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

 (*  Title:      Provers/eqsubst.ML

     ID:         $Id$

     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 list) * int * thm;

   type match =

        ((indexname * (sort * typ)) list (* type instantiations *)

         * (indexname * (typ * term)) list) (* term instantiations *)

        * (string * typ) list (* fake named type abs env *)

        * (string * typ) list (* type abs env *)

        * term (* outer term *)

   type searchinfo =

        theory (* sign for matching *)

        * int (* maxidx *)

        * BasicIsaFTerm.FcTerm (* focusterm to search under *)

   val prep_subst_in_asm :

          int (* subgoal to subst in *)

       -> thm (* target theorem with subgoals *)

       -> int (* premise to subst in *)

       -> (cterm list (* certified free var placeholders for vars *)

           * int (* premice no. to subst *)

           * int (* number of assumptions of premice *)

           * 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 (* target theorem with subgoals *)

       -> ((cterm list (* certified free var placeholders for vars *)

           * int (* premice no. to subst *)

           * int (* number of assumptions of premice *)

           * thm) (* premice as a new theorem for forward reasoning *)

          * searchinfo) list

   val apply_subst_in_asm :

       int (* subgoal *)

       -> thm (* overall theorem *)

       -> thm (* rule *)

       -> (cterm list (* certified free var placeholders for vars *)

           * int (* assump no being subst *)

           * int (* num of premises of asm *)

           * thm) (* premthm *)

       * match

       -> thm Seq.seq

   val prep_concl_subst :

          int (* subgoal *)

       -> thm (* overall goal theorem *)

       -> (cterm list * thm) * searchinfo (* (cvfs, conclthm), matchf *)

   val apply_subst_in_concl :

         int (* subgoal *)

         -> thm (* thm with all goals *)

         -> cterm list (* certified free var placeholders for vars *)

            * thm  (* trivial thm of goal concl *)

             (* possible matches/unifiers *)

         -> thm (* rule *)

         -> match

         -> thm Seq.seq (* substituted goal *)

   (* basic notion of search *)

   val searchf_tlr_unify_all :

       (searchinfo

        -> term (* lhs *)

        -> match Seq.seq Seq.seq)

   val searchf_tlr_unify_valid :

       (searchinfo

        -> term (* lhs *)

        -> match Seq.seq Seq.seq)

   (* specialise search constructor for conclusion skipping occurrences. *)

      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 list -> Proof.method

   val eqsubst_asm_tac :

       int list -> thm list -> int -> thm -> thm Seq.seq

   val eqsubst_asm_tac' :

       (* search function with skips *)

       (searchinfo -> int -> term -> match IsaPLib.skipseqT)

       -> int (* skip to *)

       -> thm (* rule *)

       -> int (* subgoal number *)

       -> thm (* tgt theorem with subgoals *)

       -> thm Seq.seq (* new theorems *)

   val eqsubst_meth : int list -> thm list -> Proof.method

   val eqsubst_tac :

       int list -> thm list -> int -> thm -> thm Seq.seq

   val eqsubst_tac' :

       (searchinfo -> term -> match Seq.seq)

       -> thm -> int -> thm -> thm Seq.seq

   val meth : (bool * int list) * 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 =

        ((indexname * (sort * typ)) list (* type instantiations *)

         * (indexname * (typ * term)) list) (* term instantiations *)

        * (string * typ) list (* fake named type abs env *)

        * (string * typ) list (* type abs env *)

        * 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 with all goals *)

         * (Thm.cterm list (* certified free var placeholders for vars *)

            * thm)  (* trivial thm of goal concl *)

             (* possible matches/unifiers *)

         * thm (* rule *)

         * (((indexname * typ) list (* type instantiations *)

               * (indexname * term) list ) (* term instantiations *)

              * (string * typ) list (* Type abs env *)

              * 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);

 (* 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 list) * int * 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, ngoalprems, 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) ((Thm.nprems_of rule) + i))

       (Tactic.dtac preelimrule i th2)

       (* (Thm.bicompose

                  false (* use unification *)

                  (true, (* elim resolution *)

                   elimrule, (2 + (Thm.nprems_of rule)) - ngoalprems)

                  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;