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

apply -

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

  fixes pi::"'x prm"

  shows "(pi\<bullet>Var a) = Var (pi\<bullet>a)"

  sorry

lemma

  fixes pi::"'x prm"

  shows "(pi\<bullet>App t1 t2) = App (pi\<bullet>t1) (pi\<bullet>t2)"

  sorry

lemma

  fixes pi::"'x prm"

  shows "(pi\<bullet>Lam a t) = Lam (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 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} *}

ML {* val consts = [@{const_name "rVar"}, @{const_name "rApp"}, @{const_name "rLam"}]; *}

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 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} *}

thm a3

ML {* val t_a = atomize_thm @{thm a3} *}

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 {* 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_c = simp_allex_prs @{context} quot t_l *}

ML {* val t_defs_sym = add_lower_defs @{context} defs *}

ML {* val t_d = repeat_eqsubst_thm @{context} t_defs_sym t_c *}

ML {* val t_b = MetaSimplifier.rewrite_rule [reps_same] t_d *}

ML {* ObjectLogic.rulify t_b *}

thm fresh_def

thm supp_def

local_setup {*

  old_make_const_def @{binding lperm} @{term "perm :: ('a \<times> 'a) list \<Rightarrow> rlam \<Rightarrow> rlam"} NoSyn @{typ "rlam"} @{typ "lam"} #> snd

*}

ML {* val consts = @{const_name perm} :: consts *}

ML {* val defs = @{thms lperm_def} @ defs *}

ML {* val t_u = MetaSimplifier.rewrite_rule @{thms fresh_def supp_def} @{thm a3} *}

ML {* val t_a = atomize_thm @{thm a3} *}

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 {* val t_l = repeat_eqsubst_thm @{context} simp_lam_prs_thms t_t *}

ML {* val t_c = simp_allex_prs @{context} quot t_l *}

ML {* val t_defs_sym = add_lower_defs @{context} defs *}

ML {* val t_d = repeat_eqsubst_thm @{context} t_defs_sym t_c *}

ML {* val t_b = MetaSimplifier.rewrite_rule [reps_same] t_d *}

ML {* val rr =  (add_lower_defs @{context} @{thms lperm_def}) *}

ML {* val rrr = @{thm eq_reflection} OF [hd (rev rr)] *}

lemma prod_fun_id: "prod_fun id id = id"

  apply (simp add: prod_fun_def)

done

lemma map_id: "map id x = x"

  apply (induct x)

  apply (simp_all add: map.simps)

done

ML {* val rrrr = repeat_eqsubst_thm @{context} @{thms prod_fun_id map_id} rrr *}

ML {* val t_b' = eqsubst_thm @{context} [rrrr] t_b *}

ML {* ObjectLogic.rulify t_b' *}

local_setup {*

  make_const_def @{binding lfresh} @{term "fresh :: 'a \<Rightarrow> rlam \<Rightarrow> bool"} NoSyn @{typ "rlam"} @{typ "lam"} #> snd #>

*}

@{const_name fresh} :: 

lfresh_def 

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 {* Toplevel.program (fn () => lift_thm_lam @{context} @{thm a3}) *}