src/Provers/quantifier1.ML
 author nipkow Mon Dec 17 14:21:59 2001 +0100 (2001-12-17) changeset 12523 0d8d5bf549b0 parent 11232 558a4feebb04 child 13480 bb72bd43c6c3 permissions -rw-r--r--
now permutations of quantifiers are allowed as well.
1 (*  Title:      Provers/quantifier1
2     ID:         \$Id\$
3     Author:     Tobias Nipkow
6 Simplification procedures for turning
8             ? x. ... & x = t & ...
9      into   ? x. x = t & ... & ...
10      where the `? x. x = t &' in the latter formula must be eliminated
11            by ordinary simplification.
13      and   ! x. (... & x = t & ...) --> P x
14      into  ! x. x = t --> (... & ...) --> P x
15      where the `!x. x=t -->' in the latter formula is eliminated
16            by ordinary simplification.
18      And analogously for t=x, but the eqn is not turned around!
20      NB Simproc is only triggered by "!x. P(x) & P'(x) --> Q(x)";
21         "!x. x=t --> P(x)" is covered by the congreunce rule for -->;
22         "!x. t=x --> P(x)" must be taken care of by an ordinary rewrite rule.
23         As must be "? x. t=x & P(x)".
26      And similarly for the bounded quantifiers.
28 Gries etc call this the "1 point rules"
29 *)
31 signature QUANTIFIER1_DATA =
32 sig
33   (*abstract syntax*)
34   val dest_eq: term -> (term*term*term)option
35   val dest_conj: term -> (term*term*term)option
36   val dest_imp:  term -> (term*term*term)option
37   val conj: term
38   val imp:  term
39   (*rules*)
40   val iff_reflection: thm (* P <-> Q ==> P == Q *)
41   val iffI:  thm
42   val iff_trans: thm
43   val conjI: thm
44   val conjE: thm
45   val impI:  thm
46   val mp:    thm
47   val exI:   thm
48   val exE:   thm
49   val uncurry: thm (* P --> Q --> R ==> P & Q --> R *)
50   val iff_allI: thm (* !!x. P x <-> Q x ==> (!x. P x) = (!x. Q x) *)
51   val iff_exI: thm (* !!x. P x <-> Q x ==> (? x. P x) = (? x. Q x) *)
52   val all_comm: thm (* (!x y. P x y) = (!y x. P x y) *)
53   val ex_comm: thm (* (? x y. P x y) = (? y x. P x y) *)
54 end;
56 signature QUANTIFIER1 =
57 sig
58   val prove_one_point_all_tac: tactic
59   val prove_one_point_ex_tac: tactic
60   val rearrange_all: Sign.sg -> thm list -> term -> thm option
61   val rearrange_ex:  Sign.sg -> thm list -> term -> thm option
62   val rearrange_ball: tactic -> Sign.sg -> thm list -> term -> thm option
63   val rearrange_bex:  tactic -> Sign.sg -> thm list -> term -> thm option
64 end;
66 functor Quantifier1Fun(Data: QUANTIFIER1_DATA): QUANTIFIER1 =
67 struct
69 open Data;
71 (* FIXME: only test! *)
72 fun def xs eq =
73   let val n = length xs
74   in case dest_eq eq of
75       Some(c,s,t) =>
76         s = Bound n andalso not(loose_bvar1(t,n)) orelse
77         t = Bound n andalso not(loose_bvar1(s,n))
78     | None => false
79   end;
81 fun extract_conj xs t = case dest_conj t of None => None
82     | Some(conj,P,Q) =>
83         (if def xs P then Some(xs,P,Q) else
84          if def xs Q then Some(xs,Q,P) else
85          (case extract_conj xs P of
86             Some(xs,eq,P') => Some(xs,eq, conj \$ P' \$ Q)
87           | None => (case extract_conj xs Q of
88                        Some(xs,eq,Q') => Some(xs,eq,conj \$ P \$ Q')
89                      | None => None)));
91 fun extract_imp xs t = case dest_imp t of None => None
92     | Some(imp,P,Q) => if def xs P then Some(xs,P,Q)
93                        else (case extract_conj xs P of
94                                Some(xs,eq,P') => Some(xs, eq, imp \$ P' \$ Q)
95                              | None => (case extract_imp xs Q of
96                                           None => None
97                                         | Some(xs,eq,Q') =>
98                                             Some(xs,eq,imp\$P\$Q')));
100 fun extract_quant extract q =
101   let fun exqu xs ((qC as Const(qa,_)) \$ Abs(x,T,Q)) =
102             if qa = q then exqu ((qC,x,T)::xs) Q else None
103         | exqu xs P = extract xs P
104   in exqu end;
106 fun prove_conv tac sg tu =
107   let val meta_eq = cterm_of sg (Logic.mk_equals tu)
108   in prove_goalw_cterm [] meta_eq (K [rtac iff_reflection 1, tac])
109      handle ERROR =>
110             error("The error(s) above occurred while trying to prove " ^
111                   string_of_cterm meta_eq)
112   end;
114 fun qcomm_tac qcomm qI i = REPEAT_DETERM (rtac qcomm i THEN rtac qI i)
116 (* Proves (? x0..xn. ... & x0 = t & ...) = (? x1..xn x0. x0 = t & ... & ...)
117    Better: instantiate exI
118 *)
119 local
120 val excomm = ex_comm RS iff_trans
121 in
122 val prove_one_point_ex_tac = qcomm_tac excomm iff_exI 1 THEN rtac iffI 1 THEN
123     ALLGOALS(EVERY'[etac exE, REPEAT_DETERM o (etac conjE), rtac exI,
124                     DEPTH_SOLVE_1 o (ares_tac [conjI])])
125 end;
127 (* Proves (! x0..xn. (... & x0 = t & ...) --> P x0) =
128           (! x1..xn x0. x0 = t --> (... & ...) --> P x0)
129 *)
130 local
131 val tac = SELECT_GOAL
132           (EVERY1[REPEAT o (dtac uncurry), REPEAT o (rtac impI), etac mp,
133                   REPEAT o (etac conjE), REPEAT o (ares_tac [conjI])])
134 val allcomm = all_comm RS iff_trans
135 in
136 val prove_one_point_all_tac =
137       EVERY1[qcomm_tac allcomm iff_allI,rtac iff_allI, rtac iffI, tac, tac]
138 end
140 fun renumber l u (Bound i) = Bound(if i < l orelse i > u then i else
141                                    if i=u then l else i+1)
142   | renumber l u (s\$t) = renumber l u s \$ renumber l u t
143   | renumber l u (Abs(x,T,t)) = Abs(x,T,renumber (l+1) (u+1) t)
144   | renumber _ _ atom = atom;
146 fun quantify qC x T xs P =
147   let fun quant [] P = P
148         | quant ((qC,x,T)::xs) P = quant xs (qC \$ Abs(x,T,P))
149       val n = length xs
150       val Q = if n=0 then P else renumber 0 n P
151   in quant xs (qC \$ Abs(x,T,Q)) end;
153 fun rearrange_all sg _ (F as (all as Const(q,_)) \$ Abs(x,T, P)) =
154      (case extract_quant extract_imp q [] P of
155         None => None
156       | Some(xs,eq,Q) =>
157           let val R = quantify all x T xs (imp \$ eq \$ Q)
158           in Some(prove_conv prove_one_point_all_tac sg (F,R)) end)
159   | rearrange_all _ _ _ = None;
161 fun rearrange_ball tac sg _ (F as Ball \$ A \$ Abs(x,T,P)) =
162      (case extract_imp [] P of
163         None => None
164       | Some(xs,eq,Q) => if not(null xs) then None else
165           let val R = imp \$ eq \$ Q
166           in Some(prove_conv tac sg (F,Ball \$ A \$ Abs(x,T,R))) end)
167   | rearrange_ball _ _ _ _ = None;
169 fun rearrange_ex sg _ (F as (ex as Const(q,_)) \$ Abs(x,T,P)) =
170      (case extract_quant extract_conj q [] P of
171         None => None
172       | Some(xs,eq,Q) =>
173           let val R = quantify ex x T xs (conj \$ eq \$ Q)
174           in Some(prove_conv prove_one_point_ex_tac sg (F,R)) end)
175   | rearrange_ex _ _ _ = None;
177 fun rearrange_bex tac sg _ (F as Bex \$ A \$ Abs(x,T,P)) =
178      (case extract_conj [] P of
179         None => None
180       | Some(xs,eq,Q) => if not(null xs) then None else
181           Some(prove_conv tac sg (F,Bex \$ A \$ Abs(x,T,conj\$eq\$Q))))
182   | rearrange_bex _ _ _ _ = None;
184 end;