src/ZF/arith_data.ML
author wenzelm
Wed, 27 Mar 2013 16:38:25 +0100
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more ambitious Goal.skip_proofs: covers Goal.prove forms as well, and do not insist in quick_and_dirty (for the sake of Isabelle/jEdit);
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(*  Title:      ZF/arith_data.ML
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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Arithmetic simplification: cancellation of common terms
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*)
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signature ARITH_DATA =
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sig
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  (*the main outcome*)
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  val nat_cancel: simproc list
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  (*tools for use in similar applications*)
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  val gen_trans_tac: thm -> thm option -> tactic
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  val prove_conv: string -> tactic list -> Proof.context -> thm list -> term * term -> thm option
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  val simplify_meta_eq: thm list -> simpset -> thm -> thm
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  (*debugging*)
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  structure EqCancelNumeralsData   : CANCEL_NUMERALS_DATA
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  structure LessCancelNumeralsData : CANCEL_NUMERALS_DATA
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  structure DiffCancelNumeralsData : CANCEL_NUMERALS_DATA
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end;
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structure ArithData: ARITH_DATA =
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struct
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val iT = Ind_Syntax.iT;
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val zero = Const(@{const_name zero}, iT);
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val succ = Const(@{const_name succ}, iT --> iT);
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fun mk_succ t = succ $ t;
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val one = mk_succ zero;
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val mk_plus = FOLogic.mk_binop @{const_name Arith.add};
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(*Thus mk_sum[t] yields t+#0; longer sums don't have a trailing zero*)
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fun mk_sum []        = zero
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  | mk_sum [t,u]     = mk_plus (t, u)
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  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
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(*this version ALWAYS includes a trailing zero*)
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fun long_mk_sum []        = zero
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  | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
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val dest_plus = FOLogic.dest_bin @{const_name Arith.add} iT;
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(* dest_sum *)
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fun dest_sum (Const(@{const_name zero},_)) = []
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  | dest_sum (Const(@{const_name succ},_) $ t) = one :: dest_sum t
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  | dest_sum (Const(@{const_name Arith.add},_) $ t $ u) = dest_sum t @ dest_sum u
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  | dest_sum tm = [tm];
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(*Apply the given rewrite (if present) just once*)
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fun gen_trans_tac th2 NONE      = all_tac
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  | gen_trans_tac th2 (SOME th) = ALLGOALS (rtac (th RS th2));
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(*Use <-> or = depending on the type of t*)
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fun mk_eq_iff(t,u) =
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  if fastype_of t = iT then FOLogic.mk_eq(t,u)
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                       else FOLogic.mk_iff(t,u);
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(*We remove equality assumptions because they confuse the simplifier and
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  because only type-checking assumptions are necessary.*)
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fun is_eq_thm th =
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    can FOLogic.dest_eq (FOLogic.dest_Trueprop (Thm.prop_of th));
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fun add_chyps chyps ct = Drule.list_implies (map cprop_of chyps, ct);
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fun prove_conv name tacs ctxt prems (t,u) =
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  if t aconv u then NONE
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  else
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  let val prems' = filter_out is_eq_thm prems
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      val goal = Logic.list_implies (map Thm.prop_of prems',
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        FOLogic.mk_Trueprop (mk_eq_iff (t, u)));
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  in SOME (prems' MRS Goal.prove ctxt [] [] goal (K (EVERY tacs)))
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      handle ERROR msg =>
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        (warning (msg ^ "\nCancellation failed: no typing information? (" ^ name ^ ")"); NONE)
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  end;
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fun prep_simproc thy (name, pats, proc) =
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  Simplifier.simproc_global thy name pats proc;
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(*** Use CancelNumerals simproc without binary numerals,
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     just for cancellation ***)
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val mk_times = FOLogic.mk_binop @{const_name Arith.mult};
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fun mk_prod [] = one
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  | mk_prod [t] = t
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  | mk_prod (t :: ts) = if t = one then mk_prod ts
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                        else mk_times (t, mk_prod ts);
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val dest_times = FOLogic.dest_bin @{const_name Arith.mult} iT;
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fun dest_prod t =
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      let val (t,u) = dest_times t
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      in  dest_prod t @ dest_prod u  end
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      handle TERM _ => [t];
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(*Dummy version: the only arguments are 0 and 1*)
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fun mk_coeff (0, t) = zero
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  | mk_coeff (1, t) = t
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  | mk_coeff _       = raise TERM("mk_coeff", []);
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(*Dummy version: the "coefficient" is always 1.
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  In the result, the factors are sorted terms*)
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fun dest_coeff t = (1, mk_prod (sort Term_Ord.term_ord (dest_prod t)));
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(*Find first coefficient-term THAT MATCHES u*)
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fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
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  | find_first_coeff past u (t::terms) =
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        let val (n,u') = dest_coeff t
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        in  if u aconv u' then (n, rev past @ terms)
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                          else find_first_coeff (t::past) u terms
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        end
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        handle TERM _ => find_first_coeff (t::past) u terms;
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(*Simplify #1*n and n*#1 to n*)
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val add_0s = [@{thm add_0_natify}, @{thm add_0_right_natify}];
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val add_succs = [@{thm add_succ}, @{thm add_succ_right}];
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val mult_1s = [@{thm mult_1_natify}, @{thm mult_1_right_natify}];
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val tc_rules = [@{thm natify_in_nat}, @{thm add_type}, @{thm diff_type}, @{thm mult_type}];
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val natifys = [@{thm natify_0}, @{thm natify_ident}, @{thm add_natify1}, @{thm add_natify2},
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               @{thm diff_natify1}, @{thm diff_natify2}];
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(*Final simplification: cancel + and **)
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fun simplify_meta_eq rules =
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  let val ss0 =
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    FOL_ss
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      delsimps @{thms iff_simps} (*these could erase the whole rule!*)
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      addsimps rules
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      |> fold Simplifier.add_eqcong [@{thm eq_cong2}, @{thm iff_cong2}]
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  in fn ss => mk_meta_eq o simplify (Simplifier.inherit_context ss ss0) end;
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val final_rules = add_0s @ mult_1s @ [@{thm mult_0}, @{thm mult_0_right}];
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structure CancelNumeralsCommon =
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  struct
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  val mk_sum            = (fn T:typ => mk_sum)
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  val dest_sum          = dest_sum
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  val mk_coeff          = mk_coeff
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  val dest_coeff        = dest_coeff
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  val find_first_coeff  = find_first_coeff []
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  val norm_ss1 = ZF_ss addsimps add_0s @ add_succs @ mult_1s @ @{thms add_ac}
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  val norm_ss2 = ZF_ss addsimps add_0s @ mult_1s @ @{thms add_ac} @
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    @{thms mult_ac} @ tc_rules @ natifys
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  fun norm_tac ss =
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    ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1))
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    THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2))
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  val numeral_simp_ss = ZF_ss addsimps add_0s @ tc_rules @ natifys
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  fun numeral_simp_tac ss =
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    ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
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  val simplify_meta_eq  = simplify_meta_eq final_rules
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  end;
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(** The functor argumnets are declared as separate structures
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    so that they can be exported to ease debugging. **)
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structure EqCancelNumeralsData =
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  struct
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  open CancelNumeralsCommon
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  val prove_conv = prove_conv "nateq_cancel_numerals"
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  val mk_bal   = FOLogic.mk_eq
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  val dest_bal = FOLogic.dest_eq
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  val bal_add1 = @{thm eq_add_iff} RS @{thm iff_trans}
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  val bal_add2 = @{thm eq_add_iff} RS @{thm iff_trans}
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  val trans_tac = gen_trans_tac @{thm iff_trans}
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  end;
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structure EqCancelNumerals = CancelNumeralsFun(EqCancelNumeralsData);
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structure LessCancelNumeralsData =
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  struct
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  open CancelNumeralsCommon
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  val prove_conv = prove_conv "natless_cancel_numerals"
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  val mk_bal   = FOLogic.mk_binrel @{const_name Ordinal.lt}
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  val dest_bal = FOLogic.dest_bin @{const_name Ordinal.lt} iT
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  val bal_add1 = @{thm less_add_iff} RS @{thm iff_trans}
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  val bal_add2 = @{thm less_add_iff} RS @{thm iff_trans}
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  val trans_tac = gen_trans_tac @{thm iff_trans}
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  end;
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structure LessCancelNumerals = CancelNumeralsFun(LessCancelNumeralsData);
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structure DiffCancelNumeralsData =
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  struct
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  open CancelNumeralsCommon
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  val prove_conv = prove_conv "natdiff_cancel_numerals"
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  val mk_bal   = FOLogic.mk_binop @{const_name Arith.diff}
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  val dest_bal = FOLogic.dest_bin @{const_name Arith.diff} iT
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  val bal_add1 = @{thm diff_add_eq} RS @{thm trans}
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  val bal_add2 = @{thm diff_add_eq} RS @{thm trans}
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  val trans_tac = gen_trans_tac @{thm trans}
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  end;
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structure DiffCancelNumerals = CancelNumeralsFun(DiffCancelNumeralsData);
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val nat_cancel =
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  map (prep_simproc @{theory})
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   [("nateq_cancel_numerals",
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     ["l #+ m = n", "l = m #+ n",
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      "l #* m = n", "l = m #* n",
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      "succ(m) = n", "m = succ(n)"],
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     (K EqCancelNumerals.proc)),
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    ("natless_cancel_numerals",
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     ["l #+ m < n", "l < m #+ n",
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      "l #* m < n", "l < m #* n",
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      "succ(m) < n", "m < succ(n)"],
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     (K LessCancelNumerals.proc)),
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    ("natdiff_cancel_numerals",
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     ["(l #+ m) #- n", "l #- (m #+ n)",
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      "(l #* m) #- n", "l #- (m #* n)",
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      "succ(m) #- n", "m #- succ(n)"],
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     (K DiffCancelNumerals.proc))];
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end;
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Addsimprocs ArithData.nat_cancel;
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(*examples:
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print_depth 22;
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set timing;
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set simp_trace;
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fun test s = (Goal s; by (Asm_simp_tac 1));
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test "x #+ y = x #+ z";
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test "y #+ x = x #+ z";
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test "x #+ y #+ z = x #+ z";
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test "y #+ (z #+ x) = z #+ x";
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test "x #+ y #+ z = (z #+ y) #+ (x #+ w)";
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test "x#*y #+ z = (z #+ y) #+ (y#*x #+ w)";
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test "x #+ succ(y) = x #+ z";
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test "x #+ succ(y) = succ(z #+ x)";
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test "succ(x) #+ succ(y) #+ z = succ(z #+ y) #+ succ(x #+ w)";
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test "(x #+ y) #- (x #+ z) = w";
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test "(y #+ x) #- (x #+ z) = dd";
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test "(x #+ y #+ z) #- (x #+ z) = dd";
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test "(y #+ (z #+ x)) #- (z #+ x) = dd";
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test "(x #+ y #+ z) #- ((z #+ y) #+ (x #+ w)) = dd";
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test "(x#*y #+ z) #- ((z #+ y) #+ (y#*x #+ w)) = dd";
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(*BAD occurrence of natify*)
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test "(x #+ succ(y)) #- (x #+ z) = dd";
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test "x #* y2 #+ y #* x2 = y #* x2 #+ x #* y2";
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test "(x #+ succ(y)) #- (succ(z #+ x)) = dd";
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test "(succ(x) #+ succ(y) #+ z) #- (succ(z #+ y) #+ succ(x #+ w)) = dd";
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(*use of typing information*)
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test "x : nat ==> x #+ y = x";
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test "x : nat --> x #+ y = x";
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test "x : nat ==> x #+ y < x";
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test "x : nat ==> x < y#+x";
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test "x : nat ==> x le succ(x)";
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(*fails: no typing information isn't visible*)
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test "x #+ y = x";
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test "x #+ y < x #+ z";
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test "y #+ x < x #+ z";
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test "x #+ y #+ z < x #+ z";
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test "y #+ z #+ x < x #+ z";
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test "y #+ (z #+ x) < z #+ x";
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test "x #+ y #+ z < (z #+ y) #+ (x #+ w)";
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test "x#*y #+ z < (z #+ y) #+ (y#*x #+ w)";
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test "x #+ succ(y) < x #+ z";
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test "x #+ succ(y) < succ(z #+ x)";
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test "succ(x) #+ succ(y) #+ z < succ(z #+ y) #+ succ(x #+ w)";
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test "x #+ succ(y) le succ(z #+ x)";
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*)