src/Provers/quantifier1.ML

tuned message;

(*  Title:      Provers/quantifier1.ML
    Author:     Tobias Nipkow
    Copyright   1997  TU Munich

Simplification procedures for turning

            ? x. ... & x = t & ...
     into   ? x. x = t & ... & ...
     where the `? x. x = t &' in the latter formula must be eliminated
           by ordinary simplification.

     and   ! x. (... & x = t & ...) --> P x
     into  ! x. x = t --> (... & ...) --> P x
     where the `!x. x=t -->' in the latter formula is eliminated
           by ordinary simplification.

     And analogously for t=x, but the eqn is not turned around!

     NB Simproc is only triggered by "!x. P(x) & P'(x) --> Q(x)";
        "!x. x=t --> P(x)" is covered by the congruence rule for -->;
        "!x. t=x --> P(x)" must be taken care of by an ordinary rewrite rule.
        As must be "? x. t=x & P(x)".

     And similarly for the bounded quantifiers.

Gries etc call this the "1 point rules"

The above also works for !x1..xn. and ?x1..xn by moving the defined
quantifier inside first, but not for nested bounded quantifiers.

For set comprehensions the basic permutations
      ... & x = t & ...  ->  x = t & (... & ...)
      ... & t = x & ...  ->  t = x & (... & ...)
are also exported.

To avoid looping, NONE is returned if the term cannot be rearranged,
esp if x=t/t=x sits at the front already.
*)

signature QUANTIFIER1_DATA =
sig
  (*abstract syntax*)
  val dest_eq: term -> (term * term) option
  val dest_conj: term -> (term * term) option
  val dest_imp: term -> (term * term) option
  val conj: term
  val imp: term
  (*rules*)
  val iff_reflection: thm (* P <-> Q ==> P == Q *)
  val iffI: thm
  val iff_trans: thm
  val conjI: thm
  val conjE: thm
  val impI: thm
  val mp: thm
  val exI: thm
  val exE: thm
  val uncurry: thm (* P --> Q --> R ==> P & Q --> R *)
  val iff_allI: thm (* !!x. P x <-> Q x ==> (!x. P x) = (!x. Q x) *)
  val iff_exI: thm (* !!x. P x <-> Q x ==> (? x. P x) = (? x. Q x) *)
  val all_comm: thm (* (!x y. P x y) = (!y x. P x y) *)
  val ex_comm: thm (* (? x y. P x y) = (? y x. P x y) *)
  val atomize: Proof.context -> conv
end;

signature QUANTIFIER1 =
sig
  val rearrange_all: Proof.context -> cterm -> thm option
  val rearrange_All: Proof.context -> cterm -> thm option
  val rearrange_Ball: (Proof.context -> tactic) -> Proof.context -> cterm -> thm option
  val rearrange_Ex: Proof.context -> cterm -> thm option
  val rearrange_Bex: (Proof.context -> tactic) -> Proof.context -> cterm -> thm option
  val rearrange_Collect: (Proof.context -> tactic) -> Proof.context -> cterm -> thm option
end;

functor Quantifier1(Data: QUANTIFIER1_DATA): QUANTIFIER1 =
struct

(* FIXME: only test! *)
fun def xs eq =
  (case Data.dest_eq eq of
    SOME (s, t) =>
      let val n = length xs in
        s = Bound n andalso not (loose_bvar1 (t, n)) orelse
        t = Bound n andalso not (loose_bvar1 (s, n))
      end
  | NONE => false);

fun extract_conj fst xs t =
  (case Data.dest_conj t of
    NONE => NONE
  | SOME (P, Q) =>
      if def xs P then (if fst then NONE else SOME (xs, P, Q))
      else if def xs Q then SOME (xs, Q, P)
      else
        (case extract_conj false xs P of
          SOME (xs, eq, P') => SOME (xs, eq, Data.conj $ P' $ Q)
        | NONE =>
            (case extract_conj false xs Q of
              SOME (xs, eq, Q') => SOME (xs, eq, Data.conj $ P $ Q')
            | NONE => NONE)));

fun extract_imp fst xs t =
  (case Data.dest_imp t of
    NONE => NONE
  | SOME (P, Q) =>
      if def xs P then (if fst then NONE else SOME (xs, P, Q))
      else
        (case extract_conj false xs P of
          SOME (xs, eq, P') => SOME (xs, eq, Data.imp $ P' $ Q)
        | NONE =>
            (case extract_imp false xs Q of
              NONE => NONE
            | SOME (xs, eq, Q') => SOME (xs, eq, Data.imp $ P $ Q'))));

fun extract_conj_from_judgment ctxt fst xs t =
  if Object_Logic.is_judgment ctxt t
  then
    let
      val judg $ t' = t
    in
      case extract_conj fst xs t' of
          NONE => NONE
        | SOME (xs, eq, P) => SOME (xs, judg $ eq, judg $ P)
    end
  else NONE;

fun extract_implies ctxt fst xs t =
  (case try Logic.dest_implies t of
    NONE => NONE
  | SOME (P, Q) =>
      if def xs P then (if fst then NONE else SOME (xs, P, Q))
      else
        (case extract_conj_from_judgment ctxt false xs P of
          SOME (xs, eq, P') => SOME (xs, eq, Logic.implies $ P' $ Q)
        | NONE =>
            (case extract_implies ctxt false xs Q of
              NONE => NONE
            | SOME (xs, eq, Q') => (SOME (xs, eq, Logic.implies $ P $ Q')))));

fun extract_quant ctxt extract q =
  let
    fun exqu xs ((qC as Const (qa, _)) $ Abs (x, T, Q)) =
          if qa = q then exqu ((qC, x, T) :: xs) Q else NONE
      | exqu xs P = extract ctxt (null xs) xs P
  in exqu [] end;

fun iff_reflection_tac ctxt =
  resolve_tac ctxt [Data.iff_reflection] 1;

fun qcomm_tac ctxt qcomm qI i =
  REPEAT_DETERM (resolve_tac ctxt [qcomm] i THEN resolve_tac ctxt [qI] i);

(* Proves (? x0..xn. ... & x0 = t & ...) = (? x1..xn x0. x0 = t & ... & ...)
   Better: instantiate exI
*)
local
  val excomm = Data.ex_comm RS Data.iff_trans;
in
  fun prove_one_point_Ex_tac ctxt =
    qcomm_tac ctxt excomm Data.iff_exI 1 THEN resolve_tac ctxt [Data.iffI] 1 THEN
    ALLGOALS
      (EVERY' [eresolve_tac ctxt [Data.exE], REPEAT_DETERM o eresolve_tac ctxt [Data.conjE],
        resolve_tac ctxt [Data.exI],
        DEPTH_SOLVE_1 o ares_tac ctxt [Data.conjI]])
end;

(* Proves (! x0..xn. (... & x0 = t & ...) --> P x0) =
          (! x1..xn x0. x0 = t --> (... & ...) --> P x0)
*)
local
  fun tac ctxt =
    SELECT_GOAL
      (EVERY1 [REPEAT o dresolve_tac ctxt [Data.uncurry],
        REPEAT o resolve_tac ctxt [Data.impI],
        eresolve_tac ctxt [Data.mp],
        REPEAT o eresolve_tac ctxt [Data.conjE],
        REPEAT o ares_tac ctxt [Data.conjI]]);
  val all_comm = Data.all_comm RS Data.iff_trans;
  val All_comm = @{thm swap_params [THEN transitive]};
in
  fun prove_one_point_All_tac ctxt =
    EVERY1 [qcomm_tac ctxt all_comm Data.iff_allI,
      resolve_tac ctxt [Data.iff_allI],
      resolve_tac ctxt [Data.iffI],
      tac ctxt,
      tac ctxt];
  fun prove_one_point_all_tac ctxt =
    EVERY1 [qcomm_tac ctxt All_comm @{thm equal_allI},
      resolve_tac ctxt [@{thm equal_allI}],
      CONVERSION (Data.atomize ctxt),
      resolve_tac ctxt [@{thm equal_intr_rule}],
      tac ctxt,
      tac ctxt];
end

fun prove_conv ctxt tu tac =
  let
    val (goal, ctxt') =
      yield_singleton (Variable.import_terms true) (Logic.mk_equals tu) ctxt;
    val thm =
      Goal.prove ctxt' [] [] goal
        (fn {context = ctxt'', ...} => tac ctxt'');
  in singleton (Variable.export ctxt' ctxt) thm end;

fun renumber l u (Bound i) =
      Bound (if i < l orelse i > u then i else if i = u then l else i + 1)
  | renumber l u (s $ t) = renumber l u s $ renumber l u t
  | renumber l u (Abs (x, T, t)) = Abs (x, T, renumber (l + 1) (u + 1) t)
  | renumber _ _ atom = atom;

fun quantify qC x T xs P =
  let
    fun quant [] P = P
      | quant ((qC, x, T) :: xs) P = quant xs (qC $ Abs (x, T, P));
    val n = length xs;
    val Q = if n = 0 then P else renumber 0 n P;
  in quant xs (qC $ Abs (x, T, Q)) end;

fun rearrange_all ctxt ct =
  (case Thm.term_of ct of
    F as (all as Const (q, _)) $ Abs (x, T, P) =>
      (case extract_quant ctxt extract_implies q P of
        NONE => NONE
      | SOME (xs, eq, Q) =>
          let val R = quantify all x T xs (Logic.implies $ eq $ Q)
            in SOME (prove_conv ctxt (F, R) prove_one_point_all_tac) end)
  | _ => NONE);

fun rearrange_All ctxt ct =
  (case Thm.term_of ct of
    F as (all as Const (q, _)) $ Abs (x, T, P) =>
      (case extract_quant ctxt (K extract_imp) q P of
        NONE => NONE
      | SOME (xs, eq, Q) =>
          let val R = quantify all x T xs (Data.imp $ eq $ Q)
          in SOME (prove_conv ctxt (F, R) (iff_reflection_tac THEN' prove_one_point_All_tac)) end)
  | _ => NONE);

fun rearrange_Ball tac ctxt ct =
  (case Thm.term_of ct of
    F as Ball $ A $ Abs (x, T, P) =>
      (case extract_imp true [] P of
        NONE => NONE
      | SOME (xs, eq, Q) =>
          if not (null xs) then NONE
          else
            let val R = Data.imp $ eq $ Q
            in SOME (prove_conv ctxt (F, Ball $ A $ Abs (x, T, R))
              (iff_reflection_tac THEN' tac THEN' prove_one_point_All_tac)) end)
  | _ => NONE);

fun rearrange_Ex ctxt ct =
  (case Thm.term_of ct of
    F as (ex as Const (q, _)) $ Abs (x, T, P) =>
      (case extract_quant ctxt (K extract_conj) q P of
        NONE => NONE
      | SOME (xs, eq, Q) =>
          let val R = quantify ex x T xs (Data.conj $ eq $ Q)
          in SOME (prove_conv ctxt (F, R) (iff_reflection_tac THEN' prove_one_point_Ex_tac)) end)
  | _ => NONE);

fun rearrange_Bex tac ctxt ct =
  (case Thm.term_of ct of
    F as Bex $ A $ Abs (x, T, P) =>
      (case extract_conj true [] P of
        NONE => NONE
      | SOME (xs, eq, Q) =>
          if not (null xs) then NONE
          else SOME (prove_conv ctxt (F, Bex $ A $ Abs (x, T, Data.conj $ eq $ Q))
            (iff_reflection_tac THEN' tac THEN' prove_one_point_Ex_tac)))
  | _ => NONE);

fun rearrange_Collect tac ctxt ct =
  (case Thm.term_of ct of
    F as Collect $ Abs (x, T, P) =>
      (case extract_conj true [] P of
        NONE => NONE
      | SOME (_, eq, Q) =>
          let val R = Collect $ Abs (x, T, Data.conj $ eq $ Q)
          in SOME (prove_conv ctxt (F, R) (iff_reflection_tac THEN' tac)) end)
  | _ => NONE);

end;