--- a/src/HOL/Tools/BNF/bnf_lift.ML Tue Dec 10 01:06:39 2019 +0100
+++ b/src/HOL/Tools/BNF/bnf_lift.ML Tue Dec 10 01:06:39 2019 +0100
@@ -1,7 +1,9 @@
(* Title: HOL/Tools/BNF/bnf_lift.ML
Author: Julian Biendarra, TU Muenchen
+ Author: Basil Fürer, ETH Zurich
+ Author: Joshua Schneider, ETH Zurich
Author: Dmitriy Traytel, ETH Zurich
- Copyright 2015
+ Copyright 2015, 2019
Lifting of BNFs through typedefs.
*)
@@ -11,6 +13,7 @@
datatype lift_bnf_option =
Plugins_Option of Proof.context -> Plugin_Name.filter
| No_Warn_Wits
+ | No_Warn_Transfer
val copy_bnf:
(((lift_bnf_option list * (binding option * (string * sort option)) list) *
string) * thm option) * (binding * binding * binding) ->
@@ -39,21 +42,350 @@
open BNF_Def
-(* typedef_bnf *)
-
datatype lift_bnf_option =
Plugins_Option of Proof.context -> Plugin_Name.filter
-| No_Warn_Wits;
+| No_Warn_Wits
+| No_Warn_Transfer;
+
+datatype lift_thm = Typedef of thm | Quotient of thm
+
+(** Util **)
+
+fun last2 [x, y] = ([], (x, y))
+ | last2 (x :: xs) = last2 xs |>> (fn y => x :: y)
+ | last2 [] = raise Match;
+
+fun strip3 thm = (case Term.strip_comb (HOLogic.dest_Trueprop (Thm.prop_of thm)) of
+ (_, [x1, x2, x3]) => (x1, x2, x3)
+ | _ => error "strip3: wrong number of arguments");
+
+val mk_Free = yield_singleton o mk_Frees;
+
+fun TWICE t = t THEN' t;
+
+fun prove lthy fvars tm tac =
+ Goal.prove_sorry lthy (map (fst o dest_Free) fvars) [] tm (fn {context, ...} => tac context);
+
+(** Term construction **)
+
+fun mk_relT aT bT = aT --> bT --> HOLogic.boolT;
+fun mk_relcompp r s = let
+ val (rT, sT) = apply2 fastype_of (r, s);
+ val ((xT, _), (_, zTs)) = apply2 dest_funT (rT, sT);
+ val T = rT --> sT --> mk_relT xT (fst (dest_funT zTs));
+ in Const (@{const_name relcompp}, T) $ r $ s end;
+val mk_OO = fold_rev mk_relcompp;
+
+(* option from sum *)
+fun mk_MaybeT T = mk_sumT (HOLogic.unitT, T);
+fun mk_Nothing T = BNF_FP_Util.mk_Inl T HOLogic.unit;
+val Just_const = BNF_FP_Util.Inr_const HOLogic.unitT;
+fun mk_Just tm = Just_const (fastype_of tm) $ tm;
+fun Maybe_map_const T =
+ let val (xT, yT) = dest_funT T in
+ Const (@{const_name map_sum}, (HOLogic.unitT --> HOLogic.unitT) --> T --> mk_MaybeT xT --> mk_MaybeT yT) $
+ HOLogic.id_const HOLogic.unitT
+ end;
+fun mk_Maybe_map tm = Maybe_map_const (fastype_of tm) $ tm;
+fun fromJust_const T = Const (@{const_name sum.projr}, mk_MaybeT T --> T)
+
+fun rel_Maybe_const T U =
+ Const (@{const_name rel_sum}, (HOLogic.unitT --> HOLogic.unitT --> HOLogic.boolT) -->
+ (T --> U --> HOLogic.boolT) --> mk_MaybeT T --> mk_MaybeT U --> HOLogic.boolT) $
+ HOLogic.eq_const HOLogic.unitT
+fun set_Maybe_const T = Const (@{const_name Basic_BNFs.setr}, mk_MaybeT T --> HOLogic.mk_setT T)
+
+fun Inf_const T = Const (@{const_name Inf}, HOLogic.mk_setT T --> T);
+
+fun Image_const T =
+ let
+ val relT = HOLogic.mk_setT (HOLogic.mk_prodT (T, T));
+ val setT = HOLogic.mk_setT T;
+ in Const (@{const_name Image}, relT --> setT --> setT) end;
+
+fun bot_const T = Const (@{const_name bot}, T);
+
+fun mk_insert x S =
+ Const (@{const_name Set.insert}, fastype_of x --> fastype_of S --> fastype_of S) $ x $ S;
+
+fun mk_vimage f s =
+ let val (xT, yT) = dest_funT (fastype_of f) in
+ Const (@{const_name vimage}, (xT --> yT) --> HOLogic.mk_setT yT --> HOLogic.mk_setT xT) $ f $ s
+ end;
+
+fun mk_case_prod (x, y) tm = let
+ val ((x, xT), (y, yT)) = apply2 dest_Free (x, y);
+ val prodT = HOLogic.mk_prodT (xT, yT);
+ in HOLogic.Collect_const prodT $ (Const (@{const_name case_prod},
+ (xT --> yT --> HOLogic.boolT) --> prodT --> HOLogic.boolT) $ absfree (x, xT)
+ (absfree (y, yT) tm)) end;
+
+fun mk_Trueprop_implies (ps, c) =
+ Logic.list_implies (map HOLogic.mk_Trueprop ps, HOLogic.mk_Trueprop c);
+
+fun mk_Collect (v, tm) = let val (n, T) = dest_Free v in
+ HOLogic.mk_Collect (n, T, tm) end;
+
+
+(** witnesses **)
+fun prepare_wits is_quotient RepT wits opts alphas wits_F var_as var_as' sets lthy =
+ let
+ fun binder_types_until_eq V T =
+ let
+ fun strip (TU as Type ("fun", [T, U])) = if V = TU then [] else T :: strip U
+ | strip T = if V = T then [] else
+ error ("Bad type for witness: " ^ quote (Syntax.string_of_typ lthy T));
+ in strip T end;
+
+ val Iwits = the_default wits_F (Option.map (map (`(map (fn T =>
+ find_index (fn U => T = U) alphas) o binder_types_until_eq RepT o fastype_of))) wits);
+
+ val wit_goals = maps (BNF_Def.mk_wit_goals var_as var_as' sets) Iwits;
+
+ val lost_wits = filter_out (fn (J, _) => exists (fn (I, _) => subset (op =) (I, J)) Iwits) wits_F;
+
+ val _ =
+ if null lost_wits orelse exists (fn No_Warn_Wits => true | _ => false) opts then ()
+ else
+ let
+ val what = (if is_quotient then "quotient type" else "typedef");
+ val (suffix1, suffix2, be) =
+ (if length lost_wits = 1 then ("", "", "was") else ("s", "es", "were"))
+ in
+ lost_wits
+ |> map (Syntax.pretty_typ lthy o fastype_of o snd)
+ |> Pretty.big_list
+ ("The following type" ^ suffix1 ^ " of nonemptiness witness" ^ suffix2 ^
+ " of the raw type's BNF " ^ be ^ " lost:")
+ |> (fn pt => Pretty.chunks [pt,
+ Pretty.para ("You can specify a liftable witness (e.g., a term of one of the above types\
+ \ that satisfies the " ^ what ^ "'s invariant)\
+ \ using the annotation [wits: <term>].")])
+ |> Pretty.string_of
+ |> warning
+ end;
+ in (Iwits, wit_goals) end;
+
+
+(** transfer theorems **)
+
+fun mk_transfer_thms' bnf_F bnf_G name consts Tss crel_def pcrel_def defs lthy = let
+
+ val live = length (#alphas Tss);
+
+ val (pcrel_tm, crel_tm) = apply2 (Thm.prop_of #> Logic.dest_equals #> fst #> head_of)
+ (pcrel_def, crel_def);
+
+ val (var_Qs, var_Rs) = fold Variable.declare_typ (#alphas Tss @ #deads Tss) lthy
+ |> mk_Frees "Q" (map2 mk_relT (#alphas Tss) (#betas Tss))
+ ||>> mk_Frees "R" (map2 mk_relT (#gammas Tss) (#deltas Tss))
+ |> fst;
+
+ (* get the "pcrel :: args_raw => rep => abs \<Rightarrow> bool" term and instantiate types *)
+ val (args_raw, (rep, abs)) = pcrel_tm
+ |> fastype_of
+ |> binder_types
+ |> last2;
+ val thy = Proof_Context.theory_of lthy;
+ val tyenv_match = Vartab.empty
+ |> Sign.typ_match thy ((rep, #rep Tss))
+ |> Sign.typ_match thy ((abs, #abs Tss));
+ val args = map (Envir.subst_type tyenv_match) args_raw;
+ val (pcrel_a, pcrel_b) = Envir.subst_term (tyenv_match, Vartab.empty) pcrel_tm
+ |> `(subst_atomic_types (#alphas Tss @ #betas Tss ~~ #gammas Tss @ #deltas Tss));
+
+ (* match "crel :: {?a F} \<Rightarrow> a G" with "rep_G :: {a F}" *)
+ val tyenv_match = Vartab.empty |> Sign.typ_match thy
+ (crel_tm |> fastype_of |> binder_types |> hd, #rep Tss);
+ val crel_b = Envir.subst_term (tyenv_match, Vartab.empty) crel_tm
+ |> subst_atomic_types (#alphas Tss ~~ #betas Tss)
+ val crel_d = subst_atomic_types (#betas Tss ~~ #deltas Tss) crel_b;
+
+ (* instantiate pcrel with Qs and Rs*)
+ val dead_args = map binder_types args
+ |> map (fn [T,U] => if T = U then SOME T else NONE | _ => NONE);
+ val parametrize = let
+ fun go (SOME T :: dTs) tms = HOLogic.eq_const T :: go dTs tms
+ | go (_ :: dTs) (tm :: tms) = tm :: go dTs tms
+ | go (_ :: dTs) tms = go dTs tms
+ | go _ _ = [];
+ in go dead_args end;
+
+ val pcrel_Qs = list_comb (pcrel_b, parametrize var_Qs);
+ val pcrel_Rs = list_comb (pcrel_a, parametrize var_Rs);
+
+ (* get the order of the dead variables right *)
+ val Ds0 = maps the_list dead_args;
+ val permute_Ds = (mk_T_of_bnf Ds0 (#betas Tss) bnf_G, nth (binder_types (type_of pcrel_Qs)) 1)
+ |> apply2 (fn Type (_, Ts) => Ts | _ => []) |> op~~ |> typ_subst_atomic;
+ val Ds0 = map permute_Ds Ds0;
+
+ (* terms for sets of the set_transfer thms *)
+ val rel_sets_Q = @{map 3} (fn aT => fn bT => fn Q =>
+ mk_rel 1 [aT] [bT] @{term rel_set} $ Q) (#alphas Tss) (#betas Tss) var_Qs;
+
+ (* rewrite rules for pcrel and BNF's relator: "pcrel Q = rel_F OO crel" *)
+ fun mk_pcr_rel_F_eq Ts Us pcrel vs crel =
+ let
+ val thm = HOLogic.mk_Trueprop (HOLogic.mk_eq (pcrel, mk_relcompp (list_comb
+ (mk_rel_of_bnf (#deads Tss) (Ts Tss) (Us Tss) bnf_F, vs)) crel));
+ fun tac ctxt = unfold_thms_tac ctxt (pcrel_def :: defs @ @{thm id_bnf_apply} ::
+ Transfer.get_relator_eq ctxt) THEN (HEADGOAL (rtac ctxt refl))
+ in prove lthy vs thm tac |> mk_abs_def end;
+
+ val pcr_rel_F_eqs =
+ [mk_pcr_rel_F_eq #alphas #betas pcrel_Qs var_Qs crel_b,
+ mk_pcr_rel_F_eq #gammas #deltas pcrel_Rs var_Rs crel_d];
+
+ (* create transfer-relations from term('s type) *)
+ fun mk_transfer_rel' i tm =
+ let
+ fun go T' (n, q) = case T' of
+ Type ("fun", [T as Type ("fun", _), U]) =>
+ go U (n+1, q) |>> mk_rel_fun (fst (go T (n, q)))
+ | Type ("fun", [T, U]) =>
+ go T (n, q) |-> (fn x => fn st => go U st |>> mk_rel_fun x)
+ | Type (@{type_name bool}, _) => (HOLogic.eq_const HOLogic.boolT, (n, q))
+ | Type (@{type_name set}, _) => (nth rel_sets_Q n, (n, q))
+ | Type _ => (if q then pcrel_Qs else pcrel_Rs, (n, not q))
+ | TFree _ => (nth (if q then var_Qs else var_Rs) n, (n, not q))
+ | _ => raise Match
+ in go (fastype_of tm) (i, true) |> fst end;
+
+ (* prove transfer rules *)
+ fun prove_transfer_thm' i vars new_tm const =
+ let
+ val tm = HOLogic.mk_Trueprop (mk_transfer_rel' i new_tm $ #raw const $ new_tm);
+ val tac = (fn ctxt => unfold_thms_tac ctxt (pcr_rel_F_eqs @ [crel_def]) THEN
+ #tac const {Rs=var_Rs,Qs=var_Qs} ctxt);
+ in prove lthy vars tm tac end;
+ val prove_transfer_thm = prove_transfer_thm' 0;
+
+ (* map transfer: "((Q ===> R) ===> pcr_G Q ===> pcr_G R) map_F' map_G" *)
+ val map_G = mk_map_of_bnf Ds0 (#betas Tss) (#deltas Tss) bnf_G;
+ val map_transfer = prove_transfer_thm (var_Qs @ var_Rs) map_G (#map consts);
+
+ (* pred_transfer: "((Q ===> (=)) ===> pcr_G Q ===> (=)) pred_F' pred_G" *)
+ val pred_G = mk_pred_of_bnf Ds0 (#betas Tss) bnf_G;
+ val pred_transfer = #pred consts |> Option.map (fn p =>
+ ("pred", [prove_transfer_thm (var_Qs @ var_Rs) pred_G p]));
+
+ (* rel_transfer: "((Q ===> R ===> (=)) ===> pcr_G Q ===> pcr_G R ===> (=)) rel_F' rel_G" *)
+ val rel_G = mk_rel_of_bnf Ds0 (#betas Tss) (#deltas Tss) bnf_G;
+ val rel_transfer = prove_transfer_thm (var_Qs @ var_Rs) rel_G (#rel consts);
+
+ (* set_transfer: "(pcr_G Q ===> rel_set Q) set_F' set_G" *)
+ val sets_G = mk_sets_of_bnf (replicate live Ds0) (replicate live (#betas Tss)) bnf_G;
+ fun mk_set_transfer i set_G raw tac = prove_transfer_thm' i var_Qs set_G {raw=raw, tac=tac};
+ val sets_transfer = @{map 4} mk_set_transfer
+ (0 upto (live-1)) sets_G (#raws (#sets consts)) (#tacs (#sets consts));
+
+ (* export transfer theorems *)
+ val transform = Morphism.thm (Morphism.thm_morphism "BNF" (unfold_thms lthy defs)) |> map;
+ val b = Binding.qualified_name name;
+ val qualify =
+ let val qs = Binding.path_of b;
+ in fold_rev (fn (s, mand) => Binding.qualify mand s) qs end;
+ fun mk_binding n = Binding.name (n ^ "_transfer_raw")
+ |> Binding.qualify true (Binding.name_of b) |> qualify;
+ val notes = [("map", [map_transfer]), ("rel", [rel_transfer])] @ the_list pred_transfer @
+ [("set", sets_transfer)] |> map (fn (n, thms) =>
+ ((mk_binding n, []), [(transform thms, @{attributes [transfer_rule]})]));
+
+ in lthy |> Local_Theory.notes notes |> snd end;
+
+(* transfer theorems for map, pred (in case of a typedef), rel and sets *)
+fun mk_transfer_thms quiet bnf_F bnf_G name consts thm Tss defs lthy = let
+
+ fun mk_crel_def quot_thm =
+ (case thm of
+ Quotient equiv => @{thm Quotient_crel_quotient} OF [quot_thm, equiv]
+ | Typedef _ => hd ([quot_thm] RL @{thms Quotient_crel_typedef Quotient_crel_typecopy}));
+ fun no_quotient _ = [Pretty.para ("No quotient theorem has been registered for " ^
+ Binding.name_of (name_of_bnf bnf_G) ^ "."),
+ Pretty.para "Use setup_lifting to register a quotient or type definition theorem."];
+ fun wrong_quotient T lthy = [Pretty.para ("A wrong quotient theorem has been registered for " ^
+ Binding.name_of (name_of_bnf bnf_G) ^ "."),
+ Pretty.para ("Expected raw type " ^
+ Pretty.string_of (Syntax.pretty_typ lthy (T_of_bnf bnf_F)) ^
+ " but the quotient theorem has raw type " ^
+ Pretty.string_of (Syntax.pretty_typ lthy T) ^ "."),
+ Pretty.para "Use setup_lifting to register a different quotient or type definition theorem."];
+ fun pcr_why _ = [Pretty.para ("The pcr_" ^ Binding.name_of (name_of_bnf bnf_G) ^
+ " relator has not been defined.")];
+ fun warn_transfer why lthy =
+ (Pretty.para "The transfer theorems can't be generated:" :: why lthy)
+ |> Pretty.chunks |> Pretty.string_of |> warning |> K lthy;
+ fun maybe_warn_transfer why = not quiet ? warn_transfer why;
+ in
+ case Lifting_Info.lookup_quotients lthy name of
+ SOME {pcr_info, quot_thm} =>
+ (let
+ val crel_def = mk_crel_def quot_thm;
+ val rty = Lifting_Util.quot_thm_rty_qty quot_thm |> fst;
+ val thy = Proof_Context.theory_of lthy;
+ in
+ if Sign.typ_instance thy (rty, T_of_bnf bnf_F) then
+ (case pcr_info of
+ SOME {pcrel_def, ...} =>
+ mk_transfer_thms' bnf_F bnf_G name consts Tss crel_def pcrel_def defs lthy
+ | _ => maybe_warn_transfer pcr_why lthy)
+ else maybe_warn_transfer (wrong_quotient rty) lthy
+ end)
+ | _ => maybe_warn_transfer no_quotient lthy
+ end;
+
+
+(** typedef_bnf **)
+
+fun mk_typedef_transfer_tacs bnf_F bnf_G thms old_defs
+ map_raw rel_raw pred_raw sets_raw = let
+
+ val live = live_of_bnf bnf_G;
+ val Abs_G_inverse = @{thm type_definition.Abs_inverse} OF [#typedef thms];
+ val Rep_G = @{thm type_definition.Rep} OF [#typedef thms];
+
+ fun common_tac addefs tac = (fn _ => fn ctxt =>
+ HEADGOAL (EVERY' [SELECT_GOAL (unfold_thms_tac ctxt addefs),
+ REPEAT_DETERM o rtac ctxt rel_funI,
+ SELECT_GOAL (unfold_thms_tac ctxt @{thms o_apply}),
+ REPEAT_DETERM o eresolve_tac ctxt @{thms relcomppE exE conjE},
+ hyp_subst_tac ctxt]) THEN tac ctxt)
+
+ fun map_tac ctxt = (HEADGOAL o EVERY')
+ [rtac ctxt @{thm relcomppI},
+ etac ctxt (mk_rel_funDN (live+1) (map_transfer_of_bnf bnf_F)),
+ REPEAT_DETERM_N live o assume_tac ctxt,
+ SELECT_GOAL (unfold_thms_tac ctxt [Abs_G_inverse OF [#map_closed thms] OF [Rep_G]]),
+ REPEAT_DETERM o rtac ctxt refl];
+ val map_tac = common_tac [#map old_defs] map_tac;
+
+ fun rel_tac ctxt =
+ HEADGOAL (etac ctxt (mk_rel_funDN (live+2) (rel_transfer_of_bnf bnf_F)) THEN'
+ REPEAT_DETERM_N (live+1) o assume_tac ctxt);
+ val rel_tac = common_tac (#rel old_defs :: @{thms vimage2p_def}) rel_tac;
+
+ fun pred_tac ctxt =
+ HEADGOAL (etac ctxt (mk_rel_funDN (live+1) (pred_transfer_of_bnf bnf_F)) THEN'
+ REPEAT_DETERM_N live o (assume_tac ctxt));
+ val pred_tac = common_tac [#pred old_defs] pred_tac;
+
+ fun set_tac set_transfer_thm ctxt =
+ HEADGOAL (etac ctxt (rel_funD OF [set_transfer_thm]));
+ fun mk_set_tac set_def set_transfer = common_tac [set_def] (set_tac set_transfer);
+ val set_tacs = map2 mk_set_tac (#sets old_defs) (set_transfer_of_bnf bnf_F);
+
+ in {map={raw=map_raw,tac=map_tac},rel={raw=rel_raw,tac=rel_tac},
+ sets={raws=sets_raw,tacs=set_tacs},pred=SOME{raw=pred_raw,tac=pred_tac}} end;
fun typedef_bnf thm wits specs map_b rel_b pred_b opts lthy =
let
val plugins =
get_first (fn Plugins_Option f => SOME (f lthy) | _ => NONE) (rev opts)
- |> the_default Plugin_Name.default_filter;
- val no_warn_wits = exists (fn No_Warn_Wits => true | _ => false) opts;
+ |> the_default Plugin_Name.default_filter;
(* extract Rep Abs F RepT AbsT *)
- val (_, [Rep_G, Abs_G, F]) = Term.strip_comb (HOLogic.dest_Trueprop (Thm.prop_of thm));
+ val (Rep_G, Abs_G, F) = strip3 thm;
val typ_Abs_G = dest_funT (fastype_of Abs_G);
val RepT = fst typ_Abs_G; (* F *)
val AbsT = snd typ_Abs_G; (* G *)
@@ -76,12 +408,12 @@
fun flatten_tyargs Ass =
map dest_TFree alpha0s
- |> filter (fn T => exists (fn Ts => member (op =) Ts T) Ass);
+ |> filter (fn T => exists (fn Ts => member op= Ts T) Ass);
val Ds0 = filter (is_none o fst) specs |> map snd;
(* get the bnf for RepT *)
- val ((bnf, (deads, alphas)),((_, unfolds), lthy)) =
+ val ((bnf_F, (deads, alphas)), ((_, unfolds), lthy)) =
bnf_of_typ true Dont_Inline (Binding.qualify true AbsT_name) flatten_tyargs []
Ds0 RepT ((empty_comp_cache, empty_unfolds), lthy);
@@ -94,33 +426,35 @@
val defs = #map_unfolds unfolds @ flat (#set_unfoldss unfolds) @ #rel_unfolds unfolds @ #pred_unfolds unfolds;
(* number of live variables *)
- val lives = length alphas;
+ val live = length alphas;
(* state the three required properties *)
val sorts = map Type.sort_of_atyp alphas;
val names_lthy = fold Variable.declare_typ (alphas @ deads) lthy;
- val (alphas', names_lthy) = mk_TFrees' sorts names_lthy;
- val (betas, names_lthy) = mk_TFrees' sorts names_lthy;
+ val (((alphas', betas), betas'), names_lthy) = names_lthy
+ |> mk_TFrees' sorts
+ ||>> mk_TFrees' sorts
+ ||>> mk_TFrees' sorts;
- val map_F = mk_map_of_bnf deads alphas betas bnf;
+ val map_F = mk_map_of_bnf deads alphas betas bnf_F;
- val (typ_fs, typ_aF) = fastype_of map_F |> strip_typeN lives ||> domain_type;
+ val (typ_fs, typ_aF) = fastype_of map_F |> strip_typeN live ||> domain_type;
val typ_pairs = map HOLogic.mk_prodT (alphas ~~ alphas');
val typ_subst_pair = typ_subst_atomic (alphas ~~ typ_pairs);
val typ_pair = typ_subst_pair RepT;
-
val subst_b = subst_atomic_types (alphas ~~ betas);
val subst_a' = subst_atomic_types (alphas ~~ alphas');
val subst_pair = subst_atomic_types (alphas ~~ typ_pairs);
val aF_set = F;
- val bF_set = subst_b F;
val aF_set' = subst_a' F;
val pairF_set = subst_pair F;
- val map_F_fst = mk_map_of_bnf deads typ_pairs alphas bnf;
- val map_F_snd = mk_map_of_bnf deads typ_pairs alphas' bnf;
+ val bF_set = subst_b F;
+ val map_F_fst = mk_map_of_bnf deads typ_pairs alphas bnf_F;
+ val map_F_snd = mk_map_of_bnf deads typ_pairs alphas' bnf_F
+ val sets_F_pairs = mk_sets_of_bnf (replicate live deads) (replicate live typ_pairs) bnf_F
val wits_F = mk_wits_of_bnf
- (replicate (nwits_of_bnf bnf) deads)
- (replicate (nwits_of_bnf bnf) alphas) bnf;
+ (replicate (nwits_of_bnf bnf_F) deads)
+ (replicate (nwits_of_bnf bnf_F) alphas) bnf_F;
(* val map_closed_F = @{term "\<And>f x. x \<in> F \<Longrightarrow> map_F f x \<in> F"}; *)
val (var_fs, names_lthy) = mk_Frees "f" typ_fs names_lthy;
@@ -131,31 +465,31 @@
val imp_map = Logic.mk_implies (mem_x, mem_map);
val map_closed_F = fold_rev Logic.all var_fs (Logic.all var_x imp_map);
- (* val zip_closed_F = @{term "\<And>z. map_F fst z \<in> F \<Longrightarrow> map_F snd z \<in> F \<Longrightarrow> z \<in> F"}; *)
- val (var_zs, names_lthy) = mk_Frees "z" [typ_pair] names_lthy;
+ (* val zip_closed_F =
+ @{term "\<And>z. \<lbrakk>map_F fst z \<in> F; map_F snd z \<in> F\<rbrakk> \<Longrightarrow>
+ \<exists>z' \<in> F. set_F z' \<subseteq> set_F z \<and> map_F fst z' = map_F fst z \<and> map_F snd z' = map_F snd z"}; *)
+ val (var_z, names_lthy) = mk_Free "z" typ_pair names_lthy;
+ val (var_z', names_lthy) = mk_Free "z'" typ_pair names_lthy;
val (pairs, names_lthy) = mk_Frees "tmp" typ_pairs names_lthy;
- val var_z = hd var_zs;
- val fsts = map (fst o Term.strip_comb o HOLogic.mk_fst) pairs;
- val snds = map (fst o Term.strip_comb o HOLogic.mk_snd) pairs;
- val map_fst = list_comb (list_comb (map_F_fst, fsts), var_zs);
+
+ fun mk_map mfs f z = Term.list_comb (mfs, map (fst o Term.strip_comb o f) pairs) $ z;
+ fun mk_set var = map (fn t => t $ var) sets_F_pairs;
+
+ val (map_fst', map_fst) = apply2 (mk_map map_F_fst HOLogic.mk_fst) (var_z', var_z);
+ val (map_snd', map_snd) = apply2 (mk_map map_F_snd HOLogic.mk_snd) (var_z', var_z);
val mem_map_fst = HOLogic.mk_Trueprop (HOLogic.mk_mem (map_fst, aF_set));
- val map_snd = list_comb (list_comb (map_F_snd, snds), var_zs);
val mem_map_snd = HOLogic.mk_Trueprop (HOLogic.mk_mem (map_snd, aF_set'));
- val mem_z = HOLogic.mk_Trueprop (HOLogic.mk_mem (var_z, pairF_set));
- val imp_zip = Logic.mk_implies (mem_map_fst, Logic.mk_implies (mem_map_snd, mem_z));
- val zip_closed_F = Logic.all var_z imp_zip;
+ val ex_conj = foldr1 HOLogic.mk_conj (map2 mk_leq (mk_set var_z') (mk_set var_z) @
+ [HOLogic.mk_eq (map_fst', map_fst), HOLogic.mk_eq (map_snd', map_snd)]);
+ val zip_concl = HOLogic.mk_Trueprop (mk_Bex pairF_set (absfree (dest_Free var_z') ex_conj));
+ val zip_closed_F = Logic.all var_z (Logic.list_implies ([mem_map_fst, mem_map_snd], zip_concl));
(* val wit_closed_F = @{term "wit_F a \<in> F"}; *)
val (var_as, names_lthy) = mk_Frees "a" alphas names_lthy;
val (var_bs, _) = mk_Frees "a" alphas names_lthy;
- fun binder_types_until_eq V T =
- let
- fun strip (TU as Type ("fun", [T, U])) = if V = TU then [] else T :: strip U
- | strip T = if V = T then [] else
- error ("Bad type for witness: " ^ quote (Syntax.string_of_typ lthy T));
- in strip T end;
- val Iwits = the_default wits_F (Option.map (map (`(map (fn T =>
- find_index (fn U => T = U) alphas) o binder_types_until_eq RepT o fastype_of))) wits);
+ val sets = mk_sets_of_bnf (replicate live deads) (replicate live alphas) bnf_F;
+ val (Iwits, wit_goals) =
+ prepare_wits false RepT wits opts alphas wits_F var_as var_bs sets lthy;
val wit_closed_Fs =
Iwits |> map (fn (I, wit_F) =>
let
@@ -163,65 +497,49 @@
val wit_a = list_comb (wit_F, vars);
in fold_rev Logic.all vars (HOLogic.mk_Trueprop (HOLogic.mk_mem (wit_a, aF_set))) end);
- val mk_wit_goals = mk_wit_goals var_as var_bs
- (mk_sets_of_bnf (replicate lives deads) (replicate lives alphas) bnf);
-
val goals = [map_closed_F, zip_closed_F] @ wit_closed_Fs @
- (case wits of NONE => [] | _ => maps mk_wit_goals Iwits);
-
- val lost_wits = filter_out (fn (J, _) => exists (fn (I, _) => I = J) Iwits) wits_F;
- val _ =
- if null lost_wits orelse no_warn_wits then ()
- else
- lost_wits
- |> map (Syntax.pretty_typ lthy o fastype_of o snd)
- |> Pretty.big_list
- "The following types of nonemptiness witnesses of the raw type's BNF were lost:"
- |> (fn pt => Pretty.chunks [pt,
- Pretty.para "You can specify a liftable witness (e.g., a term of one of the above types\
- \ that satisfies the typedef's invariant)\
- \ using the annotation [wits: <term>]."])
- |> Pretty.string_of
- |> warning;
+ (case wits of NONE => [] | _ => wit_goals);
fun after_qed ([map_closed_thm] :: [zip_closed_thm] :: wit_thmss) lthy =
let
val (wit_closed_thms, wit_thms) =
(case wits of
- NONE => (map the_single wit_thmss, wit_thms_of_bnf bnf)
+ NONE => (map the_single wit_thmss, wit_thms_of_bnf bnf_F)
| _ => chop (length wit_closed_Fs) (map the_single wit_thmss))
(* construct map set bd rel wit *)
(* val map_G = @{term "\<lambda>f. Abs_G o map_F f o Rep_G"}; *)
val Abs_Gb = subst_b Abs_G;
- val map_G =
- fold_rev HOLogic.tupled_lambda var_fs
+ val map_G = fold_rev lambda var_fs
(HOLogic.mk_comp (HOLogic.mk_comp (Abs_Gb, map_f), Rep_G));
+ val map_raw = fold_rev lambda var_fs map_f;
(* val sets_G = [@{term "set_F o Rep_G"}]; *)
- val sets_F = mk_sets_of_bnf (replicate lives deads) (replicate lives alphas) bnf;
+ val sets_F = mk_sets_of_bnf (replicate live deads) (replicate live alphas) bnf_F;
val sets_G = map (fn set_F => HOLogic.mk_comp (set_F, Rep_G)) sets_F;
(* val bd_G = @{term "bd_F"}; *)
- val bd_F = mk_bd_of_bnf deads alphas bnf;
+ val bd_F = mk_bd_of_bnf deads alphas bnf_F;
val bd_G = bd_F;
(* val rel_G = @{term "\<lambda>R. BNF_Def.vimage2p Rep_G Rep_G (rel_F R)"}; *)
- val rel_F = mk_rel_of_bnf deads alphas betas bnf;
- val (typ_Rs, _) = strip_typeN lives (fastype_of rel_F);
+ val rel_F = mk_rel_of_bnf deads alphas betas bnf_F;
+ val (typ_Rs, _) = strip_typeN live (fastype_of rel_F);
val (var_Rs, names_lthy) = mk_Frees "R" typ_Rs lthy;
val Rep_Gb = subst_b Rep_G;
val rel_G = fold_rev absfree (map dest_Free var_Rs)
(mk_vimage2p Rep_G Rep_Gb $ list_comb (rel_F, var_Rs));
+ val rel_raw = fold_rev absfree (map dest_Free var_Rs) (list_comb (rel_F, var_Rs));
(* val pred_G = @{term "\<lambda>P. pred_F P o Rep_G"}; *)
- val pred_F = mk_pred_of_bnf deads alphas bnf;
- val (typ_Ps, _) = strip_typeN lives (fastype_of pred_F);
+ val pred_F = mk_pred_of_bnf deads alphas bnf_F;
+ val (typ_Ps, _) = strip_typeN live (fastype_of pred_F);
val (var_Ps, names_lthy) = mk_Frees "P" typ_Ps names_lthy;
val pred_G = fold_rev absfree (map dest_Free var_Ps)
(HOLogic.mk_comp (list_comb (pred_F, var_Ps), Rep_G));
+ val pred_raw = fold_rev absfree (map dest_Free var_Ps) (list_comb (pred_F, var_Ps));
(* val wits_G = [@{term "Abs_G o wit_F"}]; *)
val (var_as, _) = mk_Frees "a" alphas names_lthy;
@@ -241,13 +559,13 @@
fun map_id0_tac ctxt =
HEADGOAL (EVERY' [rtac ctxt ext,
- SELECT_GOAL (unfold_thms_tac ctxt [map_id0_of_bnf bnf, id_apply, o_apply,
+ SELECT_GOAL (unfold_thms_tac ctxt [map_id0_of_bnf bnf_F, id_apply, o_apply,
Rep_inverse_thm]),
rtac ctxt refl]);
fun map_comp0_tac ctxt =
HEADGOAL (EVERY' [rtac ctxt ext,
- SELECT_GOAL (unfold_thms_tac ctxt [map_comp0_of_bnf bnf, o_apply,
+ SELECT_GOAL (unfold_thms_tac ctxt [map_comp0_of_bnf bnf_F, o_apply,
Rep_thm RS (map_closed_thm RS Abs_inverse_thm)]),
rtac ctxt refl]);
@@ -255,7 +573,7 @@
HEADGOAL (EVERY' ([SELECT_GOAL (unfold_thms_tac ctxt [o_apply]),
rtac ctxt (([Rep_thm RS map_closed_thm, Rep_thm RS map_closed_thm] MRS
Abs_inject_thm) RS iffD2),
- rtac ctxt (map_cong0_of_bnf bnf)] @ replicate lives (Goal.assume_rule_tac ctxt)));
+ rtac ctxt (map_cong0_of_bnf bnf_F)] @ replicate live (Goal.assume_rule_tac ctxt)));
val set_map0s_tac =
map (fn set_map => fn ctxt =>
@@ -263,45 +581,47 @@
SELECT_GOAL (unfold_thms_tac ctxt [set_map, o_apply,
Rep_thm RS (map_closed_thm RS Abs_inverse_thm)]),
rtac ctxt refl]))
- (set_map_of_bnf bnf);
+ (set_map_of_bnf bnf_F);
- fun card_order_bd_tac ctxt = HEADGOAL (rtac ctxt (bd_card_order_of_bnf bnf));
+ fun card_order_bd_tac ctxt = HEADGOAL (rtac ctxt (bd_card_order_of_bnf bnf_F));
- fun cinfinite_bd_tac ctxt = HEADGOAL (rtac ctxt (bd_cinfinite_of_bnf bnf));
+ fun cinfinite_bd_tac ctxt = HEADGOAL (rtac ctxt (bd_cinfinite_of_bnf bnf_F));
val set_bds_tac =
map (fn set_bd => fn ctxt =>
HEADGOAL (EVERY' [SELECT_GOAL (unfold_thms_tac ctxt [o_apply]), rtac ctxt set_bd]))
- (set_bd_of_bnf bnf);
+ (set_bd_of_bnf bnf_F);
fun le_rel_OO_tac ctxt =
HEADGOAL (EVERY' [rtac ctxt @{thm vimage2p_relcompp_mono},
- rtac ctxt ((rel_OO_of_bnf bnf RS sym) RS @{thm ord_eq_le_trans}),
+ rtac ctxt ((rel_OO_of_bnf bnf_F RS sym) RS @{thm ord_eq_le_trans}),
rtac ctxt @{thm order_refl}]);
fun rel_OO_Grp_tac ctxt =
HEADGOAL (EVERY' ([SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt ext))),
SELECT_GOAL (unfold_thms_tac ctxt [@{thm OO_Grp_alt}, mem_Collect_eq,
- o_apply, @{thm vimage2p_def}, in_rel_of_bnf bnf, Bex_def, mem_Collect_eq]),
+ o_apply, @{thm vimage2p_def}, in_rel_of_bnf bnf_F, Bex_def, mem_Collect_eq]),
rtac ctxt iffI,
- SELECT_GOAL (REPEAT_DETERM (HEADGOAL (eresolve0_tac [exE,conjE]))),
- rtac ctxt (zip_closed_thm OF (replicate 2 (Rep_thm RSN (2, @{thm ssubst_mem}))) RS
- Rep_cases_thm),
+ SELECT_GOAL (REPEAT_DETERM (HEADGOAL (eresolve_tac ctxt [exE, conjE]))),
+ forward_tac ctxt
+ [zip_closed_thm OF (replicate 2 (Rep_thm RSN (2, @{thm ssubst_mem})))],
assume_tac ctxt,
- assume_tac ctxt,
+ SELECT_GOAL (REPEAT_DETERM (HEADGOAL (eresolve_tac ctxt [bexE, conjE]))),
+ etac ctxt Rep_cases_thm,
hyp_subst_tac ctxt,
SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt exI))),
rtac ctxt conjI] @
- replicate (lives - 1) (rtac ctxt conjI THEN' assume_tac ctxt) @
- [assume_tac ctxt,
- SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt conjI))),
- REPEAT_DETERM_N 2 o
- etac ctxt (trans OF [iffD2 OF [Abs_inject_thm OF
- [map_closed_thm OF [Rep_thm], Rep_thm]], Rep_inverse_thm]),
- SELECT_GOAL (REPEAT_DETERM (HEADGOAL (eresolve0_tac [exE,conjE]))),
+ replicate live
+ (EVERY' [TRY o rtac ctxt conjI, etac ctxt @{thm order_trans}, assume_tac ctxt]) @
+ [SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt conjI))),
+ REPEAT_DETERM_N 2 o EVERY'
+ [rtac ctxt (trans OF [iffD2 OF [Abs_inject_thm OF
+ [map_closed_thm OF [Rep_thm], Rep_thm]], Rep_inverse_thm]),
+ etac ctxt trans, assume_tac ctxt],
+ SELECT_GOAL (REPEAT_DETERM (HEADGOAL (eresolve_tac ctxt [exE, conjE]))),
rtac ctxt exI,
rtac ctxt conjI] @
- replicate (lives - 1) (rtac ctxt conjI THEN' assume_tac ctxt) @
+ replicate (live-1) (rtac ctxt conjI THEN' assume_tac ctxt) @
[assume_tac ctxt,
rtac ctxt conjI,
REPEAT_DETERM_N 2 o EVERY'
@@ -310,7 +630,7 @@
fun pred_set_tac ctxt =
HEADGOAL (EVERY'
- [rtac ctxt (pred_set_of_bnf bnf RS @{thm arg_cong[of _ _ "\<lambda>f. f \<circ> _"]} RS trans),
+ [rtac ctxt (pred_set_of_bnf bnf_F RS @{thm arg_cong[of _ _ "\<lambda>f. f \<circ> _"]} RS trans),
SELECT_GOAL (unfold_thms_tac ctxt (@{thms Ball_comp_iff conj_comp_iff})),
rtac ctxt refl]);
@@ -326,16 +646,29 @@
[card_order_bd_tac, cinfinite_bd_tac] @ set_bds_tac @
[le_rel_OO_tac, rel_OO_Grp_tac, pred_set_tac];
- val (bnf, lthy) = bnf_def Dont_Inline (user_policy Note_Some) true I
+ val (bnf_G, lthy) = bnf_def Dont_Inline (user_policy Note_Some) true I
tactics wit_tac NONE map_b rel_b pred_b set_bs
(((((((Binding.empty, AbsT), map_G), sets_G), bd_G), wits_G), SOME rel_G), SOME pred_G)
lthy;
- val (bnf, lthy) =
- morph_bnf_defs (Morphism.thm_morphism "BNF" (unfold_thms lthy defs)) bnf
+ val old_defs =
+ {sets = set_defs_of_bnf bnf_G, map = map_def_of_bnf bnf_G, rel = rel_def_of_bnf bnf_G,
+ pred = pred_def_of_bnf bnf_G};
+
+ val unfold_morphism = Morphism.thm_morphism "BNF" (unfold_thms lthy defs);
+ val (bnf_G, lthy) = morph_bnf_defs unfold_morphism bnf_G
|> (fn bnf => note_bnf_defs bnf lthy);
+
+ val setup_lifting_thm = Typedef thm;
+ val quiet = exists (fn No_Warn_Transfer => true | _ => false) opts;
+
+ val transfer_consts = mk_typedef_transfer_tacs bnf_F bnf_G
+ {map_closed=map_closed_thm,typedef=thm} old_defs map_raw rel_raw pred_raw sets_F;
in
- lthy |> BNF_Def.register_bnf plugins AbsT_name bnf
+ lthy |> BNF_Def.register_bnf plugins AbsT_name bnf_G |>
+ mk_transfer_thms quiet bnf_F bnf_G AbsT_name transfer_consts setup_lifting_thm
+ {abs=typ_subst_atomic (alphas ~~ alphas') AbsT, rep=RepT, Ds0=map TFree Ds0,
+ deads = deads, alphas=alphas, betas=alphas', gammas=betas, deltas=betas'} defs
end
| after_qed _ _ = raise Match;
in
@@ -343,28 +676,1259 @@
end;
-(* main commands *)
+(** quotient_bnf **)
+
+fun mk_quotient_transfer_tacs bnf_F Tss live qthms thms set_F'_thmss old_defs
+ inst_REL_pos_distrI map_raw rel_raw sets_raw = let
+
+ fun common_tac ctxt addefs = unfold_thms_tac ctxt (#REL qthms :: addefs) THEN
+ (REPEAT_DETERM o HEADGOAL) (rtac ctxt rel_funI);
+
+ (* quotient.map_transfer tactic *)
+ val map_F_transfer = map_transfer_of_bnf bnf_F |> mk_rel_funDN (live+1);
+ fun map_transfer_q _ ctxt =
+ common_tac ctxt (#map old_defs :: @{thms o_def}) THEN
+ (HEADGOAL o EVERY') [REPEAT_DETERM o eresolve_tac ctxt @{thms relcomppE},
+ rtac ctxt @{thm relcomppI[rotated]}, hyp_subst_tac ctxt THEN' EVERY'
+ (map (rtac ctxt) [#rel_abs qthms, #map_F_rsp thms, (#rep_abs_rsp qthms), (#reflp qthms)]),
+ hyp_subst_tac ctxt, rtac ctxt map_F_transfer, REPEAT_DETERM_N (live+1) o assume_tac ctxt];
+
+ (* quotient.rel_transfer tactic *)
+ val rel_F_maps = rel_map_of_bnf bnf_F;
+ val rel_F_map_iffD2s = map (fn thm => thm RS @{thm iffD2}) rel_F_maps;
+ fun inst_REL_pos_distrI_order_refls vs aTs bTs ctxt = inst_REL_pos_distrI live vs aTs bTs ctxt
+ OF (replicate (live+1) asm_rl @ replicate live @{thm order_refl});
+ fun rel_transfer_q {Qs, Rs} ctxt = EVERY
+ [common_tac ctxt [#rel old_defs, @{thm vimage2p_def}],
+ HEADGOAL (rtac ctxt iffI),
+ (REPEAT_DETERM o ALLGOALS)
+ (eresolve_tac ctxt @{thms exE conjE relcomppE} ORELSE' hyp_subst_tac ctxt),
+ (HEADGOAL o EVERY')
+ [REPEAT_DETERM o dtac ctxt @{thm rel_fun_rel_OO1},
+ rtac ctxt (inst_REL_pos_distrI 0 (map mk_conversep Qs) (#betas Tss) (#alphas Tss) ctxt),
+ rtac ctxt @{thm relcomppI},
+ rtac ctxt (#symp qthms),
+ rtac ctxt (#map_F_rsp thms),
+ rtac ctxt (#rep_abs_rsp qthms),
+ rtac ctxt (#reflp qthms),
+ rtac ctxt @{thm relcomppI},
+ rtac ctxt (rel_flip_of_bnf bnf_F RS @{thm iffD1}),
+ rtac ctxt (nth rel_F_map_iffD2s 0),
+ rtac ctxt (nth rel_F_map_iffD2s 1),
+ etac ctxt (#rel_monoD_rotated thms)],
+ (REPEAT_DETERM_N live o HEADGOAL o EVERY')
+ [rtac ctxt @{thm predicate2I},
+ rtac ctxt @{thm conversepI},
+ rtac ctxt @{thm Basic_BNFs.rel_sum_simps(4)[THEN iffD2]},
+ etac ctxt @{thm conversepI}],
+ (HEADGOAL o EVERY')
+ [rtac ctxt (inst_REL_pos_distrI_order_refls Rs (#gammas Tss) (#deltas Tss) ctxt),
+ (SELECT_GOAL o REPEAT_DETERM o HEADGOAL) (etac ctxt @{thm relcomppI}),
+ rtac ctxt @{thm relcomppI[rotated]},
+ rtac ctxt (#map_F_rsp thms),
+ rtac ctxt (#rep_abs_rsp qthms OF [#reflp qthms]),
+ SELECT_GOAL (unfold_thms_tac ctxt (@{thms rel_sum_simps} @ rel_F_maps)),
+ assume_tac ctxt],
+ (REPEAT_DETERM_N (2*live) o HEADGOAL)
+ (rtac ctxt @{thm rel_sum_eq2_nonempty} ORELSE' rtac ctxt @{thm rel_sum_eq3_nonempty}),
+ (REPEAT_DETERM_N live)
+ (unfold_thms_tac ctxt @{thms sum.rel_compp[symmetric] eq_OO} THEN
+ HEADGOAL (etac ctxt @{thm sum.rel_mono[OF order_refl]})),
+ (HEADGOAL o EVERY')
+ [(SELECT_GOAL o REPEAT_DETERM o HEADGOAL) (dtac ctxt @{thm rel_fun_rel_OO2}),
+ rtac ctxt (inst_REL_pos_distrI 0 Qs (#alphas Tss) (#betas Tss) ctxt),
+ rtac ctxt @{thm relcomppI},
+ rtac ctxt (#reflp qthms),
+ rtac ctxt @{thm relcomppI},
+ rtac ctxt (nth rel_F_map_iffD2s 0),
+ rtac ctxt (nth rel_F_map_iffD2s 1),
+ etac ctxt (#rel_monoD_rotated thms)],
+ (REPEAT_DETERM_N live o HEADGOAL o EVERY')
+ [rtac ctxt @{thm predicate2I}, rtac ctxt @{thm rel_sum.intros(2)}, assume_tac ctxt],
+ (HEADGOAL o EVERY')
+ [rtac ctxt
+ (inst_REL_pos_distrI_order_refls (map mk_conversep Rs) (#deltas Tss) (#gammas Tss) ctxt),
+ rtac ctxt @{thm relcomppI},
+ etac ctxt (rotate_prems 1 (#transp qthms)),
+ rtac ctxt (#map_F_rsp thms),
+ rtac ctxt (#rep_abs_rsp qthms OF [#reflp qthms]),
+ etac ctxt @{thm relcomppI},
+ rtac ctxt @{thm relcomppI},
+ etac ctxt (#transp qthms),
+ rtac ctxt (#symp qthms),
+ rtac ctxt (#map_F_rsp thms),
+ rtac ctxt (#rep_abs_rsp qthms),
+ rtac ctxt (#reflp qthms),
+ rtac ctxt @{thm relcomppI[rotated]},
+ rtac ctxt (#reflp qthms),
+ rtac ctxt (rel_flip_of_bnf bnf_F RS @{thm iffD1}),
+ rtac ctxt (nth rel_F_map_iffD2s 0),
+ rtac ctxt (nth rel_F_map_iffD2s 1),
+ etac ctxt (#rel_monoD_rotated thms)],
+ (REPEAT_DETERM_N live o HEADGOAL o EVERY')
+ [rtac ctxt @{thm predicate2I},
+ rtac ctxt @{thm conversepI},
+ rtac ctxt @{thm rel_sum.intros(2)},
+ etac ctxt @{thm conversepI}],
+ (REPEAT_DETERM_N (2*live) o HEADGOAL)
+ (rtac ctxt @{thm rel_sum_eq2_nonempty} ORELSE' rtac ctxt @{thm rel_sum_eq3_nonempty}),
+ (REPEAT_DETERM_N live o EVERY)
+ [unfold_thms_tac ctxt @{thms sum.rel_compp[symmetric] eq_OO},
+ HEADGOAL (etac ctxt @{thm sum.rel_mono[OF order_refl]})]];
+
+ (* quotient.set_transfer tactics *)
+ fun set_transfer_q set_G_def set_F'_thms _ ctxt =
+ let val set_F'_rsp = mk_rel_funDN 1 (#set_F'_respect set_F'_thms) in
+ common_tac ctxt (set_G_def :: @{thms o_def}) THEN
+ (HEADGOAL o EVERY')
+ [etac ctxt @{thm relcomppE}, hyp_subst_tac ctxt,
+ SELECT_GOAL (unfold_thms_tac ctxt
+ [set_F'_rsp OF [#rep_abs qthms] OF [#reflp qthms], @{thm rel_set_def}]),
+ dtac ctxt (#rel_F_rel_F' thms), rtac ctxt conjI] THEN
+ (REPEAT_DETERM_N 2 o HEADGOAL o EVERY')
+ [SELECT_GOAL (unfold_thms_tac ctxt [#rel_F'_set thms]),
+ REPEAT_DETERM o eresolve_tac ctxt [exE, conjE],
+ REPEAT_DETERM o dtac ctxt (mk_sym set_F'_rsp),
+ SELECT_GOAL (unfold_thms_tac ctxt [#set_map_F' set_F'_thms]),
+ rtac ctxt ballI, dtac ctxt @{thm equalityD1[THEN subsetD]}, assume_tac ctxt,
+ SELECT_GOAL (unfold_thms_tac ctxt @{thms image_iff}),
+ etac ctxt bexE, dtac ctxt set_mp, assume_tac ctxt,
+ SELECT_GOAL (unfold_thms_tac ctxt @{thms mem_Collect_eq case_prod_beta}),
+ rtac ctxt bexI, hyp_subst_tac ctxt, assume_tac ctxt, etac ctxt @{thm hypsubst},
+ etac ctxt @{thm imageI}, assume_tac ctxt]
+ end;
+ in
+ {map={raw=map_raw, tac=map_transfer_q},
+ rel={raw=rel_raw, tac=rel_transfer_q},
+ sets={raws=sets_raw,tacs=map2 set_transfer_q (#sets old_defs) set_F'_thmss},
+ pred=NONE}
+ end;
+
+
+fun quotient_bnf {equiv_rel, equiv_thm, quot_thm, ...} _ wits specs map_b rel_b pred_b opts lthy =
+ let
+ (* extract rep_G and abs_G *)
+ val (_, abs_G, rep_G) = strip3 quot_thm;
+ val (repT, absT) = dest_funT (fastype_of abs_G); (* ("?'a F", "?'a G") *)
+ val absT_name = fst (dest_Type absT);
+
+ val tvs = map (fst o dest_TVar) (snd (dest_Type absT));
+ val _ = length tvs = length specs orelse
+ error ("Expected " ^ string_of_int (length tvs) ^
+ " type argument" ^ (if (length tvs) = 1 then "" else "s") ^ " to " ^ quote absT_name);
+
+ (* instantiate TVars *)
+ val alpha0s = map (TFree o snd) specs;
+ val typ_subst = typ_subst_TVars (tvs ~~ alpha0s);
+ val (repT, absT) = apply2 typ_subst (repT, absT);
+
+ (* get the bnf for RepT *)
+ val Ds0 = filter (is_none o fst) specs |> map snd;
+
+ fun flatten_tyargs Ass =
+ map dest_TFree alpha0s |> filter (fn T => exists (fn Ts => member op= Ts T) Ass);
+
+ val ((bnf_F, (deads, alphas)), ((_, unfolds), lthy)) =
+ bnf_of_typ true Dont_Inline (Binding.qualify true absT_name) flatten_tyargs
+ [] Ds0 repT ((empty_comp_cache, empty_unfolds), lthy);
+ val live = length alphas;
+ val _ = (if live = 0 then error "No live variables" else ());
+
+ val defs = #map_unfolds unfolds @ flat (#set_unfoldss unfolds) @ #rel_unfolds unfolds;
+ val set_bs =
+ map (fn T => find_index (fn U => T = U) alpha0s) alphas
+ |> map (the_default Binding.empty o fst o nth specs);
+
+ (* create and instantiate all the needed type variables *)
+ val subst = subst_TVars (tvs ~~ alpha0s);
+ val (abs_G, rep_G) = apply2 subst (abs_G, rep_G);
+
+ val sorts = map Type.sort_of_atyp alphas;
+ val (((betas, gammas), deltas), names_lthy) = fold Variable.declare_typ (alphas @ deads) lthy
+ |> mk_TFrees' sorts
+ ||>> mk_TFrees' sorts
+ ||>> mk_TFrees' sorts;
+
+ fun subst_Ts tm Ts = subst_atomic_types (alphas ~~ Ts) tm;
+ val subst_b = subst_atomic_types (alphas ~~ betas);
+ val subst_Maybe = subst_atomic_types o map (swap o `mk_MaybeT);
+ val equiv_rel_a = subst equiv_rel;
+ val map_F = mk_map_of_bnf deads alphas betas bnf_F;
+ val rel_F_ab = mk_rel_of_bnf deads alphas betas bnf_F;
+ val rel_F_bc = mk_rel_of_bnf deads betas gammas bnf_F;
+ val rel_F_ac = mk_rel_of_bnf deads alphas gammas bnf_F;
+ val rel_F_option = mk_rel_of_bnf deads (map mk_MaybeT alphas) (map mk_MaybeT betas) bnf_F;
+ val sets_F = mk_sets_of_bnf (replicate live deads) (replicate live alphas) bnf_F;
+ val wits_F = mk_wits_of_bnf
+ (replicate (nwits_of_bnf bnf_F) deads) (replicate (nwits_of_bnf bnf_F) alphas) bnf_F;
+
+ val (typ_fs, (typ_aF, typ_bF)) = strip_typeN live (fastype_of map_F) ||> dest_funT;
+ val typ_MaybeF = typ_subst_atomic (alphas ~~ map mk_MaybeT alphas) typ_aF;
+ val typ_a_sets = map HOLogic.mk_setT alphas;
+ val typ_pairs = map HOLogic.mk_prodT (alphas ~~ betas);
+ val typ_fs' = map (typ_subst_atomic (map (swap o `mk_MaybeT) betas)) typ_fs;
+
+ (* create all the needed variables *)
+ val ((((((((((((((((((((((var_Ps, var_Qs), var_Rs), var_x), var_x'), var_y), var_y'), var_mx),
+ var_As), var_As'), var_Ss), var_Bs), var_as), var_as'), var_bs), var_bs'), var_R), var_fs),
+ var_fs'), var_gs), var_gs'), var_z), var_ts) = names_lthy
+ |> mk_Frees "Ps" (map2 mk_relT alphas betas)
+ ||>> mk_Frees "Qs" (map2 mk_relT betas gammas)
+ ||>> mk_Frees "Rs" (map2 mk_relT alphas gammas)
+ ||>> mk_Free "x" typ_aF
+ ||>> mk_Free "x'" typ_aF
+ ||>> mk_Free "y" typ_bF
+ ||>> mk_Free "y'" (typ_subst_atomic (alphas ~~ gammas) typ_aF)
+ ||>> mk_Free "mx" typ_MaybeF
+ ||>> mk_Frees "As" typ_a_sets
+ ||>> mk_Frees "As'" typ_a_sets
+ ||>> mk_Frees "Ss" (map HOLogic.mk_setT typ_a_sets)
+ ||>> mk_Frees "Bs" (map HOLogic.mk_setT betas)
+ ||>> mk_Frees "as" alphas
+ ||>> mk_Frees "as'" alphas
+ ||>> mk_Frees "bs" betas
+ ||>> mk_Frees "bs'" betas
+ ||>> mk_Free "R" (typ_aF --> typ_bF --> HOLogic.boolT)
+ ||>> mk_Frees "fs" typ_fs
+ ||>> mk_Frees "fs'" typ_fs'
+ ||>> mk_Frees "gs" typ_fs
+ ||>> mk_Frees "gs'" typ_fs'
+ ||>> mk_Free "z" (typ_subst_atomic (alphas ~~ typ_pairs) typ_aF)
+ ||>> mk_Frees "ts" typ_pairs
+ |> fst;
+
+ (* create local definitions `b = tm` with n arguments *)
+ fun suffix tm s = (dest_Const tm |> fst |> Long_Name.base_name) ^ s;
+ fun define lthy b n tm =
+ let
+ val b = Binding.qualify true absT_name (Binding.qualified_name b);
+ val ((tm, (_, def)), (lthy, lthy_old)) = lthy
+ |> Local_Theory.open_target |> snd
+ |> Local_Theory.define_internal (((Binding.concealed b, NoSyn), (Binding.empty_atts, tm)))
+ ||> `Local_Theory.close_target;
+ val phi = Proof_Context.export_morphism lthy_old lthy;
+ val tm = Term.subst_atomic_types
+ (map (`(Morphism.typ phi)) (alphas @ betas @ gammas @ map TFree Ds0))
+ (Morphism.term phi tm);
+ val def = mk_unabs_def n (HOLogic.mk_obj_eq (Morphism.thm phi def));
+ in ({def=def, tm=tm}, lthy) end;
+
+ (* internally use REL, not the user-provided definition *)
+ val (REL, lthy) = define lthy "REL" 0 equiv_rel_a;
+ val REL_def = sym RS eq_reflection OF [#def REL];
+ fun REL_rewr_all ctxt thm = Conv.fconv_rule
+ (Conv.top_conv (fn _ => Conv.try_conv (Conv.rewr_conv REL_def)) ctxt) thm;
+
+ val equiv_rel_a' = equiv_rel_a;
+ val equiv_rel_a = #tm REL;
+ val (equiv_rel_b, equiv_rel_c) = apply2 (subst_Ts equiv_rel_a) (betas, gammas);
+
+ (* map_F_respect: @{term "((=) ===> REL ===> REL) map_F map_F"} *)
+ val map_F_respect = HOLogic.mk_Trueprop (fold_rev mk_rel_fun (map2 (fn xT => fn yT =>
+ HOLogic.eq_const (xT --> yT)) alphas betas @ [equiv_rel_a]) (equiv_rel_b) $ map_F $ map_F);
+
+ (* rel_pos_distr: @{term "\<And>A B.
+ A \<circ>\<circ> B \<noteq> bot \<Longrightarrow> REL \<circ>\<circ> rel_F A \<circ>\<circ> REL \<circ>\<circ> rel_F B \<circ>\<circ> REL \<le> REL \<circ>\<circ> rel_F (A \<circ>\<circ> B) \<circ>\<circ> REL"} *)
+ fun compp_not_bot comp aT cT = let
+ val T = mk_relT aT cT;
+ val mk_eq = HOLogic.eq_const T;
+ in HOLogic.mk_not (mk_eq $ comp $ bot_const T) end;
+ val ab_comps = map2 mk_relcompp var_Ps var_Qs;
+ val ne_comps = (@{map 3} compp_not_bot ab_comps alphas gammas);
+ val ab_prem = foldr1 HOLogic.mk_conj ne_comps;
+
+ val REL_pos_distrI_tm = let
+ val le_relcomps = map2 mk_leq ab_comps var_Rs;
+ val assm = mk_OO [equiv_rel_a, list_comb (rel_F_ab, var_Ps),
+ equiv_rel_b, list_comb (rel_F_bc, var_Qs)] equiv_rel_c;
+ val concl = mk_OO [equiv_rel_a, list_comb (rel_F_ac, var_Rs)] equiv_rel_c;
+ in
+ mk_Trueprop_implies
+ ([assm $ var_x $ var_y'] @ ne_comps @ le_relcomps, concl $ var_x $ var_y')
+ end;
+
+ val ab_concl = mk_leq
+ (mk_OO [list_comb (rel_F_ab, var_Ps), equiv_rel_b] (list_comb (rel_F_bc, var_Qs)))
+ (mk_OO [equiv_rel_a, list_comb (rel_F_ac, ab_comps)] (equiv_rel_c));
+ val ab_imp = Logic.mk_implies (apply2 HOLogic.mk_Trueprop (ab_prem, ab_concl));
+ val rel_pos_distr = fold_rev Logic.all (var_Ps @ var_Qs) ab_imp;
+
+ (* {(x, y) . REL x y} *)
+ fun mk_rel_pairs rel = mk_case_prod (var_x, var_x') (rel $ var_x $ var_x')
+ val rel_pairs = mk_rel_pairs equiv_rel_a;
+
+ (* rel_Inter: \<And>S. \<lbrakk> S \<noteq> {}; \<Inter>S \<noteq> {} \<rbrakk> \<Longrightarrow>
+ (\<Inter>A\<in>S. {(x, y). REL x y} `` {x. set_F x \<subseteq> A}) \<subseteq> {(x, y). REL x y} `` {x. set_F x \<subseteq> \<Inter>S} *)
+ fun rel_Inter_from_set_F (var_A, var_S) set_F = let
+
+ val typ_aset = fastype_of var_A;
+
+ (* \<Inter>S *)
+ val inter_S = Inf_const typ_aset $ var_S;
+
+ (* [S \<noteq> {}, \<Inter>S \<noteq> {}] *)
+ fun not_empty x = let val ty = fastype_of x
+ in HOLogic.mk_not (HOLogic.mk_eq (x, bot_const ty)) end;
+ val prems = map (HOLogic.mk_Trueprop o not_empty) [var_S, inter_S];
+
+ (* {x. set_F x \<subseteq> A} *)
+ val setF_sub_A = mk_in [var_A] [set_F] typ_aF;
+
+ (* {x. set_F x \<subseteq> \<Inter>S} *)
+ val setF_sub_S = mk_in [inter_S] [set_F] typ_aF;
+
+ val lhs = Inf_const (HOLogic.mk_setT typ_aF) $ (mk_image
+ (absfree (dest_Free var_A) (Image_const typ_aF $ rel_pairs $ setF_sub_A)) $ var_S);
+ val rhs = Image_const typ_aF $ rel_pairs $ setF_sub_S;
+ val concl = HOLogic.mk_Trueprop (mk_leq lhs rhs);
+
+ in Logic.all var_S (Logic.list_implies (prems, concl)) end;
+
+ val rel_Inters = map2 rel_Inter_from_set_F (var_As ~~ var_Ss) sets_F;
+
+ (* map_F_Just = map_F Just *)
+ val map_F_Just = let
+ val option_tys = map mk_MaybeT alphas;
+ val somes = map Just_const alphas;
+ in list_comb (subst_atomic_types (betas ~~ option_tys) map_F, somes) end;
+
+ fun mk_set_F'_tm typ_a set_F =
+ let
+ val typ_aset = HOLogic.mk_setT typ_a;
+
+ (* set_F' x = (\<Inter>y\<in>{y. REL (map_F Just x) y}. UNION (set_F y) set_Maybe) *)
+ val sbind = mk_UNION (subst_Maybe alphas set_F $ var_mx) (set_Maybe_const typ_a);
+ val collection = HOLogic.Collect_const typ_MaybeF $ absfree (dest_Free var_mx)
+ (subst_Maybe alphas equiv_rel_a $ (map_F_Just $ var_x) $ var_mx);
+ val set_F'_tm = lambda var_x
+ (Inf_const typ_aset $ (mk_image (absfree (dest_Free var_mx) sbind) $ collection));
+ in
+ set_F'_tm
+ end
+
+ val sets = mk_sets_of_bnf (replicate live deads) (replicate live alphas) bnf_F;
+ val sets' = map2 mk_set_F'_tm alphas sets;
+
+ val (Iwits, wit_goals) =
+ prepare_wits true repT wits opts alphas wits_F var_as var_as' sets' lthy;
+
+ val goals = map_F_respect :: rel_pos_distr :: rel_Inters @
+ (case wits of NONE => [] | _ => wit_goals);
+
+ val plugins =
+ get_first (fn Plugins_Option f => SOME (f lthy) | _ => NONE) (rev opts) |>
+ the_default Plugin_Name.default_filter;
+
+ fun after_qed thmss lthy =
+ (case thmss of [map_F_respect_thm] :: [rel_pos_distr_thm0] :: thmss =>
+ let
+ val equiv_thm' = REL_rewr_all lthy equiv_thm;
+ val rel_pos_distr_thm =
+ @{thm equivp_add_relconj} OF [equiv_thm', equiv_thm', rel_pos_distr_thm0];
+
+ val (rel_Inter_thms, wit_thmss) = apply2 (fn f => flat (f live thmss)) (take, drop);
+ val wit_thms = (case wit_thmss of
+ [] => wit_thms_of_bnf bnf_F
+ | _ => wit_thmss);
+
+ (* construct map_G, sets_G, bd_G, rel_G and wits_G *)
+
+ (* map_G f = abs_G o map_F f o rep_G *)
+ val map_G = fold_rev lambda var_fs (HOLogic.mk_comp (HOLogic.mk_comp
+ (subst_Ts abs_G betas, list_comb (map_F, var_fs)), rep_G));
+ val map_raw = fold_rev lambda var_fs (list_comb (map_F, var_fs))
+ |> subst_atomic_types (betas ~~ gammas);
+
+ (* Define set_G and the three auxiliary definitions (set_F', F_in, F_in') *)
+ fun mk_set_G var_A set_F lthy = let
+ val typ_a = HOLogic.dest_setT (fastype_of var_A);
+ val set_F'_tm = mk_set_F'_tm typ_a set_F
+
+ val (set_F', lthy) = define lthy (suffix set_F "'") 1 set_F'_tm;
+
+ (* set_G = set_F' o rep_G *)
+ val set_G = HOLogic.mk_comp (#tm set_F', rep_G);
+
+ (* F_in A = {x. set_F x \<subseteq> A} *)
+ val F_in_tm = lambda var_A (mk_in [var_A] [set_F] typ_aF);
+ val (F_in, lthy) = define lthy (suffix set_F "_in") 1 F_in_tm;
+
+ (* F_in' A = map_F Inr -` ({(x, y). REL x y} `` F_in (insert (Inl ()) (Inr ` A))) *)
+ val F_in' = lambda var_A (mk_vimage map_F_Just (Image_const typ_MaybeF $
+ subst_Maybe alphas rel_pairs $ (subst_Maybe alphas (#tm F_in) $ mk_insert
+ (mk_Nothing typ_a) (mk_image (Just_const typ_a) $ var_A))));
+ val (F_in', lthy) = define lthy (suffix set_F "_in'") 1 F_in';
+
+ in ((set_G, {set_F'=set_F', F_in=F_in, F_in'=F_in'}), lthy) end;
+
+ val ((sets_G, set_F'_aux_defs), lthy) =
+ @{fold_map 2} mk_set_G var_As sets_F lthy |>> split_list;
+
+ (* bd_G = bd_F *)
+ val bd_G = mk_bd_of_bnf deads alphas bnf_F;
+
+ (* rel_F' A =
+ BNF_Def.vimage2p (map_F Just) (map_F Just) ((\<cong>) OO rel_F (rel_Maybe A) OO (\<cong>)) *)
+ val rel_Maybes = @{map 3} (fn v => fn aT => fn bT => rel_Maybe_const aT bT $ v);
+ val rel_F'_tm = let val equiv_equiv_rel_option = subst_Ts equiv_rel_a' o map mk_MaybeT in
+ mk_vimage2p map_F_Just (subst_atomic_types (alphas ~~ betas) map_F_Just) $
+ mk_OO [equiv_equiv_rel_option alphas, list_comb (rel_F_option, rel_Maybes var_Ps alphas betas)]
+ (equiv_equiv_rel_option betas) end;
+
+ val (rel_F', lthy) =
+ define lthy (suffix rel_F_ab "'") (live+2) (fold_rev lambda var_Ps rel_F'_tm);
+
+ (* rel_G A = vimage2p rep_G rep_G (rel_F' A) *)
+ val rel_G = fold_rev lambda var_Ps (mk_vimage2p rep_G (subst_Ts rep_G betas) $ rel_F'_tm);
+ val rel_raw = fold_rev lambda var_Ps rel_F'_tm
+ |> subst_atomic_types (betas ~~ gammas);
+
+ (* val wits_G = [abs_G o wit_F] *)
+ val wits_G = map (fn (I, wit) => let val vars = (map (fn n => nth var_as n) I)
+ in fold_rev lambda vars (abs_G $ list_comb (wit, vars)) end) Iwits;
+
+ (* auxiliary lemmas *)
+ val bd_card_order = bd_card_order_of_bnf bnf_F;
+ val bd_cinfinite = bd_cinfinite_of_bnf bnf_F;
+ val in_rel = in_rel_of_bnf bnf_F;
+ val map_F_comp = map_comp_of_bnf bnf_F;
+ val map_F_comp0 = map_comp0_of_bnf bnf_F;
+ val map_F_cong = map_cong_of_bnf bnf_F;
+ val map_F_id0 = map_id0_of_bnf bnf_F;
+ val map_F_id = map_id_of_bnf bnf_F;
+ val rel_conversep = rel_conversep_of_bnf bnf_F;
+ val rel_Grp = rel_Grp_of_bnf bnf_F;
+ val rel_OO = rel_OO_of_bnf bnf_F;
+ val rel_map = rel_map_of_bnf bnf_F;
+ val rel_refl_strong = rel_refl_strong_of_bnf bnf_F;
+ val set_bd_thms = set_bd_of_bnf bnf_F;
+ val set_map_thms = set_map_of_bnf bnf_F;
+
+ val rel_funD = mk_rel_funDN (live+1);
+ val map_F_rsp = (rel_funD map_F_respect_thm) OF (replicate live refl);
+ fun map_F_rsp_of tms ctxt = (infer_instantiate' ctxt (NONE :: NONE
+ :: map (SOME o Thm.cterm_of ctxt) tms) map_F_rsp)
+
+ val qthms = let
+ fun equivp_THEN thm = REL_rewr_all lthy equiv_thm RS thm;
+ fun Quotient3_THEN thm = REL_rewr_all lthy quot_thm RS thm;
+ in
+ {abs_rep = Quotient3_THEN @{thm Quotient3_abs_rep},
+ rel_abs = Quotient3_THEN @{thm Quotient3_rel_abs},
+ rep_abs = Quotient3_THEN @{thm Quotient3_rep_abs},
+ rep_reflp = Quotient3_THEN @{thm Quotient3_rep_reflp},
+ rep_abs_rsp = Quotient3_THEN @{thm rep_abs_rsp},
+ reflp = equivp_THEN @{thm equivp_reflp},
+ symp = equivp_THEN @{thm equivp_symp},
+ transp = equivp_THEN @{thm equivp_transp},
+ REL = REL_def}
+ end;
+
+ (* lemma REL_OO_REL_left: REL OO REL OO R = REL OO R *)
+ val REL_OO_REL_left_thm = let
+ val tm = mk_Trueprop_eq
+ (mk_OO [equiv_rel_a, equiv_rel_a] var_R, mk_relcompp equiv_rel_a var_R)
+ fun tac ctxt = HEADGOAL (EVERY'
+ [rtac ctxt ext,
+ rtac ctxt ext,
+ rtac ctxt iffI,
+ TWICE (etac ctxt @{thm relcomppE}),
+ rtac ctxt @{thm relcomppI},
+ etac ctxt (#transp qthms),
+ TWICE (assume_tac ctxt),
+ etac ctxt @{thm relcomppE},
+ etac ctxt @{thm relcomppI},
+ rtac ctxt @{thm relcomppI},
+ rtac ctxt (#reflp qthms),
+ assume_tac ctxt]);
+ in prove lthy [var_R] tm tac end;
+
+ (* Generate theorems related to the setters *)
+ val map_F_fs = list_comb (map_F, var_fs);
+
+ (* aset aset asetset bset typ_b typ_b *)
+ fun mk_set_F'_thmss (((((var_A, var_A'), var_S), var_B), var_b), var_b')
+ set_F {set_F', F_in, F_in'} rel_Inter_thm set_map_F_thm (idx, vf) =
+ let
+ val (var_f, var_fs') = case vf of
+ (f :: fs) => (f, fs)
+ | _ => error "won't happen";
+
+ val typ_a = fastype_of var_f |> dest_funT |> fst;
+ val typ_b = fastype_of var_b;
+ val (typ_asetset, typ_aset) = `HOLogic.mk_setT (fastype_of var_A);
+
+ val map_F_fs_x = map_F_fs $ var_x;
+
+ (* F_in'_mono: A \<subseteq> B \<Longrightarrow> F_in' A \<subseteq> F_in' B *)
+ val F_in'_mono_tm = mk_Trueprop_implies
+ ([mk_leq var_A var_A'], mk_leq (#tm F_in' $ var_A) (#tm F_in' $ var_A'));
+ fun F_in'_mono_tac ctxt =
+ unfold_thms_tac ctxt [#def F_in', #def F_in] THEN
+ HEADGOAL (EVERY'
+ [rtac ctxt subsetI,
+ etac ctxt vimageE,
+ etac ctxt @{thm ImageE},
+ etac ctxt CollectE,
+ etac ctxt CollectE,
+ dtac ctxt @{thm case_prodD},
+ hyp_subst_tac ctxt,
+ rtac ctxt (vimageI OF [refl]),
+ rtac ctxt @{thm ImageI},
+ rtac ctxt CollectI,
+ rtac ctxt @{thm case_prodI},
+ assume_tac ctxt ORELSE' rtac ctxt refl,
+ rtac ctxt CollectI,
+ etac ctxt subset_trans,
+ etac ctxt (@{thm insert_mono} OF [@{thm image_mono}])]);
+ val F_in'_mono_thm = prove lthy [var_A, var_A'] F_in'_mono_tm F_in'_mono_tac;
+
+ (* F_in'_Inter: F_in' (\<Inter>S) = (\<Inter>A\<in>S. F_in' A) *)
+ val F_in'_Inter_tm = mk_Trueprop_eq ((#tm F_in' $ (Inf_const typ_aset $ var_S)),
+ (Inf_const (HOLogic.mk_setT typ_aF) $ (mk_image (#tm F_in') $ var_S)));
+ fun F_in'_Inter_tac ctxt =
+ Local_Defs.unfold_tac ctxt [#def F_in', #def F_in]
+ THEN HEADGOAL (rtac ctxt (infer_instantiate' ctxt
+ [SOME (Thm.cterm_of ctxt (HOLogic.mk_eq (var_S, bot_const typ_asetset)))] @{thm case_split})
+ THEN' EVERY' [
+ hyp_subst_tac ctxt,
+ SELECT_GOAL
+ (unfold_thms_tac ctxt @{thms Inter_empty INT_empty UNIV_sum_unit_conv}),
+ rtac ctxt @{thm set_eqI},
+ rtac ctxt iffI,
+ rtac ctxt UNIV_I,
+ rtac ctxt (vimageI OF [refl]),
+ rtac ctxt @{thm ImageI},
+ rtac ctxt CollectI,
+ rtac ctxt @{thm case_prodI},
+ rtac ctxt (#reflp qthms),
+ rtac ctxt CollectI,
+ rtac ctxt subset_UNIV,
+ etac ctxt @{thm exE[OF ex_in_conv[THEN iffD2]]},
+ EqSubst.eqsubst_tac ctxt [0] @{thms image_INT[of _ UNIV _ id, simplified]},
+ rtac ctxt @{thm inj_Inr},
+ assume_tac ctxt,
+ SELECT_GOAL (unfold_thms_tac ctxt @{thms INT_extend_simps vimage_INT[symmetric]}),
+ rtac ctxt @{thm arg_cong2[where f=vimage, OF refl]},
+ rtac ctxt equalityI,
+ rtac ctxt subsetI,
+ rtac ctxt @{thm InterI},
+ etac ctxt imageE,
+ etac ctxt @{thm ImageE},
+ etac ctxt CollectE,
+ etac ctxt CollectE,
+ dtac ctxt @{thm case_prodD},
+ hyp_subst_tac ctxt,
+ rtac ctxt @{thm ImageI[OF CollectI]},
+ etac ctxt @{thm case_prodI} ORELSE' (SELECT_GOAL
+ (unfold_thms_tac ctxt @{thms prod.case}) THEN' rtac ctxt @{thm refl}),
+ rtac ctxt CollectI,
+ etac ctxt subset_trans,
+ etac ctxt @{thm INT_lower[OF imageI]},
+ rtac ctxt (@{thm subset_trans} OF [asm_rl, rel_Inter_thm]),
+ K (unfold_thms_tac ctxt @{thms image_image}),
+ rtac ctxt subset_refl,
+ K (unfold_thms_tac ctxt @{thms INT_extend_simps ex_in_conv[symmetric]}),
+ rtac ctxt exI,
+ rtac ctxt imageI,
+ assume_tac ctxt,
+ rtac ctxt exI,
+ rtac ctxt @{thm InterI},
+ etac ctxt imageE,
+ hyp_subst_tac ctxt,
+ rtac ctxt @{thm insertI1}]);
+
+ val F_in'_Inter_thm = prove lthy [var_S] F_in'_Inter_tm F_in'_Inter_tac;
+
+ (* set_F'_respect: (REL ===> (=)) set_F' set_F' *)
+ val set_F'_respect_tm = HOLogic.mk_Trueprop (mk_rel_fun equiv_rel_a
+ (HOLogic.eq_const typ_aset) $ #tm set_F' $ #tm set_F');
+ fun set_F'_respect_tac ctxt = unfold_thms_tac ctxt (#def set_F' :: @{thms rel_fun_def})
+ THEN HEADGOAL (EVERY'
+ [TWICE (rtac ctxt allI),
+ rtac ctxt impI,
+ dtac ctxt (map_F_rsp_of (map Just_const alphas) ctxt),
+ rtac ctxt @{thm INF_cong},
+ rtac ctxt @{thm Collect_eqI},
+ rtac ctxt iffI,
+ etac ctxt (#transp qthms OF [#symp qthms]),
+ assume_tac ctxt,
+ etac ctxt (#transp qthms),
+ assume_tac ctxt,
+ rtac ctxt refl]);
+
+ (* F_in'_alt2: F_in' A = {x. set_F' x \<subseteq> A} *)
+ val F_in'_alt2_tm = mk_Trueprop_eq
+ (#tm F_in' $ var_A, mk_in [var_A] [#tm set_F'] typ_aF);
+ fun F_in'_alt2_tac ctxt = HEADGOAL (rtac ctxt equalityI THEN'
+ (Subgoal.FOCUS o K) (unfold_thms_tac ctxt (map #def [set_F', F_in', F_in])) ctxt
+ THEN' EVERY'
+ [rtac ctxt subsetI,
+ rtac ctxt CollectI,
+ rtac ctxt subsetI,
+ dtac ctxt vimageD,
+ etac ctxt @{thm ImageE},
+ etac ctxt CollectE,
+ etac ctxt CollectE,
+ dtac ctxt @{thm case_prodD},
+ dtac ctxt @{thm InterD},
+ rtac ctxt @{thm imageI[OF CollectI]},
+ etac ctxt (#symp qthms),
+ etac ctxt @{thm UnionE},
+ etac ctxt imageE,
+ hyp_subst_tac ctxt,
+ etac ctxt @{thm subset_lift_sum_unitD},
+ etac ctxt @{thm setr.cases},
+ hyp_subst_tac ctxt,
+ assume_tac ctxt])
+ THEN unfold_thms_tac ctxt [#def set_F'] THEN
+ (HEADGOAL o EVERY')
+ [rtac ctxt subsetI,
+ etac ctxt CollectE,
+ etac ctxt (subsetD OF [F_in'_mono_thm]),
+ EqSubst.eqsubst_tac ctxt [0] [F_in'_Inter_thm],
+ rtac ctxt @{thm InterI}] THEN
+ REPEAT_DETERM (HEADGOAL (etac ctxt @{thm imageE} THEN' hyp_subst_tac ctxt)) THEN
+ (HEADGOAL o EVERY')
+ [etac ctxt CollectE,
+ SELECT_GOAL (unfold_thms_tac ctxt (map #def [F_in', F_in])),
+ rtac ctxt @{thm vimageI[OF refl]},
+ rtac ctxt @{thm ImageI},
+ rtac ctxt CollectI,
+ rtac ctxt @{thm case_prodI},
+ etac ctxt (#symp qthms),
+ rtac ctxt CollectI,
+ rtac ctxt subsetI,
+ rtac ctxt @{thm sum_insert_Inl_unit},
+ assume_tac ctxt,
+ hyp_subst_tac ctxt,
+ rtac ctxt imageI,
+ rtac ctxt @{thm UnionI},
+ rtac ctxt imageI,
+ assume_tac ctxt,
+ rtac ctxt @{thm setr.intros[OF refl]}];
+ val F_in'_alt2_thm = prove lthy [var_A] F_in'_alt2_tm F_in'_alt2_tac;
+
+ (* set_F'_alt: set_F' x = \<Inter>{A. x \<in> F_in' A} *)
+ val set_F'_alt_tm = mk_Trueprop_eq (#tm set_F' $ var_x,
+ Inf_const typ_aset $ mk_Collect (var_A, HOLogic.mk_mem (var_x, #tm F_in' $ var_A)));
+ fun set_F'_alt_tac ctxt = unfold_thms_tac ctxt [F_in'_alt2_thm]
+ THEN HEADGOAL (EVERY'
+ [rtac ctxt @{thm set_eqI},
+ rtac ctxt iffI,
+ rtac ctxt @{thm InterI},
+ etac ctxt CollectE,
+ etac ctxt CollectE,
+ dtac ctxt subsetD,
+ assume_tac ctxt,
+ assume_tac ctxt,
+ etac ctxt @{thm InterD},
+ rtac ctxt CollectI,
+ rtac ctxt CollectI,
+ rtac ctxt subset_refl]);
+ val set_F'_alt_thm = prove lthy [var_x] set_F'_alt_tm set_F'_alt_tac;
+
+ (* map_F_in_F_in'I: x \<in> F_in' B \<Longrightarrow> map_F f x \<in> F_in' (f ` B) *)
+ val map_F_in_F_in'I_tm = mk_Trueprop_implies ([HOLogic.mk_mem (var_x, #tm F_in' $ var_A')],
+ HOLogic.mk_mem (map_F_fs_x, subst_b (#tm F_in') $ (mk_image var_f $ var_A')));
+ fun map_F_in_F_in'I_tac ctxt =
+ Local_Defs.unfold_tac ctxt ([#def F_in', #def F_in, Bex_def] @ @{thms vimage_def Image_iff}) THEN
+ HEADGOAL (EVERY'
+ [etac ctxt @{thm CollectE},
+ etac ctxt exE,
+ etac ctxt conjE,
+ etac ctxt @{thm CollectE},
+ etac ctxt @{thm CollectE},
+ dtac ctxt @{thm case_prodD},
+ rtac ctxt @{thm CollectI},
+ rtac ctxt exI,
+ rtac ctxt @{thm conjI[rotated]},
+ rtac ctxt @{thm CollectI},
+ rtac ctxt @{thm case_prodI},
+ dtac ctxt (map_F_rsp_of (map mk_Maybe_map var_fs) ctxt),
+ SELECT_GOAL (unfold_thms_tac ctxt (map_F_comp :: @{thms o_def map_sum.simps})),
+ assume_tac ctxt,
+ rtac ctxt CollectI,
+ SELECT_GOAL (unfold_thms_tac ctxt set_map_thms),
+ etac ctxt @{thm image_map_sum_unit_subset}]);
+ val map_F_in_F_in'I_thm =
+ prove lthy (var_A' :: var_x :: var_fs) map_F_in_F_in'I_tm map_F_in_F_in'I_tac;
+
+ (* REL_preimage_eq: C \<inter> range f \<noteq> {} \<Longrightarrow>
+ {(a, b). REL a b} `` {x. set_F x \<subseteq> f -` C} =
+ map_F f -` {(a, b). REL a b} `` {x. set_F x \<subseteq> C} *)
+ val REL_preimage_eq_tm = mk_Trueprop_implies ([HOLogic.mk_not (HOLogic.mk_eq
+ (HOLogic.mk_binop @{const_name inf} (var_B, mk_image var_f $ HOLogic.mk_UNIV typ_a),
+ bot_const (HOLogic.mk_setT typ_b)))], HOLogic.mk_eq (Image_const typ_aF $
+ rel_pairs $ mk_in [mk_vimage var_f var_B] [set_F] typ_aF, mk_vimage map_F_fs
+ (Image_const typ_bF $ subst_b rel_pairs $ mk_in [var_B] [subst_b set_F] typ_bF)));
+
+ (* Bs \<inter> range fs \<noteq> {} \<Longrightarrow> set_F xb \<subseteq> Bs \<Longrightarrow> REL xb (map_F fs x)
+ \<Longrightarrow> x \<in> {(x, x'). REL x x'} `` {x. set_F x \<subseteq> fs -` Bs} *)
+ fun subgoal_tac {context = ctxt, params, ...} = let
+ val (x, y) = case params of
+ [(_, x), _, (_, y)] => (x, y)
+ | _ => error "won't happen";
+ val cond = HOLogic.mk_conj (apply2 HOLogic.mk_mem ((var_b, var_B), (var_b', var_B)));
+
+ (* ["\<lambda>x y. x \<in> B \<and> y \<in> B", "(Grp UNIV f_1)\<inverse>\<inverse>"] *)
+ val cvars = var_fs |> maps (fn f => let val fT = fastype_of f in
+ map (SOME o Thm.cterm_of ctxt)
+ [if f = var_f then
+ fold_rev lambda [var_b, var_b'] cond else HOLogic.eq_const (range_type fT),
+ mk_conversep (mk_Grp (HOLogic.mk_UNIV (domain_type fT)) f)] end);
+ val rel_pos_distr_thm_inst = infer_instantiate' ctxt (cvars @ [SOME y,SOME x])
+ (@{thm predicate2D} OF [rel_pos_distr_thm]);
+
+ (* GrpI[of "map_F f1 .. fN" x, OF refl CollectI, OF "B1 \<subseteq> UNIV \<and> ... \<and> Bn \<subseteq> UNIV"] *)
+ fun subset_UNIVs n = fold (fn a => fn b => conjI OF [a, b]) (replicate (n-1)
+ @{thm subset_UNIV}) @{thm subset_UNIV};
+ val GrpI_inst = infer_instantiate' ctxt (map SOME [Thm.cterm_of ctxt map_F_fs, x])
+ @{thm GrpI} OF [refl, CollectI] OF [subset_UNIVs live];
+
+ in EVERY [
+ HEADGOAL (Method.insert_tac ctxt [rel_pos_distr_thm_inst]),
+ unfold_thms_tac ctxt [rel_conversep, rel_OO, rel_Grp],
+ HEADGOAL (etac ctxt @{thm meta_impE}),
+ REPEAT_DETERM_N (live-1) (HEADGOAL (rtac ctxt @{thm conjI[rotated]})),
+ REPEAT_DETERM_N live (HEADGOAL (etac ctxt @{thm relcompp_mem_Grp_neq_bot} ORELSE'
+ rtac ctxt @{thm relcompp_eq_Grp_neq_bot})),
+ HEADGOAL (EVERY' [etac ctxt @{thm meta_impE},
+ rtac ctxt @{thm relcomppI},
+ rtac ctxt (#reflp qthms),
+ rtac ctxt @{thm relcomppI},
+ rtac ctxt rel_refl_strong]),
+ REPEAT_DETERM_N idx (HEADGOAL (rtac ctxt refl)),
+ HEADGOAL (rtac ctxt conjI THEN' TWICE (etac ctxt subsetD THEN' assume_tac ctxt)),
+ REPEAT_DETERM_N (live-idx-1) (HEADGOAL (rtac ctxt refl)),
+ HEADGOAL (EVERY'
+ [rtac ctxt @{thm relcomppI},
+ assume_tac ctxt,
+ rtac ctxt @{thm relcomppI},
+ rtac ctxt @{thm conversepI},
+ rtac ctxt GrpI_inst,
+ rtac ctxt (#reflp qthms),
+ etac ctxt @{thm relcomppE},
+ etac ctxt @{thm relcomppE},
+ etac ctxt @{thm relcomppE},
+ etac ctxt @{thm conversepE},
+ etac ctxt @{thm GrpE},
+ hyp_subst_tac ctxt,
+ rtac ctxt @{thm ImageI},
+ rtac ctxt CollectI,
+ rtac ctxt @{thm case_prodI},
+ assume_tac ctxt,
+ EqSubst.eqsubst_asm_tac ctxt [1] rel_map,
+ EqSubst.eqsubst_asm_tac ctxt [1] [in_rel_of_bnf bnf_F],
+ etac ctxt exE,
+ etac ctxt CollectE,
+ etac ctxt conjE,
+ etac ctxt conjE,
+ etac ctxt CollectE,
+ hyp_subst_tac ctxt,
+ rtac ctxt CollectI]),
+ unfold_thms_tac ctxt set_map_thms,
+ HEADGOAL (rtac ctxt (subsetI OF [vimageI] OF [refl]) THEN'
+ etac ctxt @{thm imageE} THEN' hyp_subst_tac ctxt),
+ REPEAT_DETERM_N 6 (HEADGOAL (etac ctxt Drule.thin_rl)),
+ REPEAT_DETERM_N (live-1) (HEADGOAL (etac ctxt conjE)),
+ HEADGOAL (EVERY' [dtac ctxt subsetD, assume_tac ctxt, etac ctxt CollectE]),
+ unfold_thms_tac ctxt @{thms split_beta},
+ HEADGOAL (etac ctxt conjunct2)] end;
+
+ fun REL_preimage_eq_tac ctxt = HEADGOAL (EVERY'
+ [rtac ctxt @{thm set_eqI},
+ rtac ctxt iffI,
+ etac ctxt @{thm ImageE},
+ etac ctxt CollectE,
+ etac ctxt CollectE,
+ dtac ctxt @{thm case_prodD},
+ rtac ctxt (vimageI OF [refl]),
+ rtac ctxt @{thm ImageI},
+ rtac ctxt CollectI,
+ rtac ctxt @{thm case_prodI},
+ etac ctxt map_F_rsp,
+ rtac ctxt CollectI,
+ EqSubst.eqsubst_tac ctxt [0] [set_map_F_thm],
+ etac ctxt @{thm subset_vimage_image_subset},
+ etac ctxt vimageE,
+ etac ctxt @{thm ImageE},
+ TWICE (etac ctxt CollectE),
+ dtac ctxt @{thm case_prodD},
+ hyp_subst_tac ctxt,
+ Subgoal.FOCUS_PARAMS subgoal_tac ctxt]);
+
+ val REL_preimage_eq_thm = prove lthy (var_B :: var_fs) REL_preimage_eq_tm REL_preimage_eq_tac;
+
+ (* set_map_F': set_F' (map_F f x) = f ` set_F' x *)
+ val set_map_F'_tm = mk_Trueprop_eq (subst_b (#tm set_F')
+ $ map_F_fs_x, mk_image var_f $ (#tm set_F' $ var_x));
+ fun set_map_F'_tac ctxt = HEADGOAL (EVERY'
+ [rtac ctxt @{thm set_eqI},
+ rtac ctxt iffI,
+ EqSubst.eqsubst_asm_tac ctxt [0] [set_F'_alt_thm],
+ etac ctxt @{thm InterD},
+ rtac ctxt CollectI,
+ rtac ctxt map_F_in_F_in'I_thm,
+ SELECT_GOAL (unfold_thms_tac ctxt [F_in'_alt2_thm]),
+ rtac ctxt CollectI,
+ rtac ctxt subset_refl,
+ SELECT_GOAL (unfold_thms_tac ctxt [set_F'_alt_thm]),
+ rtac ctxt @{thm InterI},
+ etac ctxt imageE,
+ etac ctxt CollectE,
+ hyp_subst_tac ctxt,
+ etac ctxt @{thm vimageD[OF InterD]},
+ rtac ctxt CollectI]) THEN
+ (* map_F f x \<in> F_in' X \<Longrightarrow> x \<in> F_in' (f -` X) *)
+ HEADGOAL (Subgoal.FOCUS_PARAMS (fn {context = ctxt, params, ...} =>
+ let
+ val X = nth params 1 |> snd |> Thm.term_of;
+ val Inr_img = mk_image (Just_const (HOLogic.dest_setT (fastype_of X))) $ X;
+ fun cvars_of ctxt = map (SOME o Thm.cterm_of ctxt);
+ val cut_thm = infer_instantiate' ctxt (cvars_of ctxt [Inr_img, var_f])
+ @{thm insert_Inl_int_map_sum_unit};
+ val preimage_thm = infer_instantiate' ctxt (cvars_of ctxt
+ (filter (fn f => var_f <> f) var_fs |> map mk_Maybe_map))
+ (cut_thm RS REL_preimage_eq_thm);
+ in EVERY [
+ unfold_thms_tac ctxt (map #def [F_in', F_in] @ preimage_thm :: map_F_comp ::
+ @{thms lift_sum_unit_vimage_commute vimage_comp o_def map_sum.simps}),
+ unfold_thms_tac ctxt [@{thm o_def[symmetric]}, map_F_comp0],
+ Local_Defs.fold_tac ctxt @{thms vimage_comp},
+ HEADGOAL (etac ctxt (vimageI OF [refl]))] end) ctxt);
+
+ (* set_F'_subset: set_F' x \<subseteq> set_F x *)
+ val set_F'_subset_tm = HOLogic.mk_Trueprop (mk_leq (#tm set_F' $ var_x) (set_F $ var_x));
+ fun set_F'_subset_tac ctxt =
+ let val int_e_thm = infer_instantiate' ctxt
+ (replicate 3 NONE @ [SOME (Thm.cterm_of ctxt (map_F_Just $ var_x))]) @{thm INT_E};
+ in HEADGOAL (EVERY' [SELECT_GOAL (Local_Defs.unfold_tac ctxt [#def set_F']),
+ rtac ctxt subsetI,
+ etac ctxt int_e_thm,
+ SELECT_GOAL (unfold_thms_tac ctxt [set_map_F_thm]),
+ etac ctxt @{thm UN_E},
+ etac ctxt imageE,
+ hyp_subst_tac ctxt,
+ SELECT_GOAL (unfold_thms_tac ctxt @{thms sum_set_simps singleton_iff}),
+ hyp_subst_tac ctxt,
+ assume_tac ctxt,
+ etac ctxt notE,
+ rtac ctxt CollectI,
+ rtac ctxt (#reflp qthms)])
+ end;
+ in
+ ({F_in'_mono = F_in'_mono_thm,
+ F_in'_Inter = F_in'_Inter_thm,
+ set_F'_respect = prove lthy [] set_F'_respect_tm set_F'_respect_tac,
+ F_in'_alt2 = F_in'_alt2_thm,
+ set_F'_alt = set_F'_alt_thm,
+ map_F_in_F_in'I = map_F_in_F_in'I_thm,
+ set_map_F' = prove lthy (var_x :: var_fs) set_map_F'_tm set_map_F'_tac,
+ set_F'_subset = prove lthy [var_x] set_F'_subset_tm set_F'_subset_tac,
+ set_F'_def = #def set_F',
+ F_in_def = #def F_in,
+ F_in'_def = #def F_in'}, (idx + 1, var_fs'))
+ end;
+
+ val set_F'_thmss = @{fold_map 5} mk_set_F'_thmss
+ (var_As ~~ var_As' ~~ var_Ss ~~ var_Bs ~~ var_bs ~~ var_bs') sets_F set_F'_aux_defs
+ rel_Inter_thms set_map_thms (0, var_fs)
+ |> fst;
+
+ (* F_in'D: x \<in> F_in' A \<Longrightarrow> \<forall>a\<in>A. f a = g a \<Longrightarrow> REL (map_F f x) (map_F g x) *)
+ fun rel_map_F_fs_map_F_gs subst fs gs = subst equiv_rel_b $
+ (list_comb (subst map_F, fs) $ var_x) $ (list_comb (subst map_F, gs) $ var_x);
+ val F_in'D_thm = let
+ fun mk_prem var_a var_Aset {F_in', ...} var_f var_g =
+ [HOLogic.mk_mem (var_x, #tm F_in' $ var_Aset), mk_Ball var_Aset
+ ((absfree (dest_Free var_a)) (HOLogic.mk_eq (var_f $ var_a, var_g $ var_a)))];
+ val prems = @{map 5} mk_prem var_as var_As set_F'_aux_defs var_fs' var_gs';
+ val F_in'D_tm = mk_Trueprop_implies (flat prems,
+ rel_map_F_fs_map_F_gs (subst_Maybe betas) var_fs' var_gs');
+
+ fun map_F_rsp_of_case_sums fs ctxt = map_F_rsp_of
+ (@{map 2} (fn f => fn T => BNF_FP_Util.mk_case_sum
+ (Term.absdummy HOLogic.unitT (mk_Nothing T), f)) fs betas) ctxt;
+
+ fun mk_var_fgs n = take n var_gs' @ drop n var_fs';
+ fun F_in'D_tac ctxt = EVERY
+ (unfold_thms_tac ctxt
+ (maps (fn {F_in'_def, F_in_def, ...} => [F_in'_def, F_in_def]) set_F'_thmss) ::
+ map (REPEAT_DETERM_N live o HEADGOAL)
+ [etac ctxt vimageE,
+ etac ctxt @{thm ImageE},
+ etac ctxt CollectE THEN' etac ctxt CollectE,
+ dtac ctxt @{thm case_prodD}] @
+ HEADGOAL (hyp_subst_tac ctxt THEN' rotate_tac (~live)) ::
+ map (fn i => (HEADGOAL o EVERY')
+ [if i < live then rtac ctxt (#transp qthms) else K all_tac,
+ Ctr_Sugar_Tactics.select_prem_tac ctxt live (dresolve_tac ctxt [asm_rl]) i,
+ rtac ctxt (#transp qthms),
+ dtac ctxt (map_F_rsp_of_case_sums (mk_var_fgs (i-1)) ctxt),
+ SELECT_GOAL (unfold_thms_tac ctxt (map_F_comp :: @{thms case_sum_o_inj(2)})),
+ etac ctxt (#symp qthms),
+ dtac ctxt (map_F_rsp_of_case_sums (mk_var_fgs i) ctxt),
+ SELECT_GOAL (unfold_thms_tac ctxt (map_F_comp :: @{thms case_sum_o_inj(2)})),
+ EqSubst.eqsubst_tac ctxt [0] [map_F_cong OF replicate i refl @ asm_rl :: replicate (live-i) refl],
+ rtac ctxt @{thm sum.case_cong[OF refl refl]},
+ etac ctxt bspec,
+ hyp_subst_tac ctxt,
+ etac ctxt @{thm subset_lift_sum_unitD},
+ assume_tac ctxt,
+ assume_tac ctxt]) (1 upto live));
+
+ in prove lthy (var_x :: var_As @ var_fs' @ var_gs') F_in'D_tm F_in'D_tac end;
+
+ (* map_F_cong': (\<And>a. a \<in> set_F' x \<Longrightarrow> f a = g a) \<Longrightarrow> REL (map_F f x) (map_F g x) *)
+ val map_F_cong'_thm = let
+ fun mk_prem {set_F', ...} var_a var_f var_g = Logic.all var_a
+ (mk_Trueprop_implies ([HOLogic.mk_mem (var_a, #tm set_F' $ var_x)],
+ HOLogic.mk_eq (var_f $ var_a, var_g $ var_a)));
+ val map_F_cong'_tm = Logic.list_implies
+ (@{map 4} mk_prem set_F'_aux_defs var_as var_fs var_gs, HOLogic.mk_Trueprop
+ (rel_map_F_fs_map_F_gs I var_fs var_gs));
+ fun Just_o_fun bT f = HOLogic.mk_comp (Just_const bT, f);
+ fun map_F_Just_o_funs fs = list_comb
+ (subst_Maybe betas map_F, map2 Just_o_fun betas fs) $ var_x;
+ fun map_F_cong'_tac ctxt = let
+ val map_F_respect_inst = map_F_rsp
+ |> infer_instantiate' ctxt (map (SOME o Thm.cterm_of ctxt)
+ (map map_F_Just_o_funs [var_fs, var_gs] @ map fromJust_const betas))
+ |> Local_Defs.unfold ctxt (map_F_comp :: @{thms o_assoc
+ Fun.o_apply[where f=projr and g=Inr, unfolded sum.sel] id_def[symmetric]})
+ |> Local_Defs.unfold ctxt @{thms id_comp};
+ in
+ HEADGOAL (rtac ctxt map_F_respect_inst THEN' rtac ctxt F_in'D_thm) THEN
+ EVERY (map (fn {F_in'_alt2, ...} =>
+ unfold_thms_tac ctxt [F_in'_alt2] THEN
+ HEADGOAL (EVERY'
+ [rtac ctxt CollectI,
+ rtac ctxt subset_refl,
+ rtac ctxt ballI,
+ SELECT_GOAL (unfold_thms_tac ctxt [o_apply]),
+ rtac ctxt @{thm arg_cong[where f=Inr]},
+ asm_full_simp_tac ctxt])) set_F'_thmss) end;
+ in prove lthy (var_x :: var_fs @ var_gs) map_F_cong'_tm map_F_cong'_tac end;
+
+ (* rel_F'_set: "rel_F' P x y \<longleftrightarrow>
+ (\<exists>z. set_F' z \<subseteq> {(x, y). P x y} \<and> REL (map_F fst z) x \<and> REL (map_F snd z) y)" *)
+ val rel_F'_set_thm = let
+ val lhs = list_comb (#tm rel_F', var_Ps) $ var_x $ var_y;
+ fun mk_subset_A var_a var_b var_P {set_F', ...} = let
+ val collect_A = mk_case_prod (var_a, var_b) (var_P $ var_a $ var_b);
+ in mk_leq (subst_atomic_types (alphas ~~ typ_pairs) (#tm set_F') $ var_z) collect_A end;
+ val subset_As = @{map 4} mk_subset_A var_as var_bs var_Ps set_F'_aux_defs;
+ fun mk_map mfs f z =
+ Term.list_comb (mfs, map (fst o Term.strip_comb o f) var_ts) $ z;
+ val map_F_fst = mk_map_of_bnf deads typ_pairs alphas bnf_F;
+ val map_F_snd = mk_map_of_bnf deads typ_pairs betas bnf_F;
+ val map_fst = equiv_rel_a $ (mk_map map_F_fst HOLogic.mk_fst var_z) $ var_x;
+ val map_snd = equiv_rel_b $ (mk_map map_F_snd HOLogic.mk_snd var_z) $ var_y;
+ val rhs = let val (z, T) = dest_Free var_z in
+ HOLogic.mk_exists (z, T, fold_rev (fn a => fn b => HOLogic.mk_conj (a, b))
+ (subset_As @ [map_fst]) map_snd) end;
+ val rel_F'_set_tm = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs));
+
+ val maybePairsTs = map HOLogic.mk_prodT (map mk_MaybeT alphas ~~ map mk_MaybeT betas)
+ fun mk_map_prod_projr aT bT = let
+ val (mabT, (maT, mbT)) = `HOLogic.mk_prodT (apply2 mk_MaybeT (aT, bT));
+ val map_prod_const = Const (@{const_name map_prod},
+ (maT --> aT) --> (mbT --> bT) --> mabT --> HOLogic.mk_prodT (aT, bT));
+ in map_prod_const $ fromJust_const aT $ fromJust_const bT end;
+
+ fun exI_OF_tac ctxt tm = rtac ctxt
+ (infer_instantiate' ctxt (NONE :: [SOME (Thm.cterm_of ctxt tm)]) exI);
+
+ (* REL (map_F Inr x) (map_F fst z) \<Longrightarrow> REL (map_F snd z) (map_F Inr y) \<Longrightarrow>
+ set_F z \<subseteq> {(x, y). rel_sum (=) P x y} \<Longrightarrow>
+ \<exists>z. set_F' z \<subseteq> {(x, y). P x y} \<and> REL (map_F fst z) x \<and> REL (map_F snd z) y *)
+ fun subgoal1_tac {context = ctxt, params, ...} =
+ let
+ val z = (case params of
+ (_ :: _ :: (_, ct) :: _) => Thm.term_of ct
+ | _ => error "won't happen");
+ val map_F_projr_z = list_comb (mk_map_of_bnf deads maybePairsTs typ_pairs bnf_F,
+ map2 mk_map_prod_projr alphas betas) $ z;
+ in
+ HEADGOAL (exI_OF_tac ctxt map_F_projr_z) THEN
+ HEADGOAL (EVERY' (maps (fn {set_F'_subset, set_F'_respect, set_map_F', ...} =>
+ [rtac ctxt conjI,
+ dtac ctxt (set_F'_subset RS @{thm order_trans}),
+ TWICE (dtac ctxt (set_F'_respect RS @{thm rel_funD})),
+ SELECT_GOAL (unfold_thms_tac ctxt [set_map_F']),
+ etac ctxt @{thm in_rel_sum_in_image_projr},
+ TWICE (assume_tac ctxt)]) set_F'_thmss)) THEN
+ HEADGOAL (EVERY' (map (fn Ts => FIRST'
+ [dtac ctxt (map_F_rsp_of (map fromJust_const Ts) ctxt),
+ etac ctxt sym , assume_tac ctxt]) [alphas, betas])) THEN
+ unfold_thms_tac ctxt (map_F_comp ::
+ @{thms fst_comp_map_prod snd_comp_map_prod comp_projr_Inr} @ [map_F_id]) THEN
+ HEADGOAL (rtac ctxt conjI) THEN
+ HEADGOAL (etac ctxt (#symp qthms) THEN' assume_tac ctxt
+ ORELSE' (EVERY' (maps (fn Ts =>
+ [dtac ctxt (map_F_rsp_of (map fromJust_const Ts) ctxt),
+ SELECT_GOAL (unfold_thms_tac ctxt (map_F_comp ::
+ @{thms fst_comp_map_prod snd_comp_map_prod comp_projr_Inr} @ [map_F_id])),
+ assume_tac ctxt]) [alphas, betas]))) end;
+
+ (* set_F' z \<subseteq> {(x, y). P x y} \<Longrightarrow> REL (map_F fst z) x \<Longrightarrow> REL (map_F snd z) y \<Longrightarrow>
+ \<exists>b. REL (map_F Inr x) b \<and> (\<exists>ba. rel_F (rel_sum (=) P) b ba \<and> REL ba (map_F Inr y)) *)
+ fun subgoal2_tac {context = ctxt, params, ...} = let
+ val z = (case params of
+ ((_, ct) :: _) => Thm.term_of ct
+ | _ => error "won't happen");
+
+ fun exI_map_Ifs_tac mk_proj Ts = exI_OF_tac ctxt (list_comb
+ (mk_map_of_bnf deads typ_pairs (map mk_MaybeT Ts) bnf_F, @{map 3}
+ (fn var_t => fn {set_F', ...} => fn T => lambda var_t (BNF_FP_Util.mk_If
+ (HOLogic.mk_mem (var_t, subst_Ts (#tm set_F') typ_pairs $ z))
+ (mk_Just (mk_proj var_t)) (mk_Nothing T))) var_ts set_F'_aux_defs Ts) $ z)
+
+ fun mk_REL_trans_map_F n = (rotate_prems n (#transp qthms) OF
+ [rel_funD map_F_respect_thm] OF (replicate live refl @ [#symp qthms]));
+ in
+ HEADGOAL (EVERY'
+ [exI_map_Ifs_tac HOLogic.mk_fst alphas,
+ rtac ctxt conjI,
+ etac ctxt (mk_REL_trans_map_F 0)]) THEN
+ unfold_thms_tac ctxt [map_F_comp, @{thm o_def}] THEN
+ HEADGOAL (rtac ctxt map_F_cong'_thm) THEN
+ REPEAT_DETERM_N live (HEADGOAL (etac ctxt @{thm if_P[symmetric]})) THEN
+ HEADGOAL (EVERY' [exI_map_Ifs_tac HOLogic.mk_snd betas, rtac ctxt conjI]) THEN
+ unfold_thms_tac ctxt rel_map THEN
+ HEADGOAL (rtac ctxt rel_refl_strong) THEN
+ REPEAT_DETERM_N live (HEADGOAL (etac ctxt @{thm subset_rel_sumI})) THEN
+ HEADGOAL (etac ctxt (mk_REL_trans_map_F 1 OF [#symp qthms])) THEN
+ unfold_thms_tac ctxt [map_F_comp, @{thm o_def}] THEN
+ HEADGOAL (rtac ctxt map_F_cong'_thm) THEN
+ REPEAT_DETERM_N live (HEADGOAL (etac ctxt @{thm if_P})) end;
+
+ fun rel_F'_set_tac ctxt = EVERY
+ ([unfold_thms_tac ctxt (#def rel_F' :: #REL qthms :: @{thms vimage2p_def relcompp_apply}),
+ HEADGOAL (rtac ctxt iffI),
+ (HEADGOAL o TWICE) (etac ctxt exE THEN' etac ctxt conjE),
+ HEADGOAL (EVERY'
+ [dtac ctxt (in_rel RS iffD1),
+ etac ctxt exE,
+ TWICE (etac ctxt conjE),
+ etac ctxt CollectE,
+ hyp_subst_tac ctxt]),
+ (REPEAT_DETERM_N (live-1) o HEADGOAL) (etac ctxt conjE),
+ HEADGOAL (Subgoal.FOCUS_PARAMS subgoal1_tac ctxt THEN' etac ctxt exE),
+ (REPEAT_DETERM_N (live+1) o HEADGOAL) (etac ctxt conjE),
+ HEADGOAL (Subgoal.FOCUS_PARAMS subgoal2_tac ctxt)]);
+
+ in prove lthy (var_x :: var_y :: var_Ps) rel_F'_set_tm rel_F'_set_tac end;
+
+ (* tactics *)
+
+ (* map_G_id0: abs_G \<circ> map_F id \<circ> rep_G = id *)
+ fun map_G_id0_tac ctxt = HEADGOAL (EVERY' [SELECT_GOAL (unfold_thms_tac ctxt
+ [@{thm fun_eq_iff}, o_apply, map_F_id0, id_apply, #abs_rep qthms]),
+ rtac ctxt allI, rtac ctxt refl]);
+
+ (* map_G (g \<circ> f) = map_G g \<circ> map_G f *)
+ fun map_G_comp0_tac ctxt = HEADGOAL (EVERY' [rtac ctxt ext, rtac ctxt sym,
+ SELECT_GOAL (unfold_thms_tac ctxt [o_apply, map_F_comp0]), rtac ctxt (#rel_abs qthms),
+ rtac ctxt map_F_rsp, rtac ctxt (#rep_abs qthms), rtac ctxt (#reflp qthms)]);
+
+ (* map_G_cong: (\<And>z. z \<in> set_G x \<Longrightarrow> f z = g z) \<Longrightarrow> map_G f x = map_G g x *)
+ fun map_G_cong_tac ctxt = EVERY
+ [Local_Defs.fold_tac ctxt (map #set_F'_def set_F'_thmss),
+ unfold_thms_tac ctxt [o_apply],
+ HEADGOAL (rtac ctxt (#rel_abs qthms) THEN' rtac ctxt map_F_cong'_thm),
+ REPEAT_DETERM_N live (HEADGOAL (asm_full_simp_tac ctxt))];
+
+ (* set_G_map0_G: set_G \<circ> map_G f = f ` set_G *)
+ fun mk_set_G_map0_G_tac thms ctxt =
+ HEADGOAL (rtac ctxt ext) THEN
+ EVERY [unfold_thms_tac ctxt [o_apply],
+ Local_Defs.fold_tac ctxt [#set_F'_def thms]] THEN
+ HEADGOAL (EVERY' (map (rtac ctxt)
+ [trans OF [#set_map_F' thms RS sym, sym] RS sym,
+ @{thm rel_funD} OF [#set_F'_respect thms],
+ #rep_abs qthms,
+ map_F_rsp,
+ #rep_reflp qthms]));
+
+ (* bd_card_order: card_order bd_F *)
+ fun bd_card_order_tac ctxt = HEADGOAL (rtac ctxt bd_card_order);
+
+ (* bd_cinfinite: BNF_Cardinal_Arithmetic.cinfinite bd_F *)
+ fun bd_cinfinite_tac ctxt = HEADGOAL (rtac ctxt bd_cinfinite);
+
+ (*target: ordLeq3 (card_of (set_F' (rep_G x_))) bd_F*)
+ fun mk_set_G_bd_tac thms set_bd_thm ctxt = EVERY
+ [Local_Defs.fold_tac ctxt [#set_F'_def thms],
+ unfold_thms_tac ctxt [o_apply],
+ HEADGOAL (rtac ctxt (@{thm ordLeq_transitive} OF
+ [@{thm card_of_mono1} OF [#set_F'_subset thms], set_bd_thm]))];
+
+ (* rel_compp: rel_G R OO rel_G S \<le> rel_G (R OO S) *)
+ fun rel_compp_tac ctxt = EVERY
+ [unfold_thms_tac ctxt [#REL qthms],
+ HEADGOAL (TWICE (rtac ctxt @{thm vimage2p_relcompp_mono})),
+ (unfold_thms_tac ctxt (REL_OO_REL_left_thm :: @{thms relcompp_assoc})),
+ (unfold_thms_tac ctxt [Local_Defs.unfold ctxt @{thms eq_OO}
+ (infer_instantiate' ctxt [HOLogic.eq_const HOLogic.unitT |> Thm.cterm_of ctxt |> SOME]
+ @{thm sum.rel_compp})]),
+ HEADGOAL (rtac ctxt rel_pos_distr_thm),
+ unfold_thms_tac ctxt
+ @{thms fun_eq_iff bot_apply bot_bool_def not_all eq_False not_not OO_def},
+ REPEAT_DETERM (HEADGOAL (resolve_tac ctxt [exI, conjI, @{thm rel_sum.intros(1)}, refl]))];
+
+ (* rel_G R_ = (\<lambda>x y. \<exists>z. set_G z \<subseteq> {(x, y). R x y} \<and> map_G fst z = x \<and> map_G snd z = y) *)
+ fun rel_compp_Grp_tac ctxt = let
+ val _ = ()
+ in EVERY [Local_Defs.fold_tac ctxt (@{thm Grp_def} :: map #set_F'_def set_F'_thmss),
+ unfold_thms_tac ctxt
+ [o_apply, @{thm mem_Collect_eq}, @{thm OO_Grp_alt}, @{thm vimage2p_def}],
+ Local_Defs.fold_tac ctxt [Local_Defs.unfold ctxt @{thms vimage2p_def} (#def rel_F')],
+ unfold_thms_tac ctxt [rel_F'_set_thm],
+ HEADGOAL (TWICE (rtac ctxt ext)),
+ HEADGOAL (rtac ctxt iffI),
+ REPEAT_DETERM (ALLGOALS (eresolve_tac ctxt [exE, conjE])),
+ HEADGOAL (rtac ctxt exI),
+ REPEAT_FIRST (resolve_tac ctxt [conjI]),
+ HEADGOAL (EVERY' (maps (fn {set_F'_respect, ...} =>
+ [etac ctxt @{thm subset_trans[rotated]},
+ rtac ctxt equalityD1,
+ rtac ctxt (@{thm rel_funD} OF [set_F'_respect]),
+ rtac ctxt (#rep_abs qthms),
+ rtac ctxt (#reflp qthms)]) set_F'_thmss)),
+ (HEADGOAL o TWICE o EVERY')
+ [rtac ctxt (trans OF [asm_rl, #abs_rep qthms]),
+ rtac ctxt (#rel_abs qthms),
+ etac ctxt (rotate_prems 1 (#transp qthms)),
+ rtac ctxt map_F_rsp,
+ rtac ctxt (#rep_abs qthms),
+ rtac ctxt (#reflp qthms)
+ ],
+ HEADGOAL (rtac ctxt exI THEN' rtac ctxt conjI),
+ (REPEAT_DETERM_N live o HEADGOAL o EVERY')
+ [assume_tac ctxt, rtac ctxt conjI],
+ (HEADGOAL o TWICE o EVERY') [
+ hyp_subst_tac ctxt,
+ rtac ctxt (#rep_abs_rsp qthms),
+ rtac ctxt map_F_rsp,
+ rtac ctxt (#rep_reflp qthms)]]
+ end;
+
+ fun pred_G_set_G_tac ctxt = HEADGOAL (rtac ctxt refl);
+
+ val tactics = map_G_id0_tac :: map_G_comp0_tac :: map_G_cong_tac ::
+ map mk_set_G_map0_G_tac set_F'_thmss @
+ bd_card_order_tac :: bd_cinfinite_tac ::
+ map2 mk_set_G_bd_tac set_F'_thmss set_bd_thms @
+ rel_compp_tac :: rel_compp_Grp_tac :: [pred_G_set_G_tac];
+
+ fun mk_wit_tacs ({set_F'_def, set_F'_respect, ...} :: set_F'_thmss) (w :: ws) ctxt =
+ EVERY [unfold_thms_tac ctxt [@{thm o_def},
+ set_F'_respect RS @{thm rel_funD} OF [#rep_abs qthms OF [(#reflp qthms)]]],
+ unfold_thms_tac ctxt [set_F'_def],
+ HEADGOAL (etac ctxt w)]
+ THEN mk_wit_tacs set_F'_thmss ws ctxt
+ | mk_wit_tacs [] ws ctxt = mk_wit_tacs set_F'_thmss ws ctxt
+ | mk_wit_tacs _ _ _ = all_tac;
+
+ val (bnf_G, lthy) = bnf_def Dont_Inline (user_policy Note_Some) true I
+ tactics (mk_wit_tacs [] wit_thms) NONE map_b rel_b pred_b set_bs
+ (((((((Binding.empty, absT), map_G), sets_G), bd_G), wits_G), SOME rel_G), NONE) lthy;
+
+ val old_defs =
+ {sets = set_defs_of_bnf bnf_G, map = map_def_of_bnf bnf_G, rel = rel_def_of_bnf bnf_G};
+
+ val set_F'_defs = map (mk_abs_def o #set_F'_def) set_F'_thmss;
+ val unfold_morphism = Morphism.thm_morphism "BNF"
+ (unfold_thms lthy (defs @ #def REL :: set_F'_defs));
+ val (bnf_G, lthy) = morph_bnf_defs unfold_morphism bnf_G
+ |> (fn bnf => note_bnf_defs bnf lthy);
+
+ (* auxiliary lemmas transfer for transfer *)
+ val rel_monoD_rotated = rotate_prems ~1 (rel_mono_of_bnf bnf_F RS @{thm predicate2D});
+
+ val REL_pos_distrI = let
+ fun tac ctxt = EVERY
+ [HEADGOAL (dtac ctxt (rotate_prems ~1 (rel_pos_distr_thm RS @{thm predicate2D}))),
+ (REPEAT_DETERM o HEADGOAL) (rtac ctxt conjI ORELSE' assume_tac ctxt),
+ (REPEAT_DETERM o HEADGOAL) (etac ctxt @{thm relcomppE}),
+ HEADGOAL (dtac ctxt rel_monoD_rotated),
+ (REPEAT_DETERM o HEADGOAL)
+ (assume_tac ctxt ORELSE' rtac ctxt @{thm relcomppI})];
+ in prove lthy (var_x :: var_y' :: var_Ps @ var_Qs @ var_Rs) REL_pos_distrI_tm tac end;
+
+ val rel_F_rel_F' = let
+ val rel_F = mk_rel_of_bnf deads alphas betas bnf_F;
+ val rel_F_rel_F'_tm = (rel_F, #tm rel_F')
+ |> apply2 (fn R => HOLogic.mk_Trueprop (list_comb (R, var_Ps) $ var_x $ var_y))
+ |> Logic.mk_implies;
+ fun rel_F_rel_F'_tac ctxt = EVERY
+ [HEADGOAL (dtac ctxt (in_rel_of_bnf bnf_F RS iffD1)),
+ unfold_thms_tac ctxt (rel_F'_set_thm :: @{thms mem_Collect_eq}),
+ (REPEAT_DETERM o HEADGOAL) (eresolve_tac ctxt [exE, conjE]),
+ HEADGOAL (rtac ctxt exI),
+ HEADGOAL (EVERY' (maps (fn thms =>
+ [rtac ctxt conjI,
+ rtac ctxt subsetI,
+ dtac ctxt (set_mp OF [#set_F'_subset thms]),
+ dtac ctxt subsetD,
+ assume_tac ctxt, assume_tac ctxt]) set_F'_thmss)),
+ (REPEAT_DETERM o HEADGOAL)
+ (rtac ctxt conjI ORELSE' hyp_subst_tac ctxt THEN' rtac ctxt (#reflp qthms))]
+ in prove lthy (var_x :: var_y :: var_Ps) rel_F_rel_F'_tm rel_F_rel_F'_tac end;
+
+ fun inst_REL_pos_distrI n vs aTs bTs ctxt =
+ infer_instantiate' ctxt (replicate n NONE @ (rel_Maybes vs aTs bTs
+ |> map (SOME o Thm.cterm_of ctxt))) REL_pos_distrI;
+
+ val Tss = {abs = typ_subst_atomic (alphas ~~ betas) absT, rep = repT, Ds0 = map TFree Ds0,
+ deads = deads, alphas = alphas, betas = betas, gammas = gammas, deltas = deltas};
+
+ val thms =
+ {map_F_rsp = map_F_rsp,
+ rel_F'_def = #def rel_F',
+ rel_F_rel_F' = rel_F_rel_F',
+ rel_F'_set = rel_F'_set_thm,
+ rel_monoD_rotated = rel_monoD_rotated}
+
+ val transfer_consts = mk_quotient_transfer_tacs bnf_F Tss live
+ qthms thms set_F'_thmss old_defs inst_REL_pos_distrI
+ map_raw rel_raw (map (#tm o #set_F') set_F'_aux_defs);
+ val quiet = exists (fn No_Warn_Transfer => true | _ => false) opts;
+ in
+ lthy |> BNF_Def.register_bnf plugins absT_name bnf_G |>
+ mk_transfer_thms quiet bnf_F bnf_G absT_name transfer_consts (Quotient equiv_thm) Tss
+ (defs @ #def REL :: set_F'_defs)
+ end
+ | _ => raise Match);
+
+ in (goals, after_qed, #def REL :: defs, lthy) end;
+
+
+(** main commands **)
local
fun prepare_common prepare_name prepare_sort prepare_term prepare_thm
- (((((plugins, raw_specs), raw_Tname), raw_wits), xthm_opt), (map_b, rel_b, pred_b)) lthy =
+ (((((plugins, raw_specs), raw_absT_name), raw_wits), xthm_opt), (map_b, rel_b, pred_b)) lthy =
let
- val Tname = prepare_name lthy raw_Tname;
+ val absT_name = prepare_name lthy raw_absT_name;
+
val input_thm =
(case xthm_opt of
SOME xthm => prepare_thm lthy xthm
- | NONE => Typedef.get_info lthy Tname |> hd |> snd |> #type_definition);
+ | NONE => Typedef.get_info lthy absT_name |> hd |> snd |> #type_definition);
val wits = (Option.map o map) (prepare_term lthy) raw_wits;
val specs =
- map (apsnd (apsnd (the_default \<^sort>\<open>type\<close> o Option.map (prepare_sort lthy)))) raw_specs;
+ map (apsnd (apsnd (the_default @{sort type} o Option.map (prepare_sort lthy)))) raw_specs;
- val _ =
- (case HOLogic.dest_Trueprop (Thm.prop_of input_thm) of
- Const (\<^const_name>\<open>type_definition\<close>, _) $ _ $ _ $ _ => ()
- | _ => error "Unsupported type of a theorem: only type_definition is supported");
+ val which_bnf = (case Quotient_Info.lookup_quotients lthy absT_name of
+ SOME qs => quotient_bnf qs
+ | _ => typedef_bnf);
+
in
- typedef_bnf input_thm wits specs map_b rel_b pred_b plugins lthy
+ which_bnf input_thm wits specs map_b rel_b pred_b plugins lthy
end;
fun prepare_lift_bnf prepare_name prepare_sort prepare_term prepare_thm =
@@ -379,7 +1943,7 @@
fun prepare_solve prepare_name prepare_typ prepare_sort prepare_thm tacs =
(fn (goals, after_qed, definitions, lthy) =>
lthy
- |> after_qed (map2 (fn goal => fn tac => [Goal.prove lthy [] [] goal
+ |> after_qed (map2 (fn goal => fn tac => [Goal.prove_sorry lthy [] [] goal
(fn (ctxtprems as {context = ctxt, prems = _}) =>
unfold_thms_tac ctxt definitions THEN tac ctxtprems)])
goals (tacs (length goals)))) oo
@@ -395,57 +1959,63 @@
fun lift_bnf args tacs =
prepare_solve (K I) (K I) (K I) (K I) (K tacs) args;
+fun copy_bnf_tac {context = ctxt, prems = _} =
+ REPEAT_DETERM (resolve_tac ctxt [bexI, conjI, UNIV_I, refl, subset_refl] 1);
+
val copy_bnf =
apfst (apfst (rpair NONE))
#> prepare_solve (K I) (K I) (K I) (K I)
- (fn n => replicate n (fn {context = ctxt, prems = _} => rtac ctxt UNIV_I 1));
+ (fn n => replicate n copy_bnf_tac);
val copy_bnf_cmd =
apfst (apfst (rpair NONE))
#> prepare_solve
(fst o dest_Type oo Proof_Context.read_type_name {proper = true, strict = false})
Syntax.read_sort Syntax.read_term (singleton o Attrib.eval_thms)
- (fn n => replicate n (fn {context = ctxt, prems = _} => rtac ctxt UNIV_I 1));
+ (fn n => replicate n copy_bnf_tac);
end;
-
-(* outer syntax *)
+(** outer syntax **)
local
+(* parsers *)
+
val parse_wits =
- \<^keyword>\<open>[\<close> |-- (Parse.name --| \<^keyword>\<open>:\<close> -- Scan.repeat Parse.term >>
+ @{keyword "["} |-- (Parse.name --| @{keyword ":"} -- Scan.repeat Parse.term >>
(fn ("wits", Ts) => Ts
| (s, _) => error ("Unknown label " ^ quote s ^ " (expected \"wits\")"))) --|
- \<^keyword>\<open>]\<close> || Scan.succeed [];
+ @{keyword "]"} || Scan.succeed [];
-val parse_options =
- Scan.optional (\<^keyword>\<open>(\<close> |--
+fun parse_common_opts p =
+ Scan.optional (@{keyword "("} |--
Parse.list1 (Parse.group (K "option")
- (Plugin_Name.parse_filter >> Plugins_Option
- || Parse.reserved "no_warn_wits" >> K No_Warn_Wits))
- --| \<^keyword>\<open>)\<close>) [];
+ (Scan.first (p :: [Plugin_Name.parse_filter >> Plugins_Option,
+ Parse.reserved "no_warn_transfer" >> K No_Warn_Transfer])))
+ --| @{keyword ")"}) [];
-val parse_plugins =
- Scan.optional (\<^keyword>\<open>(\<close> |-- Plugin_Name.parse_filter --| \<^keyword>\<open>)\<close>)
- (K Plugin_Name.default_filter) >> Plugins_Option >> single;
+val parse_lift_opts = Parse.reserved "no_warn_wits" >> K No_Warn_Wits |> parse_common_opts;
-val parse_typedef_thm = Scan.option (Parse.reserved "via" |-- Parse.thm);
+val parse_copy_opts = parse_common_opts Scan.fail;
+
+val parse_xthm = Scan.option (Parse.reserved "via" |-- Parse.thm);
in
-val _ =
- Outer_Syntax.local_theory_to_proof \<^command_keyword>\<open>lift_bnf\<close>
- "register a subtype of a bounded natural functor (BNF) as a BNF"
- ((parse_options -- parse_type_args_named_constrained -- Parse.type_const -- parse_wits --
- parse_typedef_thm -- parse_map_rel_pred_bindings) >> lift_bnf_cmd);
+(* exposed commands *)
val _ =
- Outer_Syntax.local_theory \<^command_keyword>\<open>copy_bnf\<close>
+ Outer_Syntax.local_theory_to_proof @{command_keyword lift_bnf}
+ "register a quotient type/subtype of a bounded natural functor (BNF) as a BNF"
+ ((parse_lift_opts -- parse_type_args_named_constrained -- Parse.type_const -- parse_wits --
+ parse_xthm -- parse_map_rel_pred_bindings) >> lift_bnf_cmd);
+
+val _ =
+ Outer_Syntax.local_theory @{command_keyword copy_bnf}
"register a type copy of a bounded natural functor (BNF) as a BNF"
- ((parse_plugins -- parse_type_args_named_constrained -- Parse.type_const --
- parse_typedef_thm -- parse_map_rel_pred_bindings) >> copy_bnf_cmd);
+ ((parse_copy_opts -- parse_type_args_named_constrained -- Parse.type_const --
+ parse_xthm -- parse_map_rel_pred_bindings) >> copy_bnf_cmd);
end;