theory Abs
imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "../QuotMain" 
begin

(* lemmas that should be in Nominal \<dots>\<dots>must be cleaned *)
lemma in_permute_iff:
  shows "(p \<bullet> x) \<in> (p \<bullet> X) \<longleftrightarrow> x \<in> X"
apply(unfold mem_def permute_fun_def)[1]
apply(simp add: permute_bool_def) 
done

lemma fresh_star_permute_iff:
  shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x"
apply(simp add: fresh_star_def)
apply(auto)
apply(drule_tac x="p \<bullet> xa" in bspec)
apply(unfold mem_def permute_fun_def)[1] 
apply(simp add: eqvts)
apply(simp add: fresh_permute_iff)
apply(rule_tac ?p1="- p" in fresh_permute_iff[THEN iffD1])
apply(simp)
apply(drule_tac x="- p \<bullet> xa" in bspec)
apply(rule_tac ?p1="p" in in_permute_iff[THEN iffD1])
apply(simp)
apply(simp)
done

datatype 'a ABS_raw = Abs_raw "atom set" "'a::pt"

primrec
  Abs_raw_map
where
  "Abs_raw_map f (Abs_raw as x) = Abs_raw as (f x)"

fun
  Abs_raw_rel
where
  "Abs_raw_rel R (Abs_raw as x) (Abs_raw bs y) = R x y"

declare [[map "ABS_raw" = (Abs_raw_map, Abs_raw_rel)]]

instantiation ABS_raw :: (pt) pt
begin

primrec
  permute_ABS_raw
where
  "permute_ABS_raw p (Abs_raw as x) = Abs_raw (p \<bullet> as) (p \<bullet> x)"

instance
apply(default)
apply(case_tac [!] x)
apply(simp_all)
done

end  

fun
  alpha_abs :: "('a::pt) ABS_raw \<Rightarrow> 'a ABS_raw \<Rightarrow> bool"
where
  "alpha_abs (Abs_raw as x) (Abs_raw bs y) =
    (\<exists>pi. (supp x) - as = (supp y) - bs \<and>  ((supp x) - as) \<sharp>* pi \<and> pi \<bullet> x = y)" 

lemma alpha_reflp:
  shows "alpha_abs ab ab"
apply(induct ab)
apply(simp)
apply(rule_tac x="0" in exI)
apply(simp add: fresh_star_def fresh_zero_perm)
done

lemma alpha_symp:
  assumes a: "alpha_abs ab1 ab2"
  shows "alpha_abs ab2 ab1"
using a
apply(induct rule: alpha_abs.induct)
apply(simp)
apply(clarify)
apply(rule_tac x="- pi" in exI)
apply(auto)
apply(auto simp add: fresh_star_def)
apply(simp add: fresh_def supp_minus_perm)
done

lemma alpha_transp:
  assumes a1: "alpha_abs ab1 ab2"
  and     a2: "alpha_abs ab2 ab3"
  shows "alpha_abs ab1 ab3"
using a1 a2
apply(induct rule: alpha_abs.induct)
apply(induct rule: alpha_abs.induct)
apply(simp)
apply(clarify)
apply(rule_tac x="pia + pi" in exI)
apply(simp)
apply(auto simp add: fresh_star_def)
using supp_plus_perm
apply(simp add: fresh_def)
apply(blast)
done

lemma alpha_eqvt:
  assumes a: "alpha_abs ab1 ab2"
  shows "alpha_abs (p \<bullet> ab1) (p \<bullet> ab2)"
using a
apply(induct ab1 ab2 rule: alpha_abs.induct)
apply(simp)
apply(clarify)
apply(rule conjI)
apply(simp add: supp_eqvt[symmetric])
apply(simp add: Diff_eqvt[symmetric])
apply(rule_tac x="p \<bullet> pi" in exI)
apply(rule conjI)
apply(simp add: supp_eqvt[symmetric])
apply(simp add: Diff_eqvt[symmetric])
apply(simp only: fresh_star_permute_iff)
apply(simp add: permute_eqvt[symmetric])
done

lemma test1:
  assumes a1: "a \<notin> (supp x) - bs"
  and     a2: "b \<notin> (supp x) - bs"
  shows "alpha_abs (Abs_raw bs x) (Abs_raw ((a \<rightleftharpoons> b) \<bullet> bs) ((a \<rightleftharpoons> b) \<bullet> x))"
unfolding alpha_abs.simps
apply(rule_tac x="(a \<rightleftharpoons> b)" in exI)
apply(rule_tac conjI)
apply(simp add: supp_eqvt[symmetric])
apply(simp add: Diff_eqvt[symmetric])
using a1 a2
apply(simp add: swap_set_fresh)
apply(rule conjI)
prefer 2
apply(simp)
apply(simp add: fresh_star_def)
apply(simp add: fresh_def)
apply(subgoal_tac "supp (a \<rightleftharpoons> b) \<subseteq> {a, b}")
using a1 a2
apply -
apply(blast)
apply(simp add: supp_swap)
done

fun
  s_test
where
  "s_test (Abs_raw bs x) = (supp x) - bs"

lemma s_test_lemma:
  assumes a: "alpha_abs x y" 
  shows "s_test x = s_test y"
using a
apply(induct rule: alpha_abs.induct)
apply(simp)
done
  

quotient_type 'a ABS = "('a::pt) ABS_raw" / "alpha_abs::('a::pt) ABS_raw \<Rightarrow> 'a ABS_raw \<Rightarrow> bool"
  apply(rule equivpI)
  unfolding reflp_def symp_def transp_def
  apply(auto intro: alpha_reflp alpha_symp alpha_transp)
  done

quotient_definition
   "Abs::atom set \<Rightarrow> ('a::pt) \<Rightarrow> 'a ABS"
as
   "Abs_raw"

lemma [quot_respect]:
  shows "((op =) ===> (op =) ===> alpha_abs) Abs_raw Abs_raw"
apply(auto simp del: alpha_abs.simps)
apply(rule alpha_reflp)
done

lemma [quot_respect]:
  shows "((op =) ===> alpha_abs ===> alpha_abs) permute permute"
apply(auto)
apply(simp add: alpha_eqvt)
done

lemma [quot_respect]:
  shows "(alpha_abs ===> (op =)) s_test s_test"
apply(simp add: s_test_lemma)
done

lemma ABS_induct:
  "\<lbrakk>\<And>as (x::'a::pt). P (Abs as x)\<rbrakk> \<Longrightarrow> P t"
apply(lifting ABS_raw.induct)
done

instantiation ABS :: (pt) pt
begin

quotient_definition
  "permute_ABS::perm \<Rightarrow> ('a::pt ABS) \<Rightarrow> 'a ABS"
as
  "permute::perm \<Rightarrow> ('a::pt ABS_raw) \<Rightarrow> 'a ABS_raw"

lemma permute_ABS [simp]:
  fixes x::"'b::pt"  (* ??? has to be 'b \<dots> 'a doe not work *)
  shows "(p \<bullet> (Abs as x)) = Abs (p \<bullet> as) (p \<bullet> x)"
apply(lifting permute_ABS_raw.simps(1))
done

instance
  apply(default)
  apply(induct_tac [!] x rule: ABS_induct)
  apply(simp_all)
  done

end

lemma test1_lifted:
  assumes a1: "a \<notin> (supp x) - bs"
  and     a2: "b \<notin> (supp x) - bs"
  shows "(Abs bs x) = (Abs ((a \<rightleftharpoons> b) \<bullet> bs) ((a \<rightleftharpoons> b) \<bullet> x))"
using a1 a2
apply(lifting test1)
done

lemma Abs_supports:
  shows "((supp x) - as) supports (Abs as x) "
unfolding supports_def
apply(clarify)
apply(simp (no_asm))
apply(subst test1_lifted[symmetric])
apply(simp_all)
done

quotient_definition
  "s_test_lifted :: ('a::pt) ABS \<Rightarrow> atom \<Rightarrow> bool"
as
  "s_test::('a::pt) ABS_raw \<Rightarrow> atom \<Rightarrow> bool"

lemma s_test_lifted_simp:
  shows "s_test_lifted (Abs bs x) = (supp x) - bs"
apply(lifting s_test.simps(1))
done

lemma s_test_lifted_eqvt:
  shows "(p \<bullet> (s_test_lifted ab)) = s_test_lifted (p \<bullet> ab)"
apply(induct ab rule: ABS_induct)
apply(simp add: s_test_lifted_simp supp_eqvt Diff_eqvt)
done

lemma fresh_f_empty_supp:
  assumes a: "\<forall>p. p \<bullet> f = f"
  shows "a \<sharp> x \<Longrightarrow> a \<sharp> (f x)"
apply(simp add: fresh_def)
apply(simp add: supp_def)
apply(simp add: permute_fun_app_eq)
apply(simp add: a)
apply(rule finite_subset)
prefer 2
apply(assumption)
apply(auto)
done


lemma s_test_fresh_lemma:
  shows "(a \<sharp> Abs bs x) \<Longrightarrow> (a \<sharp> s_test_lifted (Abs bs x))"
apply(rule fresh_f_empty_supp)
apply(rule allI)
apply(subst permute_fun_def)
apply(simp add: s_test_lifted_eqvt)
apply(simp)
done

lemma supp_finite_set:
  fixes S::"atom set"
  assumes "finite S"
  shows "supp S = S"
  apply(rule finite_supp_unique)
  apply(simp add: supports_def)
  apply(auto simp add: permute_set_eq swap_atom)[1]
  apply(metis)
  apply(rule assms)
  apply(auto simp add: permute_set_eq swap_atom)[1]
done

lemma s_test_subset:
  fixes x::"'a::fs"
  shows "((supp x) - as) \<subseteq> (supp (Abs as x))"
apply(rule subsetI)
apply(rule contrapos_pp)
apply(assumption)
unfolding fresh_def[symmetric]
apply(drule_tac s_test_fresh_lemma)
apply(simp only: s_test_lifted_simp)
apply(simp add: fresh_def)
apply(subgoal_tac "finite (supp x - as)")
apply(simp add: supp_finite_set)
apply(simp add: finite_supp)
done

lemma supp_Abs:
  fixes x::"'a::fs"
  shows "supp (Abs as x) = (supp x) - as"
apply(rule subset_antisym)
apply(rule supp_is_subset)
apply(rule Abs_supports)
apply(simp add: finite_supp)
apply(rule s_test_subset)
done

instance ABS :: (fs) fs
apply(default)
apply(induct_tac x rule: ABS_induct)
apply(simp add: supp_Abs)
apply(simp add: finite_supp)
done

lemma fresh_abs:
  fixes x::"'a::fs"
  shows "a \<sharp> Abs bs x = (a \<in> bs \<or> (a \<notin> bs \<and> a \<sharp> x))"
apply(simp add: fresh_def)
apply(simp add: supp_Abs)
apply(auto)
done

lemma abs_eq:
  shows "(Abs as x) = (Abs bs y) \<longleftrightarrow> (\<exists>pi. supp x - as = supp y - bs \<and> (supp x - as) \<sharp>* pi \<and> pi \<bullet> x = y)"
apply(lifting alpha_abs.simps(1))
done

end