src/Provers/eqsubst.ML

lucas - fixed subst in assumptions to count redexes from left to right.

(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *) 
(*  Title:      Provers/eqsubst.ML
    Author:     Lucas Dixon, University of Edinburgh
                [e-mail]
    Modified:   18 Feb 2005 - Lucas - 
    Created:    29 Jan 2005
*)
(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *) 
(*  DESCRIPTION:

    A Tactic to perform a substiution using an equation.

*)
(* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- *)



(* Logic specific data stub *)
signature EQRULE_DATA =
sig

  (* to make a meta equality theorem in the current logic *)
  val prep_meta_eq : thm -> thm list

end;


(* the signature of an instance of the SQSUBST tactic *)
signature EQSUBST_TAC = 
sig

  exception eqsubst_occL_exp of 
            string * (int list) * (Thm.thm list) * int * Thm.thm;


  type match = 
       ((Term.indexname * (Term.sort * Term.typ)) list (* type instantiations *)
        * (Term.indexname * (Term.typ * Term.term)) list) (* term instantiations *)
       * (string * Term.typ) list (* fake named type abs env *)
       * (string * Term.typ) list (* type abs env *)
       * Term.term (* outer term *)

  type searchinfo = 
       Sign.sg (* sign for matching *)
       * int (* maxidx *)
       * BasicIsaFTerm.FcTerm (* focusterm to search under *)

  val prep_subst_in_asm :
         int (* subgoal to subst in *)
      -> Thm.thm (* target theorem with subgoals *)
      -> int (* premise to subst in *)
      -> (Thm.cterm list (* certified free var placeholders for vars *) 
          * int (* premice no. to subst *)
          * int (* number of assumptions of premice *)
          * Thm.thm) (* premice as a new theorem for forward reasoning *)
         * searchinfo (* search info: prem id etc *)

  val prep_subst_in_asms :
         int (* subgoal to subst in *)
      -> Thm.thm (* target theorem with subgoals *)
      -> ((Thm.cterm list (* certified free var placeholders for vars *) 
          * int (* premice no. to subst *)
          * int (* number of assumptions of premice *)
          * Thm.thm) (* premice as a new theorem for forward reasoning *)
         * searchinfo) list

  val apply_subst_in_asm :
      int (* subgoal *)
      -> Thm.thm (* overall theorem *)
      -> Thm.thm (* rule *)
      -> (Thm.cterm list (* certified free var placeholders for vars *) 
          * int (* assump no being subst *)
          * int (* num of premises of asm *) 
          * Thm.thm) (* premthm *)
      * match
      -> Thm.thm Seq.seq

  val prep_concl_subst :
         int (* subgoal *)
      -> Thm.thm (* overall goal theorem *)
      -> (Thm.cterm list * Thm.thm) * searchinfo (* (cvfs, conclthm), matchf *)

  val apply_subst_in_concl :
        int (* subgoal *)
        -> Thm.thm (* thm with all goals *)
        -> Thm.cterm list (* certified free var placeholders for vars *)
           * Thm.thm  (* trivial thm of goal concl *)
            (* possible matches/unifiers *)
        -> Thm.thm (* rule *)
        -> match
        -> Thm.thm Seq.seq (* substituted goal *)

  (* basic notion of search *)
  val searchf_tlr_unify_all : 
      (searchinfo 
       -> Term.term (* lhs *)
       -> match Seq.seq Seq.seq)
  val searchf_tlr_unify_valid : 
      (searchinfo 
       -> Term.term (* lhs *)
       -> match Seq.seq Seq.seq)

  (* specialise search constructor for conclusion skipping occurnaces. *)
     val skip_first_occs_search :
        int -> ('a -> 'b -> 'c Seq.seq Seq.seq) -> 'a -> 'b -> 'c Seq.seq
  (* specialised search constructor for assumptions using skips *)
     val skip_first_asm_occs_search :
        ('a -> 'b -> 'c Seq.seq Seq.seq) ->
        'a -> int -> 'b -> 'c IsaPLib.skipseqT

  (* tactics and methods *)
  val eqsubst_asm_meth : int list -> Thm.thm list -> Proof.method
  val eqsubst_asm_tac : 
      int list -> Thm.thm list -> int -> Thm.thm -> Thm.thm Seq.seq
  val eqsubst_asm_tac' : 
      (* search function with skips *)
      (searchinfo -> int -> Term.term -> match IsaPLib.skipseqT) 
      -> int (* skip to *)
      -> Thm.thm (* rule *)
      -> int (* subgoal number *)
      -> Thm.thm (* tgt theorem with subgoals *)
      -> Thm.thm Seq.seq (* new theorems *)

  val eqsubst_meth : int list -> Thm.thm list -> Proof.method
  val eqsubst_tac : 
      int list -> Thm.thm list -> int -> Thm.thm -> Thm.thm Seq.seq
  val eqsubst_tac' : 
      (searchinfo -> Term.term -> match Seq.seq) 
      -> Thm.thm -> int -> Thm.thm -> Thm.thm Seq.seq

  val meth : (bool * int list) * Thm.thm list -> Proof.context -> Proof.method
  val setup : (Theory.theory -> Theory.theory) list
end;


functor EQSubstTacFUN (structure EqRuleData : EQRULE_DATA) 
  : EQSUBST_TAC
= struct

  (* a type abriviation for match information *)
  type match = 
       ((Term.indexname * (Term.sort * Term.typ)) list (* type instantiations *)
        * (Term.indexname * (Term.typ * Term.term)) list) (* term instantiations *)
       * (string * Term.typ) list (* fake named type abs env *)
       * (string * Term.typ) list (* type abs env *)
       * Term.term (* outer term *)

  type searchinfo = 
       Sign.sg (* sign for matching *)
       * int (* maxidx *)
       * BasicIsaFTerm.FcTerm (* focusterm to search under *)

(* FOR DEBUGGING...
type trace_subst_errT = int (* subgoal *)
        * Thm.thm (* thm with all goals *)
        * (Thm.cterm list (* certified free var placeholders for vars *)
           * Thm.thm)  (* trivial thm of goal concl *)
            (* possible matches/unifiers *)
        * Thm.thm (* rule *)
        * (((Term.indexname * Term.typ) list (* type instantiations *)
              * (Term.indexname * Term.term) list ) (* term instantiations *)
             * (string * Term.typ) list (* Type abs env *)
             * Term.term) (* outer term *);

val trace_subst_err = (ref NONE : trace_subst_errT option ref);
val trace_subst_search = ref false;
exception trace_subst_exp of trace_subst_errT;
 *)

(* also defined in /HOL/Tools/inductive_codegen.ML, 
   maybe move this to seq.ML ? *)
infix 5 :->;
fun s :-> f = Seq.flat (Seq.map f s);

(* search from top, left to right, then down *)
fun search_tlr_all_f f ft = 
    let
      fun maux ft = 
          let val t' = (IsaFTerm.focus_of_fcterm ft) 
            (* val _ = 
                if !trace_subst_search then 
                  (writeln ("Examining: " ^ (TermLib.string_of_term t'));
                   TermLib.writeterm t'; ())
                else (); *)
          in 
          (case t' of 
            (_ $ _) => Seq.append(maux (IsaFTerm.focus_left ft), 
                       Seq.cons(f ft, 
                                  maux (IsaFTerm.focus_right ft)))
          | (Abs _) => Seq.cons(f ft, maux (IsaFTerm.focus_abs ft))
          | leaf => Seq.single (f ft)) end
    in maux ft end;

(* search from top, left to right, then down *)
fun search_tlr_valid_f f ft = 
    let
      fun maux ft = 
          let 
            val hereseq = if IsaFTerm.valid_match_start ft then f ft else Seq.empty
          in 
          (case (IsaFTerm.focus_of_fcterm ft) of 
            (_ $ _) => Seq.append(maux (IsaFTerm.focus_left ft), 
                       Seq.cons(hereseq, 
                                  maux (IsaFTerm.focus_right ft)))
          | (Abs _) => Seq.cons(hereseq, maux (IsaFTerm.focus_abs ft))
          | leaf => Seq.single (hereseq))
          end
    in maux ft end;

(* search all unifications *)
fun searchf_tlr_unify_all (sgn, maxidx, ft) lhs = 
    IsaFTerm.find_fcterm_matches 
      search_tlr_all_f 
      (IsaFTerm.clean_unify_ft sgn maxidx lhs)
      ft;

(* search only for 'valid' unifiers (non abs subterms and non vars) *)
fun searchf_tlr_unify_valid (sgn, maxidx, ft) lhs  = 
    IsaFTerm.find_fcterm_matches 
      search_tlr_valid_f 
      (IsaFTerm.clean_unify_ft sgn maxidx lhs)
      ft;


(* apply a substitution in the conclusion of the theorem th *)
(* cfvs are certified free var placeholders for goal params *)
(* conclthm is a theorem of for just the conclusion *)
(* m is instantiation/match information *)
(* rule is the equation for substitution *)
fun apply_subst_in_concl i th (cfvs, conclthm) rule m = 
    (RWInst.rw m rule conclthm)
      |> IsaND.unfix_frees cfvs
      |> RWInst.beta_eta_contract
      |> (fn r => Tactic.rtac r i th);
(*
 |> (fn r => Thm.bicompose false (false, r, Thm.nprems_of r) i th)
*)

(* substitute within the conclusion of goal i of gth, using a meta
equation rule. Note that we assume rule has var indicies zero'd *)
fun prep_concl_subst i gth = 
    let 
      val th = Thm.incr_indexes 1 gth;
      val tgt_term = Thm.prop_of th;

      val sgn = Thm.sign_of_thm th;
      val ctermify = Thm.cterm_of sgn;
      val trivify = Thm.trivial o ctermify;

      val (fixedbody, fvs) = IsaND.fix_alls_term i tgt_term;
      val cfvs = rev (map ctermify fvs);

      val conclterm = Logic.strip_imp_concl fixedbody;
      val conclthm = trivify conclterm;
      val maxidx = Term.maxidx_of_term conclterm;
      val ft = ((IsaFTerm.focus_right  
                 o IsaFTerm.focus_left
                 o IsaFTerm.fcterm_of_term 
                 o Thm.prop_of) conclthm)
    in
      ((cfvs, conclthm), (sgn, maxidx, ft))
    end;

(* substitute using an object or meta level equality *)
fun eqsubst_tac' searchf instepthm i th = 
    let 
      val (cvfsconclthm, searchinfo) = prep_concl_subst i th;
      val stepthms = 
          Seq.map Drule.zero_var_indexes 
                  (Seq.of_list (EqRuleData.prep_meta_eq instepthm));
      fun rewrite_with_thm r =
          let val (lhs,_) = Logic.dest_equals (Thm.concl_of r);
          in (searchf searchinfo lhs)
             :-> (apply_subst_in_concl i th cvfsconclthm r) end;
    in stepthms :-> rewrite_with_thm end;


(* Tactic.distinct_subgoals_tac -- fails to free type variables *)

(* custom version of distinct subgoals that works with term and 
   type variables. Asssumes th is in beta-eta normal form. 
   Also, does nothing if flexflex contraints are present. *)
fun distinct_subgoals th =
    if List.null (Thm.tpairs_of th) then
      let val (fixes,fixedthm) = IsaND.fix_vars_and_tvars th
        val (fixedthconcl,cprems) = IsaND.hide_prems fixedthm
      in
        IsaND.unfix_frees_and_tfrees 
          fixes
          (Drule.implies_intr_list 
             (Library.gen_distinct 
                (fn (x, y) => Thm.term_of x = Thm.term_of y)
                cprems) fixedthconcl)
      end
    else th;

(* General substiuttion of multiple occurances using one of 
   the given theorems*)
exception eqsubst_occL_exp of 
          string * (int list) * (Thm.thm list) * int * Thm.thm;
fun skip_first_occs_search occ srchf sinfo lhs = 
    case (IsaPLib.skipto_seqseq occ (srchf sinfo lhs)) of 
      IsaPLib.skipmore _ => Seq.empty
    | IsaPLib.skipseq ss => Seq.flat ss;

fun eqsubst_tac occL thms i th = 
    let val nprems = Thm.nprems_of th in
      if nprems < i then Seq.empty else
      let val thmseq = (Seq.of_list thms) 
        fun apply_occ occ th = 
            thmseq :-> 
                    (fn r => eqsubst_tac' (skip_first_occs_search 
                                    occ searchf_tlr_unify_valid) r
                                 (i + ((Thm.nprems_of th) - nprems))
                                 th);
        val sortedoccL = 
            Library.sort (Library.rev_order o Library.int_ord) occL;
      in
        Seq.map distinct_subgoals (Seq.EVERY (map apply_occ sortedoccL) th)
      end
    end
    handle THM _ => raise eqsubst_occL_exp ("THM",occL,thms,i,th);


(* inthms are the given arguments in Isar, and treated as eqstep with
   the first one, then the second etc *)
fun eqsubst_meth occL inthms =
    Method.METHOD 
      (fn facts =>
          HEADGOAL ( Method.insert_tac facts THEN' eqsubst_tac occL inthms ));

(* apply a substitution inside assumption j, keeps asm in the same place *)
fun apply_subst_in_asm i th rule ((cfvs, j, nprems, pth),m) = 
    let 
      val th2 = Thm.rotate_rule (j - 1) i th; (* put premice first *)
      val preelimrule = 
          (RWInst.rw m rule pth)
            |> (Seq.hd o Tactic.prune_params_tac)
            |> Thm.permute_prems 0 ~1 (* put old asm first *)
            |> IsaND.unfix_frees cfvs (* unfix any global params *)
            |> RWInst.beta_eta_contract; (* normal form *)
      val elimrule = 
          preelimrule
            |> Tactic.make_elim (* make into elim rule *)
            |> Thm.lift_rule (th2, i); (* lift into context *)
    in
      (* ~j because new asm starts at back, thus we subtract 1 *)
      Seq.map (Thm.rotate_rule (~j) (nprems + i))
              (Thm.bicompose 
                 false (* use unification *)
                 (true, (* elim resolution *)
                  elimrule, 2 + nprems) 
                 i th2)
    end;


(* prepare to substitute within the j'th premise of subgoal i of gth,
using a meta-level equation. Note that we assume rule has var indicies
zero'd. Note that we also assume that premt is the j'th premice of
subgoal i of gth. Note the repetition of work done for each
assumption, i.e. this can be made more efficient for search over
multiple assumptions.  *)
fun prep_subst_in_asm i gth j = 
    let 
      val th = Thm.incr_indexes 1 gth;
      val tgt_term = Thm.prop_of th;

      val sgn = Thm.sign_of_thm th;
      val ctermify = Thm.cterm_of sgn;
      val trivify = Thm.trivial o ctermify;

      val (fixedbody, fvs) = IsaND.fix_alls_term i tgt_term;
      val cfvs = rev (map ctermify fvs);

      val asmt = Library.nth_elem(j - 1,(Logic.strip_imp_prems fixedbody));
      val asm_nprems = length (Logic.strip_imp_prems asmt);

      val pth = trivify asmt;
      val maxidx = Term.maxidx_of_term asmt;

      val ft = ((IsaFTerm.focus_right 
                 o IsaFTerm.fcterm_of_term 
                 o Thm.prop_of) pth)
    in ((cfvs, j, asm_nprems, pth), (sgn, maxidx, ft)) end;

(* prepare subst in every possible assumption *)
fun prep_subst_in_asms i gth = 
    map (prep_subst_in_asm i gth)
        ((rev o IsaPLib.mk_num_list o length) 
           (Logic.prems_of_goal (Thm.prop_of gth) i));


(* substitute in an assumption using an object or meta level equality *)
fun eqsubst_asm_tac' searchf skipocc instepthm i th = 
    let 
      val asmpreps = prep_subst_in_asms i th;
      val stepthms = 
          Seq.map Drule.zero_var_indexes 
              (Seq.of_list (EqRuleData.prep_meta_eq instepthm))
      fun rewrite_with_thm r =
          let val (lhs,_) = Logic.dest_equals (Thm.concl_of r)
            fun occ_search occ [] = Seq.empty
              | occ_search occ ((asminfo, searchinfo)::moreasms) =
                (case searchf searchinfo occ lhs of 
                   IsaPLib.skipmore i => occ_search i moreasms
                 | IsaPLib.skipseq ss => 
                   Seq.append (Seq.map (Library.pair asminfo)
                                       (Seq.flat ss), 
                               occ_search 1 moreasms))
                              (* find later substs also *)
          in 
            (occ_search skipocc asmpreps) :-> (apply_subst_in_asm i th r)
          end;
    in stepthms :-> rewrite_with_thm end;


fun skip_first_asm_occs_search searchf sinfo occ lhs = 
    IsaPLib.skipto_seqseq occ (searchf sinfo lhs);

fun eqsubst_asm_tac occL thms i th = 
    let val nprems = Thm.nprems_of th 
    in
      if nprems < i then Seq.empty else
      let val thmseq = (Seq.of_list thms) 
        fun apply_occ occK th = 
            thmseq :-> 
                    (fn r => 
                        eqsubst_asm_tac' (skip_first_asm_occs_search 
                                            searchf_tlr_unify_valid) occK r
                                         (i + ((Thm.nprems_of th) - nprems))
                                         th);
        val sortedoccs = 
            Library.sort (Library.rev_order o Library.int_ord) occL
      in
        Seq.map distinct_subgoals
                (Seq.EVERY (map apply_occ sortedoccs) th)
      end
    end
    handle THM _ => raise eqsubst_occL_exp ("THM",occL,thms,i,th);

(* inthms are the given arguments in Isar, and treated as eqstep with
   the first one, then the second etc *)
fun eqsubst_asm_meth occL inthms =
    Method.METHOD 
      (fn facts =>
          HEADGOAL (Method.insert_tac facts THEN' eqsubst_asm_tac occL inthms ));

(* combination method that takes a flag (true indicates that subst
should be done to an assumption, false = apply to the conclusion of
the goal) as well as the theorems to use *)
fun meth ((asmflag, occL), inthms) ctxt = 
    if asmflag then eqsubst_asm_meth occL inthms else eqsubst_meth occL inthms;

(* syntax for options, given "(asm)" will give back true, without
   gives back false *)
val options_syntax =
    (Args.parens (Args.$$$ "asm") >> (K true)) ||
     (Scan.succeed false);

val ith_syntax =
    (Args.parens (Scan.repeat Args.nat))
      || (Scan.succeed [0]);

(* method syntax, first take options, then theorems *)
fun meth_syntax meth src ctxt =
    meth (snd (Method.syntax ((Scan.lift options_syntax) 
                                -- (Scan.lift ith_syntax) 
                                -- Attrib.local_thms) src ctxt)) 
         ctxt;

(* setup function for adding method to theory. *)
val setup = 
    [Method.add_method ("subst", meth_syntax meth, "Substiution with an equation. Use \"(asm)\" option to substitute in an assumption.")];

end;