author | wenzelm |
Mon, 17 Sep 2007 16:36:45 +0200 | |
changeset 24614 | a4b2eb0dd673 |
parent 23482 | 2f4be6844f7c |
child 25875 | 536dfdc25e0a |
permissions | -rwxr-xr-x |
16959 | 1 |
(* Title: Ln.thy |
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Author: Jeremy Avigad |
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ID: $Id$ |
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*) |
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header {* Properties of ln *} |
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theory Ln |
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imports Transcendental |
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begin |
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lemma exp_first_two_terms: "exp x = 1 + x + suminf (%n. |
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inverse(real (fact (n+2))) * (x ^ (n+2)))" |
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proof - |
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have "exp x = suminf (%n. inverse(real (fact n)) * (x ^ n))" |
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by (simp add: exp_def) |
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also from summable_exp have "... = (SUM n : {0..<2}. |
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inverse(real (fact n)) * (x ^ n)) + suminf (%n. |
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inverse(real (fact (n+2))) * (x ^ (n+2)))" (is "_ = ?a + _") |
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by (rule suminf_split_initial_segment) |
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also have "?a = 1 + x" |
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by (simp add: numerals) |
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finally show ?thesis . |
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qed |
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lemma exp_tail_after_first_two_terms_summable: |
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"summable (%n. inverse(real (fact (n+2))) * (x ^ (n+2)))" |
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proof - |
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note summable_exp |
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thus ?thesis |
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by (frule summable_ignore_initial_segment) |
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qed |
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lemma aux1: assumes a: "0 <= x" and b: "x <= 1" |
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shows "inverse (real (fact (n + 2))) * x ^ (n + 2) <= (x^2/2) * ((1/2)^n)" |
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proof (induct n) |
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show "inverse (real (fact (0 + 2))) * x ^ (0 + 2) <= |
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x ^ 2 / 2 * (1 / 2) ^ 0" |
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by (simp add: real_of_nat_Suc power2_eq_square) |
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next |
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fix n |
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assume c: "inverse (real (fact (n + 2))) * x ^ (n + 2) |
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<= x ^ 2 / 2 * (1 / 2) ^ n" |
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show "inverse (real (fact (Suc n + 2))) * x ^ (Suc n + 2) |
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<= x ^ 2 / 2 * (1 / 2) ^ Suc n" |
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proof - |
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have "inverse(real (fact (Suc n + 2))) <= |
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(1 / 2) *inverse (real (fact (n+2)))" |
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proof - |
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have "Suc n + 2 = Suc (n + 2)" by simp |
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then have "fact (Suc n + 2) = Suc (n + 2) * fact (n + 2)" |
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by simp |
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then have "real(fact (Suc n + 2)) = real(Suc (n + 2) * fact (n + 2))" |
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apply (rule subst) |
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apply (rule refl) |
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done |
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also have "... = real(Suc (n + 2)) * real(fact (n + 2))" |
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by (rule real_of_nat_mult) |
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finally have "real (fact (Suc n + 2)) = |
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real (Suc (n + 2)) * real (fact (n + 2))" . |
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then have "inverse(real (fact (Suc n + 2))) = |
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inverse(real (Suc (n + 2))) * inverse(real (fact (n + 2)))" |
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apply (rule ssubst) |
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apply (rule inverse_mult_distrib) |
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done |
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also have "... <= (1/2) * inverse(real (fact (n + 2)))" |
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apply (rule mult_right_mono) |
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apply (subst inverse_eq_divide) |
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apply simp |
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apply (rule inv_real_of_nat_fact_ge_zero) |
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done |
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finally show ?thesis . |
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qed |
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moreover have "x ^ (Suc n + 2) <= x ^ (n + 2)" |
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apply (simp add: mult_compare_simps) |
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apply (simp add: prems) |
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apply (subgoal_tac "0 <= x * (x * x^n)") |
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apply force |
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apply (rule mult_nonneg_nonneg, rule a)+ |
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apply (rule zero_le_power, rule a) |
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done |
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ultimately have "inverse (real (fact (Suc n + 2))) * x ^ (Suc n + 2) <= |
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(1 / 2 * inverse (real (fact (n + 2)))) * x ^ (n + 2)" |
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apply (rule mult_mono) |
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apply (rule mult_nonneg_nonneg) |
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apply simp |
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apply (subst inverse_nonnegative_iff_nonnegative) |
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apply (rule real_of_nat_fact_ge_zero) |
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apply (rule zero_le_power) |
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apply (rule a) |
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done |
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also have "... = 1 / 2 * (inverse (real (fact (n + 2))) * x ^ (n + 2))" |
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by simp |
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also have "... <= 1 / 2 * (x ^ 2 / 2 * (1 / 2) ^ n)" |
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apply (rule mult_left_mono) |
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apply (rule prems) |
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apply simp |
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done |
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also have "... = x ^ 2 / 2 * (1 / 2 * (1 / 2) ^ n)" |
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by auto |
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also have "(1::real) / 2 * (1 / 2) ^ n = (1 / 2) ^ (Suc n)" |
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by (rule realpow_Suc [THEN sym]) |
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finally show ?thesis . |
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qed |
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qed |
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||
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lemma aux2: "(%n. (x::real) ^ 2 / 2 * (1 / 2) ^ n) sums x^2" |
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proof - |
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have "(%n. (1 / 2::real)^n) sums (1 / (1 - (1/2)))" |
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apply (rule geometric_sums) |
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by (simp add: abs_less_iff) |
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also have "(1::real) / (1 - 1/2) = 2" |
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by simp |
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finally have "(%n. (1 / 2::real)^n) sums 2" . |
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then have "(%n. x ^ 2 / 2 * (1 / 2) ^ n) sums (x^2 / 2 * 2)" |
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by (rule sums_mult) |
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also have "x^2 / 2 * 2 = x^2" |
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by simp |
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finally show ?thesis . |
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qed |
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lemma exp_bound: "0 <= (x::real) ==> x <= 1 ==> exp x <= 1 + x + x^2" |
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proof - |
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assume a: "0 <= x" |
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assume b: "x <= 1" |
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have c: "exp x = 1 + x + suminf (%n. inverse(real (fact (n+2))) * |
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(x ^ (n+2)))" |
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by (rule exp_first_two_terms) |
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moreover have "suminf (%n. inverse(real (fact (n+2))) * (x ^ (n+2))) <= x^2" |
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proof - |
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have "suminf (%n. inverse(real (fact (n+2))) * (x ^ (n+2))) <= |
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suminf (%n. (x^2/2) * ((1/2)^n))" |
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apply (rule summable_le) |
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apply (auto simp only: aux1 prems) |
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apply (rule exp_tail_after_first_two_terms_summable) |
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by (rule sums_summable, rule aux2) |
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also have "... = x^2" |
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by (rule sums_unique [THEN sym], rule aux2) |
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finally show ?thesis . |
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qed |
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ultimately show ?thesis |
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by auto |
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qed |
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lemma aux4: "0 <= (x::real) ==> x <= 1 ==> exp (x - x^2) <= 1 + x" |
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proof - |
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assume a: "0 <= x" and b: "x <= 1" |
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have "exp (x - x^2) = exp x / exp (x^2)" |
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by (rule exp_diff) |
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also have "... <= (1 + x + x^2) / exp (x ^2)" |
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apply (rule divide_right_mono) |
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apply (rule exp_bound) |
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apply (rule a, rule b) |
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apply simp |
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done |
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also have "... <= (1 + x + x^2) / (1 + x^2)" |
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apply (rule divide_left_mono) |
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17013
74bc935273ea
renamed exp_ge_add_one_self2 to exp_ge_add_one_self
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parents:
16963
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apply (auto simp add: exp_ge_add_one_self_aux) |
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apply (rule add_nonneg_nonneg) |
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apply (insert prems, auto) |
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apply (rule mult_pos_pos) |
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apply auto |
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apply (rule add_pos_nonneg) |
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apply auto |
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done |
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also from a have "... <= 1 + x" |
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by(simp add:field_simps zero_compare_simps) |
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finally show ?thesis . |
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qed |
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lemma ln_one_plus_pos_lower_bound: "0 <= x ==> x <= 1 ==> |
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x - x^2 <= ln (1 + x)" |
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proof - |
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assume a: "0 <= x" and b: "x <= 1" |
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then have "exp (x - x^2) <= 1 + x" |
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by (rule aux4) |
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also have "... = exp (ln (1 + x))" |
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proof - |
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from a have "0 < 1 + x" by auto |
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thus ?thesis |
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by (auto simp only: exp_ln_iff [THEN sym]) |
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qed |
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finally have "exp (x - x ^ 2) <= exp (ln (1 + x))" . |
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thus ?thesis by (auto simp only: exp_le_cancel_iff) |
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qed |
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lemma ln_one_minus_pos_upper_bound: "0 <= x ==> x < 1 ==> ln (1 - x) <= - x" |
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proof - |
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assume a: "0 <= (x::real)" and b: "x < 1" |
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have "(1 - x) * (1 + x + x^2) = (1 - x^3)" |
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by (simp add: ring_simps power2_eq_square power3_eq_cube) |
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also have "... <= 1" |
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by (auto intro: zero_le_power simp add: a) |
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finally have "(1 - x) * (1 + x + x ^ 2) <= 1" . |
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moreover have "0 < 1 + x + x^2" |
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apply (rule add_pos_nonneg) |
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apply (insert a, auto) |
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done |
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ultimately have "1 - x <= 1 / (1 + x + x^2)" |
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by (elim mult_imp_le_div_pos) |
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also have "... <= 1 / exp x" |
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apply (rule divide_left_mono) |
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apply (rule exp_bound, rule a) |
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apply (insert prems, auto) |
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apply (rule mult_pos_pos) |
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apply (rule add_pos_nonneg) |
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apply auto |
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done |
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also have "... = exp (-x)" |
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by (auto simp add: exp_minus real_divide_def) |
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finally have "1 - x <= exp (- x)" . |
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also have "1 - x = exp (ln (1 - x))" |
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proof - |
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have "0 < 1 - x" |
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by (insert b, auto) |
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thus ?thesis |
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by (auto simp only: exp_ln_iff [THEN sym]) |
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qed |
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finally have "exp (ln (1 - x)) <= exp (- x)" . |
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thus ?thesis by (auto simp only: exp_le_cancel_iff) |
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qed |
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lemma aux5: "x < 1 ==> ln(1 - x) = - ln(1 + x / (1 - x))" |
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proof - |
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assume a: "x < 1" |
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have "ln(1 - x) = - ln(1 / (1 - x))" |
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proof - |
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have "ln(1 - x) = - (- ln (1 - x))" |
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by auto |
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also have "- ln(1 - x) = ln 1 - ln(1 - x)" |
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by simp |
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also have "... = ln(1 / (1 - x))" |
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apply (rule ln_div [THEN sym]) |
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by (insert a, auto) |
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finally show ?thesis . |
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qed |
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also have " 1 / (1 - x) = 1 + x / (1 - x)" using a by(simp add:field_simps) |
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finally show ?thesis . |
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qed |
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lemma ln_one_minus_pos_lower_bound: "0 <= x ==> x <= (1 / 2) ==> |
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- x - 2 * x^2 <= ln (1 - x)" |
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proof - |
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assume a: "0 <= x" and b: "x <= (1 / 2)" |
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from b have c: "x < 1" |
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by auto |
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then have "ln (1 - x) = - ln (1 + x / (1 - x))" |
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by (rule aux5) |
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also have "- (x / (1 - x)) <= ..." |
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proof - |
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have "ln (1 + x / (1 - x)) <= x / (1 - x)" |
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apply (rule ln_add_one_self_le_self) |
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apply (rule divide_nonneg_pos) |
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by (insert a c, auto) |
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thus ?thesis |
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by auto |
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qed |
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also have "- (x / (1 - x)) = -x / (1 - x)" |
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by auto |
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finally have d: "- x / (1 - x) <= ln (1 - x)" . |
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have "0 < 1 - x" using prems by simp |
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hence e: "-x - 2 * x^2 <= - x / (1 - x)" |
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using mult_right_le_one_le[of "x*x" "2*x"] prems |
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by(simp add:field_simps power2_eq_square) |
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16959 | 265 |
from e d show "- x - 2 * x^2 <= ln (1 - x)" |
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by (rule order_trans) |
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qed |
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||
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lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x" |
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apply (case_tac "0 <= x") |
17013
74bc935273ea
renamed exp_ge_add_one_self2 to exp_ge_add_one_self
avigad
parents:
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apply (erule exp_ge_add_one_self_aux) |
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apply (case_tac "x <= -1") |
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apply (subgoal_tac "1 + x <= 0") |
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apply (erule order_trans) |
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apply simp |
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apply simp |
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apply (subgoal_tac "1 + x = exp(ln (1 + x))") |
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apply (erule ssubst) |
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apply (subst exp_le_cancel_iff) |
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apply (subgoal_tac "ln (1 - (- x)) <= - (- x)") |
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apply simp |
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apply (rule ln_one_minus_pos_upper_bound) |
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apply auto |
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done |
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lemma ln_add_one_self_le_self2: "-1 < x ==> ln(1 + x) <= x" |
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apply (subgoal_tac "x = ln (exp x)") |
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apply (erule ssubst)back |
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apply (subst ln_le_cancel_iff) |
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apply auto |
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done |
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||
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lemma abs_ln_one_plus_x_minus_x_bound_nonneg: |
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"0 <= x ==> x <= 1 ==> abs(ln (1 + x) - x) <= x^2" |
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proof - |
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23441 | 296 |
assume x: "0 <= x" |
16959 | 297 |
assume "x <= 1" |
23441 | 298 |
from x have "ln (1 + x) <= x" |
16959 | 299 |
by (rule ln_add_one_self_le_self) |
300 |
then have "ln (1 + x) - x <= 0" |
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by simp |
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then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)" |
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by (rule abs_of_nonpos) |
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also have "... = x - ln (1 + x)" |
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by simp |
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also have "... <= x^2" |
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proof - |
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from prems have "x - x^2 <= ln (1 + x)" |
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by (intro ln_one_plus_pos_lower_bound) |
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thus ?thesis |
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by simp |
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qed |
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finally show ?thesis . |
|
314 |
qed |
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||
316 |
lemma abs_ln_one_plus_x_minus_x_bound_nonpos: |
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"-(1 / 2) <= x ==> x <= 0 ==> abs(ln (1 + x) - x) <= 2 * x^2" |
|
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proof - |
|
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assume "-(1 / 2) <= x" |
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320 |
assume "x <= 0" |
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have "abs(ln (1 + x) - x) = x - ln(1 - (-x))" |
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apply (subst abs_of_nonpos) |
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apply simp |
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apply (rule ln_add_one_self_le_self2) |
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apply (insert prems, auto) |
|
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done |
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also have "... <= 2 * x^2" |
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apply (subgoal_tac "- (-x) - 2 * (-x)^2 <= ln (1 - (-x))") |
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apply (simp add: compare_rls) |
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apply (rule ln_one_minus_pos_lower_bound) |
|
331 |
apply (insert prems, auto) |
|
332 |
done |
|
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finally show ?thesis . |
|
334 |
qed |
|
335 |
||
336 |
lemma abs_ln_one_plus_x_minus_x_bound: |
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337 |
"abs x <= 1 / 2 ==> abs(ln (1 + x) - x) <= 2 * x^2" |
|
338 |
apply (case_tac "0 <= x") |
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339 |
apply (rule order_trans) |
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340 |
apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg) |
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apply auto |
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342 |
apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos) |
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apply auto |
|
344 |
done |
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345 |
||
346 |
lemma DERIV_ln: "0 < x ==> DERIV ln x :> 1 / x" |
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347 |
apply (unfold deriv_def, unfold LIM_def, clarsimp) |
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348 |
apply (rule exI) |
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349 |
apply (rule conjI) |
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350 |
prefer 2 |
|
351 |
apply clarsimp |
|
20563 | 352 |
apply (subgoal_tac "(ln (x + xa) - ln x) / xa - (1 / x) = |
16959 | 353 |
(ln (1 + xa / x) - xa / x) / xa") |
354 |
apply (erule ssubst) |
|
355 |
apply (subst abs_divide) |
|
356 |
apply (rule mult_imp_div_pos_less) |
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357 |
apply force |
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apply (rule order_le_less_trans) |
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apply (rule abs_ln_one_plus_x_minus_x_bound) |
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apply (subst abs_divide) |
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apply (subst abs_of_pos, assumption) |
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apply (erule mult_imp_div_pos_le) |
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apply (subgoal_tac "abs xa < min (x / 2) (r * x^2 / 2)") |
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364 |
apply force |
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365 |
apply assumption |
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20432
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lin_arith_prover: splitting reverted because of performance loss
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|
366 |
apply (simp add: power2_eq_square mult_compare_simps) |
16959 | 367 |
apply (rule mult_imp_div_pos_less) |
368 |
apply (rule mult_pos_pos, assumption, assumption) |
|
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apply (subgoal_tac "xa * xa = abs xa * abs xa") |
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370 |
apply (erule ssubst) |
|
371 |
apply (subgoal_tac "abs xa * (abs xa * 2) < abs xa * (r * (x * x))") |
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372 |
apply (simp only: mult_ac) |
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apply (rule mult_strict_left_mono) |
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apply (erule conjE, assumption) |
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apply force |
|
376 |
apply simp |
|
377 |
apply (subst ln_div [THEN sym]) |
|
378 |
apply arith |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23441
diff
changeset
|
379 |
apply (auto simp add: ring_simps add_frac_eq frac_eq_eq |
16959 | 380 |
add_divide_distrib power2_eq_square) |
381 |
apply (rule mult_pos_pos, assumption)+ |
|
382 |
apply assumption |
|
383 |
done |
|
384 |
||
385 |
lemma ln_x_over_x_mono: "exp 1 <= x ==> x <= y ==> (ln y / y) <= (ln x / x)" |
|
386 |
proof - |
|
387 |
assume "exp 1 <= x" and "x <= y" |
|
388 |
have a: "0 < x" and b: "0 < y" |
|
389 |
apply (insert prems) |
|
23114 | 390 |
apply (subgoal_tac "0 < exp (1::real)") |
16959 | 391 |
apply arith |
392 |
apply auto |
|
23114 | 393 |
apply (subgoal_tac "0 < exp (1::real)") |
16959 | 394 |
apply arith |
395 |
apply auto |
|
396 |
done |
|
397 |
have "x * ln y - x * ln x = x * (ln y - ln x)" |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23441
diff
changeset
|
398 |
by (simp add: ring_simps) |
16959 | 399 |
also have "... = x * ln(y / x)" |
400 |
apply (subst ln_div) |
|
401 |
apply (rule b, rule a, rule refl) |
|
402 |
done |
|
403 |
also have "y / x = (x + (y - x)) / x" |
|
404 |
by simp |
|
23482 | 405 |
also have "... = 1 + (y - x) / x" using a prems by(simp add:field_simps) |
16959 | 406 |
also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)" |
407 |
apply (rule mult_left_mono) |
|
408 |
apply (rule ln_add_one_self_le_self) |
|
409 |
apply (rule divide_nonneg_pos) |
|
410 |
apply (insert prems a, simp_all) |
|
411 |
done |
|
23482 | 412 |
also have "... = y - x" using a by simp |
413 |
also have "... = (y - x) * ln (exp 1)" by simp |
|
16959 | 414 |
also have "... <= (y - x) * ln x" |
415 |
apply (rule mult_left_mono) |
|
416 |
apply (subst ln_le_cancel_iff) |
|
417 |
apply force |
|
418 |
apply (rule a) |
|
419 |
apply (rule prems) |
|
420 |
apply (insert prems, simp) |
|
421 |
done |
|
422 |
also have "... = y * ln x - x * ln x" |
|
423 |
by (rule left_diff_distrib) |
|
424 |
finally have "x * ln y <= y * ln x" |
|
425 |
by arith |
|
23482 | 426 |
then have "ln y <= (y * ln x) / x" using a by(simp add:field_simps) |
427 |
also have "... = y * (ln x / x)" by simp |
|
428 |
finally show ?thesis using b by(simp add:field_simps) |
|
16959 | 429 |
qed |
430 |
||
431 |
end |