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 "(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"

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