theory LamEx
imports Nominal QuotMain
begin

atom_decl name

thm abs_fresh(1)

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


function
  rfv :: "rlam \<Rightarrow> name set"
where
  rfv_var: "rfv (rVar a) = {a}"
| rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"
| rfv_lam: "rfv (rLam a t) = (rfv t) - {a}"
sorry

termination rfv sorry

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 \<notin> rfv (rLam b t)\<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

quotient_def (for lam)
  fv :: "lam \<Rightarrow> name set"
where
  "fv \<equiv> rfv"

thm fv_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

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


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 fv_var:
  shows "fv (Var a) = {a}"
sorry

lemma fv_app:
  shows "fv (App t1 t2) = (fv t1) \<union> (fv t2)"
sorry

lemma fv_lam:
  shows "fv (Lam a t) = (fv t) - {a}"
sorry

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

lemma rfv_rsp: "(alpha ===> op =) rfv rfv"
  sorry

ML {* val qty = @{typ "lam"} *}
ML {* val (rty, rel, rel_refl, rel_eqv) = lookup_quot_data @{context} qty *}
ML {* val defs = @{thms Var_def App_def Lam_def perm_lam_def fv_def} *}
ML {* val consts = lookup_quot_consts defs *}
ML {* val (trans2, reps_same, quot) = lookup_quot_thms @{context} "lam" *}
ML {* val rsp_thms = @{thms perm_rsp fresh_rsp rVar_rsp rApp_rsp rLam_rsp rfv_rsp} @ @{thms ho_all_prs ho_ex_prs} *}
ML {* fun lift_thm_lam lthy t = lift_thm lthy qty "lam" rsp_thms defs t *}

ML {* val pi_var = lift_thm_lam @{context} @{thm pi_var_com} *}
ML {* val pi_app = lift_thm_lam @{context} @{thm pi_app_com} *}
ML {* val pi_lam = lift_thm_lam @{context} @{thm pi_lam_com} *}

ML {* val fv_var = lift_thm_lam @{context} @{thm rfv_var} *}
ML {* val fv_app = lift_thm_lam @{context} @{thm rfv_app} *}
ML {* val fv_lam = lift_thm_lam @{context} @{thm rfv_lam} *}

ML {* val a1 = lift_thm_lam @{context} @{thm a1} *}
ML {* val a2 = lift_thm_lam @{context} @{thm a1} *}
ML {* val a3 = lift_thm_lam @{context} @{thm a3} *}

ML {* val alpha_cases = lift_thm_lam @{context} @{thm alpha.cases} *}

local_setup {*
  Quotient.note (@{binding "pi_var"}, pi_var) #> snd #>
  Quotient.note (@{binding "pi_app"}, pi_app) #> snd #>
  Quotient.note (@{binding "pi_lam"}, pi_lam) #> snd #>
  Quotient.note (@{binding "a1"}, a1) #> snd #>
  Quotient.note (@{binding "a2"}, a2) #> snd #>
  Quotient.note (@{binding "a3"}, a3) #> snd #>
  Quotient.note (@{binding "alpha_cases"}, alpha_cases) #> snd
*}

thm alpha.cases
thm alpha_cases

thm rlam.inject
thm pi_var

lemma var_inject:
  shows "(Var a = Var b) = (a = b)"
sorry

lemma var_supp:
  shows "supp (Var a) = ((supp a)::name set)"
  apply(simp add: supp_def)
  apply(simp add: pi_var)
  apply(simp add: var_inject)
  done

lemma var_fresh:
  fixes a::"name"
  shows "(a\<sharp>(Var b)) = (a\<sharp>b)"
  apply(simp add: fresh_def)
  apply(simp add: var_supp)
  done







ML {* val t_a = atomize_thm @{thm alpha.cases} *}
ML {* val t_r = regularize t_a rty rel rel_eqv rel_refl @{context} *}
ML {* val t_t = repabs @{context} t_r consts rty qty quot rel_refl trans2 rsp_thms *}
ML {* val abs = findabs rty (prop_of t_a); *}
ML {* val simp_lam_prs_thms = map (make_simp_lam_prs_thm @{context} quot) abs *}
ML {* val t_l = repeat_eqsubst_thm @{context} simp_lam_prs_thms t_t *}
ML {* val t_a = simp_allex_prs @{context} quot t_l *}
ML {* val defs_sym = add_lower_defs @{context} defs; *}
ML {* val t_d = repeat_eqsubst_thm @{context} defs_sym t_a *}
ML {* val t_r = MetaSimplifier.rewrite_rule [reps_same] t_d *}
ML {* ObjectLogic.rulify t_r *}



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