src/Tools/coherent.ML
author wenzelm
Mon Mar 25 17:21:26 2019 +0100 (8 weeks ago)
changeset 69981 3dced198b9ec
parent 67522 9e712280cc37
permissions -rw-r--r--
more strict AFP properties;
     1 (*  Title:      Tools/coherent.ML
     2     Author:     Stefan Berghofer, TU Muenchen
     3     Author:     Marc Bezem, Institutt for Informatikk, Universitetet i Bergen
     4 
     5 Prover for coherent logic, see e.g.
     6 
     7   Marc Bezem and Thierry Coquand, Automating Coherent Logic, LPAR 2005
     8 
     9 for a description of the algorithm.
    10 *)
    11 
    12 signature COHERENT_DATA =
    13 sig
    14   val atomize_elimL: thm
    15   val atomize_exL: thm
    16   val atomize_conjL: thm
    17   val atomize_disjL: thm
    18   val operator_names: string list
    19 end;
    20 
    21 signature COHERENT =
    22 sig
    23   val trace: bool Config.T
    24   val coherent_tac: Proof.context -> thm list -> int -> tactic
    25 end;
    26 
    27 functor Coherent(Data: COHERENT_DATA) : COHERENT =
    28 struct
    29 
    30 (** misc tools **)
    31 
    32 val (trace, trace_setup) = Attrib.config_bool \<^binding>\<open>coherent_trace\<close> (K false);
    33 fun cond_trace ctxt msg = if Config.get ctxt trace then tracing (msg ()) else ();
    34 
    35 datatype cl_prf =
    36   ClPrf of thm * (Type.tyenv * Envir.tenv) * ((indexname * typ) * term) list *
    37   int list * (term list * cl_prf) list;
    38 
    39 val is_atomic = not o exists_Const (member (op =) Data.operator_names o #1);
    40 
    41 fun rulify_elim_conv ctxt ct =
    42   if is_atomic (Logic.strip_imp_concl (Thm.term_of ct)) then Conv.all_conv ct
    43   else Conv.concl_conv (length (Logic.strip_imp_prems (Thm.term_of ct)))
    44     (Conv.rewr_conv (Thm.symmetric Data.atomize_elimL) then_conv
    45      Raw_Simplifier.rewrite ctxt true (map Thm.symmetric
    46        [Data.atomize_exL, Data.atomize_conjL, Data.atomize_disjL])) ct
    47 
    48 fun rulify_elim ctxt th = Simplifier.norm_hhf ctxt (Conv.fconv_rule (rulify_elim_conv ctxt) th);
    49 
    50 (* Decompose elimination rule of the form
    51    A1 ==> ... ==> Am ==> (!!xs1. Bs1 ==> P) ==> ... ==> (!!xsn. Bsn ==> P) ==> P
    52 *)
    53 fun dest_elim prop =
    54   let
    55     val prems = Logic.strip_imp_prems prop;
    56     val concl = Logic.strip_imp_concl prop;
    57     val (prems1, prems2) = chop_suffix (fn t => Logic.strip_assums_concl t = concl) prems;
    58   in
    59     (prems1,
    60      if null prems2 then [([], [concl])]
    61      else map (fn t =>
    62        (map snd (Logic.strip_params t), Logic.strip_assums_hyp t)) prems2)
    63   end;
    64 
    65 fun mk_rule ctxt th =
    66   let
    67     val th' = rulify_elim ctxt th;
    68     val (prems, cases) = dest_elim (Thm.prop_of th')
    69   in (th', prems, cases) end;
    70 
    71 fun mk_dom ts = fold (fn t =>
    72   Typtab.map_default (fastype_of t, []) (fn us => us @ [t])) ts Typtab.empty;
    73 
    74 val empty_env = (Vartab.empty, Vartab.empty);
    75 
    76 (* Find matcher that makes conjunction valid in given state *)
    77 fun valid_conj _ _ env [] = Seq.single (env, [])
    78   | valid_conj ctxt facts env (t :: ts) =
    79       Seq.maps (fn (u, x) => Seq.map (apsnd (cons x))
    80         (valid_conj ctxt facts
    81            (Pattern.match (Proof_Context.theory_of ctxt) (t, u) env) ts
    82          handle Pattern.MATCH => Seq.empty))
    83           (Seq.of_list (sort (int_ord o apply2 snd) (Net.unify_term facts t)));
    84 
    85 (* Instantiate variables that only occur free in conlusion *)
    86 fun inst_extra_vars ctxt dom cs =
    87   let
    88     val vs = fold Term.add_vars (maps snd cs) [];
    89     fun insts [] inst = Seq.single inst
    90       | insts ((ixn, T) :: vs') inst =
    91           Seq.maps (fn t => insts vs' (((ixn, T), t) :: inst))
    92             (Seq.of_list
    93               (case Typtab.lookup dom T of
    94                 NONE =>
    95                   error ("Unknown domain: " ^
    96                     Syntax.string_of_typ ctxt T ^ "\nfor term(s) " ^
    97                     commas (maps (map (Syntax.string_of_term ctxt) o snd) cs))
    98               | SOME ts => ts))
    99   in
   100     Seq.map (fn inst =>
   101       (inst, map (apsnd (map (subst_Vars (map (apfst fst) inst)))) cs))
   102         (insts vs [])
   103   end;
   104 
   105 (* Check whether disjunction is valid in given state *)
   106 fun is_valid_disj _ _ [] = false
   107   | is_valid_disj ctxt facts ((Ts, ts) :: ds) =
   108       let val vs = map_index (fn (i, T) => Var (("x", i), T)) Ts in
   109         (case Seq.pull (valid_conj ctxt facts empty_env
   110             (map (fn t => subst_bounds (rev vs, t)) ts)) of
   111           SOME _ => true
   112         | NONE => is_valid_disj ctxt facts ds)
   113       end;
   114 
   115 fun string_of_facts ctxt s facts =
   116   Pretty.string_of (Pretty.big_list s
   117     (map (Syntax.pretty_term ctxt) (map fst (sort (int_ord o apply2 snd) (Net.content facts)))));
   118 
   119 fun valid ctxt rules goal dom facts nfacts nparams =
   120   let
   121     val seq =
   122       Seq.of_list rules |> Seq.maps (fn (th, ps, cs) =>
   123         valid_conj ctxt facts empty_env ps |> Seq.maps (fn (env as (tye, _), is) =>
   124           let val cs' = map (fn (Ts, ts) =>
   125             (map (Envir.subst_type tye) Ts, map (Envir.subst_term env) ts)) cs
   126           in
   127             inst_extra_vars ctxt dom cs' |>
   128               Seq.map_filter (fn (inst, cs'') =>
   129                 if is_valid_disj ctxt facts cs'' then NONE
   130                 else SOME (th, env, inst, is, cs''))
   131           end));
   132   in
   133     (case Seq.pull seq of
   134       NONE =>
   135         (if Context_Position.is_visible ctxt then
   136           warning (string_of_facts ctxt "Countermodel found:" facts)
   137          else (); NONE)
   138     | SOME ((th, env, inst, is, cs), _) =>
   139         if cs = [([], [goal])] then SOME (ClPrf (th, env, inst, is, []))
   140         else
   141           (case valid_cases ctxt rules goal dom facts nfacts nparams cs of
   142             NONE => NONE
   143           | SOME prfs => SOME (ClPrf (th, env, inst, is, prfs))))
   144   end
   145 
   146 and valid_cases _ _ _ _ _ _ _ [] = SOME []
   147   | valid_cases ctxt rules goal dom facts nfacts nparams ((Ts, ts) :: ds) =
   148       let
   149         val _ =
   150           cond_trace ctxt (fn () =>
   151             Pretty.string_of (Pretty.block
   152               (Pretty.str "case" :: Pretty.brk 1 ::
   153                 Pretty.commas (map (Syntax.pretty_term ctxt) ts))));
   154 
   155         val ps = map_index (fn (i, T) => ("par" ^ string_of_int (nparams + i), T)) Ts;
   156         val (params, ctxt') = fold_map Variable.next_bound ps ctxt;
   157         val ts' = map_index (fn (i, t) => (subst_bounds (rev params, t), nfacts + i)) ts;
   158         val dom' =
   159           fold (fn (T, p) => Typtab.map_default (T, []) (fn ps => ps @ [p])) (Ts ~~ params) dom;
   160         val facts' = fold (fn (t, i) => Net.insert_term op = (t, (t, i))) ts' facts;
   161       in
   162         (case valid ctxt' rules goal dom' facts' (nfacts + length ts) (nparams + length Ts) of
   163           NONE => NONE
   164         | SOME prf =>
   165             (case valid_cases ctxt rules goal dom facts nfacts nparams ds of
   166               NONE => NONE
   167             | SOME prfs => SOME ((params, prf) :: prfs)))
   168       end;
   169 
   170 
   171 (** proof replaying **)
   172 
   173 fun thm_of_cl_prf ctxt goal asms (ClPrf (th, (tye, env), insts, is, prfs)) =
   174   let
   175     val _ =
   176       cond_trace ctxt (fn () =>
   177         Pretty.string_of (Pretty.big_list "asms:" (map (Thm.pretty_thm ctxt) asms)));
   178     val th' =
   179       Drule.implies_elim_list
   180         (Thm.instantiate
   181            (map (fn (ixn, (S, T)) => ((ixn, S), Thm.ctyp_of ctxt T)) (Vartab.dest tye),
   182             map (fn (ixn, (T, t)) =>
   183               ((ixn, Envir.subst_type tye T), Thm.cterm_of ctxt t)) (Vartab.dest env) @
   184             map (fn (ixnT, t) => (ixnT, Thm.cterm_of ctxt t)) insts) th)
   185         (map (nth asms) is);
   186     val (_, cases) = dest_elim (Thm.prop_of th');
   187   in
   188     (case (cases, prfs) of
   189       ([([], [_])], []) => th'
   190     | ([([], [_])], [([], prf)]) => thm_of_cl_prf ctxt goal (asms @ [th']) prf
   191     | _ =>
   192         Drule.implies_elim_list
   193           (Thm.instantiate (Thm.match
   194              (Drule.strip_imp_concl (Thm.cprop_of th'), goal)) th')
   195           (map (thm_of_case_prf ctxt goal asms) (prfs ~~ cases)))
   196   end
   197 
   198 and thm_of_case_prf ctxt goal asms ((params, prf), (_, asms')) =
   199   let
   200     val cparams = map (Thm.cterm_of ctxt) params;
   201     val asms'' = map (Thm.cterm_of ctxt o curry subst_bounds (rev params)) asms';
   202     val (prems'', ctxt') = fold_map Thm.assume_hyps asms'' ctxt;
   203   in
   204     Drule.forall_intr_list cparams
   205       (Drule.implies_intr_list asms'' (thm_of_cl_prf ctxt' goal (asms @ prems'') prf))
   206   end;
   207 
   208 
   209 (** external interface **)
   210 
   211 fun coherent_tac ctxt rules = SUBPROOF (fn {prems, concl, params, context = ctxt', ...} =>
   212   resolve_tac ctxt' [rulify_elim_conv ctxt' concl RS Drule.equal_elim_rule2] 1 THEN
   213   SUBPROOF (fn {prems = prems', concl, context = ctxt'', ...} =>
   214     let
   215       val xs =
   216         map (Thm.term_of o #2) params @
   217         map (fn (_, s) => Free (s, the (Variable.default_type ctxt'' s)))
   218           (rev (Variable.dest_fixes ctxt''))  (* FIXME !? *)
   219     in
   220       (case
   221         valid ctxt'' (map (mk_rule ctxt'') (prems' @ prems @ rules)) (Thm.term_of concl)
   222           (mk_dom xs) Net.empty 0 0 of
   223         NONE => no_tac
   224       | SOME prf => resolve_tac ctxt'' [thm_of_cl_prf ctxt'' concl [] prf] 1)
   225     end) ctxt' 1) ctxt;
   226 
   227 val _ = Theory.setup
   228   (trace_setup #>
   229    Method.setup \<^binding>\<open>coherent\<close>
   230     (Attrib.thms >> (fn rules => fn ctxt =>
   231         METHOD (fn facts => HEADGOAL (coherent_tac ctxt (facts @ rules)))))
   232       "prove coherent formula");
   233 
   234 end;