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} *}


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
*}

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












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

(* Not sure if it make sense or if it will be needed *)
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 qty "lam" rsp_thms defs t *}

thm a3
ML {* Toplevel.program (fn () => lift_thm_lam @{context} @{thm a3}) *}
thm a3
ML {* val t_u1 = eqsubst_thm @{context} @{thms abs_fresh(1)} (atomize_thm @{thm a3}) *}
ML {* val t_u = MetaSimplifier.rewrite_rule @{thms fresh_def supp_def} t_u1 *}

(* T_U *)

ML {* val t_a = atomize_thm @{thm rfv_var} *}
ML {* val t_r = regularize t_a rty rel rel_eqv @{context} *}
ML {* val t_t = repabs @{context} t_r consts rty qty quot rel_refl trans2 rsp_thms *}

ML {* fun r_mk_comb_tac_lam ctxt =
  r_mk_comb_tac ctxt rty quot rel_refl trans2 rsp_thms
*}

instance lam :: fs_name
apply(intro_classes)
sorry

prove asdf: {* Logic.mk_implies (concl_of t_r, (@{term "Trueprop (\<forall>t\<Colon>rlam\<in>Respects
          alpha.
         \<forall>(a\<Colon>name) b\<Colon>name.
            \<forall>s\<Colon>rlam\<in>Respects
                alpha.
               t \<approx> ([(a,
                 b)] \<bullet> s) \<longrightarrow>
               a = b \<or>
               a
               \<notin> {a\<Colon>name.
                infinite
                 {b\<Colon>name. Not
                  (([(a, b)] \<bullet>
                  s) \<approx>
                  s)}} \<longrightarrow>
               rLam a
                t \<approx> rLam
                 b s)"})) *}
apply (tactic {* full_simp_tac ((Simplifier.context @{context} HOL_ss) addsimps
        [(@{thm equiv_res_forall} OF [rel_eqv]),
         (@{thm equiv_res_exists} OF [rel_eqv])]) 1 *})
apply (rule allI)
apply (drule_tac x="t" in spec)
apply (rule allI)
apply (drule_tac x="a" in spec)
apply (rule allI)
apply (drule_tac x="b" in spec)
apply (rule allI)
apply (drule_tac x="s" in spec)
apply (rule impI)
apply (drule_tac mp)
apply (simp)
apply (simp)
apply (rule impI)
apply (rule a3)
apply (simp)
apply (simp add: abs_fresh(1))
apply (case_tac "a = b")
apply (simp)
apply (simp)
apply (auto)
apply (unfold fresh_def)
apply (unfold supp_def)
apply (simp)
prefer 2
apply (simp)
sorry

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_defs_sym = add_lower_defs @{context} defs *}

ML {* val t_r' = @{thm asdf} OF [t_r] *}
ML {* val t_t = repabs @{context} t_r' consts rty qty quot rel_refl trans2 rsp_thms *}
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 t_d = repeat_eqsubst_thm @{context} t_defs_sym t_a *}
ML {* val t_r = MetaSimplifier.rewrite_rule [reps_same] t_d *}
ML {* val tt = MetaSimplifier.rewrite_rule [symmetric @{thm supp_def}, symmetric @{thm fresh_def}] t_r *}
ML {* val rr = @{thm sym} OF @{thms abs_fresh(1)} *}
ML {* val ttt = eqsubst_thm @{context} [rr] tt *}
ML {* ObjectLogic.rulify ttt *}

lemma
  assumes a: "a \<notin> {a\<Colon>name. infinite {b\<Colon>name. \<not> ([(a, b)] \<bullet> s) \<approx> s}}"
  shows "a \<notin> {a\<Colon>name. infinite {b\<Colon>name. [(a, b)] \<bullet> s \<noteq> s}}"
  using a apply simp
  sorry (* Not true... *)