(* Title: HOL/arith_data.ML
ID: $Id$
Author: Markus Wenzel, Stefan Berghofer, and Tobias Nipkow
Basic arithmetic proof tools.
*)
(*** cancellation of common terms ***)
structure NatArithUtils =
struct
(** abstract syntax of structure nat: 0, Suc, + **)
(* mk_sum, mk_norm_sum *)
val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
fun mk_sum [] = HOLogic.zero
| mk_sum [t] = t
| mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
(*normal form of sums: Suc (... (Suc (a + (b + ...))))*)
fun mk_norm_sum ts =
let val (ones, sums) = List.partition (equal HOLogic.Suc_zero) ts in
funpow (length ones) HOLogic.mk_Suc (mk_sum sums)
end;
(* dest_sum *)
val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} HOLogic.natT;
fun dest_sum tm =
if HOLogic.is_zero tm then []
else
(case try HOLogic.dest_Suc tm of
SOME t => HOLogic.Suc_zero :: dest_sum t
| NONE =>
(case try dest_plus tm of
SOME (t, u) => dest_sum t @ dest_sum u
| NONE => [tm]));
(** generic proof tools **)
(* prove conversions *)
fun prove_conv expand_tac norm_tac ss tu = (* FIXME avoid standard *)
mk_meta_eq (standard (Goal.prove (Simplifier.the_context ss) [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq tu))
(K (EVERY [expand_tac, norm_tac ss]))));
(* rewriting *)
fun simp_all_tac rules =
let val ss0 = HOL_ss addsimps rules
in fn ss => ALLGOALS (simp_tac (Simplifier.inherit_context ss ss0)) end;
fun prep_simproc (name, pats, proc) =
Simplifier.simproc (the_context ()) name pats proc;
end;
(** ArithData **)
signature ARITH_DATA =
sig
val nat_cancel_sums_add: simproc list
val nat_cancel_sums: simproc list
val arith_data_setup: Context.generic -> Context.generic
end;
structure ArithData: ARITH_DATA =
struct
open NatArithUtils;
(** cancel common summands **)
structure Sum =
struct
val mk_sum = mk_norm_sum;
val dest_sum = dest_sum;
val prove_conv = prove_conv;
val norm_tac1 = simp_all_tac [@{thm "add_Suc"}, @{thm "add_Suc_right"},
@{thm "add_0"}, @{thm "add_0_right"}];
val norm_tac2 = simp_all_tac @{thms add_ac};
fun norm_tac ss = norm_tac1 ss THEN norm_tac2 ss;
end;
fun gen_uncancel_tac rule ct =
rtac (instantiate' [] [NONE, SOME ct] (rule RS @{thm subst_equals})) 1;
(* nat eq *)
structure EqCancelSums = CancelSumsFun
(struct
open Sum;
val mk_bal = HOLogic.mk_eq;
val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT;
val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel"};
end);
(* nat less *)
structure LessCancelSums = CancelSumsFun
(struct
open Sum;
val mk_bal = HOLogic.mk_binrel @{const_name HOL.less};
val dest_bal = HOLogic.dest_bin @{const_name HOL.less} HOLogic.natT;
val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel_less"};
end);
(* nat le *)
structure LeCancelSums = CancelSumsFun
(struct
open Sum;
val mk_bal = HOLogic.mk_binrel @{const_name HOL.less_eq};
val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} HOLogic.natT;
val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel_le"};
end);
(* nat diff *)
structure DiffCancelSums = CancelSumsFun
(struct
open Sum;
val mk_bal = HOLogic.mk_binop @{const_name HOL.minus};
val dest_bal = HOLogic.dest_bin @{const_name HOL.minus} HOLogic.natT;
val uncancel_tac = gen_uncancel_tac @{thm "diff_cancel"};
end);
(* prepare nat_cancel simprocs *)
val nat_cancel_sums_add = map prep_simproc
[("nateq_cancel_sums",
["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"],
K EqCancelSums.proc),
("natless_cancel_sums",
["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"],
K LessCancelSums.proc),
("natle_cancel_sums",
["(l::nat) + m <= n", "(l::nat) <= m + n", "Suc m <= n", "m <= Suc n"],
K LeCancelSums.proc)];
val nat_cancel_sums = nat_cancel_sums_add @
[prep_simproc ("natdiff_cancel_sums",
["((l::nat) + m) - n", "(l::nat) - (m + n)", "Suc m - n", "m - Suc n"],
K DiffCancelSums.proc)];
val arith_data_setup =
Simplifier.map_ss (fn ss => ss addsimprocs nat_cancel_sums);
(* FIXME dead code *)
(* antisymmetry:
combines x <= y (or ~(y < x)) and y <= x (or ~(x < y)) into x = y
local
val antisym = mk_meta_eq order_antisym
val not_lessD = @{thm linorder_not_less} RS iffD1
fun prp t thm = (#prop(rep_thm thm) = t)
in
fun antisym_eq prems thm =
let
val r = #prop(rep_thm thm);
in
case r of
Tr $ ((c as Const(@{const_name HOL.less_eq},T)) $ s $ t) =>
let val r' = Tr $ (c $ t $ s)
in
case Library.find_first (prp r') prems of
NONE =>
let val r' = Tr $ (HOLogic.Not $ (Const(@{const_name HOL.less},T) $ s $ t))
in case Library.find_first (prp r') prems of
NONE => []
| SOME thm' => [(thm' RS not_lessD) RS (thm RS antisym)]
end
| SOME thm' => [thm' RS (thm RS antisym)]
end
| Tr $ (Const("Not",_) $ (Const(@{const_name HOL.less},T) $ s $ t)) =>
let val r' = Tr $ (Const(@{const_name HOL.less_eq},T) $ s $ t)
in
case Library.find_first (prp r') prems of
NONE =>
let val r' = Tr $ (HOLogic.Not $ (Const(@{const_name HOL.less},T) $ t $ s))
in case Library.find_first (prp r') prems of
NONE => []
| SOME thm' =>
[(thm' RS not_lessD) RS ((thm RS not_lessD) RS antisym)]
end
| SOME thm' => [thm' RS ((thm RS not_lessD) RS antisym)]
end
| _ => []
end
handle THM _ => []
end;
*)
end;
open ArithData;