src/Provers/Arith/cancel_numerals.ML
 author wenzelm Thu Sep 15 17:16:56 2005 +0200 (2005-09-15) changeset 17412 e26cb20ef0cc parent 17223 430edc6b7826 child 20044 92cc2f4c7335 permissions -rw-r--r--
TableFun/Symtab: curried lookup and update;
1 (*  Title:      Provers/Arith/cancel_numerals.ML
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
4     Copyright   2000  University of Cambridge
6 Cancel common coefficients in balanced expressions:
8      i + #m*u + j ~~ i' + #m'*u + j'  ==  #(m-m')*u + i + j ~~ i' + j'
10 where ~~ is an appropriate balancing operation (e.g. =, <=, <, -).
12 It works by (a) massaging both sides to bring the selected term to the front:
14      #m*u + (i + j) ~~ #m'*u + (i' + j')
18      #(m-m')*u + i + j ~~ i' + j'       (if m'<=m)
20 or
22      i + j ~~ #(m'-m)*u + i' + j'       (otherwise)
23 *)
25 signature CANCEL_NUMERALS_DATA =
26 sig
27   (*abstract syntax*)
28   val mk_sum: typ -> term list -> term
29   val dest_sum: term -> term list
30   val mk_bal: term * term -> term
31   val dest_bal: term -> term * term
32   val mk_coeff: IntInf.int * term -> term
33   val dest_coeff: term -> IntInf.int * term
34   val find_first_coeff: term -> term list -> IntInf.int * term list
35   (*rules*)
38   (*proof tools*)
39   val prove_conv: tactic list -> theory ->
40                   thm list -> string list -> term * term -> thm option
41   val trans_tac: simpset -> thm option -> tactic (*applies the initial lemma*)
42   val norm_tac: simpset -> tactic                (*proves the initial lemma*)
43   val numeral_simp_tac: simpset -> tactic        (*proves the final theorem*)
44   val simplify_meta_eq: simpset -> thm -> thm    (*simplifies the final theorem*)
45 end;
48 functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA):
49   sig
50   val proc: theory -> simpset -> term -> thm option
51   end
52 =
53 struct
55 (*For t = #n*u then put u in the table*)
56 fun update_by_coeff t =
57   Termtab.update (#2 (Data.dest_coeff t), ());
59 (*a left-to-right scan of terms1, seeking a term of the form #n*u, where
60   #m*u is in terms2 for some m*)
61 fun find_common (terms1,terms2) =
62   let val tab2 = fold update_by_coeff terms2 Termtab.empty
63       fun seek [] = raise TERM("find_common", [])
64         | seek (t::terms) =
65               let val (_,u) = Data.dest_coeff t
66               in if Termtab.defined tab2 u then u else seek terms end
67   in  seek terms1 end;
69 (*the simplification procedure*)
70 fun proc thy ss t =
71   let
72       val hyps = prems_of_ss ss;
73       (*first freeze any Vars in the term to prevent flex-flex problems*)
74       val (t', xs) = Term.adhoc_freeze_vars t;
75       val (t1,t2) = Data.dest_bal t'
76       val terms1 = Data.dest_sum t1
77       and terms2 = Data.dest_sum t2
78       val u = find_common (terms1,terms2)
79       val (n1, terms1') = Data.find_first_coeff u terms1
80       and (n2, terms2') = Data.find_first_coeff u terms2
81       and T = Term.fastype_of u
82       fun newshape (i,terms) = Data.mk_sum T (Data.mk_coeff(i,u)::terms)
83       val reshape =  (*Move i*u to the front and put j*u into standard form
84                        i + #m + j + k == #m + i + (j + k) *)
85             if n1=0 orelse n2=0 then   (*trivial, so do nothing*)
86                 raise TERM("cancel_numerals", [])
87             else Data.prove_conv [Data.norm_tac ss] thy hyps xs
88                         (t',
89                          Data.mk_bal (newshape(n1,terms1'),
90                                       newshape(n2,terms2')))
91   in
92       Option.map (Data.simplify_meta_eq ss)
93        (if n2<=n1 then
94             Data.prove_conv
95                [Data.trans_tac ss reshape, rtac Data.bal_add1 1,
96                 Data.numeral_simp_tac ss] thy hyps xs
97                (t', Data.mk_bal (newshape(n1-n2,terms1'),
98                                  Data.mk_sum T terms2'))
99         else
100             Data.prove_conv
101                [Data.trans_tac ss reshape, rtac Data.bal_add2 1,
102                 Data.numeral_simp_tac ss] thy hyps xs
103                (t', Data.mk_bal (Data.mk_sum T terms1',
104                                  newshape(n2-n1,terms2'))))
105   end
106   handle TERM _ => NONE
107        | TYPE _ => NONE;   (*Typically (if thy doesn't include Numeral)
108                              Undeclared type constructor "Numeral.bin"*)
110 end;