src/Provers/Arith/cancel_numerals.ML
 author wenzelm Sun, 15 Oct 2000 19:50:35 +0200 changeset 10220 2a726de6e124 parent 9546 be095014e72f child 13484 d8f5d3391766 permissions -rw-r--r--
proper symbol markup with \isamath, \isatext; support sub/super scripts:
```
(*  Title:      Provers/Arith/cancel_numerals.ML
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright   2000  University of Cambridge

Cancel common coefficients in balanced expressions:

i + #m*u + j ~~ i' + #m'*u + j'  ==  #(m-m')*u + i + j ~~ i' + j'

where ~~ is an appropriate balancing operation (e.g. =, <=, <, -).

It works by (a) massaging both sides to bring the selected term to the front:

#m*u + (i + j) ~~ #m'*u + (i' + j')

(b) then using bal_add1 or bal_add2 to reach

#(m-m')*u + i + j ~~ i' + j'       (if m'<=m)

or

i + j ~~ #(m'-m)*u + i' + j'       (otherwise)
*)

signature CANCEL_NUMERALS_DATA =
sig
(*abstract syntax*)
val mk_sum: term list -> term
val dest_sum: term -> term list
val mk_bal: term * term -> term
val dest_bal: term -> term * term
val mk_coeff: int * term -> term
val dest_coeff: term -> int * term
val find_first_coeff: term -> term list -> int * term list
(*rules*)
(*proof tools*)
val prove_conv: tactic list -> Sign.sg ->
thm list -> term * term -> thm option
val trans_tac: thm option -> tactic (*applies the initial lemma*)
val norm_tac: tactic                (*proves the initial lemma*)
val numeral_simp_tac: tactic        (*proves the final theorem*)
val simplify_meta_eq: thm -> thm    (*simplifies the final theorem*)
end;

functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA):
sig
val proc: Sign.sg -> thm list -> term -> thm option
end
=
struct

(*For t = #n*u then put u in the table*)
fun update_by_coeff (tab, t) =
Termtab.update ((#2 (Data.dest_coeff t), ()), tab);

(*a left-to-right scan of terms1, seeking a term of the form #n*u, where
#m*u is in terms2 for some m*)
fun find_common (terms1,terms2) =
let val tab2 = foldl update_by_coeff (Termtab.empty, terms2)
fun seek [] = raise TERM("find_common", [])
| seek (t::terms) =
let val (_,u) = Data.dest_coeff t
in  if is_some (Termtab.lookup (tab2, u)) then u
else seek terms
end
in  seek terms1 end;

(*the simplification procedure*)
fun proc sg hyps t =
let (*first freeze any Vars in the term to prevent flex-flex problems*)
val rand_s = gensym"_"
fun mk_inst (var as Var((a,i),T))  =
(var,  Free((a ^ rand_s ^ string_of_int i), T))
val t' = subst_atomic (map mk_inst (term_vars t)) t
val (t1,t2) = Data.dest_bal t'
val terms1 = Data.dest_sum t1
and terms2 = Data.dest_sum t2
val u = find_common (terms1,terms2)
val (n1, terms1') = Data.find_first_coeff u terms1
and (n2, terms2') = Data.find_first_coeff u terms2
fun newshape (i,terms) = Data.mk_sum (Data.mk_coeff(i,u)::terms)
val reshape =  (*Move i*u to the front and put j*u into standard form
i + #m + j + k == #m + i + (j + k) *)
if n1=0 orelse n2=0 then   (*trivial, so do nothing*)
raise TERM("cancel_numerals", [])
else Data.prove_conv [Data.norm_tac] sg hyps
(t',
Data.mk_bal (newshape(n1,terms1'),
newshape(n2,terms2')))
in
apsome Data.simplify_meta_eq
(if n2<=n1 then
Data.prove_conv
[Data.trans_tac reshape, rtac Data.bal_add1 1,
Data.numeral_simp_tac] sg hyps
(t', Data.mk_bal (newshape(n1-n2,terms1'),
Data.mk_sum terms2'))
else
Data.prove_conv
[Data.trans_tac reshape, rtac Data.bal_add2 1,
Data.numeral_simp_tac] sg hyps
(t', Data.mk_bal (Data.mk_sum terms1',
newshape(n2-n1,terms2'))))
end
handle TERM _ => None
| TYPE _ => None;   (*Typically (if thy doesn't include Numeral)
Undeclared type constructor "Numeral.bin"*)

end;
```