theory LamEx
imports Nominal QuotMain
begin

atom_decl name

nominal_datatype rlam =
  rVar "name"
| rApp "rlam" "rlam"
| rLam "name" "rlam"

inductive 
  alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)
where
  a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"
| a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"
| a3: "\<lbrakk>t \<approx> ([(a,b)]\<bullet>s); a\<sharp>[b].s\<rbrakk> \<Longrightarrow> rLam a t \<approx> rLam b s"

quotient lam = rlam / alpha
sorry

print_quotients

quotient_def (for lam)
  Var :: "name \<Rightarrow> lam"
where
  "Var \<equiv> rVar"

quotient_def (for lam)
  App :: "lam \<Rightarrow> lam \<Rightarrow> lam"
where
  "App \<equiv> rApp"

quotient_def (for lam)
  Lam :: "name \<Rightarrow> lam \<Rightarrow> lam"
where
  "Lam \<equiv> rLam"

thm Var_def
thm App_def
thm Lam_def

(* definition of overloaded permutation function *)
(* for the lifted type lam                       *)
overloading
  perm_lam    \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"   (unchecked)
begin

quotient_def (for lam)
  perm_lam :: "'x prm \<Rightarrow> lam \<Rightarrow> lam"
where
  "perm_lam \<equiv> (perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam)"

end

thm perm_lam_def

(* lemmas that need to lift *)
lemma pi_var_com:
  fixes pi::"'x prm"
  shows "(pi\<bullet>rVar a) \<approx> rVar (pi\<bullet>a)"
  sorry

lemma pi_app_com:
  fixes pi::"'x prm"
  shows "(pi\<bullet>rApp t1 t2) \<approx> rApp (pi\<bullet>t1) (pi\<bullet>t2)"
  sorry

lemma pi_lam_com:
  fixes pi::"'x prm"
  shows "(pi\<bullet>rLam a t) \<approx> rLam (pi\<bullet>a) (pi\<bullet>t)"
  sorry

lemma real_alpha:
  assumes "t = ([(a,b)]\<bullet>s)" "a\<sharp>s"
  shows "Lam a t = Lam b s"
sorry





(* Construction Site code *)

lemma perm_rsp: "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"
  apply(auto)
  (* this is propably true if some type conditions are imposed ;o) *)
  sorry

lemma fresh_rsp: "(op = ===> alpha ===> op =) fresh fresh" 
  apply(auto)
  (* this is probably only true if some type conditions are imposed *)
  sorry

lemma rVar_rsp: "(op = ===> alpha) rVar rVar"
  apply(auto)
  apply(rule a1)
  apply(simp)
  done

lemma rApp_rsp: "(alpha ===> alpha ===> alpha) rApp rApp"
  apply(auto)
  apply(rule a2)
  apply (assumption)
  apply (assumption)
  done

lemma rLam_rsp: "(op = ===> alpha ===> alpha) rLam rLam"
  apply(auto)
  apply(rule a3)
  apply(rule_tac t="[(x,x)]\<bullet>y" and s="y" in subst)
  apply(rule sym)
  apply(rule trans)
  apply(rule pt_name3)
  apply(rule at_ds1[OF at_name_inst])
  apply(simp add: pt_name1)
  apply(assumption)
  apply(simp add: abs_fresh)
  done

ML {* val defs = @{thms Var_def App_def Lam_def perm_lam_def} *}
ML {* val consts = [@{const_name "rVar"}, @{const_name "rApp"}, @{const_name "rLam"}, @{const_name "perm"}]; *}

ML {* val rty = @{typ "rlam"} *}
ML {* val qty = @{typ "lam"} *}
ML {* val rel = @{term "alpha"} *}
ML {* val rel_eqv = (#equiv_thm o hd) (quotdata_lookup @{context}) *}
ML {* val rel_refl = @{thm EQUIV_REFL} OF [rel_eqv] *}
ML {* val quot = @{thm QUOTIENT_lam} *}
ML {* val rsp_thms = @{thms perm_rsp fresh_rsp rVar_rsp rApp_rsp rLam_rsp} @ @{thms ho_all_prs ho_ex_prs} *}
ML {* val trans2 = @{thm QUOT_TYPE_I_lam.R_trans2} *}
ML {* val reps_same = @{thm QUOT_TYPE_I_lam.REPS_same} *}

ML {* add_lower_defs @{context} @{thms perm_lam_def} *}
ML {* val rr = @{thm eq_reflection} OF [hd (rev (add_lower_defs @{context} @{thms perm_lam_def}))] *}

ML {* val rrr = repeat_eqsubst_thm @{context} @{thms prod_fun_id map_id} rr *}
ML {*
fun lift_thm_lam lthy t =
  lift_thm lthy consts rty qty rel rel_eqv rel_refl quot rsp_thms trans2 reps_same defs t
*}

ML {* lift_thm_lam @{context} @{thm pi_var_com} *}
ML {* lift_thm_lam @{context} @{thm pi_app_com} *}
ML {* lift_thm_lam @{context} @{thm pi_lam_com} *}

fun
  option_map::"('a \<Rightarrow> 'b) \<Rightarrow> ('a noption) \<Rightarrow> ('b noption)"
where
  "option_map f (nSome x) = nSome (f x)"
| "option_map f nNone = nNone"

fun
  option_rel
where
  "option_rel r (nSome x) (nSome y) = r x y"
| "option_rel r _ _ = False"

declare [[map noption = (option_map, option_rel)]]

lemma OPT_QUOTIENT:
  assumes q: "QUOTIENT R Abs Rep"
  shows "QUOTIENT (option_rel R) (option_map Abs) (option_map Rep)"
  apply (unfold QUOTIENT_def)
  apply (auto)
  using q
  apply (unfold QUOTIENT_def)
  apply (case_tac "a :: 'b noption")
  apply (simp)
  apply (simp)
  apply (case_tac "a :: 'b noption")
  apply (simp only: option_map.simps)
  apply (subst option_rel.simps)
  (* Simp starts hanging so don't know how to continue *)
  sorry

(* Christian: Does it make sense? *)
lemma abs_fun_rsp: "(op = ===> alpha ===> op = ===> op =) abs_fun abs_fun"
sorry

(* Should not be needed *)
lemma eq_rsp2: "((op = ===> op =) ===> (op = ===> op =) ===> op =) op = op ="
apply auto
apply (rule ext)
apply auto
apply (rule ext)
apply auto
done

(* Should not be needed *)
lemma perm_rsp_eq: "(op = ===> (op = ===> op =) ===> op = ===> op =) op \<bullet> op \<bullet>"
apply auto
thm arg_cong2
apply (rule_tac f="perm x" in arg_cong2)
apply (auto)
apply (rule ext)
apply (auto)
done

(* Should not be needed *)
lemma fresh_rsp_eq: "(op = ===> (op = ===> op =) ===> op =) fresh fresh"
apply (simp add: FUN_REL.simps)
apply (metis ext)
done

(* It is just a test, it doesn't seem true... *)
lemma quotient_cheat: "QUOTIENT op = (option_map ABS_lam) (option_map REP_lam)"
sorry


ML {* val rsp_thms = @{thms abs_fun_rsp OPT_QUOTIENT eq_rsp2 quotient_cheat perm_rsp_eq fresh_rsp_eq} @ rsp_thms *}
ML {*
fun lift_thm_lam lthy t =
  lift_thm lthy consts rty qty rel rel_eqv rel_refl quot rsp_thms trans2 reps_same defs t
*}

thm a3
ML {* Toplevel.program (fn () => lift_thm_lam @{context} @{thm a3}) *}
ML {* val t_u = MetaSimplifier.rewrite_rule @{thms fresh_def supp_def} @{thm a3} *}
ML {* Toplevel.program (fn () => lift_thm_lam @{context} t_u) *}

ML t_u
ML {* val t_a = atomize_thm t_u *}
ML {* val t_r = regularize t_a rty rel rel_eqv @{context} *}
ML {* val t = (fst o Thm.dest_comb o snd o Thm.dest_comb) (cprop_of t_u) *}
ML {* val t = (snd o Thm.dest_comb o snd o Thm.dest_comb) t *}
ML {* val t = (snd o Thm.dest_comb o snd o Thm.dest_comb) t *}
ML {* val t = (snd o (Thm.dest_abs NONE) o snd o Thm.dest_comb) t *}
ML {* val t = (snd o Thm.dest_comb o snd o Thm.dest_comb) t *}
ML {* val t = (snd o (Thm.dest_abs NONE) o snd o Thm.dest_comb) t *}
ML {* val t = (fst o Thm.dest_comb o snd o Thm.dest_comb) t *}
ML {* term_of t *}
term "[b].(s::rlam)"
thm abs_fun_def
ML {* val t_t = repabs @{context} t_r consts rty qty quot rel_refl trans2 rsp_thms *}