src/HOL/Tools/arith_data.ML
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use long names for old-style fold combinators;
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(*  Title:      HOL/arith_data.ML
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    Author:     Markus Wenzel, Stefan Berghofer, and Tobias Nipkow
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Basic arithmetic proof tools.
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*)
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signature ARITH_DATA =
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sig
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  val prove_conv: tactic -> (simpset -> tactic) -> simpset -> term * term -> thm
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  val simp_all_tac: thm list -> simpset -> tactic
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  val mk_sum: term list -> term
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  val mk_norm_sum: term list -> term
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  val dest_sum: term -> term list
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  val nat_cancel_sums_add: simproc list
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  val nat_cancel_sums: simproc list
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  val setup: Context.generic -> Context.generic
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end;
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structure ArithData: ARITH_DATA =
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struct
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(** generic proof tools **)
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(* prove conversions *)
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fun prove_conv expand_tac norm_tac ss tu =  (* FIXME avoid standard *)
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  mk_meta_eq (standard (Goal.prove (Simplifier.the_context ss) [] []
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      (HOLogic.mk_Trueprop (HOLogic.mk_eq tu))
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    (K (EVERY [expand_tac, norm_tac ss]))));
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(* rewriting *)
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fun simp_all_tac rules =
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  let val ss0 = HOL_ss addsimps rules
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  in fn ss => ALLGOALS (simp_tac (Simplifier.inherit_context ss ss0)) end;
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(** abstract syntax of structure nat: 0, Suc, + **)
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local
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val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
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val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} HOLogic.natT;
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in
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fun mk_sum [] = HOLogic.zero
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  | mk_sum [t] = t
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  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
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(*normal form of sums: Suc (... (Suc (a + (b + ...))))*)
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fun mk_norm_sum ts =
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  let val (ones, sums) = List.partition (equal HOLogic.Suc_zero) ts in
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    funpow (length ones) HOLogic.mk_Suc (mk_sum sums)
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  end;
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fun dest_sum tm =
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  if HOLogic.is_zero tm then []
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  else
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    (case try HOLogic.dest_Suc tm of
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      SOME t => HOLogic.Suc_zero :: dest_sum t
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    | NONE =>
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        (case try dest_plus tm of
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          SOME (t, u) => dest_sum t @ dest_sum u
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        | NONE => [tm]));
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end;
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(** cancel common summands **)
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structure Sum =
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struct
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  val mk_sum = mk_norm_sum;
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  val dest_sum = dest_sum;
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  val prove_conv = prove_conv;
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  val norm_tac1 = simp_all_tac [@{thm "add_Suc"}, @{thm "add_Suc_right"},
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    @{thm "add_0"}, @{thm "add_0_right"}];
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  val norm_tac2 = simp_all_tac @{thms add_ac};
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  fun norm_tac ss = norm_tac1 ss THEN norm_tac2 ss;
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end;
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fun gen_uncancel_tac rule ct =
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  rtac (instantiate' [] [NONE, SOME ct] (rule RS @{thm subst_equals})) 1;
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(* nat eq *)
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structure EqCancelSums = CancelSumsFun
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(struct
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  open Sum;
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  val mk_bal = HOLogic.mk_eq;
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  val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT;
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  val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel"};
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end);
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(* nat less *)
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structure LessCancelSums = CancelSumsFun
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(struct
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  open Sum;
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  val mk_bal = HOLogic.mk_binrel @{const_name HOL.less};
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  val dest_bal = HOLogic.dest_bin @{const_name HOL.less} HOLogic.natT;
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  val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel_less"};
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end);
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(* nat le *)
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structure LeCancelSums = CancelSumsFun
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(struct
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  open Sum;
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  val mk_bal = HOLogic.mk_binrel @{const_name HOL.less_eq};
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  val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} HOLogic.natT;
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  val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel_le"};
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end);
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(* nat diff *)
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structure DiffCancelSums = CancelSumsFun
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(struct
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  open Sum;
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  val mk_bal = HOLogic.mk_binop @{const_name HOL.minus};
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  val dest_bal = HOLogic.dest_bin @{const_name HOL.minus} HOLogic.natT;
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  val uncancel_tac = gen_uncancel_tac @{thm "diff_cancel"};
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end);
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(* prepare nat_cancel simprocs *)
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val nat_cancel_sums_add =
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  [Simplifier.simproc (the_context ()) "nateq_cancel_sums"
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     ["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"]
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     (K EqCancelSums.proc),
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   Simplifier.simproc (the_context ()) "natless_cancel_sums"
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     ["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"]
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     (K LessCancelSums.proc),
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   Simplifier.simproc (the_context ()) "natle_cancel_sums"
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     ["(l::nat) + m <= n", "(l::nat) <= m + n", "Suc m <= n", "m <= Suc n"]
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     (K LeCancelSums.proc)];
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val nat_cancel_sums = nat_cancel_sums_add @
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  [Simplifier.simproc (the_context ()) "natdiff_cancel_sums"
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    ["((l::nat) + m) - n", "(l::nat) - (m + n)", "Suc m - n", "m - Suc n"]
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    (K DiffCancelSums.proc)];
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val setup =
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  Simplifier.map_ss (fn ss => ss addsimprocs nat_cancel_sums);
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end;