--- a/src/HOL/Real/Hyperreal/HyperPow.ML Thu Dec 21 10:16:07 2000 +0100
+++ b/src/HOL/Real/Hyperreal/HyperPow.ML Thu Dec 21 10:16:33 2000 +0100
@@ -20,18 +20,17 @@
by (induct_tac "n" 1);
by (Auto_tac);
by (forw_inst_tac [("n","n")] hrealpow_not_zero 1);
-by (auto_tac (claset() addDs [inverse_mult_eq],
- simpset()));
+by (auto_tac (claset(), simpset() addsimps [inverse_mult_eq]));
qed_spec_mp "hrealpow_inverse";
Goal "abs (r::hypreal) ^ n = abs (r ^ n)";
by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset() addsimps [hrabs_mult,hrabs_one]));
+by (auto_tac (claset(), simpset() addsimps [hrabs_mult,hrabs_one]));
qed "hrealpow_hrabs";
Goal "(r::hypreal) ^ (n + m) = (r ^ n) * (r ^ m)";
by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset() addsimps hypreal_mult_ac));
+by (auto_tac (claset(), simpset() addsimps hypreal_mult_ac));
qed "hrealpow_add";
Goal "(r::hypreal) ^ 1 = r";
@@ -46,33 +45,34 @@
Goal "(0::hypreal) < r --> 0 <= r ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset() addDs [hypreal_less_imp_le]
- addIs [hypreal_le_mult_order],simpset() addsimps
- [hypreal_zero_less_one RS hypreal_less_imp_le]));
+ addIs [hypreal_le_mult_order],
+ simpset() addsimps [hypreal_zero_less_one RS hypreal_less_imp_le]));
qed_spec_mp "hrealpow_ge_zero";
Goal "(0::hypreal) < r --> 0 < r ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset() addIs [hypreal_mult_order],
- simpset() addsimps [hypreal_zero_less_one]));
+ simpset() addsimps [hypreal_zero_less_one]));
qed_spec_mp "hrealpow_gt_zero";
Goal "(0::hypreal) <= r --> 0 <= r ^ n";
by (induct_tac "n" 1);
-by (auto_tac (claset() addIs [hypreal_le_mult_order],simpset()
- addsimps [hypreal_zero_less_one RS hypreal_less_imp_le]));
+by (auto_tac (claset() addIs [hypreal_le_mult_order],
+ simpset() addsimps [hypreal_zero_less_one RS hypreal_less_imp_le]));
qed_spec_mp "hrealpow_ge_zero2";
Goal "(0::hypreal) < x & x <= y --> x ^ n <= y ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset() addSIs [hypreal_mult_le_mono],
- simpset() addsimps [hypreal_le_refl]));
+ simpset() addsimps [hypreal_le_refl]));
by (asm_simp_tac (simpset() addsimps [hrealpow_ge_zero]) 1);
qed_spec_mp "hrealpow_le";
Goal "(0::hypreal) < x & x < y & 0 < n --> x ^ n < y ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset() addIs [hypreal_mult_less_mono,gr0I]
- addDs [hrealpow_gt_zero],simpset()));
+ addDs [hrealpow_gt_zero],
+ simpset()));
qed "hrealpow_less";
Goal "1hr ^ n = 1hr";
@@ -82,21 +82,20 @@
Addsimps [hrealpow_eq_one];
Goal "abs(-(1hr ^ n)) = 1hr";
-by (simp_tac (simpset() addsimps
- [hrabs_minus_cancel,hrabs_one]) 1);
+by (simp_tac (simpset() addsimps [hrabs_one]) 1);
qed "hrabs_minus_hrealpow_one";
Addsimps [hrabs_minus_hrealpow_one];
Goal "abs((-1hr) ^ n) = 1hr";
by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset() addsimps [hrabs_mult,
- hrabs_minus_cancel,hrabs_one]));
+by (auto_tac (claset(),
+ simpset() addsimps [hrabs_mult, hrabs_one]));
qed "hrabs_hrealpow_minus_one";
Addsimps [hrabs_hrealpow_minus_one];
Goal "((r::hypreal) * s) ^ n = (r ^ n) * (s ^ n)";
by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset() addsimps hypreal_mult_ac));
+by (auto_tac (claset(), simpset() addsimps hypreal_mult_ac));
qed "hrealpow_mult";
Goal "(0::hypreal) <= r ^ 2";
@@ -105,12 +104,12 @@
Addsimps [hrealpow_two_le];
Goal "(0::hypreal) <= u ^ 2 + v ^ 2";
-by (auto_tac (claset() addIs [hypreal_le_add_order],simpset()));
+by (auto_tac (claset() addIs [hypreal_le_add_order], simpset()));
qed "hrealpow_two_le_add_order";
Addsimps [hrealpow_two_le_add_order];
Goal "(0::hypreal) <= u ^ 2 + v ^ 2 + w ^ 2";
-by (auto_tac (claset() addSIs [hypreal_le_add_order],simpset()));
+by (auto_tac (claset() addSIs [hypreal_le_add_order], simpset()));
qed "hrealpow_two_le_add_order2";
Addsimps [hrealpow_two_le_add_order2];
@@ -132,12 +131,12 @@
Addsimps [hrealpow_two_hrabs];
Goal "!!r. 1hr < r ==> 1hr < r ^ 2";
-by (auto_tac (claset(),simpset() addsimps [hrealpow_two]));
+by (auto_tac (claset(), simpset() addsimps [hrealpow_two]));
by (cut_facts_tac [hypreal_zero_less_one] 1);
by (forw_inst_tac [("R1.0","0")] hypreal_less_trans 1);
by (assume_tac 1);
by (dres_inst_tac [("z","r"),("x","1hr")] hypreal_mult_less_mono1 1);
-by (auto_tac (claset() addIs [hypreal_less_trans],simpset()));
+by (auto_tac (claset() addIs [hypreal_less_trans], simpset()));
qed "hrealpow_two_gt_one";
Goal "!!r. 1hr <= r ==> 1hr <= r ^ 2";
@@ -151,7 +150,8 @@
by (forw_inst_tac [("n1","n")]
((hypreal_not_refl2 RS not_sym) RS hrealpow_not_zero RS not_sym) 1);
by (auto_tac (claset() addSDs [hypreal_le_imp_less_or_eq]
- addIs [hypreal_mult_order],simpset()));
+ addIs [hypreal_mult_order],
+ simpset()));
qed "hrealpow_Suc_gt_zero";
Goal "!!r. (0::hypreal) <= r ==> 0 <= r ^ Suc n";
@@ -170,8 +170,9 @@
Goal "hypreal_of_nat n < (1hr + 1hr) ^ n";
by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset() addsimps [hypreal_of_nat_Suc,hypreal_of_nat_zero,
- hypreal_zero_less_one,hypreal_add_mult_distrib,hypreal_of_nat_one]));
+by (auto_tac (claset(),
+ simpset() addsimps [hypreal_of_nat_Suc,hypreal_of_nat_zero,
+ hypreal_zero_less_one,hypreal_add_mult_distrib,hypreal_of_nat_one]));
by (blast_tac (claset() addSIs [hypreal_add_less_le_mono,
two_hrealpow_ge_one]) 1);
qed "two_hrealpow_gt";
@@ -195,21 +196,21 @@
Goal "(0::hypreal) < r & r < 1hr --> r ^ Suc n < r ^ n";
by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset() addsimps
- [hypreal_mult_less_mono2]));
+by (auto_tac (claset(),
+ simpset() addsimps [hypreal_mult_less_mono2]));
qed_spec_mp "hrealpow_Suc_less";
Goal "(0::hypreal) <= r & r < 1hr --> r ^ Suc n <= r ^ n";
by (induct_tac "n" 1);
-by (auto_tac (claset() addIs [hypreal_less_imp_le] addSDs
- [hypreal_le_imp_less_or_eq,hrealpow_Suc_less],simpset()
- addsimps [hypreal_le_refl,hypreal_mult_less_mono2]));
+by (auto_tac (claset() addIs [hypreal_less_imp_le]
+ addSDs [hypreal_le_imp_less_or_eq,hrealpow_Suc_less],
+ simpset() addsimps [hypreal_le_refl,hypreal_mult_less_mono2]));
qed_spec_mp "hrealpow_Suc_le";
(* a few more theorems needed here *)
Goal "1hr <= r --> r ^ n <= r ^ Suc n";
by (induct_tac "n" 1);
-by (auto_tac (claset() addSIs [hypreal_mult_le_le_mono1],simpset()));
+by (auto_tac (claset() addSIs [hypreal_mult_le_le_mono1], simpset()));
by (rtac ccontr 1 THEN dtac not_hypreal_leE 1);
by (dtac hypreal_le_less_trans 1 THEN assume_tac 1);
by (etac (hypreal_zero_less_one RS hypreal_less_asym) 1);
@@ -217,8 +218,8 @@
Goal "Abs_hypreal(hyprel^^{%n. X n}) ^ m = Abs_hypreal(hyprel^^{%n. (X n) ^ m})";
by (nat_ind_tac "m" 1);
-by (auto_tac (claset(),simpset() addsimps
- [hypreal_one_def,hypreal_mult]));
+by (auto_tac (claset(),
+ simpset() addsimps [hypreal_one_def,hypreal_mult]));
qed "hrealpow";
Goal "(x + (y::hypreal)) ^ 2 = \
@@ -238,24 +239,23 @@
"[|(0::hypreal) <= x;0 <= y;x ^ Suc n <= y ^ Suc n |] ==> x <= y";
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps
- [hrealpow,hypreal_le,hypreal_mult]));
-by (ultra_tac (claset() addIs [realpow_increasing],simpset()) 1);
+by (auto_tac (claset(),
+ simpset() addsimps [hrealpow,hypreal_le,hypreal_mult]));
+by (ultra_tac (claset() addIs [realpow_increasing], simpset()) 1);
qed "hrealpow_increasing";
goalw HyperPow.thy [hypreal_zero_def]
"!!x. [|(0::hypreal) <= x;0 <= y;x ^ Suc n = y ^ Suc n |] ==> x = y";
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps
- [hrealpow,hypreal_mult,hypreal_le]));
+by (auto_tac (claset(), simpset() addsimps [hrealpow,hypreal_mult,hypreal_le]));
by (ultra_tac (claset() addIs [realpow_Suc_cancel_eq],
simpset()) 1);
qed "hrealpow_Suc_cancel_eq";
Goal "x : HFinite --> x ^ n : HFinite";
by (induct_tac "n" 1);
-by (auto_tac (claset() addIs [HFinite_mult],simpset()));
+by (auto_tac (claset() addIs [HFinite_mult], simpset()));
qed_spec_mp "hrealpow_HFinite";
(*---------------------------------------------------------------
@@ -281,8 +281,7 @@
Goalw [hypreal_zero_def,hypnat_one_def]
"(0::hypreal) pow (n + 1hn) = 0";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
-by (auto_tac (claset(),simpset() addsimps
- [hyperpow,hypnat_add]));
+by (auto_tac (claset(), simpset() addsimps [hyperpow,hypnat_add]));
qed "hyperpow_zero";
Addsimps [hyperpow_zero];
@@ -290,7 +289,7 @@
"r ~= (0::hypreal) --> r pow n ~= 0";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [hyperpow]));
+by (auto_tac (claset(), simpset() addsimps [hyperpow]));
by (dtac FreeUltrafilterNat_Compl_mem 1);
by (fuf_empty_tac (claset() addIs [realpow_not_zero RS notE],
simpset()) 1);
@@ -304,57 +303,56 @@
simpset() addsimps [hypreal_inverse,hyperpow]));
by (rtac FreeUltrafilterNat_subset 1);
by (auto_tac (claset() addDs [realpow_not_zero]
- addIs [realpow_inverse],simpset()));
+ addIs [realpow_inverse],
+ simpset()));
qed "hyperpow_inverse";
Goal "abs r pow n = abs (r pow n)";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [hypreal_hrabs,
- hyperpow,realpow_abs]));
+by (auto_tac (claset(),
+ simpset() addsimps [hypreal_hrabs, hyperpow,realpow_abs]));
qed "hyperpow_hrabs";
Goal "r pow (n + m) = (r pow n) * (r pow m)";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","m")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [hyperpow,hypnat_add,
- hypreal_mult,realpow_add]));
+by (auto_tac (claset(),
+ simpset() addsimps [hyperpow,hypnat_add, hypreal_mult,realpow_add]));
qed "hyperpow_add";
Goalw [hypnat_one_def] "r pow 1hn = r";
by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [hyperpow]));
+by (auto_tac (claset(), simpset() addsimps [hyperpow]));
qed "hyperpow_one";
Addsimps [hyperpow_one];
Goalw [hypnat_one_def]
"r pow (1hn + 1hn) = r * r";
by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [hyperpow,hypnat_add,
- hypreal_mult,realpow_two]));
+by (auto_tac (claset(),
+ simpset() addsimps [hyperpow,hypnat_add, hypreal_mult,realpow_two]));
qed "hyperpow_two";
Goalw [hypreal_zero_def]
"(0::hypreal) < r --> 0 < r pow n";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
-by (auto_tac (claset() addSEs [FreeUltrafilterNat_subset,
- realpow_gt_zero],simpset() addsimps [hyperpow,hypreal_less,
- hypreal_le]));
+by (auto_tac (claset() addSEs [FreeUltrafilterNat_subset, realpow_gt_zero],
+ simpset() addsimps [hyperpow,hypreal_less, hypreal_le]));
qed_spec_mp "hyperpow_gt_zero";
Goal "(0::hypreal) < r --> 0 <= r pow n";
-by (blast_tac (claset() addSIs [hyperpow_gt_zero,
- hypreal_less_imp_le]) 1);
+by (blast_tac (claset() addSIs [hyperpow_gt_zero, hypreal_less_imp_le]) 1);
qed_spec_mp "hyperpow_ge_zero";
Goalw [hypreal_zero_def]
"(0::hypreal) <= r --> 0 <= r pow n";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
-by (auto_tac (claset() addSEs [FreeUltrafilterNat_subset,
- realpow_ge_zero2],simpset() addsimps [hyperpow,hypreal_le]));
+by (auto_tac (claset() addSEs [FreeUltrafilterNat_subset, realpow_ge_zero2],
+ simpset() addsimps [hyperpow,hypreal_le]));
qed "hyperpow_ge_zero2";
Goalw [hypreal_zero_def]
@@ -369,23 +367,23 @@
Goalw [hypreal_one_def] "1hr pow n = 1hr";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
-by (auto_tac (claset(),simpset() addsimps [hyperpow]));
+by (auto_tac (claset(), simpset() addsimps [hyperpow]));
qed "hyperpow_eq_one";
Addsimps [hyperpow_eq_one];
Goalw [hypreal_one_def]
"abs(-(1hr pow n)) = 1hr";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
-by (auto_tac (claset(),simpset() addsimps [abs_one,
- hrabs_minus_cancel,hyperpow,hypreal_hrabs]));
+by (auto_tac (claset(),
+ simpset() addsimps [abs_one, hyperpow,hypreal_hrabs]));
qed "hrabs_minus_hyperpow_one";
Addsimps [hrabs_minus_hyperpow_one];
Goalw [hypreal_one_def]
"abs((-1hr) pow n) = 1hr";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
-by (auto_tac (claset(),simpset() addsimps
- [hyperpow,hypreal_minus,hypreal_hrabs]));
+by (auto_tac (claset(),
+ simpset() addsimps [hyperpow,hypreal_minus,hypreal_hrabs]));
qed "hrabs_hyperpow_minus_one";
Addsimps [hrabs_hyperpow_minus_one];
@@ -393,8 +391,8 @@
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
by (res_inst_tac [("z","s")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [hyperpow,
- hypreal_mult,realpow_mult]));
+by (auto_tac (claset(),
+ simpset() addsimps [hyperpow, hypreal_mult,realpow_mult]));
qed "hyperpow_mult";
Goal "(0::hypreal) <= r pow (1hn + 1hn)";
@@ -413,13 +411,13 @@
Addsimps [hyperpow_two_hrabs];
Goal "!!r. 1hr < r ==> 1hr < r pow (1hn + 1hn)";
-by (auto_tac (claset(),simpset() addsimps [hyperpow_two]));
+by (auto_tac (claset(), simpset() addsimps [hyperpow_two]));
by (cut_facts_tac [hypreal_zero_less_one] 1);
by (forw_inst_tac [("R1.0","0")] hypreal_less_trans 1);
by (assume_tac 1);
by (dres_inst_tac [("z","r"),("x","1hr")]
hypreal_mult_less_mono1 1);
-by (auto_tac (claset() addIs [hypreal_less_trans],simpset()));
+by (auto_tac (claset() addIs [hypreal_less_trans], simpset()));
qed "hyperpow_two_gt_one";
Goal "!!r. 1hr <= r ==> 1hr <= r pow (1hn + 1hn)";
@@ -433,8 +431,8 @@
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
by (auto_tac (claset() addSEs [FreeUltrafilterNat_subset]
- addDs [realpow_Suc_gt_zero],simpset() addsimps [hyperpow,
- hypreal_less,hypnat_add]));
+ addDs [realpow_Suc_gt_zero],
+ simpset() addsimps [hyperpow, hypreal_less,hypnat_add]));
qed "hyperpow_Suc_gt_zero";
Goal "!!r. (0::hypreal) <= r ==> 0 <= r pow (n + 1hn)";
@@ -456,8 +454,8 @@
Goalw [hypreal_one_def]
"(-1hr) pow ((1hn + 1hn)*n) = 1hr";
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
-by (auto_tac (claset(),simpset() addsimps [hyperpow,
- hypnat_add,hypreal_minus]));
+by (auto_tac (claset(),
+ simpset() addsimps [hyperpow, hypnat_add,hypreal_minus]));
qed "hyperpow_minus_one2";
Addsimps [hyperpow_minus_one2];
@@ -488,8 +486,8 @@
by (res_inst_tac [("z","n")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","N")] eq_Abs_hypnat 1);
by (res_inst_tac [("z","r")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [hyperpow,
- hypreal_le,hypreal_less,hypnat_less]));
+by (auto_tac (claset(),
+ simpset() addsimps [hyperpow, hypreal_le,hypreal_less,hypnat_less]));
by (etac (FreeUltrafilterNat_Int RS FreeUltrafilterNat_subset) 1);
by (etac FreeUltrafilterNat_Int 1);
by (auto_tac (claset() addSDs [conjI RS realpow_less_le],
@@ -510,12 +508,12 @@
Goalw [hypreal_of_real_def,hypnat_of_nat_def]
"(hypreal_of_real r) pow (hypnat_of_nat n) = hypreal_of_real (r ^ n)";
-by (auto_tac (claset(),simpset() addsimps [hyperpow]));
+by (auto_tac (claset(), simpset() addsimps [hyperpow]));
qed "hyperpow_realpow";
Goalw [SReal_def]
"(hypreal_of_real r) pow (hypnat_of_nat n) : SReal";
-by (auto_tac (claset(),simpset() addsimps [hyperpow_realpow]));
+by (auto_tac (claset(), simpset() addsimps [hyperpow_realpow]));
qed "hyperpow_SReal";
Addsimps [hyperpow_SReal];
@@ -563,8 +561,7 @@
"(x ^ n : Infinitesimal) = \
\ (x pow (hypnat_of_nat n) : Infinitesimal)";
by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [hrealpow,
- hyperpow]));
+by (auto_tac (claset(), simpset() addsimps [hrealpow, hyperpow]));
qed "hrealpow_hyperpow_Infinitesimal_iff";
goal HyperPow.thy
@@ -572,8 +569,8 @@
\ ==> x ^ n : Infinitesimal";
by (auto_tac (claset() addSIs [Infinitesimal_hyperpow],
simpset() addsimps [hrealpow_hyperpow_Infinitesimal_iff,
- hypnat_of_nat_less_iff,hypnat_of_nat_zero]
- delsimps [hypnat_of_nat_less_iff RS sym]));
+ hypnat_of_nat_less_iff,hypnat_of_nat_zero]
+ delsimps [hypnat_of_nat_less_iff RS sym]));
qed "Infinitesimal_hrealpow";