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src/HOLCF/Tools/adm_tac.ML

author | wenzelm |

Wed, 11 Jun 2008 18:02:50 +0200 | |

changeset 27155 | 30e3bdfbbef1 |

parent 26626 | c6231d64d264 |

child 27331 | 5c66afff695e |

permissions | -rw-r--r-- |

Drule.types_sorts;

(* ID: $Id$ 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: (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 List.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 List.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 [] = true | eq_terms (ts as (t, _) :: _) = forall (fn (t2, _) => t2 aconv t) ts; (*figure out internal names*) val chfin_pcpoS = Sign.intern_sort (the_context ()) ["chfin", "pcpo"]; val cont_name = Sign.intern_const (the_context ()) "cont"; val adm_name = Sign.intern_const (the_context ()) "adm"; (*** check whether type of terms in list is chain finite ***) fun is_chfin sign T params ((t, _)::_) = let val parTs = map snd (rev params) in Sign.of_sort sign (fastype_of1 (T::parTs, t), chfin_pcpoS) 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_name, contT)) $ (Abs(s, T, t))); val t' = mk_all params (Logic.list_implies (prems, t)); val thm = Goal.prove (ProofContext.init sign) [] [] 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 sign = 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 = valOf 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 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 = Sign.typ_match sign (tT, #T (rep_cterm tt)) Vartab.empty; val tye' = Sign.typ_match sign (PT, #T (rep_cterm Pt)) tye; val ctye = map (fn (ixn, (S, T)) => (ctyp_of sign (TVar (ixn, S)), ctyp_of sign T)) (Vartab.dest tye'); val tv = cterm_of sign (Var (("t", j), Envir.typ_subst_TVars tye' tT)); val Pv = cterm_of sign (Var (("P", j), Envir.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 = List.nth (prems_of thm, i-1); (*** the admissibility tactic ***) fun try_dest_adm (Const _ $ (Const (name, _) $ Abs abs)) = if name = adm_name then SOME abs else NONE | try_dest_adm _ = NONE; fun adm_tac tac i state = state |> let val goali = nth_subgoal i state in (case try_dest_adm (Logic.strip_assums_concl goali) of NONE => no_tac | SOME (s, T, t) => let val sign = Thm.theory_of_thm state; val prems = Logic.strip_assums_hyp goali; val params = Logic.strip_params goali; val ts = find_subterms t 0 []; val ts' = List.filter eq_terms ts; val ts'' = List.filter (is_chfin sign T params) ts'; val thms = Library.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 @{thm adm_chfin} i end | [] => no_tac) end) end; end; open Adm;