theory NewParser
imports "../Nominal-General/Nominal2_Base"
"../Nominal-General/Nominal2_Eqvt"
"../Nominal-General/Nominal2_Supp"
"Perm" "Tacs" "Equivp"
begin
section{* Interface for nominal_datatype *}
ML {*
(* attributes *)
val eqvt_attrib = Attrib.internal (K Nominal_ThmDecls.eqvt_add)
val rsp_attrib = Attrib.internal (K Quotient_Info.rsp_rules_add)
*}
ML {* print_depth 50 *}
ML {*
fun get_cnstrs dts =
map (fn (_, _, _, constrs) => constrs) dts
fun get_typed_cnstrs dts =
flat (map (fn (_, bn, _, constrs) =>
(map (fn (bn', _, _) => (Binding.name_of bn, Binding.name_of bn')) constrs)) dts)
fun get_cnstr_strs dts =
map (fn (bn, _, _) => Binding.name_of bn) (flat (get_cnstrs dts))
fun get_bn_fun_strs bn_funs =
map (fn (bn_fun, _, _) => Binding.name_of bn_fun) bn_funs
*}
text {* Infrastructure for adding "_raw" to types and terms *}
ML {*
fun add_raw s = s ^ "_raw"
fun add_raws ss = map add_raw ss
fun raw_bind bn = Binding.suffix_name "_raw" bn
fun replace_str ss s =
case (AList.lookup (op=) ss s) of
SOME s' => s'
| NONE => s
fun replace_typ ty_ss (Type (a, Ts)) = Type (replace_str ty_ss a, map (replace_typ ty_ss) Ts)
| replace_typ ty_ss T = T
fun raw_dts ty_ss dts =
let
fun raw_dts_aux1 (bind, tys, mx) =
(raw_bind bind, map (replace_typ ty_ss) tys, mx)
fun raw_dts_aux2 (ty_args, bind, mx, constrs) =
(ty_args, raw_bind bind, mx, map raw_dts_aux1 constrs)
in
map raw_dts_aux2 dts
end
fun replace_aterm trm_ss (Const (a, T)) = Const (replace_str trm_ss a, T)
| replace_aterm trm_ss (Free (a, T)) = Free (replace_str trm_ss a, T)
| replace_aterm trm_ss trm = trm
fun replace_term trm_ss ty_ss trm =
trm |> Term.map_aterms (replace_aterm trm_ss) |> map_types (replace_typ ty_ss)
*}
ML {*
fun rawify_dts dt_names dts dts_env =
let
val raw_dts = raw_dts dts_env dts
val raw_dt_names = add_raws dt_names
in
(raw_dt_names, raw_dts)
end
*}
ML {*
fun rawify_bn_funs dts_env cnstrs_env bn_fun_env bn_funs bn_eqs =
let
val bn_funs' = map (fn (bn, ty, mx) =>
(raw_bind bn, SOME (replace_typ dts_env ty), mx)) bn_funs
val bn_eqs' = map (fn (attr, trm) =>
(attr, replace_term (cnstrs_env @ bn_fun_env) dts_env trm)) bn_eqs
in
(bn_funs', bn_eqs')
end
*}
ML {*
fun rawify_bclauses dts_env cnstrs_env bn_fun_env bclauses =
let
fun rawify_bnds bnds =
map (apfst (Option.map (replace_term (cnstrs_env @ bn_fun_env) dts_env))) bnds
fun rawify_bclause (BC (mode, bnds, bdys)) = BC (mode, rawify_bnds bnds, bdys)
in
map (map (map rawify_bclause)) bclauses
end
*}
(* strip_bn_fun takes a rhs of a bn function: this can only contain unions or
appends of elements; in case of recursive calls it retruns also the applied
bn function *)
ML {*
fun strip_bn_fun lthy args t =
let
fun aux t =
case t of
Const (@{const_name sup}, _) $ l $ r => aux l @ aux r
| Const (@{const_name append}, _) $ l $ r => aux l @ aux r
| Const (@{const_name insert}, _) $ (Const (@{const_name atom}, _) $ (x as Var _)) $ y =>
(find_index (equal x) args, NONE) :: aux y
| Const (@{const_name Cons}, _) $ (Const (@{const_name atom}, _) $ (x as Var _)) $ y =>
(find_index (equal x) args, NONE) :: aux y
| Const (@{const_name bot}, _) => []
| Const (@{const_name Nil}, _) => []
| (f as Const _) $ (x as Var _) => [(find_index (equal x) args, SOME f)]
| _ => error ("Unsupported binding function: " ^ (Syntax.string_of_term lthy t))
in
aux t
end
*}
ML {*
(** definition of the raw binding functions **)
(* TODO: needs cleaning *)
fun find [] _ = error ("cannot find element")
| find ((x, z)::xs) y = if (Long_Name.base_name x) = y then z else find xs y
fun prep_bn_info lthy dt_names dts eqs =
let
fun aux eq =
let
val (lhs, rhs) = eq
|> HOLogic.dest_Trueprop
|> HOLogic.dest_eq
val (bn_fun, [cnstr]) = strip_comb lhs
val (_, ty) = dest_Const bn_fun
val (ty_name, _) = dest_Type (domain_type ty)
val dt_index = find_index (fn x => x = ty_name) dt_names
val (cnstr_head, cnstr_args) = strip_comb cnstr
val rhs_elements = strip_bn_fun lthy cnstr_args rhs
in
(dt_index, (bn_fun, (cnstr_head, rhs_elements)))
end
fun order dts i ts =
let
val dt = nth dts i
val cts = map (fn (x, _, _) => Binding.name_of x) ((fn (_, _, _, x) => x) dt)
val ts' = map (fn (x, y) => (fst (dest_Const x), y)) ts
in
map (find ts') cts
end
val unordered = AList.group (op=) (map aux eqs)
val unordered' = map (fn (x, y) => (x, AList.group (op=) y)) unordered
val ordered = map (fn (x, y) => (x, map (fn (v, z) => (v, order dts x z)) y)) unordered'
val ordered' = flat (map (fn (ith, l) => map (fn (bn, data) => (bn, ith, data)) l) ordered)
(*val _ = tracing ("eqs\n" ^ cat_lines (map (Syntax.string_of_term lthy) eqs))*)
(*val _ = tracing ("map eqs\n" ^ @{make_string} (map aux2 eqs))*)
(*val _ = tracing ("ordered'\n" ^ @{make_string} ordered')*)
in
ordered'
end
fun define_raw_bns dt_names dts raw_bn_funs raw_bn_eqs constr_thms size_thms lthy =
if null raw_bn_funs
then ([], [], [], [], lthy)
else
let
val (_, lthy1) = Function.add_function raw_bn_funs raw_bn_eqs
Function_Common.default_config (pat_completeness_simp constr_thms) lthy
val (info, lthy2) = prove_termination size_thms (Local_Theory.restore lthy1)
val {fs, simps, inducts, ...} = info
val raw_bn_induct = (the inducts)
val raw_bn_eqs = the simps
val raw_bn_info =
prep_bn_info lthy dt_names dts (map prop_of raw_bn_eqs)
in
(fs, raw_bn_eqs, raw_bn_info, raw_bn_induct, lthy2)
end
*}
ML {*
fun define_raw_dts dts bn_funs bn_eqs binds lthy =
let
val thy = Local_Theory.exit_global lthy
val thy_name = Context.theory_name thy
val dt_names = map (fn (_, s, _, _) => Binding.name_of s) dts
val dt_full_names = map (Long_Name.qualify thy_name) dt_names
val dt_full_names' = add_raws dt_full_names
val dts_env = dt_full_names ~~ dt_full_names'
val cnstrs = get_cnstr_strs dts
val cnstrs_ty = get_typed_cnstrs dts
val cnstrs_full_names = map (Long_Name.qualify thy_name) cnstrs
val cnstrs_full_names' = map (fn (x, y) => Long_Name.qualify thy_name
(Long_Name.qualify (add_raw x) (add_raw y))) cnstrs_ty
val cnstrs_env = cnstrs_full_names ~~ cnstrs_full_names'
val bn_fun_strs = get_bn_fun_strs bn_funs
val bn_fun_strs' = add_raws bn_fun_strs
val bn_fun_env = bn_fun_strs ~~ bn_fun_strs'
val bn_fun_full_env = map (pairself (Long_Name.qualify thy_name))
(bn_fun_strs ~~ bn_fun_strs')
val (raw_dt_names, raw_dts) = rawify_dts dt_names dts dts_env
val (raw_bn_funs, raw_bn_eqs) = rawify_bn_funs dts_env cnstrs_env bn_fun_env bn_funs bn_eqs
val raw_bclauses = rawify_bclauses dts_env cnstrs_env bn_fun_full_env binds
val (raw_dt_full_names, thy1) =
Datatype.add_datatype Datatype.default_config raw_dt_names raw_dts thy
val lthy1 = Named_Target.theory_init thy1
in
(raw_dt_full_names, raw_dts, raw_bclauses, raw_bn_funs, raw_bn_eqs, lthy1)
end
*}
ML {*
(* for testing porposes - to exit the procedure early *)
exception TEST of Proof.context
val (STEPS, STEPS_setup) = Attrib.config_int "STEPS" (K 0);
fun get_STEPS ctxt = Config.get ctxt STEPS
*}
setup STEPS_setup
ML {*
fun nominal_datatype2 thm_name dts bn_funs bn_eqs bclauses lthy =
let
(* definition of the raw datatypes *)
val _ = warning "Definition of raw datatypes";
val (raw_dt_names, raw_dts, raw_bclauses, raw_bn_funs, raw_bn_eqs, lthy0) =
if get_STEPS lthy > 0
then define_raw_dts dts bn_funs bn_eqs bclauses lthy
else raise TEST lthy
val dtinfo = Datatype.the_info (ProofContext.theory_of lthy0) (hd raw_dt_names)
val {descr, sorts, ...} = dtinfo
val raw_tys = all_dtyps descr sorts
val raw_full_ty_names = map (fst o dest_Type) raw_tys
val tvs = hd raw_tys
|> snd o dest_Type
|> map dest_TFree
val dtinfos = map (Datatype.the_info (ProofContext.theory_of lthy0)) raw_full_ty_names
val raw_cns_info = all_dtyp_constrs_types descr sorts
val raw_constrs = flat (map (map (fn (c, _, _, _) => c)) raw_cns_info)
val raw_inject_thms = flat (map #inject dtinfos)
val raw_distinct_thms = flat (map #distinct dtinfos)
val raw_induct_thm = #induct dtinfo
val raw_induct_thms = #inducts dtinfo
val raw_exhaust_thms = map #exhaust dtinfos
val raw_size_trms = map size_const raw_tys
val raw_size_thms = Size.size_thms (ProofContext.theory_of lthy0) (hd raw_dt_names)
|> `(fn thms => (length thms) div 2)
|> uncurry drop
(* definitions of raw permutations by primitive recursion *)
val _ = warning "Definition of raw permutations";
val ((raw_perm_funs, raw_perm_simps, raw_perm_laws), lthy2a) =
if get_STEPS lthy0 > 1
then define_raw_perms raw_full_ty_names raw_tys tvs raw_constrs raw_induct_thm lthy0
else raise TEST lthy0
(* noting the raw permutations as eqvt theorems *)
val (_, lthy3) = Local_Theory.note ((Binding.empty, [eqvt_attrib]), raw_perm_simps) lthy2a
(* definition of raw fv_functions *)
val _ = warning "Definition of raw fv-functions";
val (raw_bns, raw_bn_defs, raw_bn_info, raw_bn_induct, lthy3a) =
if get_STEPS lthy3 > 2
then define_raw_bns raw_full_ty_names raw_dts raw_bn_funs raw_bn_eqs
(raw_inject_thms @ raw_distinct_thms) raw_size_thms lthy3
else raise TEST lthy3
val (raw_fvs, raw_fv_bns, raw_fv_defs, raw_fv_bns_induct, lthy3b) =
if get_STEPS lthy3a > 3
then define_raw_fvs raw_full_ty_names raw_tys raw_cns_info raw_bn_info raw_bclauses
(raw_inject_thms @ raw_distinct_thms) raw_size_thms lthy3a
else raise TEST lthy3a
(* definition of raw alphas *)
val _ = warning "Definition of alphas";
val (alpha_trms, alpha_bn_trms, alpha_intros, alpha_cases, alpha_induct, lthy4) =
if get_STEPS lthy3b > 4
then define_raw_alpha raw_full_ty_names raw_tys raw_cns_info raw_bn_info raw_bclauses raw_fvs lthy3b
else raise TEST lthy3b
val alpha_tys = map (domain_type o fastype_of) alpha_trms
(* definition of alpha-distinct lemmas *)
val _ = warning "Distinct theorems";
val alpha_distincts =
mk_alpha_distincts lthy4 alpha_cases raw_distinct_thms alpha_trms raw_tys
(* definition of alpha_eq_iff lemmas *)
(* they have a funny shape for the simplifier *)
val _ = warning "Eq-iff theorems";
val (alpha_eq_iff_simps, alpha_eq_iff) =
if get_STEPS lthy > 5
then mk_alpha_eq_iff lthy4 alpha_intros raw_distinct_thms raw_inject_thms alpha_cases
else raise TEST lthy4
(* proving equivariance lemmas for bns, fvs, size and alpha *)
val _ = warning "Proving equivariance";
val raw_bn_eqvt =
if get_STEPS lthy > 6
then raw_prove_eqvt raw_bns raw_bn_induct (raw_bn_defs @ raw_perm_simps) lthy4
else raise TEST lthy4
(* noting the raw_bn_eqvt lemmas in a temprorary theory *)
val lthy_tmp = snd (Local_Theory.note ((Binding.empty, [eqvt_attrib]), raw_bn_eqvt) lthy4)
val raw_fv_eqvt =
if get_STEPS lthy > 7
then raw_prove_eqvt (raw_fvs @ raw_fv_bns) raw_fv_bns_induct (raw_fv_defs @ raw_perm_simps)
(Local_Theory.restore lthy_tmp)
else raise TEST lthy4
val raw_size_eqvt =
if get_STEPS lthy > 8
then raw_prove_eqvt raw_size_trms raw_induct_thms (raw_size_thms @ raw_perm_simps)
(Local_Theory.restore lthy_tmp)
|> map (rewrite_rule @{thms permute_nat_def[THEN eq_reflection]})
|> map (fn thm => thm RS @{thm sym})
else raise TEST lthy4
val lthy5 = snd (Local_Theory.note ((Binding.empty, [eqvt_attrib]), raw_fv_eqvt) lthy_tmp)
val (alpha_eqvt, lthy6) =
if get_STEPS lthy > 9
then Nominal_Eqvt.equivariance true (alpha_trms @ alpha_bn_trms) alpha_induct alpha_intros lthy5
else raise TEST lthy4
(* proving alpha equivalence *)
val _ = warning "Proving equivalence"
val alpha_refl_thms =
if get_STEPS lthy > 10
then raw_prove_refl alpha_trms alpha_bn_trms alpha_intros raw_induct_thm lthy6
else raise TEST lthy6
val alpha_sym_thms =
if get_STEPS lthy > 11
then raw_prove_sym (alpha_trms @ alpha_bn_trms) alpha_intros alpha_induct lthy6
else raise TEST lthy6
val alpha_trans_thms =
if get_STEPS lthy > 12
then raw_prove_trans (alpha_trms @ alpha_bn_trms) (raw_distinct_thms @ raw_inject_thms)
alpha_intros alpha_induct alpha_cases lthy6
else raise TEST lthy6
val (alpha_equivp_thms, alpha_bn_equivp_thms) =
if get_STEPS lthy > 13
then raw_prove_equivp alpha_trms alpha_bn_trms alpha_refl_thms alpha_sym_thms
alpha_trans_thms lthy6
else raise TEST lthy6
(* proving alpha implies alpha_bn *)
val _ = warning "Proving alpha implies bn"
val alpha_bn_imp_thms =
if get_STEPS lthy > 14
then raw_prove_bn_imp alpha_trms alpha_bn_trms alpha_intros alpha_induct lthy6
else raise TEST lthy6
(* respectfulness proofs *)
val raw_funs_rsp_aux = raw_fv_bn_rsp_aux alpha_trms alpha_bn_trms raw_fvs
raw_bns raw_fv_bns alpha_induct (raw_bn_defs @ raw_fv_defs) lthy6
val raw_funs_rsp = map mk_funs_rsp raw_funs_rsp_aux
val raw_size_rsp = raw_size_rsp_aux (alpha_trms @ alpha_bn_trms) alpha_induct
(raw_size_thms @ raw_size_eqvt) lthy6
|> map mk_funs_rsp
val raw_constrs_rsp = raw_constrs_rsp raw_constrs alpha_trms alpha_intros
(alpha_bn_imp_thms @ raw_funs_rsp_aux) lthy6
val alpha_permute_rsp = map mk_alpha_permute_rsp alpha_eqvt
val alpha_bn_rsp = raw_alpha_bn_rsp alpha_bn_equivp_thms alpha_bn_imp_thms
(* noting the quot_respects lemmas *)
val (_, lthy6a) =
if get_STEPS lthy > 15
then Local_Theory.note ((Binding.empty, [rsp_attrib]),
raw_constrs_rsp @ raw_funs_rsp @ raw_size_rsp @ alpha_permute_rsp @ alpha_bn_rsp) lthy6
else raise TEST lthy6
(* defining the quotient type *)
val _ = warning "Declaring the quotient types"
val qty_descr = map (fn (vs, bind, mx, _) => (vs, bind, mx)) dts
val (qty_infos, lthy7) =
if get_STEPS lthy > 16
then define_qtypes qty_descr alpha_tys alpha_trms alpha_equivp_thms lthy6a
else raise TEST lthy6a
val qtys = map #qtyp qty_infos
val qty_full_names = map (fst o dest_Type) qtys
val qty_names = map Long_Name.base_name qty_full_names
(* defining of quotient term-constructors, binding functions, free vars functions *)
val _ = warning "Defining the quotient constants"
val qconstrs_descr =
flat (map (fn (_, _, _, cs) => map (fn (b, _, mx) => (Name.of_binding b, mx)) cs) dts)
|> map2 (fn t => fn (b, mx) => (b, t, mx)) raw_constrs
val qbns_descr =
map2 (fn (b, _, mx) => fn t => (Name.of_binding b, t, mx)) bn_funs raw_bns
val qfvs_descr =
map2 (fn n => fn t => ("fv_" ^ n, t, NoSyn)) qty_names raw_fvs
val qfv_bns_descr =
map2 (fn (b, _, _) => fn t => ("fv_" ^ Name.of_binding b, t, NoSyn)) bn_funs raw_fv_bns
val qalpha_bns_descr =
map2 (fn (b, _, _) => fn t => ("alpha_" ^ Name.of_binding b, t, NoSyn)) bn_funs alpha_bn_trms
val qperm_descr =
map2 (fn n => fn t => ("permute_" ^ n, Type.legacy_freeze t, NoSyn)) qty_names raw_perm_funs
val qsize_descr =
map2 (fn n => fn t => ("size_" ^ n, t, NoSyn)) qty_names raw_size_trms
val (((((qconstrs_info, qbns_info), qfvs_info), qfv_bns_info), qalpha_bns_info), lthy8) =
if get_STEPS lthy > 17
then
lthy7
|> define_qconsts qtys qconstrs_descr
||>> define_qconsts qtys qbns_descr
||>> define_qconsts qtys qfvs_descr
||>> define_qconsts qtys qfv_bns_descr
||>> define_qconsts qtys qalpha_bns_descr
else raise TEST lthy7
(* definition of the quotient permfunctions and pt-class *)
val lthy9 =
if get_STEPS lthy > 18
then define_qperms qtys qty_full_names tvs qperm_descr raw_perm_laws lthy8
else raise TEST lthy8
val lthy9a =
if get_STEPS lthy > 19
then define_qsizes qtys qty_full_names tvs qsize_descr lthy9
else raise TEST lthy9
val qconstrs = map #qconst qconstrs_info
val qbns = map #qconst qbns_info
val qfvs = map #qconst qfvs_info
val qfv_bns = map #qconst qfv_bns_info
val qalpha_bns = map #qconst qalpha_bns_info
(* lifting of the theorems *)
val _ = warning "Lifting of Theorems"
val eq_iff_simps = @{thms alphas permute_prod.simps prod_fv.simps prod_alpha_def prod_rel.simps
prod.cases}
val ((((((qdistincts, qeq_iffs), qfv_defs), qbn_defs), qperm_simps), qfv_qbn_eqvts), lthyA) =
if get_STEPS lthy > 20
then
lthy9a
|> lift_thms qtys [] alpha_distincts
||>> lift_thms qtys eq_iff_simps alpha_eq_iff
||>> lift_thms qtys [] raw_fv_defs
||>> lift_thms qtys [] raw_bn_defs
||>> lift_thms qtys [] raw_perm_simps
||>> lift_thms qtys [] (raw_fv_eqvt @ raw_bn_eqvt)
else raise TEST lthy9a
val (((qsize_eqvt, [qinduct]), qexhausts), lthyB) =
if get_STEPS lthy > 20
then
lthyA
|> lift_thms qtys [] raw_size_eqvt
||>> lift_thms qtys [] [raw_induct_thm]
||>> lift_thms qtys [] raw_exhaust_thms
else raise TEST lthyA
(* temporary theorem bindings *)
val (_, lthy9') = lthyB
|> Local_Theory.note ((@{binding "distinct"}, []), qdistincts)
||>> Local_Theory.note ((@{binding "eq_iff"}, []), qeq_iffs)
||>> Local_Theory.note ((@{binding "fv_defs"}, []), qfv_defs)
||>> Local_Theory.note ((@{binding "bn_defs"}, []), qbn_defs)
||>> Local_Theory.note ((@{binding "perm_simps"}, []), qperm_simps)
||>> Local_Theory.note ((@{binding "fv_bn_eqvt"}, []), qfv_qbn_eqvts)
||>> Local_Theory.note ((@{binding "size_eqvt"}, []), qsize_eqvt)
||>> Local_Theory.note ((@{binding "induct"}, []), [qinduct])
||>> Local_Theory.note ((@{binding "exhaust"}, []), qexhausts)
val _ =
if get_STEPS lthy > 21
then true else raise TEST lthy9'
(* old stuff *)
val thy = ProofContext.theory_of lthy9'
val thy_name = Context.theory_name thy
val qty_full_names = map (Long_Name.qualify thy_name) qty_names
val _ = warning "Proving respects";
val bn_nos = map (fn (_, i, _) => i) raw_bn_info;
val bns = raw_bns ~~ bn_nos;
val bns_rsp_pre' = build_fvbv_rsps alpha_trms alpha_induct raw_bn_defs (map fst bns) lthy9';
val (bns_rsp_pre, lthy9) = fold_map (
fn (bn_t, _) => prove_const_rsp qtys Binding.empty [bn_t] (fn _ =>
resolve_tac bns_rsp_pre' 1)) bns lthy9';
val bns_rsp = flat (map snd bns_rsp_pre);
fun fv_rsp_tac _ = fvbv_rsp_tac alpha_induct raw_fv_defs lthy9' 1;
val fv_alpha_all = combine_fv_alpha_bns (raw_fvs, raw_fv_bns) (alpha_trms, alpha_bn_trms) bn_nos
val fv_rsps = prove_fv_rsp fv_alpha_all alpha_trms fv_rsp_tac lthy9;
val (fv_rsp_pre, lthy10) = fold_map
(fn fv => fn ctxt => prove_const_rsp qtys Binding.empty [fv]
(fn _ => asm_simp_tac (HOL_ss addsimps fv_rsps) 1) ctxt) (raw_fvs @ raw_fv_bns) lthy9;
val fv_rsp = flat (map snd fv_rsp_pre);
val (perms_rsp, lthy11) = prove_const_rsp qtys Binding.empty raw_perm_funs
(fn _ => asm_simp_tac (HOL_ss addsimps alpha_eqvt) 1) lthy10;
fun alpha_bn_rsp_tac _ = let val alpha_bn_rsp_pre = prove_alpha_bn_rsp alpha_trms alpha_induct (alpha_eq_iff @ alpha_distincts) alpha_equivp_thms raw_exhaust_thms alpha_bn_trms lthy11 in asm_simp_tac (HOL_ss addsimps alpha_bn_rsp_pre) 1 end;
val (alpha_bn_rsps, lthy11a) = fold_map (fn cnst => prove_const_rsp qtys Binding.empty [cnst]
alpha_bn_rsp_tac) alpha_bn_trms lthy11
fun const_rsp_tac _ =
let val alpha_alphabn = prove_alpha_alphabn alpha_trms alpha_induct alpha_eq_iff alpha_bn_trms lthy11a
in constr_rsp_tac alpha_eq_iff (fv_rsp @ bns_rsp @ alpha_refl_thms @ alpha_alphabn) 1 end
val (const_rsps, lthy12) = fold_map (fn cnst => prove_const_rsp qtys Binding.empty [cnst]
const_rsp_tac) raw_constrs lthy11a
val qfv_names = map (unsuffix "_raw" o Long_Name.base_name o fst o dest_Const) (raw_fvs @ raw_fv_bns)
val dd = map2 (fn x => fn y => (x, y, NoSyn)) qfv_names (raw_fvs @ raw_fv_bns)
val (qfv_info, lthy12a) = define_qconsts qtys dd lthy12;
val qfv_ts = map #qconst qfv_info
val qfv_defs = map #def qfv_info
val (qfv_ts_nobn, qfv_ts_bn) = chop (length raw_perm_funs) qfv_ts;
val qbn_names = map (fn (b, _ , _) => Name.of_binding b) bn_funs
val dd = map2 (fn x => fn y => (x, y, NoSyn)) qbn_names raw_bns
val (qbn_info, lthy12b) = define_qconsts qtys dd lthy12a;
val qbn_ts = map #qconst qbn_info
val qbn_defs = map #def qbn_info
val qalpha_bn_names = map (unsuffix "_raw" o Long_Name.base_name o fst o dest_Const) alpha_bn_trms
val dd = map2 (fn x => fn y => (x, y, NoSyn)) qalpha_bn_names alpha_bn_trms
val (qalpha_info, lthy12c) = define_qconsts qtys dd lthy12b;
val qalpha_bn_trms = map #qconst qalpha_info
val qalphabn_defs = map #def qalpha_info
val _ = warning "Lifting permutations";
val perm_names = map (fn x => "permute_" ^ x) qty_names
val dd = map2 (fn x => fn y => (x, y, NoSyn)) perm_names raw_perm_funs
val lthy13 = define_qperms qtys qty_full_names [] dd raw_perm_laws lthy12c
val q_name = space_implode "_" qty_names;
fun suffix_bind s = Binding.qualify true q_name (Binding.name s);
val _ = warning "Lifting induction";
val constr_names = map (Long_Name.base_name o fst o dest_Const) [];
val q_induct = Rule_Cases.name constr_names (the_single (fst (lift_thms qtys [] [raw_induct_thm] lthy13)));
fun note_suffix s th ctxt =
snd (Local_Theory.note ((suffix_bind s, []), th) ctxt);
fun note_simp_suffix s th ctxt =
snd (Local_Theory.note ((suffix_bind s, [Attrib.internal (K Simplifier.simp_add)]), th) ctxt);
val (_, lthy14) = Local_Theory.note ((suffix_bind "induct",
[Attrib.internal (K (Rule_Cases.case_names constr_names))]),
[Rule_Cases.name constr_names q_induct]) lthy13;
val q_inducts = Project_Rule.projects lthy13 (1 upto (length raw_fvs)) q_induct
val (_, lthy14a) = Local_Theory.note ((suffix_bind "inducts", []), q_inducts) lthy14;
val q_perm = fst (lift_thms qtys [] raw_perm_simps lthy14);
val lthy15 = note_simp_suffix "perm" q_perm lthy14a;
val q_fv = fst (lift_thms qtys [] raw_fv_defs lthy15);
val lthy16 = note_simp_suffix "fv" q_fv lthy15;
val q_bn = fst (lift_thms qtys [] raw_bn_defs lthy16);
val lthy17 = note_simp_suffix "bn" q_bn lthy16;
val _ = warning "Lifting eq-iff";
(*val _ = map tracing (map PolyML.makestring alpha_eq_iff);*)
val eq_iff_unfolded0 = map (Local_Defs.unfold lthy17 @{thms alphas}) alpha_eq_iff
val eq_iff_unfolded1 = map (Local_Defs.unfold lthy17 @{thms Pair_eqvt}) eq_iff_unfolded0
val q_eq_iff_pre0 = fst (lift_thms qtys [] eq_iff_unfolded1 lthy17);
val q_eq_iff_pre1 = map (Local_Defs.fold lthy17 @{thms Pair_eqvt}) q_eq_iff_pre0
val q_eq_iff_pre2 = map (Local_Defs.fold lthy17 @{thms alphas}) q_eq_iff_pre1
val q_eq_iff = map (Local_Defs.unfold lthy17 (Quotient_Info.id_simps_get lthy17)) q_eq_iff_pre2
val (_, lthy18) = Local_Theory.note ((suffix_bind "eq_iff", []), q_eq_iff) lthy17;
val q_dis = fst (lift_thms qtys [] alpha_distincts lthy18);
val lthy19 = note_simp_suffix "distinct" q_dis lthy18;
val q_eqvt = fst (lift_thms qtys [] (raw_bn_eqvt @ raw_fv_eqvt) lthy19);
val (_, lthy20) = Local_Theory.note ((Binding.empty,
[Attrib.internal (fn _ => Nominal_ThmDecls.eqvt_add)]), q_eqvt) lthy19;
val _ = warning "Supports";
val supports = map (prove_supports lthy20 q_perm) [];
val fin_supp = HOLogic.conj_elims (prove_fs lthy20 q_induct supports qtys);
val thy3 = Local_Theory.exit_global lthy20;
val _ = warning "Instantiating FS";
val lthy21 = Class.instantiation (qty_full_names, [], @{sort fs}) thy3;
fun tac _ = Class.intro_classes_tac [] THEN (ALLGOALS (resolve_tac fin_supp))
val lthy22 = Class.prove_instantiation_instance tac lthy21
val fv_alpha_all = combine_fv_alpha_bns (qfv_ts_nobn, qfv_ts_bn) (alpha_trms, qalpha_bn_trms) bn_nos;
val (names, supp_eq_t) = supp_eq fv_alpha_all;
val _ = warning "Support Equations";
fun supp_eq_tac' _ = supp_eq_tac q_induct q_fv q_perm q_eq_iff lthy22 1;
val q_supp = HOLogic.conj_elims (Goal.prove lthy22 names [] supp_eq_t supp_eq_tac') handle e =>
let val _ = warning ("Support eqs failed") in [] end;
val lthy23 = note_suffix "supp" q_supp lthy22;
in
(0, lthy23)
end handle TEST ctxt => (0, ctxt)
*}
section {* Preparing and parsing of the specification *}
ML {*
(* generates the parsed datatypes and
declares the constructors
*)
fun prepare_dts dt_strs thy =
let
fun inter_fs_sort thy (a, S) =
(a, Type.inter_sort (Sign.tsig_of thy) (@{sort fs}, S))
fun mk_type tname sorts (cname, cargs, mx) =
let
val full_tname = Sign.full_name thy tname
val ty = Type (full_tname, map (TFree o inter_fs_sort thy) sorts)
in
(cname, cargs ---> ty, mx)
end
fun prep_constr (cname, cargs, mx, _) (constrs, sorts) =
let
val (cargs', sorts') =
fold_map (Datatype.read_typ thy) (map snd cargs) sorts
|>> map (map_type_tfree (TFree o inter_fs_sort thy))
in
(constrs @ [(cname, cargs', mx)], sorts')
end
fun prep_dts (tvs, tname, mx, constrs) (constr_trms, dts, sorts) =
let
val (constrs', sorts') =
fold prep_constr constrs ([], sorts)
val constr_trms' =
map (mk_type tname (rev sorts')) constrs'
in
(constr_trms @ constr_trms', dts @ [(tvs, tname, mx, constrs')], sorts')
end
val (constr_trms, dts, sorts) = fold prep_dts dt_strs ([], [], []);
in
thy
|> Sign.add_consts_i constr_trms
|> pair dts
end
*}
ML {*
(* parsing the binding function specification and *)
(* declaring the functions in the local theory *)
fun prepare_bn_funs bn_fun_strs bn_eq_strs thy =
let
val lthy = Named_Target.theory_init thy
val ((bn_funs, bn_eqs), lthy') =
Specification.read_spec bn_fun_strs bn_eq_strs lthy
fun prep_bn_fun ((bn, T), mx) = (bn, T, mx)
val bn_funs' = map prep_bn_fun bn_funs
in
(Local_Theory.exit_global lthy')
|> Sign.add_consts_i bn_funs'
|> pair (bn_funs', bn_eqs)
end
*}
text {* associates every SOME with the index in the list; drops NONEs *}
ML {*
fun indexify xs =
let
fun mapp _ [] = []
| mapp i (NONE :: xs) = mapp (i + 1) xs
| mapp i (SOME x :: xs) = (x, i) :: mapp (i + 1) xs
in
mapp 0 xs
end
fun index_lookup xs x =
case AList.lookup (op=) xs x of
SOME x => x
| NONE => error ("Cannot find " ^ x ^ " as argument annotation.");
*}
ML {*
fun prepare_bclauses dt_strs thy =
let
val annos_bclauses =
get_cnstrs dt_strs
|> map (map (fn (_, antys, _, bns) => (map fst antys, bns)))
fun prep_binder env bn_str =
case (Syntax.read_term_global thy bn_str) of
Free (x, _) => (NONE, index_lookup env x)
| Const (a, T) $ Free (x, _) => (SOME (Const (a, T)), index_lookup env x)
| _ => error ("The term " ^ bn_str ^ " is not allowed as binding function.")
fun prep_body env bn_str = index_lookup env bn_str
fun prep_bclause env (mode, binders, bodies) =
let
val binders' = map (prep_binder env) binders
val bodies' = map (prep_body env) bodies
in
BC (mode, binders', bodies')
end
fun prep_bclauses (annos, bclause_strs) =
let
val env = indexify annos (* for every label, associate the index *)
in
map (prep_bclause env) bclause_strs
end
in
(map (map prep_bclauses) annos_bclauses, thy)
end
*}
text {*
adds an empty binding clause for every argument
that is not already part of a binding clause
*}
ML {*
fun included i bcs =
let
fun incl (BC (_, bns, bds)) =
member (op =) (map snd bns) i orelse member (op =) bds i
in
exists incl bcs
end
*}
ML {*
fun complete dt_strs bclauses =
let
val args =
get_cnstrs dt_strs
|> map (map (fn (_, antys, _, _) => length antys))
fun complt n bcs =
let
fun add bcs i = (if included i bcs then [] else [BC (Lst, [], [i])])
in
bcs @ (flat (map_range (add bcs) n))
end
in
map2 (map2 complt) args bclauses
end
*}
ML {*
fun nominal_datatype2_cmd (opt_thm_name, dt_strs, bn_fun_strs, bn_eq_strs) lthy =
let
val (pre_typ_names, pre_typs) = split_list
(map (fn (tvs, tname, mx, _) =>
(Binding.name_of tname, (tname, length tvs, mx))) dt_strs)
(* this theory is used just for parsing *)
val thy = ProofContext.theory_of lthy
val tmp_thy = Theory.copy thy
val (((dts, (bn_funs, bn_eqs)), bclauses), tmp_thy') =
tmp_thy
|> Sign.add_types pre_typs
|> prepare_dts dt_strs
||>> prepare_bn_funs bn_fun_strs bn_eq_strs
||>> prepare_bclauses dt_strs
val bclauses' = complete dt_strs bclauses
val thm_name =
the_default (Binding.name (space_implode "_" pre_typ_names)) opt_thm_name
in
timeit (fn () => nominal_datatype2 thm_name dts bn_funs bn_eqs bclauses' lthy |> snd)
end
*}
ML {*
(* nominal datatype parser *)
local
structure P = Parse;
structure S = Scan
fun triple ((x, y), z) = (x, y, z)
fun tuple1 ((x, y, z), u) = (x, y, z, u)
fun tuple2 (((x, y), z), u) = (x, y, u, z)
fun tuple3 ((x, y), (z, u)) = (x, y, z, u)
in
val _ = Keyword.keyword "bind"
val opt_name = Scan.option (P.binding --| Args.colon)
val anno_typ = S.option (P.name --| P.$$$ "::") -- P.typ
val bind_mode = P.$$$ "bind" |--
S.optional (Args.parens
(Args.$$$ "list" >> K Lst || Args.$$$ "set" >> K Set || Args.$$$ "res" >> K Res)) Lst
val bind_clauses =
P.enum "," (bind_mode -- S.repeat1 P.term -- (P.$$$ "in" |-- S.repeat1 P.name) >> triple)
val cnstr_parser =
P.binding -- S.repeat anno_typ -- bind_clauses -- P.opt_mixfix >> tuple2
(* datatype parser *)
val dt_parser =
(P.type_args -- P.binding -- P.opt_mixfix >> triple) --
(P.$$$ "=" |-- P.enum1 "|" cnstr_parser) >> tuple1
(* binding function parser *)
val bnfun_parser =
S.optional (P.$$$ "binder" |-- P.fixes -- Parse_Spec.where_alt_specs) ([], [])
(* main parser *)
val main_parser =
opt_name -- P.and_list1 dt_parser -- bnfun_parser >> tuple3
end
(* Command Keyword *)
val _ = Outer_Syntax.local_theory "nominal_datatype" "test" Keyword.thy_decl
(main_parser >> nominal_datatype2_cmd)
*}
text {*
nominal_datatype2 does the following things in order:
Parser.thy/raw_nominal_decls
1) define the raw datatype
2) define the raw binding functions
Perm.thy/define_raw_perms
3) define permutations of the raw datatype and show that the raw type is
in the pt typeclass
Lift.thy/define_fv_alpha_export, Fv.thy/define_fv & define_alpha
4) define fv and fv_bn
5) define alpha and alpha_bn
Perm.thy/distinct_rel
6) prove alpha_distincts (C1 x \<notsimeq> C2 y ...) (Proof by cases; simp)
Tacs.thy/build_rel_inj
6) prove alpha_eq_iff (C1 x = C2 y \<leftrightarrow> P x y ...)
(left-to-right by intro rule, right-to-left by cases; simp)
Equivp.thy/prove_eqvt
7) prove bn_eqvt (common induction on the raw datatype)
8) prove fv_eqvt (common induction on the raw datatype with help of above)
Rsp.thy/build_alpha_eqvts
9) prove alpha_eqvt and alpha_bn_eqvt
(common alpha-induction, unfolding alpha_gen, permute of #* and =)
Equivp.thy/build_alpha_refl & Equivp.thy/build_equivps
10) prove that alpha and alpha_bn are equivalence relations
(common induction and application of 'compose' lemmas)
Lift.thy/define_quotient_types
11) define quotient types
Rsp.thy/build_fvbv_rsps
12) prove bn respects (common induction and simp with alpha_gen)
Rsp.thy/prove_const_rsp
13) prove fv respects (common induction and simp with alpha_gen)
14) prove permute respects (unfolds to alpha_eqvt)
Rsp.thy/prove_alpha_bn_rsp
15) prove alpha_bn respects
(alpha_induct then cases then sym and trans of the relations)
Rsp.thy/prove_alpha_alphabn
16) show that alpha implies alpha_bn (by unduction, needed in following step)
Rsp.thy/prove_const_rsp
17) prove respects for all datatype constructors
(unfold eq_iff and alpha_gen; introduce zero permutations; simp)
Perm.thy/quotient_lift_consts_export
18) define lifted constructors, fv, bn, alpha_bn, permutations
Perm.thy/define_lifted_perms
19) lift permutation zero and add properties to show that quotient type is in the pt typeclass
Lift.thy/lift_thm
20) lift permutation simplifications
21) lift induction
22) lift fv
23) lift bn
24) lift eq_iff
25) lift alpha_distincts
26) lift fv and bn eqvts
Equivp.thy/prove_supports
27) prove that union of arguments supports constructors
Equivp.thy/prove_fs
28) show that the lifted type is in fs typeclass (* by q_induct, supports *)
Equivp.thy/supp_eq
29) prove supp = fv
*}
end