theory Term5
imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv" "Rsp" "../Attic/Prove"
begin

atom_decl name

datatype rtrm5 =
  rVr5 "name"
| rAp5 "rtrm5" "rtrm5"
| rLt5 "rlts" "rtrm5" --"bind (bv5 lts) in (rtrm5)"
and rlts =
  rLnil
| rLcons "name" "rtrm5" "rlts"

primrec
  rbv5
where
  "rbv5 rLnil = {}"
| "rbv5 (rLcons n t ltl) = {atom n} \<union> (rbv5 ltl)"


setup {* snd o define_raw_perms (Datatype.the_info @{theory} "Term5.rtrm5") 2 *}
print_theorems

local_setup {* snd o define_fv_alpha (Datatype.the_info @{theory} "Term5.rtrm5")
  [[[], [], [(SOME (@{term rbv5}, true), 0, 1)]], [[], []]] [(@{term rbv5}, 1, [[], [0, 2]])] *}
print_theorems

notation
  alpha_rtrm5 ("_ \<approx>5 _" [100, 100] 100) and
  alpha_rlts ("_ \<approx>l _" [100, 100] 100)
thm alpha_rtrm5_alpha_rlts_alpha_rbv5.intros

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha5_inj}, []), (build_alpha_inj @{thms alpha_rtrm5_alpha_rlts_alpha_rbv5.intros} @{thms rtrm5.distinct rtrm5.inject rlts.distinct rlts.inject} @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} ctxt)) ctxt)) *}
thm alpha5_inj

lemma rbv5_eqvt[eqvt]:
  "pi \<bullet> (rbv5 x) = rbv5 (pi \<bullet> x)"
  apply (induct x)
  apply (simp_all add: eqvts atom_eqvt)
  done

lemma fv_rtrm5_rlts_eqvt[eqvt]:
  "pi \<bullet> (fv_rtrm5 x) = fv_rtrm5 (pi \<bullet> x)"
  "pi \<bullet> (fv_rlts l) = fv_rlts (pi \<bullet> l)"
  apply (induct x and l)
  apply (simp_all add: eqvts atom_eqvt)
  done

(*lemma alpha5_eqvt:
  "(xa \<approx>5 y \<longrightarrow> (p \<bullet> xa) \<approx>5 (p \<bullet> y)) \<and>
  (xb \<approx>l ya \<longrightarrow> (p \<bullet> xb) \<approx>l (p \<bullet> ya)) \<and>
  (alpha_rbv5 b c \<longrightarrow> alpha_rbv5 (p \<bullet> b) (p \<bullet> c))"
apply (tactic {* alpha_eqvt_tac @{thm alpha_rtrm5_alpha_rlts_alpha_rbv5.induct} @{thms alpha5_inj permute_rtrm5_permute_rlts.simps} @{context} 1 *})
done*)

local_setup {*
(fn ctxt => snd (Local_Theory.note ((@{binding alpha5_eqvt}, []),
build_alpha_eqvts [@{term alpha_rtrm5}, @{term alpha_rlts}, @{term alpha_rbv5}] (fn _ => alpha_eqvt_tac  @{thm alpha_rtrm5_alpha_rlts_alpha_rbv5.induct} @{thms alpha5_inj permute_rtrm5_permute_rlts.simps} ctxt 1) ctxt) ctxt)) *}
print_theorems

lemma alpha5_reflp:
"y \<approx>5 y \<and> (x \<approx>l x \<and> alpha_rbv5 x x)"
apply (rule rtrm5_rlts.induct)
apply (simp_all add: alpha5_inj)
apply (rule_tac x="0::perm" in exI)
apply (simp add: eqvts alpha_gen fresh_star_def fresh_zero_perm)
done

lemma alpha5_symp:
"(a \<approx>5 b \<longrightarrow> b \<approx>5 a) \<and>
(x \<approx>l y \<longrightarrow> y \<approx>l x) \<and>
(alpha_rbv5 x y \<longrightarrow> alpha_rbv5 y x)"
apply (tactic {* symp_tac @{thm alpha_rtrm5_alpha_rlts_alpha_rbv5.induct} @{thms alpha5_inj} @{thms alpha5_eqvt} @{context} 1 *})
(*
apply (rule alpha_rtrm5_alpha_rlts_alpha_rbv5.induct)
apply (simp_all add: alpha5_inj)
apply (erule exE)
apply (rule_tac x="- pi" in exI)
apply (simp add: alpha_gen)
  apply(simp add: fresh_star_def fresh_minus_perm)
apply clarify
apply (rule conjI)
apply (rotate_tac 3)
apply (frule_tac p="- pi" in alpha5_eqvt(2))
apply simp
apply (rule conjI)
apply (rotate_tac 5)
apply (frule_tac p="- pi" in alpha5_eqvt(1))
apply simp
apply (rotate_tac 6)
apply simp
apply (drule_tac p1="- pi" in permute_eq_iff[symmetric,THEN iffD1])
apply (simp)*)
done

lemma alpha5_transp:
"(a \<approx>5 b \<longrightarrow> (\<forall>c. b \<approx>5 c \<longrightarrow> a \<approx>5 c)) \<and>
(x \<approx>l y \<longrightarrow> (\<forall>z. y \<approx>l z \<longrightarrow> x \<approx>l z)) \<and>
(alpha_rbv5 k l \<longrightarrow> (\<forall>m. alpha_rbv5 l m \<longrightarrow> alpha_rbv5 k m))"
(*apply (tactic {* transp_tac @{context} @{thm alpha_rtrm5_alpha_rlts_alpha_rbv5.induct} @{thms alpha5_inj} @{thms rtrm5.distinct rtrm5.inject rlts.distinct rlts.inject} [] @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{thms alpha5_eqvt} 1 *})*)
apply (rule alpha_rtrm5_alpha_rlts_alpha_rbv5.induct)
apply (rule_tac [!] allI)
apply (tactic {* (imp_elim_tac @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{context}) 1 *})
apply (simp_all add: alpha5_inj)
apply (tactic {* (imp_elim_tac @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{context}) 1 *})
apply (simp_all add: alpha5_inj)
apply (tactic {* (imp_elim_tac @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{context}) 1 *})
apply (simp_all add: alpha5_inj)
defer
apply (tactic {* (imp_elim_tac @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{context}) 1 *})
apply (simp_all add: alpha5_inj)
apply (tactic {* (imp_elim_tac @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases} @{context}) 1 *})
apply (simp_all add: alpha5_inj)
apply (tactic {* eetac @{thm exi_sum} @{context} 1 *})
apply (simp add: alpha_gen)
apply clarify
apply(simp add: fresh_star_plus)
apply (rule conjI)
apply (erule_tac x="- pi \<bullet> rltsaa" in allE)
apply (rotate_tac 5)
apply (drule_tac p="- pi" in alpha5_eqvt(2))
apply simp
apply (drule_tac p="pi" in alpha5_eqvt(2))
apply simp
apply (erule_tac x="- pi \<bullet> rtrm5aa" in allE)
apply (rotate_tac 7)
apply (drule_tac p="- pi" in alpha5_eqvt(1))
apply simp
apply (rotate_tac 3)
apply (drule_tac p="pi" in alpha5_eqvt(1))
apply simp
done

lemma alpha5_equivp:
  "equivp alpha_rtrm5"
  "equivp alpha_rlts"
  unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def
  apply (simp_all only: alpha5_reflp)
  apply (meson alpha5_symp alpha5_transp)
  apply (meson alpha5_symp alpha5_transp)
  done

quotient_type
  trm5 = rtrm5 / alpha_rtrm5
and
  lts = rlts / alpha_rlts
  by (auto intro: alpha5_equivp)

local_setup {*
(fn ctxt => ctxt
 |> snd o (Quotient_Def.quotient_lift_const ("Vr5", @{term rVr5}))
 |> snd o (Quotient_Def.quotient_lift_const ("Ap5", @{term rAp5}))
 |> snd o (Quotient_Def.quotient_lift_const ("Lt5", @{term rLt5}))
 |> snd o (Quotient_Def.quotient_lift_const ("Lnil", @{term rLnil}))
 |> snd o (Quotient_Def.quotient_lift_const ("Lcons", @{term rLcons}))
 |> snd o (Quotient_Def.quotient_lift_const ("fv_trm5", @{term fv_rtrm5}))
 |> snd o (Quotient_Def.quotient_lift_const ("fv_lts", @{term fv_rlts}))
 |> snd o (Quotient_Def.quotient_lift_const ("bv5", @{term rbv5}))
 |> snd o (Quotient_Def.quotient_lift_const ("alpha_bv5", @{term alpha_rbv5})))
*}
print_theorems

lemma alpha5_rfv:
  "(t \<approx>5 s \<Longrightarrow> fv_rtrm5 t = fv_rtrm5 s)"
  "(l \<approx>l m \<Longrightarrow> fv_rlts l = fv_rlts m)"
  "(alpha_rbv5 b c \<Longrightarrow> True)"
  apply(induct rule: alpha_rtrm5_alpha_rlts_alpha_rbv5.inducts)
  apply(simp_all add: eqvts)
  apply(simp add: alpha_gen)
  apply(clarify)
  apply blast
done

lemma bv_list_rsp:
  shows "x \<approx>l y \<Longrightarrow> rbv5 x = rbv5 y"
  apply(induct rule: alpha_rtrm5_alpha_rlts_alpha_rbv5.inducts(2))
  apply(simp_all)
  apply(clarify)
  apply simp
  done

local_setup {* snd o Local_Theory.note ((@{binding alpha_dis}, []), (flat (map (distinct_rel @{context} @{thms alpha_rtrm5.cases alpha_rlts.cases alpha_rbv5.cases}) [(@{thms rtrm5.distinct}, @{term alpha_rtrm5}), (@{thms rlts.distinct}, @{term alpha_rlts}), (@{thms rlts.distinct}, @{term alpha_rbv5})]))) *}
print_theorems

local_setup {* snd o Local_Theory.note ((@{binding alpha_bn_rsp}, []), prove_alpha_bn_rsp [@{term alpha_rtrm5}, @{term alpha_rlts}] @{thms alpha_rtrm5_alpha_rlts_alpha_rbv5.inducts} @{thms rtrm5.exhaust rlts.exhaust} @{thms alpha5_inj alpha_dis} @{thms alpha5_equivp} @{context} (@{term alpha_rbv5}, 1)) *}
thm alpha_bn_rsp

lemma [quot_respect]:
  "(alpha_rlts ===> op =) fv_rlts fv_rlts"
  "(alpha_rtrm5 ===> op =) fv_rtrm5 fv_rtrm5"
  "(alpha_rlts ===> op =) rbv5 rbv5"
  "(op = ===> alpha_rtrm5) rVr5 rVr5"
  "(alpha_rtrm5 ===> alpha_rtrm5 ===> alpha_rtrm5) rAp5 rAp5"
  "(alpha_rlts ===> alpha_rtrm5 ===> alpha_rtrm5) rLt5 rLt5"
  "(op = ===> alpha_rtrm5 ===> alpha_rlts ===> alpha_rlts) rLcons rLcons"
  "(op = ===> alpha_rtrm5 ===> alpha_rtrm5) permute permute"
  "(op = ===> alpha_rlts ===> alpha_rlts) permute permute"
  "(alpha_rlts ===> alpha_rlts ===> op =) alpha_rbv5 alpha_rbv5"
  apply (simp_all add: alpha5_inj alpha5_rfv alpha5_eqvt bv_list_rsp alpha_bn_rsp)
  apply (clarify)
  apply (rule_tac x="0" in exI)
  apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)
done


lemma
  shows "(alpha_rlts ===> op =) rbv5 rbv5"
  by (simp add: bv_list_rsp)

lemmas trm5_lts_inducts = rtrm5_rlts.inducts[quot_lifted]

instantiation trm5 and lts :: pt
begin

quotient_definition
  "permute_trm5 :: perm \<Rightarrow> trm5 \<Rightarrow> trm5"
is
  "permute :: perm \<Rightarrow> rtrm5 \<Rightarrow> rtrm5"

quotient_definition
  "permute_lts :: perm \<Rightarrow> lts \<Rightarrow> lts"
is
  "permute :: perm \<Rightarrow> rlts \<Rightarrow> rlts"

instance by default
  (simp_all add: permute_rtrm5_permute_rlts_zero[quot_lifted] permute_rtrm5_permute_rlts_append[quot_lifted])

end

lemmas permute_trm5_lts = permute_rtrm5_permute_rlts.simps[quot_lifted]
lemmas bv5[simp] = rbv5.simps[quot_lifted]
lemmas fv_trm5_lts[simp] = fv_rtrm5_fv_rlts.simps[quot_lifted]
lemmas alpha5_INJ = alpha5_inj[unfolded alpha_gen2, unfolded alpha_gen, quot_lifted, folded alpha_gen2, folded alpha_gen]
lemmas alpha5_DIS = alpha_dis[quot_lifted]

(* why is this not in Isabelle? *)
lemma set_sub: "{a, b} - {b} = {a} - {b}"
by auto

lemma lets_bla:
  "x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Lt5 (Lcons x (Vr5 y) Lnil) (Vr5 x)) \<noteq> (Lt5 (Lcons x (Vr5 z) Lnil) (Vr5 x))"
apply (simp only: alpha5_INJ bv5)
apply simp
apply (rule allI)
apply (simp_all add: alpha_gen)
apply (simp add: permute_trm5_lts fresh_star_def alpha5_INJ eqvts)
apply (rule impI)
apply (rule impI)
apply (rule impI)
apply (simp add: set_sub)
done

lemma lets_ok:
  "(Lt5 (Lcons x (Vr5 x) Lnil) (Vr5 x)) = (Lt5 (Lcons y (Vr5 y) Lnil) (Vr5 y))"
thm alpha5_INJ
apply (simp only: alpha5_INJ)
apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
apply (simp_all add: alpha_gen)
apply (simp add: permute_trm5_lts fresh_star_def)
apply (simp add: eqvts)
done

lemma lets_ok3:
  "x \<noteq> y \<Longrightarrow>
   (Lt5 (Lcons x (Ap5 (Vr5 y) (Vr5 x)) (Lcons y (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) \<noteq>
   (Lt5 (Lcons y (Ap5 (Vr5 x) (Vr5 y)) (Lcons x (Vr5 x) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"
apply (simp add: permute_trm5_lts alpha_gen alpha5_INJ)
done


lemma lets_not_ok1:
  "x \<noteq> y \<Longrightarrow>
   (Lt5 (Lcons x (Vr5 x) (Lcons y (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) \<noteq>
   (Lt5 (Lcons y (Vr5 x) (Lcons x (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"
apply (simp add: alpha5_INJ alpha_gen)
apply (simp add: permute_trm5_lts eqvts)
apply (simp add: alpha5_INJ)
done

lemma lets_nok:
  "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>
   (Lt5 (Lcons x (Ap5 (Vr5 z) (Vr5 z)) (Lcons y (Vr5 z) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) \<noteq>
   (Lt5 (Lcons y (Vr5 z) (Lcons x (Ap5 (Vr5 z) (Vr5 z)) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"
apply (simp only: alpha5_INJ(3) alpha5_INJ(5) alpha_gen permute_trm5_lts fresh_star_def)
apply (simp add: alpha5_DIS alpha5_INJ permute_trm5_lts)
done

end