# File ‹~~/src/Provers/Arith/cancel_numerals.ML›

```(*  Title:      Provers/Arith/cancel_numerals.ML
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

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')

#(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: typ -> 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 -> Proof.context -> thm list -> term * term -> thm option
val trans_tac: Proof.context -> thm option -> tactic            (*applies the initial lemma*)
val norm_tac: Proof.context -> tactic          (*proves the initial lemma*)
val numeral_simp_tac: Proof.context -> tactic  (*proves the final theorem*)
val simplify_meta_eq: Proof.context -> thm -> thm (*simplifies the final theorem*)
end;

signature CANCEL_NUMERALS =
sig
val proc: Proof.context -> cterm -> thm option
end;

functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA): CANCEL_NUMERALS =
struct

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

(*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 = fold update_by_coeff terms2 Termtab.empty
fun seek [] = raise TERM("find_common", [])
| seek (t::terms) =
let val (_,u) = Data.dest_coeff t
in if Termtab.defined tab2 u then u else seek terms end
in  seek terms1 end;

(*the simplification procedure*)
fun proc ctxt ct =
let
val prems = Simplifier.prems_of ctxt
val t = Thm.term_of ct
val (t', ctxt') = yield_singleton (Variable.import_terms true) t ctxt

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
and T = Term.fastype_of u

fun newshape (i,terms) = Data.mk_sum T (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 ctxt'] ctxt' prems
(t', Data.mk_bal (newshape(n1,terms1'), newshape(n2,terms2')))
in
(if n2 <= n1 then
Data.prove_conv
[Data.trans_tac ctxt' reshape, resolve_tac ctxt' [Data.bal_add1] 1,
Data.numeral_simp_tac ctxt'] ctxt' prems
(t', Data.mk_bal (newshape(n1-n2,terms1'), Data.mk_sum T terms2'))
else
Data.prove_conv
[Data.trans_tac ctxt' reshape, resolve_tac ctxt' [Data.bal_add2] 1,
Data.numeral_simp_tac ctxt'] ctxt' prems
(t', Data.mk_bal (Data.mk_sum T terms1', newshape(n2-n1,terms2'))))
|> Option.map
(Data.simplify_meta_eq ctxt' #>
singleton (Variable.export ctxt' ctxt))
end
(* FIXME avoid handling of generic exceptions *)
handle TERM _ => NONE
| TYPE _ => NONE;   (*Typically (if thy doesn't include Numeral)
Undeclared type constructor "Numeral.bin"*)
end;
```