src/Provers/IsaPlanner/rw_inst.ML

tuned;

(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *) 
(*  Title:      Pure/IsaPlanner/rw_inst.ML
    ID:         $Id$
    Author:     Lucas Dixon, University of Edinburgh
                [e-mail]
    Created:    25 Aug 2004
*)
(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *) 
(*  DESCRIPTION:

    rewriting using a conditional meta-equality theorem which supports 
    schematic variable instantiation.

*)   
(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *)
signature RW_INST =
sig

  (* Rewrite: give it instantiation infromation, a rule, and the
  target thm, and it will return the rewritten target thm *)
  val rw :
      ((Term.indexname * (Term.sort * Term.typ)) list *  (* type var instantiations *)
       (Term.indexname * (Term.typ * Term.term)) list)  (* schematic var instantiations *)
      * (string * Term.typ) list           (* Fake named bounds + types *)
      * (string * Term.typ) list           (* names of bound + types *)
      * Term.term ->                       (* outer term for instantiation *)
      Thm.thm ->                           (* rule with indexies lifted *)
      Thm.thm ->                           (* target thm *)
      Thm.thm                              (* rewritten theorem possibly 
                                              with additional premises for 
                                              rule conditions *)

  (* used tools *)
  val mk_abstractedrule :
      (string * Term.typ) list (* faked outer bound *)
      -> (string * Term.typ) list (* hopeful name of outer bounds *)
      -> Thm.thm -> Thm.cterm list * Thm.thm
  val mk_fixtvar_tyinsts :
      (Term.indexname * (Term.sort * Term.typ)) list ->
      Term.term list -> ((string * int) * (Term.sort * Term.typ)) list 
                        * (string * Term.sort) list
  val mk_renamings :
      Term.term -> Thm.thm -> (((string * int) * Term.typ) * Term.term) list
  val new_tfree :
      ((string * int) * Term.sort) *
      (((string * int) * (Term.sort * Term.typ)) list * string list) ->
      ((string * int) * (Term.sort * Term.typ)) list * string list
  val cross_inst : (Term.indexname * (Term.typ * Term.term)) list 
                   -> (Term.indexname *(Term.typ * Term.term)) list
  val cross_inst_typs : (Term.indexname * (Term.sort * Term.typ)) list 
                   -> (Term.indexname * (Term.sort * Term.typ)) list

  val beta_contract : Thm.thm -> Thm.thm
  val beta_eta_contract : Thm.thm -> Thm.thm

end;

structure RWInst 
: RW_INST
= struct


(* beta contract the theorem *)
fun beta_contract thm = 
    equal_elim (Thm.beta_conversion true (Thm.cprop_of thm)) thm;

(* beta-eta contract the theorem *)
fun beta_eta_contract thm = 
    let
      val thm2 = equal_elim (Thm.beta_conversion true (Thm.cprop_of thm)) thm
      val thm3 = equal_elim (Thm.eta_conversion (Thm.cprop_of thm2)) thm2
    in thm3 end;


(* to get the free names of a theorem (including hyps and flexes) *)
fun usednames_of_thm th =
    let val rep = Thm.rep_thm th
      val hyps = #hyps rep
      val (tpairl,tpairr) = Library.split_list (#tpairs rep)
      val prop = #prop rep
    in
      List.foldr Term.add_term_names [] (prop :: (tpairl @ (tpairr @ hyps)))
    end;

(* Given a list of variables that were bound, and a that has been
instantiated with free variable placeholders for the bound vars, it
creates an abstracted version of the theorem, with local bound vars as
lambda-params:

Ts: 
("x", ty)

rule::
C :x ==> P :x = Q :x

results in:
("!! x. C x", (%x. p x = %y. p y) [!! x. C x])

note: assumes rule is instantiated
*)
(* Note, we take abstraction in the order of last abstraction first *)
fun mk_abstractedrule TsFake Ts rule = 
    let 
      val ctermify = Thm.cterm_of (Thm.sign_of_thm rule);

      (* now we change the names of temporary free vars that represent 
         bound vars with binders outside the redex *)
      val prop = Thm.prop_of rule;
      val names = usednames_of_thm rule;
      val (fromnames,tonames,names2,Ts') = 
          Library.foldl (fn ((rnf,rnt,names, Ts''),((faken,_),(n,ty))) => 
                    let val n2 = Term.variant names n in
                      (ctermify (Free(faken,ty)) :: rnf,
                       ctermify (Free(n2,ty)) :: rnt, 
                       n2 :: names,
                       (n2,ty) :: Ts'')
                    end)
                (([],[],names, []), TsFake~~Ts);

      (* rename conflicting free's in the rule to avoid cconflicts
      with introduced vars from bounds outside in redex *)
      val rule' = rule |> Drule.forall_intr_list fromnames
                       |> Drule.forall_elim_list tonames;
      
      (* make unconditional rule and prems *)
      val (uncond_rule, cprems) = IsaND.allify_conditions ctermify (rev Ts') 
                                                          rule';

      (* using these names create lambda-abstracted version of the rule *)
      val abstractions = rev (Ts' ~~ tonames);
      val abstract_rule = Library.foldl (fn (th,((n,ty),ct)) => 
                                    Thm.abstract_rule n ct th)
                                (uncond_rule, abstractions);
    in (cprems, abstract_rule) end;


(* given names to avoid, and vars that need to be fixed, it gives
unique new names to the vars so that they can be fixed as free
variables *)
(* make fixed unique free variable instantiations for non-ground vars *)
(* Create a table of vars to be renamed after instantiation - ie
      other uninstantiated vars in the hyps of the rule 
      ie ?z in C ?z ?x ==> A ?x ?y = B ?x ?y *)
fun mk_renamings tgt rule_inst = 
    let
      val rule_conds = Thm.prems_of rule_inst
      val names = foldr Term.add_term_names [] (tgt :: rule_conds);
      val (conds_tyvs,cond_vs) = 
          Library.foldl (fn ((tyvs, vs), t) => 
                    (Library.union
                       (Term.term_tvars t, tyvs),
                     Library.union 
                       (map Term.dest_Var (Term.term_vars t), vs))) 
                (([],[]), rule_conds);
      val termvars = map Term.dest_Var (Term.term_vars tgt); 
      val vars_to_fix = Library.union (termvars, cond_vs);
      val (renamings, names2) = 
          foldr (fn (((n,i),ty), (vs, names')) => 
                    let val n' = Term.variant names' n in
                      ((((n,i),ty), Free (n', ty)) :: vs, n'::names')
                    end)
                ([], names) vars_to_fix;
    in renamings end;

(* make a new fresh typefree instantiation for the given tvar *)
fun new_tfree (tv as (ix,sort), (pairs,used)) =
      let val v = variant used (string_of_indexname ix)
      in  ((ix,(sort,TFree(v,sort)))::pairs, v::used)  end;


(* make instantiations to fix type variables that are not 
   already instantiated (in ignore_ixs) from the list of terms. *)
fun mk_fixtvar_tyinsts ignore_insts ts = 
    let 
      val ignore_ixs = map fst ignore_insts;
      val (tvars, tfrees) = 
            foldr (fn (t, (varixs, tfrees)) => 
                      (Term.add_term_tvars (t,varixs),
                       Term.add_term_tfrees (t,tfrees)))
                  ([],[]) ts;
        val unfixed_tvars = 
            List.filter (fn (ix,s) => not (ix mem ignore_ixs)) tvars;
        val (fixtyinsts, _) = foldr new_tfree ([], map fst tfrees) unfixed_tvars
    in (fixtyinsts, tfrees) end;


(* cross-instantiate the instantiations - ie for each instantiation
replace all occurances in other instantiations - no loops are possible
and thus only one-parsing of the instantiations is necessary. *)
fun cross_inst insts = 
    let 
      fun instL (ix, (ty,t)) = 
          map (fn (ix2,(ty2,t2)) => 
                  (ix2, (ty2,Term.subst_vars ([], [(ix, t)]) t2)));

      fun cross_instL ([], l) = rev l
        | cross_instL ((ix, t) :: insts, l) = 
          cross_instL (instL (ix, t) insts, (ix, t) :: (instL (ix, t) l));

    in cross_instL (insts, []) end;

(* as above but for types -- I don't know if this is needed, will we ever incur mixed up types? *)
fun cross_inst_typs insts = 
    let 
      fun instL (ix, (srt,ty)) = 
          map (fn (ix2,(srt2,ty2)) => 
                  (ix2, (srt2,Term.typ_subst_TVars [(ix, ty)] ty2)));

      fun cross_instL ([], l) = rev l
        | cross_instL ((ix, t) :: insts, l) = 
          cross_instL (instL (ix, t) insts, (ix, t) :: (instL (ix, t) l));

    in cross_instL (insts, []) end;


(* assume that rule and target_thm have distinct var names. THINK:
efficient version with tables for vars for: target vars, introduced
vars, and rule vars, for quicker instantiation?  The outerterm defines
which part of the target_thm was modified.  Note: we take Ts in the
upterm order, ie last abstraction first., and with an outeterm where
the abstracted subterm has the arguments in the revered order, ie
first abstraction first.  FakeTs has abstractions using the fake name
- ie the name distinct from all other abstractions. *)

fun rw ((nonfixed_typinsts, unprepinsts), FakeTs, Ts, outerterm) rule target_thm = 
    let 
      (* general signature info *)
      val target_sign = (Thm.sign_of_thm target_thm);
      val ctermify = Thm.cterm_of target_sign;
      val ctypeify = Thm.ctyp_of target_sign;

      (* fix all non-instantiated tvars *)
      val (fixtyinsts, othertfrees) = 
          mk_fixtvar_tyinsts nonfixed_typinsts
                             [Thm.prop_of rule, Thm.prop_of target_thm];
      val new_fixed_typs = map (fn ((s,i),(srt,ty)) => (Term.dest_TFree ty))
                               fixtyinsts;
      val typinsts = cross_inst_typs (nonfixed_typinsts @ fixtyinsts);

      (* certified instantiations for types *)
      val ctyp_insts = 
          map (fn (ix,(s,ty)) => (ctypeify (TVar (ix,s)), ctypeify ty)) 
              typinsts;

      (* type instantiated versions *)
      val tgt_th_tyinst = Thm.instantiate (ctyp_insts,[]) target_thm;
      val rule_tyinst =  Thm.instantiate (ctyp_insts,[]) rule;

      val term_typ_inst = map (fn (ix,(srt,ty)) => (ix,ty)) typinsts;
      (* type instanitated outer term *)
      val outerterm_tyinst = Term.subst_TVars term_typ_inst outerterm;

      val FakeTs_tyinst = map (apsnd (Term.typ_subst_TVars term_typ_inst)) 
                              FakeTs;
      val Ts_tyinst = map (apsnd (Term.typ_subst_TVars term_typ_inst)) 
                          Ts;

      (* type-instantiate the var instantiations *)
      val insts_tyinst = foldr (fn ((ix,(ty,t)),insts_tyinst) => 
                            (ix, (Term.typ_subst_TVars term_typ_inst ty, 
                                  Term.subst_TVars term_typ_inst t))
                            :: insts_tyinst)
                        [] unprepinsts;

      (* cross-instantiate *)
      val insts_tyinst_inst = cross_inst insts_tyinst;

      (* create certms of instantiations *)
      val cinsts_tyinst = 
          map (fn (ix,(ty,t)) => (ctermify (Var (ix, ty)), 
                                  ctermify t)) insts_tyinst_inst;

      (* The instantiated rule *)
      val rule_inst = rule_tyinst |> Thm.instantiate ([], cinsts_tyinst);

      (* Create a table of vars to be renamed after instantiation - ie
      other uninstantiated vars in the hyps the *instantiated* rule 
      ie ?z in C ?z ?x ==> A ?x ?y = B ?x ?y *)
      val renamings = mk_renamings (Thm.prop_of tgt_th_tyinst) 
                                   rule_inst;
      val cterm_renamings = 
          map (fn (x,y) => (ctermify (Var x), ctermify y)) renamings;

      (* Create the specific version of the rule for this target application *)
      val outerterm_inst = 
          outerterm_tyinst 
            |> Term.subst_Vars (map (fn (ix,(ty,t)) => (ix,t)) insts_tyinst_inst)
            |> Term.subst_Vars (map (fn ((ix,ty),t) => (ix,t)) renamings);
      val couter_inst = Thm.reflexive (ctermify outerterm_inst);
      val (cprems, abstract_rule_inst) = 
          rule_inst |> Thm.instantiate ([], cterm_renamings)
                    |> mk_abstractedrule FakeTs_tyinst Ts_tyinst;
      val specific_tgt_rule = 
          beta_eta_contract
            (Thm.combination couter_inst abstract_rule_inst);

      (* create an instantiated version of the target thm *)
      val tgt_th_inst = 
          tgt_th_tyinst |> Thm.instantiate ([], cinsts_tyinst)
                        |> Thm.instantiate ([], cterm_renamings);

      val (vars,frees_of_fixed_vars) = Library.split_list cterm_renamings;

    in
      (beta_eta_contract tgt_th_inst)
        |> Thm.equal_elim specific_tgt_rule
        |> Drule.implies_intr_list cprems
        |> Drule.forall_intr_list frees_of_fixed_vars
        |> Drule.forall_elim_list vars
        |> Thm.varifyT' othertfrees
        |-> K Drule.zero_var_indexes
    end;


end; (* struct *)