(* Title : Log.thy Author : Jacques D. Fleuriot Additional contributions by Jeremy Avigad Copyright : 2000,2001 University of Edinburgh *) header{*Logarithms: Standard Version*} theory Log imports Transcendental begin constdefs powr :: "[real,real] => real" (infixr "powr" 80) --{*exponentation with real exponent*} "x powr a == exp(a * ln x)" log :: "[real,real] => real" --{*logarithm of @{term x} to base @{term a}*} "log a x == ln x / ln a" lemma powr_one_eq_one [simp]: "1 powr a = 1" by (simp add: powr_def) lemma powr_zero_eq_one [simp]: "x powr 0 = 1" by (simp add: powr_def) lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)" by (simp add: powr_def) declare powr_one_gt_zero_iff [THEN iffD2, simp] lemma powr_mult: "[| 0 < x; 0 < y |] ==> (x * y) powr a = (x powr a) * (y powr a)" by (simp add: powr_def exp_add [symmetric] ln_mult right_distrib) lemma powr_gt_zero [simp]: "0 < x powr a" by (simp add: powr_def) lemma powr_ge_pzero [simp]: "0 <= x powr y" by (rule order_less_imp_le, rule powr_gt_zero) lemma powr_not_zero [simp]: "x powr a ≠ 0" by (simp add: powr_def) lemma powr_divide: "[| 0 < x; 0 < y |] ==> (x / y) powr a = (x powr a)/(y powr a)" apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult) apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse) done lemma powr_divide2: "x powr a / x powr b = x powr (a - b)" apply (simp add: powr_def) apply (subst exp_diff [THEN sym]) apply (simp add: left_diff_distrib) done lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)" by (simp add: powr_def exp_add [symmetric] left_distrib) lemma powr_powr: "(x powr a) powr b = x powr (a * b)" by (simp add: powr_def) lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a" by (simp add: powr_powr real_mult_commute) lemma powr_minus: "x powr (-a) = inverse (x powr a)" by (simp add: powr_def exp_minus [symmetric]) lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)" by (simp add: divide_inverse powr_minus) lemma powr_less_mono: "[| a < b; 1 < x |] ==> x powr a < x powr b" by (simp add: powr_def) lemma powr_less_cancel: "[| x powr a < x powr b; 1 < x |] ==> a < b" by (simp add: powr_def) lemma powr_less_cancel_iff [simp]: "1 < x ==> (x powr a < x powr b) = (a < b)" by (blast intro: powr_less_cancel powr_less_mono) lemma powr_le_cancel_iff [simp]: "1 < x ==> (x powr a ≤ x powr b) = (a ≤ b)" by (simp add: linorder_not_less [symmetric]) lemma log_ln: "ln x = log (exp(1)) x" by (simp add: log_def) lemma powr_log_cancel [simp]: "[| 0 < a; a ≠ 1; 0 < x |] ==> a powr (log a x) = x" by (simp add: powr_def log_def) lemma log_powr_cancel [simp]: "[| 0 < a; a ≠ 1 |] ==> log a (a powr y) = y" by (simp add: log_def powr_def) lemma log_mult: "[| 0 < a; a ≠ 1; 0 < x; 0 < y |] ==> log a (x * y) = log a x + log a y" by (simp add: log_def ln_mult divide_inverse left_distrib) lemma log_eq_div_ln_mult_log: "[| 0 < a; a ≠ 1; 0 < b; b ≠ 1; 0 < x |] ==> log a x = (ln b/ln a) * log b x" by (simp add: log_def divide_inverse) text{*Base 10 logarithms*} lemma log_base_10_eq1: "0 < x ==> log 10 x = (ln (exp 1) / ln 10) * ln x" by (simp add: log_def) lemma log_base_10_eq2: "0 < x ==> log 10 x = (log 10 (exp 1)) * ln x" by (simp add: log_def) lemma log_one [simp]: "log a 1 = 0" by (simp add: log_def) lemma log_eq_one [simp]: "[| 0 < a; a ≠ 1 |] ==> log a a = 1" by (simp add: log_def) lemma log_inverse: "[| 0 < a; a ≠ 1; 0 < x |] ==> log a (inverse x) = - log a x" apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1]) apply (simp add: log_mult [symmetric]) done lemma log_divide: "[|0 < a; a ≠ 1; 0 < x; 0 < y|] ==> log a (x/y) = log a x - log a y" by (simp add: log_mult divide_inverse log_inverse) lemma log_less_cancel_iff [simp]: "[| 1 < a; 0 < x; 0 < y |] ==> (log a x < log a y) = (x < y)" apply safe apply (rule_tac [2] powr_less_cancel) apply (drule_tac a = "log a x" in powr_less_mono, auto) done lemma log_le_cancel_iff [simp]: "[| 1 < a; 0 < x; 0 < y |] ==> (log a x ≤ log a y) = (x ≤ y)" by (simp add: linorder_not_less [symmetric]) lemma powr_realpow: "0 < x ==> x powr (real n) = x^n" apply (induct n, simp) apply (subgoal_tac "real(Suc n) = real n + 1") apply (erule ssubst) apply (subst powr_add, simp, simp) done lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))" apply (case_tac "x = 0", simp, simp) apply (rule powr_realpow [THEN sym], simp) done lemma ln_pwr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x" by (unfold powr_def, simp) lemma ln_bound: "1 <= x ==> ln x <= x" apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1") apply simp apply (rule ln_add_one_self_le_self, simp) done lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b" apply (case_tac "x = 1", simp) apply (case_tac "a = b", simp) apply (rule order_less_imp_le) apply (rule powr_less_mono, auto) done lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a" apply (subst powr_zero_eq_one [THEN sym]) apply (rule powr_mono, assumption+) done lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a < y powr a" apply (unfold powr_def) apply (rule exp_less_mono) apply (rule mult_strict_left_mono) apply (subst ln_less_cancel_iff, assumption) apply (rule order_less_trans) prefer 2 apply assumption+ done lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a < x powr a" apply (unfold powr_def) apply (rule exp_less_mono) apply (rule mult_strict_left_mono_neg) apply (subst ln_less_cancel_iff) apply assumption apply (rule order_less_trans) prefer 2 apply assumption+ done lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a" apply (case_tac "a = 0", simp) apply (case_tac "x = y", simp) apply (rule order_less_imp_le) apply (rule powr_less_mono2, auto) done lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a" apply (rule mult_imp_le_div_pos) apply (assumption) apply (subst mult_commute) apply (subst ln_pwr [THEN sym]) apply auto apply (rule ln_bound) apply (erule ge_one_powr_ge_zero) apply (erule order_less_imp_le) done lemma ln_powr_bound2: "1 < x ==> 0 < a ==> (ln x) powr a <= (a powr a) * x" proof - assume "1 < x" and "0 < a" then have "ln x <= (x powr (1 / a)) / (1 / a)" apply (intro ln_powr_bound) apply (erule order_less_imp_le) apply (rule divide_pos_pos) apply simp_all done also have "... = a * (x powr (1 / a))" by simp finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a" apply (intro powr_mono2) apply (rule order_less_imp_le, rule prems) apply (rule ln_gt_zero) apply (rule prems) apply assumption done also have "... = (a powr a) * ((x powr (1 / a)) powr a)" apply (rule powr_mult) apply (rule prems) apply (rule powr_gt_zero) done also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)" by (rule powr_powr) also have "... = x" apply simp apply (subgoal_tac "a ~= 0") apply (insert prems, auto) done finally show ?thesis . qed lemma LIMSEQ_neg_powr: "0 < s ==> (%x. (real x) powr - s) ----> 0" apply (unfold LIMSEQ_def) apply clarsimp apply (rule_tac x = "natfloor(r powr (1 / - s)) + 1" in exI) apply clarify proof - fix r fix n assume "0 < s" and "0 < r" and "natfloor (r powr (1 / - s)) + 1 <= n" have "r powr (1 / - s) < real(natfloor(r powr (1 / - s))) + 1" by (rule real_natfloor_add_one_gt) also have "... = real(natfloor(r powr (1 / -s)) + 1)" by simp also have "... <= real n" apply (subst real_of_nat_le_iff) apply (rule prems) done finally have "r powr (1 / - s) < real n". then have "real n powr (- s) < (r powr (1 / - s)) powr - s" apply (intro powr_less_mono2_neg) apply (auto simp add: prems) done also have "... = r" by (simp add: powr_powr prems less_imp_neq [THEN not_sym]) finally show "real n powr - s < r" . qed ML {* val powr_one_eq_one = thm "powr_one_eq_one"; val powr_zero_eq_one = thm "powr_zero_eq_one"; val powr_one_gt_zero_iff = thm "powr_one_gt_zero_iff"; val powr_mult = thm "powr_mult"; val powr_gt_zero = thm "powr_gt_zero"; val powr_not_zero = thm "powr_not_zero"; val powr_divide = thm "powr_divide"; val powr_add = thm "powr_add"; val powr_powr = thm "powr_powr"; val powr_powr_swap = thm "powr_powr_swap"; val powr_minus = thm "powr_minus"; val powr_minus_divide = thm "powr_minus_divide"; val powr_less_mono = thm "powr_less_mono"; val powr_less_cancel = thm "powr_less_cancel"; val powr_less_cancel_iff = thm "powr_less_cancel_iff"; val powr_le_cancel_iff = thm "powr_le_cancel_iff"; val log_ln = thm "log_ln"; val powr_log_cancel = thm "powr_log_cancel"; val log_powr_cancel = thm "log_powr_cancel"; val log_mult = thm "log_mult"; val log_eq_div_ln_mult_log = thm "log_eq_div_ln_mult_log"; val log_base_10_eq1 = thm "log_base_10_eq1"; val log_base_10_eq2 = thm "log_base_10_eq2"; val log_one = thm "log_one"; val log_eq_one = thm "log_eq_one"; val log_inverse = thm "log_inverse"; val log_divide = thm "log_divide"; val log_less_cancel_iff = thm "log_less_cancel_iff"; val log_le_cancel_iff = thm "log_le_cancel_iff"; *} end
lemma powr_one_eq_one:
1 powr a = 1
lemma powr_zero_eq_one:
x powr 0 = 1
lemma powr_one_gt_zero_iff:
(x powr 1 = x) = (0 < x)
lemma powr_mult:
[| 0 < x; 0 < y |] ==> (x * y) powr a = x powr a * y powr a
lemma powr_gt_zero:
0 < x powr a
lemma powr_ge_pzero:
0 ≤ x powr y
lemma powr_not_zero:
x powr a ≠ 0
lemma powr_divide:
[| 0 < x; 0 < y |] ==> (x / y) powr a = x powr a / y powr a
lemma powr_divide2:
x powr a / x powr b = x powr (a - b)
lemma powr_add:
x powr (a + b) = x powr a * x powr b
lemma powr_powr:
(x powr a) powr b = x powr (a * b)
lemma powr_powr_swap:
(x powr a) powr b = (x powr b) powr a
lemma powr_minus:
x powr - a = inverse (x powr a)
lemma powr_minus_divide:
x powr - a = 1 / x powr a
lemma powr_less_mono:
[| a < b; 1 < x |] ==> x powr a < x powr b
lemma powr_less_cancel:
[| x powr a < x powr b; 1 < x |] ==> a < b
lemma powr_less_cancel_iff:
1 < x ==> (x powr a < x powr b) = (a < b)
lemma powr_le_cancel_iff:
1 < x ==> (x powr a ≤ x powr b) = (a ≤ b)
lemma log_ln:
ln x = log (exp 1) x
lemma powr_log_cancel:
[| 0 < a; a ≠ 1; 0 < x |] ==> a powr log a x = x
lemma log_powr_cancel:
[| 0 < a; a ≠ 1 |] ==> log a (a powr y) = y
lemma log_mult:
[| 0 < a; a ≠ 1; 0 < x; 0 < y |] ==> log a (x * y) = log a x + log a y
lemma log_eq_div_ln_mult_log:
[| 0 < a; a ≠ 1; 0 < b; b ≠ 1; 0 < x |] ==> log a x = ln b / ln a * log b x
lemma log_base_10_eq1:
0 < x ==> log 10 x = ln (exp 1) / ln 10 * ln x
lemma log_base_10_eq2:
0 < x ==> log 10 x = log 10 (exp 1) * ln x
lemma log_one:
log a 1 = 0
lemma log_eq_one:
[| 0 < a; a ≠ 1 |] ==> log a a = 1
lemma log_inverse:
[| 0 < a; a ≠ 1; 0 < x |] ==> log a (inverse x) = - log a x
lemma log_divide:
[| 0 < a; a ≠ 1; 0 < x; 0 < y |] ==> log a (x / y) = log a x - log a y
lemma log_less_cancel_iff:
[| 1 < a; 0 < x; 0 < y |] ==> (log a x < log a y) = (x < y)
lemma log_le_cancel_iff:
[| 1 < a; 0 < x; 0 < y |] ==> (log a x ≤ log a y) = (x ≤ y)
lemma powr_realpow:
0 < x ==> x powr real n = x ^ n
lemma powr_realpow2:
[| 0 ≤ x; 0 < n |] ==> x ^ n = (if x = 0 then 0 else x powr real n)
lemma ln_pwr:
[| 0 < x; 0 < y |] ==> ln (x powr y) = y * ln x
lemma ln_bound:
1 ≤ x ==> ln x ≤ x
lemma powr_mono:
[| a ≤ b; 1 ≤ x |] ==> x powr a ≤ x powr b
lemma ge_one_powr_ge_zero:
[| 1 ≤ x; 0 ≤ a |] ==> 1 ≤ x powr a
lemma powr_less_mono2:
[| 0 < a; 0 < x; x < y |] ==> x powr a < y powr a
lemma powr_less_mono2_neg:
[| a < 0; 0 < x; x < y |] ==> y powr a < x powr a
lemma powr_mono2:
[| 0 ≤ a; 0 < x; x ≤ y |] ==> x powr a ≤ y powr a
lemma ln_powr_bound:
[| 1 ≤ x; 0 < a |] ==> ln x ≤ x powr a / a
lemma ln_powr_bound2:
[| 1 < x; 0 < a |] ==> ln x powr a ≤ a powr a * x
lemma LIMSEQ_neg_powr:
0 < s ==> (%x. real x powr - s) ----> 0