(* 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) *)
end;
signature QUANTIFIER1 =
sig
val prove_one_point_all_tac: tactic
val prove_one_point_ex_tac: tactic
val rearrange_all: Proof.context -> cterm -> thm option
val rearrange_ex: Proof.context -> cterm -> thm option
val rearrange_ball: (Proof.context -> tactic) -> 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_quant 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 (null xs) xs P
in exqu [] 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'', ...} => rtac Data.iff_reflection 1 THEN tac ctxt'');
in singleton (Variable.export ctxt' ctxt) thm end;
fun qcomm_tac qcomm qI i = REPEAT_DETERM (rtac qcomm i THEN rtac 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
val prove_one_point_ex_tac =
qcomm_tac excomm Data.iff_exI 1 THEN rtac Data.iffI 1 THEN
ALLGOALS
(EVERY' [etac Data.exE, REPEAT_DETERM o etac Data.conjE, rtac Data.exI,
DEPTH_SOLVE_1 o ares_tac [Data.conjI]])
end;
(* Proves (! x0..xn. (... & x0 = t & ...) --> P x0) =
(! x1..xn x0. x0 = t --> (... & ...) --> P x0)
*)
local
val tac =
SELECT_GOAL
(EVERY1 [REPEAT o dtac Data.uncurry, REPEAT o rtac Data.impI, etac Data.mp,
REPEAT o etac Data.conjE, REPEAT o ares_tac [Data.conjI]]);
val allcomm = Data.all_comm RS Data.iff_trans;
in
val prove_one_point_all_tac =
EVERY1 [qcomm_tac allcomm Data.iff_allI, rtac Data.iff_allI, rtac Data.iffI, tac, tac];
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 term_of ct of
F as (all as Const (q, _)) $ Abs (x, T, P) =>
(case extract_quant 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) (K prove_one_point_all_tac)) end)
| _ => NONE);
fun rearrange_ball tac ctxt ct =
(case 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)) tac) end)
| _ => NONE);
fun rearrange_ex ctxt ct =
(case term_of ct of
F as (ex as Const (q, _)) $ Abs (x, T, P) =>
(case extract_quant 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) (K prove_one_point_ex_tac)) end)
| _ => NONE);
fun rearrange_bex tac ctxt ct =
(case 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)) tac))
| _ => NONE);
fun rearrange_Collect tac ctxt ct =
(case 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) tac) end)
| _ => NONE);
end;