src/Pure/drule.ML
author wenzelm
Sun Nov 11 20:31:46 2012 +0100 (2012-11-11)
changeset 50081 9b92ee8dec98
parent 48127 d30957198bbb
child 52131 366fa32ee2a3
permissions -rw-r--r--
tuned;
     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 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 implies_elim_list: thm -> thm list -> thm
    24   val implies_intr_list: cterm list -> thm -> thm
    25   val instantiate_normalize: (ctyp * ctyp) list * (cterm * cterm) list -> thm -> thm
    26   val zero_var_indexes_list: thm list -> thm list
    27   val zero_var_indexes: thm -> thm
    28   val implies_intr_hyps: thm -> thm
    29   val rotate_prems: int -> thm -> thm
    30   val rearrange_prems: int list -> thm -> thm
    31   val RSN: thm * (int * thm) -> thm
    32   val RS: thm * thm -> thm
    33   val RLN: thm list * (int * thm list) -> thm list
    34   val RL: thm list * thm list -> thm list
    35   val MRS: thm list * thm -> thm
    36   val OF: thm * thm list -> thm
    37   val compose: thm * int * thm -> thm list
    38   val COMP: thm * thm -> thm
    39   val INCR_COMP: thm * thm -> thm
    40   val COMP_INCR: thm * thm -> thm
    41   val cterm_instantiate: (cterm * cterm) list -> thm -> thm
    42   val size_of_thm: thm -> int
    43   val reflexive_thm: thm
    44   val symmetric_thm: thm
    45   val transitive_thm: thm
    46   val extensional: thm -> thm
    47   val asm_rl: thm
    48   val cut_rl: thm
    49   val revcut_rl: thm
    50   val thin_rl: thm
    51   val instantiate': ctyp option list -> cterm option list -> thm -> thm
    52 end;
    53 
    54 signature DRULE =
    55 sig
    56   include BASIC_DRULE
    57   val generalize: string list * string list -> thm -> thm
    58   val list_comb: cterm * cterm list -> cterm
    59   val strip_comb: cterm -> cterm * cterm list
    60   val strip_type: ctyp -> ctyp list * ctyp
    61   val beta_conv: cterm -> cterm -> cterm
    62   val types_sorts: thm -> (indexname-> typ option) * (indexname-> sort option)
    63   val flexflex_unique: thm -> thm
    64   val export_without_context: thm -> thm
    65   val export_without_context_open: thm -> thm
    66   val store_thm: binding -> thm -> thm
    67   val store_standard_thm: binding -> thm -> thm
    68   val store_thm_open: binding -> thm -> thm
    69   val store_standard_thm_open: binding -> thm -> thm
    70   val multi_resolve: thm list -> thm -> thm Seq.seq
    71   val multi_resolves: thm list -> thm list -> thm Seq.seq
    72   val compose_single: thm * int * thm -> thm
    73   val equals_cong: thm
    74   val imp_cong: thm
    75   val swap_prems_eq: thm
    76   val imp_cong_rule: thm -> thm -> thm
    77   val arg_cong_rule: cterm -> thm -> thm
    78   val binop_cong_rule: cterm -> thm -> thm -> thm
    79   val fun_cong_rule: thm -> cterm -> thm
    80   val beta_eta_conversion: cterm -> thm
    81   val eta_long_conversion: cterm -> thm
    82   val eta_contraction_rule: thm -> thm
    83   val norm_hhf_eq: thm
    84   val norm_hhf_eqs: thm list
    85   val is_norm_hhf: term -> bool
    86   val norm_hhf: theory -> term -> term
    87   val norm_hhf_cterm: cterm -> cterm
    88   val protect: cterm -> cterm
    89   val protectI: thm
    90   val protectD: thm
    91   val protect_cong: thm
    92   val implies_intr_protected: cterm list -> thm -> thm
    93   val termI: thm
    94   val mk_term: cterm -> thm
    95   val dest_term: thm -> cterm
    96   val cterm_rule: (thm -> thm) -> cterm -> cterm
    97   val dummy_thm: thm
    98   val sort_constraintI: thm
    99   val sort_constraint_eq: thm
   100   val with_subgoal: int -> (thm -> thm) -> thm -> thm
   101   val comp_no_flatten: thm * int -> int -> thm -> thm
   102   val rename_bvars: (string * string) list -> thm -> thm
   103   val rename_bvars': string option list -> thm -> thm
   104   val incr_indexes: thm -> thm -> thm
   105   val incr_indexes2: thm -> thm -> thm -> thm
   106   val triv_forall_equality: thm
   107   val distinct_prems_rl: thm
   108   val equal_intr_rule: thm
   109   val equal_elim_rule1: thm
   110   val equal_elim_rule2: thm
   111   val remdups_rl: thm
   112   val abs_def: thm -> thm
   113 end;
   114 
   115 structure Drule: DRULE =
   116 struct
   117 
   118 
   119 (** some cterm->cterm operations: faster than calling cterm_of! **)
   120 
   121 (* A1==>...An==>B  goes to  [A1,...,An], where B is not an implication *)
   122 fun strip_imp_prems ct =
   123   let val (cA, cB) = Thm.dest_implies ct
   124   in cA :: strip_imp_prems cB end
   125   handle TERM _ => [];
   126 
   127 (* A1==>...An==>B  goes to B, where B is not an implication *)
   128 fun strip_imp_concl ct =
   129   (case Thm.term_of ct of
   130     Const ("==>", _) $ _ $ _ => strip_imp_concl (Thm.dest_arg ct)
   131   | _ => ct);
   132 
   133 (*The premises of a theorem, as a cterm list*)
   134 val cprems_of = strip_imp_prems o cprop_of;
   135 
   136 fun cterm_fun f ct = Thm.cterm_of (Thm.theory_of_cterm ct) (f (Thm.term_of ct));
   137 fun ctyp_fun f cT = Thm.ctyp_of (Thm.theory_of_ctyp cT) (f (Thm.typ_of cT));
   138 
   139 fun certify t = Thm.cterm_of (Context.the_theory (Context.the_thread_data ())) t;
   140 
   141 val implies = certify Logic.implies;
   142 fun mk_implies (A, B) = Thm.apply (Thm.apply implies A) B;
   143 
   144 (*cterm version of list_implies: [A1,...,An], B  goes to [|A1;==>;An|]==>B *)
   145 fun list_implies([], B) = B
   146   | list_implies(A::AS, B) = mk_implies (A, list_implies(AS,B));
   147 
   148 (*cterm version of list_comb: maps  (f, [t1,...,tn])  to  f(t1,...,tn) *)
   149 fun list_comb (f, []) = f
   150   | list_comb (f, t::ts) = list_comb (Thm.apply f t, ts);
   151 
   152 (*cterm version of strip_comb: maps  f(t1,...,tn)  to  (f, [t1,...,tn]) *)
   153 fun strip_comb ct =
   154   let
   155     fun stripc (p as (ct, cts)) =
   156       let val (ct1, ct2) = Thm.dest_comb ct
   157       in stripc (ct1, ct2 :: cts) end handle CTERM _ => p
   158   in stripc (ct, []) end;
   159 
   160 (* cterm version of strip_type: maps  [T1,...,Tn]--->T  to   ([T1,T2,...,Tn], T) *)
   161 fun strip_type cT = (case Thm.typ_of cT of
   162     Type ("fun", _) =>
   163       let
   164         val [cT1, cT2] = Thm.dest_ctyp cT;
   165         val (cTs, cT') = strip_type cT2
   166       in (cT1 :: cTs, cT') end
   167   | _ => ([], cT));
   168 
   169 (*Beta-conversion for cterms, where x is an abstraction. Simply returns the rhs
   170   of the meta-equality returned by the beta_conversion rule.*)
   171 fun beta_conv x y =
   172   Thm.dest_arg (cprop_of (Thm.beta_conversion false (Thm.apply x y)));
   173 
   174 
   175 
   176 (*** Find the type (sort) associated with a (T)Var or (T)Free in a term
   177      Used for establishing default types (of variables) and sorts (of
   178      type variables) when reading another term.
   179      Index -1 indicates that a (T)Free rather than a (T)Var is wanted.
   180 ***)
   181 
   182 fun types_sorts thm =
   183   let
   184     val vars = Thm.fold_terms Term.add_vars thm [];
   185     val frees = Thm.fold_terms Term.add_frees thm [];
   186     val tvars = Thm.fold_terms Term.add_tvars thm [];
   187     val tfrees = Thm.fold_terms Term.add_tfrees thm [];
   188     fun types (a, i) =
   189       if i < 0 then AList.lookup (op =) frees a else AList.lookup (op =) vars (a, i);
   190     fun sorts (a, i) =
   191       if i < 0 then AList.lookup (op =) tfrees a else AList.lookup (op =) tvars (a, i);
   192   in (types, sorts) end;
   193 
   194 
   195 
   196 
   197 (** Standardization of rules **)
   198 
   199 (*Generalization over a list of variables*)
   200 val forall_intr_list = fold_rev Thm.forall_intr;
   201 
   202 (*Generalization over Vars -- canonical order*)
   203 fun forall_intr_vars th =
   204   fold Thm.forall_intr
   205     (map (Thm.cterm_of (Thm.theory_of_thm th) o Var) (Thm.fold_terms Term.add_vars th [])) th;
   206 
   207 fun outer_params t =
   208   let val vs = Term.strip_all_vars t
   209   in Name.variant_list [] (map (Name.clean o #1) vs) ~~ map #2 vs end;
   210 
   211 (*generalize outermost parameters*)
   212 fun gen_all th =
   213   let
   214     val thy = Thm.theory_of_thm th;
   215     val {prop, maxidx, ...} = Thm.rep_thm th;
   216     val cert = Thm.cterm_of thy;
   217     fun elim (x, T) = Thm.forall_elim (cert (Var ((x, maxidx + 1), T)));
   218   in fold elim (outer_params prop) th end;
   219 
   220 (*lift vars wrt. outermost goal parameters
   221   -- reverses the effect of gen_all modulo higher-order unification*)
   222 fun lift_all goal th =
   223   let
   224     val thy = Theory.merge (Thm.theory_of_cterm goal, Thm.theory_of_thm th);
   225     val cert = Thm.cterm_of thy;
   226     val maxidx = Thm.maxidx_of th;
   227     val ps = outer_params (Thm.term_of goal)
   228       |> map (fn (x, T) => Var ((x, maxidx + 1), Logic.incr_tvar (maxidx + 1) T));
   229     val Ts = map Term.fastype_of ps;
   230     val inst = Thm.fold_terms Term.add_vars th [] |> map (fn (xi, T) =>
   231       (cert (Var (xi, T)), cert (Term.list_comb (Var (xi, Ts ---> T), ps))));
   232   in
   233     th |> Thm.instantiate ([], inst)
   234     |> fold_rev (Thm.forall_intr o cert) ps
   235   end;
   236 
   237 (*direct generalization*)
   238 fun generalize names th = Thm.generalize names (Thm.maxidx_of th + 1) th;
   239 
   240 (*specialization over a list of cterms*)
   241 val forall_elim_list = fold Thm.forall_elim;
   242 
   243 (*maps A1,...,An |- B  to  [| A1;...;An |] ==> B*)
   244 val implies_intr_list = fold_rev Thm.implies_intr;
   245 
   246 (*maps [| A1;...;An |] ==> B and [A1,...,An]  to  B*)
   247 fun implies_elim_list impth ths = fold Thm.elim_implies ths impth;
   248 
   249 (*Reset Var indexes to zero, renaming to preserve distinctness*)
   250 fun zero_var_indexes_list [] = []
   251   | zero_var_indexes_list ths =
   252       let
   253         val thy = Theory.merge_list (map Thm.theory_of_thm ths);
   254         val certT = Thm.ctyp_of thy and cert = Thm.cterm_of thy;
   255         val (instT, inst) = Term_Subst.zero_var_indexes_inst (map Thm.full_prop_of ths);
   256         val cinstT = map (fn (v, T) => (certT (TVar v), certT T)) instT;
   257         val cinst = map (fn (v, t) => (cert (Var v), cert t)) inst;
   258       in map (Thm.adjust_maxidx_thm ~1 o Thm.instantiate (cinstT, cinst)) ths end;
   259 
   260 val zero_var_indexes = singleton zero_var_indexes_list;
   261 
   262 
   263 (** Standard form of object-rule: no hypotheses, flexflex constraints,
   264     Frees, or outer quantifiers; all generality expressed by Vars of index 0.**)
   265 
   266 (*Discharge all hypotheses.*)
   267 fun implies_intr_hyps th =
   268   fold Thm.implies_intr (#hyps (Thm.crep_thm th)) th;
   269 
   270 (*Squash a theorem's flexflex constraints provided it can be done uniquely.
   271   This step can lose information.*)
   272 fun flexflex_unique th =
   273   if null (Thm.tpairs_of th) then th else
   274     case distinct Thm.eq_thm (Seq.list_of (Thm.flexflex_rule th)) of
   275       [th] => th
   276     | []   => raise THM("flexflex_unique: impossible constraints", 0, [th])
   277     |  _   => raise THM("flexflex_unique: multiple unifiers", 0, [th]);
   278 
   279 
   280 (* old-style export without context *)
   281 
   282 val export_without_context_open =
   283   implies_intr_hyps
   284   #> Thm.forall_intr_frees
   285   #> `Thm.maxidx_of
   286   #-> (fn maxidx =>
   287     Thm.forall_elim_vars (maxidx + 1)
   288     #> Thm.strip_shyps
   289     #> zero_var_indexes
   290     #> Thm.varifyT_global);
   291 
   292 val export_without_context =
   293   flexflex_unique
   294   #> export_without_context_open
   295   #> Thm.close_derivation;
   296 
   297 
   298 (*Rotates a rule's premises to the left by k*)
   299 fun rotate_prems 0 = I
   300   | rotate_prems k = Thm.permute_prems 0 k;
   301 
   302 fun with_subgoal i f = rotate_prems (i - 1) #> f #> rotate_prems (1 - i);
   303 
   304 (*Permute prems, where the i-th position in the argument list (counting from 0)
   305   gives the position within the original thm to be transferred to position i.
   306   Any remaining trailing positions are left unchanged.*)
   307 val rearrange_prems =
   308   let
   309     fun rearr new [] thm = thm
   310       | rearr new (p :: ps) thm =
   311           rearr (new + 1)
   312             (map (fn q => if new <= q andalso q < p then q + 1 else q) ps)
   313             (Thm.permute_prems (new + 1) (new - p) (Thm.permute_prems new (p - new) thm))
   314   in rearr 0 end;
   315 
   316 
   317 (*Resolution: multiple arguments, multiple results*)
   318 local
   319   fun res th i rule =
   320     Thm.biresolution false [(false, th)] i rule handle THM _ => Seq.empty;
   321 
   322   fun multi_res _ [] rule = Seq.single rule
   323     | multi_res i (th :: ths) rule = Seq.maps (res th i) (multi_res (i + 1) ths rule);
   324 in
   325   val multi_resolve = multi_res 1;
   326   fun multi_resolves facts rules = Seq.maps (multi_resolve facts) (Seq.of_list rules);
   327 end;
   328 
   329 (*Resolution: exactly one resolvent must be produced*)
   330 fun tha RSN (i, thb) =
   331   (case Seq.chop 2 (Thm.biresolution false [(false, tha)] i thb) of
   332     ([th], _) => th
   333   | ([], _) => raise THM ("RSN: no unifiers", i, [tha, thb])
   334   | _ => raise THM ("RSN: multiple unifiers", i, [tha, thb]));
   335 
   336 (*Resolution: P==>Q, Q==>R gives P==>R*)
   337 fun tha RS thb = tha RSN (1,thb);
   338 
   339 (*For joining lists of rules*)
   340 fun thas RLN (i, thbs) =
   341   let val resolve = Thm.biresolution false (map (pair false) thas) i
   342       fun resb thb = Seq.list_of (resolve thb) handle THM _ => []
   343   in maps resb thbs end;
   344 
   345 fun thas RL thbs = thas RLN (1, thbs);
   346 
   347 (*Isar-style multi-resolution*)
   348 fun bottom_rl OF rls =
   349   (case Seq.chop 2 (multi_resolve rls bottom_rl) of
   350     ([th], _) => th
   351   | ([], _) => raise THM ("OF: no unifiers", 0, bottom_rl :: rls)
   352   | _ => raise THM ("OF: multiple unifiers", 0, bottom_rl :: rls));
   353 
   354 (*Resolve a list of rules against bottom_rl from right to left;
   355   makes proof trees*)
   356 fun rls MRS bottom_rl = bottom_rl OF rls;
   357 
   358 (*compose Q and [...,Qi,Q(i+1),...]==>R to [...,Q(i+1),...]==>R
   359   with no lifting or renaming!  Q may contain ==> or meta-quants
   360   ALWAYS deletes premise i *)
   361 fun compose(tha,i,thb) =
   362   distinct Thm.eq_thm (Seq.list_of (Thm.bicompose false (false,tha,0) i thb));
   363 
   364 fun compose_single (tha,i,thb) =
   365   (case compose (tha,i,thb) of
   366     [th] => th
   367   | _ => raise THM ("compose: unique result expected", i, [tha,thb]));
   368 
   369 (*compose Q and [Q1,Q2,...,Qk]==>R to [Q2,...,Qk]==>R getting unique result*)
   370 fun tha COMP thb =
   371   (case compose(tha, 1, thb) of
   372     [th] => th
   373   | _ => raise THM ("COMP", 1, [tha, thb]));
   374 
   375 
   376 (** theorem equality **)
   377 
   378 (*Useful "distance" function for BEST_FIRST*)
   379 val size_of_thm = size_of_term o Thm.full_prop_of;
   380 
   381 
   382 
   383 (*** Meta-Rewriting Rules ***)
   384 
   385 val read_prop = certify o Simple_Syntax.read_prop;
   386 
   387 fun store_thm name th =
   388   Context.>>> (Context.map_theory_result (Global_Theory.store_thm (name, th)));
   389 
   390 fun store_thm_open name th =
   391   Context.>>> (Context.map_theory_result (Global_Theory.store_thm_open (name, th)));
   392 
   393 fun store_standard_thm name th = store_thm name (export_without_context th);
   394 fun store_standard_thm_open name thm = store_thm_open name (export_without_context_open thm);
   395 
   396 val reflexive_thm =
   397   let val cx = certify (Var(("x",0),TVar(("'a",0),[])))
   398   in store_standard_thm_open (Binding.name "reflexive") (Thm.reflexive cx) end;
   399 
   400 val symmetric_thm =
   401   let
   402     val xy = read_prop "x::'a == y::'a";
   403     val thm = Thm.implies_intr xy (Thm.symmetric (Thm.assume xy));
   404   in store_standard_thm_open (Binding.name "symmetric") thm end;
   405 
   406 val transitive_thm =
   407   let
   408     val xy = read_prop "x::'a == y::'a";
   409     val yz = read_prop "y::'a == z::'a";
   410     val xythm = Thm.assume xy;
   411     val yzthm = Thm.assume yz;
   412     val thm = Thm.implies_intr yz (Thm.transitive xythm yzthm);
   413   in store_standard_thm_open (Binding.name "transitive") thm end;
   414 
   415 fun extensional eq =
   416   let val eq' =
   417     Thm.abstract_rule "x" (Thm.dest_arg (fst (Thm.dest_equals (cprop_of eq)))) eq
   418   in Thm.equal_elim (Thm.eta_conversion (cprop_of eq')) eq' end;
   419 
   420 val equals_cong =
   421   store_standard_thm_open (Binding.name "equals_cong")
   422     (Thm.reflexive (read_prop "x::'a == y::'a"));
   423 
   424 val imp_cong =
   425   let
   426     val ABC = read_prop "A ==> B::prop == C::prop"
   427     val AB = read_prop "A ==> B"
   428     val AC = read_prop "A ==> C"
   429     val A = read_prop "A"
   430   in
   431     store_standard_thm_open (Binding.name "imp_cong") (Thm.implies_intr ABC (Thm.equal_intr
   432       (Thm.implies_intr AB (Thm.implies_intr A
   433         (Thm.equal_elim (Thm.implies_elim (Thm.assume ABC) (Thm.assume A))
   434           (Thm.implies_elim (Thm.assume AB) (Thm.assume A)))))
   435       (Thm.implies_intr AC (Thm.implies_intr A
   436         (Thm.equal_elim (Thm.symmetric (Thm.implies_elim (Thm.assume ABC) (Thm.assume A)))
   437           (Thm.implies_elim (Thm.assume AC) (Thm.assume A)))))))
   438   end;
   439 
   440 val swap_prems_eq =
   441   let
   442     val ABC = read_prop "A ==> B ==> C"
   443     val BAC = read_prop "B ==> A ==> C"
   444     val A = read_prop "A"
   445     val B = read_prop "B"
   446   in
   447     store_standard_thm_open (Binding.name "swap_prems_eq")
   448       (Thm.equal_intr
   449         (Thm.implies_intr ABC (Thm.implies_intr B (Thm.implies_intr A
   450           (Thm.implies_elim (Thm.implies_elim (Thm.assume ABC) (Thm.assume A)) (Thm.assume B)))))
   451         (Thm.implies_intr BAC (Thm.implies_intr A (Thm.implies_intr B
   452           (Thm.implies_elim (Thm.implies_elim (Thm.assume BAC) (Thm.assume B)) (Thm.assume A))))))
   453   end;
   454 
   455 val imp_cong_rule = Thm.combination o Thm.combination (Thm.reflexive implies);
   456 
   457 fun arg_cong_rule ct th = Thm.combination (Thm.reflexive ct) th;    (*AP_TERM in LCF/HOL*)
   458 fun fun_cong_rule th ct = Thm.combination th (Thm.reflexive ct);    (*AP_THM in LCF/HOL*)
   459 fun binop_cong_rule ct th1 th2 = Thm.combination (arg_cong_rule ct th1) th2;
   460 
   461 local
   462   val dest_eq = Thm.dest_equals o cprop_of
   463   val rhs_of = snd o dest_eq
   464 in
   465 fun beta_eta_conversion t =
   466   let val thm = Thm.beta_conversion true t
   467   in Thm.transitive thm (Thm.eta_conversion (rhs_of thm)) end
   468 end;
   469 
   470 fun eta_long_conversion ct =
   471   Thm.transitive
   472     (beta_eta_conversion ct)
   473     (Thm.symmetric (beta_eta_conversion (cterm_fun (Pattern.eta_long []) ct)));
   474 
   475 (*Contract all eta-redexes in the theorem, lest they give rise to needless abstractions*)
   476 fun eta_contraction_rule th =
   477   Thm.equal_elim (Thm.eta_conversion (cprop_of th)) th;
   478 
   479 
   480 (* abs_def *)
   481 
   482 (*
   483    f ?x1 ... ?xn == u
   484   --------------------
   485    f == %x1 ... xn. u
   486 *)
   487 
   488 local
   489 
   490 fun contract_lhs th =
   491   Thm.transitive (Thm.symmetric (beta_eta_conversion
   492     (fst (Thm.dest_equals (cprop_of th))))) th;
   493 
   494 fun var_args ct =
   495   (case try Thm.dest_comb ct of
   496     SOME (f, arg) =>
   497       (case Thm.term_of arg of
   498         Var ((x, _), _) => update (eq_snd (op aconvc)) (x, arg) (var_args f)
   499       | _ => [])
   500   | NONE => []);
   501 
   502 in
   503 
   504 fun abs_def th =
   505   let
   506     val th' = contract_lhs th;
   507     val args = var_args (Thm.lhs_of th');
   508   in contract_lhs (fold (uncurry Thm.abstract_rule) args th') end;
   509 
   510 end;
   511 
   512 
   513 
   514 (*** Some useful meta-theorems ***)
   515 
   516 (*The rule V/V, obtains assumption solving for eresolve_tac*)
   517 val asm_rl = store_standard_thm_open (Binding.name "asm_rl") (Thm.trivial (read_prop "?psi"));
   518 
   519 (*Meta-level cut rule: [| V==>W; V |] ==> W *)
   520 val cut_rl =
   521   store_standard_thm_open (Binding.name "cut_rl")
   522     (Thm.trivial (read_prop "?psi ==> ?theta"));
   523 
   524 (*Generalized elim rule for one conclusion; cut_rl with reversed premises:
   525      [| PROP V;  PROP V ==> PROP W |] ==> PROP W *)
   526 val revcut_rl =
   527   let
   528     val V = read_prop "V";
   529     val VW = read_prop "V ==> W";
   530   in
   531     store_standard_thm_open (Binding.name "revcut_rl")
   532       (Thm.implies_intr V (Thm.implies_intr VW (Thm.implies_elim (Thm.assume VW) (Thm.assume V))))
   533   end;
   534 
   535 (*for deleting an unwanted assumption*)
   536 val thin_rl =
   537   let
   538     val V = read_prop "V";
   539     val W = read_prop "W";
   540     val thm = Thm.implies_intr V (Thm.implies_intr W (Thm.assume W));
   541   in store_standard_thm_open (Binding.name "thin_rl") thm end;
   542 
   543 (* (!!x. PROP ?V) == PROP ?V       Allows removal of redundant parameters*)
   544 val triv_forall_equality =
   545   let
   546     val V = read_prop "V";
   547     val QV = read_prop "!!x::'a. V";
   548     val x = certify (Free ("x", Term.aT []));
   549   in
   550     store_standard_thm_open (Binding.name "triv_forall_equality")
   551       (Thm.equal_intr (Thm.implies_intr QV (Thm.forall_elim x (Thm.assume QV)))
   552         (Thm.implies_intr V (Thm.forall_intr x (Thm.assume V))))
   553   end;
   554 
   555 (* (PROP ?Phi ==> PROP ?Phi ==> PROP ?Psi) ==>
   556    (PROP ?Phi ==> PROP ?Psi)
   557 *)
   558 val distinct_prems_rl =
   559   let
   560     val AAB = read_prop "Phi ==> Phi ==> Psi";
   561     val A = read_prop "Phi";
   562   in
   563     store_standard_thm_open (Binding.name "distinct_prems_rl")
   564       (implies_intr_list [AAB, A] (implies_elim_list (Thm.assume AAB) [Thm.assume A, Thm.assume A]))
   565   end;
   566 
   567 (* [| PROP ?phi ==> PROP ?psi; PROP ?psi ==> PROP ?phi |]
   568    ==> PROP ?phi == PROP ?psi
   569    Introduction rule for == as a meta-theorem.
   570 *)
   571 val equal_intr_rule =
   572   let
   573     val PQ = read_prop "phi ==> psi";
   574     val QP = read_prop "psi ==> phi";
   575   in
   576     store_standard_thm_open (Binding.name "equal_intr_rule")
   577       (Thm.implies_intr PQ (Thm.implies_intr QP (Thm.equal_intr (Thm.assume PQ) (Thm.assume QP))))
   578   end;
   579 
   580 (* PROP ?phi == PROP ?psi ==> PROP ?phi ==> PROP ?psi *)
   581 val equal_elim_rule1 =
   582   let
   583     val eq = read_prop "phi::prop == psi::prop";
   584     val P = read_prop "phi";
   585   in
   586     store_standard_thm_open (Binding.name "equal_elim_rule1")
   587       (Thm.equal_elim (Thm.assume eq) (Thm.assume P) |> implies_intr_list [eq, P])
   588   end;
   589 
   590 (* PROP ?psi == PROP ?phi ==> PROP ?phi ==> PROP ?psi *)
   591 val equal_elim_rule2 =
   592   store_standard_thm_open (Binding.name "equal_elim_rule2")
   593     (symmetric_thm RS equal_elim_rule1);
   594 
   595 (* PROP ?phi ==> PROP ?phi ==> PROP ?psi ==> PROP ?psi *)
   596 val remdups_rl =
   597   let
   598     val P = read_prop "phi";
   599     val Q = read_prop "psi";
   600     val thm = implies_intr_list [P, P, Q] (Thm.assume Q);
   601   in store_standard_thm_open (Binding.name "remdups_rl") thm end;
   602 
   603 
   604 
   605 (** embedded terms and types **)
   606 
   607 local
   608   val A = certify (Free ("A", propT));
   609   val axiom = Thm.unvarify_global o Thm.axiom (Context.the_theory (Context.the_thread_data ()));
   610   val prop_def = axiom "Pure.prop_def";
   611   val term_def = axiom "Pure.term_def";
   612   val sort_constraint_def = axiom "Pure.sort_constraint_def";
   613   val C = Thm.lhs_of sort_constraint_def;
   614   val T = Thm.dest_arg C;
   615   val CA = mk_implies (C, A);
   616 in
   617 
   618 (* protect *)
   619 
   620 val protect = Thm.apply (certify Logic.protectC);
   621 
   622 val protectI =
   623   store_standard_thm (Binding.conceal (Binding.name "protectI"))
   624     (Thm.equal_elim (Thm.symmetric prop_def) (Thm.assume A));
   625 
   626 val protectD =
   627   store_standard_thm (Binding.conceal (Binding.name "protectD"))
   628     (Thm.equal_elim prop_def (Thm.assume (protect A)));
   629 
   630 val protect_cong =
   631   store_standard_thm_open (Binding.name "protect_cong") (Thm.reflexive (protect A));
   632 
   633 fun implies_intr_protected asms th =
   634   let val asms' = map protect asms in
   635     implies_elim_list
   636       (implies_intr_list asms th)
   637       (map (fn asm' => Thm.assume asm' RS protectD) asms')
   638     |> implies_intr_list asms'
   639   end;
   640 
   641 
   642 (* term *)
   643 
   644 val termI =
   645   store_standard_thm (Binding.conceal (Binding.name "termI"))
   646     (Thm.equal_elim (Thm.symmetric term_def) (Thm.forall_intr A (Thm.trivial A)));
   647 
   648 fun mk_term ct =
   649   let
   650     val thy = Thm.theory_of_cterm ct;
   651     val cert = Thm.cterm_of thy;
   652     val certT = Thm.ctyp_of thy;
   653     val T = Thm.typ_of (Thm.ctyp_of_term ct);
   654     val a = certT (TVar (("'a", 0), []));
   655     val x = cert (Var (("x", 0), T));
   656   in Thm.instantiate ([(a, certT T)], [(x, ct)]) termI end;
   657 
   658 fun dest_term th =
   659   let val cprop = strip_imp_concl (Thm.cprop_of th) in
   660     if can Logic.dest_term (Thm.term_of cprop) then
   661       Thm.dest_arg cprop
   662     else raise THM ("dest_term", 0, [th])
   663   end;
   664 
   665 fun cterm_rule f = dest_term o f o mk_term;
   666 
   667 val dummy_thm = mk_term (certify Term.dummy_prop);
   668 
   669 
   670 (* sort_constraint *)
   671 
   672 val sort_constraintI =
   673   store_standard_thm (Binding.conceal (Binding.name "sort_constraintI"))
   674     (Thm.equal_elim (Thm.symmetric sort_constraint_def) (mk_term T));
   675 
   676 val sort_constraint_eq =
   677   store_standard_thm (Binding.conceal (Binding.name "sort_constraint_eq"))
   678     (Thm.equal_intr
   679       (Thm.implies_intr CA (Thm.implies_elim (Thm.assume CA)
   680         (Thm.unvarify_global sort_constraintI)))
   681       (implies_intr_list [A, C] (Thm.assume A)));
   682 
   683 end;
   684 
   685 
   686 (* HHF normalization *)
   687 
   688 (* (PROP ?phi ==> (!!x. PROP ?psi x)) == (!!x. PROP ?phi ==> PROP ?psi x) *)
   689 val norm_hhf_eq =
   690   let
   691     val aT = TFree ("'a", []);
   692     val x = Free ("x", aT);
   693     val phi = Free ("phi", propT);
   694     val psi = Free ("psi", aT --> propT);
   695 
   696     val cx = certify x;
   697     val cphi = certify phi;
   698     val lhs = certify (Logic.mk_implies (phi, Logic.all x (psi $ x)));
   699     val rhs = certify (Logic.all x (Logic.mk_implies (phi, psi $ x)));
   700   in
   701     Thm.equal_intr
   702       (Thm.implies_elim (Thm.assume lhs) (Thm.assume cphi)
   703         |> Thm.forall_elim cx
   704         |> Thm.implies_intr cphi
   705         |> Thm.forall_intr cx
   706         |> Thm.implies_intr lhs)
   707       (Thm.implies_elim
   708           (Thm.assume rhs |> Thm.forall_elim cx) (Thm.assume cphi)
   709         |> Thm.forall_intr cx
   710         |> Thm.implies_intr cphi
   711         |> Thm.implies_intr rhs)
   712     |> store_standard_thm_open (Binding.name "norm_hhf_eq")
   713   end;
   714 
   715 val norm_hhf_prop = Logic.dest_equals (Thm.prop_of norm_hhf_eq);
   716 val norm_hhf_eqs = [norm_hhf_eq, sort_constraint_eq];
   717 
   718 fun is_norm_hhf (Const ("Pure.sort_constraint", _)) = false
   719   | is_norm_hhf (Const ("==>", _) $ _ $ (Const ("all", _) $ _)) = false
   720   | is_norm_hhf (Abs _ $ _) = false
   721   | is_norm_hhf (t $ u) = is_norm_hhf t andalso is_norm_hhf u
   722   | is_norm_hhf (Abs (_, _, t)) = is_norm_hhf t
   723   | is_norm_hhf _ = true;
   724 
   725 fun norm_hhf thy t =
   726   if is_norm_hhf t then t
   727   else Pattern.rewrite_term thy [norm_hhf_prop] [] t;
   728 
   729 fun norm_hhf_cterm ct =
   730   if is_norm_hhf (Thm.term_of ct) then ct
   731   else cterm_fun (Pattern.rewrite_term (Thm.theory_of_cterm ct) [norm_hhf_prop] []) ct;
   732 
   733 
   734 (* var indexes *)
   735 
   736 fun incr_indexes th = Thm.incr_indexes (Thm.maxidx_of th + 1);
   737 
   738 fun incr_indexes2 th1 th2 =
   739   Thm.incr_indexes (Int.max (Thm.maxidx_of th1, Thm.maxidx_of th2) + 1);
   740 
   741 fun th1 INCR_COMP th2 = incr_indexes th2 th1 COMP th2;
   742 fun th1 COMP_INCR th2 = th1 COMP incr_indexes th1 th2;
   743 
   744 fun comp_no_flatten (th, n) i rule =
   745   (case distinct Thm.eq_thm (Seq.list_of
   746       (Thm.compose_no_flatten false (th, n) i (incr_indexes th rule))) of
   747     [th'] => th'
   748   | [] => raise THM ("comp_no_flatten", i, [th, rule])
   749   | _ => raise THM ("comp_no_flatten: unique result expected", i, [th, rule]));
   750 
   751 
   752 
   753 (** variations on Thm.instantiate **)
   754 
   755 fun instantiate_normalize instpair th =
   756   Thm.adjust_maxidx_thm ~1 (Thm.instantiate instpair th COMP_INCR asm_rl);
   757 
   758 (*Left-to-right replacements: tpairs = [..., (vi, ti), ...].
   759   Instantiates distinct Vars by terms, inferring type instantiations.*)
   760 local
   761   fun add_types (ct, cu) (thy, tye, maxidx) =
   762     let
   763       val {t, T, maxidx = maxt, ...} = Thm.rep_cterm ct;
   764       val {t = u, T = U, maxidx = maxu, ...} = Thm.rep_cterm cu;
   765       val maxi = Int.max (maxidx, Int.max (maxt, maxu));
   766       val thy' = Theory.merge (thy, Theory.merge (Thm.theory_of_cterm ct, Thm.theory_of_cterm cu));
   767       val (tye', maxi') = Sign.typ_unify thy' (T, U) (tye, maxi)
   768         handle Type.TUNIFY => raise TYPE ("Ill-typed instantiation:\nType\n" ^
   769           Syntax.string_of_typ_global thy' (Envir.norm_type tye T) ^
   770           "\nof variable " ^
   771           Syntax.string_of_term_global thy' (Term.map_types (Envir.norm_type tye) t) ^
   772           "\ncannot be unified with type\n" ^
   773           Syntax.string_of_typ_global thy' (Envir.norm_type tye U) ^ "\nof term " ^
   774           Syntax.string_of_term_global thy' (Term.map_types (Envir.norm_type tye) u),
   775           [T, U], [t, u])
   776     in (thy', tye', maxi') end;
   777 in
   778 
   779 fun cterm_instantiate [] th = th
   780   | cterm_instantiate ctpairs th =
   781       let
   782         val (thy, tye, _) = fold_rev add_types ctpairs (Thm.theory_of_thm th, Vartab.empty, 0);
   783         val certT = ctyp_of thy;
   784         val instT =
   785           Vartab.fold (fn (xi, (S, T)) =>
   786             cons (certT (TVar (xi, S)), certT (Envir.norm_type tye T))) tye [];
   787         val inst = map (pairself (Thm.instantiate_cterm (instT, []))) ctpairs;
   788       in instantiate_normalize (instT, inst) th end
   789       handle TERM (msg, _) => raise THM (msg, 0, [th])
   790         | TYPE (msg, _, _) => raise THM (msg, 0, [th]);
   791 end;
   792 
   793 
   794 (* instantiate by left-to-right occurrence of variables *)
   795 
   796 fun instantiate' cTs cts thm =
   797   let
   798     fun err msg =
   799       raise TYPE ("instantiate': " ^ msg,
   800         map_filter (Option.map Thm.typ_of) cTs,
   801         map_filter (Option.map Thm.term_of) cts);
   802 
   803     fun inst_of (v, ct) =
   804       (Thm.cterm_of (Thm.theory_of_cterm ct) (Var v), ct)
   805         handle TYPE (msg, _, _) => err msg;
   806 
   807     fun tyinst_of (v, cT) =
   808       (Thm.ctyp_of (Thm.theory_of_ctyp cT) (TVar v), cT)
   809         handle TYPE (msg, _, _) => err msg;
   810 
   811     fun zip_vars xs ys =
   812       zip_options xs ys handle ListPair.UnequalLengths =>
   813         err "more instantiations than variables in thm";
   814 
   815     (*instantiate types first!*)
   816     val thm' =
   817       if forall is_none cTs then thm
   818       else Thm.instantiate
   819         (map tyinst_of (zip_vars (rev (Thm.fold_terms Term.add_tvars thm [])) cTs), []) thm;
   820     val thm'' =
   821       if forall is_none cts then thm'
   822       else Thm.instantiate
   823         ([], map inst_of (zip_vars (rev (Thm.fold_terms Term.add_vars thm' [])) cts)) thm';
   824     in thm'' end;
   825 
   826 
   827 
   828 (** renaming of bound variables **)
   829 
   830 (* replace bound variables x_i in thm by y_i *)
   831 (* where vs = [(x_1, y_1), ..., (x_n, y_n)]  *)
   832 
   833 fun rename_bvars [] thm = thm
   834   | rename_bvars vs thm =
   835       let
   836         val cert = Thm.cterm_of (Thm.theory_of_thm thm);
   837         fun ren (Abs (x, T, t)) = Abs (AList.lookup (op =) vs x |> the_default x, T, ren t)
   838           | ren (t $ u) = ren t $ ren u
   839           | ren t = t;
   840       in Thm.equal_elim (Thm.reflexive (cert (ren (Thm.prop_of thm)))) thm end;
   841 
   842 
   843 (* renaming in left-to-right order *)
   844 
   845 fun rename_bvars' xs thm =
   846   let
   847     val cert = Thm.cterm_of (Thm.theory_of_thm thm);
   848     val prop = Thm.prop_of thm;
   849     fun rename [] t = ([], t)
   850       | rename (x' :: xs) (Abs (x, T, t)) =
   851           let val (xs', t') = rename xs t
   852           in (xs', Abs (the_default x x', T, t')) end
   853       | rename xs (t $ u) =
   854           let
   855             val (xs', t') = rename xs t;
   856             val (xs'', u') = rename xs' u
   857           in (xs'', t' $ u') end
   858       | rename xs t = (xs, t);
   859   in case rename xs prop of
   860       ([], prop') => Thm.equal_elim (Thm.reflexive (cert prop')) thm
   861     | _ => error "More names than abstractions in theorem"
   862   end;
   863 
   864 end;
   865 
   866 structure Basic_Drule: BASIC_DRULE = Drule;
   867 open Basic_Drule;