(* Title: HOL/Codatatype/Tools/bnf_comp.ML
Author: Dmitriy Traytel, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Composition of bounded natural functors.
*)
signature BNF_COMP =
sig
type unfold_thms
val empty_unfold: unfold_thms
val map_unfolds_of: unfold_thms -> thm list
val set_unfoldss_of: unfold_thms -> thm list list
val rel_unfolds_of: unfold_thms -> thm list
val pred_unfolds_of: unfold_thms -> thm list
val bnf_of_typ: BNF_Def.const_policy -> binding -> (binding -> binding) ->
((string * sort) list list -> (string * sort) list) -> typ -> unfold_thms * Proof.context ->
(BNF_Def.BNF * (typ list * typ list)) * (unfold_thms * Proof.context)
val bnf_of_typ_cmd: binding * string -> Proof.context -> Proof.context
val default_comp_sort: (string * sort) list list -> (string * sort) list
val normalize_bnfs: (int -> binding -> binding) -> ''a list list -> ''a list ->
(''a list list -> ''a list) -> BNF_Def.BNF list -> unfold_thms -> Proof.context ->
(int list list * ''a list) * (BNF_Def.BNF list * (unfold_thms * Proof.context))
val seal_bnf: unfold_thms -> binding -> typ list -> BNF_Def.BNF -> Proof.context ->
BNF_Def.BNF * local_theory
end;
structure BNF_Comp : BNF_COMP =
struct
open BNF_Def
open BNF_Util
open BNF_Tactics
open BNF_Comp_Tactics
type unfold_thms = {
map_unfolds: thm list,
set_unfoldss: thm list list,
rel_unfolds: thm list,
pred_unfolds: thm list
};
fun add_to_thms thms NONE = thms
| add_to_thms thms (SOME new) = if Thm.is_reflexive new then thms else insert Thm.eq_thm new thms;
fun adds_to_thms thms NONE = thms
| adds_to_thms thms (SOME news) = insert (eq_set Thm.eq_thm) (filter_refl news) thms;
fun mk_unfold_thms maps setss rels preds =
{map_unfolds = maps, set_unfoldss = setss, rel_unfolds = rels, pred_unfolds = preds};
val empty_unfold = mk_unfold_thms [] [] [] [];
fun add_to_unfold_opt map_opt sets_opt rel_opt pred_opt
{map_unfolds = maps, set_unfoldss = setss, rel_unfolds = rels, pred_unfolds = preds} = {
map_unfolds = add_to_thms maps map_opt,
set_unfoldss = adds_to_thms setss sets_opt,
rel_unfolds = add_to_thms rels rel_opt,
pred_unfolds = add_to_thms preds pred_opt};
fun add_to_unfold map sets rel pred =
add_to_unfold_opt (SOME map) (SOME sets) (SOME rel) (SOME pred);
val map_unfolds_of = #map_unfolds;
val set_unfoldss_of = #set_unfoldss;
val rel_unfolds_of = #rel_unfolds;
val pred_unfolds_of = #pred_unfolds;
val bdTN = "bdT";
val compN = "comp_"
fun mk_killN n = "kill" ^ string_of_int n ^ "_";
fun mk_liftN n = "lift" ^ string_of_int n ^ "_";
fun mk_permuteN src dest =
"permute_" ^ implode (map string_of_int src) ^ "_" ^ implode (map string_of_int dest) ^ "_";
val no_thm = refl;
val Collect_split_box_equals = box_equals RS @{thm Collect_split_cong};
val abs_pred_sym = sym RS @{thm abs_pred_def};
val abs_pred_sym_pred_abs = abs_pred_sym RS @{thm pred_def_abs};
(*copied from Envir.expand_term_free*)
fun expand_term_const defs =
let
val eqs = map ((fn ((x, U), u) => (x, (U, u))) o apfst dest_Const) defs;
val get = fn Const (x, _) => AList.lookup (op =) eqs x | _ => NONE;
in Envir.expand_term get end;
fun clean_compose_bnf const_policy qualify b outer inners (unfold, lthy) =
let
val olive = live_of_bnf outer;
val onwits = nwits_of_bnf outer;
val odead = dead_of_bnf outer;
val inner = hd inners;
val ilive = live_of_bnf inner;
val ideads = map dead_of_bnf inners;
val inwitss = map nwits_of_bnf inners;
(* TODO: check olive = length inners > 0,
forall inner from inners. ilive = live,
forall inner from inners. idead = dead *)
val (oDs, lthy1) = apfst (map TFree)
(Variable.invent_types (replicate odead HOLogic.typeS) lthy);
val (Dss, lthy2) = apfst (map (map TFree))
(fold_map Variable.invent_types (map (fn n => replicate n HOLogic.typeS) ideads) lthy1);
val (Ass, lthy3) = apfst (replicate ilive o map TFree)
(Variable.invent_types (replicate ilive HOLogic.typeS) lthy2);
val As = if ilive > 0 then hd Ass else [];
val Ass_repl = replicate olive As;
val (Bs, _(*lthy4*)) = apfst (map TFree)
(Variable.invent_types (replicate ilive HOLogic.typeS) lthy3);
val Bss_repl = replicate olive Bs;
val (((fs', Asets), xs), _(*names_lthy*)) = lthy
|> apfst snd o mk_Frees' "f" (map2 (curry (op -->)) As Bs)
||>> mk_Frees "A" (map (HOLogic.mk_setT) As)
||>> mk_Frees "x" As;
val CAs = map3 mk_T_of_bnf Dss Ass_repl inners;
val CCA = mk_T_of_bnf oDs CAs outer;
val CBs = map3 mk_T_of_bnf Dss Bss_repl inners;
val outer_sets = mk_sets_of_bnf (replicate olive oDs) (replicate olive CAs) outer;
val inner_setss = map3 mk_sets_of_bnf (map (replicate ilive) Dss) (replicate olive Ass) inners;
val inner_bds = map3 mk_bd_of_bnf Dss Ass_repl inners;
val outer_bd = mk_bd_of_bnf oDs CAs outer;
(*%f1 ... fn. outer.map (inner_1.map f1 ... fn) ... (inner_m.map f1 ... fn)*)
val comp_map = fold_rev Term.abs fs'
(Term.list_comb (mk_map_of_bnf oDs CAs CBs outer,
map2 (fn Ds => (fn f => Term.list_comb (f, map Bound ((ilive - 1) downto 0))) o
mk_map_of_bnf Ds As Bs) Dss inners));
(*Union o collect {outer.set_1 ... outer.set_m} o outer.map inner_1.set_i ... inner_m.set_i*)
(*Union o collect {image inner_1.set_i o outer.set_1 ... image inner_m.set_i o outer.set_m}*)
fun mk_comp_set i =
let
val (setTs, T) = `(replicate olive o HOLogic.mk_setT) (nth As i);
val outer_set = mk_collect
(mk_sets_of_bnf (replicate olive oDs) (replicate olive setTs) outer)
(mk_T_of_bnf oDs setTs outer --> HOLogic.mk_setT T);
val inner_sets = map (fn sets => nth sets i) inner_setss;
val outer_map = mk_map_of_bnf oDs CAs setTs outer;
val map_inner_sets = Term.list_comb (outer_map, inner_sets);
val collect_image = mk_collect
(map2 (fn f => fn set => HOLogic.mk_comp (mk_image f, set)) inner_sets outer_sets)
(CCA --> HOLogic.mk_setT T);
in
(Library.foldl1 HOLogic.mk_comp [mk_Union T, outer_set, map_inner_sets],
HOLogic.mk_comp (mk_Union T, collect_image))
end;
val (comp_sets, comp_sets_alt) = map_split mk_comp_set (0 upto ilive - 1);
(*(inner_1.bd +c ... +c inner_m.bd) *c outer.bd*)
val comp_bd = Term.absdummy CCA (mk_cprod
(Library.foldr1 (uncurry mk_csum) inner_bds) outer_bd);
fun comp_map_id_tac {context = ctxt, ...} =
let
(*order the theorems by reverse size to prevent bad interaction with nonconfluent rewrite
rules*)
val thms = (map map_id_of_bnf inners
|> map (`(Term.size_of_term o Thm.prop_of))
|> sort (rev_order o int_ord o pairself fst)
|> map snd) @ [map_id_of_bnf outer];
in
(EVERY' (map (fn thm => subst_tac ctxt [thm]) thms) THEN' rtac refl) 1
end;
fun comp_map_comp_tac _ =
mk_comp_map_comp_tac (map_comp_of_bnf outer) (map_cong_of_bnf outer)
(map map_comp_of_bnf inners);
fun mk_single_comp_set_natural_tac i _ =
mk_comp_set_natural_tac (map_comp_of_bnf outer) (map_cong_of_bnf outer)
(collect_set_natural_of_bnf outer)
(map ((fn thms => nth thms i) o set_natural_of_bnf) inners);
val comp_set_natural_tacs = map mk_single_comp_set_natural_tac (0 upto ilive - 1);
fun comp_bd_card_order_tac _ =
mk_comp_bd_card_order_tac (map bd_card_order_of_bnf inners) (bd_card_order_of_bnf outer);
fun comp_bd_cinfinite_tac _ =
mk_comp_bd_cinfinite_tac (bd_cinfinite_of_bnf inner) (bd_cinfinite_of_bnf outer);
val comp_set_alt_thms =
if ! quick_and_dirty then
replicate ilive no_thm
else
map (fn goal => Skip_Proof.prove lthy [] [] goal
(fn {context, ...} => (mk_comp_set_alt_tac context (collect_set_natural_of_bnf outer))))
(map2 (curry (HOLogic.mk_Trueprop o HOLogic.mk_eq)) comp_sets comp_sets_alt);
fun comp_map_cong_tac _ =
mk_comp_map_cong_tac comp_set_alt_thms (map_cong_of_bnf outer) (map map_cong_of_bnf inners);
val comp_set_bd_tacs =
if ! quick_and_dirty then
replicate (length comp_set_alt_thms) (K all_tac)
else
let
val outer_set_bds = set_bd_of_bnf outer;
val inner_set_bdss = map set_bd_of_bnf inners;
val inner_bd_Card_orders = map bd_Card_order_of_bnf inners;
fun comp_single_set_bd_thm i j =
@{thm comp_single_set_bd} OF [nth inner_bd_Card_orders j, nth (nth inner_set_bdss j) i,
nth outer_set_bds j]
val single_set_bd_thmss =
map ((fn f => map f (0 upto olive - 1)) o comp_single_set_bd_thm) (0 upto ilive - 1);
in
map2 (fn comp_set_alt => fn single_set_bds => fn {context, ...} =>
mk_comp_set_bd_tac context comp_set_alt single_set_bds)
comp_set_alt_thms single_set_bd_thmss
end;
val comp_in_alt_thm =
if ! quick_and_dirty then
no_thm
else
let
val comp_in = mk_in Asets comp_sets CCA;
val comp_in_alt = mk_in (map2 (mk_in Asets) inner_setss CAs) outer_sets CCA;
val goal =
fold_rev Logic.all Asets
(HOLogic.mk_Trueprop (HOLogic.mk_eq (comp_in, comp_in_alt)));
in
Skip_Proof.prove lthy [] [] goal
(fn {context, ...} => mk_comp_in_alt_tac context comp_set_alt_thms)
end;
fun comp_in_bd_tac _ =
mk_comp_in_bd_tac comp_in_alt_thm (map in_bd_of_bnf inners) (in_bd_of_bnf outer)
(map bd_Cinfinite_of_bnf inners) (bd_Card_order_of_bnf outer);
fun comp_map_wpull_tac _ =
mk_map_wpull_tac comp_in_alt_thm (map map_wpull_of_bnf inners) (map_wpull_of_bnf outer);
val tacs = [comp_map_id_tac, comp_map_comp_tac, comp_map_cong_tac] @ comp_set_natural_tacs @
[comp_bd_card_order_tac, comp_bd_cinfinite_tac] @ comp_set_bd_tacs @
[comp_in_bd_tac, comp_map_wpull_tac];
val outer_wits = mk_wits_of_bnf (replicate onwits oDs) (replicate onwits CAs) outer;
val inner_witss = map (map (fn (I, wit) => Term.list_comb (wit, map (nth xs) I)))
(map3 (fn Ds => fn n => mk_wits_of_bnf (replicate n Ds) (replicate n As))
Dss inwitss inners);
val inner_witsss = map (map (nth inner_witss) o fst) outer_wits;
val comp_wits = (inner_witsss, (map (single o snd) outer_wits))
|-> map2 (fold (map_product (fn iwit => fn owit => owit $ iwit)))
|> flat
|> map (`(fn t => Term.add_frees t []))
|> minimize_wits
|> map (fn (frees, t) => fold absfree frees t);
fun wit_tac {context = ctxt, ...} =
mk_comp_wit_tac ctxt (wit_thms_of_bnf outer) (collect_set_natural_of_bnf outer)
(maps wit_thms_of_bnf inners);
val (bnf', lthy') =
add_bnf const_policy (K Derive_Some_Facts) qualify tacs wit_tac (SOME (oDs @ flat Dss))
((((b, comp_map), comp_sets), comp_bd), comp_wits) lthy;
val outer_rel_Gr = rel_Gr_of_bnf outer RS sym;
val outer_rel_cong = rel_cong_of_bnf outer;
val comp_rel_unfold_thm =
trans OF [rel_def_of_bnf bnf',
trans OF [comp_in_alt_thm RS @{thm subst_rel_def},
trans OF [@{thm arg_cong2[of _ _ _ _ relcomp]} OF
[trans OF [outer_rel_Gr RS @{thm arg_cong[of _ _ converse]},
rel_converse_of_bnf outer RS sym], outer_rel_Gr],
trans OF [rel_O_of_bnf outer RS sym, outer_rel_cong OF
(map (fn bnf => rel_def_of_bnf bnf RS sym) inners)]]]];
val comp_pred_unfold_thm = Collect_split_box_equals OF [comp_rel_unfold_thm,
pred_def_of_bnf bnf' RS abs_pred_sym,
trans OF [outer_rel_cong OF (map (fn bnf => pred_def_of_bnf bnf RS abs_pred_sym) inners),
pred_def_of_bnf outer RS abs_pred_sym]];
val unfold' = add_to_unfold (map_def_of_bnf bnf') (set_defs_of_bnf bnf')
comp_rel_unfold_thm comp_pred_unfold_thm unfold;
in
(bnf', (unfold', lthy'))
end;
(* Killing live variables *)
fun killN_bnf qualify n bnf (unfold, lthy) = if n = 0 then (bnf, (unfold, lthy)) else
let
val b = Binding.prefix_name (mk_killN n) (name_of_bnf bnf);
val live = live_of_bnf bnf;
val dead = dead_of_bnf bnf;
val nwits = nwits_of_bnf bnf;
(* TODO: check 0 < n <= live *)
val (Ds, lthy1) = apfst (map TFree)
(Variable.invent_types (replicate dead HOLogic.typeS) lthy);
val ((killedAs, As), lthy2) = apfst (`(take n) o map TFree)
(Variable.invent_types (replicate live HOLogic.typeS) lthy1);
val (Bs, _(*lthy3*)) = apfst (append killedAs o map TFree)
(Variable.invent_types (replicate (live - n) HOLogic.typeS) lthy2);
val ((Asets, lives), _(*names_lthy*)) = lthy
|> mk_Frees "A" (map (HOLogic.mk_setT) (drop n As))
||>> mk_Frees "x" (drop n As);
val xs = map (fn T => HOLogic.choice_const T $ absdummy T @{term True}) killedAs @ lives;
val T = mk_T_of_bnf Ds As bnf;
(*bnf.map id ... id*)
val killN_map = Term.list_comb (mk_map_of_bnf Ds As Bs bnf, map HOLogic.id_const killedAs);
val bnf_sets = mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf;
val killN_sets = drop n bnf_sets;
(*(|UNIV :: A1 set| +c ... +c |UNIV :: An set|) *c bnf.bd*)
val bnf_bd = mk_bd_of_bnf Ds As bnf;
val killN_bd = mk_cprod
(Library.foldr1 (uncurry mk_csum) (map (mk_card_of o HOLogic.mk_UNIV) killedAs)) bnf_bd;
fun killN_map_id_tac _ = rtac (map_id_of_bnf bnf) 1;
fun killN_map_comp_tac {context, ...} =
Local_Defs.unfold_tac context ((map_comp_of_bnf bnf RS sym) :: @{thms o_assoc id_o o_id}) THEN
rtac refl 1;
fun killN_map_cong_tac {context, ...} =
mk_killN_map_cong_tac context n (live - n) (map_cong_of_bnf bnf);
val killN_set_natural_tacs =
map (fn thm => fn _ => rtac thm 1) (drop n (set_natural_of_bnf bnf));
fun killN_bd_card_order_tac _ = mk_killN_bd_card_order_tac n (bd_card_order_of_bnf bnf);
fun killN_bd_cinfinite_tac _ = mk_killN_bd_cinfinite_tac (bd_Cinfinite_of_bnf bnf);
val killN_set_bd_tacs =
map (fn thm => fn _ => mk_killN_set_bd_tac (bd_Card_order_of_bnf bnf) thm)
(drop n (set_bd_of_bnf bnf));
val killN_in_alt_thm =
if ! quick_and_dirty then
no_thm
else
let
val killN_in = mk_in Asets killN_sets T;
val killN_in_alt = mk_in (map HOLogic.mk_UNIV killedAs @ Asets) bnf_sets T;
val goal =
fold_rev Logic.all Asets (HOLogic.mk_Trueprop (HOLogic.mk_eq (killN_in, killN_in_alt)));
in
Skip_Proof.prove lthy [] [] goal (K killN_in_alt_tac)
end;
fun killN_in_bd_tac _ =
mk_killN_in_bd_tac n (live > n) killN_in_alt_thm (in_bd_of_bnf bnf)
(bd_Card_order_of_bnf bnf) (bd_Cinfinite_of_bnf bnf) (bd_Cnotzero_of_bnf bnf);
fun killN_map_wpull_tac _ =
mk_map_wpull_tac killN_in_alt_thm [] (map_wpull_of_bnf bnf);
val tacs = [killN_map_id_tac, killN_map_comp_tac, killN_map_cong_tac] @ killN_set_natural_tacs @
[killN_bd_card_order_tac, killN_bd_cinfinite_tac] @ killN_set_bd_tacs @
[killN_in_bd_tac, killN_map_wpull_tac];
val wits = mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf;
val killN_wits = map (fn t => fold absfree (Term.add_frees t []) t)
(map (fn (I, wit) => Term.list_comb (wit, map (nth xs) I)) wits);
fun wit_tac _ = mk_simple_wit_tac (wit_thms_of_bnf bnf);
val (bnf', lthy') =
add_bnf Smart_Inline (K Derive_Some_Facts) qualify tacs wit_tac (SOME (killedAs @ Ds))
((((b, killN_map), killN_sets), Term.absdummy T killN_bd), killN_wits) lthy;
val rel_Gr = rel_Gr_of_bnf bnf RS sym;
val killN_rel_unfold_thm =
trans OF [rel_def_of_bnf bnf',
trans OF [killN_in_alt_thm RS @{thm subst_rel_def},
trans OF [@{thm arg_cong2[of _ _ _ _ relcomp]} OF
[trans OF [rel_Gr RS @{thm arg_cong[of _ _ converse]}, rel_converse_of_bnf bnf RS sym],
rel_Gr],
trans OF [rel_O_of_bnf bnf RS sym, rel_cong_of_bnf bnf OF
(replicate n @{thm trans[OF Gr_UNIV_id[OF refl] Id_alt[symmetric]]} @
replicate (live - n) @{thm Gr_fst_snd})]]]];
val killN_pred_unfold_thm = Collect_split_box_equals OF
[Local_Defs.unfold lthy' @{thms Id_def'} killN_rel_unfold_thm,
pred_def_of_bnf bnf' RS abs_pred_sym, pred_def_of_bnf bnf RS abs_pred_sym];
val unfold' = add_to_unfold (map_def_of_bnf bnf') (set_defs_of_bnf bnf')
killN_rel_unfold_thm killN_pred_unfold_thm unfold;
in
(bnf', (unfold', lthy'))
end;
(* Adding dummy live variables *)
fun liftN_bnf qualify n bnf (unfold, lthy) = if n = 0 then (bnf, (unfold, lthy)) else
let
val b = Binding.prefix_name (mk_liftN n) (name_of_bnf bnf);
val live = live_of_bnf bnf;
val dead = dead_of_bnf bnf;
val nwits = nwits_of_bnf bnf;
(* TODO: check 0 < n *)
val (Ds, lthy1) = apfst (map TFree)
(Variable.invent_types (replicate dead HOLogic.typeS) lthy);
val ((newAs, As), lthy2) = apfst (chop n o map TFree)
(Variable.invent_types (replicate (n + live) HOLogic.typeS) lthy1);
val ((newBs, Bs), _(*lthy3*)) = apfst (chop n o map TFree)
(Variable.invent_types (replicate (n + live) HOLogic.typeS) lthy2);
val (Asets, _(*names_lthy*)) = lthy
|> mk_Frees "A" (map (HOLogic.mk_setT) (newAs @ As));
val T = mk_T_of_bnf Ds As bnf;
(*%f1 ... fn. bnf.map*)
val liftN_map =
fold_rev Term.absdummy (map2 (curry (op -->)) newAs newBs) (mk_map_of_bnf Ds As Bs bnf);
val bnf_sets = mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf;
val liftN_sets = map (fn A => absdummy T (HOLogic.mk_set A [])) newAs @ bnf_sets;
val liftN_bd = mk_bd_of_bnf Ds As bnf;
fun liftN_map_id_tac _ = rtac (map_id_of_bnf bnf) 1;
fun liftN_map_comp_tac {context, ...} =
Local_Defs.unfold_tac context ((map_comp_of_bnf bnf RS sym) :: @{thms o_assoc id_o o_id}) THEN
rtac refl 1;
fun liftN_map_cong_tac {context, ...} =
rtac (map_cong_of_bnf bnf) 1 THEN REPEAT_DETERM_N live (Goal.assume_rule_tac context 1);
val liftN_set_natural_tacs =
if ! quick_and_dirty then
replicate (n + live) (K all_tac)
else
replicate n (K empty_natural_tac) @
map (fn thm => fn _ => rtac thm 1) (set_natural_of_bnf bnf);
fun liftN_bd_card_order_tac _ = rtac (bd_card_order_of_bnf bnf) 1;
fun liftN_bd_cinfinite_tac _ = rtac (bd_cinfinite_of_bnf bnf) 1;
val liftN_set_bd_tacs =
if ! quick_and_dirty then
replicate (n + live) (K all_tac)
else
replicate n (K (mk_liftN_set_bd_tac (bd_Card_order_of_bnf bnf))) @
(map (fn thm => fn _ => rtac thm 1) (set_bd_of_bnf bnf));
val liftN_in_alt_thm =
if ! quick_and_dirty then
no_thm
else
let
val liftN_in = mk_in Asets liftN_sets T;
val liftN_in_alt = mk_in (drop n Asets) bnf_sets T;
val goal =
fold_rev Logic.all Asets (HOLogic.mk_Trueprop (HOLogic.mk_eq (liftN_in, liftN_in_alt)))
in
Skip_Proof.prove lthy [] [] goal (K liftN_in_alt_tac)
end;
fun liftN_in_bd_tac _ =
mk_liftN_in_bd_tac n liftN_in_alt_thm (in_bd_of_bnf bnf) (bd_Card_order_of_bnf bnf);
fun liftN_map_wpull_tac _ =
mk_map_wpull_tac liftN_in_alt_thm [] (map_wpull_of_bnf bnf);
val tacs = [liftN_map_id_tac, liftN_map_comp_tac, liftN_map_cong_tac] @ liftN_set_natural_tacs @
[liftN_bd_card_order_tac, liftN_bd_cinfinite_tac] @ liftN_set_bd_tacs @
[liftN_in_bd_tac, liftN_map_wpull_tac];
val liftN_wits = map snd (mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf);
fun wit_tac _ = mk_simple_wit_tac (wit_thms_of_bnf bnf);
val (bnf', lthy') =
add_bnf Smart_Inline (K Derive_Some_Facts) qualify tacs wit_tac (SOME Ds)
((((b, liftN_map), liftN_sets), Term.absdummy T liftN_bd), liftN_wits) lthy;
val liftN_rel_unfold_thm =
trans OF [rel_def_of_bnf bnf',
trans OF [liftN_in_alt_thm RS @{thm subst_rel_def}, rel_def_of_bnf bnf RS sym]];
val liftN_pred_unfold_thm = Collect_split_box_equals OF [liftN_rel_unfold_thm,
pred_def_of_bnf bnf' RS abs_pred_sym, pred_def_of_bnf bnf RS abs_pred_sym];
val unfold' = add_to_unfold (map_def_of_bnf bnf') (set_defs_of_bnf bnf')
liftN_rel_unfold_thm liftN_pred_unfold_thm unfold;
in
(bnf', (unfold', lthy'))
end;
(* Changing the order of live variables *)
fun permute_bnf qualify src dest bnf (unfold, lthy) = if src = dest then (bnf, (unfold, lthy)) else
let
val b = Binding.prefix_name (mk_permuteN src dest) (name_of_bnf bnf);
val live = live_of_bnf bnf;
val dead = dead_of_bnf bnf;
val nwits = nwits_of_bnf bnf;
fun permute xs = mk_permute src dest xs;
fun permute_rev xs = mk_permute dest src xs;
val (Ds, lthy1) = apfst (map TFree)
(Variable.invent_types (replicate dead HOLogic.typeS) lthy);
val (As, lthy2) = apfst (map TFree)
(Variable.invent_types (replicate live HOLogic.typeS) lthy1);
val (Bs, _(*lthy3*)) = apfst (map TFree)
(Variable.invent_types (replicate live HOLogic.typeS) lthy2);
val (Asets, _(*names_lthy*)) = lthy
|> mk_Frees "A" (map (HOLogic.mk_setT) (permute As));
val T = mk_T_of_bnf Ds As bnf;
(*%f(1) ... f(n). bnf.map f\<sigma>(1) ... f\<sigma>(n)*)
val permute_map = fold_rev Term.absdummy (permute (map2 (curry op -->) As Bs))
(Term.list_comb (mk_map_of_bnf Ds As Bs bnf,
permute_rev (map Bound ((live - 1) downto 0))));
val bnf_sets = mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf;
val permute_sets = permute bnf_sets;
val permute_bd = mk_bd_of_bnf Ds As bnf;
fun permute_map_id_tac _ = rtac (map_id_of_bnf bnf) 1;
fun permute_map_comp_tac _ = rtac (map_comp_of_bnf bnf) 1;
fun permute_map_cong_tac {context, ...} =
rtac (map_cong_of_bnf bnf) 1 THEN REPEAT_DETERM_N live (Goal.assume_rule_tac context 1);
val permute_set_natural_tacs =
permute (map (fn thm => fn _ => rtac thm 1) (set_natural_of_bnf bnf));
fun permute_bd_card_order_tac _ = rtac (bd_card_order_of_bnf bnf) 1;
fun permute_bd_cinfinite_tac _ = rtac (bd_cinfinite_of_bnf bnf) 1;
val permute_set_bd_tacs = permute (map (fn thm => fn _ => rtac thm 1) (set_bd_of_bnf bnf));
val permute_in_alt_thm =
if ! quick_and_dirty then
no_thm
else
let
val permute_in = mk_in Asets permute_sets T;
val permute_in_alt = mk_in (permute_rev Asets) bnf_sets T;
val goal =
fold_rev Logic.all Asets
(HOLogic.mk_Trueprop (HOLogic.mk_eq (permute_in, permute_in_alt)));
in
Skip_Proof.prove lthy [] [] goal (K (mk_permute_in_alt_tac src dest))
end;
fun permute_in_bd_tac _ =
mk_permute_in_bd_tac src dest permute_in_alt_thm (in_bd_of_bnf bnf)
(bd_Card_order_of_bnf bnf);
fun permute_map_wpull_tac _ =
mk_map_wpull_tac permute_in_alt_thm [] (map_wpull_of_bnf bnf);
val tacs = [permute_map_id_tac, permute_map_comp_tac, permute_map_cong_tac] @
permute_set_natural_tacs @ [permute_bd_card_order_tac, permute_bd_cinfinite_tac] @
permute_set_bd_tacs @ [permute_in_bd_tac, permute_map_wpull_tac];
val permute_wits = map snd (mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf);
fun wit_tac _ = mk_simple_wit_tac (wit_thms_of_bnf bnf);
val (bnf', lthy') =
add_bnf Smart_Inline (K Derive_Some_Facts) qualify tacs wit_tac (SOME Ds)
((((b, permute_map), permute_sets), Term.absdummy T permute_bd), permute_wits) lthy;
val permute_rel_unfold_thm =
trans OF [rel_def_of_bnf bnf',
trans OF [permute_in_alt_thm RS @{thm subst_rel_def}, rel_def_of_bnf bnf RS sym]];
val permute_pred_unfold_thm = Collect_split_box_equals OF [permute_rel_unfold_thm,
pred_def_of_bnf bnf' RS abs_pred_sym, pred_def_of_bnf bnf RS abs_pred_sym];
val unfold' = add_to_unfold (map_def_of_bnf bnf') (set_defs_of_bnf bnf')
permute_rel_unfold_thm permute_pred_unfold_thm unfold;
in
(bnf', (unfold', lthy'))
end;
(* Composition pipeline *)
fun permute_and_kill qualify n src dest bnf =
bnf
|> permute_bnf qualify src dest
#> uncurry (killN_bnf qualify n);
fun lift_and_permute qualify n src dest bnf =
bnf
|> liftN_bnf qualify n
#> uncurry (permute_bnf qualify src dest);
fun normalize_bnfs qualify Ass Ds sort bnfs unfold lthy =
let
val before_kill_src = map (fn As => 0 upto (length As - 1)) Ass;
val kill_poss = map (find_indices Ds) Ass;
val live_poss = map2 (subtract (op =)) kill_poss before_kill_src;
val before_kill_dest = map2 append kill_poss live_poss;
val kill_ns = map length kill_poss;
val (inners', (unfold', lthy')) =
fold_map5 (fn i => permute_and_kill (qualify i))
(if length bnfs = 1 then [0] else (1 upto length bnfs))
kill_ns before_kill_src before_kill_dest bnfs (unfold, lthy);
val Ass' = map2 (map o nth) Ass live_poss;
val As = sort Ass';
val after_lift_dest = replicate (length Ass') (0 upto (length As - 1));
val old_poss = map (map (fn x => find_index (fn y => x = y) As)) Ass';
val new_poss = map2 (subtract (op =)) old_poss after_lift_dest;
val after_lift_src = map2 append new_poss old_poss;
val lift_ns = map (fn xs => length As - length xs) Ass';
in
((kill_poss, As), fold_map5 (fn i => lift_and_permute (qualify i))
(if length bnfs = 1 then [0] else (1 upto length bnfs))
lift_ns after_lift_src after_lift_dest inners' (unfold', lthy'))
end;
fun default_comp_sort Ass =
Library.sort (Term_Ord.typ_ord o pairself TFree) (fold (fold (insert (op =))) Ass []);
fun compose_bnf const_policy qualify' b sort outer inners oDs Dss tfreess (unfold, lthy) =
let
val name = Binding.name_of b;
fun qualify i bind =
let val namei = if i > 0 then name ^ string_of_int i else name;
in
if member (op =) (#2 (Binding.dest bind)) (namei, true) then qualify' bind
else qualify' (Binding.prefix_name namei bind)
end;
val Ass = map (map dest_TFree) tfreess;
val Ds = fold (fold Term.add_tfreesT) (oDs :: Dss) [];
val ((kill_poss, As), (inners', (unfold', lthy'))) =
normalize_bnfs qualify Ass Ds sort inners unfold lthy;
val Ds = oDs @ flat (map3 (append oo map o nth) tfreess kill_poss Dss);
val As = map TFree As;
in
apfst (rpair (Ds, As)) (clean_compose_bnf const_policy I b outer inners' (unfold', lthy'))
end;
(* Hide the type of the bound (optimization) and unfold the definitions (nicer to the user) *)
fun seal_bnf unfold b Ds bnf lthy =
let
val live = live_of_bnf bnf;
val nwits = nwits_of_bnf bnf;
val (As, lthy1) = apfst (map TFree)
(Variable.invent_types (replicate live HOLogic.typeS) (fold Variable.declare_typ Ds lthy));
val (Bs, _) = apfst (map TFree)
(Variable.invent_types (replicate live HOLogic.typeS) lthy1);
val map_unfolds = filter_refl (map_unfolds_of unfold);
val set_unfoldss = map filter_refl (set_unfoldss_of unfold);
val expand_maps = fold expand_term_const (map (single o Logic.dest_equals o Thm.prop_of)
map_unfolds);
val expand_sets = fold expand_term_const (map (map (Logic.dest_equals o Thm.prop_of))
set_unfoldss);
val unfold_maps = fold (Local_Defs.unfold lthy o single) map_unfolds;
val unfold_sets = fold (Local_Defs.unfold lthy) set_unfoldss;
val unfold_defs = unfold_sets o unfold_maps;
val bnf_map = expand_maps (mk_map_of_bnf Ds As Bs bnf);
val bnf_sets = map (expand_maps o expand_sets)
(mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf);
val bnf_bd = mk_bd_of_bnf Ds As bnf;
val T = mk_T_of_bnf Ds As bnf;
(*bd should only depend on dead type variables!*)
val bd_repT = fst (dest_relT (fastype_of bnf_bd));
val bdT_bind = Binding.suffix_name ("_" ^ bdTN) b;
val params = fold Term.add_tfreesT Ds [];
val ((bdT_name, (bdT_glob_info, bdT_loc_info)), (lthy', lthy)) =
lthy
|> Typedef.add_typedef true NONE (bdT_bind, params, NoSyn)
(HOLogic.mk_UNIV bd_repT) NONE (EVERY' [rtac exI, rtac UNIV_I] 1)
||> `Local_Theory.restore;
val phi = Proof_Context.export_morphism lthy lthy';
val bnf_bd' = mk_dir_image bnf_bd
(Const (#Abs_name bdT_glob_info, bd_repT --> Type (bdT_name, map TFree params)))
val set_def = Morphism.thm phi (the (#set_def bdT_loc_info));
val Abs_inject = Morphism.thm phi (#Abs_inject bdT_loc_info);
val Abs_cases = Morphism.thm phi (#Abs_cases bdT_loc_info);
val Abs_bdT_inj = mk_Abs_inj_thm set_def Abs_inject;
val Abs_bdT_bij = mk_Abs_bij_thm lthy' set_def Abs_inject Abs_cases;
val bd_ordIso = @{thm dir_image} OF [Abs_bdT_inj, bd_Card_order_of_bnf bnf];
val bd_card_order =
@{thm card_order_dir_image} OF [Abs_bdT_bij, bd_card_order_of_bnf bnf];
val bd_cinfinite =
(@{thm Cinfinite_cong} OF [bd_ordIso, bd_Cinfinite_of_bnf bnf]) RS conjunct1;
val set_bds =
map (fn thm => @{thm ordLeq_ordIso_trans} OF [thm, bd_ordIso]) (set_bd_of_bnf bnf);
val in_bd =
@{thm ordLeq_ordIso_trans} OF [in_bd_of_bnf bnf,
@{thm cexp_cong2_Cnotzero} OF [bd_ordIso, if live = 0 then
@{thm ctwo_Cnotzero} else @{thm ctwo_Cnotzero} RS @{thm csum_Cnotzero2},
bd_Card_order_of_bnf bnf]];
fun mk_tac thm {context = ctxt, prems = _} = (rtac (unfold_defs thm) THEN'
SOLVE o REPEAT_DETERM o (atac ORELSE' Goal.assume_rule_tac ctxt)) 1;
val tacs =
map mk_tac ([map_id_of_bnf bnf, map_comp_of_bnf bnf, map_cong_of_bnf bnf] @
set_natural_of_bnf bnf) @
map K [rtac bd_card_order 1, rtac bd_cinfinite 1] @
map mk_tac (set_bds @ [in_bd, map_wpull_of_bnf bnf]);
val bnf_wits = map snd (mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf);
fun wit_tac _ = mk_simple_wit_tac (map unfold_defs (wit_thms_of_bnf bnf));
val (bnf', lthy') = add_bnf Hardly_Inline (K Derive_All_Facts) I tacs wit_tac NONE
((((b, bnf_map), bnf_sets), Term.absdummy T bnf_bd'), bnf_wits) lthy;
val defs' = filter_refl (map_def_of_bnf bnf' :: set_defs_of_bnf bnf');
val unfold_defs' = unfold_defs o Local_Defs.unfold lthy' defs';
val rel_def = unfold_defs' (rel_def_of_bnf bnf');
val rel_unfold = Local_Defs.unfold lthy'
(map unfold_defs (filter_refl (rel_unfolds_of unfold))) rel_def;
val unfold_defs'' = unfold_defs' o Local_Defs.unfold lthy' (filter_refl [rel_def_of_bnf bnf']);
val pred_def = unfold_defs'' (pred_def_of_bnf bnf' RS abs_pred_sym_pred_abs);
val pred_unfold = Local_Defs.unfold lthy'
(map unfold_defs (filter_refl (pred_unfolds_of unfold))) pred_def;
fun note thmN thms =
snd o Local_Theory.note
((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), thms);
in
(bnf', lthy'
|> note rel_unfoldN [rel_unfold]
|> note pred_unfoldN [pred_unfold])
end;
fun bnf_of_typ _ _ _ _ (T as TFree _) (unfold, lthy) =
((Basic_BNFs.ID_bnf, ([], [T])), (add_to_unfold_opt NONE NONE
(SOME Basic_BNFs.ID_rel_def) (SOME Basic_BNFs.ID_pred_def) unfold, lthy))
| bnf_of_typ _ _ _ _ (TVar _) _ = error "Unexpected schematic variable"
| bnf_of_typ const_policy b qualify' sort (T as Type (C, Ts)) (unfold, lthy) =
let val tfrees = Term.add_tfreesT T [];
in
if null tfrees then
((Basic_BNFs.DEADID_bnf, ([T], [])), (unfold, lthy))
else if forall (can Term.dest_TFree) Ts andalso length Ts = length tfrees then let
val bnf = the (bnf_of lthy (Long_Name.base_name C));
val T' = T_of_bnf bnf;
val deads = deads_of_bnf bnf;
val lives = lives_of_bnf bnf;
val tvars' = Term.add_tvarsT T' [];
val deads_lives =
pairself (map (Term.typ_subst_TVars (map fst tvars' ~~ map TFree tfrees)))
(deads, lives);
val rel_def = rel_def_of_bnf bnf;
val unfold' = add_to_unfold_opt NONE NONE (SOME (rel_def RS sym))
(SOME (Local_Defs.unfold lthy [rel_def] (pred_def_of_bnf bnf) RS sym)) unfold;
in ((bnf, deads_lives), (unfold', lthy)) end
else
let
(* FIXME: we should allow several BNFs with the same base name *)
val Tname = Long_Name.base_name C;
val name = Binding.name_of b ^ "_" ^ Tname;
fun qualify i bind =
let val namei = if i > 0 then name ^ string_of_int i else name;
in
if member (op =) (#2 (Binding.dest bind)) (namei, true) then qualify' bind
else qualify' (Binding.prefix_name namei bind)
end;
val outer = the (bnf_of lthy Tname);
val odead = dead_of_bnf outer;
val olive = live_of_bnf outer;
val oDs_pos = find_indices [TFree ("dead", [])]
(snd (dest_Type
(mk_T_of_bnf (replicate odead (TFree ("dead", []))) (replicate olive dummyT) outer)));
val oDs = map (nth Ts) oDs_pos;
val Ts' = map (nth Ts) (subtract (op =) oDs_pos (0 upto length Ts - 1));
val ((inners, (Dss, Ass)), (unfold', lthy')) =
apfst (apsnd split_list o split_list)
(fold_map2 (fn i =>
bnf_of_typ Smart_Inline (Binding.name (name ^ string_of_int i)) (qualify i) sort)
(if length Ts' = 1 then [0] else (1 upto length Ts'))
Ts' (unfold, lthy));
in
compose_bnf const_policy (qualify 0) b sort outer inners oDs Dss Ass (unfold', lthy')
end
end;
fun bnf_of_typ_cmd (b, rawT) lthy =
let
val ((bnf, (Ds, As)), (unfold, lthy')) =
bnf_of_typ Smart_Inline b I default_comp_sort (Syntax.read_typ lthy rawT)
(empty_unfold, lthy);
in
snd (seal_bnf unfold b Ds bnf lthy')
end;
val _ =
Outer_Syntax.local_theory @{command_spec "bnf_of_typ"} "parse a type as composition of BNFs"
(((Parse.binding --| Parse.$$$ "=") -- Parse.typ) >> bnf_of_typ_cmd);
end;