author | blanchet |
Tue, 14 Sep 2010 13:24:18 +0200 | |
changeset 39359 | 6f49c7fbb1b1 |
parent 36777 | be5461582d0f |
child 40864 | 4abaaadfdaf2 |
permissions | -rw-r--r-- |
16959 | 1 |
(* Title: Ln.thy |
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Author: Jeremy Avigad |
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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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19765 | 15 |
by (simp add: exp_def) |
16959 | 16 |
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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32038 | 34 |
shows "inverse (real (fact ((n::nat) + 2))) * x ^ (n + 2) <= (x^2/2) * ((1/2)^n)" |
16959 | 35 |
proof (induct n) |
32038 | 36 |
show "inverse (real (fact ((0::nat) + 2))) * x ^ (0 + 2) <= |
16959 | 37 |
x ^ 2 / 2 * (1 / 2) ^ 0" |
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by (simp add: real_of_nat_Suc power2_eq_square) |
16959 | 39 |
next |
32038 | 40 |
fix n :: nat |
16959 | 41 |
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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use real_of_nat_ge_zero instead of real_of_nat_fact_ge_zero
huffman
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apply (rule real_of_nat_ge_zero) |
16959 | 88 |
apply (rule zero_le_power) |
23441 | 89 |
apply (rule a) |
16959 | 90 |
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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declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
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by (rule power_Suc [THEN sym]) |
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finally show ?thesis . |
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qed |
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qed |
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lemma aux2: "(%n. (x::real) ^ 2 / 2 * (1 / 2) ^ n) sums x^2" |
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proof - |
20692 | 108 |
have "(%n. (1 / 2::real)^n) sums (1 / (1 - (1/2)))" |
16959 | 109 |
apply (rule geometric_sums) |
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by (simp add: abs_less_iff) |
16959 | 111 |
also have "(1::real) / (1 - 1/2) = 2" |
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by simp |
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20692 | 113 |
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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23114 | 121 |
lemma exp_bound: "0 <= (x::real) ==> x <= 1 ==> exp x <= 1 + x + x^2" |
16959 | 122 |
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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||
23114 | 144 |
lemma aux4: "0 <= (x::real) ==> x <= 1 ==> exp (x - x^2) <= 1 + x" |
16959 | 145 |
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
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renamed exp_ge_add_one_self2 to exp_ge_add_one_self
avigad
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16963
diff
changeset
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apply (auto simp add: exp_ge_add_one_self_aux) |
16959 | 158 |
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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23482 | 166 |
by(simp add:field_simps zero_compare_simps) |
16959 | 167 |
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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29667 | 190 |
by (simp add: algebra_simps power2_eq_square power3_eq_cube) |
16959 | 191 |
also have "... <= 1" |
25875 | 192 |
by (auto simp add: a) |
16959 | 193 |
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 divide_inverse) |
16959 | 210 |
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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23482 | 236 |
also have " 1 / (1 - x) = 1 + x / (1 - x)" using a by(simp add:field_simps) |
16959 | 237 |
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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23482 | 260 |
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 | 264 |
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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||
23114 | 268 |
lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x" |
16959 | 269 |
apply (case_tac "0 <= x") |
17013
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renamed exp_ge_add_one_self2 to exp_ge_add_one_self
avigad
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apply (erule exp_ge_add_one_self_aux) |
16959 | 271 |
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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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 - |
|
23441 | 295 |
assume x: "0 <= x" |
16959 | 296 |
assume "x <= 1" |
23441 | 297 |
from x have "ln (1 + x) <= x" |
16959 | 298 |
by (rule ln_add_one_self_le_self) |
299 |
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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302 |
by (rule abs_of_nonpos) |
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also have "... = x - ln (1 + x)" |
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304 |
by simp |
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305 |
also have "... <= x^2" |
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306 |
proof - |
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307 |
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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310 |
by simp |
|
311 |
qed |
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312 |
finally show ?thesis . |
|
313 |
qed |
|
314 |
||
315 |
lemma abs_ln_one_plus_x_minus_x_bound_nonpos: |
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316 |
"-(1 / 2) <= x ==> x <= 0 ==> abs(ln (1 + x) - x) <= 2 * x^2" |
|
317 |
proof - |
|
318 |
assume "-(1 / 2) <= x" |
|
319 |
assume "x <= 0" |
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320 |
have "abs(ln (1 + x) - x) = x - ln(1 - (-x))" |
|
321 |
apply (subst abs_of_nonpos) |
|
322 |
apply simp |
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323 |
apply (rule ln_add_one_self_le_self2) |
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324 |
apply (insert prems, auto) |
|
325 |
done |
|
326 |
also have "... <= 2 * x^2" |
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327 |
apply (subgoal_tac "- (-x) - 2 * (-x)^2 <= ln (1 - (-x))") |
|
29667 | 328 |
apply (simp add: algebra_simps) |
16959 | 329 |
apply (rule ln_one_minus_pos_lower_bound) |
330 |
apply (insert prems, auto) |
|
29667 | 331 |
done |
16959 | 332 |
finally show ?thesis . |
333 |
qed |
|
334 |
||
335 |
lemma abs_ln_one_plus_x_minus_x_bound: |
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336 |
"abs x <= 1 / 2 ==> abs(ln (1 + x) - x) <= 2 * x^2" |
|
337 |
apply (case_tac "0 <= x") |
|
338 |
apply (rule order_trans) |
|
339 |
apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg) |
|
340 |
apply auto |
|
341 |
apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos) |
|
342 |
apply auto |
|
343 |
done |
|
344 |
||
345 |
lemma ln_x_over_x_mono: "exp 1 <= x ==> x <= y ==> (ln y / y) <= (ln x / x)" |
|
346 |
proof - |
|
347 |
assume "exp 1 <= x" and "x <= y" |
|
348 |
have a: "0 < x" and b: "0 < y" |
|
349 |
apply (insert prems) |
|
23114 | 350 |
apply (subgoal_tac "0 < exp (1::real)") |
16959 | 351 |
apply arith |
352 |
apply auto |
|
23114 | 353 |
apply (subgoal_tac "0 < exp (1::real)") |
16959 | 354 |
apply arith |
355 |
apply auto |
|
356 |
done |
|
357 |
have "x * ln y - x * ln x = x * (ln y - ln x)" |
|
29667 | 358 |
by (simp add: algebra_simps) |
16959 | 359 |
also have "... = x * ln(y / x)" |
360 |
apply (subst ln_div) |
|
361 |
apply (rule b, rule a, rule refl) |
|
362 |
done |
|
363 |
also have "y / x = (x + (y - x)) / x" |
|
364 |
by simp |
|
23482 | 365 |
also have "... = 1 + (y - x) / x" using a prems by(simp add:field_simps) |
16959 | 366 |
also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)" |
367 |
apply (rule mult_left_mono) |
|
368 |
apply (rule ln_add_one_self_le_self) |
|
369 |
apply (rule divide_nonneg_pos) |
|
370 |
apply (insert prems a, simp_all) |
|
371 |
done |
|
23482 | 372 |
also have "... = y - x" using a by simp |
373 |
also have "... = (y - x) * ln (exp 1)" by simp |
|
16959 | 374 |
also have "... <= (y - x) * ln x" |
375 |
apply (rule mult_left_mono) |
|
376 |
apply (subst ln_le_cancel_iff) |
|
377 |
apply force |
|
378 |
apply (rule a) |
|
379 |
apply (rule prems) |
|
380 |
apply (insert prems, simp) |
|
381 |
done |
|
382 |
also have "... = y * ln x - x * ln x" |
|
383 |
by (rule left_diff_distrib) |
|
384 |
finally have "x * ln y <= y * ln x" |
|
385 |
by arith |
|
23482 | 386 |
then have "ln y <= (y * ln x) / x" using a by(simp add:field_simps) |
387 |
also have "... = y * (ln x / x)" by simp |
|
388 |
finally show ?thesis using b by(simp add:field_simps) |
|
16959 | 389 |
qed |
390 |
||
391 |
end |