(******************* admissibility tactic ***********************
checks whether adm_subst theorem is applicable to the
current proof state. "t" is instantiated with a term of chain-
finite type, so that adm_chain_finite can be applied.
example of usage:
by (adm_tac cont_tacRs 1);
*****************************************************************)
local
(*** find_subterms t 0 []
returns lists of terms with the following properties:
1. all terms in the list are disjoint subterms of t
2. all terms contain the variable which is bound at level 0
3. all occurences of the variable which is bound at level 0
are "covered" by a term in the list
a list of integers is associated with every term which describes
the "path" leading to the subterm (required for instantiation of
the adm_subst theorem (see functions mk_term, inst_adm_subst_thm))
***)
fun find_subterms (Bound i) lev path =
if i = lev then [[(Bound 0, path)]]
else []
| find_subterms (t as (Abs (_, _, t2))) lev path =
if filter (fn x => x<=lev)
(add_loose_bnos (t, 0, [])) = [lev] then
[(incr_bv (~lev, 0, t), path)]::
(find_subterms t2 (lev+1) (0::path))
else find_subterms t2 (lev+1) (0::path)
| find_subterms (t as (t1 $ t2)) lev path =
let val ts1 = find_subterms t1 lev (0::path);
val ts2 = find_subterms t2 lev (1::path);
fun combine [] y = []
| combine (x::xs) ys =
(map (fn z => x @ z) ys) @ (combine xs ys)
in
(if filter (fn x => x<=lev)
(add_loose_bnos (t, 0, [])) = [lev] then
[[(incr_bv (~lev, 0, t), path)]]
else []) @
(if ts1 = [] then ts2
else if ts2 = [] then ts1
else combine ts1 ts2)
end
| find_subterms _ _ _ = [];
(*** make term for instantiation of predicate "P" in adm_subst theorem ***)
fun make_term t path paths lev =
if path mem paths then Bound lev
else case t of
(Abs (s, T, t1)) => Abs (s, T, make_term t1 (0::path) paths (lev+1))
| (t1 $ t2) => (make_term t1 (0::path) paths lev) $
(make_term t2 (1::path) paths lev)
| t1 => t1;
(*** check whether all terms in list are equal ***)
fun eq_terms (ts as ((t, _)::_)) = forall (fn (t2, _) => t2 aconv t) ts;
(*** NOTE: when the following two functions are called, all terms in the list
are equal (only their "paths" differ!)
***)
(*** check whether type of terms in list is chain finite ***)
fun is_chfin sign T params ((t, _)::_) =
let val {tsig, ...} = Sign.rep_sg sign;
val parTs = map snd (rev params)
in Type.of_sort tsig (fastype_of1 (T::parTs, t), ["chfin", "pcpo"]) end;
(*** try to prove that terms in list are continuous
if successful, add continuity theorem to list l ***)
fun prove_cont tac sign s T prems params (l, ts as ((t, _)::_)) =
(let val parTs = map snd (rev params);
val contT = (T --> (fastype_of1 (T::parTs, t))) --> HOLogic.boolT;
fun mk_all [] t = t
| mk_all ((a,T)::Ts) t = (all T) $ (Abs (a, T, mk_all Ts t));
val t = HOLogic.mk_Trueprop ((Const ("cont", contT)) $ (Abs (s, T, t)));
val t' = mk_all params (Logic.list_implies (prems, t));
val thm = prove_goalw_cterm [] (cterm_of sign t')
(fn ps => [cut_facts_tac ps 1, tac 1])
in (ts, thm)::l end) handle _ => l;
(*** instantiation of adm_subst theorem (a bit tricky)
NOTE: maybe unnecessary (if "cont_thm RS adm_subst" works properly) ***)
fun inst_adm_subst_thm state i params s T subt t paths =
let val {sign, maxidx, ...} = rep_thm state;
val j = maxidx+1;
val {tsig, ...} = Sign.rep_sg sign;
val parTs = map snd (rev params);
val rule = lift_rule (state, i) adm_subst;
val types = the o (fst (types_sorts rule));
val tT = types ("t", j);
val PT = types ("P", j);
fun mk_abs [] t = t
| mk_abs ((a,T)::Ts) t = Abs (a, T, mk_abs Ts t);
val tt = cterm_of sign (mk_abs (params @ [(s, T)]) subt);
val Pt = cterm_of sign (mk_abs (params @ [(s, fastype_of1 (T::parTs, subt))])
(make_term t [] paths 0));
val tye = Type.typ_match tsig ([], (tT, #T (rep_cterm tt)));
val tye' = Type.typ_match tsig (tye, (PT, #T (rep_cterm Pt)));
val ctye = map (fn (x, y) => (x, ctyp_of sign y)) tye';
val tv = cterm_of sign (Var (("t", j), typ_subst_TVars tye' tT));
val Pv = cterm_of sign (Var (("P", j), typ_subst_TVars tye' PT));
val rule' = instantiate (ctye, [(tv, tt), (Pv, Pt)]) rule
in rule' end;
(*** extract subgoal i from proof state ***)
fun nth_subgoal i thm = nth_elem (i-1, prems_of thm);
in
(*** the admissibility tactic
NOTE:
(compose_tac (false, rule, 2) i) THEN
(rtac cont_thm i) THEN ...
could probably be replaced by
(rtac (cont_thm RS adm_subst) 1) THEN ...
***)
fun adm_tac tac i state =
state |>
let val goali = nth_subgoal i state
in
case Logic.strip_assums_concl goali of
((Const _) $ ((Const ("adm", _)) $ (Abs (s, T, t)))) =>
let val {sign, ...} = rep_thm state;
val prems = Logic.strip_assums_hyp goali;
val params = Logic.strip_params goali;
val ts = find_subterms t 0 [];
val ts' = filter eq_terms ts;
val ts'' = filter (is_chfin sign T params) ts';
val thms = foldl (prove_cont tac sign s T prems params) ([], ts'')
in case thms of
((ts as ((t', _)::_), cont_thm)::_) =>
let val paths = map snd ts;
val rule = inst_adm_subst_thm state i params s T t' t paths;
in
(compose_tac (false, rule, 2) i) THEN
(rtac cont_thm i) THEN
(REPEAT (assume_tac i)) THEN
(rtac adm_chain_finite i)
end
| [] => no_tac
end
| _ => no_tac
end;
end;