|
(* Title: HOL/Library/Cancellation/cancel_data.ML
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Author: Mathias Fleury, MPII
Datastructure for the cancelation simprocs.
*)
signature CANCEL_DATA =
sig
val mk_sum : typ -> term list -> term
val dest_sum : term -> term list
val mk_coeff : int * term -> term
val dest_coeff : term -> int * term
val find_first_coeff : term -> term list -> int * term list
val trans_tac : Proof.context -> thm option -> tactic
val norm_ss1 : simpset
val norm_ss2: simpset
val norm_tac: Proof.context -> tactic
val numeral_simp_tac : Proof.context -> tactic
val simplify_meta_eq : Proof.context -> thm -> thm
val prove_conv : tactic list -> Proof.context -> thm list -> term * term -> thm option
end;
structure Cancel_Data : CANCEL_DATA =
struct
(*** Utilities ***)
(*No reordering of the arguments.*)
fun fast_mk_iterate_add (n, mset) =
let val T = fastype_of mset
in
Const (\<^const_name>\<open>iterate_add\<close>, \<^typ>\<open>nat\<close> --> T --> T) $ n $ mset
end;
(*iterate_add is not symmetric, unlike multiplication over natural numbers.*)
fun mk_iterate_add (t, u) =
(if fastype_of t = \<^typ>\<open>nat\<close> then (t, u) else (u, t))
|> fast_mk_iterate_add;
(*Maps n to #n for n = 1, 2*)
val numeral_syms =
map (fn th => th RS sym) @{thms numeral_One numeral_2_eq_2 numeral_1_eq_Suc_0};
val numeral_sym_ss =
simpset_of (put_simpset HOL_basic_ss \<^context> addsimps numeral_syms);
fun mk_number 1 = HOLogic.numeral_const HOLogic.natT $ HOLogic.one_const
| mk_number n = HOLogic.mk_number HOLogic.natT n;
fun dest_number t = Int.max (0, snd (HOLogic.dest_number t));
fun find_first_numeral past (t::terms) =
((dest_number t, t, rev past @ terms)
handle TERM _ => find_first_numeral (t::past) terms)
| find_first_numeral _ [] = raise TERM("find_first_numeral", []);
fun typed_zero T = Const (\<^const_name>\<open>Groups.zero\<close>, T);
fun typed_one T = HOLogic.numeral_const T $ HOLogic.one_const
val mk_plus = HOLogic.mk_binop \<^const_name>\<open>Groups.plus\<close>;
(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero.*)
fun mk_sum T [] = typed_zero T
| mk_sum _ [t,u] = mk_plus (t, u)
| mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
val dest_plus = HOLogic.dest_bin \<^const_name>\<open>Groups.plus\<close> dummyT;
(*** Other simproc items ***)
val bin_simps =
(@{thm numeral_One} RS sym) ::
@{thms add_numeral_left diff_nat_numeral diff_0_eq_0 mult_numeral_left(1)
if_True if_False not_False_eq_True nat_0 nat_numeral nat_neg_numeral iterate_add_simps(1)
iterate_add_empty arith_simps rel_simps of_nat_numeral};
(*** CancelNumerals simprocs ***)
val one = mk_number 1;
fun mk_prod T [] = typed_one T
| mk_prod _ [t] = t
| mk_prod T (t :: ts) = if t = one then mk_prod T ts else mk_iterate_add (t, mk_prod T ts);
val dest_iterate_add = HOLogic.dest_bin \<^const_name>\<open>iterate_add\<close> dummyT;
fun dest_iterate_adds t =
let val (t,u) = dest_iterate_add t in
t :: dest_iterate_adds u end
handle TERM _ => [t];
fun mk_coeff (k,t) = mk_iterate_add (mk_number k, t);
(*Express t as a product of (possibly) a numeral with other factors, sorted*)
fun dest_coeff t =
let
val T = fastype_of t
val ts = sort Term_Ord.term_ord (dest_iterate_adds t);
val (n, _, ts') =
find_first_numeral [] ts
handle TERM _ => (1, one, ts);
in (n, mk_prod T ts') end;
(*Find first coefficient-term THAT MATCHES u*)
fun find_first_coeff _ _ [] = 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;
(*
Split up a sum into the list of its constituent terms.
*)
fun dest_summation (t, ts) =
let val (t1,t2) = dest_plus t in
dest_summation (t1, dest_summation (t2, ts)) end
handle TERM _ => t :: ts;
fun dest_sum t = dest_summation (t, []);
val rename_numerals = simplify (put_simpset numeral_sym_ss \<^context>) o Thm.transfer \<^theory>;
(*Simplify \<open>iterate_add (Suc 0) n\<close>, \<open>iterate_add n (Suc 0)\<close>, \<open>n+0\<close>, and \<open>0+n\<close> to \<open>n\<close>*)
val add_0s = map rename_numerals @{thms monoid_add_class.add_0_left monoid_add_class.add_0_right};
val mult_1s = map rename_numerals @{thms iterate_add_1 iterate_add_simps(2)[of 0]};
(*And these help the simproc return False when appropriate. We use the same list as the
simproc for natural numbers, but adapted.*)
fun contra_rules ctxt =
@{thms le_zero_eq} @ Named_Theorems.get ctxt \<^named_theorems>\<open>cancelation_simproc_eq_elim\<close>;
fun simplify_meta_eq ctxt =
Arith_Data.simplify_meta_eq
(@{thms numeral_1_eq_Suc_0 Nat.add_0_right
mult_0 mult_0_right mult_1 mult_1_right iterate_add_Numeral1 of_nat_numeral
monoid_add_class.add_0_left iterate_add_simps(1) monoid_add_class.add_0_right
iterate_add_Numeral1} @
contra_rules ctxt) ctxt;
val mk_sum = 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 trans_tac = Numeral_Simprocs.trans_tac;
val norm_ss1 =
simpset_of (put_simpset Numeral_Simprocs.num_ss \<^context> addsimps
numeral_syms @ add_0s @ mult_1s @ @{thms ac_simps iterate_add_simps});
val norm_ss2 =
simpset_of (put_simpset Numeral_Simprocs.num_ss \<^context> addsimps
bin_simps @
@{thms ac_simps});
fun norm_tac ctxt =
ALLGOALS (simp_tac (put_simpset norm_ss1 ctxt))
THEN ALLGOALS (simp_tac (put_simpset norm_ss2 ctxt));
val mset_simps_ss =
simpset_of (put_simpset HOL_basic_ss \<^context> addsimps bin_simps);
fun numeral_simp_tac ctxt = ALLGOALS (simp_tac (put_simpset mset_simps_ss ctxt));
val simplify_meta_eq = simplify_meta_eq;
val prove_conv = Arith_Data.prove_conv;
end
¤ Dauer der Verarbeitung: 0.32 Sekunden
(vorverarbeitet)
¤
|
Haftungshinweis
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
|
