src/Provers/quantifier1.ML
 author nipkow Fri May 17 11:25:07 2002 +0200 (2002-05-17 ago) changeset 13157 4a4599f78f18 parent 12523 0d8d5bf549b0 child 13480 bb72bd43c6c3 permissions -rw-r--r--
allowed more general split rules to cope with div/mod 2
```     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: 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;
```
```    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 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;
```
```   113
```
```   114 fun qcomm_tac qcomm qI i = REPEAT_DETERM (rtac qcomm i THEN rtac qI i)
```
```   115
```
```   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;
```
```   126
```
```   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
```
```   139
```
```   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;
```
```   145
```
```   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;
```
```   152
```
```   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;
```
```   160
```
```   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;
```
```   168
```
```   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;
```
```   176
```
```   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;
```
```   183
```
```   184 end;
```