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_a = atomize_thm t_u *}

ML {* val t_r = regularize t_a rty rel rel_eqv @{context} *}