File ‹arith_data.ML›

(*  Title:      ZF/arith_data.ML
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

Arithmetic simplification: cancellation of common terms

signature ARITH_DATA =
  (*the main outcome*)
  val nateq_cancel_numerals_proc: Simplifier.proc
  val natless_cancel_numerals_proc: Simplifier.proc
  val natdiff_cancel_numerals_proc: Simplifier.proc
  (*tools for use in similar applications*)
  val gen_trans_tac: Proof.context -> thm -> thm option -> tactic
  val prove_conv: string -> tactic list -> Proof.context -> thm list -> term * term -> thm option
  val simplify_meta_eq: thm list -> Proof.context -> thm -> thm
  structure EqCancelNumeralsData   : CANCEL_NUMERALS_DATA
  structure LessCancelNumeralsData : CANCEL_NUMERALS_DATA
  structure DiffCancelNumeralsData : CANCEL_NUMERALS_DATA

structure ArithData: ARITH_DATA =

val zero = Constzero;
val succ = Constsucc;
fun mk_succ t = succ $ t;
val one = mk_succ zero;

fun mk_plus (t, u) = ConstArith.add for t u;

(*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);

(* dest_sum *)

fun dest_sum Const_zero = []
  | dest_sum Const_succ for t = one :: dest_sum t
  | dest_sum Const_Arith.add for t u = dest_sum t @ dest_sum u
  | dest_sum tm = [tm];

(*Apply the given rewrite (if present) just once*)
fun gen_trans_tac _ _ NONE = all_tac
  | gen_trans_tac ctxt th2 (SOME th) = ALLGOALS (resolve_tac ctxt [th RS th2]);

(*Use <-> or = depending on the type of t*)
fun mk_eq_iff(t,u) =
  if fastype_of t = Typei
  then ConstIFOL.eq Typei for t u
  else ConstIFOL.iff for 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 (dest_judgment (Thm.prop_of th));

fun prove_conv name tacs ctxt prems (t,u) =
  if t aconv u then NONE
  let val prems' = filter_out is_eq_thm prems
      val goal = Logic.list_implies (map Thm.prop_of prems', make_judgment (mk_eq_iff (t, u)));
  in SOME (prems' MRS Goal.prove ctxt [] [] goal (K (EVERY tacs)))
      handle ERROR msg =>
        (warning (msg ^ "\nCancellation failed: no typing information? (" ^ name ^ ")"); NONE)

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

fun mk_times (t, u) = ConstArith.mult for t u;

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);

fun dest_prod tm =
  let val (t,u) = Const_fnArith.mult for t u => (t, u) tm
  in  dest_prod t @ dest_prod u  end
  handle TERM _ => [tm];

(*Dummy version: the only arguments are 0 and 1*)
fun mk_coeff (0, 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, mk_prod (sort Term_Ord.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
        handle TERM _ => find_first_coeff (t::past) u terms;

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

(*Final simplification: cancel + and **)
fun simplify_meta_eq rules ctxt =
  let val ctxt' =
    put_simpset FOL_ss ctxt
      delsimps @{thms iff_simps} (*these could erase the whole rule!*)
      addsimps rules
      |> fold Simplifier.add_eqcong [@{thm eq_cong2}, @{thm iff_cong2}]
  in mk_meta_eq o simplify ctxt' end;

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

structure CancelNumeralsCommon =
  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_ss1 =
    simpset_of (put_simpset ZF_ss context addsimps add_0s @ add_succs @ mult_1s @ @{thms add_ac})
  val norm_ss2 =
    simpset_of (put_simpset ZF_ss context addsimps add_0s @ mult_1s @ @{thms add_ac} @
      @{thms mult_ac} @ tc_rules @ natifys)
  fun norm_tac ctxt =
    ALLGOALS (asm_simp_tac (put_simpset norm_ss1 ctxt))
    THEN ALLGOALS (asm_simp_tac (put_simpset norm_ss2 ctxt))
  val numeral_simp_ss =
    simpset_of (put_simpset ZF_ss context addsimps add_0s @ tc_rules @ natifys)
  fun numeral_simp_tac ctxt =
    ALLGOALS (asm_simp_tac (put_simpset numeral_simp_ss ctxt))
  val simplify_meta_eq  = simplify_meta_eq final_rules

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

structure EqCancelNumeralsData =
  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 = @{thm eq_add_iff [THEN iff_trans]}
  val bal_add2 = @{thm eq_add_iff [THEN iff_trans]}
  fun trans_tac ctxt = gen_trans_tac ctxt @{thm iff_trans}

structure EqCancelNumerals = CancelNumeralsFun(EqCancelNumeralsData);

structure LessCancelNumeralsData =
  open CancelNumeralsCommon
  val prove_conv = prove_conv "natless_cancel_numerals"
  fun mk_bal (t, u) = for t u
  val dest_bal = for t u => (t, u)
  val bal_add1 = @{thm less_add_iff [THEN iff_trans]}
  val bal_add2 = @{thm less_add_iff [THEN iff_trans]}
  fun trans_tac ctxt = gen_trans_tac ctxt @{thm iff_trans}

structure LessCancelNumerals = CancelNumeralsFun(LessCancelNumeralsData);

structure DiffCancelNumeralsData =
  open CancelNumeralsCommon
  val prove_conv = prove_conv "natdiff_cancel_numerals"
  fun mk_bal (t, u) = ConstArith.diff for t u
  val dest_bal = Const_fnArith.diff for t u => (t, u)
  val bal_add1 = @{thm diff_add_eq [THEN trans]}
  val bal_add2 = @{thm diff_add_eq [THEN trans]}
  fun trans_tac ctxt = gen_trans_tac ctxt @{thm trans}

structure DiffCancelNumerals = CancelNumeralsFun(DiffCancelNumeralsData);

val nateq_cancel_numerals_proc = EqCancelNumerals.proc;
val natless_cancel_numerals_proc = LessCancelNumerals.proc;
val natdiff_cancel_numerals_proc = DiffCancelNumerals.proc;