--- a/src/HOL/Hyperreal/HyperBin.ML Thu Nov 01 21:12:13 2001 +0100
+++ b/src/HOL/Hyperreal/HyperBin.ML Fri Nov 02 17:55:24 2001 +0100
@@ -13,13 +13,13 @@
qed "hypreal_number_of";
Addsimps [hypreal_number_of];
-Goalw [hypreal_number_of_def] "(0::hypreal) = Numeral0";
-by (simp_tac (simpset() addsimps [hypreal_of_real_zero RS sym]) 1);
-qed "zero_eq_numeral_0";
+Goalw [hypreal_number_of_def] "Numeral0 = (0::hypreal)";
+by (Simp_tac 1);
+qed "hypreal_numeral_0_eq_0";
-Goalw [hypreal_number_of_def] "(1::hypreal) = Numeral1";
-by (simp_tac (simpset() addsimps [hypreal_of_real_one RS sym]) 1);
-qed "one_eq_numeral_1";
+Goalw [hypreal_number_of_def] "Numeral1 = (1::hypreal)";
+by (Simp_tac 1);
+qed "hypreal_numeral_1_eq_1";
(** Addition **)
@@ -57,15 +57,15 @@
qed "mult_hypreal_number_of";
Addsimps [mult_hypreal_number_of];
-Goal "(2::hypreal) = Numeral1 + Numeral1";
-by (Simp_tac 1);
+Goal "(2::hypreal) = 1 + 1";
+by (simp_tac (simpset() addsimps [hypreal_numeral_1_eq_1 RS sym]) 1);
val lemma = result();
(*For specialist use: NOT as default simprules*)
Goal "2 * z = (z+z::hypreal)";
by (simp_tac (simpset ()
addsimps [lemma, hypreal_add_mult_distrib,
- one_eq_numeral_1 RS sym]) 1);
+ hypreal_numeral_1_eq_1]) 1);
qed "hypreal_mult_2";
Goal "z * 2 = (z+z::hypreal)";
@@ -107,46 +107,20 @@
(*** New versions of existing theorems involving 0, 1 ***)
-Goal "- Numeral1 = (-1::hypreal)";
-by (Simp_tac 1);
-qed "minus_numeral_one";
+Goal "- 1 = (-1::hypreal)";
+by (simp_tac (simpset() addsimps [hypreal_numeral_1_eq_1 RS sym]) 1);
+qed "hypreal_minus_1_eq_m1";
-(*Maps 0 to Numeral0 and (1::hypreal) to Numeral1 and -(Numeral1) to -1*)
+(*Maps 0 to Numeral0 and 1 to Numeral1 and -(Numeral1) to -1*)
val hypreal_numeral_ss =
- real_numeral_ss addsimps [zero_eq_numeral_0, one_eq_numeral_1,
- minus_numeral_one];
+ real_numeral_ss addsimps [hypreal_numeral_0_eq_0 RS sym,
+ hypreal_numeral_1_eq_1 RS sym,
+ hypreal_minus_1_eq_m1];
fun rename_numerals th =
asm_full_simplify hypreal_numeral_ss (Thm.transfer (the_context ()) th);
-(*Now insert some identities previously stated for 0 and 1*)
-
-(** HyperDef **)
-
-Addsimps (map rename_numerals
- [hypreal_minus_zero, hypreal_minus_zero_iff,
- hypreal_add_zero_left, hypreal_add_zero_right,
- hypreal_diff_zero, hypreal_diff_zero_right,
- hypreal_mult_0_right, hypreal_mult_0,
- hypreal_mult_1_right, hypreal_mult_1,
- hypreal_inverse_1, hypreal_minus_zero_less_iff]);
-
-bind_thm ("hypreal_0_less_mult_iff",
- rename_numerals hypreal_zero_less_mult_iff);
-bind_thm ("hypreal_0_le_mult_iff",
- rename_numerals hypreal_zero_le_mult_iff);
-bind_thm ("hypreal_mult_less_0_iff",
- rename_numerals hypreal_mult_less_zero_iff);
-bind_thm ("hypreal_mult_le_0_iff",
- rename_numerals hypreal_mult_le_zero_iff);
-
-bind_thm ("hypreal_inverse_less_0", rename_numerals hypreal_inverse_less_zero);
-bind_thm ("hypreal_inverse_gt_0", rename_numerals hypreal_inverse_gt_zero);
-
-Addsimps [zero_eq_numeral_0,one_eq_numeral_1];
-
-
(** Simplification of arithmetic when nested to the right **)
Goal "number_of v + (number_of w + z) = (number_of(bin_add v w) + z::hypreal)";
@@ -176,15 +150,15 @@
(** Combining of literal coefficients in sums of products **)
-Goal "(x < y) = (x-y < (Numeral0::hypreal))";
+Goal "(x < y) = (x-y < (0::hypreal))";
by (simp_tac (simpset() addsimps [hypreal_diff_less_eq]) 1);
qed "hypreal_less_iff_diff_less_0";
-Goal "(x = y) = (x-y = (Numeral0::hypreal))";
+Goal "(x = y) = (x-y = (0::hypreal))";
by (simp_tac (simpset() addsimps [hypreal_diff_eq_eq]) 1);
qed "hypreal_eq_iff_diff_eq_0";
-Goal "(x <= y) = (x-y <= (Numeral0::hypreal))";
+Goal "(x <= y) = (x-y <= (0::hypreal))";
by (simp_tac (simpset() addsimps [hypreal_diff_le_eq]) 1);
qed "hypreal_le_iff_diff_le_0";
@@ -209,7 +183,7 @@
Goal "!!i::hypreal. (i*u + m = j*u + n) = ((i-j)*u + m = n)";
by (asm_simp_tac
(simpset() addsimps [hypreal_diff_def, hypreal_add_mult_distrib]@
- hypreal_add_ac@rel_iff_rel_0_rls) 1);
+ hypreal_add_ac@rel_iff_rel_0_rls) 1);
qed "hypreal_eq_add_iff1";
Goal "!!i::hypreal. (i*u + m = j*u + n) = (m = (j-i)*u + n)";
@@ -242,26 +216,31 @@
hypreal_add_ac@rel_iff_rel_0_rls) 1);
qed "hypreal_le_add_iff2";
+Goal "-1 * (z::hypreal) = -z";
+by (simp_tac (simpset() addsimps [hypreal_minus_1_eq_m1 RS sym,
+ hypreal_mult_minus_1]) 1);
+qed "hypreal_mult_minus1";
+Addsimps [hypreal_mult_minus1];
+
Goal "(z::hypreal) * -1 = -z";
-by (stac (minus_numeral_one RS sym) 1);
-by (stac (hypreal_minus_mult_eq2 RS sym) 1);
-by Auto_tac;
-qed "hypreal_mult_minus_1_right";
-Addsimps [hypreal_mult_minus_1_right];
+by (stac hypreal_mult_commute 1 THEN rtac hypreal_mult_minus1 1);
+qed "hypreal_mult_minus1_right";
+Addsimps [hypreal_mult_minus1_right];
-Goal "-1 * (z::hypreal) = -z";
-by (simp_tac (simpset() addsimps [hypreal_mult_commute]) 1);
-qed "hypreal_mult_minus_1";
-Addsimps [hypreal_mult_minus_1];
+Addsimps [hypreal_numeral_0_eq_0, hypreal_numeral_1_eq_1];
structure Hyperreal_Numeral_Simprocs =
struct
+(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in simprocs
+ isn't complicated by the abstract 0 and 1.*)
+val numeral_syms = [hypreal_numeral_0_eq_0 RS sym,
+ hypreal_numeral_1_eq_1 RS sym];
+
(*Utilities*)
-
val hyprealT = Type("HyperDef.hypreal",[]);
fun mk_numeral n = HOLogic.number_of_const hyprealT $ HOLogic.mk_bin n;
@@ -271,11 +250,11 @@
val find_first_numeral = Real_Numeral_Simprocs.find_first_numeral;
val zero = mk_numeral 0;
-val mk_plus = HOLogic.mk_binop "op +";
+val mk_plus = Real_Numeral_Simprocs.mk_plus;
val uminus_const = Const ("uminus", hyprealT --> hyprealT);
-(*Thus mk_sum[t] yields t+Numeral0; longer sums don't have a trailing zero*)
+(*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);
@@ -335,24 +314,23 @@
handle TERM _ => find_first_coeff (t::past) u terms;
-(*Simplify Numeral1*n and n*Numeral1 to n*)
+(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*)
val add_0s = map rename_numerals
[hypreal_add_zero_left, hypreal_add_zero_right];
-val mult_plus_1s = map rename_numerals
- [hypreal_mult_1, hypreal_mult_1_right];
-val mult_minus_1s = map rename_numerals
- [hypreal_mult_minus_1, hypreal_mult_minus_1_right];
-val mult_1s = mult_plus_1s @ mult_minus_1s;
+val mult_1s = map rename_numerals [hypreal_mult_1, hypreal_mult_1_right] @
+ [hypreal_mult_minus1, hypreal_mult_minus1_right];
(*To perform binary arithmetic*)
val bin_simps =
- [add_hypreal_number_of, hypreal_add_number_of_left,
- minus_hypreal_number_of, diff_hypreal_number_of, mult_hypreal_number_of,
+ [hypreal_numeral_0_eq_0 RS sym, hypreal_numeral_1_eq_1 RS sym,
+ add_hypreal_number_of, hypreal_add_number_of_left,
+ minus_hypreal_number_of,
+ diff_hypreal_number_of, mult_hypreal_number_of,
hypreal_mult_number_of_left] @ bin_arith_simps @ bin_rel_simps;
(*To evaluate binary negations of coefficients*)
val hypreal_minus_simps = NCons_simps @
- [minus_hypreal_number_of,
+ [hypreal_minus_1_eq_m1, minus_hypreal_number_of,
bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,
bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
@@ -374,24 +352,6 @@
val hypreal_mult_minus_simps =
[hypreal_mult_assoc, hypreal_minus_mult_eq1, hypreal_minus_mult_eq_1_to_2];
-(*Apply the given rewrite (if present) just once*)
-fun trans_tac None = all_tac
- | trans_tac (Some th) = ALLGOALS (rtac (th RS trans));
-
-fun prove_conv name tacs sg (hyps: thm list) (t,u) =
- if t aconv u then None
- else
- let val ct = cterm_of sg (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u)))
- in Some
- (prove_goalw_cterm [] ct (K tacs)
- handle ERROR => error
- ("The error(s) above occurred while trying to prove " ^
- string_of_cterm ct ^ "\nInternal failure of simproc " ^ name))
- end;
-
-(*version without the hyps argument*)
-fun prove_conv_nohyps name tacs sg = prove_conv name tacs sg [];
-
(*Final simplification: cancel + and * *)
val simplify_meta_eq =
Int_Numeral_Simprocs.simplify_meta_eq
@@ -409,7 +369,7 @@
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff 1
val find_first_coeff = find_first_coeff []
- val trans_tac = trans_tac
+ val trans_tac = Real_Numeral_Simprocs.trans_tac
val norm_tac =
ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
hypreal_minus_simps@hypreal_add_ac))
@@ -424,7 +384,7 @@
structure EqCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
- val prove_conv = prove_conv "hyprealeq_cancel_numerals"
+ val prove_conv = Real_Numeral_Simprocs.prove_conv "hyprealeq_cancel_numerals"
val mk_bal = HOLogic.mk_eq
val dest_bal = HOLogic.dest_bin "op =" hyprealT
val bal_add1 = hypreal_eq_add_iff1 RS trans
@@ -433,7 +393,7 @@
structure LessCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
- val prove_conv = prove_conv "hyprealless_cancel_numerals"
+ val prove_conv = Real_Numeral_Simprocs.prove_conv "hyprealless_cancel_numerals"
val mk_bal = HOLogic.mk_binrel "op <"
val dest_bal = HOLogic.dest_bin "op <" hyprealT
val bal_add1 = hypreal_less_add_iff1 RS trans
@@ -442,7 +402,7 @@
structure LeCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
- val prove_conv = prove_conv "hyprealle_cancel_numerals"
+ val prove_conv = Real_Numeral_Simprocs.prove_conv "hyprealle_cancel_numerals"
val mk_bal = HOLogic.mk_binrel "op <="
val dest_bal = HOLogic.dest_bin "op <=" hyprealT
val bal_add1 = hypreal_le_add_iff1 RS trans
@@ -476,8 +436,9 @@
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff 1
val left_distrib = left_hypreal_add_mult_distrib RS trans
- val prove_conv = prove_conv_nohyps "hypreal_combine_numerals"
- val trans_tac = trans_tac
+ val prove_conv = Real_Numeral_Simprocs.prove_conv_nohyps
+ "hypreal_combine_numerals"
+ val trans_tac = Real_Numeral_Simprocs.trans_tac
val norm_tac =
ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
hypreal_minus_simps@hypreal_add_ac))
@@ -516,8 +477,52 @@
[hypreal_mult_1, hypreal_mult_1_right]
(([th, cancel_th]) MRS trans);
+(*** Making constant folding work for 0 and 1 too ***)
+
+structure HyperrealAbstractNumeralsData =
+ struct
+ val dest_eq = HOLogic.dest_eq o HOLogic.dest_Trueprop o concl_of
+ val is_numeral = Bin_Simprocs.is_numeral
+ val numeral_0_eq_0 = hypreal_numeral_0_eq_0
+ val numeral_1_eq_1 = hypreal_numeral_1_eq_1
+ val prove_conv = Real_Numeral_Simprocs.prove_conv_nohyps
+ "hypreal_abstract_numerals"
+ fun norm_tac simps = ALLGOALS (simp_tac (HOL_ss addsimps simps))
+ val simplify_meta_eq = Bin_Simprocs.simplify_meta_eq
+ end
+
+structure HyperrealAbstractNumerals =
+ AbstractNumeralsFun (HyperrealAbstractNumeralsData)
+
+(*For addition, we already have rules for the operand 0.
+ Multiplication is omitted because there are already special rules for
+ both 0 and 1 as operands. Unary minus is trivial, just have - 1 = -1.
+ For the others, having three patterns is a compromise between just having
+ one (many spurious calls) and having nine (just too many!) *)
+val eval_numerals =
+ map prep_simproc
+ [("hypreal_add_eval_numerals",
+ prep_pats ["(m::hypreal) + 1", "(m::hypreal) + number_of v"],
+ HyperrealAbstractNumerals.proc add_hypreal_number_of),
+ ("hypreal_diff_eval_numerals",
+ prep_pats ["(m::hypreal) - 1", "(m::hypreal) - number_of v"],
+ HyperrealAbstractNumerals.proc diff_hypreal_number_of),
+ ("hypreal_eq_eval_numerals",
+ prep_pats ["(m::hypreal) = 0", "(m::hypreal) = 1",
+ "(m::hypreal) = number_of v"],
+ HyperrealAbstractNumerals.proc eq_hypreal_number_of),
+ ("hypreal_less_eval_numerals",
+ prep_pats ["(m::hypreal) < 0", "(m::hypreal) < 1",
+ "(m::hypreal) < number_of v"],
+ HyperrealAbstractNumerals.proc less_hypreal_number_of),
+ ("hypreal_le_eval_numerals",
+ prep_pats ["(m::hypreal) <= 0", "(m::hypreal) <= 1",
+ "(m::hypreal) <= number_of v"],
+ HyperrealAbstractNumerals.proc le_hypreal_number_of_eq_not_less)]
+
end;
+Addsimprocs Hyperreal_Numeral_Simprocs.eval_numerals;
Addsimprocs Hyperreal_Numeral_Simprocs.cancel_numerals;
Addsimprocs [Hyperreal_Numeral_Simprocs.combine_numerals];
@@ -581,8 +586,6 @@
Addsimprocs [Hyperreal_Times_Assoc.conv];
-Addsimps [rename_numerals hypreal_of_real_zero_iff];
-
(*Simplification of x-y < 0, etc.*)
AddIffs [hypreal_less_iff_diff_less_0 RS sym];
AddIffs [hypreal_eq_iff_diff_eq_0 RS sym];
@@ -603,6 +606,12 @@
qed "number_of_le_hypreal_of_real_iff";
Addsimps [number_of_le_hypreal_of_real_iff];
+Goal "(hypreal_of_real z = number_of w) = (z = number_of w)";
+by (stac (hypreal_of_real_eq_iff RS sym) 1);
+by (Simp_tac 1);
+qed "hypreal_of_real_eq_number_of_iff";
+Addsimps [hypreal_of_real_eq_number_of_iff];
+
Goal "(hypreal_of_real z < number_of w) = (z < number_of w)";
by (stac (hypreal_of_real_less_iff RS sym) 1);
by (Simp_tac 1);
@@ -615,6 +624,20 @@
qed "hypreal_of_real_le_number_of_iff";
Addsimps [hypreal_of_real_le_number_of_iff];
+(*As above, for the special case of zero*)
+Addsimps
+ (map (simplify (simpset()) o inst "w" "Pls")
+ [hypreal_of_real_eq_number_of_iff,
+ hypreal_of_real_le_number_of_iff, hypreal_of_real_less_number_of_iff,
+ number_of_le_hypreal_of_real_iff, number_of_less_hypreal_of_real_iff]);
+
+(*As above, for the special case of one*)
+Addsimps
+ (map (simplify (simpset()) o inst "w" "Pls BIT True")
+ [hypreal_of_real_eq_number_of_iff,
+ hypreal_of_real_le_number_of_iff, hypreal_of_real_less_number_of_iff,
+ number_of_le_hypreal_of_real_iff, number_of_less_hypreal_of_real_iff]);
+
(** <= monotonicity results: needed for arithmetic **)
Goal "[| i <= j; (0::hypreal) <= k |] ==> i*k <= j*k";
@@ -634,3 +657,4 @@
by (assume_tac 1);
qed "hypreal_mult_le_mono";
+Addsimps [hypreal_minus_1_eq_m1];