src/HOL/Hyperreal/Ln.thy
author avigad
Wed Aug 03 14:48:42 2005 +0200 (2005-08-03)
changeset 17013 74bc935273ea
parent 16963 32626fb8ae49
child 19765 dfe940911617
permissions -rwxr-xr-x
renamed exp_ge_add_one_self2 to exp_ge_add_one_self
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(*  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 (unfold exp_def, simp)
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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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    apply (simp add: power2_eq_square)
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    apply (subgoal_tac "real (Suc (Suc 0)) = 2")
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    apply (erule ssubst)
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    apply simp
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    apply simp
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    done
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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 assumption
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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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lemma aux2: "(%n. x ^ 2 / 2 * (1 / 2) ^ n) sums x^2"
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proof -
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  have "(%n. (1 / 2)^n) sums (1 / (1 - (1/2)))"
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    apply (rule geometric_sums)
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    by (simp add: abs_interval_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)^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 ==> 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 aux3: "(0::real) <= x ==> (1 + x + x^2)/(1 + x^2) <= 1 + x"
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  apply (subst pos_divide_le_eq)
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  apply (simp add: zero_compare_simps)
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  apply (simp add: ring_eq_simps zero_compare_simps)
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done
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lemma aux4: "0 <= x ==> 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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    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 (rule aux3)
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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_eq_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)"
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  proof -
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    have "1 / (1 - x) = (1 - x + x) / (1 - x)"
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      by auto
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    also have "... = (1 - x) / (1 - x) + x / (1 - x)"
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      by (rule add_divide_distrib)
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    also have "... = 1 + x / (1-x)"
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      apply (subst add_right_cancel)
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      apply (insert a, simp)
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      done
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    finally show ?thesis .
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  qed
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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 e: "-x - 2 * x^2 <= - x / (1 - x)"
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    apply (rule mult_imp_le_div_pos)
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    apply (insert prems, force)
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    apply (auto simp add: ring_eq_simps power2_eq_square)
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    apply (subgoal_tac "- (x * x) + x * (x * (x * 2)) = x^2 * (2 * x - 1)")
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    apply (erule ssubst)
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    apply (rule mult_nonneg_nonpos)
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    apply auto
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    apply (auto simp add: ring_eq_simps power2_eq_square)
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    done
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  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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lemma exp_ge_add_one_self [simp]: "1 + x <= exp x"
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  apply (case_tac "0 <= x")
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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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  apply (rule sym) 
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  apply (subst exp_ln_iff)
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  apply auto
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done
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avigad@16959
   318
lemma ln_add_one_self_le_self2: "-1 < x ==> ln(1 + x) <= x"
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   319
  apply (subgoal_tac "x = ln (exp x)")
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   320
  apply (erule ssubst)back
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   321
  apply (subst ln_le_cancel_iff)
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   322
  apply auto
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   323
done
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   324
avigad@16959
   325
lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
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   326
    "0 <= x ==> x <= 1 ==> abs(ln (1 + x) - x) <= x^2"
avigad@16959
   327
proof -
avigad@16959
   328
  assume "0 <= x"
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   329
  assume "x <= 1"
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   330
  have "ln (1 + x) <= x"
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   331
    by (rule ln_add_one_self_le_self)
avigad@16959
   332
  then have "ln (1 + x) - x <= 0" 
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   333
    by simp
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   334
  then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
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   335
    by (rule abs_of_nonpos)
avigad@16959
   336
  also have "... = x - ln (1 + x)" 
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   337
    by simp
avigad@16959
   338
  also have "... <= x^2"
avigad@16959
   339
  proof -
avigad@16959
   340
    from prems have "x - x^2 <= ln (1 + x)"
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   341
      by (intro ln_one_plus_pos_lower_bound)
avigad@16959
   342
    thus ?thesis
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   343
      by simp
avigad@16959
   344
  qed
avigad@16959
   345
  finally show ?thesis .
avigad@16959
   346
qed
avigad@16959
   347
avigad@16959
   348
lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
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   349
    "-(1 / 2) <= x ==> x <= 0 ==> abs(ln (1 + x) - x) <= 2 * x^2"
avigad@16959
   350
proof -
avigad@16959
   351
  assume "-(1 / 2) <= x"
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   352
  assume "x <= 0"
avigad@16959
   353
  have "abs(ln (1 + x) - x) = x - ln(1 - (-x))" 
avigad@16959
   354
    apply (subst abs_of_nonpos)
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   355
    apply simp
avigad@16959
   356
    apply (rule ln_add_one_self_le_self2)
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   357
    apply (insert prems, auto)
avigad@16959
   358
    done
avigad@16959
   359
  also have "... <= 2 * x^2"
avigad@16959
   360
    apply (subgoal_tac "- (-x) - 2 * (-x)^2 <= ln (1 - (-x))")
avigad@16959
   361
    apply (simp add: compare_rls)
avigad@16959
   362
    apply (rule ln_one_minus_pos_lower_bound)
avigad@16959
   363
    apply (insert prems, auto)
avigad@16959
   364
    done 
avigad@16959
   365
  finally show ?thesis .
avigad@16959
   366
qed
avigad@16959
   367
avigad@16959
   368
lemma abs_ln_one_plus_x_minus_x_bound:
avigad@16959
   369
    "abs x <= 1 / 2 ==> abs(ln (1 + x) - x) <= 2 * x^2"
avigad@16959
   370
  apply (case_tac "0 <= x")
avigad@16959
   371
  apply (rule order_trans)
avigad@16959
   372
  apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
avigad@16959
   373
  apply auto
avigad@16959
   374
  apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
avigad@16959
   375
  apply auto
avigad@16959
   376
done
avigad@16959
   377
avigad@16959
   378
lemma DERIV_ln: "0 < x ==> DERIV ln x :> 1 / x"
avigad@16959
   379
  apply (unfold deriv_def, unfold LIM_def, clarsimp)
avigad@16959
   380
  apply (rule exI)
avigad@16959
   381
  apply (rule conjI)
avigad@16959
   382
  prefer 2
avigad@16959
   383
  apply clarsimp
avigad@16959
   384
  apply (subgoal_tac "(ln (x + xa) + - ln x) / xa + - (1 / x) = 
avigad@16959
   385
      (ln (1 + xa / x) - xa / x) / xa")
avigad@16959
   386
  apply (erule ssubst)
avigad@16959
   387
  apply (subst abs_divide)
avigad@16959
   388
  apply (rule mult_imp_div_pos_less)
avigad@16959
   389
  apply force
avigad@16959
   390
  apply (rule order_le_less_trans)
avigad@16959
   391
  apply (rule abs_ln_one_plus_x_minus_x_bound)
avigad@16959
   392
  apply (subst abs_divide)
avigad@16959
   393
  apply (subst abs_of_pos, assumption)
avigad@16959
   394
  apply (erule mult_imp_div_pos_le)
avigad@16959
   395
  apply (subgoal_tac "abs xa < min (x / 2) (r * x^2 / 2)")
avigad@16959
   396
  apply force
avigad@16959
   397
  apply assumption
avigad@16959
   398
  apply (simp add: power2_eq_square mult_compare_simps)
avigad@16959
   399
  apply (rule mult_imp_div_pos_less)
avigad@16959
   400
  apply (rule mult_pos_pos, assumption, assumption)
avigad@16959
   401
  apply (subgoal_tac "xa * xa = abs xa * abs xa")
avigad@16959
   402
  apply (erule ssubst)
avigad@16959
   403
  apply (subgoal_tac "abs xa * (abs xa * 2) < abs xa * (r * (x * x))")
avigad@16959
   404
  apply (simp only: mult_ac)
avigad@16959
   405
  apply (rule mult_strict_left_mono)
avigad@16959
   406
  apply (erule conjE, assumption)
avigad@16959
   407
  apply force
avigad@16959
   408
  apply simp
avigad@16959
   409
  apply (subst diff_minus [THEN sym])+
avigad@16959
   410
  apply (subst ln_div [THEN sym])
avigad@16959
   411
  apply arith
avigad@16959
   412
  apply (auto simp add: ring_eq_simps add_frac_eq frac_eq_eq 
avigad@16959
   413
    add_divide_distrib power2_eq_square)
avigad@16959
   414
  apply (rule mult_pos_pos, assumption)+
avigad@16959
   415
  apply assumption
avigad@16959
   416
done
avigad@16959
   417
avigad@16959
   418
lemma ln_x_over_x_mono: "exp 1 <= x ==> x <= y ==> (ln y / y) <= (ln x / x)"  
avigad@16959
   419
proof -
avigad@16959
   420
  assume "exp 1 <= x" and "x <= y"
avigad@16959
   421
  have a: "0 < x" and b: "0 < y"
avigad@16959
   422
    apply (insert prems)
avigad@16959
   423
    apply (subgoal_tac "0 < exp 1")
avigad@16959
   424
    apply arith
avigad@16959
   425
    apply auto
avigad@16959
   426
    apply (subgoal_tac "0 < exp 1")
avigad@16959
   427
    apply arith
avigad@16959
   428
    apply auto
avigad@16959
   429
    done
avigad@16959
   430
  have "x * ln y - x * ln x = x * (ln y - ln x)"
avigad@16959
   431
    by (simp add: ring_eq_simps)
avigad@16959
   432
  also have "... = x * ln(y / x)"
avigad@16959
   433
    apply (subst ln_div)
avigad@16959
   434
    apply (rule b, rule a, rule refl)
avigad@16959
   435
    done
avigad@16959
   436
  also have "y / x = (x + (y - x)) / x"
avigad@16959
   437
    by simp
avigad@16959
   438
  also have "... = 1 + (y - x) / x"
avigad@16959
   439
    apply (simp only: add_divide_distrib)
avigad@16959
   440
    apply (simp add: prems)
avigad@16959
   441
    apply (insert a, arith)
avigad@16959
   442
    done
avigad@16959
   443
  also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
avigad@16959
   444
    apply (rule mult_left_mono)
avigad@16959
   445
    apply (rule ln_add_one_self_le_self)
avigad@16959
   446
    apply (rule divide_nonneg_pos)
avigad@16959
   447
    apply (insert prems a, simp_all) 
avigad@16959
   448
    done
avigad@16959
   449
  also have "... = y - x"
avigad@16959
   450
    by (insert a, simp)
avigad@16959
   451
  also have "... = (y - x) * ln (exp 1)"
avigad@16959
   452
    by simp
avigad@16959
   453
  also have "... <= (y - x) * ln x"
avigad@16959
   454
    apply (rule mult_left_mono)
avigad@16959
   455
    apply (subst ln_le_cancel_iff)
avigad@16959
   456
    apply force
avigad@16959
   457
    apply (rule a)
avigad@16959
   458
    apply (rule prems)
avigad@16959
   459
    apply (insert prems, simp)
avigad@16959
   460
    done
avigad@16959
   461
  also have "... = y * ln x - x * ln x"
avigad@16959
   462
    by (rule left_diff_distrib)
avigad@16959
   463
  finally have "x * ln y <= y * ln x"
avigad@16959
   464
    by arith
avigad@16959
   465
  then have "ln y <= (y * ln x) / x"
avigad@16959
   466
    apply (subst pos_le_divide_eq)
avigad@16959
   467
    apply (rule a)
avigad@16959
   468
    apply (simp add: mult_ac)
avigad@16959
   469
    done
avigad@16959
   470
  also have "... = y * (ln x / x)"
avigad@16959
   471
    by simp
avigad@16959
   472
  finally show ?thesis 
avigad@16959
   473
    apply (subst pos_divide_le_eq)
avigad@16959
   474
    apply (rule b)
avigad@16959
   475
    apply (simp add: mult_ac)
avigad@16959
   476
    done
avigad@16959
   477
qed
avigad@16959
   478
avigad@16959
   479
end
avigad@16959
   480