src/Pure/drule.ML
author wenzelm
Mon Jun 27 15:03:55 2011 +0200 (2011-06-27)
changeset 43559 c1966f322105
parent 43333 2bdec7f430d3
child 44117 88a5a8f44d15
permissions -rw-r--r--
old gensym is now legacy -- global state is out of fashion, and its result is not guaranteed to be fresh;
     1 (*  Title:      Pure/drule.ML
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3 
     4 Derived rules and other operations on theorems.
     5 *)
     6 
     7 infix 0 RS RSN RL RLN MRS MRL OF COMP INCR_COMP COMP_INCR;
     8 
     9 signature BASIC_DRULE =
    10 sig
    11   val mk_implies: cterm * cterm -> cterm
    12   val list_implies: cterm list * cterm -> cterm
    13   val strip_imp_prems: cterm -> cterm list
    14   val strip_imp_concl: cterm -> cterm
    15   val cprems_of: thm -> cterm list
    16   val cterm_fun: (term -> term) -> (cterm -> cterm)
    17   val ctyp_fun: (typ -> typ) -> (ctyp -> ctyp)
    18   val forall_intr_list: cterm list -> thm -> thm
    19   val forall_intr_vars: thm -> thm
    20   val forall_elim_list: cterm list -> thm -> thm
    21   val gen_all: thm -> thm
    22   val lift_all: cterm -> thm -> thm
    23   val legacy_freeze_thaw: thm -> thm * (thm -> thm)
    24   val legacy_freeze_thaw_robust: thm -> thm * (int -> thm -> thm)
    25   val implies_elim_list: thm -> thm list -> thm
    26   val implies_intr_list: cterm list -> thm -> thm
    27   val instantiate_normalize: (ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm
    28   val zero_var_indexes_list: thm list -> thm list
    29   val zero_var_indexes: thm -> thm
    30   val implies_intr_hyps: thm -> thm
    31   val rotate_prems: int -> thm -> thm
    32   val rearrange_prems: int list -> thm -> thm
    33   val RSN: thm * (int * thm) -> thm
    34   val RS: thm * thm -> thm
    35   val RLN: thm list * (int * thm list) -> thm list
    36   val RL: thm list * thm list -> thm list
    37   val MRS: thm list * thm -> thm
    38   val MRL: thm list list * thm list -> thm list
    39   val OF: thm * thm list -> thm
    40   val compose: thm * int * thm -> thm list
    41   val COMP: thm * thm -> thm
    42   val INCR_COMP: thm * thm -> thm
    43   val COMP_INCR: thm * thm -> thm
    44   val cterm_instantiate: (cterm*cterm)list -> thm -> thm
    45   val size_of_thm: thm -> int
    46   val reflexive_thm: thm
    47   val symmetric_thm: thm
    48   val transitive_thm: thm
    49   val symmetric_fun: thm -> thm
    50   val extensional: thm -> thm
    51   val equals_cong: thm
    52   val imp_cong: thm
    53   val swap_prems_eq: thm
    54   val asm_rl: thm
    55   val cut_rl: thm
    56   val revcut_rl: thm
    57   val thin_rl: thm
    58   val triv_forall_equality: thm
    59   val distinct_prems_rl: thm
    60   val swap_prems_rl: thm
    61   val equal_intr_rule: thm
    62   val equal_elim_rule1: thm
    63   val equal_elim_rule2: thm
    64   val instantiate': ctyp option list -> cterm option list -> thm -> thm
    65 end;
    66 
    67 signature DRULE =
    68 sig
    69   include BASIC_DRULE
    70   val generalize: string list * string list -> thm -> thm
    71   val list_comb: cterm * cterm list -> cterm
    72   val strip_comb: cterm -> cterm * cterm list
    73   val strip_type: ctyp -> ctyp list * ctyp
    74   val beta_conv: cterm -> cterm -> cterm
    75   val types_sorts: thm -> (indexname-> typ option) * (indexname-> sort option)
    76   val flexflex_unique: thm -> thm
    77   val export_without_context: thm -> thm
    78   val export_without_context_open: thm -> thm
    79   val store_thm: binding -> thm -> thm
    80   val store_standard_thm: binding -> thm -> thm
    81   val store_thm_open: binding -> thm -> thm
    82   val store_standard_thm_open: binding -> thm -> thm
    83   val compose_single: thm * int * thm -> thm
    84   val imp_cong_rule: thm -> thm -> thm
    85   val arg_cong_rule: cterm -> thm -> thm
    86   val binop_cong_rule: cterm -> thm -> thm -> thm
    87   val fun_cong_rule: thm -> cterm -> thm
    88   val beta_eta_conversion: cterm -> thm
    89   val eta_long_conversion: cterm -> thm
    90   val eta_contraction_rule: thm -> thm
    91   val norm_hhf_eq: thm
    92   val norm_hhf_eqs: thm list
    93   val is_norm_hhf: term -> bool
    94   val norm_hhf: theory -> term -> term
    95   val norm_hhf_cterm: cterm -> cterm
    96   val protect: cterm -> cterm
    97   val protectI: thm
    98   val protectD: thm
    99   val protect_cong: thm
   100   val implies_intr_protected: cterm list -> thm -> thm
   101   val termI: thm
   102   val mk_term: cterm -> thm
   103   val dest_term: thm -> cterm
   104   val cterm_rule: (thm -> thm) -> cterm -> cterm
   105   val term_rule: theory -> (thm -> thm) -> term -> term
   106   val dummy_thm: thm
   107   val sort_constraintI: thm
   108   val sort_constraint_eq: thm
   109   val with_subgoal: int -> (thm -> thm) -> thm -> thm
   110   val comp_no_flatten: thm * int -> int -> thm -> thm
   111   val rename_bvars: (string * string) list -> thm -> thm
   112   val rename_bvars': string option list -> thm -> thm
   113   val incr_indexes: thm -> thm -> thm
   114   val incr_indexes2: thm -> thm -> thm -> thm
   115   val remdups_rl: thm
   116   val multi_resolve: thm list -> thm -> thm Seq.seq
   117   val multi_resolves: thm list -> thm list -> thm Seq.seq
   118   val abs_def: thm -> thm
   119 end;
   120 
   121 structure Drule: DRULE =
   122 struct
   123 
   124 
   125 (** some cterm->cterm operations: faster than calling cterm_of! **)
   126 
   127 (* A1==>...An==>B  goes to  [A1,...,An], where B is not an implication *)
   128 fun strip_imp_prems ct =
   129   let val (cA, cB) = Thm.dest_implies ct
   130   in cA :: strip_imp_prems cB end
   131   handle TERM _ => [];
   132 
   133 (* A1==>...An==>B  goes to B, where B is not an implication *)
   134 fun strip_imp_concl ct =
   135   (case Thm.term_of ct of
   136     Const ("==>", _) $ _ $ _ => strip_imp_concl (Thm.dest_arg ct)
   137   | _ => ct);
   138 
   139 (*The premises of a theorem, as a cterm list*)
   140 val cprems_of = strip_imp_prems o cprop_of;
   141 
   142 fun cterm_fun f ct = Thm.cterm_of (Thm.theory_of_cterm ct) (f (Thm.term_of ct));
   143 fun ctyp_fun f cT = Thm.ctyp_of (Thm.theory_of_ctyp cT) (f (Thm.typ_of cT));
   144 
   145 fun certify t = Thm.cterm_of (Context.the_theory (Context.the_thread_data ())) t;
   146 
   147 val implies = certify Logic.implies;
   148 fun mk_implies (A, B) = Thm.capply (Thm.capply implies A) B;
   149 
   150 (*cterm version of list_implies: [A1,...,An], B  goes to [|A1;==>;An|]==>B *)
   151 fun list_implies([], B) = B
   152   | list_implies(A::AS, B) = mk_implies (A, list_implies(AS,B));
   153 
   154 (*cterm version of list_comb: maps  (f, [t1,...,tn])  to  f(t1,...,tn) *)
   155 fun list_comb (f, []) = f
   156   | list_comb (f, t::ts) = list_comb (Thm.capply f t, ts);
   157 
   158 (*cterm version of strip_comb: maps  f(t1,...,tn)  to  (f, [t1,...,tn]) *)
   159 fun strip_comb ct =
   160   let
   161     fun stripc (p as (ct, cts)) =
   162       let val (ct1, ct2) = Thm.dest_comb ct
   163       in stripc (ct1, ct2 :: cts) end handle CTERM _ => p
   164   in stripc (ct, []) end;
   165 
   166 (* cterm version of strip_type: maps  [T1,...,Tn]--->T  to   ([T1,T2,...,Tn], T) *)
   167 fun strip_type cT = (case Thm.typ_of cT of
   168     Type ("fun", _) =>
   169       let
   170         val [cT1, cT2] = Thm.dest_ctyp cT;
   171         val (cTs, cT') = strip_type cT2
   172       in (cT1 :: cTs, cT') end
   173   | _ => ([], cT));
   174 
   175 (*Beta-conversion for cterms, where x is an abstraction. Simply returns the rhs
   176   of the meta-equality returned by the beta_conversion rule.*)
   177 fun beta_conv x y =
   178   Thm.dest_arg (cprop_of (Thm.beta_conversion false (Thm.capply x y)));
   179 
   180 
   181 
   182 (*** Find the type (sort) associated with a (T)Var or (T)Free in a term
   183      Used for establishing default types (of variables) and sorts (of
   184      type variables) when reading another term.
   185      Index -1 indicates that a (T)Free rather than a (T)Var is wanted.
   186 ***)
   187 
   188 fun types_sorts thm =
   189   let
   190     val vars = Thm.fold_terms Term.add_vars thm [];
   191     val frees = Thm.fold_terms Term.add_frees thm [];
   192     val tvars = Thm.fold_terms Term.add_tvars thm [];
   193     val tfrees = Thm.fold_terms Term.add_tfrees thm [];
   194     fun types (a, i) =
   195       if i < 0 then AList.lookup (op =) frees a else AList.lookup (op =) vars (a, i);
   196     fun sorts (a, i) =
   197       if i < 0 then AList.lookup (op =) tfrees a else AList.lookup (op =) tvars (a, i);
   198   in (types, sorts) end;
   199 
   200 
   201 
   202 
   203 (** Standardization of rules **)
   204 
   205 (*Generalization over a list of variables*)
   206 val forall_intr_list = fold_rev Thm.forall_intr;
   207 
   208 (*Generalization over Vars -- canonical order*)
   209 fun forall_intr_vars th =
   210   fold Thm.forall_intr
   211     (map (Thm.cterm_of (Thm.theory_of_thm th) o Var) (Thm.fold_terms Term.add_vars th [])) th;
   212 
   213 fun outer_params t =
   214   let val vs = Term.strip_all_vars t
   215   in Name.variant_list [] (map (Name.clean o #1) vs) ~~ map #2 vs end;
   216 
   217 (*generalize outermost parameters*)
   218 fun gen_all th =
   219   let
   220     val thy = Thm.theory_of_thm th;
   221     val {prop, maxidx, ...} = Thm.rep_thm th;
   222     val cert = Thm.cterm_of thy;
   223     fun elim (x, T) = Thm.forall_elim (cert (Var ((x, maxidx + 1), T)));
   224   in fold elim (outer_params prop) th end;
   225 
   226 (*lift vars wrt. outermost goal parameters
   227   -- reverses the effect of gen_all modulo higher-order unification*)
   228 fun lift_all goal th =
   229   let
   230     val thy = Theory.merge (Thm.theory_of_cterm goal, Thm.theory_of_thm th);
   231     val cert = Thm.cterm_of thy;
   232     val maxidx = Thm.maxidx_of th;
   233     val ps = outer_params (Thm.term_of goal)
   234       |> map (fn (x, T) => Var ((x, maxidx + 1), Logic.incr_tvar (maxidx + 1) T));
   235     val Ts = map Term.fastype_of ps;
   236     val inst = Thm.fold_terms Term.add_vars th [] |> map (fn (xi, T) =>
   237       (cert (Var (xi, T)), cert (Term.list_comb (Var (xi, Ts ---> T), ps))));
   238   in
   239     th |> Thm.instantiate ([], inst)
   240     |> fold_rev (Thm.forall_intr o cert) ps
   241   end;
   242 
   243 (*direct generalization*)
   244 fun generalize names th = Thm.generalize names (Thm.maxidx_of th + 1) th;
   245 
   246 (*specialization over a list of cterms*)
   247 val forall_elim_list = fold Thm.forall_elim;
   248 
   249 (*maps A1,...,An |- B  to  [| A1;...;An |] ==> B*)
   250 val implies_intr_list = fold_rev Thm.implies_intr;
   251 
   252 (*maps [| A1;...;An |] ==> B and [A1,...,An]  to  B*)
   253 fun implies_elim_list impth ths = fold Thm.elim_implies ths impth;
   254 
   255 (*Reset Var indexes to zero, renaming to preserve distinctness*)
   256 fun zero_var_indexes_list [] = []
   257   | zero_var_indexes_list ths =
   258       let
   259         val thy = Theory.merge_list (map Thm.theory_of_thm ths);
   260         val certT = Thm.ctyp_of thy and cert = Thm.cterm_of thy;
   261         val (instT, inst) = Term_Subst.zero_var_indexes_inst (map Thm.full_prop_of ths);
   262         val cinstT = map (fn (v, T) => (certT (TVar v), certT T)) instT;
   263         val cinst = map (fn (v, t) => (cert (Var v), cert t)) inst;
   264       in map (Thm.adjust_maxidx_thm ~1 o Thm.instantiate (cinstT, cinst)) ths end;
   265 
   266 val zero_var_indexes = singleton zero_var_indexes_list;
   267 
   268 
   269 (** Standard form of object-rule: no hypotheses, flexflex constraints,
   270     Frees, or outer quantifiers; all generality expressed by Vars of index 0.**)
   271 
   272 (*Discharge all hypotheses.*)
   273 fun implies_intr_hyps th =
   274   fold Thm.implies_intr (#hyps (Thm.crep_thm th)) th;
   275 
   276 (*Squash a theorem's flexflex constraints provided it can be done uniquely.
   277   This step can lose information.*)
   278 fun flexflex_unique th =
   279   if null (Thm.tpairs_of th) then th else
   280     case distinct Thm.eq_thm (Seq.list_of (Thm.flexflex_rule th)) of
   281       [th] => th
   282     | []   => raise THM("flexflex_unique: impossible constraints", 0, [th])
   283     |  _   => raise THM("flexflex_unique: multiple unifiers", 0, [th]);
   284 
   285 
   286 (* old-style export without context *)
   287 
   288 val export_without_context_open =
   289   implies_intr_hyps
   290   #> Thm.forall_intr_frees
   291   #> `Thm.maxidx_of
   292   #-> (fn maxidx =>
   293     Thm.forall_elim_vars (maxidx + 1)
   294     #> Thm.strip_shyps
   295     #> zero_var_indexes
   296     #> Thm.varifyT_global);
   297 
   298 val export_without_context =
   299   flexflex_unique
   300   #> export_without_context_open
   301   #> Thm.close_derivation;
   302 
   303 
   304 (*Convert all Vars in a theorem to Frees.  Also return a function for
   305   reversing that operation.  DOES NOT WORK FOR TYPE VARIABLES.
   306   Similar code in type/freeze_thaw*)
   307 
   308 fun legacy_freeze_thaw_robust th =
   309  let val fth = Thm.legacy_freezeT th
   310      val thy = Thm.theory_of_thm fth
   311      val {prop, tpairs, ...} = rep_thm fth
   312  in
   313    case List.foldr OldTerm.add_term_vars [] (prop :: Thm.terms_of_tpairs tpairs) of
   314        [] => (fth, fn i => fn x => x)   (*No vars: nothing to do!*)
   315      | vars =>
   316          let fun newName (Var(ix,_)) = (ix, legacy_gensym (string_of_indexname ix))
   317              val alist = map newName vars
   318              fun mk_inst (Var(v,T)) =
   319                  (cterm_of thy (Var(v,T)),
   320                   cterm_of thy (Free(((the o AList.lookup (op =) alist) v), T)))
   321              val insts = map mk_inst vars
   322              fun thaw i th' = (*i is non-negative increment for Var indexes*)
   323                  th' |> forall_intr_list (map #2 insts)
   324                      |> forall_elim_list (map (Thm.incr_indexes_cterm i o #1) insts)
   325          in  (Thm.instantiate ([],insts) fth, thaw)  end
   326  end;
   327 
   328 (*Basic version of the function above. No option to rename Vars apart in thaw.
   329   The Frees created from Vars have nice names.*)
   330 fun legacy_freeze_thaw th =
   331  let val fth = Thm.legacy_freezeT th
   332      val thy = Thm.theory_of_thm fth
   333      val {prop, tpairs, ...} = rep_thm fth
   334  in
   335    case List.foldr OldTerm.add_term_vars [] (prop :: Thm.terms_of_tpairs tpairs) of
   336        [] => (fth, fn x => x)
   337      | vars =>
   338          let fun newName (Var(ix,_), (pairs,used)) =
   339                    let val v = singleton (Name.variant_list used) (string_of_indexname ix)
   340                    in  ((ix,v)::pairs, v::used)  end;
   341              val (alist, _) = List.foldr newName ([], Library.foldr OldTerm.add_term_names
   342                (prop :: Thm.terms_of_tpairs tpairs, [])) vars
   343              fun mk_inst (Var(v,T)) =
   344                  (cterm_of thy (Var(v,T)),
   345                   cterm_of thy (Free(((the o AList.lookup (op =) alist) v), T)))
   346              val insts = map mk_inst vars
   347              fun thaw th' =
   348                  th' |> forall_intr_list (map #2 insts)
   349                      |> forall_elim_list (map #1 insts)
   350          in  (Thm.instantiate ([],insts) fth, thaw)  end
   351  end;
   352 
   353 (*Rotates a rule's premises to the left by k*)
   354 fun rotate_prems 0 = I
   355   | rotate_prems k = Thm.permute_prems 0 k;
   356 
   357 fun with_subgoal i f = rotate_prems (i - 1) #> f #> rotate_prems (1 - i);
   358 
   359 (*Permute prems, where the i-th position in the argument list (counting from 0)
   360   gives the position within the original thm to be transferred to position i.
   361   Any remaining trailing positions are left unchanged.*)
   362 val rearrange_prems =
   363   let
   364     fun rearr new [] thm = thm
   365       | rearr new (p :: ps) thm =
   366           rearr (new + 1)
   367             (map (fn q => if new <= q andalso q < p then q + 1 else q) ps)
   368             (Thm.permute_prems (new + 1) (new - p) (Thm.permute_prems new (p - new) thm))
   369   in rearr 0 end;
   370 
   371 (*Resolution: exactly one resolvent must be produced.*)
   372 fun tha RSN (i,thb) =
   373   case Seq.chop 2 (Thm.biresolution false [(false,tha)] i thb) of
   374       ([th],_) => th
   375     | ([],_)   => raise THM("RSN: no unifiers", i, [tha,thb])
   376     |      _   => raise THM("RSN: multiple unifiers", i, [tha,thb]);
   377 
   378 (*resolution: P==>Q, Q==>R gives P==>R. *)
   379 fun tha RS thb = tha RSN (1,thb);
   380 
   381 (*For joining lists of rules*)
   382 fun thas RLN (i,thbs) =
   383   let val resolve = Thm.biresolution false (map (pair false) thas) i
   384       fun resb thb = Seq.list_of (resolve thb) handle THM _ => []
   385   in maps resb thbs end;
   386 
   387 fun thas RL thbs = thas RLN (1,thbs);
   388 
   389 (*Resolve a list of rules against bottom_rl from right to left;
   390   makes proof trees*)
   391 fun rls MRS bottom_rl =
   392   let fun rs_aux i [] = bottom_rl
   393         | rs_aux i (rl::rls) = rl RSN (i, rs_aux (i+1) rls)
   394   in  rs_aux 1 rls  end;
   395 
   396 (*As above, but for rule lists*)
   397 fun rlss MRL bottom_rls =
   398   let fun rs_aux i [] = bottom_rls
   399         | rs_aux i (rls::rlss) = rls RLN (i, rs_aux (i+1) rlss)
   400   in  rs_aux 1 rlss  end;
   401 
   402 (*A version of MRS with more appropriate argument order*)
   403 fun bottom_rl OF rls = rls MRS bottom_rl;
   404 
   405 (*compose Q and [...,Qi,Q(i+1),...]==>R to [...,Q(i+1),...]==>R
   406   with no lifting or renaming!  Q may contain ==> or meta-quants
   407   ALWAYS deletes premise i *)
   408 fun compose(tha,i,thb) =
   409     distinct Thm.eq_thm (Seq.list_of (Thm.bicompose false (false,tha,0) i thb));
   410 
   411 fun compose_single (tha,i,thb) =
   412   case compose (tha,i,thb) of
   413     [th] => th
   414   | _ => raise THM ("compose: unique result expected", i, [tha,thb]);
   415 
   416 (*compose Q and [Q1,Q2,...,Qk]==>R to [Q2,...,Qk]==>R getting unique result*)
   417 fun tha COMP thb =
   418     case compose(tha,1,thb) of
   419         [th] => th
   420       | _ =>   raise THM("COMP", 1, [tha,thb]);
   421 
   422 
   423 (** theorem equality **)
   424 
   425 (*Useful "distance" function for BEST_FIRST*)
   426 val size_of_thm = size_of_term o Thm.full_prop_of;
   427 
   428 
   429 
   430 (*** Meta-Rewriting Rules ***)
   431 
   432 val read_prop = certify o Simple_Syntax.read_prop;
   433 
   434 fun store_thm name th =
   435   Context.>>> (Context.map_theory_result (Global_Theory.store_thm (name, th)));
   436 
   437 fun store_thm_open name th =
   438   Context.>>> (Context.map_theory_result (Global_Theory.store_thm_open (name, th)));
   439 
   440 fun store_standard_thm name th = store_thm name (export_without_context th);
   441 fun store_standard_thm_open name thm = store_thm_open name (export_without_context_open thm);
   442 
   443 val reflexive_thm =
   444   let val cx = certify (Var(("x",0),TVar(("'a",0),[])))
   445   in store_standard_thm_open (Binding.name "reflexive") (Thm.reflexive cx) end;
   446 
   447 val symmetric_thm =
   448   let
   449     val xy = read_prop "x::'a == y::'a";
   450     val thm = Thm.implies_intr xy (Thm.symmetric (Thm.assume xy));
   451   in store_standard_thm_open (Binding.name "symmetric") thm end;
   452 
   453 val transitive_thm =
   454   let
   455     val xy = read_prop "x::'a == y::'a";
   456     val yz = read_prop "y::'a == z::'a";
   457     val xythm = Thm.assume xy;
   458     val yzthm = Thm.assume yz;
   459     val thm = Thm.implies_intr yz (Thm.transitive xythm yzthm);
   460   in store_standard_thm_open (Binding.name "transitive") thm end;
   461 
   462 fun symmetric_fun thm = thm RS symmetric_thm;
   463 
   464 fun extensional eq =
   465   let val eq' =
   466     Thm.abstract_rule "x" (Thm.dest_arg (fst (Thm.dest_equals (cprop_of eq)))) eq
   467   in Thm.equal_elim (Thm.eta_conversion (cprop_of eq')) eq' end;
   468 
   469 val equals_cong =
   470   store_standard_thm_open (Binding.name "equals_cong")
   471     (Thm.reflexive (read_prop "x::'a == y::'a"));
   472 
   473 val imp_cong =
   474   let
   475     val ABC = read_prop "A ==> B::prop == C::prop"
   476     val AB = read_prop "A ==> B"
   477     val AC = read_prop "A ==> C"
   478     val A = read_prop "A"
   479   in
   480     store_standard_thm_open (Binding.name "imp_cong") (Thm.implies_intr ABC (Thm.equal_intr
   481       (Thm.implies_intr AB (Thm.implies_intr A
   482         (Thm.equal_elim (Thm.implies_elim (Thm.assume ABC) (Thm.assume A))
   483           (Thm.implies_elim (Thm.assume AB) (Thm.assume A)))))
   484       (Thm.implies_intr AC (Thm.implies_intr A
   485         (Thm.equal_elim (Thm.symmetric (Thm.implies_elim (Thm.assume ABC) (Thm.assume A)))
   486           (Thm.implies_elim (Thm.assume AC) (Thm.assume A)))))))
   487   end;
   488 
   489 val swap_prems_eq =
   490   let
   491     val ABC = read_prop "A ==> B ==> C"
   492     val BAC = read_prop "B ==> A ==> C"
   493     val A = read_prop "A"
   494     val B = read_prop "B"
   495   in
   496     store_standard_thm_open (Binding.name "swap_prems_eq")
   497       (Thm.equal_intr
   498         (Thm.implies_intr ABC (Thm.implies_intr B (Thm.implies_intr A
   499           (Thm.implies_elim (Thm.implies_elim (Thm.assume ABC) (Thm.assume A)) (Thm.assume B)))))
   500         (Thm.implies_intr BAC (Thm.implies_intr A (Thm.implies_intr B
   501           (Thm.implies_elim (Thm.implies_elim (Thm.assume BAC) (Thm.assume B)) (Thm.assume A))))))
   502   end;
   503 
   504 val imp_cong_rule = Thm.combination o Thm.combination (Thm.reflexive implies);
   505 
   506 fun arg_cong_rule ct th = Thm.combination (Thm.reflexive ct) th;    (*AP_TERM in LCF/HOL*)
   507 fun fun_cong_rule th ct = Thm.combination th (Thm.reflexive ct);    (*AP_THM in LCF/HOL*)
   508 fun binop_cong_rule ct th1 th2 = Thm.combination (arg_cong_rule ct th1) th2;
   509 
   510 local
   511   val dest_eq = Thm.dest_equals o cprop_of
   512   val rhs_of = snd o dest_eq
   513 in
   514 fun beta_eta_conversion t =
   515   let val thm = Thm.beta_conversion true t
   516   in Thm.transitive thm (Thm.eta_conversion (rhs_of thm)) end
   517 end;
   518 
   519 fun eta_long_conversion ct =
   520   Thm.transitive
   521     (beta_eta_conversion ct)
   522     (Thm.symmetric (beta_eta_conversion (cterm_fun (Pattern.eta_long []) ct)));
   523 
   524 (*Contract all eta-redexes in the theorem, lest they give rise to needless abstractions*)
   525 fun eta_contraction_rule th =
   526   Thm.equal_elim (Thm.eta_conversion (cprop_of th)) th;
   527 
   528 
   529 (* abs_def *)
   530 
   531 (*
   532    f ?x1 ... ?xn == u
   533   --------------------
   534    f == %x1 ... xn. u
   535 *)
   536 
   537 local
   538 
   539 fun contract_lhs th =
   540   Thm.transitive (Thm.symmetric (beta_eta_conversion
   541     (fst (Thm.dest_equals (cprop_of th))))) th;
   542 
   543 fun var_args ct =
   544   (case try Thm.dest_comb ct of
   545     SOME (f, arg) =>
   546       (case Thm.term_of arg of
   547         Var ((x, _), _) => update (eq_snd (op aconvc)) (x, arg) (var_args f)
   548       | _ => [])
   549   | NONE => []);
   550 
   551 in
   552 
   553 fun abs_def th =
   554   let
   555     val th' = contract_lhs th;
   556     val args = var_args (Thm.lhs_of th');
   557   in contract_lhs (fold (uncurry Thm.abstract_rule) args th') end;
   558 
   559 end;
   560 
   561 
   562 
   563 (*** Some useful meta-theorems ***)
   564 
   565 (*The rule V/V, obtains assumption solving for eresolve_tac*)
   566 val asm_rl = store_standard_thm_open (Binding.name "asm_rl") (Thm.trivial (read_prop "?psi"));
   567 
   568 (*Meta-level cut rule: [| V==>W; V |] ==> W *)
   569 val cut_rl =
   570   store_standard_thm_open (Binding.name "cut_rl")
   571     (Thm.trivial (read_prop "?psi ==> ?theta"));
   572 
   573 (*Generalized elim rule for one conclusion; cut_rl with reversed premises:
   574      [| PROP V;  PROP V ==> PROP W |] ==> PROP W *)
   575 val revcut_rl =
   576   let
   577     val V = read_prop "V";
   578     val VW = read_prop "V ==> W";
   579   in
   580     store_standard_thm_open (Binding.name "revcut_rl")
   581       (Thm.implies_intr V (Thm.implies_intr VW (Thm.implies_elim (Thm.assume VW) (Thm.assume V))))
   582   end;
   583 
   584 (*for deleting an unwanted assumption*)
   585 val thin_rl =
   586   let
   587     val V = read_prop "V";
   588     val W = read_prop "W";
   589     val thm = Thm.implies_intr V (Thm.implies_intr W (Thm.assume W));
   590   in store_standard_thm_open (Binding.name "thin_rl") thm end;
   591 
   592 (* (!!x. PROP ?V) == PROP ?V       Allows removal of redundant parameters*)
   593 val triv_forall_equality =
   594   let
   595     val V = read_prop "V";
   596     val QV = read_prop "!!x::'a. V";
   597     val x = certify (Free ("x", Term.aT []));
   598   in
   599     store_standard_thm_open (Binding.name "triv_forall_equality")
   600       (Thm.equal_intr (Thm.implies_intr QV (Thm.forall_elim x (Thm.assume QV)))
   601         (Thm.implies_intr V (Thm.forall_intr x (Thm.assume V))))
   602   end;
   603 
   604 (* (PROP ?Phi ==> PROP ?Phi ==> PROP ?Psi) ==>
   605    (PROP ?Phi ==> PROP ?Psi)
   606 *)
   607 val distinct_prems_rl =
   608   let
   609     val AAB = read_prop "Phi ==> Phi ==> Psi";
   610     val A = read_prop "Phi";
   611   in
   612     store_standard_thm_open (Binding.name "distinct_prems_rl")
   613       (implies_intr_list [AAB, A] (implies_elim_list (Thm.assume AAB) [Thm.assume A, Thm.assume A]))
   614   end;
   615 
   616 (* (PROP ?PhiA ==> PROP ?PhiB ==> PROP ?Psi) ==>
   617    (PROP ?PhiB ==> PROP ?PhiA ==> PROP ?Psi)
   618    `thm COMP swap_prems_rl' swaps the first two premises of `thm'
   619 *)
   620 val swap_prems_rl =
   621   let
   622     val cmajor = read_prop "PhiA ==> PhiB ==> Psi";
   623     val major = Thm.assume cmajor;
   624     val cminor1 = read_prop "PhiA";
   625     val minor1 = Thm.assume cminor1;
   626     val cminor2 = read_prop "PhiB";
   627     val minor2 = Thm.assume cminor2;
   628   in
   629     store_standard_thm_open (Binding.name "swap_prems_rl")
   630       (Thm.implies_intr cmajor (Thm.implies_intr cminor2 (Thm.implies_intr cminor1
   631         (Thm.implies_elim (Thm.implies_elim major minor1) minor2))))
   632   end;
   633 
   634 (* [| PROP ?phi ==> PROP ?psi; PROP ?psi ==> PROP ?phi |]
   635    ==> PROP ?phi == PROP ?psi
   636    Introduction rule for == as a meta-theorem.
   637 *)
   638 val equal_intr_rule =
   639   let
   640     val PQ = read_prop "phi ==> psi";
   641     val QP = read_prop "psi ==> phi";
   642   in
   643     store_standard_thm_open (Binding.name "equal_intr_rule")
   644       (Thm.implies_intr PQ (Thm.implies_intr QP (Thm.equal_intr (Thm.assume PQ) (Thm.assume QP))))
   645   end;
   646 
   647 (* PROP ?phi == PROP ?psi ==> PROP ?phi ==> PROP ?psi *)
   648 val equal_elim_rule1 =
   649   let
   650     val eq = read_prop "phi::prop == psi::prop";
   651     val P = read_prop "phi";
   652   in
   653     store_standard_thm_open (Binding.name "equal_elim_rule1")
   654       (Thm.equal_elim (Thm.assume eq) (Thm.assume P) |> implies_intr_list [eq, P])
   655   end;
   656 
   657 (* PROP ?psi == PROP ?phi ==> PROP ?phi ==> PROP ?psi *)
   658 val equal_elim_rule2 =
   659   store_standard_thm_open (Binding.name "equal_elim_rule2")
   660     (symmetric_thm RS equal_elim_rule1);
   661 
   662 (* PROP ?phi ==> PROP ?phi ==> PROP ?psi ==> PROP ?psi *)
   663 val remdups_rl =
   664   let
   665     val P = read_prop "phi";
   666     val Q = read_prop "psi";
   667     val thm = implies_intr_list [P, P, Q] (Thm.assume Q);
   668   in store_standard_thm_open (Binding.name "remdups_rl") thm end;
   669 
   670 
   671 
   672 (** embedded terms and types **)
   673 
   674 local
   675   val A = certify (Free ("A", propT));
   676   val axiom = Thm.unvarify_global o Thm.axiom (Context.the_theory (Context.the_thread_data ()));
   677   val prop_def = axiom "Pure.prop_def";
   678   val term_def = axiom "Pure.term_def";
   679   val sort_constraint_def = axiom "Pure.sort_constraint_def";
   680   val C = Thm.lhs_of sort_constraint_def;
   681   val T = Thm.dest_arg C;
   682   val CA = mk_implies (C, A);
   683 in
   684 
   685 (* protect *)
   686 
   687 val protect = Thm.capply (certify Logic.protectC);
   688 
   689 val protectI =
   690   store_standard_thm (Binding.conceal (Binding.name "protectI"))
   691     (Thm.equal_elim (Thm.symmetric prop_def) (Thm.assume A));
   692 
   693 val protectD =
   694   store_standard_thm (Binding.conceal (Binding.name "protectD"))
   695     (Thm.equal_elim prop_def (Thm.assume (protect A)));
   696 
   697 val protect_cong =
   698   store_standard_thm_open (Binding.name "protect_cong") (Thm.reflexive (protect A));
   699 
   700 fun implies_intr_protected asms th =
   701   let val asms' = map protect asms in
   702     implies_elim_list
   703       (implies_intr_list asms th)
   704       (map (fn asm' => Thm.assume asm' RS protectD) asms')
   705     |> implies_intr_list asms'
   706   end;
   707 
   708 
   709 (* term *)
   710 
   711 val termI =
   712   store_standard_thm (Binding.conceal (Binding.name "termI"))
   713     (Thm.equal_elim (Thm.symmetric term_def) (Thm.forall_intr A (Thm.trivial A)));
   714 
   715 fun mk_term ct =
   716   let
   717     val thy = Thm.theory_of_cterm ct;
   718     val cert = Thm.cterm_of thy;
   719     val certT = Thm.ctyp_of thy;
   720     val T = Thm.typ_of (Thm.ctyp_of_term ct);
   721     val a = certT (TVar (("'a", 0), []));
   722     val x = cert (Var (("x", 0), T));
   723   in Thm.instantiate ([(a, certT T)], [(x, ct)]) termI end;
   724 
   725 fun dest_term th =
   726   let val cprop = strip_imp_concl (Thm.cprop_of th) in
   727     if can Logic.dest_term (Thm.term_of cprop) then
   728       Thm.dest_arg cprop
   729     else raise THM ("dest_term", 0, [th])
   730   end;
   731 
   732 fun cterm_rule f = dest_term o f o mk_term;
   733 fun term_rule thy f t = Thm.term_of (cterm_rule f (Thm.cterm_of thy t));
   734 
   735 val dummy_thm = mk_term (certify (Term.dummy_pattern propT));
   736 
   737 
   738 (* sort_constraint *)
   739 
   740 val sort_constraintI =
   741   store_standard_thm (Binding.conceal (Binding.name "sort_constraintI"))
   742     (Thm.equal_elim (Thm.symmetric sort_constraint_def) (mk_term T));
   743 
   744 val sort_constraint_eq =
   745   store_standard_thm (Binding.conceal (Binding.name "sort_constraint_eq"))
   746     (Thm.equal_intr
   747       (Thm.implies_intr CA (Thm.implies_elim (Thm.assume CA)
   748         (Thm.unvarify_global sort_constraintI)))
   749       (implies_intr_list [A, C] (Thm.assume A)));
   750 
   751 end;
   752 
   753 
   754 (* HHF normalization *)
   755 
   756 (* (PROP ?phi ==> (!!x. PROP ?psi(x))) == (!!x. PROP ?phi ==> PROP ?psi(x)) *)
   757 val norm_hhf_eq =
   758   let
   759     val aT = TFree ("'a", []);
   760     val all = Term.all aT;
   761     val x = Free ("x", aT);
   762     val phi = Free ("phi", propT);
   763     val psi = Free ("psi", aT --> propT);
   764 
   765     val cx = certify x;
   766     val cphi = certify phi;
   767     val lhs = certify (Logic.mk_implies (phi, all $ Abs ("x", aT, psi $ Bound 0)));
   768     val rhs = certify (all $ Abs ("x", aT, Logic.mk_implies (phi, psi $ Bound 0)));
   769   in
   770     Thm.equal_intr
   771       (Thm.implies_elim (Thm.assume lhs) (Thm.assume cphi)
   772         |> Thm.forall_elim cx
   773         |> Thm.implies_intr cphi
   774         |> Thm.forall_intr cx
   775         |> Thm.implies_intr lhs)
   776       (Thm.implies_elim
   777           (Thm.assume rhs |> Thm.forall_elim cx) (Thm.assume cphi)
   778         |> Thm.forall_intr cx
   779         |> Thm.implies_intr cphi
   780         |> Thm.implies_intr rhs)
   781     |> store_standard_thm_open (Binding.name "norm_hhf_eq")
   782   end;
   783 
   784 val norm_hhf_prop = Logic.dest_equals (Thm.prop_of norm_hhf_eq);
   785 val norm_hhf_eqs = [norm_hhf_eq, sort_constraint_eq];
   786 
   787 fun is_norm_hhf (Const ("Pure.sort_constraint", _)) = false
   788   | is_norm_hhf (Const ("==>", _) $ _ $ (Const ("all", _) $ _)) = false
   789   | is_norm_hhf (Abs _ $ _) = false
   790   | is_norm_hhf (t $ u) = is_norm_hhf t andalso is_norm_hhf u
   791   | is_norm_hhf (Abs (_, _, t)) = is_norm_hhf t
   792   | is_norm_hhf _ = true;
   793 
   794 fun norm_hhf thy t =
   795   if is_norm_hhf t then t
   796   else Pattern.rewrite_term thy [norm_hhf_prop] [] t;
   797 
   798 fun norm_hhf_cterm ct =
   799   if is_norm_hhf (Thm.term_of ct) then ct
   800   else cterm_fun (Pattern.rewrite_term (Thm.theory_of_cterm ct) [norm_hhf_prop] []) ct;
   801 
   802 
   803 (* var indexes *)
   804 
   805 fun incr_indexes th = Thm.incr_indexes (Thm.maxidx_of th + 1);
   806 
   807 fun incr_indexes2 th1 th2 =
   808   Thm.incr_indexes (Int.max (Thm.maxidx_of th1, Thm.maxidx_of th2) + 1);
   809 
   810 fun th1 INCR_COMP th2 = incr_indexes th2 th1 COMP th2;
   811 fun th1 COMP_INCR th2 = th1 COMP incr_indexes th1 th2;
   812 
   813 fun comp_no_flatten (th, n) i rule =
   814   (case distinct Thm.eq_thm (Seq.list_of
   815       (Thm.compose_no_flatten false (th, n) i (incr_indexes th rule))) of
   816     [th'] => th'
   817   | [] => raise THM ("comp_no_flatten", i, [th, rule])
   818   | _ => raise THM ("comp_no_flatten: unique result expected", i, [th, rule]));
   819 
   820 
   821 
   822 (*** Instantiate theorem th, reading instantiations in theory thy ****)
   823 
   824 fun instantiate_normalize instpair th =
   825   Thm.adjust_maxidx_thm ~1 (Thm.instantiate instpair th COMP_INCR asm_rl);
   826 
   827 (*Left-to-right replacements: tpairs = [...,(vi,ti),...].
   828   Instantiates distinct Vars by terms, inferring type instantiations. *)
   829 local
   830   fun add_types ((ct,cu), (thy,tye,maxidx)) =
   831     let
   832         val thyt = Thm.theory_of_cterm ct;
   833         val thyu = Thm.theory_of_cterm cu;
   834         val {t, T, maxidx = maxt, ...} = Thm.rep_cterm ct;
   835         val {t = u, T = U, maxidx = maxu, ...} = Thm.rep_cterm cu;
   836         val maxi = Int.max(maxidx, Int.max(maxt, maxu));
   837         val thy' = Theory.merge(thy, Theory.merge(thyt, thyu))
   838         val (tye',maxi') = Sign.typ_unify thy' (T, U) (tye, maxi)
   839           handle Type.TUNIFY => raise TYPE ("Ill-typed instantiation:\nType\n" ^
   840             Syntax.string_of_typ_global thy' (Envir.norm_type tye T) ^
   841             "\nof variable " ^
   842             Syntax.string_of_term_global thy' (Term.map_types (Envir.norm_type tye) t) ^
   843             "\ncannot be unified with type\n" ^
   844             Syntax.string_of_typ_global thy' (Envir.norm_type tye U) ^ "\nof term " ^
   845             Syntax.string_of_term_global thy' (Term.map_types (Envir.norm_type tye) u),
   846             [T, U], [t, u])
   847     in  (thy', tye', maxi')  end;
   848 in
   849 fun cterm_instantiate [] th = th
   850   | cterm_instantiate ctpairs0 th =
   851   let val (thy,tye,_) = List.foldr add_types (Thm.theory_of_thm th, Vartab.empty, 0) ctpairs0
   852       fun instT(ct,cu) =
   853         let val inst = cterm_of thy o Term.map_types (Envir.norm_type tye) o term_of
   854         in (inst ct, inst cu) end
   855       fun ctyp2 (ixn, (S, T)) = (ctyp_of thy (TVar (ixn, S)), ctyp_of thy (Envir.norm_type tye T))
   856   in  instantiate_normalize (map ctyp2 (Vartab.dest tye), map instT ctpairs0) th  end
   857   handle TERM _ =>
   858            raise THM("cterm_instantiate: incompatible theories",0,[th])
   859        | TYPE (msg, _, _) => raise THM(msg, 0, [th])
   860 end;
   861 
   862 
   863 
   864 (** variations on instantiate **)
   865 
   866 (* instantiate by left-to-right occurrence of variables *)
   867 
   868 fun instantiate' cTs cts thm =
   869   let
   870     fun err msg =
   871       raise TYPE ("instantiate': " ^ msg,
   872         map_filter (Option.map Thm.typ_of) cTs,
   873         map_filter (Option.map Thm.term_of) cts);
   874 
   875     fun inst_of (v, ct) =
   876       (Thm.cterm_of (Thm.theory_of_cterm ct) (Var v), ct)
   877         handle TYPE (msg, _, _) => err msg;
   878 
   879     fun tyinst_of (v, cT) =
   880       (Thm.ctyp_of (Thm.theory_of_ctyp cT) (TVar v), cT)
   881         handle TYPE (msg, _, _) => err msg;
   882 
   883     fun zip_vars xs ys =
   884       zip_options xs ys handle ListPair.UnequalLengths =>
   885         err "more instantiations than variables in thm";
   886 
   887     (*instantiate types first!*)
   888     val thm' =
   889       if forall is_none cTs then thm
   890       else Thm.instantiate
   891         (map tyinst_of (zip_vars (rev (Thm.fold_terms Term.add_tvars thm [])) cTs), []) thm;
   892     val thm'' =
   893       if forall is_none cts then thm'
   894       else Thm.instantiate
   895         ([], map inst_of (zip_vars (rev (Thm.fold_terms Term.add_vars thm' [])) cts)) thm';
   896     in thm'' end;
   897 
   898 
   899 
   900 (** renaming of bound variables **)
   901 
   902 (* replace bound variables x_i in thm by y_i *)
   903 (* where vs = [(x_1, y_1), ..., (x_n, y_n)]  *)
   904 
   905 fun rename_bvars [] thm = thm
   906   | rename_bvars vs thm =
   907       let
   908         val cert = Thm.cterm_of (Thm.theory_of_thm thm);
   909         fun ren (Abs (x, T, t)) = Abs (AList.lookup (op =) vs x |> the_default x, T, ren t)
   910           | ren (t $ u) = ren t $ ren u
   911           | ren t = t;
   912       in Thm.equal_elim (Thm.reflexive (cert (ren (Thm.prop_of thm)))) thm end;
   913 
   914 
   915 (* renaming in left-to-right order *)
   916 
   917 fun rename_bvars' xs thm =
   918   let
   919     val cert = Thm.cterm_of (Thm.theory_of_thm thm);
   920     val prop = Thm.prop_of thm;
   921     fun rename [] t = ([], t)
   922       | rename (x' :: xs) (Abs (x, T, t)) =
   923           let val (xs', t') = rename xs t
   924           in (xs', Abs (the_default x x', T, t')) end
   925       | rename xs (t $ u) =
   926           let
   927             val (xs', t') = rename xs t;
   928             val (xs'', u') = rename xs' u
   929           in (xs'', t' $ u') end
   930       | rename xs t = (xs, t);
   931   in case rename xs prop of
   932       ([], prop') => Thm.equal_elim (Thm.reflexive (cert prop')) thm
   933     | _ => error "More names than abstractions in theorem"
   934   end;
   935 
   936 
   937 
   938 (** multi_resolve **)
   939 
   940 local
   941 
   942 fun res th i rule =
   943   Thm.biresolution false [(false, th)] i rule handle THM _ => Seq.empty;
   944 
   945 fun multi_res _ [] rule = Seq.single rule
   946   | multi_res i (th :: ths) rule = Seq.maps (res th i) (multi_res (i + 1) ths rule);
   947 
   948 in
   949 
   950 val multi_resolve = multi_res 1;
   951 fun multi_resolves facts rules = Seq.maps (multi_resolve facts) (Seq.of_list rules);
   952 
   953 end;
   954 
   955 end;
   956 
   957 structure Basic_Drule: BASIC_DRULE = Drule;
   958 open Basic_Drule;