theory Lambda

imports "../Nominal2" 

begin

atom_decl name

nominal_datatype lam =

  Var "name"

| App "lam" "lam"

| Lam x::"name" l::"lam"  bind x in l ("Lam [_]. _" [100, 100] 100)

inductive 

  triv :: "lam \<Rightarrow> nat \<Rightarrow> bool"

where

  Var: "triv (Var x) n"

| App: "\<lbrakk>triv t1 n; triv t2 n\<rbrakk> \<Longrightarrow> triv (App t1 t2) n"

lemma 

  "p \<bullet> (triv t x) = triv (p \<bullet> t) (p \<bullet> x)"

unfolding triv_def

apply(perm_simp)

apply(rule refl)

oops

(*apply(perm_simp)*)

ML {*

  Inductive.the_inductive @{context} "Lambda.triv"

*}

thm triv_def

equivariance triv

nominal_inductive triv avoids Var: "{}::name set"

apply(auto simp add: fresh_star_def) 

done

inductive 

  triv2 :: "lam \<Rightarrow> nat \<Rightarrow> bool" 

where

  Var1: "triv2 (Var x) 0"

| Var2: "triv2 (Var x) (n + n)"

| Var3: "triv2 (Var x) n"

equivariance triv2

nominal_inductive triv2 .

lemma Abs1_eq_fdest:

  fixes x y :: "'a :: at_base"

    and S T :: "'b :: fs"

  assumes "(Abs_lst [atom x] T) = (Abs_lst [atom y] S)"

  and "x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom x \<sharp> f x T"

  and "x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom y \<sharp> f x T"

  and "x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> T = S \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> (f x T) = f y S"

  and "sort_of (atom x) = sort_of (atom y)"

  shows "f x T = f y S"

using assms apply -

apply (subst (asm) Abs1_eq_iff')

apply simp_all

apply (elim conjE disjE)

apply simp

apply(rule trans)

apply (rule_tac p="(atom x \<rightleftharpoons> atom y)" in supp_perm_eq[symmetric])

apply(rule fresh_star_supp_conv)

apply(simp add: supp_swap fresh_star_def)

apply(simp add: swap_commute)

done

text {* height function *}

nominal_primrec

  height :: "lam \<Rightarrow> int"

where

  "height (Var x) = 1"

| "height (App t1 t2) = max (height t1) (height t2) + 1"

| "height (Lam [x].t) = height t + 1"

  unfolding eqvt_def height_graph_def

  apply (rule, perm_simp, rule)

apply(rule_tac y="x" in lam.exhaust)

apply(auto simp add: lam.distinct lam.eq_iff)

apply (erule Abs1_eq_fdest)

apply(simp_all add: fresh_def pure_supp eqvt_at_def)

done

termination

  by (relation "measure size") (simp_all add: lam.size)

thm height.simps

text {* free name function - returns atom lists *}

nominal_primrec 

  frees_lst :: "lam \<Rightarrow> atom list"

where

  "frees_lst (Var x) = [atom x]"

| "frees_lst (App t1 t2) = frees_lst t1 @ frees_lst t2"

| "frees_lst (Lam [x]. t) = removeAll (atom x) (frees_lst t)"

  unfolding eqvt_def frees_lst_graph_def

  apply (rule, perm_simp, rule)

apply(rule_tac y="x" in lam.exhaust)

apply(auto)

apply (erule Abs1_eq_fdest)

apply(simp add: supp_removeAll fresh_def)

apply(drule supp_eqvt_at)

apply(simp add: finite_supp)

apply(auto simp add: fresh_def supp_removeAll eqvts eqvt_at_def)

done

termination

  by (relation "measure size") (simp_all add: lam.size)

text {* a small test lemma *}

lemma

  shows "supp t = set (frees_lst t)"

apply(induct t rule: frees_lst.induct)

apply(simp_all add: lam.supp supp_at_base)

done

text {* capture - avoiding substitution *}

nominal_primrec

  subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"  ("_ [_ ::= _]" [90, 90, 90] 90)

where

  "(Var x)[y ::= s] = (if x = y then s else (Var x))"

| "(App t1 t2)[y ::= s] = App (t1[y ::= s]) (t2[y ::= s])"

| "atom x \<sharp> (y, s) \<Longrightarrow> (Lam [x]. t)[y ::= s] = Lam [x].(t[y ::= s])"

  unfolding eqvt_def subst_graph_def

  apply (rule, perm_simp, rule)

apply(auto simp add: lam.distinct lam.eq_iff)

apply(rule_tac y="a" and c="(aa, b)" in lam.strong_exhaust)

apply(blast)+

apply(simp_all add: fresh_star_def fresh_Pair_elim)

apply (erule Abs1_eq_fdest)

apply(simp_all add: Abs_fresh_iff)

apply(drule_tac a="atom (xa)" in fresh_eqvt_at)

apply(simp_all add: finite_supp fresh_Pair)

apply(subgoal_tac "(atom x \<rightleftharpoons> atom xa) \<bullet> sa = sa")

apply(subgoal_tac "(atom x \<rightleftharpoons> atom xa) \<bullet> ya = ya")

apply(simp add: eqvt_at_def)

apply(rule perm_supp_eq,simp add: fresh_star_def fresh_Pair supp_swap)+

done

termination

  by (relation "measure (\<lambda>(t,_,_). size t)") (simp_all add: lam.size)

lemma subst_eqvt[eqvt]:

  shows "(p \<bullet> t[x ::= s]) = (p \<bullet> t)[(p \<bullet> x) ::= (p \<bullet> s)]"

by (induct t x s rule: subst.induct) (simp_all)

lemma forget:

  shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"

apply(nominal_induct t avoiding: x s rule: lam.strong_induct)

apply(auto simp add: lam.fresh fresh_at_base)

done

text {* same lemma but with subst.induction *}

lemma forget2:

  shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"

apply(induct t x s rule: subst.induct)

apply(auto simp add: lam.fresh fresh_at_base fresh_Pair)

done

lemma fresh_fact:

  fixes z::"name"

  assumes a: "atom z \<sharp> s"

  and b: "z = y \<or> atom z \<sharp> t"

  shows "atom z \<sharp> t[y ::= s]"

using a b

apply (nominal_induct t avoiding: z y s rule: lam.strong_induct)

apply (auto simp add: lam.fresh fresh_at_base)

done

lemma substitution_lemma:  

  assumes a: "x \<noteq> y" "atom x \<sharp> u"

  shows "t[x ::= s][y ::= u] = t[y ::= u][x ::= s[y ::= u]]"

using a 

by (nominal_induct t avoiding: x y s u rule: lam.strong_induct)

   (auto simp add: fresh_fact forget)

lemma subst_rename: 

  assumes a: "atom y \<sharp> t"

  shows "t[x ::= s] = ((y \<leftrightarrow> x) \<bullet>t)[y ::= s]"

using a 

apply (nominal_induct t avoiding: x y s rule: lam.strong_induct)

apply (auto simp add: lam.fresh fresh_at_base)

done

lemma height_ge_one:

  shows "1 \<le> (height e)"

by (induct e rule: lam.induct) (simp_all)

theorem height_subst:

  shows "height (e[x::=e']) \<le> ((height e) - 1) + (height e')"

proof (nominal_induct e avoiding: x e' rule: lam.strong_induct)

  case (Var y)

  have "1 \<le> height e'" by (rule height_ge_one)

  then show "height (Var y[x::=e']) \<le> height (Var y) - 1 + height e'" by simp

next

  case (Lam y e1)

  hence ih: "height (e1[x::=e']) \<le> ((height e1) - 1) + (height e')" by simp

  moreover

  have vc: "atom y\<sharp>x" "atom y\<sharp>e'" by fact+ (* usual variable convention *)

  ultimately show "height ((Lam [y]. e1)[x::=e']) \<le> height (Lam [y]. e1) - 1 + height e'" by simp

next

  case (App e1 e2)

  hence ih1: "height (e1[x::=e']) \<le> ((height e1) - 1) + (height e')"

    and ih2: "height (e2[x::=e']) \<le> ((height e2) - 1) + (height e')" by simp_all

  then show "height ((App e1 e2)[x::=e']) \<le> height (App e1 e2) - 1 + height e'"  by simp

qed

subsection {* single-step beta-reduction *}

inductive 

  beta :: "lam \<Rightarrow> lam \<Rightarrow> bool" (" _ \<longrightarrow>b _" [80,80] 80)

where

  b1[intro]: "t1 \<longrightarrow>b t2 \<Longrightarrow> App t1 s \<longrightarrow>b App t2 s"

| b2[intro]: "s1 \<longrightarrow>b s2 \<Longrightarrow> App t s1 \<longrightarrow>b App t s2"

| b3[intro]: "t1 \<longrightarrow>b t2 \<Longrightarrow> Lam [x]. t1 \<longrightarrow>b Lam [x]. t2"

| b4[intro]: "atom x \<sharp> s \<Longrightarrow> App (Lam [x]. t) s \<longrightarrow>b t[x ::= s]"

equivariance beta

nominal_inductive beta

  avoids b4: "x"

  by (simp_all add: fresh_star_def fresh_Pair lam.fresh fresh_fact)

text {* One-Reduction *}

inductive 

  One :: "lam \<Rightarrow> lam \<Rightarrow> bool" (" _ \<longrightarrow>1 _" [80,80] 80)

where

  o1[intro]: "Var x \<longrightarrow>1 Var x"

| o2[intro]: "\<lbrakk>t1 \<longrightarrow>1 t2; s1 \<longrightarrow>1 s2\<rbrakk> \<Longrightarrow> App t1 s1 \<longrightarrow>1 App t2 s2"

| o3[intro]: "t1 \<longrightarrow>1 t2 \<Longrightarrow> Lam [x].t1 \<longrightarrow>1 Lam [x].t2"

| o4[intro]: "\<lbrakk>atom x \<sharp> (s1, s2); t1 \<longrightarrow>1 t2; s1 \<longrightarrow>1 s2\<rbrakk> \<Longrightarrow> App (Lam [x].t1) s1 \<longrightarrow>1 t2[x ::= s2]"

equivariance One

nominal_inductive One 

  avoids o3: "x"

      |  o4: "x"

  by (simp_all add: fresh_star_def fresh_Pair lam.fresh fresh_fact)

lemma One_refl:

  shows "t \<longrightarrow>1 t"

by (nominal_induct t rule: lam.strong_induct) (auto)

lemma One_subst: 

  assumes a: "t1 \<longrightarrow>1 t2" "s1 \<longrightarrow>1 s2"

  shows "t1[x ::= s1] \<longrightarrow>1 t2[x ::= s2]" 

using a 

apply(nominal_induct t1 t2 avoiding: s1 s2 x rule: One.strong_induct)

apply(auto simp add: substitution_lemma fresh_at_base fresh_fact fresh_Pair)

done

lemma better_o4_intro:

  assumes a: "t1 \<longrightarrow>1 t2" "s1 \<longrightarrow>1 s2"

  shows "App (Lam [x]. t1) s1 \<longrightarrow>1 t2[ x ::= s2]"

proof -

  obtain y::"name" where fs: "atom y \<sharp> (x, t1, s1, t2, s2)" by (rule obtain_fresh)

  have "App (Lam [x]. t1) s1 = App (Lam [y]. ((y \<leftrightarrow> x) \<bullet> t1)) s1" using fs

    by (auto simp add: lam.eq_iff Abs1_eq_iff' flip_def fresh_Pair fresh_at_base)

  also have "\<dots> \<longrightarrow>1 ((y \<leftrightarrow> x) \<bullet> t2)[y ::= s2]" using fs a by (auto simp add: One.eqvt)

  also have "\<dots> = t2[x ::= s2]" using fs by (simp add: subst_rename[symmetric])

  finally show "App (Lam [x].t1) s1 \<longrightarrow>1 t2[x ::= s2]" by simp

qed

section {* Locally Nameless Terms *}

nominal_datatype ln = 

  LNBnd nat

| LNVar name

| LNApp ln ln

| LNLam ln

fun

  lookup :: "name list \<Rightarrow> nat \<Rightarrow> name \<Rightarrow> ln" 

where

  "lookup [] n x = LNVar x"

| "lookup (y # ys) n x = (if x = y then LNBnd n else (lookup ys (n + 1) x))"

lemma [eqvt]:

  shows "(p \<bullet> lookup xs n x) = lookup (p \<bullet> xs) (p \<bullet> n) (p \<bullet> x)"

  by (induct xs arbitrary: n) (simp_all add: permute_pure)

nominal_primrec

  trans :: "lam \<Rightarrow> name list \<Rightarrow> ln"

where

  "trans (Var x) xs = lookup xs 0 x"

| "trans (App t1 t2) xs = LNApp (trans t1 xs) (trans t2 xs)"

| "atom x \<sharp> xs \<Longrightarrow> trans (Lam [x]. t) xs = LNLam (trans t (x # xs))"

  unfolding eqvt_def trans_graph_def

  apply (rule, perm_simp, rule)

apply(case_tac x)

apply(simp)

apply(rule_tac y="a" and c="b" in lam.strong_exhaust)

apply(simp_all add: fresh_star_def)[3]

apply(blast)

apply(blast)

apply(simp_all add: lam.distinct lam.eq_iff)

apply(elim conjE)

apply clarify

apply (erule Abs1_eq_fdest)

apply (simp_all add: ln.fresh)

prefer 2

apply(drule supp_eqvt_at)

apply (auto simp add: finite_supp supp_Pair fresh_def supp_Cons supp_at_base)[2]

prefer 2

apply (subgoal_tac "(atom x \<rightleftharpoons> atom xa) \<bullet> xsa = xsa")

apply (simp add: eqvt_at_def)

apply (metis atom_name_def swap_fresh_fresh)

oops

(* lemma helpr: "atom x \<sharp> ta \<Longrightarrow> Lam [xa]. ta = Lam [x]. ((xa \<leftrightarrow> x) \<bullet> ta)"

  apply (case_tac "x = xa")

  apply simp

  apply (simp add: lam.eq_iff Abs1_eq_iff flip_def[symmetric])

  by (metis atom_eqvt flip_at_simps(2) fresh_permute_iff)

lemma supp_lookup: "supp (lookup l n name) = {atom name} - supp l"

  apply (induct l arbitrary: n)

  apply (simp_all add: ln.supp supp_at_base supp_Nil supp_Cons pure_supp)

  done

lemma trans_eqvt[eqvt]: "p \<bullet> (trans t l) = trans (p \<bullet> t) (p \<bullet> l)"

  apply (induct t l rule: trans.induct)

  apply simp_all

  apply (simp add: eqvts permute_pure)

  done

lemma diff_un: "a - (b \<union> c) = a - b - c"

  by blast

lemma supp_trans: "supp (trans t l) = supp t - supp l"

  apply (induct t arbitrary: l rule: lam.induct)

  apply (simp_all add: lam.supp supp_at_base supp_lookup ln.supp)

  apply blast

  apply (rule_tac x="(lam, l)" and ?'a="name" in obtain_fresh)

  apply (simp add: fresh_Pair)

  apply clarify

  apply (subgoal_tac "supp (Lambda.trans (Lam [a]. ((name \<leftrightarrow> a) \<bullet> lam)) l) =

    supp lam - {atom name} - supp l")

  using helpr

  apply simp

  apply (simp add: ln.supp)

  apply (subgoal_tac "supp ((name \<leftrightarrow> a) \<bullet> (Lambda.trans lam ((name \<leftrightarrow> a) \<bullet> (a # l)))) = supp lam - {atom name} - supp l")

  apply (simp add: trans_eqvt)

  apply (simp add: supp_eqvt[symmetric])

  apply (simp add: Diff_eqvt)

  apply (simp add: supp_eqvt supp_Cons  union_eqvt)

  apply (simp add: diff_un)

  apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1*})

  apply (tactic {* Cong_Tac.cong_tac @{thm cong} 1*})

  apply rule

  prefer 2

  apply rule

  apply (simp add: supp_at_base)

  apply (subgoal_tac "(name \<leftrightarrow> a) \<bullet> (supp lam - {atom name}) = supp lam - {atom name}")

  apply (simp add: eqvts)

  unfolding flip_def

  apply (rule swap_fresh_fresh)

apply (metis fresh_at_base fresh_def fresh_minus_atom_set lam.fsupp supp_at_base)

by (metis fresh_def fresh_finite_atom_set fresh_minus_atom_set lam.fsupp)

lemma "atom x \<sharp> trans_sumC (t, x # xsa)"

  by (simp add: fresh_def meta_eq_to_obj_eq[OF trans_def, symmetric, unfolded fun_eq_iff] supp_trans supp_Cons supp_at_base)

*)

nominal_datatype db = 

  DBVar nat

| DBApp db db

| DBLam db

fun dbapp_in where

  "dbapp_in None _ = None"

| "dbapp_in (Some _ ) None = None"

| "dbapp_in (Some x) (Some y) = Some (DBApp x y)"

fun dblam_in where

  "dblam_in None = None"

| "dblam_in (Some x) = Some (DBLam x)"

lemma db_in_eqvt[eqvt]:

  "p \<bullet> (dbapp_in x y) = dbapp_in (p \<bullet> x) (p \<bullet> y)"

  "p \<bullet> (dblam_in x) = dblam_in (p \<bullet> x)"

  apply (case_tac [!] x)

  apply (simp_all add: eqvts)

  apply (case_tac y)

  apply (simp_all add: eqvts)

  done

instance db :: pure

  apply default

  apply (induct_tac x rule: db.induct)

  apply (simp_all add: permute_pure)

  done

lemma fresh_at_list: "atom x \<sharp> xs \<longleftrightarrow> x \<notin> set xs"

  unfolding fresh_def supp_set[symmetric]

  apply (induct xs)

  apply (simp add: supp_set_empty)

  apply simp

  apply auto

  apply (simp_all add: insert_absorb UnI2 finite_set supp_of_finite_insert supp_at_base)

  done

fun

  vindex :: "name list \<Rightarrow> name \<Rightarrow> nat \<Rightarrow> db option" 

where

  "vindex [] v n = None"

| "vindex (h # t) v n = (if v = h then (Some (DBVar n)) else (vindex t v (Suc n)))"

lemma vindex_eqvt[eqvt]:

  "(p \<bullet> vindex l v n) = vindex (p \<bullet> l) (p \<bullet> v) (p \<bullet> n)"

  by (induct l arbitrary: n) (simp_all add: permute_pure)

nominal_primrec

  trans :: "lam \<Rightarrow> name list \<Rightarrow> db option"

where

  "trans (Var x) l = vindex l x 0"

| "trans (App t1 t2) xs = dbapp_in (trans t1 xs) (trans t2 xs)"

| "x \<notin> set xs \<Longrightarrow> trans (Lam [x].t) xs = dblam_in (trans t (x # xs))"

  unfolding eqvt_def trans_graph_def

  apply (rule, perm_simp, rule)

  apply (case_tac x)

  apply (rule_tac y="a" and c="b" in lam.strong_exhaust)

  apply (auto simp add: fresh_star_def fresh_at_list)

  apply (rule_tac f="dblam_in" in arg_cong)

  apply (erule Abs1_eq_fdest)

  apply (simp_all add: pure_fresh)

  apply (subgoal_tac "(atom x \<rightleftharpoons> atom xa) \<bullet> xsa = xsa")

  apply (simp add: eqvt_at_def)

  apply (metis atom_name_def swap_fresh_fresh fresh_at_list)

  done

termination

  by (relation "measure (\<lambda>(t,_). size t)") (simp_all add: lam.size)

lemma trans_eqvt[eqvt]:

  "p \<bullet> trans t l = trans (p \<bullet>t) (p \<bullet>l)"

  apply (nominal_induct t avoiding: l p rule: lam.strong_induct)

  apply (simp add: vindex_eqvt)

  apply (simp_all add: permute_pure)

  apply (simp add: fresh_at_list)

  apply (subst trans.simps)

  apply (simp add: fresh_at_list[symmetric])

  apply (drule_tac x="name # l" in meta_spec)

  apply auto

  done

(*

lemma db_trans_test:

  assumes a: "y \<noteq> x"

  shows "trans (Lam [x]. Lam [y]. App (Var x) (Var y)) [] 0 = Some (DBLam (DBLam (DBApp (DBVar 1) (DBVar 0))))"

  using a by simp

*)

abbreviation

  mbind :: "'a option => ('a => 'b option) => 'b option"  ("_ \<guillemotright>= _" [65,65] 65) 

where  

  "c \<guillemotright>= f \<equiv> case c of None => None | (Some v) => f v"

lemma mbind_eqvt:

  fixes c::"'a::pt option"

  shows "(p \<bullet> (c \<guillemotright>= f)) = ((p \<bullet> c) \<guillemotright>= (p \<bullet> f))"

apply(cases c)

apply(simp_all)

apply(perm_simp)

apply(rule refl)

done

lemma mbind_eqvt_raw[eqvt_raw]:

  shows "(p \<bullet> option_case) \<equiv> option_case"

apply(rule eq_reflection)

apply(rule ext)+

apply(case_tac xb)

apply(simp_all)

apply(rule_tac p="-p" in permute_boolE)

apply(perm_simp add: permute_minus_cancel)

apply(simp)

apply(rule_tac p="-p" in permute_boolE)

apply(perm_simp add: permute_minus_cancel)

apply(simp)

done