(*  Title:      Provers/eqsubst.ML

     Author:     Lucas Dixon, University of Edinburgh

 A proof method to perform a substiution using an equation.

 *)

 signature EQSUBST =

 sig

   (* a type abbreviation 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 =

        theory

        * int (* maxidx *)

        * Zipper.T (* focusterm to search under *)

     exception eqsubst_occL_exp of

        string * int list * Thm.thm list * int * Thm.thm

     (* low level substitution functions *)

     val apply_subst_in_asm :

        int ->

        Thm.thm ->

        Thm.thm ->

        (Thm.cterm list * int * 'a * Thm.thm) * match -> Thm.thm Seq.seq

     val apply_subst_in_concl :

        int ->

        Thm.thm ->

        Thm.cterm list * Thm.thm ->

        Thm.thm -> match -> Thm.thm Seq.seq

     (* matching/unification within zippers *)

     val clean_match_z :

        Context.theory -> Term.term -> Zipper.T -> match option

     val clean_unify_z :

        Context.theory -> int -> Term.term -> Zipper.T -> match Seq.seq

     (* skipping things in seq seq's *)

    (* skipping non-empty sub-sequences but when we reach the end

       of the seq, remembering how much we have left to skip. *)

     datatype 'a skipseq = SkipMore of int

       | SkipSeq of 'a Seq.seq Seq.seq;

     val skip_first_asm_occs_search :

        ('a -> 'b -> 'c Seq.seq Seq.seq) ->

        'a -> int -> 'b -> 'c skipseq

     val skip_first_occs_search :

        int -> ('a -> 'b -> 'c Seq.seq Seq.seq) -> 'a -> 'b -> 'c Seq.seq

     val skipto_skipseq : int -> 'a Seq.seq Seq.seq -> 'a skipseq

     (* tactics *)

     val eqsubst_asm_tac :

        Proof.context ->

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

     val eqsubst_asm_tac' :

        Proof.context ->

        (searchinfo -> int -> Term.term -> match skipseq) ->

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

     val eqsubst_tac :

        Proof.context ->

        int list -> (* list of occurences to rewrite, use [0] for any *)

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

     val eqsubst_tac' :

        Proof.context -> (* proof context *)

        (searchinfo -> Term.term -> match Seq.seq) (* search function *)

        -> Thm.thm (* equation theorem to rewrite with *)

        -> int (* subgoal number in goal theorem *)

        -> Thm.thm (* goal theorem *)

        -> Thm.thm Seq.seq (* rewritten goal theorem *)

     val fakefree_badbounds :

        (string * Term.typ) list ->

        Term.term ->

        (string * Term.typ) list * (string * Term.typ) list * Term.term

     val mk_foo_match :

        (Term.term -> Term.term) ->

        ('a * Term.typ) list -> Term.term -> Term.term

     (* preparing substitution *)

     val prep_meta_eq : Proof.context -> Thm.thm -> Thm.thm list

     val prep_concl_subst :

        int -> Thm.thm -> (Thm.cterm list * Thm.thm) * searchinfo

     val prep_subst_in_asm :

        int -> Thm.thm -> int ->

        (Thm.cterm list * int * int * Thm.thm) * searchinfo

     val prep_subst_in_asms :

        int -> Thm.thm ->

        ((Thm.cterm list * int * int * Thm.thm) * searchinfo) list

     val prep_zipper_match :

        Zipper.T -> Term.term * ((string * Term.typ) list * (string * Term.typ) list * Term.term)

     (* search for substitutions *)

     val valid_match_start : Zipper.T -> bool

     val search_lr_all : Zipper.T -> Zipper.T Seq.seq

     val search_lr_valid : (Zipper.T -> bool) -> Zipper.T -> Zipper.T Seq.seq

     val searchf_lr_unify_all :

        searchinfo -> Term.term -> match Seq.seq Seq.seq

     val searchf_lr_unify_valid :

        searchinfo -> Term.term -> match Seq.seq Seq.seq

     val searchf_bt_unify_valid :

        searchinfo -> Term.term -> match Seq.seq Seq.seq

     (* syntax tools *)

     val ith_syntax : Args.T list -> int list * Args.T list

     val options_syntax : Args.T list -> bool * Args.T list

     (* Isar level hooks *)

     val eqsubst_asm_meth : Proof.context -> int list -> Thm.thm list -> Proof.method

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

     val subst_meth : Method.src -> Proof.context -> Proof.method

     val setup : theory -> theory

 end;

 structure EqSubst

 : EQSUBST

 = struct

 structure Z = Zipper;

 (* changes object "=" to meta "==" which prepares a given rewrite rule *)

 fun prep_meta_eq ctxt =

   let val (_, {mk_rews = {mk, ...}, ...}) = Simplifier.rep_ss (Simplifier.local_simpset_of ctxt)

   in mk #> map Drule.zero_var_indexes end;

   (* 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 =

        theory

        * int (* maxidx *)

        * Zipper.T (* focusterm to search under *)

 (* skipping non-empty sub-sequences but when we reach the end

    of the seq, remembering how much we have left to skip. *)

 datatype 'a skipseq = SkipMore of int

   | SkipSeq of 'a Seq.seq Seq.seq;

 (* given a seqseq, skip the first m non-empty seq's, note deficit *)

 fun skipto_skipseq m s = 

     let 

       fun skip_occs n sq = 

           case Seq.pull sq of 

             NONE => SkipMore n

           | SOME (h,t) => 

             (case Seq.pull h of NONE => skip_occs n t

              | SOME _ => if n <= 1 then SkipSeq (Seq.cons h t)

                          else skip_occs (n - 1) t)

     in (skip_occs m s) end;

 (* note: outerterm is the taget with the match replaced by a bound 

          variable : ie: "P lhs" beocmes "%x. P x" 

          insts is the types of instantiations of vars in lhs

          and typinsts is the type instantiations of types in the lhs

          Note: Final rule is the rule lifted into the ontext of the 

          taget thm. *)

 fun mk_foo_match mkuptermfunc Ts t = 

     let 

       val ty = Term.type_of t

       val bigtype = (rev (map snd Ts)) ---> ty

       fun mk_foo 0 t = t

         | mk_foo i t = mk_foo (i - 1) (t $ (Bound (i - 1)))

       val num_of_bnds = (length Ts)

       (* foo_term = "fooabs y0 ... yn" where y's are local bounds *)

       val foo_term = mk_foo num_of_bnds (Bound num_of_bnds)

     in Abs("fooabs", bigtype, mkuptermfunc foo_term) end;

 (* T is outer bound vars, n is number of locally bound vars *)

 (* THINK: is order of Ts correct...? or reversed? *)

 fun fakefree_badbounds Ts t = 

     let val (FakeTs,Ts,newnames) = 

             List.foldr (fn ((n,ty),(FakeTs,Ts,usednames)) => 

                            let val newname = Name.variant usednames n

                            in ((RWTools.mk_fake_bound_name newname,ty)::FakeTs,

                                (newname,ty)::Ts, 

                                newname::usednames) end)

                        ([],[],[])

                        Ts

     in (FakeTs, Ts, Term.subst_bounds (map Free FakeTs, t)) end;

 (* before matching we need to fake the bound vars that are missing an

 abstraction. In this function we additionally construct the

 abstraction environment, and an outer context term (with the focus

 abstracted out) for use in rewriting with RWInst.rw *)

 fun prep_zipper_match z = 

     let 

       val t = Z.trm z  

       val c = Z.ctxt z

       val Ts = Z.C.nty_ctxt c

       val (FakeTs', Ts', t') = fakefree_badbounds Ts t

       val absterm = mk_foo_match (Z.C.apply c) Ts' t'

     in

       (t', (FakeTs', Ts', absterm))

     end;

 (* Matching and Unification with exception handled *)

 fun clean_match thy (a as (pat, t)) =

   let val (tyenv, tenv) = Pattern.match thy a (Vartab.empty, Vartab.empty)

   in SOME (Vartab.dest tyenv, Vartab.dest tenv)

   end handle Pattern.MATCH => NONE;

 (* given theory, max var index, pat, tgt; returns Seq of instantiations *)

 fun clean_unify thry ix (a as (pat, tgt)) =

     let

       (* type info will be re-derived, maybe this can be cached

          for efficiency? *)

       val pat_ty = Term.type_of pat;

       val tgt_ty = Term.type_of tgt;

       (* is it OK to ignore the type instantiation info?

          or should I be using it? *)

       val typs_unify =

           SOME (Sign.typ_unify thry (pat_ty, tgt_ty) (Vartab.empty, ix))

             handle Type.TUNIFY => NONE;

     in

       case typs_unify of

         SOME (typinsttab, ix2) =>

         let

       (* is it right to throw away the flexes?

          or should I be using them somehow? *)

           fun mk_insts env =

             (Vartab.dest (Envir.type_env env),

              Envir.alist_of env);

           val initenv = Envir.Envir {asol = Vartab.empty,

                                      iTs = typinsttab, maxidx = ix2};

           val useq = Unify.smash_unifiers thry [a] initenv

 	            handle UnequalLengths => Seq.empty

 		               | Term.TERM _ => Seq.empty;

           fun clean_unify' useq () =

               (case (Seq.pull useq) of

                  NONE => NONE

                | SOME (h,t) => SOME (mk_insts h, Seq.make (clean_unify' t)))

 	            handle UnequalLengths => NONE

                    | Term.TERM _ => NONE

         in

           (Seq.make (clean_unify' useq))

         end

       | NONE => Seq.empty

     end;

 (* Matching and Unification for zippers *)

 (* Note: Ts is a modified version of the original names of the outer

 bound variables. New names have been introduced to make sure they are

 unique w.r.t all names in the term and each other. usednames' is

 oldnames + new names. *)

 fun clean_match_z thy pat z = 

     let val (t, (FakeTs,Ts,absterm)) = prep_zipper_match z in

       case clean_match thy (pat, t) of 

         NONE => NONE 

       | SOME insts => SOME (insts, FakeTs, Ts, absterm) end;

 (* ix = max var index *)

 fun clean_unify_z sgn ix pat z = 

     let val (t, (FakeTs, Ts,absterm)) = prep_zipper_match z in

     Seq.map (fn insts => (insts, FakeTs, Ts, absterm)) 

             (clean_unify sgn ix (t, pat)) end;

 (* 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;

 *)

 fun bot_left_leaf_of (l $ r) = bot_left_leaf_of l

   | bot_left_leaf_of (Abs(s,ty,t)) = bot_left_leaf_of t

   | bot_left_leaf_of x = x;

 (* Avoid considering replacing terms which have a var at the head as

    they always succeed trivially, and uninterestingly. *)

 fun valid_match_start z =

     (case bot_left_leaf_of (Z.trm z) of 

       Var _ => false 

       | _ => true);

 (* search from top, left to right, then down *)

 val search_lr_all = ZipperSearch.all_bl_ur;

 (* search from top, left to right, then down *)

 fun search_lr_valid validf =

     let 

       fun sf_valid_td_lr z = 

           let val here = if validf z then [Z.Here z] else [] in

             case Z.trm z 

              of _ $ _ => [Z.LookIn (Z.move_down_left z)] 

                          @ here 

                          @ [Z.LookIn (Z.move_down_right z)]

               | Abs _ => here @ [Z.LookIn (Z.move_down_abs z)]

               | _ => here

           end;

     in Z.lzy_search sf_valid_td_lr end;

 (* search from bottom to top, left to right *)

 fun search_bt_valid validf =

     let 

       fun sf_valid_td_lr z = 

           let val here = if validf z then [Z.Here z] else [] in

             case Z.trm z 

              of _ $ _ => [Z.LookIn (Z.move_down_left z), 

                           Z.LookIn (Z.move_down_right z)] @ here

               | Abs _ => [Z.LookIn (Z.move_down_abs z)] @ here

               | _ => here

           end;

     in Z.lzy_search sf_valid_td_lr end;

 fun searchf_unify_gen f (sgn, maxidx, z) lhs =

     Seq.map (clean_unify_z sgn maxidx lhs) 

             (Z.limit_apply f z);

 (* search all unifications *)

 val searchf_lr_unify_all =

     searchf_unify_gen search_lr_all;

 (* search only for 'valid' unifiers (non abs subterms and non vars) *)

 val searchf_lr_unify_valid = 

     searchf_unify_gen (search_lr_valid valid_match_start);

 val searchf_bt_unify_valid =

     searchf_unify_gen (search_bt_valid valid_match_start);

 (* 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.theory_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 = Thm.maxidx_of th;

       val ft = ((Z.move_down_right (* ==> *)

                  o Z.move_down_left (* Trueprop *)

                  o Z.mktop

                  o Thm.prop_of) conclthm)

     in

       ((cfvs, conclthm), (sgn, maxidx, ft))

     end;

 (* substitute using an object or meta level equality *)

 fun eqsubst_tac' ctxt searchf instepthm i th =

     let

       val (cvfsconclthm, searchinfo) = prep_concl_subst i th;

       val stepthms = Seq.of_list (prep_meta_eq ctxt instepthm);

       fun rewrite_with_thm r =

           let val (lhs,_) = Logic.dest_equals (Thm.concl_of r);

           in searchf searchinfo lhs

              |> Seq.maps (apply_subst_in_concl i th cvfsconclthm r) end;

     in stepthms |> Seq.maps rewrite_with_thm end;

 (* distinct subgoals *)

 fun distinct_subgoals th =

   the_default th (SINGLE distinct_subgoals_tac th);

 (* General substitution 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 (skipto_skipseq occ (srchf sinfo lhs)) of

       SkipMore _ => Seq.empty

     | SkipSeq ss => Seq.flat ss;

 (* The occL is a list of integers indicating which occurence

 w.r.t. the search order, to rewrite. Backtracking will also find later

 occurences, but all earlier ones are skipped. Thus you can use [0] to

 just find all rewrites. *)

 fun eqsubst_tac ctxt 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 |> Seq.maps

                     (fn r => eqsubst_tac' 

                                ctxt 

                                (skip_first_occs_search

                                   occ searchf_lr_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 ctxt occL inthms =

     Method.SIMPLE_METHOD' (eqsubst_tac ctxt 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 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.theory_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 = nth (Logic.strip_imp_prems fixedbody) (j - 1);

       val asm_nprems = length (Logic.strip_imp_prems asmt);

       val pth = trivify asmt;

       val maxidx = Thm.maxidx_of th;

       val ft = ((Z.move_down_right (* trueprop *)

                  o Z.mktop

                  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)

         ((fn l => Library.upto (1, length l))

            (Logic.prems_of_goal (Thm.prop_of gth) i));

 (* substitute in an assumption using an object or meta level equality *)

 fun eqsubst_asm_tac' ctxt searchf skipocc instepthm i th =

     let

       val asmpreps = prep_subst_in_asms i th;

       val stepthms = Seq.of_list (prep_meta_eq ctxt 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

                    SkipMore i => occ_search i moreasms

                  | 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 |> Seq.maps (apply_subst_in_asm i th r)

           end;

     in stepthms |> Seq.maps rewrite_with_thm end;

 fun skip_first_asm_occs_search searchf sinfo occ lhs =

     skipto_skipseq occ (searchf sinfo lhs);

 fun eqsubst_asm_tac ctxt 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 |> Seq.maps

                     (fn r =>

                         eqsubst_asm_tac' ctxt (skip_first_asm_occs_search

                                             searchf_lr_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 ctxt occL inthms =

     Method.SIMPLE_METHOD' (eqsubst_asm_tac ctxt 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 =

     Scan.optional (Args.parens (Scan.repeat OuterParse.nat)) [0];

 (* 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 subst_meth src =

   Method.syntax ((Scan.lift options_syntax) -- (Scan.lift ith_syntax) -- Attrib.thms) src

   #> (fn (((asmflag, occL), inthms), ctxt) =>

     (if asmflag then eqsubst_asm_meth else eqsubst_meth) ctxt occL inthms);

 val setup =

   Method.add_method ("subst", subst_meth, "single-step substitution");

 end;