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

(*  Title:      Provers/eqsubst.ML

    Author:     Lucas Dixon, University of Edinburgh

                lucas.dixon@ed.ac.uk

    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;