src/Provers/Arith/cancel_div_mod.ML

moved theory Interval from the AFP

(*  Title:      Provers/Arith/cancel_div_mod.ML
    Author:     Tobias Nipkow, TU Muenchen

Cancel div and mod terms:

  A + n*(m div n) + B + (m mod n) + C  ==  A + B + C + m

FIXME: Is parameterized but assumes for simplicity that + and * are named
as in HOL
*)

signature CANCEL_DIV_MOD_DATA =
sig
  (*abstract syntax*)
  val div_name: string
  val mod_name: string
  val mk_binop: string -> term * term -> term
  val mk_sum: typ -> term list -> term
  val dest_sum: term -> term list
  (*logic*)
  val div_mod_eqs: thm list
  (* (n*(m div n) + m mod n) + k == m + k and
     ((m div n)*n + m mod n) + k == m + k *)
  val prove_eq_sums: Proof.context -> term * term -> thm
  (* must prove ac0-equivalence of sums *)
end;

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


functor Cancel_Div_Mod(Data: CANCEL_DIV_MOD_DATA): CANCEL_DIV_MOD =
struct

fun coll_div_mod (Const (\<^const_name>\<open>Groups.plus\<close>, _) $ s $ t) dms =
      coll_div_mod t (coll_div_mod s dms)
  | coll_div_mod (Const (\<^const_name>\<open>Groups.times\<close>, _) $ m $ (Const (d, _) $ s $ n))
                 (dms as (divs, mods)) =
      if d = Data.div_name andalso m = n then ((s, n) :: divs, mods) else dms
  | coll_div_mod (Const (\<^const_name>\<open>Groups.times\<close>, _) $ (Const (d, _) $ s $ n) $ m)
                 (dms as (divs, mods)) =
      if d = Data.div_name andalso m = n then ((s, n) :: divs, mods) else dms
  | coll_div_mod (Const (m, _) $ s $ n) (dms as (divs, mods)) =
      if m = Data.mod_name then (divs, (s, n) :: mods) else dms
  | coll_div_mod _ dms = dms;


(* Proof principle:
   1. (...div...)+(...mod...) == (div + mod) + rest
      in function rearrange
   2. (div + mod) + ?x = d + ?x
      Data.div_mod_eq
   ==> thesis by transitivity
*)

val mk_plus = Data.mk_binop \<^const_name>\<open>Groups.plus\<close>;
val mk_times = Data.mk_binop \<^const_name>\<open>Groups.times\<close>;

fun rearrange T t pq =
  let
    val ts = Data.dest_sum t;
    val dpq = Data.mk_binop Data.div_name pq;
    val d1 = mk_times (snd pq, dpq) and d2 = mk_times (dpq, snd pq);
    val d = if member (op =) ts d1 then d1 else d2;
    val m = Data.mk_binop Data.mod_name pq;
  in mk_plus (mk_plus (d, m), Data.mk_sum T (ts |> remove (op =) d |> remove (op =) m)) end

fun cancel ctxt t pq =
  let
    val teqt' = Data.prove_eq_sums ctxt (t, rearrange (fastype_of t) t pq);
  in hd (Data.div_mod_eqs RL [teqt' RS transitive_thm]) end;

fun proc ctxt ct =
  let
    val t = Thm.term_of ct;
    val (divs, mods) = coll_div_mod t ([], []);
  in
    if null divs orelse null mods then NONE
    else
      (case inter (op =) mods divs of
        pq :: _ => SOME (cancel ctxt t pq)
      | [] => NONE)
  end;

end