File casesplit.ML (Isabelle2005: October 2005)

# File casesplit.ML

    (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *)
    (*  Title:      TFL/casesplit.ML
        Author:     Lucas Dixon, University of Edinburgh
                    lucas.dixon@ed.ac.uk
        Date:       17 Aug 2004
    *)
    (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *)
    (*  DESCRIPTION:

        A structure that defines a tactic to program case splits.

        casesplit_free :
          string * typ -> int -> thm -> thm Seq.seq

        casesplit_name :
          string -> int -> thm -> thm Seq.seq

        These use the induction theorem associated with the recursive data
        type to be split.

        The structure includes a function to try and recursively split a
        conjecture into a list sub-theorems:

        splitto : thm list -> thm -> thm
    *)
    (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *)

    (* logic-specific *)
    signature CASE_SPLIT_DATA =
    sig
      val dest_Trueprop : term -> term
      val mk_Trueprop : term -> term

      val localize : thm list
      val local_impI : thm
      val atomize : thm list
      val rulify1 : thm list
      val rulify2 : thm list

    end;

    (* for HOL *)
    structure CaseSplitData_HOL : CASE_SPLIT_DATA =
    struct
    val dest_Trueprop = HOLogic.dest_Trueprop;
    val mk_Trueprop = HOLogic.mk_Trueprop;

    val localize = [Thm.symmetric (thm "induct_implies_def")];
    val local_impI = thm "induct_impliesI";
    val atomize = thms "induct_atomize";
    val rulify1 = thms "induct_rulify1";
    val rulify2 = thms "induct_rulify2";

    end;

    signature CASE_SPLIT =
    sig
      (* failure to find a free to split on *)
      exception find_split_exp of string

      (* getting a case split thm from the induction thm *)
      val case_thm_of_ty : theory -> typ -> thm
      val cases_thm_of_induct_thm : thm -> thm

      (* case split tactics *)
      val casesplit_free :
          string * typ -> int -> thm -> thm Seq.seq
      val casesplit_name : string -> int -> thm -> thm Seq.seq

      (* finding a free var to split *)
      val find_term_split :
          term * term -> (string * typ) option
      val find_thm_split :
          thm -> int -> thm -> (string * typ) option
      val find_thms_split :
          thm list -> int -> thm -> (string * typ) option

      (* try to recursively split conjectured thm to given list of thms *)
      val splitto : thm list -> thm -> thm

      (* for use with the recdef package *)
      val derive_init_eqs :
          theory ->
          (thm * int) list -> term list -> (thm * int) list
    end;

    functor CaseSplitFUN(Data : CASE_SPLIT_DATA) =
    struct

    val rulify_goals = Tactic.rewrite_goals_rule Data.rulify1;
    val atomize_goals = Tactic.rewrite_goals_rule Data.atomize;

    (*
    val localize = Tactic.norm_hhf_rule o Tactic.simplify false Data.localize;
    fun atomize_term sg =
      ObjectLogic.drop_judgment sg o MetaSimplifier.rewrite_term sg Data.atomize [];
    val rulify_tac =  Tactic.rewrite_goal_tac Data.rulify1;
    val atomize_tac =  Tactic.rewrite_goal_tac Data.atomize;
    Tactic.simplify_goal
    val rulify_tac =
      Tactic.rewrite_goal_tac Data.rulify1 THEN'
      Tactic.rewrite_goal_tac Data.rulify2 THEN'
      Tactic.norm_hhf_tac;
    val atomize = Tactic.norm_hhf_rule o Tactic.simplify true Data.atomize;
    *)

    (* 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;

    (* make a casethm from an induction thm *)
    val cases_thm_of_induct_thm =
         Seq.hd o (ALLGOALS (fn i => REPEAT (etac Drule.thin_rl i)));

    (* get the case_thm (my version) from a type *)
    fun case_thm_of_ty sgn ty  =
        let
          val dtypestab = DatatypePackage.get_datatypes sgn;
          val ty_str = case ty of
                         Type(ty_str, _) => ty_str
                       | TFree(s,_)  => raise ERROR_MESSAGE
                                                ("Free type: " ^ s)
                       | TVar((s,i),_) => raise ERROR_MESSAGE
                                                ("Free variable: " ^ s)
          val dt = case Symtab.lookup dtypestab ty_str
                    of SOME dt => dt
                     | NONE => raise ERROR_MESSAGE ("Not a Datatype: " ^ ty_str)
        in
          cases_thm_of_induct_thm (#induction dt)
        end;

    (*
     val ty = (snd o hd o map Term.dest_Free o Term.term_frees) t;
    *)

    (* for use when there are no prems to the subgoal *)
    (* does a case split on the given variable *)
    fun mk_casesplit_goal_thm sgn (vstr,ty) gt =
        let
          val x = Free(vstr,ty)
          val abst = Abs(vstr, ty, Term.abstract_over (x, gt));

          val ctermify = Thm.cterm_of sgn;
          val ctypify = Thm.ctyp_of sgn;
          val case_thm = case_thm_of_ty sgn ty;

          val abs_ct = ctermify abst;
          val free_ct = ctermify x;

          val casethm_vars = rev (Term.term_vars (Thm.concl_of case_thm));

          val casethm_tvars = Term.term_tvars (Thm.concl_of case_thm);
          val (Pv, Dv, type_insts) =
              case (Thm.concl_of case_thm) of
                (_ $ ((Pv as Var(P,Pty)) $ (Dv as Var(D, Dty)))) =>
                (Pv, Dv,
                 Sign.typ_match sgn (Dty, ty) Vartab.empty)
              | _ => raise ERROR_MESSAGE ("not a valid case thm");
          val type_cinsts = map (fn (ixn, (S, T)) => (ctypify (TVar (ixn, S)), ctypify T))
            (Vartab.dest type_insts);
          val cPv = ctermify (Envir.subst_TVars type_insts Pv);
          val cDv = ctermify (Envir.subst_TVars type_insts Dv);
        in
          (beta_eta_contract
             (case_thm
                |> Thm.instantiate (type_cinsts, [])
                |> Thm.instantiate ([], [(cPv, abs_ct), (cDv, free_ct)])))
        end;

    (* for use when there are no prems to the subgoal *)
    (* does a case split on the given variable (Free fv) *)
    fun casesplit_free fv i th =
        let
          val (subgoalth, exp) = IsaND.fix_alls i th;
          val subgoalth' = atomize_goals subgoalth;
          val gt = Data.dest_Trueprop (Logic.get_goal (Thm.prop_of subgoalth') 1);
          val sgn = Thm.sign_of_thm th;

          val splitter_thm = mk_casesplit_goal_thm sgn fv gt;
          val nsplits = Thm.nprems_of splitter_thm;

          val split_goal_th = splitter_thm RS subgoalth';
          val rulified_split_goal_th = rulify_goals split_goal_th;
        in
          IsaND.export_back exp rulified_split_goal_th
        end;

    (* for use when there are no prems to the subgoal *)
    (* does a case split on the given variable *)
    fun casesplit_name vstr i th =
        let
          val (subgoalth, exp) = IsaND.fix_alls i th;
          val subgoalth' = atomize_goals subgoalth;
          val gt = Data.dest_Trueprop (Logic.get_goal (Thm.prop_of subgoalth') 1);

          val freets = Term.term_frees gt;
          fun getter x =
              let val (n,ty) = Term.dest_Free x in
                (if vstr = n orelse vstr = Syntax.dest_skolem n
                 then SOME (n,ty) else NONE )
                handle Fail _ => NONE (* dest_skolem *)
              end;
          val (n,ty) = case Library.get_first getter freets
                    of SOME (n, ty) => (n, ty)
                     | _ => raise ERROR_MESSAGE ("no such variable " ^ vstr);
          val sgn = Thm.sign_of_thm th;

          val splitter_thm = mk_casesplit_goal_thm sgn (n,ty) gt;
          val nsplits = Thm.nprems_of splitter_thm;

          val split_goal_th = splitter_thm RS subgoalth';

          val rulified_split_goal_th = rulify_goals split_goal_th;
        in
          IsaND.export_back exp rulified_split_goal_th
        end;

    (* small example:
    Goal "P (x :: nat) & (C y --> Q (y :: nat))";
    by (rtac (thm "conjI") 1);
    val th = topthm();
    val i = 2;
    val vstr = "y";

    by (casesplit_name "y" 2);

    val th = topthm();
    val i = 1;
    val th' = casesplit_name "x" i th;
    *)

    (* the find_XXX_split functions are simply doing a lightwieght (I
    think) term matching equivalent to find where to do the next split *)

    (* assuming two twems are identical except for a free in one at a
    subterm, or constant in another, ie assume that one term is a plit of
    another, then gives back the free variable that has been split. *)
    exception find_split_exp of string
    fun find_term_split (Free v, _ $ _) = SOME v
      | find_term_split (Free v, Const _) = SOME v
      | find_term_split (Free v, Abs _) = SOME v (* do we really want this case? *)
      | find_term_split (Free v, Var _) = NONE (* keep searching *)
      | find_term_split (a $ b, a2 $ b2) =
        (case find_term_split (a, a2) of
           NONE => find_term_split (b,b2)
         | vopt => vopt)
      | find_term_split (Abs(_,ty,t1), Abs(_,ty2,t2)) =
        find_term_split (t1, t2)
      | find_term_split (Const (x,ty), Const(x2,ty2)) =
        if x = x2 then NONE else (* keep searching *)
        raise find_split_exp (* stop now *)
                "Terms are not identical upto a free varaible! (Consts)"
      | find_term_split (Bound i, Bound j) =
        if i = j then NONE else (* keep searching *)
        raise find_split_exp (* stop now *)
                "Terms are not identical upto a free varaible! (Bound)"
      | find_term_split (a, b) =
        raise find_split_exp (* stop now *)
                "Terms are not identical upto a free varaible! (Other)";

    (* assume that "splitth" is a case split form of subgoal i of "genth",
    then look for a free variable to split, breaking the subgoal closer to
    splitth. *)
    fun find_thm_split splitth i genth =
        find_term_split (Logic.get_goal (Thm.prop_of genth) i,
                         Thm.concl_of splitth) handle find_split_exp _ => NONE;

    (* as above but searches "splitths" for a theorem that suggest a case split *)
    fun find_thms_split splitths i genth =
        Library.get_first (fn sth => find_thm_split sth i genth) splitths;

    (* split the subgoal i of "genth" until we get to a member of
    splitths. Assumes that genth will be a general form of splitths, that
    can be case-split, as needed. Otherwise fails. Note: We assume that
    all of "splitths" are split to the same level, and thus it doesn't
    matter which one we choose to look for the next split. Simply add
    search on splitthms and split variable, to change this.  *)
    (* Note: possible efficiency measure: when a case theorem is no longer
    useful, drop it? *)
    (* Note: This should not be a separate tactic but integrated into the
    case split done during recdef's case analysis, this would avoid us
    having to (re)search for variables to split. *)
    fun splitto splitths genth =
        let
          val _ = assert (not (null splitths)) "splitto: no given splitths";
          val sgn = Thm.sign_of_thm genth;

          (* check if we are a member of splitths - FIXME: quicker and
          more flexible with discrim net. *)
          fun solve_by_splitth th split =
              Thm.biresolution false [(false,split)] 1 th;

          fun split th =
              (case find_thms_split splitths 1 th of
                 NONE =>
                 (writeln "th:";
                  Display.print_thm th; writeln "split ths:";
                  Display.print_thms splitths; writeln "\n--";
                  raise ERROR_MESSAGE "splitto: cannot find variable to split on")
                | SOME v =>
                 let
                   val gt = Data.dest_Trueprop (List.nth(Thm.prems_of th, 0));
                   val split_thm = mk_casesplit_goal_thm sgn v gt;
                   val (subthms, expf) = IsaND.fixed_subgoal_thms split_thm;
                 in
                   expf (map recsplitf subthms)
                 end)

          and recsplitf th =
              (* note: multiple unifiers! we only take the first element,
                 probably fine -- there is probably only one anyway. *)
              (case Library.get_first (Seq.pull o solve_by_splitth th) splitths of
                 NONE => split th
               | SOME (solved_th, more) => solved_th)
        in
          recsplitf genth
        end;

    (* Note: We dont do this if wf conditions fail to be solved, as each
    case may have a different wf condition - we could group the conditions
    togeather and say that they must be true to solve the general case,
    but that would hide from the user which sub-case they were related
    to. Probably this is not important, and it would work fine, but I
    prefer leaving more fine grain control to the user. *)

    (* derive eqs, assuming strict, ie the rules have no assumptions = all
       the well-foundness conditions have been solved. *)
    local
      fun get_related_thms i =
          List.mapPartial ((fn (r,x) => if x = i then SOME r else NONE));

      fun solve_eq (th, [], i) =
          raise ERROR_MESSAGE "derive_init_eqs: missing rules"
        | solve_eq (th, [a], i) = (a, i)
        | solve_eq (th, splitths as (_ :: _), i) = (splitto splitths th,i);
    in
    fun derive_init_eqs sgn rules eqs =
        let
          val eqths = map (Thm.trivial o (Thm.cterm_of sgn) o Data.mk_Trueprop)
                          eqs
        in
          (rev o map solve_eq)
            (Library.foldln
               (fn (e,i) =>
                   (curry (op ::)) (e, (get_related_thms (i - 1) rules), i - 1))
               eqths [])
        end;
    end;
    (*
        val (rs_hwfc, unhidefs) = Library.split_list (map hide_prems rules)
        (map2 (op |>) (ths, expfs))
    *)

    end;

    structure CaseSplit = CaseSplitFUN(CaseSplitData_HOL);