src/Provers/Arith/combine_numerals.ML
 author huffman Mon May 21 19:05:37 2007 +0200 (2007-05-21) changeset 23058 c722004c5a22 parent 20114 a1bb4bc68ff3 child 35762 af3ff2ba4c54 permissions -rw-r--r--
generalize CombineNumerals functor to allow coefficients with types other than IntInf.int
```     1 (*  Title:      Provers/Arith/combine_numerals.ML
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   2000  University of Cambridge
```
```     5
```
```     6 Combine coefficients in expressions:
```
```     7
```
```     8      i + #m*u + j ... + #n*u + k  ==  #(m+n)*u + (i + (j + k))
```
```     9
```
```    10 It works by (a) massaging the sum to bring the selected terms to the front:
```
```    11
```
```    12      #m*u + (#n*u + (i + (j + k)))
```
```    13
```
```    14 (b) then using left_distrib to reach
```
```    15
```
```    16      #(m+n)*u + (i + (j + k))
```
```    17 *)
```
```    18
```
```    19 signature COMBINE_NUMERALS_DATA =
```
```    20 sig
```
```    21   (*abstract syntax*)
```
```    22   eqtype coeff
```
```    23   val iszero: coeff -> bool
```
```    24   val add: coeff * coeff -> coeff     (*addition (or multiplication) *)
```
```    25   val mk_sum: typ -> term list -> term
```
```    26   val dest_sum: term -> term list
```
```    27   val mk_coeff: coeff * term -> term
```
```    28   val dest_coeff: term -> coeff * term
```
```    29   (*rules*)
```
```    30   val left_distrib: thm
```
```    31   (*proof tools*)
```
```    32   val prove_conv: tactic list -> Proof.context -> term * term -> thm option
```
```    33   val trans_tac: simpset -> thm option -> tactic (*applies the initial lemma*)
```
```    34   val norm_tac: simpset -> tactic                (*proves the initial lemma*)
```
```    35   val numeral_simp_tac: simpset -> tactic        (*proves the final theorem*)
```
```    36   val simplify_meta_eq: simpset -> thm -> thm    (*simplifies the final theorem*)
```
```    37 end;
```
```    38
```
```    39
```
```    40 functor CombineNumeralsFun(Data: COMBINE_NUMERALS_DATA):
```
```    41   sig
```
```    42   val proc: simpset -> term -> thm option
```
```    43   end
```
```    44 =
```
```    45 struct
```
```    46
```
```    47 (*Remove the first occurrence of #m*u from the term list*)
```
```    48 fun remove (_, _, []) = (*impossible, since #m*u was found by find_repeated*)
```
```    49       raise TERM("combine_numerals: remove", [])
```
```    50   | remove (m, u, t::terms) =
```
```    51       case try Data.dest_coeff t of
```
```    52           SOME(n,v) => if m=n andalso u aconv v then terms
```
```    53                        else t :: remove (m, u, terms)
```
```    54         | NONE      =>  t :: remove (m, u, terms);
```
```    55
```
```    56 (*a left-to-right scan of terms, seeking another term of the form #n*u, where
```
```    57   #m*u is already in terms for some m*)
```
```    58 fun find_repeated (tab, _, []) = raise TERM("find_repeated", [])
```
```    59   | find_repeated (tab, past, t::terms) =
```
```    60       case try Data.dest_coeff t of
```
```    61           SOME(n,u) =>
```
```    62               (case Termtab.lookup tab u of
```
```    63                   SOME m => (u, m, n, rev (remove (m,u,past)) @ terms)
```
```    64                 | NONE => find_repeated (Termtab.update (u, n) tab,
```
```    65                                          t::past,  terms))
```
```    66         | NONE => find_repeated (tab, t::past, terms);
```
```    67
```
```    68 (*the simplification procedure*)
```
```    69 fun proc ss t =
```
```    70   let
```
```    71     val ctxt = Simplifier.the_context ss;
```
```    72     val ([t'], ctxt') = Variable.import_terms true [t] ctxt
```
```    73     val export = singleton (Variable.export ctxt' ctxt)
```
```    74
```
```    75     val (u,m,n,terms) = find_repeated (Termtab.empty, [], Data.dest_sum t')
```
```    76     val T = Term.fastype_of u
```
```    77
```
```    78     val reshape =  (*Move i*u to the front and put j*u into standard form
```
```    79                        i + #m + j + k == #m + i + (j + k) *)
```
```    80       if Data.iszero m orelse Data.iszero n then   (*trivial, so do nothing*)
```
```    81         raise TERM("combine_numerals", [])
```
```    82       else Data.prove_conv [Data.norm_tac ss] ctxt
```
```    83         (t', Data.mk_sum T ([Data.mk_coeff (m, u), Data.mk_coeff (n, u)] @ terms))
```
```    84   in
```
```    85     Option.map (export o Data.simplify_meta_eq ss)
```
```    86       (Data.prove_conv
```
```    87          [Data.trans_tac ss reshape, rtac Data.left_distrib 1,
```
```    88           Data.numeral_simp_tac ss] ctxt
```
```    89          (t', Data.mk_sum T (Data.mk_coeff(Data.add(m,n), u) :: terms)))
```
```    90   end
```
```    91   (* FIXME avoid handling of generic exceptions *)
```
```    92   handle TERM _ => NONE
```
```    93        | TYPE _ => NONE;   (*Typically (if thy doesn't include Numeral)
```
```    94                              Undeclared type constructor "Numeral.bin"*)
```
```    95
```
```    96 end;
```