src/Provers/quantifier1.ML
 author nipkow Thu Mar 29 13:59:54 2001 +0200 (2001-03-29) changeset 11232 558a4feebb04 parent 11221 60c6e91f6079 child 12523 0d8d5bf549b0 permissions -rw-r--r--
generalization of 1 point rules for ALL
```     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 conjI: thm
```
```    43   val conjE: thm
```
```    44   val impI:  thm
```
```    45   val mp:    thm
```
```    46   val exI:   thm
```
```    47   val exE:   thm
```
```    48   val uncurry: thm (* P --> Q --> R ==> P & Q --> R *)
```
```    49   val iff_allI: thm (* !!x. P x <-> Q x ==> (!x. P x) = (!x. Q x) *)
```
```    50 end;
```
```    51
```
```    52 signature QUANTIFIER1 =
```
```    53 sig
```
```    54   val prove_one_point_all_tac: tactic
```
```    55   val prove_one_point_ex_tac: tactic
```
```    56   val rearrange_all: Sign.sg -> thm list -> term -> thm option
```
```    57   val rearrange_ex:  Sign.sg -> thm list -> term -> thm option
```
```    58   val rearrange_ball: tactic -> Sign.sg -> thm list -> term -> thm option
```
```    59   val rearrange_bex:  tactic -> Sign.sg -> thm list -> term -> thm option
```
```    60 end;
```
```    61
```
```    62 functor Quantifier1Fun(Data: QUANTIFIER1_DATA): QUANTIFIER1 =
```
```    63 struct
```
```    64
```
```    65 open Data;
```
```    66
```
```    67 (* FIXME: only test! *)
```
```    68 fun def eq = case dest_eq eq of
```
```    69       Some(c,s,t) =>
```
```    70         s = Bound 0 andalso not(loose_bvar1(t,0)) orelse
```
```    71         t = Bound 0 andalso not(loose_bvar1(s,0))
```
```    72     | None => false;
```
```    73
```
```    74 fun extract_conj t = case dest_conj t of None => None
```
```    75     | Some(conj,P,Q) =>
```
```    76         (if def P then Some(P,Q) else
```
```    77          if def Q then Some(Q,P) else
```
```    78          (case extract_conj P of
```
```    79             Some(eq,P') => Some(eq, conj \$ P' \$ Q)
```
```    80           | None => (case extract_conj Q of
```
```    81                        Some(eq,Q') => Some(eq,conj \$ P \$ Q')
```
```    82                      | None => None)));
```
```    83
```
```    84 fun extract_imp t = case dest_imp t of None => None
```
```    85     | Some(imp,P,Q) => if def P then Some(P,Q)
```
```    86                        else (case extract_conj P of
```
```    87                                Some(eq,P') => Some(eq, imp \$ P' \$ Q)
```
```    88                              | None => (case extract_imp Q of
```
```    89                                           None => None
```
```    90                                         | Some(eq,Q') => Some(eq, imp\$P\$Q')));
```
```    91
```
```    92
```
```    93 fun prove_conv tac sg tu =
```
```    94   let val meta_eq = cterm_of sg (Logic.mk_equals tu)
```
```    95   in prove_goalw_cterm [] meta_eq (K [rtac iff_reflection 1, tac])
```
```    96      handle ERROR =>
```
```    97             error("The error(s) above occurred while trying to prove " ^
```
```    98                   string_of_cterm meta_eq)
```
```    99   end;
```
```   100
```
```   101 (* Proves (? x. ... & x = t & ...) = (? x. x = t & ... & ...)
```
```   102    Better: instantiate exI
```
```   103 *)
```
```   104 val prove_one_point_ex_tac = rtac iffI 1 THEN
```
```   105     ALLGOALS(EVERY'[etac exE, REPEAT_DETERM o (etac conjE), rtac exI,
```
```   106                     DEPTH_SOLVE_1 o (ares_tac [conjI])]);
```
```   107
```
```   108 (* Proves (! x. (... & x = t & ...) --> P x) =
```
```   109           (! x. x = t --> (... & ...) --> P x)
```
```   110 *)
```
```   111 local
```
```   112 val tac = SELECT_GOAL
```
```   113           (EVERY1[REPEAT o (dtac uncurry), REPEAT o (rtac impI), etac mp,
```
```   114                   REPEAT o (etac conjE), REPEAT o (ares_tac [conjI])])
```
```   115 in
```
```   116 val prove_one_point_all_tac = EVERY1[rtac iff_allI, rtac iffI, tac, tac]
```
```   117 end
```
```   118
```
```   119 fun rearrange_all sg _ (F as all \$ Abs(x,T, P)) =
```
```   120      (case extract_imp P of
```
```   121         None => None
```
```   122       | Some(eq,Q) =>
```
```   123           let val R = imp \$ eq \$ Q
```
```   124           in Some(prove_conv prove_one_point_all_tac sg (F,all\$Abs(x,T,R))) end)
```
```   125   | rearrange_all _ _ _ = None;
```
```   126
```
```   127 fun rearrange_ball tac sg _ (F as Ball \$ A \$ Abs(x,T,P)) =
```
```   128      (case extract_imp P of
```
```   129         None => None
```
```   130       | Some(eq,Q) =>
```
```   131           let val R = imp \$ eq \$ Q
```
```   132           in Some(prove_conv tac sg (F,Ball \$ A \$ Abs(x,T,R))) end)
```
```   133   | rearrange_ball _ _ _ _ = None;
```
```   134
```
```   135 fun rearrange_ex sg _ (F as ex \$ Abs(x,T,P)) =
```
```   136      (case extract_conj P of
```
```   137         None => None
```
```   138       | Some(eq,Q) =>
```
```   139           Some(prove_conv prove_one_point_ex_tac sg (F,ex \$ Abs(x,T,conj\$eq\$Q))))
```
```   140   | rearrange_ex _ _ _ = None;
```
```   141
```
```   142 fun rearrange_bex tac sg _ (F as Bex \$ A \$ Abs(x,T,P)) =
```
```   143      (case extract_conj P of
```
```   144         None => None
```
```   145       | Some(eq,Q) =>
```
```   146           Some(prove_conv tac sg (F,Bex \$ A \$ Abs(x,T,conj\$eq\$Q))))
```
```   147   | rearrange_bex _ _ _ _ = None;
```
```   148
```
```   149 end;
```