theory Term4
imports "../Abs" "../Perm" "../Fv" "../Rsp" "../Lift" "Quotient_List"
begin

atom_decl name

section {*** lam with indirect list recursion ***}

datatype rtrm4 =
  rVr4 "name"
| rAp4 "rtrm4" "rtrm4 list"
| rLm4 "name" "rtrm4"  --"bind (name) in (trm)"
print_theorems

thm rtrm4.recs

(* there cannot be a clause for lists, as *)
(* permutations are  already defined in Nominal (also functions, options, and so on) *)
setup {* snd o define_raw_perms (Datatype.the_info @{theory} "Term4.rtrm4") 1 *}
print_theorems

(* "repairing" of the permute function *)
lemma repaired:
  fixes ts::"rtrm4 list"
  shows "permute_rtrm4_list p ts = p \<bullet> ts"
  apply(induct ts)
  apply(simp_all)
  done

thm permute_rtrm4_permute_rtrm4_list.simps
lemmas rawperm=permute_rtrm4_permute_rtrm4_list.simps[simplified repaired]


local_setup {* snd o define_fv_alpha_export (Datatype.the_info @{theory} "Term4.rtrm4")
  [[[], [], [(NONE, 0, 1, AlphaGen)]], [[], []] ] [] *}
print_theorems

lemma fix2: "alpha_rtrm4_list = list_rel alpha_rtrm4"
apply (rule ext)+
apply (induct_tac x xa rule: list_induct2')
apply (simp_all add: alpha_rtrm4_alpha_rtrm4_list.intros)
apply clarify apply (erule alpha_rtrm4_list.cases) apply(simp_all)
apply clarify apply (erule alpha_rtrm4_list.cases) apply(simp_all)
apply rule
apply (erule alpha_rtrm4_list.cases)
apply simp_all
apply (rule alpha_rtrm4_alpha_rtrm4_list.intros)
apply simp_all
done

(* We need sth like:
lemma fix3: "fv_rtrm4_list = set o map fv_rtrm4" *)

notation
  alpha_rtrm4 ("_ \<approx>4 _" [100, 100] 100) and
  alpha_rtrm4_list ("_ \<approx>4l _" [100, 100] 100)
thm alpha_rtrm4_alpha_rtrm4_list.intros[simplified fix2]

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha4_inj}, []), (build_rel_inj @{thms alpha_rtrm4_alpha_rtrm4_list.intros[simplified fix2]} @{thms rtrm4.distinct rtrm4.inject list.distinct list.inject} @{thms alpha_rtrm4.cases[simplified fix2] alpha_rtrm4_list.cases[simplified fix2]} ctxt)) ctxt)) *}
thm alpha4_inj

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha4_inj_no}, []), (build_rel_inj @{thms alpha_rtrm4_alpha_rtrm4_list.intros} @{thms rtrm4.distinct rtrm4.inject list.distinct list.inject} @{thms alpha_rtrm4.cases alpha_rtrm4_list.cases} ctxt)) ctxt)) *}
thm alpha4_inj_no

local_setup {* snd o (prove_eqvt [@{typ rtrm4},@{typ "rtrm4 list"}] @{thm rtrm4.induct} @{thms permute_rtrm4_permute_rtrm4_list.simps[simplified repaired] fv_rtrm4_fv_rtrm4_list.simps} [@{term fv_rtrm4}, @{term fv_rtrm4_list}]) *}
thm eqvts(1-2)

local_setup {*
(fn ctxt => snd (Local_Theory.note ((@{binding alpha4_eqvt_no}, []),
  build_alpha_eqvts [@{term alpha_rtrm4}, @{term alpha_rtrm4_list}] (fn _ => alpha_eqvt_tac @{thm alpha_rtrm4_alpha_rtrm4_list.induct} @{thms permute_rtrm4_permute_rtrm4_list.simps[simplified repaired] alpha4_inj_no} ctxt 1) ctxt) ctxt))
*}
lemmas alpha4_eqvt = alpha4_eqvt_no[simplified fix2]

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha4_reflp}, []),
  build_alpha_refl [((0, @{term alpha_rtrm4}), 0), ((0, @{term alpha_rtrm4_list}), 0)] [@{term alpha_rtrm4}, @{term alpha_rtrm4_list}] @{thm rtrm4.induct} @{thms alpha4_inj_no} ctxt) ctxt)) *}
thm alpha4_reflp
ML build_equivps

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha4_equivp_no}, []),
  (build_equivps [@{term alpha_rtrm4}, @{term alpha_rtrm4_list}] @{thms alpha4_reflp} @{thm alpha_rtrm4_alpha_rtrm4_list.induct} @{thms rtrm4.inject list.inject} @{thms alpha4_inj_no} @{thms rtrm4.distinct list.distinct} @{thms alpha_rtrm4_list.cases alpha_rtrm4.cases} @{thms alpha4_eqvt_no} ctxt)) ctxt)) *}
lemmas alpha4_equivp = alpha4_equivp_no[simplified fix2]

quotient_type 
  trm4 = rtrm4 / alpha_rtrm4
(*and
  trm4list = "rtrm4 list" / alpha_rtrm4_list*)
  by (simp_all add: alpha4_equivp)

local_setup {*
(fn ctxt => ctxt
 |> snd o (Quotient_Def.quotient_lift_const [] ("Vr4", @{term rVr4}))
 |> snd o (Quotient_Def.quotient_lift_const [@{typ "trm4"}] ("Ap4", @{term rAp4}))
 |> snd o (Quotient_Def.quotient_lift_const [] ("Lm4", @{term rLm4}))
 |> snd o (Quotient_Def.quotient_lift_const [] ("fv_trm4", @{term fv_rtrm4})))
*}
print_theorems



lemma fv_rtrm4_rsp:
  "xa \<approx>4 ya \<Longrightarrow> fv_rtrm4 xa = fv_rtrm4 ya"
  "x \<approx>4l y \<Longrightarrow> fv_rtrm4_list x = fv_rtrm4_list y"
  apply (induct rule: alpha_rtrm4_alpha_rtrm4_list.inducts)
  apply (simp_all add: alpha_gen)
done

local_setup {* snd o prove_const_rsp [] @{binding fv_rtrm4_rsp'} [@{term fv_rtrm4}]
  (fn _ => asm_full_simp_tac (@{simpset} addsimps @{thms fv_rtrm4_rsp}) 1) *}
print_theorems

local_setup {* snd o prove_const_rsp [] @{binding rVr4_rsp} [@{term rVr4}]
  (fn _ => constr_rsp_tac @{thms alpha4_inj} @{thms fv_rtrm4_rsp alpha4_equivp} 1) *}
local_setup {* snd o prove_const_rsp [] @{binding rLm4_rsp} [@{term rLm4}]
  (fn _ => constr_rsp_tac @{thms alpha4_inj} @{thms fv_rtrm4_rsp alpha4_equivp} 1) *}

lemma [quot_respect]:
  "(alpha_rtrm4 ===> list_rel alpha_rtrm4 ===> alpha_rtrm4) rAp4 rAp4"
  by (simp add: alpha4_inj)

(* Maybe also need: @{term "permute :: perm \<Rightarrow> rtrm4 list \<Rightarrow> rtrm4 list"} *)
local_setup {* snd o prove_const_rsp [] @{binding permute_rtrm4_rsp}
  [@{term "permute :: perm \<Rightarrow> rtrm4 \<Rightarrow> rtrm4"}]
  (fn _ => asm_simp_tac (HOL_ss addsimps @{thms alpha4_eqvt}) 1) *}
print_theorems

setup {* define_lifted_perms [@{typ trm4}] ["Term4.trm4"] [("permute_trm4", @{term "permute :: perm \<Rightarrow> rtrm4 \<Rightarrow> rtrm4"})] @{thms permute_rtrm4_permute_rtrm4_list_zero permute_rtrm4_permute_rtrm4_list_append} *}
print_theorems



lemma bla: "(Ap4 trm4 list = Ap4 trm4a lista) =
  (trm4 = trm4a \<and> list_rel (op =) list lista)"
  by (lifting alpha4_inj(2))

thm bla[simplified list_rel_eq]

ML {* lift_thm [@{typ trm4}] @{context} @{thm alpha4_inj(1)} *}
ML {* lift_thm [@{typ trm4}] @{context} @{thm alpha4_inj(2)} *}
ML {* lift_thm [@{typ trm4}] @{context} @{thm alpha4_inj(3)[unfolded alpha_gen]} *}
ML {* lift_thm [@{typ trm4}] @{context} @{thm rtrm4.induct} *}
.

(*lemmas trm1_bp_induct = rtrm4.induct[quot_lifted]*)

end