(* Title: Provers/quantifier1
ID: $Id$
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 congreunce 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"
*)
signature QUANTIFIER1_DATA =
sig
(*abstract syntax*)
val dest_eq: term -> (term*term*term)option
val dest_conj: term -> (term*term*term)option
val dest_imp: term -> (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: theory -> simpset -> term -> thm option
val rearrange_ex: theory -> simpset -> term -> thm option
val rearrange_ball: (simpset -> tactic) -> theory -> simpset -> term -> thm option
val rearrange_bex: (simpset -> tactic) -> theory -> simpset -> term -> thm option
end;
functor Quantifier1Fun(Data: QUANTIFIER1_DATA): QUANTIFIER1 =
struct
open Data;
(* FIXME: only test! *)
fun def xs eq =
let val n = length xs
in case dest_eq eq of
SOME(c,s,t) =>
s = Bound n andalso not(loose_bvar1(t,n)) orelse
t = Bound n andalso not(loose_bvar1(s,n))
| NONE => false
end;
fun extract_conj xs t = case dest_conj t of NONE => NONE
| SOME(conj,P,Q) =>
(if def xs P then SOME(xs,P,Q) else
if def xs Q then SOME(xs,Q,P) else
(case extract_conj xs P of
SOME(xs,eq,P') => SOME(xs,eq, conj $ P' $ Q)
| NONE => (case extract_conj xs Q of
SOME(xs,eq,Q') => SOME(xs,eq,conj $ P $ Q')
| NONE => NONE)));
fun extract_imp xs t = case dest_imp t of NONE => NONE
| SOME(imp,P,Q) => if def xs P then SOME(xs,P,Q)
else (case extract_conj xs P of
SOME(xs,eq,P') => SOME(xs, eq, imp $ P' $ Q)
| NONE => (case extract_imp xs Q of
NONE => NONE
| SOME(xs,eq,Q') =>
SOME(xs,eq,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 xs P
in exqu end;
fun prove_conv tac thy tu =
Goal.prove (ProofContext.init thy) [] [] (Logic.mk_equals tu)
(K (rtac iff_reflection 1 THEN tac));
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 = ex_comm RS iff_trans
in
val prove_one_point_ex_tac = qcomm_tac excomm iff_exI 1 THEN rtac iffI 1 THEN
ALLGOALS(EVERY'[etac exE, REPEAT_DETERM o (etac conjE), rtac exI,
DEPTH_SOLVE_1 o (ares_tac [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 uncurry), REPEAT o (rtac impI), etac mp,
REPEAT o (etac conjE), REPEAT o (ares_tac [conjI])])
val allcomm = all_comm RS iff_trans
in
val prove_one_point_all_tac =
EVERY1[qcomm_tac allcomm iff_allI,rtac iff_allI, rtac 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 thy _ (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 (imp $ eq $ Q)
in SOME(prove_conv prove_one_point_all_tac thy (F,R)) end)
| rearrange_all _ _ _ = NONE;
fun rearrange_ball tac thy ss (F as Ball $ A $ Abs(x,T,P)) =
(case extract_imp [] P of
NONE => NONE
| SOME(xs,eq,Q) => if not(null xs) then NONE else
let val R = imp $ eq $ Q
in SOME(prove_conv (tac ss) thy (F,Ball $ A $ Abs(x,T,R))) end)
| rearrange_ball _ _ _ _ = NONE;
fun rearrange_ex thy _ (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 (conj $ eq $ Q)
in SOME(prove_conv prove_one_point_ex_tac thy (F,R)) end)
| rearrange_ex _ _ _ = NONE;
fun rearrange_bex tac thy ss (F as Bex $ A $ Abs(x,T,P)) =
(case extract_conj [] P of
NONE => NONE
| SOME(xs,eq,Q) => if not(null xs) then NONE else
SOME(prove_conv (tac ss) thy (F,Bex $ A $ Abs(x,T,conj$eq$Q))))
| rearrange_bex _ _ _ _ = NONE;
end;