src/Provers/quantifier1.ML
 author wenzelm Thu Aug 09 23:53:51 2007 +0200 (2007-08-09) changeset 24209 8a2c8d623e43 parent 20049 f48c4a3a34bc child 31166 a90fe83f58ea permissions -rw-r--r--
schedule: misc cleanup, more precise task model;
```     1 (*  Title:      Provers/quantifier1
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow
```
```     4     Copyright   1997  TU Munich
```
```     5
```
```     6 Simplification procedures for turning
```
```     7
```
```     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.
```
```    12
```
```    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.
```
```    17
```
```    18      And analogously for t=x, but the eqn is not turned around!
```
```    19
```
```    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)".
```
```    24
```
```    25
```
```    26      And similarly for the bounded quantifiers.
```
```    27
```
```    28 Gries etc call this the "1 point rules"
```
```    29 *)
```
```    30
```
```    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;
```
```    55
```
```    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: theory -> simpset -> term -> thm option
```
```    61   val rearrange_ex:  theory -> simpset -> term -> thm option
```
```    62   val rearrange_ball: (simpset -> tactic) -> theory -> simpset -> term -> thm option
```
```    63   val rearrange_bex:  (simpset -> tactic) -> theory -> simpset -> term -> thm option
```
```    64 end;
```
```    65
```
```    66 functor Quantifier1Fun(Data: QUANTIFIER1_DATA): QUANTIFIER1 =
```
```    67 struct
```
```    68
```
```    69 open Data;
```
```    70
```
```    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;
```
```    80
```
```    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)));
```
```    90
```
```    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')));
```
```    99
```
```   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;
```
```   105
```
```   106 fun prove_conv tac thy tu =
```
```   107   Goal.prove (ProofContext.init thy) [] [] (Logic.mk_equals tu)
```
```   108     (K (rtac iff_reflection 1 THEN tac));
```
```   109
```
```   110 fun qcomm_tac qcomm qI i = REPEAT_DETERM (rtac qcomm i THEN rtac qI i)
```
```   111
```
```   112 (* Proves (? x0..xn. ... & x0 = t & ...) = (? x1..xn x0. x0 = t & ... & ...)
```
```   113    Better: instantiate exI
```
```   114 *)
```
```   115 local
```
```   116 val excomm = ex_comm RS iff_trans
```
```   117 in
```
```   118 val prove_one_point_ex_tac = qcomm_tac excomm iff_exI 1 THEN rtac iffI 1 THEN
```
```   119     ALLGOALS(EVERY'[etac exE, REPEAT_DETERM o (etac conjE), rtac exI,
```
```   120                     DEPTH_SOLVE_1 o (ares_tac [conjI])])
```
```   121 end;
```
```   122
```
```   123 (* Proves (! x0..xn. (... & x0 = t & ...) --> P x0) =
```
```   124           (! x1..xn x0. x0 = t --> (... & ...) --> P x0)
```
```   125 *)
```
```   126 local
```
```   127 val tac = SELECT_GOAL
```
```   128           (EVERY1[REPEAT o (dtac uncurry), REPEAT o (rtac impI), etac mp,
```
```   129                   REPEAT o (etac conjE), REPEAT o (ares_tac [conjI])])
```
```   130 val allcomm = all_comm RS iff_trans
```
```   131 in
```
```   132 val prove_one_point_all_tac =
```
```   133       EVERY1[qcomm_tac allcomm iff_allI,rtac iff_allI, rtac iffI, tac, tac]
```
```   134 end
```
```   135
```
```   136 fun renumber l u (Bound i) = Bound(if i < l orelse i > u then i else
```
```   137                                    if i=u then l else i+1)
```
```   138   | renumber l u (s\$t) = renumber l u s \$ renumber l u t
```
```   139   | renumber l u (Abs(x,T,t)) = Abs(x,T,renumber (l+1) (u+1) t)
```
```   140   | renumber _ _ atom = atom;
```
```   141
```
```   142 fun quantify qC x T xs P =
```
```   143   let fun quant [] P = P
```
```   144         | quant ((qC,x,T)::xs) P = quant xs (qC \$ Abs(x,T,P))
```
```   145       val n = length xs
```
```   146       val Q = if n=0 then P else renumber 0 n P
```
```   147   in quant xs (qC \$ Abs(x,T,Q)) end;
```
```   148
```
```   149 fun rearrange_all thy _ (F as (all as Const(q,_)) \$ Abs(x,T, P)) =
```
```   150      (case extract_quant extract_imp q [] P of
```
```   151         NONE => NONE
```
```   152       | SOME(xs,eq,Q) =>
```
```   153           let val R = quantify all x T xs (imp \$ eq \$ Q)
```
```   154           in SOME(prove_conv prove_one_point_all_tac thy (F,R)) end)
```
```   155   | rearrange_all _ _ _ = NONE;
```
```   156
```
```   157 fun rearrange_ball tac thy ss (F as Ball \$ A \$ Abs(x,T,P)) =
```
```   158      (case extract_imp [] P of
```
```   159         NONE => NONE
```
```   160       | SOME(xs,eq,Q) => if not(null xs) then NONE else
```
```   161           let val R = imp \$ eq \$ Q
```
```   162           in SOME(prove_conv (tac ss) thy (F,Ball \$ A \$ Abs(x,T,R))) end)
```
```   163   | rearrange_ball _ _ _ _ = NONE;
```
```   164
```
```   165 fun rearrange_ex thy _ (F as (ex as Const(q,_)) \$ Abs(x,T,P)) =
```
```   166      (case extract_quant extract_conj q [] P of
```
```   167         NONE => NONE
```
```   168       | SOME(xs,eq,Q) =>
```
```   169           let val R = quantify ex x T xs (conj \$ eq \$ Q)
```
```   170           in SOME(prove_conv prove_one_point_ex_tac thy (F,R)) end)
```
```   171   | rearrange_ex _ _ _ = NONE;
```
```   172
```
```   173 fun rearrange_bex tac thy ss (F as Bex \$ A \$ Abs(x,T,P)) =
```
```   174      (case extract_conj [] P of
```
```   175         NONE => NONE
```
```   176       | SOME(xs,eq,Q) => if not(null xs) then NONE else
```
```   177           SOME(prove_conv (tac ss) thy (F,Bex \$ A \$ Abs(x,T,conj\$eq\$Q))))
```
```   178   | rearrange_bex _ _ _ _ = NONE;
```
```   179
```
```   180 end;
```