src/HOL/Ln.thy

 header {* Properties of ln *}
 
 theory Ln
 imports Transcendental
 begin
-
-lemma exp_first_two_terms: "exp x = 1 + x + suminf (%n. 
-  inverse(fact (n+2)) * (x ^ (n+2)))"
-proof -
-  have "exp x = suminf (%n. inverse(fact n) * (x ^ n))"
-    by (simp add: exp_def)
-  also from summable_exp have "... = (SUM n::nat : {0..<2}. 
-      inverse(fact n) * (x ^ n)) + suminf (%n.
-      inverse(fact(n+2)) * (x ^ (n+2)))" (is "_ = ?a + _")
-    by (rule suminf_split_initial_segment)
-  also have "?a = 1 + x"
-    by (simp add: numeral_2_eq_2)
-  finally show ?thesis .
-qed
-
-lemma exp_bound: "0 <= (x::real) ==> x <= 1 ==> exp x <= 1 + x + x^2"
-proof -
-  assume a: "0 <= x"
-  assume b: "x <= 1"
-  { fix n :: nat
-    have "2 * 2 ^ n \<le> fact (n + 2)"
-      by (induct n, simp, simp)
-    hence "real ((2::nat) * 2 ^ n) \<le> real (fact (n + 2))"
-      by (simp only: real_of_nat_le_iff)
-    hence "2 * 2 ^ n \<le> real (fact (n + 2))"
-      by simp
-    hence "inverse (fact (n + 2)) \<le> inverse (2 * 2 ^ n)"
-      by (rule le_imp_inverse_le) simp
-    hence "inverse (fact (n + 2)) \<le> 1/2 * (1/2)^n"
-      by (simp add: inverse_mult_distrib power_inverse)
-    hence "inverse (fact (n + 2)) * (x^n * x\<twosuperior>) \<le> 1/2 * (1/2)^n * (1 * x\<twosuperior>)"
-      by (rule mult_mono)
-        (rule mult_mono, simp_all add: power_le_one a b mult_nonneg_nonneg)
-    hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<twosuperior>/2) * ((1/2)^n)"
-      unfolding power_add by (simp add: mult_ac del: fact_Suc) }
-  note aux1 = this
-  have "(\<lambda>n. x\<twosuperior> / 2 * (1 / 2) ^ n) sums (x\<twosuperior> / 2 * (1 / (1 - 1 / 2)))"
-    by (intro sums_mult geometric_sums, simp)
-  hence aux2: "(\<lambda>n. (x::real) ^ 2 / 2 * (1 / 2) ^ n) sums x^2"
-    by simp
-  have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <= x^2"
-  proof -
-    have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <=
-        suminf (%n. (x^2/2) * ((1/2)^n))"
-      apply (rule summable_le)
-      apply (rule allI, rule aux1)
-      apply (rule summable_exp [THEN summable_ignore_initial_segment])
-      by (rule sums_summable, rule aux2)
-    also have "... = x^2"
-      by (rule sums_unique [THEN sym], rule aux2)
-    finally show ?thesis .
-  qed
-  thus ?thesis unfolding exp_first_two_terms by auto
-qed
 
 lemma ln_one_plus_pos_lower_bound: "0 <= x ==> x <= 1 ==> 
     x - x^2 <= ln (1 + x)"
 proof -
   assume a: "0 <= x" and b: "x <= 1"

     from a have "0 < 1 + x" by auto
     thus ?thesis
       by (auto simp only: exp_ln_iff [THEN sym])
   qed
   finally have "exp (x - x ^ 2) <= exp (ln (1 + x))" .
-  thus ?thesis by (auto simp only: exp_le_cancel_iff)
-qed
-
-lemma ln_one_minus_pos_upper_bound: "0 <= x ==> x < 1 ==> ln (1 - x) <= - x"
-proof -
-  assume a: "0 <= (x::real)" and b: "x < 1"
-  have "(1 - x) * (1 + x + x^2) = (1 - x^3)"
-    by (simp add: algebra_simps power2_eq_square power3_eq_cube)
-  also have "... <= 1"
-    by (auto simp add: a)
-  finally have "(1 - x) * (1 + x + x ^ 2) <= 1" .
-  moreover have c: "0 < 1 + x + x\<twosuperior>"
-    by (simp add: add_pos_nonneg a)
-  ultimately have "1 - x <= 1 / (1 + x + x^2)"
-    by (elim mult_imp_le_div_pos)
-  also have "... <= 1 / exp x"
-    apply (rule divide_left_mono)
-    apply (rule exp_bound, rule a)
-    apply (rule b [THEN less_imp_le])
-    apply simp
-    apply (rule mult_pos_pos)
-    apply (rule c)
-    apply simp
-    done
-  also have "... = exp (-x)"
-    by (auto simp add: exp_minus divide_inverse)
-  finally have "1 - x <= exp (- x)" .
-  also have "1 - x = exp (ln (1 - x))"
-  proof -
-    have "0 < 1 - x"
-      by (insert b, auto)
-    thus ?thesis
-      by (auto simp only: exp_ln_iff [THEN sym])
-  qed
-  finally have "exp (ln (1 - x)) <= exp (- x)" .
   thus ?thesis by (auto simp only: exp_le_cancel_iff)
 qed
 
 lemma aux5: "x < 1 ==> ln(1 - x) = - ln(1 + x / (1 - x))"
 proof -

     using mult_right_le_one_le[of "x*x" "2*x"] a b
     by (simp add:field_simps power2_eq_square)
   from e d show "- x - 2 * x^2 <= ln (1 - x)"
     by (rule order_trans)
 qed
-
-lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
-  apply (case_tac "0 <= x")
-  apply (erule exp_ge_add_one_self_aux)
-  apply (case_tac "x <= -1")
-  apply (subgoal_tac "1 + x <= 0")
-  apply (erule order_trans)
-  apply simp
-  apply simp
-  apply (subgoal_tac "1 + x = exp(ln (1 + x))")
-  apply (erule ssubst)
-  apply (subst exp_le_cancel_iff)
-  apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
-  apply simp
-  apply (rule ln_one_minus_pos_upper_bound) 
-  apply auto
-done
 
 lemma ln_add_one_self_le_self2: "-1 < x ==> ln(1 + x) <= x"
   apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
   apply (subst ln_le_cancel_iff)
   apply auto