(* Title: HOL/Probability/measurable.ML
Author: Johannes Hölzl <hoelzl@in.tum.de>
Measurability prover.
*)
signature MEASURABLE =
sig
type preprocessor = thm -> Proof.context -> (thm list * Proof.context)
datatype level = Concrete | Generic
val app_thm_attr : attribute context_parser
val dest_thm_attr : attribute context_parser
val cong_thm_attr : attribute context_parser
val measurable_thm_attr : bool * (bool * level) -> attribute
val add_del_cong_thm : bool -> thm -> Context.generic -> Context.generic ;
val get_all : Context.generic -> thm list
val get_dest : Context.generic -> thm list
val get_cong : Context.generic -> thm list
val measurable_tac : Proof.context -> thm list -> tactic
val simproc : Proof.context -> cterm -> thm option
val add_preprocessor : string -> preprocessor -> Context.generic -> Context.generic
val del_preprocessor : string -> Context.generic -> Context.generic
val add_local_cong : thm -> Proof.context -> Proof.context
end ;
structure Measurable : MEASURABLE =
struct
type preprocessor = thm -> Proof.context -> (thm list * Proof.context)
datatype level = Concrete | Generic;
fun eq_measurable_thms ((th1, d1), (th2, d2)) =
d1 = d2 andalso Thm.eq_thm_prop (th1, th2) ;
fun merge_dups (xs:(string * preprocessor) list) ys =
xs @ (filter (fn (name, _) => is_none (find_first (fn (name', _) => name' = name) xs)) ys)
structure Data = Generic_Data
(
type T = {
measurable_thms : (thm * (bool * level)) Item_Net.T,
dest_thms : thm Item_Net.T,
app_thms : thm Item_Net.T,
cong_thms : thm Item_Net.T,
preprocessors : (string * preprocessor) list }
val empty = {
measurable_thms = Item_Net.init eq_measurable_thms (single o Thm.prop_of o fst),
dest_thms = Thm.full_rules,
app_thms = Thm.full_rules,
cong_thms = Thm.full_rules,
preprocessors = [] };
val extend = I;
fun merge ({measurable_thms = t1, dest_thms = dt1, app_thms = at1, cong_thms = ct1, preprocessors = i1 },
{measurable_thms = t2, dest_thms = dt2, app_thms = at2, cong_thms = ct2, preprocessors = i2 }) = {
measurable_thms = Item_Net.merge (t1, t2),
dest_thms = Item_Net.merge (dt1, dt2),
app_thms = Item_Net.merge (at1, at2),
cong_thms = Item_Net.merge (ct1, ct2),
preprocessors = merge_dups i1 i2
};
);
val debug =
Attrib.setup_config_bool @{binding measurable_debug} (K false)
val split =
Attrib.setup_config_bool @{binding measurable_split} (K true)
fun map_data f1 f2 f3 f4 f5
{measurable_thms = t1, dest_thms = t2, app_thms = t3, cong_thms = t4, preprocessors = t5 } =
{measurable_thms = f1 t1, dest_thms = f2 t2, app_thms = f3 t3, cong_thms = f4 t4, preprocessors = f5 t5}
fun map_measurable_thms f = map_data f I I I I
fun map_dest_thms f = map_data I f I I I
fun map_app_thms f = map_data I I f I I
fun map_cong_thms f = map_data I I I f I
fun map_preprocessors f = map_data I I I I f
fun generic_add_del map =
Scan.lift
(Args.add >> K Item_Net.update || Args.del >> K Item_Net.remove || Scan.succeed Item_Net.update) >>
(fn f => Thm.declaration_attribute (Data.map o map o f))
val app_thm_attr = generic_add_del map_app_thms
val dest_thm_attr = generic_add_del map_dest_thms
val cong_thm_attr = generic_add_del map_cong_thms
fun del_thm th net =
let
val thms = net |> Item_Net.content |> filter (fn (th', _) => Thm.eq_thm (th, th'))
in fold Item_Net.remove thms net end ;
fun measurable_thm_attr (do_add, d) = Thm.declaration_attribute
(Data.map o map_measurable_thms o (if do_add then Item_Net.update o rpair d else del_thm))
val get_dest = Item_Net.content o #dest_thms o Data.get;
val get_app = Item_Net.content o #app_thms o Data.get;
val get_cong = Item_Net.content o #cong_thms o Data.get;
val add_cong = Data.map o map_cong_thms o Item_Net.update;
val del_cong = Data.map o map_cong_thms o Item_Net.remove;
fun add_del_cong_thm true = add_cong
| add_del_cong_thm false = del_cong
fun add_preprocessor name f = Data.map (map_preprocessors (fn xs => xs @ [(name, f)]))
fun del_preprocessor name = Data.map (map_preprocessors (filter (fn (n, _) => n <> name)))
val add_local_cong = Context.proof_map o add_cong
val get_preprocessors = Context.Proof #> Data.get #> #preprocessors ;
fun is_too_generic thm =
let
val concl = concl_of thm
val concl' = HOLogic.dest_Trueprop concl handle TERM _ => concl
in is_Var (head_of concl') end
val get_thms = Data.get #> #measurable_thms #> Item_Net.content ;
val get_all = get_thms #> map fst ;
fun debug_tac ctxt msg f =
if Config.get ctxt debug then print_tac ctxt (msg ()) THEN f else f
fun nth_hol_goal thm i =
HOLogic.dest_Trueprop (Logic.strip_imp_concl (strip_all_body (nth (prems_of thm) (i - 1))))
fun dest_measurable_fun t =
(case t of
(Const (@{const_name "Set.member"}, _) $ f $ (Const (@{const_name "measurable"}, _) $ _ $ _)) => f
| _ => raise (TERM ("not a measurability predicate", [t])))
fun not_measurable_prop n thm = if length (prems_of thm) < n then false else
(case nth_hol_goal thm n of
(Const (@{const_name "Set.member"}, _) $ _ $ (Const (@{const_name "sets"}, _) $ _)) => false
| (Const (@{const_name "Set.member"}, _) $ _ $ (Const (@{const_name "measurable"}, _) $ _ $ _)) => false
| _ => true)
handle TERM _ => true;
fun indep (Bound i) t b = i < b orelse t <= i
| indep (f $ t) top bot = indep f top bot andalso indep t top bot
| indep (Abs (_,_,t)) top bot = indep t (top + 1) (bot + 1)
| indep _ _ _ = true;
fun cnt_prefixes ctxt (Abs (n, T, t)) = let
fun is_countable t = Type.of_sort (Proof_Context.tsig_of ctxt) (t, @{sort countable})
fun cnt_walk (Abs (ns, T, t)) Ts =
map (fn (t', t'') => (Abs (ns, T, t'), t'')) (cnt_walk t (T::Ts))
| cnt_walk (f $ g) Ts = let
val n = length Ts - 1
in
map (fn (f', t) => (f' $ g, t)) (cnt_walk f Ts) @
map (fn (g', t) => (f $ g', t)) (cnt_walk g Ts) @
(if is_countable (type_of1 (Ts, g)) andalso loose_bvar1 (g, n)
andalso indep g n 0 andalso g <> Bound n
then [(f $ Bound (n + 1), incr_boundvars (~ n) g)]
else [])
end
| cnt_walk _ _ = []
in map (fn (t1, t2) => let
val T1 = type_of1 ([T], t2)
val T2 = type_of1 ([T], t)
in ([SOME (Abs (n, T1, Abs (n, T, t1))), NONE, NONE, SOME (Abs (n, T, t2))],
[SOME T1, SOME T, SOME T2])
end) (cnt_walk t [T])
end
| cnt_prefixes _ _ = []
fun measurable_tac ctxt facts =
let
fun debug_fact msg thm () =
msg ^ " " ^ Pretty.str_of (Syntax.pretty_term ctxt (prop_of thm))
fun IF' c t i = COND (c i) (t i) no_tac
fun r_tac msg =
if Config.get ctxt debug
then FIRST' o
map (fn thm => resolve_tac [thm]
THEN' K (debug_tac ctxt (debug_fact (msg ^ "resolved using") thm) all_tac))
else resolve_tac
val elem_congI = @{lemma "A = B \<Longrightarrow> x \<in> B \<Longrightarrow> x \<in> A" by simp}
val dests = get_dest (Context.Proof ctxt)
fun prep_dest thm =
(if is_too_generic thm then [] else [thm] @ map_filter (try (fn th' => thm RS th')) dests) ;
val preprocessors = (("std", prep_dest #> pair) :: get_preprocessors ctxt) ;
fun preprocess_thm (thm, raw) =
if raw then pair [thm] else fold_map (fn (_, proc) => proc thm) preprocessors #>> flat
fun sel lv (th, (raw, lv')) = if lv = lv' then SOME (th, raw) else NONE ;
fun get lv = ctxt |> Context.Proof |> get_thms |> rev |> map_filter (sel lv) ;
val pre_thms = map (Simplifier.norm_hhf ctxt #> rpair false) facts @ get Concrete @ get Generic
val (thms, ctxt) = fold_map preprocess_thm pre_thms ctxt |>> flat
fun is_sets_eq (Const (@{const_name "HOL.eq"}, _) $
(Const (@{const_name "sets"}, _) $ _) $
(Const (@{const_name "sets"}, _) $ _)) = true
| is_sets_eq (Const (@{const_name "HOL.eq"}, _) $
(Const (@{const_name "measurable"}, _) $ _ $ _) $
(Const (@{const_name "measurable"}, _) $ _ $ _)) = true
| is_sets_eq _ = false
val cong_thms = get_cong (Context.Proof ctxt) @
filter (fn thm => concl_of thm |> HOLogic.dest_Trueprop |> is_sets_eq handle TERM _ => false) facts
fun sets_cong_tac i =
Subgoal.FOCUS (fn {context = ctxt', prems = prems, ...} => (
let
val ctxt'' = Simplifier.add_prems prems ctxt'
in
r_tac "cong intro" [elem_congI]
THEN' SOLVED' (fn i => REPEAT_DETERM (
((r_tac "cong solve" (cong_thms @ [@{thm refl}])
ORELSE' IF' (fn i => fn thm => nprems_of thm > i)
(SOLVED' (asm_full_simp_tac ctxt''))) i)))
end) 1) ctxt i
THEN flexflex_tac ctxt
val simp_solver_tac =
IF' not_measurable_prop (debug_tac ctxt (K "simp ") o SOLVED' (asm_full_simp_tac ctxt))
val split_countable_tac =
Subgoal.FOCUS (fn {context = ctxt, ...} => SUBGOAL (fn (t, i) =>
let
val f = dest_measurable_fun (HOLogic.dest_Trueprop t)
fun cert f = map (Option.map (f (Proof_Context.theory_of ctxt)))
fun inst (ts, Ts) =
Drule.instantiate' (cert ctyp_of Ts) (cert cterm_of ts) @{thm measurable_compose_countable}
in r_tac "split countable" (cnt_prefixes ctxt f |> map inst) i end
handle TERM _ => no_tac) 1)
val splitter = if Config.get ctxt split then split_countable_tac ctxt else K no_tac
val split_app_tac =
Subgoal.FOCUS (fn {context = ctxt, ...} => SUBGOAL (fn (t, i) =>
let
fun app_prefixes (Abs (n, T, (f $ g))) = let
val ps = (if not (loose_bvar1 (g, 0)) then [(f, g)] else [])
in map (fn (f, c) => (Abs (n, T, f), c, T, type_of c, type_of1 ([T], f $ c))) ps end
| app_prefixes _ = []
fun dest_app (Abs (_, T, t as ((f $ Bound 0) $ c))) = (f, c, T, type_of c, type_of1 ([T], t))
| dest_app t = raise (TERM ("not a measurability predicate of an application", [t]))
val thy = Proof_Context.theory_of ctxt
val tunify = Sign.typ_unify thy
val thms = map
(fn thm => (thm, dest_app (dest_measurable_fun (HOLogic.dest_Trueprop (concl_of thm)))))
(get_app (Context.Proof ctxt))
fun cert f = map (fn (t, t') => (f thy t, f thy t'))
fun inst (f, c, T, Tc, Tf) (thm, (thmf, thmc, thmT, thmTc, thmTf)) =
let
val inst =
(Vartab.empty, ~1)
|> tunify (T, thmT)
|> tunify (Tf, thmTf)
|> tunify (Tc, thmTc)
|> Vartab.dest o fst
val subst = subst_TVars (map (apsnd snd) inst)
in
Thm.instantiate (cert ctyp_of (map (fn (n, (s, T)) => (TVar (n, s), T)) inst),
cert cterm_of [(subst thmf, f), (subst thmc, c)]) thm
end
val cps = map_product inst (app_prefixes (dest_measurable_fun (HOLogic.dest_Trueprop t))) thms
in if null cps then no_tac
else r_tac "split app" cps i ORELSE debug_tac ctxt (fn () => "split app fun FAILED") no_tac end
handle TERM t => debug_tac ctxt (fn () => "TERM " ^ fst t ^ Pretty.str_of (Pretty.list "[" "]" (map (Syntax.pretty_term ctxt) (snd t)))) no_tac
handle Type.TUNIFY => debug_tac ctxt (fn () => "TUNIFY") no_tac) 1)
val single_step_tac =
simp_solver_tac
ORELSE' r_tac "step" thms
ORELSE' (split_app_tac ctxt)
ORELSE' splitter
ORELSE' (CHANGED o sets_cong_tac)
ORELSE' (K (debug_tac ctxt (K "backtrack") no_tac))
in debug_tac ctxt (K "start") (REPEAT (single_step_tac 1)) end;
fun simproc ctxt redex =
let
val t = HOLogic.mk_Trueprop (term_of redex);
fun tac {context = ctxt, prems = _ } =
SOLVE (measurable_tac ctxt (Simplifier.prems_of ctxt));
in try (fn () => Goal.prove ctxt [] [] t tac RS @{thm Eq_TrueI}) () end;
end