all examples for strong exhausts work; recursive binders need to be treated differently; still unclean version with lots of diagnostic code
theory Nominal2
imports
Nominal2_Base Nominal2_Eqvt Nominal2_Abs
uses ("nominal_dt_rawfuns.ML")
("nominal_dt_alpha.ML")
("nominal_dt_quot.ML")
begin
use "nominal_dt_rawfuns.ML"
ML {* open Nominal_Dt_RawFuns *}
use "nominal_dt_alpha.ML"
ML {* open Nominal_Dt_Alpha *}
use "nominal_dt_quot.ML"
ML {* open Nominal_Dt_Quot *}
text {* TEST *}
ML {*
fun strip_outer_params (Const("all", _) $ Abs (a, T, t)) = strip_outer_params t |>> cons (a, T)
| strip_outer_params B = ([], B)
fun strip_params_prems_concl trm =
let
val (params, body) = strip_outer_params trm
val (prems, concl) = Logic.strip_horn body
in
(params, prems, concl)
end
fun list_params_prems_concl params prems concl =
Logic.list_implies (prems, concl)
|> fold_rev mk_all params
fun mk_binop_env tys c (t, u) =
let val ty = fastype_of1 (tys, t) in
Const (c, [ty, ty] ---> ty) $ t $ u
end
fun mk_union_env tys (t1, @{term "{}::atom set"}) = t1
| mk_union_env tys (@{term "{}::atom set"}, t2) = t2
| mk_union_env tys (t1, @{term "set ([]::atom list)"}) = t1
| mk_union_env tys (@{term "set ([]::atom list)"}, t2) = t2
| mk_union_env tys (t1, t2) = mk_binop_env tys @{const_name "sup"} (t1, t2)
fun fold_union_env tys trms = fold_rev (curry (mk_union_env tys)) trms @{term "{}::atom set"}
*}
ML {*
fun process_ecase lthy c (params, prems, concl) bclauses =
let
val tys = map snd params
val binders = get_all_binders bclauses
fun prep_binder (opt, i) =
let
val t = Bound (length tys - i - 1)
in
case opt of
NONE => setify_ty lthy (nth tys i) t
| SOME bn => to_set_ty (fastype_of1 (tys, bn $ t)) (bn $ t)
end
val fresh_prem =
case binders of
[] => [] (* case: no binders *)
| _ => binders (* case: binders *)
|> map prep_binder
|> fold_union_env tys
|> (fn t => mk_fresh_star t c)
|> HOLogic.mk_Trueprop
|> single
in
list_params_prems_concl params (fresh_prem @ prems) concl
end
*}
ML {*
(* derives the freshness theorem that there exists a p, such that
(p o as) #* (c, t1,\<dots>, tn) *)
fun fresh_thm ctxt c parms binders bn_finite_thms =
let
fun prep_binder (opt, i) =
case opt of
NONE => setify ctxt (nth parms i)
| SOME bn => to_set (bn $ (nth parms i))
fun prep_binder2 (opt, i) =
case opt of
NONE => atomify ctxt (nth parms i)
| SOME bn => bn $ (nth parms i)
val rhs = HOLogic.mk_tuple ([c] @ parms @ (map prep_binder2 binders))
val lhs = binders
|> map prep_binder
|> fold_union
|> mk_perm (Bound 0)
val goal = mk_fresh_star lhs rhs
|> (fn t => HOLogic.mk_exists ("p", @{typ perm}, t))
|> HOLogic.mk_Trueprop
val ss = bn_finite_thms @ @{thms supp_Pair finite_supp finite_sets_supp}
@ @{thms finite.intros finite_Un finite_set finite_fset}
in
Goal.prove ctxt [] [] goal
(K (HEADGOAL (rtac @{thm at_set_avoiding1}
THEN_ALL_NEW (simp_tac (HOL_ss addsimps ss)))))
end
*}
ML {*
(* derives abs_eq theorems of the form Exists s. [as].t = [p o as].s *)
fun abs_eq_thm ctxt fprops p parms bn_finite_thms bn_eqvt permute_bns
(bclause as (BC (bmode, binders, bodies))) =
case binders of
[] => []
| _ =>
let
val binder_trm = comb_binders ctxt bmode parms binders
val body_trm = foldl1 HOLogic.mk_prod (map (nth parms) bodies)
val body_ty = fastype_of body_trm
val (abs_name, binder_ty, abs_ty) =
case bmode of
Lst => (@{const_name "Abs_lst"}, @{typ "atom list"}, @{type_name abs_lst})
| Set => (@{const_name "Abs_set"}, @{typ "atom set"}, @{type_name abs_set})
| Res => (@{const_name "Abs_res"}, @{typ "atom set"}, @{type_name abs_res})
val abs = Const (abs_name, [binder_ty, body_ty] ---> Type (abs_ty, [body_ty]))
val abs_lhs = abs $ binder_trm $ body_trm
val abs_rhs = abs $ mk_perm p binder_trm $ mk_perm (Bound 0) body_trm
val abs_rhs' = abs $ mk_perm (Bound 0) binder_trm $ mk_perm (Bound 0) body_trm
val abs_eq = HOLogic.mk_eq (abs_lhs, abs_rhs)
val abs_eq' = HOLogic.mk_eq (abs_lhs, abs_rhs')
val eq = HOLogic.mk_eq (mk_perm (Bound 0) binder_trm, mk_perm p binder_trm)
val goal = HOLogic.mk_conj (abs_eq, eq)
|> (fn t => HOLogic.mk_exists ("q", @{typ "perm"}, t))
|> HOLogic.mk_Trueprop
val goal' = HOLogic.mk_conj (abs_eq', eq)
|> (fn t => HOLogic.mk_exists ("q", @{typ "perm"}, t))
|> HOLogic.mk_Trueprop
val ss = fprops @ bn_finite_thms @ @{thms set.simps set_append union_eqvt}
@ @{thms fresh_star_Un fresh_star_Pair fresh_star_list fresh_star_singleton fresh_star_fset
fresh_star_set} @ @{thms finite.intros finite_fset}
in
if is_recursive_binder bclause
then
(tracing "recursive";
[ Goal.prove ctxt [] [] goal'
(K (HEADGOAL (resolve_tac @{thms Abs_rename_set' Abs_rename_res' Abs_rename_lst'}
THEN_ALL_NEW (simp_tac (HOL_basic_ss addsimps ss) THEN' TRY o simp_tac HOL_ss))))
|> Nominal_Permeq.eqvt_strict_rule ctxt bn_eqvt []
])
else
(tracing "non-recursive";
[ Goal.prove ctxt [] [] goal
(K (HEADGOAL (resolve_tac @{thms Abs_rename_set Abs_rename_res Abs_rename_lst}
THEN_ALL_NEW (simp_tac (HOL_basic_ss addsimps ss) THEN' TRY o simp_tac HOL_ss))))
|> Nominal_Permeq.eqvt_strict_rule ctxt permute_bns []
])
end
*}
ML {*
fun conj_tac tac i =
let
fun select (t, i) =
case t of
@{term "Trueprop"} $ t' => select (t', i)
| @{term "op \<and>"} $ _ $ _ => (rtac @{thm conjI} THEN' RANGE [conj_tac tac, conj_tac tac]) i
| _ => tac i
in
SUBGOAL select i
end
*}
ML {*
fun is_abs_eq thm =
let
fun is_abs trm =
case (head_of trm) of
Const (@{const_name "Abs_set"}, _) => true
| Const (@{const_name "Abs_lst"}, _) => true
| Const (@{const_name "Abs_res"}, _) => true
| _ => false
in
thm |> prop_of
|> HOLogic.dest_Trueprop
|> HOLogic.dest_eq
|> fst
|> is_abs
end
*}
lemma setify:
shows "xs = ys \<Longrightarrow> set xs = set ys"
by simp
ML {*
fun case_tac ctxt c bn_finite_thms eq_iff_thms bn_eqvt permute_bns perm_bn_alphas
(prems, bclausess) qexhaust_thm =
let
fun aux_tac prem bclauses =
case (get_all_binders bclauses) of
[] => EVERY' [rtac prem, atac]
| binders => Subgoal.SUBPROOF (fn {params, prems, concl, context = ctxt, ...} =>
let
val parms = map (term_of o snd) params
val fthm = fresh_thm ctxt c parms binders bn_finite_thms
val ss = @{thms fresh_star_Pair union_eqvt fresh_star_Un}
val (([(_, fperm)], fprops), ctxt') = Obtain.result
(K (EVERY1 [etac exE,
full_simp_tac (HOL_basic_ss addsimps ss),
REPEAT o (etac @{thm conjE})])) [fthm] ctxt
val abs_eq_thms = flat
(map (abs_eq_thm ctxt fprops (term_of fperm) parms bn_finite_thms bn_eqvt permute_bns) bclauses)
val ((_, eqs), ctxt'') = Obtain.result
(K (EVERY1
[ REPEAT o (etac @{thm exE}),
REPEAT o (etac @{thm conjE}),
REPEAT o (dtac @{thm setify}),
full_simp_tac (HOL_basic_ss addsimps @{thms set_append set.simps})])) abs_eq_thms ctxt'
val (abs_eqs, peqs) = split_filter is_abs_eq eqs
val fprops' = map (Nominal_Permeq.eqvt_strict_rule ctxt permute_bns []) fprops
val fprops'' = map (Nominal_Permeq.eqvt_strict_rule ctxt bn_eqvt []) fprops
val _ = tracing ("prem:\n" ^ (Syntax.string_of_term ctxt'' o prop_of) prem)
val _ = tracing ("prems:\n" ^ cat_lines (map (Syntax.string_of_term ctxt'' o prop_of) prems))
val _ = tracing ("fprop':\n" ^ cat_lines (map (Syntax.string_of_term ctxt'' o prop_of) fprops'))
val _ = tracing ("fprop'':\n" ^ cat_lines (map (Syntax.string_of_term ctxt'' o prop_of) fprops''))
val _ = tracing ("abseq:\n" ^ cat_lines (map (Syntax.string_of_term ctxt'' o prop_of) abs_eqs))
val _ = tracing ("peqs:\n" ^ cat_lines (map (Syntax.string_of_term ctxt'' o prop_of) peqs))
val tac1 = EVERY'
[ simp_tac (HOL_basic_ss addsimps peqs),
rewrite_goal_tac (@{thms fresh_star_Un[THEN eq_reflection]}),
K (print_tac "before solving freshness"),
conj_tac (TRY o DETERM o resolve_tac (fprops' @ fprops'')) ]
val tac2 = EVERY'
[ rtac (@{thm ssubst} OF prems),
rewrite_goal_tac (map safe_mk_equiv eq_iff_thms),
K (print_tac "before substituting"),
rewrite_goal_tac (map safe_mk_equiv abs_eqs),
K (print_tac "after substituting"),
conj_tac (TRY o DETERM o resolve_tac (@{thms refl} @ perm_bn_alphas)),
K (print_tac "end")
]
val side_thm = Goal.prove ctxt'' [] [] (term_of concl)
(fn _ => (* Skip_Proof.cheat_tac (ProofContext.theory_of ctxt'') *)
EVERY
[ rtac prem 1,
print_tac "after applying prem",
RANGE [SOLVED' tac1, SOLVED' tac2] 1,
print_tac "final" ] )
|> singleton (ProofContext.export ctxt'' ctxt)
val _ = tracing ("side_thm:\n" ^ (Syntax.string_of_term ctxt o prop_of) side_thm)
in
rtac side_thm 1
end) ctxt
in
rtac qexhaust_thm THEN' RANGE (map2 aux_tac prems bclausess)
end
*}
ML {*
fun prove_strong_exhausts lthy exhausts qtrms bclausesss bn_finite_thms eq_iff_thms bn_eqvt permute_bns
perm_bn_alphas =
let
val ((_, exhausts'), lthy') = Variable.import true exhausts lthy
val ([c, a], lthy'') = Variable.variant_fixes ["c", "'a"] lthy'
val c = Free (c, TFree (a, @{sort fs}))
val (ecases, main_concls) = exhausts' (* ecases or of the form (params, prems, concl) *)
|> map prop_of
|> map Logic.strip_horn
|> split_list
|>> (map o map) strip_params_prems_concl
val prems = map2 (process_ecase lthy'' c) (flat ecases) (flat bclausesss)
in
Goal.prove_multi lthy'' [] prems main_concls
(fn {prems:thm list, context} =>
let
val prems' = partitions prems (map length bclausesss)
in
EVERY1 [Goal.conjunction_tac,
RANGE (map2 (case_tac context c bn_finite_thms eq_iff_thms bn_eqvt permute_bns perm_bn_alphas)
(prems' ~~ bclausesss) exhausts')]
end)
end
*}
ML {*
val eqvt_attr = Attrib.internal (K Nominal_ThmDecls.eqvt_add)
val rsp_attr = Attrib.internal (K Quotient_Info.rsp_rules_add)
val simp_attr = Attrib.internal (K Simplifier.simp_add)
*}
section{* Interface for nominal_datatype *}
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 o map o map) rawify_bclause bclauses
end
*}
ML {*
fun define_raw_dts dts bn_funs bn_eqs bclauses 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 bclauses
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 100);
fun get_STEPS ctxt = Config.get ctxt STEPS
*}
setup STEPS_setup
ML {*
fun nominal_datatype2 opt_thms_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 = (map o 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 HOLogic.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 > 0
then define_raw_perms raw_full_ty_names raw_tys tvs (flat 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_attr]), raw_perm_simps) lthy2a
(* definition of raw fv and bn functions *)
val _ = warning "Definition of raw fv- and bn-functions";
val (raw_bns, raw_bn_defs, raw_bn_info, raw_bn_induct, lthy3a) =
if get_STEPS lthy3 > 1
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
(* defining the permute_bn functions *)
val (raw_perm_bns, raw_perm_bn_simps, lthy3b) =
if get_STEPS lthy3a > 2
then define_raw_bn_perms raw_tys raw_bn_info raw_cns_info
(raw_inject_thms @ raw_distinct_thms) raw_size_thms lthy3a
else raise TEST lthy3a
val (raw_fvs, raw_fv_bns, raw_fv_defs, raw_fv_bns_induct, lthy3c) =
if get_STEPS lthy3b > 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 lthy3b
else raise TEST lthy3b
(* 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 lthy3c > 4
then define_raw_alpha raw_full_ty_names raw_tys raw_cns_info raw_bn_info raw_bclauses raw_fvs lthy3c
else raise TEST lthy3c
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 *)
val _ = warning "Eq-iff theorems";
val 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_attr]), 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_attr]), 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 =
if get_STEPS lthy > 15
then 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
else raise TEST lthy6
val raw_funs_rsp =
if get_STEPS lthy > 16
then map mk_funs_rsp raw_funs_rsp_aux
else raise TEST lthy6
val raw_size_rsp =
if get_STEPS lthy > 17
then
raw_size_rsp_aux (alpha_trms @ alpha_bn_trms) alpha_induct
(raw_size_thms @ raw_size_eqvt) lthy6
|> map mk_funs_rsp
else raise TEST lthy6
val raw_constrs_rsp =
if get_STEPS lthy > 18
then raw_constrs_rsp (flat raw_constrs) alpha_trms alpha_intros
(alpha_bn_imp_thms @ raw_funs_rsp_aux) lthy6
else raise TEST lthy6
val alpha_permute_rsp =
if get_STEPS lthy > 19
then map mk_alpha_permute_rsp alpha_eqvt
else raise TEST lthy6
val alpha_bn_rsp =
if get_STEPS lthy > 20
then raw_alpha_bn_rsp alpha_bn_trms alpha_bn_equivp_thms alpha_bn_imp_thms
else raise TEST lthy6
val raw_perm_bn_rsp =
if get_STEPS lthy > 21
then raw_perm_bn_rsp (alpha_trms @ alpha_bn_trms) raw_perm_bns alpha_induct
alpha_intros raw_perm_bn_simps lthy6
else raise TEST lthy6
(* noting the quot_respects lemmas *)
val (_, lthy6a) =
if get_STEPS lthy > 22
then Local_Theory.note ((Binding.empty, [rsp_attr]),
raw_constrs_rsp @ raw_funs_rsp @ raw_size_rsp @ alpha_permute_rsp @
alpha_bn_rsp @ raw_perm_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 > 23
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_descrs =
(map2 o map2) (fn (b, _, mx) => fn t => (Name.of_binding b, t, mx)) (get_cnstrs dts) 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 qperm_bn_descr =
map2 (fn (b, _, _) => fn t => ("permute_" ^ Name.of_binding b, t, NoSyn)) bn_funs raw_perm_bns
val ((((((qconstrs_infos, qbns_info), qfvs_info), qfv_bns_info), qalpha_bns_info), qperm_bns_info),
lthy8) =
if get_STEPS lthy > 24
then
lthy7
|> fold_map (define_qconsts qtys) qconstrs_descrs
||>> define_qconsts qtys qbns_descr
||>> define_qconsts qtys qfvs_descr
||>> define_qconsts qtys qfv_bns_descr
||>> define_qconsts qtys qalpha_bns_descr
||>> define_qconsts qtys qperm_bn_descr
else raise TEST lthy7
(* definition of the quotient permfunctions and pt-class *)
val lthy9 =
if get_STEPS lthy > 25
then define_qperms qtys qty_full_names tvs qperm_descr raw_perm_laws lthy8
else raise TEST lthy8
val lthy9a =
if get_STEPS lthy > 26
then define_qsizes qtys qty_full_names tvs qsize_descr lthy9
else raise TEST lthy9
val qtrms = (map o map) #qconst qconstrs_infos
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
val qperm_bns = map #qconst qperm_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_def
prod.cases}
val ((((((qdistincts, qeq_iffs), qfv_defs), qbn_defs), qperm_simps), qfv_qbn_eqvts), lthyA) =
if get_STEPS lthy > 27
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), qsize_simps), qperm_bn_simps), qalpha_refl_thms), lthyB) =
if get_STEPS lthy > 28
then
lthyA
|> lift_thms qtys [] raw_size_eqvt
||>> lift_thms qtys [] [raw_induct_thm]
||>> lift_thms qtys [] raw_exhaust_thms
||>> lift_thms qtys [] raw_size_thms
||>> lift_thms qtys [] raw_perm_bn_simps
||>> lift_thms qtys [] alpha_refl_thms
else raise TEST lthyA
val qinducts = Project_Rule.projections lthyA qinduct
(* supports lemmas *)
val _ = warning "Proving Supports Lemmas and fs-Instances"
val qsupports_thms =
if get_STEPS lthy > 29
then prove_supports lthyB qperm_simps (flat qtrms)
else raise TEST lthyB
(* finite supp lemmas *)
val qfsupp_thms =
if get_STEPS lthy > 30
then prove_fsupp lthyB qtys qinduct qsupports_thms
else raise TEST lthyB
(* fs instances *)
val lthyC =
if get_STEPS lthy > 31
then fs_instance qtys qty_full_names tvs qfsupp_thms lthyB
else raise TEST lthyB
(* fv - supp equality *)
val _ = warning "Proving Equality between fv and supp"
val qfv_supp_thms =
if get_STEPS lthy > 32
then prove_fv_supp qtys (flat qtrms) qfvs qfv_bns qalpha_bns qfv_defs qeq_iffs
qperm_simps qfv_qbn_eqvts qinduct (flat raw_bclauses) lthyC
else []
(* postprocessing of eq and fv theorems *)
val qeq_iffs' = qeq_iffs
|> map (simplify (HOL_basic_ss addsimps qfv_supp_thms))
|> map (simplify (HOL_basic_ss addsimps @{thms prod_fv_supp prod_alpha_eq Abs_eq_iff[symmetric]}))
val qsupp_constrs = qfv_defs
|> map (simplify (HOL_basic_ss addsimps (take (length qfvs) qfv_supp_thms)))
val transform_thm = @{lemma "x = y \<Longrightarrow> a \<notin> x \<longleftrightarrow> a \<notin> y" by simp}
val transform_thms =
[ @{lemma "a \<notin> (S \<union> T) \<longleftrightarrow> a \<notin> S \<and> a \<notin> T" by simp},
@{lemma "a \<notin> (S - T) \<longleftrightarrow> a \<notin> S \<or> a \<in> T" by simp},
@{lemma "(lhs = (a \<notin> {})) \<longleftrightarrow> lhs" by simp},
@{thm fresh_def[symmetric]}]
val qfresh_constrs = qsupp_constrs
|> map (fn thm => thm RS transform_thm)
|> map (simplify (HOL_basic_ss addsimps transform_thms))
(* proving that the qbn result is finite *)
val qbn_finite_thms =
if get_STEPS lthy > 33
then prove_bns_finite qtys qbns qinduct qbn_defs lthyC
else []
(* proving that perm_bns preserve alpha *)
val qperm_bn_alpha_thms =
if get_STEPS lthy > 33
then prove_perm_bn_alpha_thms qtys qperm_bns qalpha_bns qinduct qperm_bn_simps qeq_iffs'
qalpha_refl_thms lthyC
else []
(* proving the relationship of bn and permute_bn *)
val qpermute_bn_thms =
if get_STEPS lthy > 33
then prove_permute_bn_thms qtys qbns qperm_bns qinduct qperm_bn_simps qbn_defs qfv_qbn_eqvts lthyC
else []
val qstrong_exhaust_thms = prove_strong_exhausts lthyC qexhausts qtrms bclauses qbn_finite_thms qeq_iffs'
qfv_qbn_eqvts qpermute_bn_thms qperm_bn_alpha_thms
(* noting the theorems *)
(* generating the prefix for the theorem names *)
val thms_name =
the_default (Binding.name (space_implode "_" qty_names)) opt_thms_name
fun thms_suffix s = Binding.qualified true s thms_name
val (_, lthy9') = lthyC
|> Local_Theory.note ((thms_suffix "distinct", []), qdistincts)
||>> Local_Theory.note ((thms_suffix "eq_iff", []), qeq_iffs')
||>> Local_Theory.note ((thms_suffix "fv_defs", []), qfv_defs)
||>> Local_Theory.note ((thms_suffix "bn_defs", []), qbn_defs)
||>> Local_Theory.note ((thms_suffix "perm_simps", [eqvt_attr, simp_attr]), qperm_simps)
||>> Local_Theory.note ((thms_suffix "fv_bn_eqvt", []), qfv_qbn_eqvts)
||>> Local_Theory.note ((thms_suffix "size", []), qsize_simps)
||>> Local_Theory.note ((thms_suffix "size_eqvt", []), qsize_eqvt)
||>> Local_Theory.note ((thms_suffix "induct", []), [qinduct])
||>> Local_Theory.note ((thms_suffix "inducts", []), qinducts)
||>> Local_Theory.note ((thms_suffix "exhaust", []), qexhausts)
||>> Local_Theory.note ((thms_suffix "supports", []), qsupports_thms)
||>> Local_Theory.note ((thms_suffix "fsupp", []), qfsupp_thms)
||>> Local_Theory.note ((thms_suffix "supp", []), qsupp_constrs)
||>> Local_Theory.note ((thms_suffix "fresh", []), qfresh_constrs)
||>> Local_Theory.note ((thms_suffix "raw_alpha", []), alpha_intros)
||>> Local_Theory.note ((thms_suffix "perm_bn_simps", []), qperm_bn_simps)
||>> Local_Theory.note ((thms_suffix "bn_finite", []), qbn_finite_thms)
||>> Local_Theory.note ((thms_suffix "perm_bn_alpha", []), qperm_bn_alpha_thms)
||>> Local_Theory.note ((thms_suffix "permute_bn", []), qpermute_bn_thms)
in
lthy9'
end handle TEST ctxt => 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 o 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 o 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 o 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 o map2) complt args bclauses
end
*}
ML {*
fun nominal_datatype2_cmd (opt_thms_name, dt_strs, bn_fun_strs, bn_eq_strs) lthy =
let
val pre_typs =
map (fn (tvs, tname, mx, _) => (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
in
timeit (fn () => nominal_datatype2 opt_thms_name dts bn_funs bn_eqs bclauses' lthy)
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)
*}
end