apply (case_tac a) apply simp

  apply (frule app_some) apply (simp add: lam.size)

  apply (case_tac a) apply simp

  apply (frule app_some) apply (simp add: lam.size)

done

lemma subst_eqs:

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

  "subst y s (App l r) = App (subst y s l) (subst y s r)"

  "subst y s (Lam x t) =

    (let n = next_name ((y, s), Lam x t) in Lam n (subst y s ((x \<leftrightarrow> n) \<bullet> t)))"

  apply (subst subst.simps)

  apply (simp only: match_Var_simps option.simps)

  apply (subst subst.simps)

  apply (simp only: match_App_simps option.simps prod.simps match_Var_simps)

  apply (subst subst.simps)

  apply (simp only: match_App_simps option.simps prod.simps match_Var_simps match_Lam_simps Let_def)

  done

lemma subst_eqvt:

  "p \<bullet> (subst v s t) = subst (p \<bullet> v) (p \<bullet> s) (p \<bullet> t)"

  proof (induct v s t rule: subst.induct)

    case (1 v s t)

    show ?case proof (cases t rule: lam.exhaust)

      fix n

      assume "t = Var n"

      then show ?thesis by (simp add: match_Var_simps)

    next

      fix l r

      assume "t = App l r"

      then show ?thesis

        apply (simp only: subst_eqs lam.perm_simps)

        apply (subst 1(2)[of "(l, r)" "l" "r"])

        apply (simp add: match_Var_simps)

        apply (simp add: match_App_simps)

        apply (rule refl)

        apply (subst 1(3)[of "(l, r)" "l" "r"])

        apply (simp add: match_Var_simps)

        apply (simp add: match_App_simps)

        apply (rule refl)

        apply (rule refl)

        done

    next

      fix n t'

      assume "t = Lam n t'"

      then show ?thesis

        apply (simp only: subst_eqs lam.perm_simps Let_def)

        apply (subst 1(1))

        apply (simp add: match_Var_simps)

        apply (simp add: match_App_simps)

        apply (simp add: match_Lam_simps Let_def)

        apply (rule refl)

        (* next_name is not equivariant :( *)

        apply (simp only: lam_eq)

        sorry

    qed

  qed

lemma subst_eqvt':

  "p \<bullet> (subst v s t) = subst (p \<bullet> v) (p \<bullet> s) (p \<bullet> t)"

  apply (induct t arbitrary: v s rule: lam.induct)

  apply (simp only: subst_eqs lam.perm_simps eqvts)

  apply (simp only: subst_eqs lam.perm_simps)

  apply (simp only: subst_eqs lam.perm_simps Let_def)

  apply (simp only: lam_eq)

  sorry

lemma subst_eq3:

  "atom x \<sharp> (y, s) \<Longrightarrow> subst y s (Lam x t) = Lam x (subst y s t)"

  apply (simp only: subst_eqs Let_def)

  apply (simp only: lam_eq)

  (* p # y   p # s   subst_eqvt *)

  (* p \<bullet> -p \<bullet> (subst y s t) = subst y s t *)

  sorry

end