theory Beta
imports 
  "../Nominal2"
begin


atom_decl name

nominal_datatype lam =
  Var "name"
| App "lam" "lam"
| Lam x::"name" l::"lam"  binds x in l ("Lam [_]. _" [100, 100] 100)

section {* 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])"
| "\<lbrakk>atom x \<sharp> y; atom x \<sharp> s\<rbrakk> \<Longrightarrow> (Lam [x]. t)[y ::= s] = Lam [x].(t[y ::= s])"
  unfolding eqvt_def subst_graph_def
  apply (rule, perm_simp, rule)
  apply(rule TrueI)
  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_tac c="(ya,sa)" in Abs_lst1_fcb2)
  apply(simp_all add: Abs_fresh_iff)
  apply(simp add: fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
done

termination (eqvt)
  by lexicographic_order

lemma forget:
  shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"
  by (nominal_induct t avoiding: x s rule: lam.strong_induct)
     (auto simp add: lam.fresh fresh_at_base)

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
  by (nominal_induct t avoiding: z y s rule: lam.strong_induct)
     (auto simp add: lam.fresh fresh_at_base)

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)

inductive
  equ :: "lam \<Rightarrow> lam \<Rightarrow> bool" ("_ \<approx> _")
where
  equ_beta:  "atom x \<sharp> s \<Longrightarrow> App (Lam [x].t) s \<approx> t[x ::= s]"
| equ_refl:  "t \<approx> t"
| equ_sym:   "t \<approx> s \<Longrightarrow> s \<approx> t"
| equ_trans: "\<lbrakk>t1 \<approx> t2; t2 \<approx> t3\<rbrakk> \<Longrightarrow> t1 \<approx> t3"
| equ_Lam:   "t \<approx> s \<Longrightarrow> Lam [x].t \<approx> Lam [x].s"
| equ_App1:  "t \<approx> s \<Longrightarrow> App t u \<approx> App s u"
| equ_App2:  "t \<approx> s \<Longrightarrow> App u t \<approx> App u s"

equivariance equ

nominal_inductive equ
  avoids equ_beta: "x" 
       | equ_Lam: "x"
by (simp_all add: fresh_star_def fresh_Pair lam.fresh fresh_fact)

lemma [quot_respect]:
  shows "(op = ===> equ) Var Var"
  and   "(equ ===> equ ===> equ) App App"
  and   "(op = ===> equ ===> equ) Beta.Lam Beta.Lam"
unfolding fun_rel_def
by (auto intro: equ.intros)

lemma equ_subst1:
  assumes "t \<approx> s"
  shows "t[x ::= u] \<approx> s[x ::= u]"
using assms
apply(nominal_induct avoiding: x u rule: equ.strong_induct)
apply(simp)
apply(rule equ_trans)
apply(rule equ_beta)
apply(simp add: fresh_fact)
apply(subst (2) substitution_lemma)
apply(simp add: fresh_at_base)
apply(simp)
apply(rule equ_refl)
apply(rule equ_refl)
apply(auto intro: equ_sym)[1]
apply(blast intro: equ_trans)
apply(simp add: equ_Lam)
apply(simp add: equ_App1)
apply(simp add: equ_App2)
done

lemma equ_subst2:
  assumes "t \<approx> s"
  shows "u[x ::= t] \<approx> u[x ::= s]"
using assms
apply(nominal_induct u avoiding: x t s rule: lam.strong_induct)
apply(simp add: equ_refl)
apply(simp)
apply(smt equ_App1 equ_App2 equ_trans)
apply(simp)
apply(metis equ_Lam)
done

lemma [quot_respect]:
  shows "(equ ===> op = ===> equ ===> equ) subst subst"
unfolding fun_rel_def
by (metis equ_subst1 equ_subst2 equ_trans)

lemma [quot_respect]:
  shows "(op = ===> equ ===> equ) permute permute"
unfolding fun_rel_def
by (auto intro: eqvt)

quotient_type qlam = lam / equ
apply(rule equivpI)
apply(rule reflpI)
apply(simp add: equ_refl)
apply(rule sympI)
apply(simp add: equ_sym)
apply(rule transpI)
apply(auto intro: equ_trans)
done

quotient_definition "QVar::name \<Rightarrow> qlam" is Var
quotient_definition "QApp::qlam \<Rightarrow> qlam \<Rightarrow> qlam" is App
quotient_definition QLam ("QLam [_]._") 
  where "QLam::name \<Rightarrow> qlam \<Rightarrow> qlam" is Beta.Lam

lemmas qlam_strong_induct = lam.strong_induct[quot_lifted]
lemmas qlam_induct = lam.induct[quot_lifted]

instantiation qlam :: pt
begin

quotient_definition "permute_qlam::perm \<Rightarrow> qlam \<Rightarrow> qlam" 
  is "permute::perm \<Rightarrow> lam \<Rightarrow> lam"

instance
apply default
apply(descending)
apply(simp)
apply(rule equ_refl)
apply(descending)
apply(simp)
apply(rule equ_refl)
done

end

lemma [eqvt]: "(p \<bullet> abs_qlam t) = abs_qlam (p \<bullet> t)"
 apply (subst fun_cong[OF fun_cong[OF meta_eq_to_obj_eq[OF permute_qlam_def]], of p, simplified])
 apply (subst Quotient_rel[OF Quotient_qlam, simplified equivp_reflp[OF qlam_equivp], simplified])
 by (metis Quotient_qlam equ_refl eqvt rep_abs_rsp_left)

lemma supports_abs_qlam:
  "(supp t) supports (abs_qlam t)"
unfolding supports_def
unfolding fresh_def[symmetric]
apply(auto)
apply(perm_simp)
apply(simp add: swap_fresh_fresh)
done

lemma rep_qlam_inverse:
  "abs_qlam (rep_qlam t) = t"
by (metis Quotient_abs_rep Quotient_qlam)

lemma supports_rep_qlam:
  "supp (rep_qlam t) supports t"
apply(subst (2) rep_qlam_inverse[symmetric])
apply(rule supports_abs_qlam)
done

 lemma supports_qvar:
  "{atom x} supports (QVar x)"
apply(subgoal_tac "supp (Var x) supports (abs_qlam (Var x))")
apply(simp add: lam.supp supp_at_base)
defer
apply(rule supports_abs_qlam)
apply(simp add: QVar_def)
done

 lemma supports_qapp:
  "supp (t, s) supports (QApp t s)"
apply(subgoal_tac "supp (t, s) supports (abs_qlam (App (rep_qlam t) (rep_qlam s)))")
apply(simp add: QApp_def)
apply(subgoal_tac "supp (App (rep_qlam t) (rep_qlam s)) supports (abs_qlam (App (rep_qlam t) (rep_qlam s)))")
prefer 2
apply(rule supports_abs_qlam)
apply(simp add: lam.supp)
oops

lemma "(p \<bullet> rep_qlam t) \<approx> rep_qlam (p \<bullet> t)"
 apply (subst fun_cong[OF fun_cong[OF meta_eq_to_obj_eq[OF permute_qlam_def]], of p, simplified])
 apply (rule rep_abs_rsp[OF Quotient_qlam])
 apply (rule equ_refl)
 done

section {* Supp *}

definition
  "Supp t = \<Inter>{supp s | s. s \<approx> t}"

lemma [quot_respect]:
  shows "(equ ===> op=) Supp Supp"
unfolding fun_rel_def
unfolding Supp_def
apply(rule allI)+
apply(rule impI)
apply(rule_tac f="Inter" in arg_cong)
apply(auto)
apply (metis equ_trans)
by (metis equivp_def qlam_equivp)

quotient_definition "supp_qlam::qlam \<Rightarrow> atom set" 
  is "Supp::lam \<Rightarrow> atom set"


lemma
  fixes t::"qlam"
  shows "(supp x) supports (QVar x)"
apply(rule_tac x="QVar x" in Abs_qlam_cases)
apply(auto)
thm QVar_def
apply(descending)
oops


 
section {* Size *}

definition
  "Size t = Least {size s | s. s \<approx> t}" 

lemma [quot_respect]:
  shows "(equ ===> op=) Size Size"
unfolding fun_rel_def
unfolding Size_def
apply(auto)
apply(rule_tac f="Least" in arg_cong)
apply(auto)
apply (metis equ_trans)
by (metis equivp_def qlam_equivp)

instantiation qlam :: size
begin

quotient_definition "size_qlam::qlam \<Rightarrow> nat" 
  is "Size::lam \<Rightarrow> nat"

instance
apply default
done

end

thm lam.size

lemma
  "size (QVar x) = 0"
apply(descending)
apply(simp add: Size_def)
apply(rule Least_equality)
apply(auto)
apply(simp add: Collect_def)
apply(rule_tac x="Var x" in exI)
apply(auto intro: equ_refl)
done

lemma
  "size (QLam [x].t) = Suc (size t)"
apply(descending)
apply(simp add: Size_def)
apply(rule_tac n="Suc (size t)" and m="size t" in Least_Suc2)
apply(simp add: Collect_def)
apply(rule_tac x="Lam [x].t" in exI)
apply(auto intro: equ_refl)[1]
apply(simp add: Collect_def)
apply(rule_tac x="t" in exI)
apply(auto intro: equ_refl)[1]
apply(simp add: Collect_def)
defer
apply(simp add: Collect_def)
apply(auto)[1]

apply(auto)
done

term rep_qlam
lemmas qlam_size_def = Size_def[quot_lifted]

lemma [quot_preserve]:
  assumes "Quotient equ Abs Rep"
  shows "(id ---> Rep ---> id) fresh = fresh"
using assms
unfolding Quotient_def
apply(simp add: map_fun_def)
apply(simp add: comp_def fun_eq_iff)

sorry

lemma [simp]:
  shows "(QVar x)[y :::= s] = (if x = y then s else (QVar x))"
  and "(QApp t1 t2)[y :::= s] = QApp (t1[y :::= s]) (t2[y :::= s])"
  and "\<lbrakk>atom x \<sharp> y; atom x \<sharp> s\<rbrakk> \<Longrightarrow> (QLam [x]. t)[y :::= s] = QLam [x].(t[y :::= s])"
apply(lifting subst.simps(1))
apply(lifting subst.simps(2))
apply(lifting subst.simps(3))
done



end