# File arith_data.ML

    (*  Title:      ZF/arith_data.ML
        ID:         $Id: arith_data.ML,v 1.16 2005/08/01 17:20:26 wenzelm Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   2000  University of Cambridge

    Arithmetic simplification: cancellation of common terms
    *)

    signature ARITH_DATA =
    sig
      (*the main outcome*)
      val nat_cancel: simproc list
      (*tools for use in similar applications*)
      val gen_trans_tac: thm -> thm option -> tactic
      val prove_conv: string -> tactic list -> Sign.sg ->
                      thm list -> string list -> term * term -> thm option
      val simplify_meta_eq: thm list -> simpset -> thm -> thm
      (*debugging*)
      structure EqCancelNumeralsData   : CANCEL_NUMERALS_DATA
      structure LessCancelNumeralsData : CANCEL_NUMERALS_DATA
      structure DiffCancelNumeralsData : CANCEL_NUMERALS_DATA
    end;

    structure ArithData: ARITH_DATA =
    struct

    val iT = Ind_Syntax.iT;

    val zero = Const("0", iT);
    val succ = Const("succ", iT --> iT);
    fun mk_succ t = succ $ t;
    val one = mk_succ zero;

    val mk_plus = FOLogic.mk_binop "Arith.add";

    (*Thus mk_sum[t] yields t+#0; longer sums don't have a trailing zero*)
    fun mk_sum []        = zero
      | mk_sum [t,u]     = mk_plus (t, u)
      | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);

    (*this version ALWAYS includes a trailing zero*)
    fun long_mk_sum []        = zero
      | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);

    val dest_plus = FOLogic.dest_bin "Arith.add" iT;

    (* dest_sum *)

    fun dest_sum (Const("0",_)) = []
      | dest_sum (Const("succ",_) $ t) = one :: dest_sum t
      | dest_sum (Const("Arith.add",_) $ t $ u) = dest_sum t @ dest_sum u
      | dest_sum tm = [tm];

    (*Apply the given rewrite (if present) just once*)
    fun gen_trans_tac th2 NONE      = all_tac
      | gen_trans_tac th2 (SOME th) = ALLGOALS (rtac (th RS th2));

    (*Use <-> or = depending on the type of t*)
    fun mk_eq_iff(t,u) =
      if fastype_of t = iT then FOLogic.mk_eq(t,u)
                           else FOLogic.mk_iff(t,u);

    (*We remove equality assumptions because they confuse the simplifier and
      because only type-checking assumptions are necessary.*)
    fun is_eq_thm th =
        can FOLogic.dest_eq (FOLogic.dest_Trueprop (#prop (rep_thm th)));

    fun add_chyps chyps ct = Drule.list_implies (map cprop_of chyps, ct);

    fun prove_conv name tacs sg hyps xs (t,u) =
      if t aconv u then NONE
      else
      let val hyps' = List.filter (not o is_eq_thm) hyps
          val goal = Logic.list_implies (map (#prop o Thm.rep_thm) hyps',
            FOLogic.mk_Trueprop (mk_eq_iff (t, u)));
      in SOME (hyps' MRS Tactic.prove sg xs [] goal (K (EVERY tacs)))
          handle ERROR_MESSAGE msg =>
            (warning (msg ^ "\nCancellation failed: no typing information? (" ^ name ^ ")"); NONE)
      end;

    fun prep_simproc (name, pats, proc) =
      Simplifier.simproc (Theory.sign_of (the_context ())) name pats proc;

    (*** Use CancelNumerals simproc without binary numerals,
         just for cancellation ***)

    val mk_times = FOLogic.mk_binop "Arith.mult";

    fun mk_prod [] = one
      | mk_prod [t] = t
      | mk_prod (t :: ts) = if t = one then mk_prod ts
                            else mk_times (t, mk_prod ts);

    val dest_times = FOLogic.dest_bin "Arith.mult" iT;

    fun dest_prod t =
          let val (t,u) = dest_times t
          in  dest_prod t @ dest_prod u  end
          handle TERM _ => [t];

    (*Dummy version: the only arguments are 0 and 1*)
    fun mk_coeff (0: IntInf.int, t) = zero
      | mk_coeff (1, t) = t
      | mk_coeff _       = raise TERM("mk_coeff", []);

    (*Dummy version: the "coefficient" is always 1.
      In the result, the factors are sorted terms*)
    fun dest_coeff t = (1 : IntInf.int, mk_prod (sort Term.term_ord (dest_prod t)));

    (*Find first coefficient-term THAT MATCHES u*)
    fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
      | find_first_coeff past u (t::terms) =
            let val (n,u') = dest_coeff t
            in  if u aconv u' then (n, rev past @ terms)
                              else find_first_coeff (t::past) u terms
            end
            handle TERM _ => find_first_coeff (t::past) u terms;

    (*Simplify #1*n and n*#1 to n*)
    val add_0s = [add_0_natify, add_0_right_natify];
    val add_succs = [add_succ, add_succ_right];
    val mult_1s = [mult_1_natify, mult_1_right_natify];
    val tc_rules = [natify_in_nat, add_type, diff_type, mult_type];
    val natifys = [natify_0, natify_ident, add_natify1, add_natify2,
                   diff_natify1, diff_natify2];

    (*Final simplification: cancel + and **)
    fun simplify_meta_eq rules ss =
        mk_meta_eq o
        simplify (Simplifier.inherit_bounds ss FOL_ss addeqcongs[eq_cong2,iff_cong2]
                         delsimps iff_simps (*these could erase the whole rule!*)
                         addsimps rules);

    val final_rules = add_0s @ mult_1s @ [mult_0, mult_0_right];

    structure CancelNumeralsCommon =
      struct
      val mk_sum            = (fn T:typ => mk_sum)
      val dest_sum          = dest_sum
      val mk_coeff          = mk_coeff
      val dest_coeff        = dest_coeff
      val find_first_coeff  = find_first_coeff []
      val norm_tac_ss1      = ZF_ss addsimps add_0s @ add_succs @ mult_1s @ add_ac
      val norm_tac_ss2      = ZF_ss addsimps add_0s @ mult_1s @ add_ac @ mult_ac @ tc_rules @ natifys
      fun norm_tac ss =
        ALLGOALS (asm_simp_tac (Simplifier.inherit_bounds ss norm_tac_ss1))
        THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_bounds ss norm_tac_ss2))
      val numeral_simp_tac_ss = ZF_ss addsimps add_0s @ tc_rules @ natifys
      fun numeral_simp_tac ss =
        ALLGOALS (asm_simp_tac (Simplifier.inherit_bounds ss numeral_simp_tac_ss))
      val simplify_meta_eq  = simplify_meta_eq final_rules
      end;

    (** The functor argumnets are declared as separate structures
        so that they can be exported to ease debugging. **)

    structure EqCancelNumeralsData =
      struct
      open CancelNumeralsCommon
      val prove_conv = prove_conv "nateq_cancel_numerals"
      val mk_bal   = FOLogic.mk_eq
      val dest_bal = FOLogic.dest_eq
      val bal_add1 = eq_add_iff RS iff_trans
      val bal_add2 = eq_add_iff RS iff_trans
      fun trans_tac _ = gen_trans_tac iff_trans
      end;

    structure EqCancelNumerals = CancelNumeralsFun(EqCancelNumeralsData);

    structure LessCancelNumeralsData =
      struct
      open CancelNumeralsCommon
      val prove_conv = prove_conv "natless_cancel_numerals"
      val mk_bal   = FOLogic.mk_binrel "Ordinal.lt"
      val dest_bal = FOLogic.dest_bin "Ordinal.lt" iT
      val bal_add1 = less_add_iff RS iff_trans
      val bal_add2 = less_add_iff RS iff_trans
      fun trans_tac _ = gen_trans_tac iff_trans
      end;

    structure LessCancelNumerals = CancelNumeralsFun(LessCancelNumeralsData);

    structure DiffCancelNumeralsData =
      struct
      open CancelNumeralsCommon
      val prove_conv = prove_conv "natdiff_cancel_numerals"
      val mk_bal   = FOLogic.mk_binop "Arith.diff"
      val dest_bal = FOLogic.dest_bin "Arith.diff" iT
      val bal_add1 = diff_add_eq RS trans
      val bal_add2 = diff_add_eq RS trans
      fun trans_tac _ = gen_trans_tac trans
      end;

    structure DiffCancelNumerals = CancelNumeralsFun(DiffCancelNumeralsData);

    val nat_cancel =
      map prep_simproc
       [("nateq_cancel_numerals",
         ["l #+ m = n", "l = m #+ n",
          "l #* m = n", "l = m #* n",
          "succ(m) = n", "m = succ(n)"],
         EqCancelNumerals.proc),
        ("natless_cancel_numerals",
         ["l #+ m < n", "l < m #+ n",
          "l #* m < n", "l < m #* n",
          "succ(m) < n", "m < succ(n)"],
         LessCancelNumerals.proc),
        ("natdiff_cancel_numerals",
         ["(l #+ m) #- n", "l #- (m #+ n)",
          "(l #* m) #- n", "l #- (m #* n)",
          "succ(m) #- n", "m #- succ(n)"],
         DiffCancelNumerals.proc)];

    end;

    Addsimprocs ArithData.nat_cancel;

    (*examples:
    print_depth 22;
    set timing;
    set trace_simp;
    fun test s = (Goal s; by (Asm_simp_tac 1));

    test "x #+ y = x #+ z";
    test "y #+ x = x #+ z";
    test "x #+ y #+ z = x #+ z";
    test "y #+ (z #+ x) = z #+ x";
    test "x #+ y #+ z = (z #+ y) #+ (x #+ w)";
    test "x#*y #+ z = (z #+ y) #+ (y#*x #+ w)";

    test "x #+ succ(y) = x #+ z";
    test "x #+ succ(y) = succ(z #+ x)";
    test "succ(x) #+ succ(y) #+ z = succ(z #+ y) #+ succ(x #+ w)";

    test "(x #+ y) #- (x #+ z) = w";
    test "(y #+ x) #- (x #+ z) = dd";
    test "(x #+ y #+ z) #- (x #+ z) = dd";
    test "(y #+ (z #+ x)) #- (z #+ x) = dd";
    test "(x #+ y #+ z) #- ((z #+ y) #+ (x #+ w)) = dd";
    test "(x#*y #+ z) #- ((z #+ y) #+ (y#*x #+ w)) = dd";

    (*BAD occurrence of natify*)
    test "(x #+ succ(y)) #- (x #+ z) = dd";

    test "x #* y2 #+ y #* x2 = y #* x2 #+ x #* y2";

    test "(x #+ succ(y)) #- (succ(z #+ x)) = dd";
    test "(succ(x) #+ succ(y) #+ z) #- (succ(z #+ y) #+ succ(x #+ w)) = dd";

    (*use of typing information*)
    test "x : nat ==> x #+ y = x";
    test "x : nat --> x #+ y = x";
    test "x : nat ==> x #+ y < x";
    test "x : nat ==> x < y#+x";
    test "x : nat ==> x le succ(x)";

    (*fails: no typing information isn't visible*)
    test "x #+ y = x";

    test "x #+ y < x #+ z";
    test "y #+ x < x #+ z";
    test "x #+ y #+ z < x #+ z";
    test "y #+ z #+ x < x #+ z";
    test "y #+ (z #+ x) < z #+ x";
    test "x #+ y #+ z < (z #+ y) #+ (x #+ w)";
    test "x#*y #+ z < (z #+ y) #+ (y#*x #+ w)";

    test "x #+ succ(y) < x #+ z";
    test "x #+ succ(y) < succ(z #+ x)";
    test "succ(x) #+ succ(y) #+ z < succ(z #+ y) #+ succ(x #+ w)";

    test "x #+ succ(y) le succ(z #+ x)";
    *)