(* Author: Stefan Berghofer, TU Muenchen Admissibility tactic. Checks whether adm_subst theorem is applicable to the current proof state: cont t ==> adm P ==> adm (%x. P (t x)) "t" is instantiated with a term of chain-finite type, so that adm_chfin can be applied: adm (P::'a::{chfin,pcpo} => bool) *) signature ADM = sig val adm_tac: Proof.context -> (int -> tactic) -> int -> tactic end; structure Adm: ADM = struct (*** 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 member (op =) paths path 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 [] = true | eq_terms (ts as (t, _) :: _) = forall (fn (t2, _) => t2 aconv t) ts; (*** check whether type of terms in list is chain finite ***) fun is_chfin thy T params ((t, _)::_) = let val parTs = map snd (rev params) in Sign.of_sort thy (fastype_of1 (T::parTs, t), @{sort "{chfin,pcpo}"}) end; (*** try to prove that terms in list are continuous if successful, add continuity theorem to list l ***) fun prove_cont ctxt tac s T prems params (ts as ((t, _)::_)) l = (* FIXME proper context *) 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 = Term.all T $ (Abs (a, T, mk_all Ts t)); val t = HOLogic.mk_Trueprop (Const (@{const_name cont}, contT) $ Abs (s, T, t)); val t' = mk_all params (Logic.list_implies (prems, t)); val thm = Goal.prove ctxt [] [] t' (K (tac 1)); in (ts, thm)::l end handle ERROR _ => l; (*** instantiation of adm_subst theorem (a bit tricky) ***) fun inst_adm_subst_thm state i params s T subt t paths = let val thy = Thm.theory_of_thm state; val j = Thm.maxidx_of state + 1; val parTs = map snd (rev params); val rule = Thm.lift_rule (Thm.cprem_of state i) @{thm adm_subst}; val types = the o fst (Drule.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 thy (mk_abs (params @ [(s, T)]) subt); val Pt = cterm_of thy (mk_abs (params @ [(s, fastype_of1 (T::parTs, subt))]) (make_term t [] paths 0)); val tye = Sign.typ_match thy (tT, #T (rep_cterm tt)) Vartab.empty; val tye' = Sign.typ_match thy (PT, #T (rep_cterm Pt)) tye; val ctye = map (fn (ixn, (S, T)) => (ctyp_of thy (TVar (ixn, S)), ctyp_of thy T)) (Vartab.dest tye'); val tv = cterm_of thy (Var (("t", j), Envir.typ_subst_TVars tye' tT)); val Pv = cterm_of thy (Var (("P", j), Envir.typ_subst_TVars tye' PT)); val rule' = instantiate (ctye, [(tv, tt), (Pv, Pt)]) rule in rule' end; (*** the admissibility tactic ***) fun try_dest_adm (Const _ $ (Const (@{const_name adm}, _) $ Abs abs)) = SOME abs | try_dest_adm _ = NONE; fun adm_tac ctxt tac i state = (i, state) |-> SUBGOAL (fn (goali, _) => (case try_dest_adm (Logic.strip_assums_concl goali) of NONE => no_tac | SOME (s, T, t) => let val thy = ProofContext.theory_of ctxt; 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 thy T params) ts'; val thms = fold (prove_cont ctxt tac 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 resolve_tac [cont_thm] i THEN REPEAT (assume_tac i) THEN resolve_tac [@{thm adm_chfin}] i end | [] => no_tac) end)); end;