src/Provers/hypsubst.ML
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(*  Title:      Provers/hypsubst.ML
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    Authors:    Martin D Coen, Tobias Nipkow and Lawrence C Paulson
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    Copyright   1995  University of Cambridge
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Basic equational reasoning: hyp_subst_tac and methods "hypsubst", "subst".
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Tactic to substitute using (at least) the assumption x=t in the rest
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of the subgoal, and to delete (at least) that assumption.  Original
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version due to Martin Coen.
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This version uses the simplifier, and requires it to be already present.
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Test data:
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Goal "!!x.[| Q(x,y,z); y=x; a=x; z=y; P(y) |] ==> P(z)";
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Goal "!!x.[| Q(x,y,z); z=f(x); x=z |] ==> P(z)";
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Goal "!!y. [| ?x=y; P(?x) |] ==> y = a";
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Goal "!!z. [| ?x=y; P(?x) |] ==> y = a";
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Goal "!!x a. [| x = f(b); g(a) = b |] ==> P(x)";
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by (bound_hyp_subst_tac 1);
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by (hyp_subst_tac 1);
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Here hyp_subst_tac goes wrong; harder still to prove P(f(f(a))) & P(f(a))
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Goal "P(a) --> (EX y. a=y --> P(f(a)))";
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Goal "!!x. [| Q(x,h1); P(a,h2); R(x,y,h3); R(y,z,h4); x=f(y); \
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\                 P(x,h5); P(y,h6); K(x,h7) |] ==> Q(x,c)";
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by (blast_hyp_subst_tac true 1);
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*)
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signature HYPSUBST_DATA =
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sig
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  val dest_Trueprop    : term -> term
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  val dest_eq          : term -> term * term
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  val dest_imp         : term -> term * term
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  val eq_reflection    : thm               (* a=b ==> a==b *)
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  val rev_eq_reflection: thm               (* a==b ==> a=b *)
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  val imp_intr         : thm               (* (P ==> Q) ==> P-->Q *)
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  val rev_mp           : thm               (* [| P;  P-->Q |] ==> Q *)
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  val subst            : thm               (* [| a=b;  P(a) |] ==> P(b) *)
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  val sym              : thm               (* a=b ==> b=a *)
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  val thin_refl        : thm               (* [|x=x; PROP W|] ==> PROP W *)
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end;
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signature HYPSUBST =
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sig
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  val bound_hyp_subst_tac    : int -> tactic
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  val hyp_subst_tac          : int -> tactic
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  val blast_hyp_subst_tac    : bool -> int -> tactic
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  val stac                   : thm -> int -> tactic
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  val hypsubst_setup         : theory -> theory
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end;
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functor Hypsubst(Data: HYPSUBST_DATA): HYPSUBST =
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struct
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exception EQ_VAR;
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(*Simplifier turns Bound variables to special Free variables:
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  change it back (any Bound variable will do)*)
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fun contract t =
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  (case Envir.eta_contract t of
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    Free (a, T) => if Name.is_bound a then Bound 0 else Free (a, T)
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  | t' => t');
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val has_vars = Term.exists_subterm Term.is_Var;
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val has_tvars = Term.exists_type (Term.exists_subtype Term.is_TVar);
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(*If novars then we forbid Vars in the equality.
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  If bnd then we only look for Bound variables to eliminate.
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  When can we safely delete the equality?
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    Not if it equates two constants; consider 0=1.
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    Not if it resembles x=t[x], since substitution does not eliminate x.
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    Not if it resembles ?x=0; consider ?x=0 ==> ?x=1 or even ?x=0 ==> P
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    Not if it involves a variable free in the premises,
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        but we can't check for this -- hence bnd and bound_hyp_subst_tac
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  Prefer to eliminate Bound variables if possible.
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  Result:  true = use as is,  false = reorient first *)
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fun inspect_pair bnd novars (t, u) =
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  if novars andalso (has_tvars t orelse has_tvars u)
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  then raise Match   (*variables in the type!*)
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  else
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    (case (contract t, contract u) of
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      (Bound i, _) =>
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        if loose_bvar1 (u, i) orelse novars andalso has_vars u
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        then raise Match
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        else true                (*eliminates t*)
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    | (_, Bound i) =>
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        if loose_bvar1 (t, i) orelse novars andalso has_vars t
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        then raise Match
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        else false               (*eliminates u*)
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    | (t' as Free _, _) =>
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        if bnd orelse Logic.occs (t', u) orelse novars andalso has_vars u
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        then raise Match
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        else true                (*eliminates t*)
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    | (_, u' as Free _) =>
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        if bnd orelse Logic.occs (u', t) orelse novars andalso has_vars t
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        then raise Match
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        else false               (*eliminates u*)
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    | _ => raise Match);
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(*Locates a substitutable variable on the left (resp. right) of an equality
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   assumption.  Returns the number of intervening assumptions. *)
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fun eq_var bnd novars =
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  let fun eq_var_aux k (Const("all",_) $ Abs(_,_,t)) = eq_var_aux k t
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        | eq_var_aux k (Const("==>",_) $ A $ B) =
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              ((k, inspect_pair bnd novars
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                    (Data.dest_eq (Data.dest_Trueprop A)))
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               handle TERM _ => eq_var_aux (k+1) B
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                 | Match => eq_var_aux (k+1) B)
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        | eq_var_aux k _ = raise EQ_VAR
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  in  eq_var_aux 0  end;
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(*For the simpset.  Adds ALL suitable equalities, even if not first!
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  No vars are allowed here, as simpsets are built from meta-assumptions*)
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fun mk_eqs bnd th =
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    [ if inspect_pair bnd false (Data.dest_eq (Data.dest_Trueprop (Thm.prop_of th)))
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      then th RS Data.eq_reflection
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      else Thm.symmetric(th RS Data.eq_reflection) (*reorient*) ]
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    handle TERM _ => [] | Match => [];
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local
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in
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  (*Select a suitable equality assumption; substitute throughout the subgoal
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    If bnd is true, then it replaces Bound variables only. *)
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  fun gen_hyp_subst_tac bnd =
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    let fun tac i st = SUBGOAL (fn (Bi, _) =>
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      let
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        val (k, _) = eq_var bnd true Bi
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        val hyp_subst_ss = Simplifier.global_context (Thm.theory_of_thm st) empty_ss
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          |> Simplifier.set_mksimps (K (mk_eqs bnd))
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      in EVERY [rotate_tac k i, asm_lr_simp_tac hyp_subst_ss i,
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        etac thin_rl i, rotate_tac (~k) i]
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      end handle THM _ => no_tac | EQ_VAR => no_tac) i st
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    in REPEAT_DETERM1 o tac end;
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end;
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val ssubst = Drule.export_without_context (Data.sym RS Data.subst);
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fun inst_subst_tac b rl = CSUBGOAL (fn (cBi, i) =>
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  case try (Logic.strip_assums_hyp #> hd #>
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      Data.dest_Trueprop #> Data.dest_eq #> pairself contract) (Thm.term_of cBi) of
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    SOME (t, t') =>
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      let
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        val Bi = Thm.term_of cBi;
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        val ps = Logic.strip_params Bi;
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        val U = Term.fastype_of1 (rev (map snd ps), t);
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        val Q = Data.dest_Trueprop (Logic.strip_assums_concl Bi);
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        val rl' = Thm.lift_rule cBi rl;
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        val Var (ixn, T) = Term.head_of (Data.dest_Trueprop
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          (Logic.strip_assums_concl (Thm.prop_of rl')));
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        val (v1, v2) = Data.dest_eq (Data.dest_Trueprop
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          (Logic.strip_assums_concl (hd (Thm.prems_of rl'))));
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        val (Ts, V) = split_last (Term.binder_types T);
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        val u = list_abs (ps @ [("x", U)], case (if b then t else t') of
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            Bound j => subst_bounds (map Bound
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              ((1 upto j) @ 0 :: (j + 2 upto length ps)), Q)
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          | t => Term.abstract_over (t, Term.incr_boundvars 1 Q));
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        val thy = Thm.theory_of_thm rl';
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        val (instT, _) = Thm.match (pairself (cterm_of thy o Logic.mk_type) (V, U));
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      in compose_tac (true, Drule.instantiate_normalize (instT,
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        map (pairself (cterm_of thy))
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          [(Var (ixn, Ts ---> U --> body_type T), u),
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           (Var (fst (dest_Var (head_of v1)), Ts ---> U), list_abs (ps, t)),
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           (Var (fst (dest_Var (head_of v2)), Ts ---> U), list_abs (ps, t'))]) rl',
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        nprems_of rl) i
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      end
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  | NONE => no_tac);
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val imp_intr_tac = rtac Data.imp_intr;
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(* FIXME: "etac Data.rev_mp i" will not behave as expected if goal has *)
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(* premises containing meta-implications or quantifiers                *)
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(*Old version of the tactic above -- slower but the only way
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  to handle equalities containing Vars.*)
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fun vars_gen_hyp_subst_tac bnd = SUBGOAL(fn (Bi,i) =>
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      let val n = length(Logic.strip_assums_hyp Bi) - 1
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          val (k,symopt) = eq_var bnd false Bi
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      in
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         DETERM
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           (EVERY [REPEAT_DETERM_N k (etac Data.rev_mp i),
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                   rotate_tac 1 i,
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                   REPEAT_DETERM_N (n-k) (etac Data.rev_mp i),
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                   inst_subst_tac symopt (if symopt then ssubst else Data.subst) i,
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                   REPEAT_DETERM_N n (imp_intr_tac i THEN rotate_tac ~1 i)])
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      end
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      handle THM _ => no_tac | EQ_VAR => no_tac);
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(*Substitutes for Free or Bound variables*)
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val hyp_subst_tac = FIRST' [ematch_tac [Data.thin_refl],
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        gen_hyp_subst_tac false, vars_gen_hyp_subst_tac false];
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(*Substitutes for Bound variables only -- this is always safe*)
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val bound_hyp_subst_tac =
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    gen_hyp_subst_tac true ORELSE' vars_gen_hyp_subst_tac true;
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(** Version for Blast_tac.  Hyps that are affected by the substitution are
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    moved to the front.  Defect: even trivial changes are noticed, such as
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    substitutions in the arguments of a function Var. **)
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(*final re-reversal of the changed assumptions*)
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fun reverse_n_tac 0 i = all_tac
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  | reverse_n_tac 1 i = rotate_tac ~1 i
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  | reverse_n_tac n i =
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      REPEAT_DETERM_N n (rotate_tac ~1 i THEN etac Data.rev_mp i) THEN
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      REPEAT_DETERM_N n (imp_intr_tac i THEN rotate_tac ~1 i);
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(*Use imp_intr, comparing the old hyps with the new ones as they come out.*)
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fun all_imp_intr_tac hyps i =
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  let
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    fun imptac (r, []) st = reverse_n_tac r i st
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      | imptac (r, hyp::hyps) st =
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          let
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            val (hyp', _) =
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              term_of (Thm.cprem_of st i)
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              |> Logic.strip_assums_concl
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              |> Data.dest_Trueprop |> Data.dest_imp;
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            val (r', tac) =
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              if Pattern.aeconv (hyp, hyp')
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              then (r, imp_intr_tac i THEN rotate_tac ~1 i)
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              else (*leave affected hyps at end*) (r + 1, imp_intr_tac i);
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          in
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            (case Seq.pull (tac st) of
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              NONE => Seq.single st
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            | SOME (st', _) => imptac (r', hyps) st')
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          end
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  in imptac (0, rev hyps) end;
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fun blast_hyp_subst_tac trace = SUBGOAL(fn (Bi,i) =>
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      let val (k,symopt) = eq_var false false Bi
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          val hyps0 = map Data.dest_Trueprop (Logic.strip_assums_hyp Bi)
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          (*omit selected equality, returning other hyps*)
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          val hyps = List.take(hyps0, k) @ List.drop(hyps0, k+1)
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          val n = length hyps
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      in
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         if trace then tracing "Substituting an equality" else ();
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         DETERM
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           (EVERY [REPEAT_DETERM_N k (etac Data.rev_mp i),
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                   rotate_tac 1 i,
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                   REPEAT_DETERM_N (n-k) (etac Data.rev_mp i),
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                   inst_subst_tac symopt (if symopt then ssubst else Data.subst) i,
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                   all_imp_intr_tac hyps i])
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      end
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      handle THM _ => no_tac | EQ_VAR => no_tac);
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(*apply an equality or definition ONCE;
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  fails unless the substitution has an effect*)
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fun stac th =
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  let val th' = th RS Data.rev_eq_reflection handle THM _ => th
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  in CHANGED_GOAL (rtac (th' RS ssubst)) end;
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(* theory setup *)
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val hypsubst_setup =
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  Method.setup @{binding hypsubst}
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    (Scan.succeed (K (SIMPLE_METHOD' (CHANGED_PROP o hyp_subst_tac))))
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    "substitution using an assumption (improper)" #>
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  Method.setup @{binding simplesubst} (Attrib.thm >> (fn th => K (SIMPLE_METHOD' (stac th))))
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    "simple substitution";
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end;