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-(*  Title:      eqsubst.ML
-    Author:     Lucas Dixon, University of Edinburgh
-    Modified:   Joshua Chen, University of Innsbruck
-
-Perform a substitution using an equation.
-
-This code is slightly modified from the original at Tools/eqsubst..ML,
-to incorporate auto-typechecking for type theory.
-*)
-
-signature EQSUBST =
-sig
-  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 =
-    Proof.context
-    * int (* maxidx *)
-    * Zipper.T (* focusterm to search under *)
-
-  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 list -> int -> tactic
-  val eqsubst_asm_tac': Proof.context ->
-    (searchinfo -> int -> term -> match skipseq) -> int -> thm -> int -> tactic
-  val eqsubst_tac: Proof.context ->
-    int list -> (* list of occurrences to rewrite, use [0] for any *)
-    thm list -> int -> tactic
-  val eqsubst_tac': Proof.context ->
-    (searchinfo -> term -> match Seq.seq) (* search function *)
-    -> thm (* equation theorem to rewrite with *)
-    -> int (* subgoal number in goal theorem *)
-    -> thm (* goal theorem *)
-    -> thm Seq.seq (* rewritten goal theorem *)
-
-  (* 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 -> match Seq.seq Seq.seq
-  val searchf_lr_unify_valid: searchinfo -> term -> match Seq.seq Seq.seq
-  val searchf_bt_unify_valid: searchinfo -> term -> match Seq.seq Seq.seq
-end;
-
-structure EqSubst: EQSUBST =
-struct
-
-(* changes object "=" to meta "==" which prepares a given rewrite rule *)
-fun prep_meta_eq ctxt =
-  Simplifier.mksimps ctxt #> map Drule.zero_var_indexes;
-
-(* make free vars into schematic vars with index zero *)
-fun unfix_frees frees =
-   fold (K (Thm.forall_elim_var 0)) frees o Drule.forall_intr_list frees;
-
-
-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 =
-  Proof.context
-  * 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 mk_fake_bound_name n = ":b_" ^ n;
-fun fakefree_badbounds Ts t =
-  let val (FakeTs, Ts, newnames) =
-    fold_rev (fn (n, ty) => fn (FakeTs, Ts, usednames) =>
-      let
-        val newname = singleton (Name.variant_list usednames) n
-      in
-        ((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 RW_Inst.rw *)
-fun prep_zipper_match z =
-  let
-    val t = Zipper.trm z
-    val c = Zipper.ctxt z
-    val Ts = Zipper.C.nty_ctxt c
-    val (FakeTs', Ts', t') = fakefree_badbounds Ts t
-    val absterm = mk_foo_match (Zipper.C.apply c) Ts' t'
-  in
-    (t', (FakeTs', Ts', absterm))
-  end;
-
-(* Unification with exception handled *)
-(* given context, max var index, pat, tgt; returns Seq of instantiations *)
-fun clean_unify ctxt 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;
-    (* FIXME is it OK to ignore the type instantiation info?
-       or should I be using it? *)
-    val typs_unify =
-      SOME (Sign.typ_unify (Proof_Context.theory_of ctxt) (pat_ty, tgt_ty) (Vartab.empty, ix))
-        handle Type.TUNIFY => NONE;
-  in
-    (case typs_unify of
-      SOME (typinsttab, ix2) =>
-        let
-          (* FIXME 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),
-             Vartab.dest (Envir.term_env env));
-          val initenv =
-            Envir.Envir {maxidx = ix2, tenv = Vartab.empty, tyenv = typinsttab};
-          val useq = Unify.smash_unifiers (Context.Proof ctxt) [a] initenv
-            handle ListPair.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 ListPair.UnequalLengths => NONE
-              | Term.TERM _ => NONE;
-        in
-          (Seq.make (clean_unify' useq))
-        end
-    | NONE => Seq.empty)
-  end;
-
-(* 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_unify_z ctxt maxidx pat z =
-  let val (t, (FakeTs, Ts, absterm)) = prep_zipper_match z in
-    Seq.map (fn insts => (insts, FakeTs, Ts, absterm))
-      (clean_unify ctxt maxidx (t, pat))
-  end;
-
-
-fun bot_left_leaf_of (l $ _) = bot_left_leaf_of l
-  | bot_left_leaf_of (Abs (_, _, 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 (Zipper.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 [Zipper.Here z] else [] in
-        (case Zipper.trm z of
-          _ $ _ =>
-            [Zipper.LookIn (Zipper.move_down_left z)] @ here @
-            [Zipper.LookIn (Zipper.move_down_right z)]
-        | Abs _ => here @ [Zipper.LookIn (Zipper.move_down_abs z)]
-        | _ => here)
-      end;
-  in Zipper.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 [Zipper.Here z] else [] in
-        (case Zipper.trm z of
-          _ $ _ =>
-            [Zipper.LookIn (Zipper.move_down_left z),
-             Zipper.LookIn (Zipper.move_down_right z)] @ here
-        | Abs _ => [Zipper.LookIn (Zipper.move_down_abs z)] @ here
-        | _ => here)
-      end;
-  in Zipper.lzy_search sf_valid_td_lr end;
-
-fun searchf_unify_gen f (ctxt, maxidx, z) lhs =
-  Seq.map (clean_unify_z ctxt maxidx lhs) (Zipper.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 *)
-(* 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 ctxt i st (cfvs, conclthm) rule m =
-  RW_Inst.rw ctxt m rule conclthm
-  |> unfix_frees cfvs
-  |> Conv.fconv_rule Drule.beta_eta_conversion
-  |> (fn r => resolve_tac ctxt [r] i st);
-
-(* 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 ctxt i gth =
-  let
-    val th = Thm.incr_indexes 1 gth;
-    val tgt_term = Thm.prop_of th;
-
-    val (fixedbody, fvs) = IsaND.fix_alls_term ctxt i tgt_term;
-    val cfvs = rev (map (Thm.cterm_of ctxt) fvs);
-
-    val conclterm = Logic.strip_imp_concl fixedbody;
-    val conclthm = Thm.trivial (Thm.cterm_of ctxt conclterm);
-    val maxidx = Thm.maxidx_of th;
-    val ft =
-      (Zipper.move_down_right (* ==> *)
-       o Zipper.move_down_left (* Trueprop *)
-       o Zipper.mktop
-       o Thm.prop_of) conclthm
-  in
-    ((cfvs, conclthm), (ctxt, maxidx, ft))
-  end;
-
-(* substitute using an object or meta level equality *)
-fun eqsubst_tac' ctxt searchf instepthm i st =
-  let
-    val (cvfsconclthm, searchinfo) = prep_concl_subst ctxt i st;
-    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 ctxt i st cvfsconclthm r)
-      end;
-  in stepthms |> Seq.maps rewrite_with_thm end;
-
-
-(* General substitution of multiple occurrences using one of
-   the given theorems *)
-
-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 "occs" argument is a list of integers indicating which occurrence
-w.r.t. the search order, to rewrite. Backtracking will also find later
-occurrences, but all earlier ones are skipped. Thus you can use [0] to
-just find all rewrites. *)
-
-fun eqsubst_tac ctxt occs thms i st =
-  let val nprems = Thm.nprems_of st in
-    if nprems < i then Seq.empty else
-    let
-      val thmseq = Seq.of_list thms;
-      fun apply_occ occ st =
-        thmseq |> Seq.maps (fn r =>
-          eqsubst_tac' ctxt
-            (skip_first_occs_search occ searchf_lr_unify_valid) r
-            (i + (Thm.nprems_of st - nprems)) st);
-      val sorted_occs = Library.sort (rev_order o int_ord) occs;
-    in
-      Seq.maps distinct_subgoals_tac (Seq.EVERY (map apply_occ sorted_occs) st)
-    end
-  end;
-
-
-(* apply a substitution inside assumption j, keeps asm in the same place *)
-fun apply_subst_in_asm ctxt i st rule ((cfvs, j, _, pth),m) =
-  let
-    val st2 = Thm.rotate_rule (j - 1) i st; (* put premice first *)
-    val preelimrule =
-      RW_Inst.rw ctxt m rule pth
-      |> (Seq.hd o prune_params_tac ctxt)
-      |> Thm.permute_prems 0 ~1 (* put old asm first *)
-      |> unfix_frees cfvs (* unfix any global params *)
-      |> Conv.fconv_rule Drule.beta_eta_conversion; (* normal form *)
-  in
-    (* ~j because new asm starts at back, thus we subtract 1 *)
-    Seq.map (Thm.rotate_rule (~ j) (Thm.nprems_of rule + i))
-      (dresolve_tac ctxt [preelimrule] i st2)
-  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 ctxt i gth j =
-  let
-    val th = Thm.incr_indexes 1 gth;
-    val tgt_term = Thm.prop_of th;
-
-    val (fixedbody, fvs) = IsaND.fix_alls_term ctxt i tgt_term;
-    val cfvs = rev (map (Thm.cterm_of ctxt) fvs);
-
-    val asmt = nth (Logic.strip_imp_prems fixedbody) (j - 1);
-    val asm_nprems = length (Logic.strip_imp_prems asmt);
-
-    val pth = Thm.trivial ((Thm.cterm_of ctxt) asmt);
-    val maxidx = Thm.maxidx_of th;
-
-    val ft =
-      (Zipper.move_down_right (* trueprop *)
-         o Zipper.mktop
-         o Thm.prop_of) pth
-  in ((cfvs, j, asm_nprems, pth), (ctxt, maxidx, ft)) end;
-
-(* prepare subst in every possible assumption *)
-fun prep_subst_in_asms ctxt i gth =
-  map (prep_subst_in_asm ctxt 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 st =
-  let
-    val asmpreps = prep_subst_in_asms ctxt i st;
-    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 ctxt i st 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 occs thms i st =
-  let val nprems = Thm.nprems_of st in
-    if nprems < i then Seq.empty
-    else
-      let
-        val thmseq = Seq.of_list thms;
-        fun apply_occ occ st =
-          thmseq |> Seq.maps (fn r =>
-            eqsubst_asm_tac' ctxt
-              (skip_first_asm_occs_search searchf_lr_unify_valid) occ r
-              (i + (Thm.nprems_of st - nprems)) st);
-        val sorted_occs = Library.sort (rev_order o int_ord) occs;
-      in
-        Seq.maps distinct_subgoals_tac (Seq.EVERY (map apply_occ sorted_occs) st)
-      end
-  end;
-
-(* 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 *)
-val _ =
-  Theory.setup
-    (Method.setup \<^binding>\<open>sub\<close>
-      (Scan.lift (Args.mode "asm" -- Scan.optional (Args.parens (Scan.repeat Parse.nat)) [0]) --
-        Attrib.thms >> (fn ((asm, occs), inthms) => fn ctxt =>
-          SIMPLE_METHOD' ((if asm then eqsubst_asm_tac else eqsubst_tac) ctxt occs inthms)))
-      "single-step substitution"
-    #>
-    (Method.setup \<^binding>\<open>subst\<close>
-      (Scan.lift (Args.mode "asm" -- Scan.optional (Args.parens (Scan.repeat Parse.nat)) [0]) --
-        Attrib.thms >> (fn ((asm, occs), inthms) => fn ctxt =>
-          SIMPLE_METHOD' (SIDE_CONDS
-            ((if asm then eqsubst_asm_tac else eqsubst_tac) ctxt occs inthms)
-            ctxt)))
-      "single-step substitution with auto-typechecking"))
-
-end;