theory Nominal2_FSet
imports "../Nominal-General/Nominal2_Supp"
        "../Nominal-General/Nominal2_Atoms" 
        "../Nominal-General/Nominal2_Eqvt" 
        FSet 
begin

lemma "p \<bullet> {} = {}"
apply(perm_simp)
by simp

lemma permute_rsp_fset[quot_respect]:
  "(op = ===> list_eq ===> list_eq) permute permute"
  apply (simp add: eqvts[symmetric])
  apply clarify
  apply (subst permute_minus_cancel(1)[symmetric, of "xb"])
  apply (subst mem_eqvt[symmetric])
  apply (subst (2) permute_minus_cancel(1)[symmetric, of "xb"])
  apply (subst mem_eqvt[symmetric])
  apply (erule_tac x="- x \<bullet> xb" in allE)
  apply simp
  done

instantiation fset :: (pt) pt
begin

quotient_definition
  "permute_fset :: perm \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
is
  "permute :: perm \<Rightarrow> 'a list \<Rightarrow> 'a list"

instance 
proof
  fix x :: "'a fset" and p q :: "perm"
  show "0 \<bullet> x = x"
    by (lifting permute_zero [where 'a="'a list"])
  show "(p + q) \<bullet> x = p \<bullet> q \<bullet> x"
    by (lifting permute_plus [where 'a="'a list"])
qed

end

lemma "p \<bullet> {} = {}"
apply(perm_simp)
by simp

lemma permute_fset[simp]:
  fixes S::"('a::pt) fset"
  shows "(p \<bullet> {||}) = ({||} ::('a::pt) fset)"
  and "p \<bullet> finsert x S = finsert (p \<bullet> x) (p \<bullet> S)"
  by (lifting permute_list.simps)

lemma "p \<bullet> {} = {}"
apply(perm_simp)
by simp

ML {* @{term "{}"} ; @{term "{||}"} *}

declare permute_fset[eqvt]

lemma "p \<bullet> {} = {}"
apply(perm_simp)
by simp


lemma fmap_eqvt[eqvt]: 
  shows "p \<bullet> (fmap f S) = fmap (p \<bullet> f) (p \<bullet> S)"
  by (lifting map_eqvt)

lemma fset_to_set_eqvt[eqvt]: 
  shows "p \<bullet> (fset_to_set S) = fset_to_set (p \<bullet> S)"
  by (lifting set_eqvt)

lemma fin_fset_to_set[simp]:
  shows "finite (fset_to_set S)"
  by (induct S) (simp_all)

lemma supp_fset_to_set:
  shows "supp (fset_to_set S) = supp S"
  apply (simp add: supp_def)
  apply (simp add: eqvts)
  apply (simp add: fset_cong)
  done

lemma supp_finsert:
  fixes x::"'a::fs"
  shows "supp (finsert x S) = supp x \<union> supp S"
  apply(subst supp_fset_to_set[symmetric])
  apply(simp add: supp_fset_to_set)
  apply(simp add: supp_of_fin_insert)
  apply(simp add: supp_fset_to_set)
  done

lemma supp_fempty:
  shows "supp {||} = {}"
  unfolding supp_def
  by simp

instance fset :: (fs) fs
  apply (default)
  apply (induct_tac x rule: fset_induct)
  apply (simp add: supp_fempty)
  apply (simp add: supp_finsert)
  apply (simp add: finite_supp)
  done

lemma atom_fmap_cong:
  shows "fmap atom x = fmap atom y \<longleftrightarrow> x = y"
  apply(rule inj_fmap_eq_iff)
  apply(simp add: inj_on_def)
  done

lemma supp_fmap_atom:
  shows "supp (fmap atom S) = supp S"
  unfolding supp_def
  apply(perm_simp)
  apply(simp add: atom_fmap_cong)
  done

lemma supp_at_fset:
  fixes S::"('a::at_base) fset"
  shows "supp S = fset_to_set (fmap atom S)"
  apply (induct S)
  apply (simp add: supp_fempty)
  apply (simp add: supp_finsert)
  apply (simp add: supp_at_base)
  done

lemma fresh_star_atom:
  "fset_to_set s \<sharp>* (a :: _ :: at_base) \<Longrightarrow> atom a \<sharp> fset_to_set s"
  apply (induct s)
  apply (simp add: fresh_set_empty)
  apply simp
  apply (unfold fresh_def)
  apply (simp add: supp_atom_insert)
  apply (rule conjI)
  apply (unfold fresh_star_def)
  apply simp
  apply (unfold fresh_def)
  apply (simp add: supp_at_base supp_atom)
  apply clarify
  apply auto
  done

lemma "p \<bullet> {} = {}"
apply(perm_simp)
by simp

end