(* Title: HOL/Tools/BNF/bnf_def.ML
Author: Dmitriy Traytel, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Definition of bounded natural functors.
*)
signature BNF_DEF =
sig
type bnf
type nonemptiness_witness = {I: int list, wit: term, prop: thm list}
val morph_bnf: morphism -> bnf -> bnf
val eq_bnf: bnf * bnf -> bool
val bnf_of: Proof.context -> string -> bnf option
val register_bnf: string -> (bnf * local_theory) -> (bnf * local_theory)
val name_of_bnf: bnf -> binding
val T_of_bnf: bnf -> typ
val live_of_bnf: bnf -> int
val lives_of_bnf: bnf -> typ list
val dead_of_bnf: bnf -> int
val deads_of_bnf: bnf -> typ list
val nwits_of_bnf: bnf -> int
val mapN: string
val relN: string
val setN: string
val mk_setN: int -> string
val mk_witN: int -> string
val map_of_bnf: bnf -> term
val sets_of_bnf: bnf -> term list
val rel_of_bnf: bnf -> term
val mk_T_of_bnf: typ list -> typ list -> bnf -> typ
val mk_bd_of_bnf: typ list -> typ list -> bnf -> term
val mk_map_of_bnf: typ list -> typ list -> typ list -> bnf -> term
val mk_rel_of_bnf: typ list -> typ list -> typ list -> bnf -> term
val mk_sets_of_bnf: typ list list -> typ list list -> bnf -> term list
val mk_wits_of_bnf: typ list list -> typ list list -> bnf -> (int list * term) list
val bd_Card_order_of_bnf: bnf -> thm
val bd_Cinfinite_of_bnf: bnf -> thm
val bd_Cnotzero_of_bnf: bnf -> thm
val bd_card_order_of_bnf: bnf -> thm
val bd_cinfinite_of_bnf: bnf -> thm
val collect_set_map_of_bnf: bnf -> thm
val in_bd_of_bnf: bnf -> thm
val in_cong_of_bnf: bnf -> thm
val in_mono_of_bnf: bnf -> thm
val in_rel_of_bnf: bnf -> thm
val map_comp0_of_bnf: bnf -> thm
val map_comp_of_bnf: bnf -> thm
val map_cong0_of_bnf: bnf -> thm
val map_cong_of_bnf: bnf -> thm
val map_def_of_bnf: bnf -> thm
val map_id0_of_bnf: bnf -> thm
val map_id_of_bnf: bnf -> thm
val map_transfer_of_bnf: bnf -> thm
val le_rel_OO_of_bnf: bnf -> thm
val rel_def_of_bnf: bnf -> thm
val rel_Grp_of_bnf: bnf -> thm
val rel_OO_of_bnf: bnf -> thm
val rel_OO_Grp_of_bnf: bnf -> thm
val rel_cong_of_bnf: bnf -> thm
val rel_conversep_of_bnf: bnf -> thm
val rel_mono_of_bnf: bnf -> thm
val rel_mono_strong_of_bnf: bnf -> thm
val rel_eq_of_bnf: bnf -> thm
val rel_flip_of_bnf: bnf -> thm
val set_bd_of_bnf: bnf -> thm list
val set_defs_of_bnf: bnf -> thm list
val set_map0_of_bnf: bnf -> thm list
val set_map_of_bnf: bnf -> thm list
val wit_thms_of_bnf: bnf -> thm list
val wit_thmss_of_bnf: bnf -> thm list list
val mk_map: int -> typ list -> typ list -> term -> term
val mk_rel: int -> typ list -> typ list -> term -> term
val build_map: Proof.context -> (typ * typ -> term) -> typ * typ -> term
val build_rel: Proof.context -> (typ * typ -> term) -> typ * typ -> term
val flatten_type_args_of_bnf: bnf -> 'a -> 'a list -> 'a list
val map_flattened_map_args: Proof.context -> string -> (term list -> 'a list) -> term list ->
'a list
val mk_witness: int list * term -> thm list -> nonemptiness_witness
val minimize_wits: (''a list * 'b) list -> (''a list * 'b) list
val wits_of_bnf: bnf -> nonemptiness_witness list
val zip_axioms: 'a -> 'a -> 'a -> 'a list -> 'a -> 'a -> 'a list -> 'a -> 'a -> 'a list
datatype const_policy = Dont_Inline | Hardly_Inline | Smart_Inline | Do_Inline
datatype fact_policy = Dont_Note | Note_Some | Note_All
val bnf_note_all: bool Config.T
val bnf_timing: bool Config.T
val user_policy: fact_policy -> Proof.context -> fact_policy
val note_bnf_thms: fact_policy -> (binding -> binding) -> binding -> bnf -> Proof.context ->
Proof.context
val print_bnfs: Proof.context -> unit
val prepare_def: const_policy -> (Proof.context -> fact_policy) -> (binding -> binding) ->
(Proof.context -> 'a -> typ) -> (Proof.context -> 'b -> term) -> typ list option ->
binding -> binding -> binding list ->
(((((binding * 'a) * 'b) * 'b list) * 'b) * 'b list) * 'b option -> Proof.context ->
string * term list *
((thm list -> {context: Proof.context, prems: thm list} -> tactic) option * term list list) *
((thm list -> thm list list) -> thm list list -> Proof.context -> bnf * local_theory) *
local_theory * thm list
val define_bnf_consts: const_policy -> fact_policy -> typ list option ->
binding -> binding -> binding list ->
(((((binding * typ) * term) * term list) * term) * term list) * term option -> local_theory ->
((typ list * typ list * typ list * typ) *
(term * term list * term * (int list * term) list * term) *
(thm * thm list * thm * thm list * thm) *
((typ list -> typ list -> typ list -> term) *
(typ list -> typ list -> term -> term) *
(typ list -> typ list -> typ -> typ) *
(typ list -> typ list -> typ list -> term) *
(typ list -> typ list -> typ list -> term))) * local_theory
val bnf_def: const_policy -> (Proof.context -> fact_policy) -> (binding -> binding) ->
({prems: thm list, context: Proof.context} -> tactic) list ->
({prems: thm list, context: Proof.context} -> tactic) -> typ list option -> binding ->
binding -> binding list ->
(((((binding * typ) * term) * term list) * term) * term list) * term option ->
local_theory -> bnf * local_theory
end;
structure BNF_Def : BNF_DEF =
struct
open BNF_Util
open BNF_Tactics
open BNF_Def_Tactics
val fundefcong_attrs = @{attributes [fundef_cong]};
type axioms = {
map_id0: thm,
map_comp0: thm,
map_cong0: thm,
set_map0: thm list,
bd_card_order: thm,
bd_cinfinite: thm,
set_bd: thm list,
le_rel_OO: thm,
rel_OO_Grp: thm
};
fun mk_axioms' ((((((((id, comp), cong), map), c_o), cinf), set_bd), le_rel_OO), rel) =
{map_id0 = id, map_comp0 = comp, map_cong0 = cong, set_map0 = map, bd_card_order = c_o,
bd_cinfinite = cinf, set_bd = set_bd, le_rel_OO = le_rel_OO, rel_OO_Grp = rel};
fun dest_cons [] = raise List.Empty
| dest_cons (x :: xs) = (x, xs);
fun mk_axioms n thms = thms
|> map the_single
|> dest_cons
||>> dest_cons
||>> dest_cons
||>> chop n
||>> dest_cons
||>> dest_cons
||>> chop n
||>> dest_cons
||> the_single
|> mk_axioms';
fun zip_axioms mid mcomp mcong smap bdco bdinf sbd le_rel_OO rel =
[mid, mcomp, mcong] @ smap @ [bdco, bdinf] @ sbd @ [le_rel_OO, rel];
fun dest_axioms {map_id0, map_comp0, map_cong0, set_map0, bd_card_order, bd_cinfinite, set_bd,
le_rel_OO, rel_OO_Grp} =
zip_axioms map_id0 map_comp0 map_cong0 set_map0 bd_card_order bd_cinfinite set_bd le_rel_OO
rel_OO_Grp;
fun map_axioms f {map_id0, map_comp0, map_cong0, set_map0, bd_card_order, bd_cinfinite, set_bd,
le_rel_OO, rel_OO_Grp} =
{map_id0 = f map_id0,
map_comp0 = f map_comp0,
map_cong0 = f map_cong0,
set_map0 = map f set_map0,
bd_card_order = f bd_card_order,
bd_cinfinite = f bd_cinfinite,
set_bd = map f set_bd,
le_rel_OO = f le_rel_OO,
rel_OO_Grp = f rel_OO_Grp};
val morph_axioms = map_axioms o Morphism.thm;
type defs = {
map_def: thm,
set_defs: thm list,
rel_def: thm
}
fun mk_defs map sets rel = {map_def = map, set_defs = sets, rel_def = rel};
fun map_defs f {map_def, set_defs, rel_def} =
{map_def = f map_def, set_defs = map f set_defs, rel_def = f rel_def};
val morph_defs = map_defs o Morphism.thm;
type facts = {
bd_Card_order: thm,
bd_Cinfinite: thm,
bd_Cnotzero: thm,
collect_set_map: thm lazy,
in_bd: thm lazy,
in_cong: thm lazy,
in_mono: thm lazy,
in_rel: thm lazy,
map_comp: thm lazy,
map_cong: thm lazy,
map_id: thm lazy,
map_transfer: thm lazy,
rel_eq: thm lazy,
rel_flip: thm lazy,
set_map: thm lazy list,
rel_cong: thm lazy,
rel_mono: thm lazy,
rel_mono_strong: thm lazy,
rel_Grp: thm lazy,
rel_conversep: thm lazy,
rel_OO: thm lazy
};
fun mk_facts bd_Card_order bd_Cinfinite bd_Cnotzero collect_set_map in_bd in_cong in_mono in_rel
map_comp map_cong map_id map_transfer rel_eq rel_flip set_map rel_cong rel_mono
rel_mono_strong rel_Grp rel_conversep rel_OO = {
bd_Card_order = bd_Card_order,
bd_Cinfinite = bd_Cinfinite,
bd_Cnotzero = bd_Cnotzero,
collect_set_map = collect_set_map,
in_bd = in_bd,
in_cong = in_cong,
in_mono = in_mono,
in_rel = in_rel,
map_comp = map_comp,
map_cong = map_cong,
map_id = map_id,
map_transfer = map_transfer,
rel_eq = rel_eq,
rel_flip = rel_flip,
set_map = set_map,
rel_cong = rel_cong,
rel_mono = rel_mono,
rel_mono_strong = rel_mono_strong,
rel_Grp = rel_Grp,
rel_conversep = rel_conversep,
rel_OO = rel_OO};
fun map_facts f {
bd_Card_order,
bd_Cinfinite,
bd_Cnotzero,
collect_set_map,
in_bd,
in_cong,
in_mono,
in_rel,
map_comp,
map_cong,
map_id,
map_transfer,
rel_eq,
rel_flip,
set_map,
rel_cong,
rel_mono,
rel_mono_strong,
rel_Grp,
rel_conversep,
rel_OO} =
{bd_Card_order = f bd_Card_order,
bd_Cinfinite = f bd_Cinfinite,
bd_Cnotzero = f bd_Cnotzero,
collect_set_map = Lazy.map f collect_set_map,
in_bd = Lazy.map f in_bd,
in_cong = Lazy.map f in_cong,
in_mono = Lazy.map f in_mono,
in_rel = Lazy.map f in_rel,
map_comp = Lazy.map f map_comp,
map_cong = Lazy.map f map_cong,
map_id = Lazy.map f map_id,
map_transfer = Lazy.map f map_transfer,
rel_eq = Lazy.map f rel_eq,
rel_flip = Lazy.map f rel_flip,
set_map = map (Lazy.map f) set_map,
rel_cong = Lazy.map f rel_cong,
rel_mono = Lazy.map f rel_mono,
rel_mono_strong = Lazy.map f rel_mono_strong,
rel_Grp = Lazy.map f rel_Grp,
rel_conversep = Lazy.map f rel_conversep,
rel_OO = Lazy.map f rel_OO};
val morph_facts = map_facts o Morphism.thm;
type nonemptiness_witness = {
I: int list,
wit: term,
prop: thm list
};
fun mk_witness (I, wit) prop = {I = I, wit = wit, prop = prop};
fun map_witness f g {I, wit, prop} = {I = I, wit = f wit, prop = map g prop};
fun morph_witness phi = map_witness (Morphism.term phi) (Morphism.thm phi);
datatype bnf = BNF of {
name: binding,
T: typ,
live: int,
lives: typ list, (*source type variables of map*)
lives': typ list, (*target type variables of map*)
dead: int,
deads: typ list,
map: term,
sets: term list,
bd: term,
axioms: axioms,
defs: defs,
facts: facts,
nwits: int,
wits: nonemptiness_witness list,
rel: term
};
(* getters *)
fun rep_bnf (BNF bnf) = bnf;
val name_of_bnf = #name o rep_bnf;
val T_of_bnf = #T o rep_bnf;
fun mk_T_of_bnf Ds Ts bnf =
let val bnf_rep = rep_bnf bnf
in Term.typ_subst_atomic ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts)) (#T bnf_rep) end;
val live_of_bnf = #live o rep_bnf;
val lives_of_bnf = #lives o rep_bnf;
val dead_of_bnf = #dead o rep_bnf;
val deads_of_bnf = #deads o rep_bnf;
val axioms_of_bnf = #axioms o rep_bnf;
val facts_of_bnf = #facts o rep_bnf;
val nwits_of_bnf = #nwits o rep_bnf;
val wits_of_bnf = #wits o rep_bnf;
fun flatten_type_args_of_bnf bnf dead_x xs =
let
val Type (_, Ts) = T_of_bnf bnf;
val lives = lives_of_bnf bnf;
val deads = deads_of_bnf bnf;
in
permute_like (op =) (deads @ lives) Ts (replicate (length deads) dead_x @ xs)
end;
(*terms*)
val map_of_bnf = #map o rep_bnf;
val sets_of_bnf = #sets o rep_bnf;
fun mk_map_of_bnf Ds Ts Us bnf =
let val bnf_rep = rep_bnf bnf;
in
Term.subst_atomic_types
((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts) @ (#lives' bnf_rep ~~ Us)) (#map bnf_rep)
end;
fun mk_sets_of_bnf Dss Tss bnf =
let val bnf_rep = rep_bnf bnf;
in
map2 (fn (Ds, Ts) => Term.subst_atomic_types
((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts))) (Dss ~~ Tss) (#sets bnf_rep)
end;
val bd_of_bnf = #bd o rep_bnf;
fun mk_bd_of_bnf Ds Ts bnf =
let val bnf_rep = rep_bnf bnf;
in Term.subst_atomic_types ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts)) (#bd bnf_rep) end;
fun mk_wits_of_bnf Dss Tss bnf =
let
val bnf_rep = rep_bnf bnf;
val wits = map (fn x => (#I x, #wit x)) (#wits bnf_rep);
in
map2 (fn (Ds, Ts) => apsnd (Term.subst_atomic_types
((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts)))) (Dss ~~ Tss) wits
end;
val rel_of_bnf = #rel o rep_bnf;
fun mk_rel_of_bnf Ds Ts Us bnf =
let val bnf_rep = rep_bnf bnf;
in
Term.subst_atomic_types
((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts) @ (#lives' bnf_rep ~~ Us)) (#rel bnf_rep)
end;
(*thms*)
val bd_card_order_of_bnf = #bd_card_order o #axioms o rep_bnf;
val bd_cinfinite_of_bnf = #bd_cinfinite o #axioms o rep_bnf;
val bd_Card_order_of_bnf = #bd_Card_order o #facts o rep_bnf;
val bd_Cinfinite_of_bnf = #bd_Cinfinite o #facts o rep_bnf;
val bd_Cnotzero_of_bnf = #bd_Cnotzero o #facts o rep_bnf;
val collect_set_map_of_bnf = Lazy.force o #collect_set_map o #facts o rep_bnf;
val in_bd_of_bnf = Lazy.force o #in_bd o #facts o rep_bnf;
val in_cong_of_bnf = Lazy.force o #in_cong o #facts o rep_bnf;
val in_mono_of_bnf = Lazy.force o #in_mono o #facts o rep_bnf;
val in_rel_of_bnf = Lazy.force o #in_rel o #facts o rep_bnf;
val map_def_of_bnf = #map_def o #defs o rep_bnf;
val map_id0_of_bnf = #map_id0 o #axioms o rep_bnf;
val map_id_of_bnf = Lazy.force o #map_id o #facts o rep_bnf;
val map_comp0_of_bnf = #map_comp0 o #axioms o rep_bnf;
val map_comp_of_bnf = Lazy.force o #map_comp o #facts o rep_bnf;
val map_cong0_of_bnf = #map_cong0 o #axioms o rep_bnf;
val map_cong_of_bnf = Lazy.force o #map_cong o #facts o rep_bnf;
val map_transfer_of_bnf = Lazy.force o #map_transfer o #facts o rep_bnf;
val le_rel_OO_of_bnf = #le_rel_OO o #axioms o rep_bnf;
val rel_def_of_bnf = #rel_def o #defs o rep_bnf;
val rel_eq_of_bnf = Lazy.force o #rel_eq o #facts o rep_bnf;
val rel_flip_of_bnf = Lazy.force o #rel_flip o #facts o rep_bnf;
val set_bd_of_bnf = #set_bd o #axioms o rep_bnf;
val set_defs_of_bnf = #set_defs o #defs o rep_bnf;
val set_map0_of_bnf = #set_map0 o #axioms o rep_bnf;
val set_map_of_bnf = map Lazy.force o #set_map o #facts o rep_bnf;
val rel_cong_of_bnf = Lazy.force o #rel_cong o #facts o rep_bnf;
val rel_mono_of_bnf = Lazy.force o #rel_mono o #facts o rep_bnf;
val rel_mono_strong_of_bnf = Lazy.force o #rel_mono_strong o #facts o rep_bnf;
val rel_Grp_of_bnf = Lazy.force o #rel_Grp o #facts o rep_bnf;
val rel_conversep_of_bnf = Lazy.force o #rel_conversep o #facts o rep_bnf;
val rel_OO_of_bnf = Lazy.force o #rel_OO o #facts o rep_bnf;
val rel_OO_Grp_of_bnf = #rel_OO_Grp o #axioms o rep_bnf;
val wit_thms_of_bnf = maps #prop o wits_of_bnf;
val wit_thmss_of_bnf = map #prop o wits_of_bnf;
fun mk_bnf name T live lives lives' dead deads map sets bd axioms defs facts wits rel =
BNF {name = name, T = T,
live = live, lives = lives, lives' = lives', dead = dead, deads = deads,
map = map, sets = sets, bd = bd,
axioms = axioms, defs = defs, facts = facts,
nwits = length wits, wits = wits, rel = rel};
fun morph_bnf phi (BNF {name = name, T = T, live = live, lives = lives, lives' = lives',
dead = dead, deads = deads, map = map, sets = sets, bd = bd,
axioms = axioms, defs = defs, facts = facts,
nwits = nwits, wits = wits, rel = rel}) =
BNF {name = Morphism.binding phi name, T = Morphism.typ phi T,
live = live, lives = List.map (Morphism.typ phi) lives,
lives' = List.map (Morphism.typ phi) lives',
dead = dead, deads = List.map (Morphism.typ phi) deads,
map = Morphism.term phi map, sets = List.map (Morphism.term phi) sets,
bd = Morphism.term phi bd,
axioms = morph_axioms phi axioms,
defs = morph_defs phi defs,
facts = morph_facts phi facts,
nwits = nwits,
wits = List.map (morph_witness phi) wits,
rel = Morphism.term phi rel};
fun eq_bnf (BNF {T = T1, live = live1, dead = dead1, ...},
BNF {T = T2, live = live2, dead = dead2, ...}) =
Type.could_unify (T1, T2) andalso live1 = live2 andalso dead1 = dead2;
structure Data = Generic_Data
(
type T = bnf Symtab.table;
val empty = Symtab.empty;
val extend = I;
val merge = Symtab.merge eq_bnf;
);
fun bnf_of ctxt =
Symtab.lookup (Data.get (Context.Proof ctxt))
#> Option.map (morph_bnf (Morphism.transfer_morphism (Proof_Context.theory_of ctxt)));
(* Utilities *)
fun normalize_set insts instA set =
let
val (T, T') = dest_funT (fastype_of set);
val A = fst (Term.dest_TVar (HOLogic.dest_setT T'));
val params = Term.add_tvar_namesT T [];
in Term.subst_TVars ((A :: params) ~~ (instA :: insts)) set end;
fun normalize_rel ctxt instTs instA instB rel =
let
val thy = Proof_Context.theory_of ctxt;
val tyenv =
Sign.typ_match thy (fastype_of rel, Library.foldr (op -->) (instTs, mk_pred2T instA instB))
Vartab.empty;
in Envir.subst_term (tyenv, Vartab.empty) rel end
handle Type.TYPE_MATCH => error "Bad relator";
fun normalize_wit insts CA As wit =
let
fun strip_param (Ts, T as Type (@{type_name fun}, [T1, T2])) =
if Type.raw_instance (CA, T) then (Ts, T) else strip_param (T1 :: Ts, T2)
| strip_param x = x;
val (Ts, T) = strip_param ([], fastype_of wit);
val subst = Term.add_tvar_namesT T [] ~~ insts;
fun find y = find_index (fn x => x = y) As;
in
(map (find o Term.typ_subst_TVars subst) (rev Ts), Term.subst_TVars subst wit)
end;
fun minimize_wits wits =
let
fun minimize done [] = done
| minimize done ((I, wit) :: todo) =
if exists (fn (J, _) => subset (op =) (J, I)) (done @ todo)
then minimize done todo
else minimize ((I, wit) :: done) todo;
in minimize [] wits end;
fun mk_map live Ts Us t =
let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
end;
fun mk_rel live Ts Us t =
let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
end;
fun build_map_or_rel mk const of_bnf dest ctxt build_simple =
let
fun build (TU as (T, U)) =
if T = U then
const T
else
(case TU of
(Type (s, Ts), Type (s', Us)) =>
if s = s' then
let
val bnf = the (bnf_of ctxt s);
val live = live_of_bnf bnf;
val mapx = mk live Ts Us (of_bnf bnf);
val TUs' = map dest (fst (strip_typeN live (fastype_of mapx)));
in Term.list_comb (mapx, map build TUs') end
else
build_simple TU
| _ => build_simple TU);
in build end;
val build_map = build_map_or_rel mk_map HOLogic.id_const map_of_bnf dest_funT;
val build_rel = build_map_or_rel mk_rel HOLogic.eq_const rel_of_bnf dest_pred2T;
fun map_flattened_map_args ctxt s map_args fs =
let
val flat_fs = flatten_type_args_of_bnf (the (bnf_of ctxt s)) Term.dummy fs;
val flat_fs' = map_args flat_fs;
in
permute_like (op aconv) flat_fs fs flat_fs'
end;
(* Names *)
val mapN = "map";
val setN = "set";
fun mk_setN i = setN ^ nonzero_string_of_int i;
val bdN = "bd";
val witN = "wit";
fun mk_witN i = witN ^ nonzero_string_of_int i;
val relN = "rel";
val bd_card_orderN = "bd_card_order";
val bd_cinfiniteN = "bd_cinfinite";
val bd_Card_orderN = "bd_Card_order";
val bd_CinfiniteN = "bd_Cinfinite";
val bd_CnotzeroN = "bd_Cnotzero";
val collect_set_mapN = "collect_set_map";
val in_bdN = "in_bd";
val in_monoN = "in_mono";
val in_relN = "in_rel";
val map_id0N = "map_id0";
val map_idN = "map_id";
val map_comp0N = "map_comp0";
val map_compN = "map_comp";
val map_cong0N = "map_cong0";
val map_congN = "map_cong";
val map_transferN = "map_transfer";
val rel_eqN = "rel_eq";
val rel_flipN = "rel_flip";
val set_map0N = "set_map0";
val set_mapN = "set_map";
val set_bdN = "set_bd";
val rel_GrpN = "rel_Grp";
val rel_conversepN = "rel_conversep";
val rel_monoN = "rel_mono"
val rel_mono_strongN = "rel_mono_strong"
val rel_comppN = "rel_compp";
val rel_compp_GrpN = "rel_compp_Grp";
datatype const_policy = Dont_Inline | Hardly_Inline | Smart_Inline | Do_Inline;
datatype fact_policy = Dont_Note | Note_Some | Note_All;
val bnf_note_all = Attrib.setup_config_bool @{binding bnf_note_all} (K false);
val bnf_timing = Attrib.setup_config_bool @{binding bnf_timing} (K false);
fun user_policy policy ctxt = if Config.get ctxt bnf_note_all then Note_All else policy;
val smart_max_inline_size = 25; (*FUDGE*)
fun note_bnf_thms fact_policy qualify' bnf_b bnf =
let
val axioms = axioms_of_bnf bnf;
val facts = facts_of_bnf bnf;
val wits = wits_of_bnf bnf;
val qualify =
let val (_, qs, _) = Binding.dest bnf_b;
in fold_rev (fn (s, mand) => Binding.qualify mand s) qs #> qualify' end;
in
(if fact_policy = Note_All then
let
val witNs = if length wits = 1 then [witN] else map mk_witN (1 upto length wits);
val notes =
[(bd_card_orderN, [#bd_card_order axioms]),
(bd_cinfiniteN, [#bd_cinfinite axioms]),
(bd_Card_orderN, [#bd_Card_order facts]),
(bd_CinfiniteN, [#bd_Cinfinite facts]),
(bd_CnotzeroN, [#bd_Cnotzero facts]),
(collect_set_mapN, [Lazy.force (#collect_set_map facts)]),
(in_bdN, [Lazy.force (#in_bd facts)]),
(in_monoN, [Lazy.force (#in_mono facts)]),
(in_relN, [Lazy.force (#in_rel facts)]),
(map_comp0N, [#map_comp0 axioms]),
(map_id0N, [#map_id0 axioms]),
(map_transferN, [Lazy.force (#map_transfer facts)]),
(rel_mono_strongN, [Lazy.force (#rel_mono_strong facts)]),
(set_map0N, #set_map0 axioms),
(set_bdN, #set_bd axioms)] @
(witNs ~~ wit_thmss_of_bnf bnf)
|> map (fn (thmN, thms) =>
((qualify (Binding.qualify true (Binding.name_of bnf_b) (Binding.name thmN)), []),
[(thms, [])]));
in
Local_Theory.notes notes #> snd
end
else
I)
#> (if fact_policy <> Dont_Note then
let
val notes =
[(map_compN, [Lazy.force (#map_comp facts)], []),
(map_cong0N, [#map_cong0 axioms], []),
(map_congN, [Lazy.force (#map_cong facts)], fundefcong_attrs),
(map_idN, [Lazy.force (#map_id facts)], []),
(rel_comppN, [Lazy.force (#rel_OO facts)], []),
(rel_compp_GrpN, no_refl [#rel_OO_Grp axioms], []),
(rel_conversepN, [Lazy.force (#rel_conversep facts)], []),
(rel_eqN, [Lazy.force (#rel_eq facts)], []),
(rel_flipN, [Lazy.force (#rel_flip facts)], []),
(rel_GrpN, [Lazy.force (#rel_Grp facts)], []),
(rel_monoN, [Lazy.force (#rel_mono facts)], []),
(set_mapN, map Lazy.force (#set_map facts), [])]
|> filter_out (null o #2)
|> map (fn (thmN, thms, attrs) =>
((qualify (Binding.qualify true (Binding.name_of bnf_b) (Binding.name thmN)),
attrs), [(thms, [])]));
in
Local_Theory.notes notes #> snd
end
else
I)
end;
(* Define new BNFs *)
fun define_bnf_consts const_policy fact_policy Ds_opt map_b rel_b set_bs
((((((bnf_b, T_rhs), map_rhs), set_rhss), bd_rhs), wit_rhss), rel_rhs_opt) no_defs_lthy =
let
val live = length set_rhss;
val def_qualify = Binding.conceal o Binding.qualify false (Binding.name_of bnf_b);
fun mk_prefix_binding pre = Binding.prefix_name (pre ^ "_") bnf_b;
fun maybe_define user_specified (b, rhs) lthy =
let
val inline =
(user_specified orelse fact_policy = Dont_Note) andalso
(case const_policy of
Dont_Inline => false
| Hardly_Inline => Term.is_Free rhs orelse Term.is_Const rhs
| Smart_Inline => Term.size_of_term rhs <= smart_max_inline_size
| Do_Inline => true)
in
if inline then
((rhs, Drule.reflexive_thm), lthy)
else
let val b = b () in
apfst (apsnd snd) (Local_Theory.define ((b, NoSyn), ((Thm.def_binding b, []), rhs))
lthy)
end
end;
fun maybe_restore lthy_old lthy =
lthy |> not (pointer_eq (lthy_old, lthy)) ? Local_Theory.restore;
val map_bind_def =
(fn () => def_qualify (if Binding.is_empty map_b then mk_prefix_binding mapN else map_b),
map_rhs);
val set_binds_defs =
let
fun set_name i get_b =
(case try (nth set_bs) (i - 1) of
SOME b => if Binding.is_empty b then get_b else K b
| NONE => get_b) #> def_qualify;
val bs = if live = 1 then [set_name 1 (fn () => mk_prefix_binding setN)]
else map (fn i => set_name i (fn () => mk_prefix_binding (mk_setN i))) (1 upto live);
in bs ~~ set_rhss end;
val bd_bind_def = (fn () => def_qualify (mk_prefix_binding bdN), bd_rhs);
val ((((bnf_map_term, raw_map_def),
(bnf_set_terms, raw_set_defs)),
(bnf_bd_term, raw_bd_def)), (lthy, lthy_old)) =
no_defs_lthy
|> maybe_define true map_bind_def
||>> apfst split_list o fold_map (maybe_define true) set_binds_defs
||>> maybe_define true bd_bind_def
||> `(maybe_restore no_defs_lthy);
val phi = Proof_Context.export_morphism lthy_old lthy;
val bnf_map_def = Morphism.thm phi raw_map_def;
val bnf_set_defs = map (Morphism.thm phi) raw_set_defs;
val bnf_bd_def = Morphism.thm phi raw_bd_def;
val bnf_map = Morphism.term phi bnf_map_term;
(*TODO: handle errors*)
(*simple shape analysis of a map function*)
val ((alphas, betas), (Calpha, _)) =
fastype_of bnf_map
|> strip_typeN live
|>> map_split dest_funT
||> dest_funT
handle TYPE _ => error "Bad map function";
val Calpha_params = map TVar (Term.add_tvarsT Calpha []);
val bnf_T = Morphism.typ phi T_rhs;
val bad_args = Term.add_tfreesT bnf_T [];
val _ = if null bad_args then () else error ("Locally fixed type arguments " ^
commas_quote (map (Syntax.string_of_typ no_defs_lthy o TFree) bad_args));
val bnf_sets =
map2 (normalize_set Calpha_params) alphas (map (Morphism.term phi) bnf_set_terms);
val bnf_bd =
Term.subst_TVars (Term.add_tvar_namesT bnf_T [] ~~ Calpha_params)
(Morphism.term phi bnf_bd_term);
(*TODO: assert Ds = (TVars of bnf_map) \ (alphas @ betas) as sets*)
val deads = (case Ds_opt of
NONE => subtract (op =) (alphas @ betas) (map TVar (Term.add_tvars bnf_map []))
| SOME Ds => map (Morphism.typ phi) Ds);
(*TODO: further checks of type of bnf_map*)
(*TODO: check types of bnf_sets*)
(*TODO: check type of bnf_bd*)
(*TODO: check type of bnf_rel*)
fun mk_bnf_map Ds As' Bs' =
Term.subst_atomic_types ((deads ~~ Ds) @ (alphas ~~ As') @ (betas ~~ Bs')) bnf_map;
fun mk_bnf_t Ds As' = Term.subst_atomic_types ((deads ~~ Ds) @ (alphas ~~ As'));
fun mk_bnf_T Ds As' = Term.typ_subst_atomic ((deads ~~ Ds) @ (alphas ~~ As'));
val (((As, Bs), Ds), names_lthy) = lthy
|> mk_TFrees live
||>> mk_TFrees live
||>> mk_TFrees (length deads);
val RTs = map2 (curry HOLogic.mk_prodT) As Bs;
val pred2RTs = map2 mk_pred2T As Bs;
val (Rs, Rs') = names_lthy |> mk_Frees' "R" pred2RTs |> fst
val CA = mk_bnf_T Ds As Calpha;
val CR = mk_bnf_T Ds RTs Calpha;
val setRs =
map3 (fn R => fn T => fn U =>
HOLogic.Collect_const (HOLogic.mk_prodT (T, U)) $ HOLogic.mk_split R) Rs As Bs;
(*Grp (in (Collect (split R1) .. Collect (split Rn))) (map fst .. fst)^--1 OO
Grp (in (Collect (split R1) .. Collect (split Rn))) (map snd .. snd)*)
val OO_Grp =
let
val map1 = Term.list_comb (mk_bnf_map Ds RTs As, map fst_const RTs);
val map2 = Term.list_comb (mk_bnf_map Ds RTs Bs, map snd_const RTs);
val bnf_in = mk_in setRs (map (mk_bnf_t Ds RTs) bnf_sets) CR;
in
mk_rel_compp (mk_conversep (mk_Grp bnf_in map1), mk_Grp bnf_in map2)
|> fold_rev Term.absfree Rs'
end;
val rel_rhs = the_default OO_Grp rel_rhs_opt;
val rel_bind_def =
(fn () => def_qualify (if Binding.is_empty rel_b then mk_prefix_binding relN else rel_b),
rel_rhs);
val wit_rhss =
if null wit_rhss then
[fold_rev Term.absdummy As (Term.list_comb (mk_bnf_map Ds As As,
map2 (fn T => fn i => Term.absdummy T (Bound i)) As (live downto 1)) $
Const (@{const_name undefined}, CA))]
else wit_rhss;
val nwits = length wit_rhss;
val wit_binds_defs =
let
val bs = if nwits = 1 then [fn () => def_qualify (mk_prefix_binding witN)]
else map (fn i => fn () => def_qualify (mk_prefix_binding (mk_witN i))) (1 upto nwits);
in bs ~~ wit_rhss end;
val (((bnf_rel_term, raw_rel_def), (bnf_wit_terms, raw_wit_defs)), (lthy, lthy_old)) =
lthy
|> maybe_define (is_some rel_rhs_opt) rel_bind_def
||>> apfst split_list o fold_map (maybe_define (not (null wit_rhss))) wit_binds_defs
||> `(maybe_restore lthy);
val phi = Proof_Context.export_morphism lthy_old lthy;
val bnf_rel_def = Morphism.thm phi raw_rel_def;
val bnf_rel = Morphism.term phi bnf_rel_term;
fun mk_bnf_rel Ds As' Bs' =
normalize_rel lthy (map2 mk_pred2T As' Bs') (mk_bnf_T Ds As' Calpha) (mk_bnf_T Ds Bs' Calpha)
bnf_rel;
val bnf_wit_defs = map (Morphism.thm phi) raw_wit_defs;
val bnf_wits =
map (normalize_wit Calpha_params Calpha alphas o Morphism.term phi) bnf_wit_terms;
fun mk_OO_Grp Ds' As' Bs' =
Term.subst_atomic_types ((Ds ~~ Ds') @ (As ~~ As') @ (Bs ~~ Bs')) OO_Grp;
in
(((alphas, betas, deads, Calpha),
(bnf_map, bnf_sets, bnf_bd, bnf_wits, bnf_rel),
(bnf_map_def, bnf_set_defs, bnf_bd_def, bnf_wit_defs, bnf_rel_def),
(mk_bnf_map, mk_bnf_t, mk_bnf_T, mk_bnf_rel, mk_OO_Grp)), lthy)
end;
fun prepare_def const_policy mk_fact_policy qualify prep_typ prep_term Ds_opt map_b rel_b set_bs
((((((raw_bnf_b, raw_bnf_T), raw_map), raw_sets), raw_bd), raw_wits), raw_rel_opt)
no_defs_lthy =
let
val fact_policy = mk_fact_policy no_defs_lthy;
val bnf_b = qualify raw_bnf_b;
val live = length raw_sets;
val T_rhs = prep_typ no_defs_lthy raw_bnf_T;
val map_rhs = prep_term no_defs_lthy raw_map;
val set_rhss = map (prep_term no_defs_lthy) raw_sets;
val bd_rhs = prep_term no_defs_lthy raw_bd;
val wit_rhss = map (prep_term no_defs_lthy) raw_wits;
val rel_rhs_opt = Option.map (prep_term no_defs_lthy) raw_rel_opt;
fun err T =
error ("Trying to register the type " ^ quote (Syntax.string_of_typ no_defs_lthy T) ^
" as unnamed BNF");
val (bnf_b, key) =
if Binding.eq_name (bnf_b, Binding.empty) then
(case T_rhs of
Type (C, Ts) => if forall (can dest_TFree) Ts
then (Binding.qualified_name C, C) else err T_rhs
| T => err T)
else (bnf_b, Local_Theory.full_name no_defs_lthy bnf_b);
val (((alphas, betas, deads, Calpha),
(bnf_map, bnf_sets, bnf_bd, bnf_wits, bnf_rel),
(bnf_map_def, bnf_set_defs, bnf_bd_def, bnf_wit_defs, bnf_rel_def),
(mk_bnf_map_Ds, mk_bnf_t_Ds, mk_bnf_T_Ds, _, mk_OO_Grp)), lthy) =
define_bnf_consts const_policy fact_policy Ds_opt map_b rel_b set_bs
((((((bnf_b, T_rhs), map_rhs), set_rhss), bd_rhs), wit_rhss), rel_rhs_opt) no_defs_lthy;
val dead = length deads;
val ((((((As', Bs'), Cs), Ds), B1Ts), B2Ts), (Ts, T)) = lthy
|> mk_TFrees live
||>> mk_TFrees live
||>> mk_TFrees live
||>> mk_TFrees dead
||>> mk_TFrees live
||>> mk_TFrees live
||> fst o mk_TFrees 1
||> the_single
||> `(replicate live);
val mk_bnf_map = mk_bnf_map_Ds Ds;
val mk_bnf_t = mk_bnf_t_Ds Ds;
val mk_bnf_T = mk_bnf_T_Ds Ds;
val pred2RTs = map2 mk_pred2T As' Bs';
val pred2RTsAsCs = map2 mk_pred2T As' Cs;
val pred2RTsBsCs = map2 mk_pred2T Bs' Cs;
val pred2RT's = map2 mk_pred2T Bs' As';
val self_pred2RTs = map2 mk_pred2T As' As';
val transfer_domRTs = map2 mk_pred2T As' B1Ts;
val transfer_ranRTs = map2 mk_pred2T Bs' B2Ts;
val CA' = mk_bnf_T As' Calpha;
val CB' = mk_bnf_T Bs' Calpha;
val CC' = mk_bnf_T Cs Calpha;
val CB1 = mk_bnf_T B1Ts Calpha;
val CB2 = mk_bnf_T B2Ts Calpha;
val bnf_map_AsAs = mk_bnf_map As' As';
val bnf_map_AsBs = mk_bnf_map As' Bs';
val bnf_map_AsCs = mk_bnf_map As' Cs;
val bnf_map_BsCs = mk_bnf_map Bs' Cs;
val bnf_sets_As = map (mk_bnf_t As') bnf_sets;
val bnf_sets_Bs = map (mk_bnf_t Bs') bnf_sets;
val bnf_bd_As = mk_bnf_t As' bnf_bd;
fun mk_bnf_rel RTs CA CB = normalize_rel lthy RTs CA CB bnf_rel;
val pre_names_lthy = lthy;
val (((((((((((((((fs, gs), hs), x), y), zs), ys), As),
As_copy), bs), Rs), Rs_copy), Ss),
transfer_domRs), transfer_ranRs), names_lthy) = pre_names_lthy
|> mk_Frees "f" (map2 (curry op -->) As' Bs')
||>> mk_Frees "g" (map2 (curry op -->) Bs' Cs)
||>> mk_Frees "h" (map2 (curry op -->) As' Ts)
||>> yield_singleton (mk_Frees "x") CA'
||>> yield_singleton (mk_Frees "y") CB'
||>> mk_Frees "z" As'
||>> mk_Frees "y" Bs'
||>> mk_Frees "A" (map HOLogic.mk_setT As')
||>> mk_Frees "A" (map HOLogic.mk_setT As')
||>> mk_Frees "b" As'
||>> mk_Frees "R" pred2RTs
||>> mk_Frees "R" pred2RTs
||>> mk_Frees "S" pred2RTsBsCs
||>> mk_Frees "R" transfer_domRTs
||>> mk_Frees "S" transfer_ranRTs;
val fs_copy = map2 (retype_free o fastype_of) fs gs;
val x_copy = retype_free CA' y;
val rel = mk_bnf_rel pred2RTs CA' CB';
val relAsAs = mk_bnf_rel self_pred2RTs CA' CA';
val bnf_wit_As = map (apsnd (mk_bnf_t As')) bnf_wits;
val map_id0_goal =
let val bnf_map_app_id = Term.list_comb (bnf_map_AsAs, map HOLogic.id_const As') in
mk_Trueprop_eq (bnf_map_app_id, HOLogic.id_const CA')
end;
val map_comp0_goal =
let
val bnf_map_app_comp = Term.list_comb (bnf_map_AsCs, map2 (curry HOLogic.mk_comp) gs fs);
val comp_bnf_map_app = HOLogic.mk_comp
(Term.list_comb (bnf_map_BsCs, gs), Term.list_comb (bnf_map_AsBs, fs));
in
fold_rev Logic.all (fs @ gs) (mk_Trueprop_eq (bnf_map_app_comp, comp_bnf_map_app))
end;
fun mk_map_cong_prem x z set f f_copy =
Logic.all z (Logic.mk_implies
(HOLogic.mk_Trueprop (HOLogic.mk_mem (z, set $ x)),
mk_Trueprop_eq (f $ z, f_copy $ z)));
val map_cong0_goal =
let
val prems = map4 (mk_map_cong_prem x) zs bnf_sets_As fs fs_copy;
val eq = mk_Trueprop_eq (Term.list_comb (bnf_map_AsBs, fs) $ x,
Term.list_comb (bnf_map_AsBs, fs_copy) $ x);
in
fold_rev Logic.all (x :: fs @ fs_copy) (Logic.list_implies (prems, eq))
end;
val set_map0s_goal =
let
fun mk_goal setA setB f =
let
val set_comp_map =
HOLogic.mk_comp (setB, Term.list_comb (bnf_map_AsBs, fs));
val image_comp_set = HOLogic.mk_comp (mk_image f, setA);
in
fold_rev Logic.all fs (mk_Trueprop_eq (set_comp_map, image_comp_set))
end;
in
map3 mk_goal bnf_sets_As bnf_sets_Bs fs
end;
val card_order_bd_goal = HOLogic.mk_Trueprop (mk_card_order bnf_bd_As);
val cinfinite_bd_goal = HOLogic.mk_Trueprop (mk_cinfinite bnf_bd_As);
val set_bds_goal =
let
fun mk_goal set =
Logic.all x (HOLogic.mk_Trueprop (mk_ordLeq (mk_card_of (set $ x)) bnf_bd_As));
in
map mk_goal bnf_sets_As
end;
val relAsCs = mk_bnf_rel pred2RTsAsCs CA' CC';
val relBsCs = mk_bnf_rel pred2RTsBsCs CB' CC';
val rel_OO_lhs = Term.list_comb (relAsCs, map2 (curry mk_rel_compp) Rs Ss);
val rel_OO_rhs = mk_rel_compp (Term.list_comb (rel, Rs), Term.list_comb (relBsCs, Ss));
val le_rel_OO_goal =
fold_rev Logic.all (Rs @ Ss) (HOLogic.mk_Trueprop (mk_leq rel_OO_rhs rel_OO_lhs));
val rel_OO_Grp_goal = fold_rev Logic.all Rs (mk_Trueprop_eq (Term.list_comb (rel, Rs),
Term.list_comb (mk_OO_Grp Ds As' Bs', Rs)));
val goals = zip_axioms map_id0_goal map_comp0_goal map_cong0_goal set_map0s_goal
card_order_bd_goal cinfinite_bd_goal set_bds_goal le_rel_OO_goal rel_OO_Grp_goal;
fun mk_wit_goals (I, wit) =
let
val xs = map (nth bs) I;
fun wit_goal i =
let
val z = nth zs i;
val set_wit = nth bnf_sets_As i $ Term.list_comb (wit, xs);
val concl = HOLogic.mk_Trueprop
(if member (op =) I i then HOLogic.mk_eq (z, nth bs i)
else @{term False});
in
fold_rev Logic.all (z :: xs)
(Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (z, set_wit)), concl))
end;
in
map wit_goal (0 upto live - 1)
end;
val triv_wit_tac = mk_trivial_wit_tac bnf_wit_defs;
val wit_goalss =
(if null raw_wits then SOME triv_wit_tac else NONE, map mk_wit_goals bnf_wit_As);
fun after_qed mk_wit_thms thms lthy =
let
val (axioms, nontriv_wit_thms) = apfst (mk_axioms live) (chop (length goals) thms);
val bd_Card_order = #bd_card_order axioms RS @{thm conjunct2[OF card_order_on_Card_order]};
val bd_Cinfinite = @{thm conjI} OF [#bd_cinfinite axioms, bd_Card_order];
val bd_Cnotzero = bd_Cinfinite RS @{thm Cinfinite_Cnotzero};
fun mk_collect_set_map () =
let
val defT = mk_bnf_T Ts Calpha --> HOLogic.mk_setT T;
val collect_map = HOLogic.mk_comp
(mk_collect (map (mk_bnf_t Ts) bnf_sets) defT,
Term.list_comb (mk_bnf_map As' Ts, hs));
val image_collect = mk_collect
(map2 (fn h => fn set => HOLogic.mk_comp (mk_image h, set)) hs bnf_sets_As)
defT;
(*collect {set1 ... setm} o map f1 ... fm = collect {f1` o set1 ... fm` o setm}*)
val goal = fold_rev Logic.all hs (mk_Trueprop_eq (collect_map, image_collect));
in
Goal.prove_sorry lthy [] [] goal (K (mk_collect_set_map_tac (#set_map0 axioms)))
|> Thm.close_derivation
end;
val collect_set_map = Lazy.lazy mk_collect_set_map;
fun mk_in_mono () =
let
val prems_mono = map2 (HOLogic.mk_Trueprop oo mk_leq) As As_copy;
val in_mono_goal =
fold_rev Logic.all (As @ As_copy)
(Logic.list_implies (prems_mono, HOLogic.mk_Trueprop
(mk_leq (mk_in As bnf_sets_As CA') (mk_in As_copy bnf_sets_As CA'))));
in
Goal.prove_sorry lthy [] [] in_mono_goal (K (mk_in_mono_tac live))
|> Thm.close_derivation
end;
val in_mono = Lazy.lazy mk_in_mono;
fun mk_in_cong () =
let
val prems_cong = map2 (curry mk_Trueprop_eq) As As_copy;
val in_cong_goal =
fold_rev Logic.all (As @ As_copy)
(Logic.list_implies (prems_cong,
mk_Trueprop_eq (mk_in As bnf_sets_As CA', mk_in As_copy bnf_sets_As CA')));
in
Goal.prove_sorry lthy [] [] in_cong_goal
(K ((TRY o hyp_subst_tac lthy THEN' rtac refl) 1))
|> Thm.close_derivation
end;
val in_cong = Lazy.lazy mk_in_cong;
val map_id = Lazy.lazy (fn () => mk_map_id (#map_id0 axioms));
val map_comp = Lazy.lazy (fn () => mk_map_comp (#map_comp0 axioms));
fun mk_map_cong () =
let
val prem0 = mk_Trueprop_eq (x, x_copy);
val prems = map4 (mk_map_cong_prem x_copy) zs bnf_sets_As fs fs_copy;
val eq = mk_Trueprop_eq (Term.list_comb (bnf_map_AsBs, fs) $ x,
Term.list_comb (bnf_map_AsBs, fs_copy) $ x_copy);
val goal = fold_rev Logic.all (x :: x_copy :: fs @ fs_copy)
(Logic.list_implies (prem0 :: prems, eq));
in
Goal.prove_sorry lthy [] [] goal (fn _ => mk_map_cong_tac lthy (#map_cong0 axioms))
|> Thm.close_derivation
end;
val map_cong = Lazy.lazy mk_map_cong;
val set_map = map (fn thm => Lazy.lazy (fn () => mk_set_map thm)) (#set_map0 axioms);
val wit_thms =
if null nontriv_wit_thms then mk_wit_thms (map Lazy.force set_map) else nontriv_wit_thms;
fun mk_in_bd () =
let
val bdT = fst (dest_relT (fastype_of bnf_bd_As));
val bdTs = replicate live bdT;
val bd_bnfT = mk_bnf_T bdTs Calpha;
val surj_imp_ordLeq_inst = (if live = 0 then TrueI else
let
val ranTs = map (fn AT => mk_sumT (AT, HOLogic.unitT)) As';
val funTs = map (fn T => bdT --> T) ranTs;
val ran_bnfT = mk_bnf_T ranTs Calpha;
val (revTs, Ts) = `rev (bd_bnfT :: funTs);
val cTs = map (SOME o certifyT lthy) [ran_bnfT, Library.foldr1 HOLogic.mk_prodT Ts];
val tinst = fold (fn T => fn t => HOLogic.mk_split (Term.absdummy T t)) (tl revTs)
(Term.absdummy (hd revTs) (Term.list_comb (mk_bnf_map bdTs ranTs,
map Bound (live - 1 downto 0)) $ Bound live));
val cts = [NONE, SOME (certify lthy tinst)];
in
Drule.instantiate' cTs cts @{thm surj_imp_ordLeq}
end);
val bd = mk_cexp
(if live = 0 then ctwo
else mk_csum (Library.foldr1 (uncurry mk_csum) (map mk_card_of As)) ctwo)
(mk_csum bnf_bd_As (mk_card_of (HOLogic.mk_UNIV bd_bnfT)));
val in_bd_goal =
fold_rev Logic.all As
(HOLogic.mk_Trueprop (mk_ordLeq (mk_card_of (mk_in As bnf_sets_As CA')) bd));
in
Goal.prove_sorry lthy [] [] in_bd_goal
(mk_in_bd_tac live surj_imp_ordLeq_inst
(Lazy.force map_comp) (Lazy.force map_id) (#map_cong0 axioms)
(map Lazy.force set_map) (#set_bd axioms) (#bd_card_order axioms)
bd_Card_order bd_Cinfinite bd_Cnotzero)
|> Thm.close_derivation
end;
val in_bd = Lazy.lazy mk_in_bd;
val rel_OO_Grp = #rel_OO_Grp axioms;
val rel_OO_Grps = no_refl [rel_OO_Grp];
fun mk_rel_Grp () =
let
val lhs = Term.list_comb (rel, map2 mk_Grp As fs);
val rhs = mk_Grp (mk_in As bnf_sets_As CA') (Term.list_comb (bnf_map_AsBs, fs));
val goal = fold_rev Logic.all (As @ fs) (mk_Trueprop_eq (lhs, rhs));
in
Goal.prove_sorry lthy [] [] goal
(mk_rel_Grp_tac rel_OO_Grps (#map_id0 axioms) (#map_cong0 axioms) (Lazy.force map_id)
(Lazy.force map_comp) (map Lazy.force set_map))
|> Thm.close_derivation
end;
val rel_Grp = Lazy.lazy mk_rel_Grp;
fun mk_rel_prems f = map2 (HOLogic.mk_Trueprop oo f) Rs Rs_copy
fun mk_rel_concl f = HOLogic.mk_Trueprop
(f (Term.list_comb (rel, Rs), Term.list_comb (rel, Rs_copy)));
fun mk_rel_mono () =
let
val mono_prems = mk_rel_prems mk_leq;
val mono_concl = mk_rel_concl (uncurry mk_leq);
in
Goal.prove_sorry lthy [] []
(fold_rev Logic.all (Rs @ Rs_copy) (Logic.list_implies (mono_prems, mono_concl)))
(K (mk_rel_mono_tac rel_OO_Grps (Lazy.force in_mono)))
|> Thm.close_derivation
end;
fun mk_rel_cong () =
let
val cong_prems = mk_rel_prems (curry HOLogic.mk_eq);
val cong_concl = mk_rel_concl HOLogic.mk_eq;
in
Goal.prove_sorry lthy [] []
(fold_rev Logic.all (Rs @ Rs_copy) (Logic.list_implies (cong_prems, cong_concl)))
(fn _ => (TRY o hyp_subst_tac lthy THEN' rtac refl) 1)
|> Thm.close_derivation
end;
val rel_mono = Lazy.lazy mk_rel_mono;
val rel_cong = Lazy.lazy mk_rel_cong;
fun mk_rel_eq () =
Goal.prove_sorry lthy [] []
(mk_Trueprop_eq (Term.list_comb (relAsAs, map HOLogic.eq_const As'),
HOLogic.eq_const CA'))
(K (mk_rel_eq_tac live (Lazy.force rel_Grp) (Lazy.force rel_cong) (#map_id0 axioms)))
|> Thm.close_derivation;
val rel_eq = Lazy.lazy mk_rel_eq;
fun mk_rel_conversep () =
let
val relBsAs = mk_bnf_rel pred2RT's CB' CA';
val lhs = Term.list_comb (relBsAs, map mk_conversep Rs);
val rhs = mk_conversep (Term.list_comb (rel, Rs));
val le_goal = fold_rev Logic.all Rs (HOLogic.mk_Trueprop (mk_leq lhs rhs));
val le_thm = Goal.prove_sorry lthy [] [] le_goal
(mk_rel_conversep_le_tac rel_OO_Grps (Lazy.force rel_eq) (#map_cong0 axioms)
(Lazy.force map_comp) (map Lazy.force set_map))
|> Thm.close_derivation
val goal = fold_rev Logic.all Rs (mk_Trueprop_eq (lhs, rhs));
in
Goal.prove_sorry lthy [] [] goal
(K (mk_rel_conversep_tac le_thm (Lazy.force rel_mono)))
|> Thm.close_derivation
end;
val rel_conversep = Lazy.lazy mk_rel_conversep;
fun mk_rel_OO () =
Goal.prove_sorry lthy [] []
(fold_rev Logic.all (Rs @ Ss) (HOLogic.mk_Trueprop (mk_leq rel_OO_lhs rel_OO_rhs)))
(mk_rel_OO_le_tac rel_OO_Grps (Lazy.force rel_eq) (#map_cong0 axioms)
(Lazy.force map_comp) (map Lazy.force set_map))
|> Thm.close_derivation
|> (fn thm => @{thm antisym} OF [thm, #le_rel_OO axioms]);
val rel_OO = Lazy.lazy mk_rel_OO;
fun mk_in_rel () = trans OF [rel_OO_Grp, @{thm OO_Grp_alt}] RS @{thm predicate2_eqD};
val in_rel = Lazy.lazy mk_in_rel;
fun mk_rel_flip () =
let
val rel_conversep_thm = Lazy.force rel_conversep;
val cts = map (SOME o certify lthy) Rs;
val rel_conversep_thm' = cterm_instantiate_pos cts rel_conversep_thm;
in
unfold_thms lthy @{thms conversep_iff} (rel_conversep_thm' RS @{thm predicate2_eqD})
|> singleton (Proof_Context.export names_lthy pre_names_lthy)
end;
val rel_flip = Lazy.lazy mk_rel_flip;
fun mk_rel_mono_strong () =
let
fun mk_prem setA setB R S a b =
HOLogic.mk_Trueprop
(mk_Ball (setA $ x) (Term.absfree (dest_Free a)
(mk_Ball (setB $ y) (Term.absfree (dest_Free b)
(HOLogic.mk_imp (R $ a $ b, S $ a $ b))))));
val prems = HOLogic.mk_Trueprop (Term.list_comb (rel, Rs) $ x $ y) ::
map6 mk_prem bnf_sets_As bnf_sets_Bs Rs Rs_copy zs ys;
val concl = HOLogic.mk_Trueprop (Term.list_comb (rel, Rs_copy) $ x $ y);
in
Goal.prove_sorry lthy [] []
(fold_rev Logic.all (x :: y :: Rs @ Rs_copy) (Logic.list_implies (prems, concl)))
(mk_rel_mono_strong_tac (Lazy.force in_rel) (map Lazy.force set_map))
|> Thm.close_derivation
end;
val rel_mono_strong = Lazy.lazy mk_rel_mono_strong;
fun mk_map_transfer () =
let
val rels = map2 mk_fun_rel transfer_domRs transfer_ranRs;
val rel = mk_fun_rel
(Term.list_comb (mk_bnf_rel transfer_domRTs CA' CB1, transfer_domRs))
(Term.list_comb (mk_bnf_rel transfer_ranRTs CB' CB2, transfer_ranRs));
val concl = HOLogic.mk_Trueprop
(fold_rev mk_fun_rel rels rel $ bnf_map_AsBs $ mk_bnf_map B1Ts B2Ts);
in
Goal.prove_sorry lthy [] []
(fold_rev Logic.all (transfer_domRs @ transfer_ranRs) concl)
(mk_map_transfer_tac (Lazy.force rel_mono) (Lazy.force in_rel)
(map Lazy.force set_map) (#map_cong0 axioms) (Lazy.force map_comp))
|> Thm.close_derivation
end;
val map_transfer = Lazy.lazy mk_map_transfer;
val defs = mk_defs bnf_map_def bnf_set_defs bnf_rel_def;
val facts = mk_facts bd_Card_order bd_Cinfinite bd_Cnotzero collect_set_map in_bd in_cong
in_mono in_rel map_comp map_cong map_id map_transfer rel_eq rel_flip set_map
rel_cong rel_mono rel_mono_strong rel_Grp rel_conversep rel_OO;
val wits = map2 mk_witness bnf_wits wit_thms;
val bnf_rel =
Term.subst_atomic_types ((Ds ~~ deads) @ (As' ~~ alphas) @ (Bs' ~~ betas)) rel;
val bnf = mk_bnf bnf_b Calpha live alphas betas dead deads bnf_map bnf_sets bnf_bd axioms
defs facts wits bnf_rel;
in
(bnf, lthy |> note_bnf_thms fact_policy qualify bnf_b bnf)
end;
val one_step_defs =
no_reflexive (bnf_map_def :: bnf_bd_def :: bnf_set_defs @ bnf_wit_defs @ [bnf_rel_def]);
in
(key, goals, wit_goalss, after_qed, lthy, one_step_defs)
end;
fun register_bnf key (bnf, lthy) =
(bnf, Local_Theory.declaration {syntax = false, pervasive = true}
(fn phi => Data.map (Symtab.default (key, morph_bnf phi bnf))) lthy);
fun bnf_def const_policy fact_policy qualify tacs wit_tac Ds map_b rel_b set_bs =
(fn (_, goals, (triv_tac_opt, wit_goalss), after_qed, lthy, one_step_defs) =>
let
fun mk_wits_tac set_maps =
K (TRYALL Goal.conjunction_tac) THEN'
(case triv_tac_opt of
SOME tac => tac set_maps
| NONE => fn {context = ctxt, prems} =>
unfold_thms_tac ctxt one_step_defs THEN wit_tac {context = ctxt, prems = prems});
val wit_goals = map Logic.mk_conjunction_balanced wit_goalss;
fun mk_wit_thms set_maps =
Goal.prove_sorry lthy [] [] (Logic.mk_conjunction_balanced wit_goals) (mk_wits_tac set_maps)
|> Conjunction.elim_balanced (length wit_goals)
|> map2 (Conjunction.elim_balanced o length) wit_goalss
|> map (map (Thm.close_derivation o Thm.forall_elim_vars 0));
in
map2 (Thm.close_derivation oo Goal.prove_sorry lthy [] [])
goals (map (fn tac => fn {context = ctxt, prems} =>
unfold_thms_tac ctxt one_step_defs THEN tac {context = ctxt, prems = prems}) tacs)
|> (fn thms => after_qed mk_wit_thms (map single thms) lthy)
end) oo prepare_def const_policy fact_policy qualify (K I) (K I) Ds map_b rel_b set_bs;
val bnf_cmd = (fn (key, goals, (triv_tac_opt, wit_goalss), after_qed, lthy, defs) =>
let
val wit_goals = map Logic.mk_conjunction_balanced wit_goalss;
fun mk_triv_wit_thms tac set_maps =
Goal.prove_sorry lthy [] [] (Logic.mk_conjunction_balanced wit_goals)
(K (TRYALL Goal.conjunction_tac) THEN' tac set_maps)
|> Conjunction.elim_balanced (length wit_goals)
|> map2 (Conjunction.elim_balanced o length) wit_goalss
|> map (map (Thm.close_derivation o Thm.forall_elim_vars 0));
val (mk_wit_thms, nontriv_wit_goals) =
(case triv_tac_opt of
NONE => (fn _ => [], map (map (rpair [])) wit_goalss)
| SOME tac => (mk_triv_wit_thms tac, []));
in
Proof.unfolding ([[(defs, [])]])
(Proof.theorem NONE (snd o register_bnf key oo after_qed mk_wit_thms)
(map (single o rpair []) goals @ nontriv_wit_goals) lthy)
end) oo prepare_def Do_Inline (user_policy Note_Some) I Syntax.read_typ Syntax.read_term NONE
Binding.empty Binding.empty [];
fun print_bnfs ctxt =
let
fun pretty_set sets i = Pretty.block
[Pretty.str (mk_setN (i + 1) ^ ":"), Pretty.brk 1,
Pretty.quote (Syntax.pretty_term ctxt (nth sets i))];
fun pretty_bnf (key, BNF {T = T, map = map, sets = sets, bd = bd,
live = live, lives = lives, dead = dead, deads = deads, ...}) =
Pretty.big_list
(Pretty.string_of (Pretty.block [Pretty.str key, Pretty.str ":", Pretty.brk 1,
Pretty.quote (Syntax.pretty_typ ctxt T)]))
([Pretty.block [Pretty.str "live:", Pretty.brk 1, Pretty.str (string_of_int live),
Pretty.brk 3, Pretty.list "[" "]" (List.map (Syntax.pretty_typ ctxt) lives)],
Pretty.block [Pretty.str "dead:", Pretty.brk 1, Pretty.str (string_of_int dead),
Pretty.brk 3, Pretty.list "[" "]" (List.map (Syntax.pretty_typ ctxt) deads)],
Pretty.block [Pretty.str (mapN ^ ":"), Pretty.brk 1,
Pretty.quote (Syntax.pretty_term ctxt map)]] @
List.map (pretty_set sets) (0 upto length sets - 1) @
[Pretty.block [Pretty.str (bdN ^ ":"), Pretty.brk 1,
Pretty.quote (Syntax.pretty_term ctxt bd)]]);
in
Pretty.big_list "BNFs:" (map pretty_bnf (Symtab.dest (Data.get (Context.Proof ctxt))))
|> Pretty.writeln
end;
val _ =
Outer_Syntax.improper_command @{command_spec "print_bnfs"}
"print all bounded natural functors"
(Scan.succeed (Toplevel.keep (print_bnfs o Toplevel.context_of)));
val _ =
Outer_Syntax.local_theory_to_proof @{command_spec "bnf"}
"register a type as a bounded natural functor"
(parse_opt_binding_colon -- Parse.typ --|
(Parse.reserved "map" -- @{keyword ":"}) -- Parse.term --
(Scan.option ((Parse.reserved "sets" -- @{keyword ":"}) |--
Scan.repeat1 (Scan.unless (Parse.reserved "bd") Parse.term)) >> the_default []) --|
(Parse.reserved "bd" -- @{keyword ":"}) -- Parse.term --
(Scan.option ((Parse.reserved "wits" -- @{keyword ":"}) |--
Scan.repeat1 (Scan.unless (Parse.reserved "rel") Parse.term)) >> the_default []) --
Scan.option ((Parse.reserved "rel" -- @{keyword ":"}) |-- Parse.term)
>> bnf_cmd);
end;