(* Title : RealPow.ML
ID : $Id$
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : Natural Powers of reals theory
*)
Goal "0r ^ (Suc n) = 0r";
by (Auto_tac);
qed "realpow_zero";
Addsimps [realpow_zero];
Goal "r ~= 0r --> r ^ n ~= 0r";
by (induct_tac "n" 1);
by (auto_tac (claset() addIs [real_mult_not_zeroE],
simpset() addsimps [real_zero_not_eq_one RS not_sym]));
qed_spec_mp "realpow_not_zero";
Goal "r ^ n = 0r ==> r = 0r";
by (blast_tac (claset() addIs [ccontr]
addDs [realpow_not_zero]) 1);
qed "realpow_zero_zero";
Goal "r ~= 0r --> rinv(r ^ n) = (rinv r) ^ n";
by (induct_tac "n" 1);
by (Auto_tac);
by (forw_inst_tac [("n","n")] realpow_not_zero 1);
by (auto_tac (claset() addDs [real_rinv_distrib],
simpset()));
qed_spec_mp "realpow_rinv";
Goal "rabs r ^ n = rabs (r ^ n)";
by (induct_tac "n" 1);
by (auto_tac (claset(),simpset() addsimps
[rabs_mult,rabs_one]));
qed "realpow_rabs";
Goal "(r::real) ^ (n + m) = (r ^ n) * (r ^ m)";
by (induct_tac "n" 1);
by (auto_tac (claset(),simpset() addsimps real_mult_ac));
qed "realpow_add";
Goal "(r::real) ^ 1 = r";
by (Simp_tac 1);
qed "realpow_one";
Addsimps [realpow_one];
Goal "(r::real) ^ 2 = r * r";
by (Simp_tac 1);
qed "realpow_two";
Goal "0r < r --> 0r <= r ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset() addDs [real_less_imp_le]
addIs [real_le_mult_order],simpset()));
qed_spec_mp "realpow_ge_zero";
Goal "0r < r --> 0r < r ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset() addIs [real_mult_order],
simpset() addsimps [real_zero_less_one]));
qed_spec_mp "realpow_gt_zero";
Goal "0r <= r --> 0r <= r ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset() addIs [real_le_mult_order],
simpset()));
qed_spec_mp "realpow_ge_zero2";
Goal "0r < x & x <= y --> x ^ n <= y ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset() addSIs [real_mult_le_mono],
simpset()));
by (asm_simp_tac (simpset() addsimps [realpow_ge_zero]) 1);
qed_spec_mp "realpow_le";
Goal "0r <= x & x <= y --> x ^ n <= y ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset() addSIs [real_mult_le_mono4],
simpset()));
by (asm_simp_tac (simpset() addsimps [realpow_ge_zero2]) 1);
qed_spec_mp "realpow_le2";
Goal "0r < x & x < y & 0 < n --> x ^ n < y ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset() addIs [real_mult_less_mono,
gr0I] addDs [realpow_gt_zero],simpset()));
qed_spec_mp "realpow_less";
Goal "1r ^ n = 1r";
by (induct_tac "n" 1);
by (Auto_tac);
qed "realpow_eq_one";
Addsimps [realpow_eq_one];
(** New versions using #0 and #1 instead of 0r and 1r
REMOVE AFTER CONVERTING THIS FILE TO USE #0 AND #1 **)
Addsimps (map (rename_numerals thy) [realpow_zero, realpow_eq_one]);
Goal "rabs(-(1r ^ n)) = 1r";
by (simp_tac (simpset() addsimps
[rabs_minus_cancel,rabs_one]) 1);
qed "rabs_minus_realpow_one";
Addsimps [rabs_minus_realpow_one];
Goal "rabs((-1r) ^ n) = 1r";
by (induct_tac "n" 1);
by (auto_tac (claset(),simpset() addsimps [rabs_mult,
rabs_minus_cancel,rabs_one]));
qed "rabs_realpow_minus_one";
Addsimps [rabs_realpow_minus_one];
Goal "((r::real) * s) ^ n = (r ^ n) * (s ^ n)";
by (induct_tac "n" 1);
by (auto_tac (claset(),simpset() addsimps real_mult_ac));
qed "realpow_mult";
Goal "0r <= r ^ 2";
by (simp_tac (simpset() addsimps [realpow_two]) 1);
qed "realpow_two_le";
Addsimps [realpow_two_le];
Goal "rabs(x ^ 2) = x ^ 2";
by (simp_tac (simpset() addsimps [rabs_eqI1]) 1);
qed "rabs_realpow_two";
Addsimps [rabs_realpow_two];
Goal "rabs(x) ^ 2 = x ^ 2";
by (simp_tac (simpset() addsimps
[realpow_rabs,rabs_eqI1] delsimps [realpow_Suc]) 1);
qed "realpow_two_rabs";
Addsimps [realpow_two_rabs];
Goal "1r < r ==> 1r < r ^ 2";
by (auto_tac (claset(),simpset() addsimps [realpow_two]));
by (cut_facts_tac [real_zero_less_one] 1);
by (forw_inst_tac [("R1.0","0r")] real_less_trans 1);
by (assume_tac 1);
by (dres_inst_tac [("z","r"),("x","1r")] real_mult_less_mono1 1);
by (auto_tac (claset() addIs [real_less_trans],simpset()));
qed "realpow_two_gt_one";
Goal "1r < r --> 1r <= r ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset() addSDs [real_le_imp_less_or_eq],
simpset()));
by (dtac (real_zero_less_one RS real_mult_less_mono) 1);
by (auto_tac (claset() addSIs [real_less_imp_le],
simpset() addsimps [real_zero_less_one]));
qed_spec_mp "realpow_ge_one";
Goal "1r < r ==> 1r < r ^ (Suc n)";
by (forw_inst_tac [("n","n")] realpow_ge_one 1);
by (dtac real_le_imp_less_or_eq 1 THEN Step_tac 1);
by (dtac sym 2);
by (forward_tac [real_zero_less_one RS real_less_trans] 1);
by (dres_inst_tac [("y","r ^ n")] real_mult_less_mono2 1);
by (auto_tac (claset() addDs [real_less_trans],
simpset()));
qed "realpow_Suc_gt_one";
Goal "1r <= r ==> 1r <= r ^ n";
by (dtac real_le_imp_less_or_eq 1);
by (auto_tac (claset() addDs [realpow_ge_one], simpset()));
qed "realpow_ge_one2";
Goal "0r < r ==> 0r < r ^ Suc n";
by (forw_inst_tac [("n","n")] realpow_ge_zero 1);
by (forw_inst_tac [("n1","n")]
((real_not_refl2 RS not_sym) RS realpow_not_zero RS not_sym) 1);
by (auto_tac (claset() addSDs [real_le_imp_less_or_eq]
addIs [real_mult_order],simpset()));
qed "realpow_Suc_gt_zero";
Goal "0r <= r ==> 0r <= r ^ Suc n";
by (etac (real_le_imp_less_or_eq RS disjE) 1);
by (etac (realpow_ge_zero) 1);
by (Asm_simp_tac 1);
qed "realpow_Suc_ge_zero";
Goal "(#1::real) <= #2 ^ n";
by (res_inst_tac [("j","#1 ^ n")] real_le_trans 1);
by (rtac realpow_le 2);
by (auto_tac (claset() addIs [real_less_imp_le],
simpset() addsimps [zero_eq_numeral_0]));
qed "two_realpow_ge_one";
Goal "real_of_nat n < #2 ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset(),
simpset() addsimps [zero_eq_numeral_0, one_eq_numeral_1,
real_mult_2,
real_of_nat_Suc, real_of_nat_zero,
real_add_less_le_mono, two_realpow_ge_one]));
qed "two_realpow_gt";
Addsimps [two_realpow_gt,two_realpow_ge_one];
Goal "(-1r) ^ (2*n) = 1r";
by (induct_tac "n" 1);
by (Auto_tac);
qed "realpow_minus_one";
Addsimps [realpow_minus_one];
Goal "(-1r) ^ (n + n) = 1r";
by (induct_tac "n" 1);
by (Auto_tac);
qed "realpow_minus_one2";
Addsimps [realpow_minus_one2];
Goal "(-1r) ^ Suc (n + n) = -1r";
by (induct_tac "n" 1);
by (Auto_tac);
qed "realpow_minus_one_odd";
Addsimps [realpow_minus_one_odd];
Goal "(-1r) ^ Suc (Suc (n + n)) = 1r";
by (induct_tac "n" 1);
by (Auto_tac);
qed "realpow_minus_one_even";
Addsimps [realpow_minus_one_even];
Goal "0r < r & r < 1r --> r ^ Suc n < r ^ n";
by (induct_tac "n" 1);
by (Auto_tac);
qed_spec_mp "realpow_Suc_less";
Goal "0r <= r & r < 1r --> r ^ Suc n <= r ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset() addIs [real_less_imp_le] addSDs
[real_le_imp_less_or_eq],simpset()));
qed_spec_mp "realpow_Suc_le";
Goal "0r <= 0r ^ n";
by (cases_tac "n" 1);
by (Auto_tac);
qed "realpow_zero_le";
Addsimps [realpow_zero_le];
Goal "0r < r & r < 1r --> r ^ Suc n <= r ^ n";
by (blast_tac (claset() addSIs [real_less_imp_le,
realpow_Suc_less]) 1);
qed_spec_mp "realpow_Suc_le2";
Goal "[| 0r <= r; r < 1r |] ==> r ^ Suc n <= r ^ n";
by (etac (real_le_imp_less_or_eq RS disjE) 1);
by (rtac realpow_Suc_le2 1);
by (Auto_tac);
qed "realpow_Suc_le3";
Goal "0r <= r & r < 1r & n < N --> r ^ N <= r ^ n";
by (induct_tac "N" 1);
by (Auto_tac);
by (ALLGOALS(forw_inst_tac [("n","na")] realpow_ge_zero2));
by (ALLGOALS(dtac real_mult_le_mono3));
by (REPEAT(assume_tac 1));
by (REPEAT(assume_tac 3));
by (auto_tac (claset(),simpset() addsimps
[less_Suc_eq]));
qed_spec_mp "realpow_less_le";
Goal "[| 0r <= r; r < 1r; n <= N |] ==> r ^ N <= r ^ n";
by (dres_inst_tac [("n","N")] le_imp_less_or_eq 1);
by (auto_tac (claset() addIs [realpow_less_le],
simpset()));
qed "realpow_le_le";
Goal "[| 0r < r; r < 1r |] ==> r ^ Suc n <= r";
by (dres_inst_tac [("n","1"),("N","Suc n")]
(real_less_imp_le RS realpow_le_le) 1);
by (Auto_tac);
qed "realpow_Suc_le_self";
Goal "[| 0r < r; r < 1r |] ==> r ^ Suc n < 1r";
by (blast_tac (claset() addIs [realpow_Suc_le_self,
real_le_less_trans]) 1);
qed "realpow_Suc_less_one";
Goal "1r <= r --> r ^ n <= r ^ Suc n";
by (induct_tac "n" 1);
by (auto_tac (claset() addSIs [real_mult_le_le_mono1],simpset()));
by (rtac ccontr 1 THEN dtac not_real_leE 1);
by (dtac real_le_less_trans 1 THEN assume_tac 1);
by (etac (real_zero_less_one RS real_less_asym) 1);
qed_spec_mp "realpow_le_Suc";
Goal "1r < r --> r ^ n < r ^ Suc n";
by (induct_tac "n" 1);
by (auto_tac (claset() addSIs [real_mult_less_mono2],simpset()));
by (rtac ccontr 1 THEN dtac real_leI 1);
by (dtac real_less_le_trans 1 THEN assume_tac 1);
by (etac (real_zero_less_one RS real_less_asym) 1);
qed_spec_mp "realpow_less_Suc";
Goal "1r < r --> r ^ n <= r ^ Suc n";
by (blast_tac (claset() addSIs [real_less_imp_le,
realpow_less_Suc]) 1);
qed_spec_mp "realpow_le_Suc2";
Goal "1r < r & n < N --> r ^ n <= r ^ N";
by (induct_tac "N" 1);
by (Auto_tac);
by (ALLGOALS(forw_inst_tac [("n","na")] realpow_ge_one));
by (ALLGOALS(dtac real_mult_self_le));
by (assume_tac 1);
by (assume_tac 2);
by (auto_tac (claset() addIs [real_le_trans],
simpset() addsimps [less_Suc_eq]));
qed_spec_mp "realpow_gt_ge";
Goal "1r <= r & n < N --> r ^ n <= r ^ N";
by (induct_tac "N" 1);
by (Auto_tac);
by (ALLGOALS(forw_inst_tac [("n","na")] realpow_ge_one2));
by (ALLGOALS(dtac real_mult_self_le2));
by (assume_tac 1);
by (assume_tac 2);
by (auto_tac (claset() addIs [real_le_trans],
simpset() addsimps [less_Suc_eq]));
qed_spec_mp "realpow_gt_ge2";
Goal "[| 1r < r; n <= N |] ==> r ^ n <= r ^ N";
by (dres_inst_tac [("n","N")] le_imp_less_or_eq 1);
by (auto_tac (claset() addIs [realpow_gt_ge],simpset()));
qed "realpow_ge_ge";
Goal "[| 1r <= r; n <= N |] ==> r ^ n <= r ^ N";
by (dres_inst_tac [("n","N")] le_imp_less_or_eq 1);
by (auto_tac (claset() addIs [realpow_gt_ge2],simpset()));
qed "realpow_ge_ge2";
Goal "1r < r ==> r <= r ^ Suc n";
by (dres_inst_tac [("n","1"),("N","Suc n")]
realpow_ge_ge 1);
by (Auto_tac);
qed_spec_mp "realpow_Suc_ge_self";
Goal "1r <= r ==> r <= r ^ Suc n";
by (dres_inst_tac [("n","1"),("N","Suc n")]
realpow_ge_ge2 1);
by (Auto_tac);
qed_spec_mp "realpow_Suc_ge_self2";
Goal "[| 1r < r; 0 < n |] ==> r <= r ^ n";
by (dtac (less_not_refl2 RS not0_implies_Suc) 1);
by (auto_tac (claset() addSIs
[realpow_Suc_ge_self],simpset()));
qed "realpow_ge_self";
Goal "[| 1r <= r; 0 < n |] ==> r <= r ^ n";
by (dtac (less_not_refl2 RS not0_implies_Suc) 1);
by (auto_tac (claset() addSIs [realpow_Suc_ge_self2],simpset()));
qed "realpow_ge_self2";
Goal "0 < n --> (x::real) ^ (n - 1) * x = x ^ n";
by (induct_tac "n" 1);
by (auto_tac (claset(),simpset()
addsimps [real_mult_commute]));
qed_spec_mp "realpow_minus_mult";
Addsimps [realpow_minus_mult];
Goal "r ~= 0r ==> r * rinv(r) ^ 2 = rinv r";
by (asm_simp_tac (simpset() addsimps [realpow_two,
real_mult_assoc RS sym]) 1);
qed "realpow_two_mult_rinv";
Addsimps [realpow_two_mult_rinv];
(** New versions using #0 and #1 instead of 0r and 1r
REMOVE AFTER CONVERTING THIS FILE TO USE #0 AND #1 **)
Addsimps (map (rename_numerals thy)
[realpow_two_le, realpow_zero_le,
rabs_minus_realpow_one, rabs_realpow_minus_one,
realpow_minus_one, realpow_minus_one2, realpow_minus_one_odd]);