src/HOL/Decision_Procs/Approximation.thy

     hence "0 < real x" using `0 \<le> real x` by auto
     hence prem: "0 < 1 / (0 * 2 + (1::nat)) * real x ^ (0 * 2 + 1)" by auto
 
     have "\<bar> real x \<bar> \<le> 1"  using `0 \<le> real x` `real x \<le> 1` by auto
     from mp[OF summable_Leibniz(2)[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] prem, THEN spec, of m, unfolded `2 * m = n`]
-    show ?thesis unfolding arctan_series[OF `\<bar> real x \<bar> \<le> 1`] Suc_eq_plus1  .
+    show ?thesis unfolding arctan_series[OF `\<bar> real x \<bar> \<le> 1`] Suc_eq_plus1 atLeast0LessThan .
   qed auto
   note arctan_bounds = this[unfolded atLeastAtMost_iff]
 
   have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto
 

   proof -
     have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
     also have "\<dots> =
       (\<Sum> j = 0 ..< n. -1 ^ ((2 * j) div 2) / (real (fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto
     also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then -1 ^ (i div 2) / (real (fact i)) * x ^ i else 0)"
-      unfolding sum_split_even_odd ..
+      unfolding sum_split_even_odd atLeast0LessThan ..
     also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then -1 ^ (i div 2) / (real (fact i)) else 0) * x ^ i)"
       by (rule setsum_cong2) auto
     finally show ?thesis by assumption
   qed } note morph_to_if_power = this
 

     obtain t where "0 < t" and "t < real x" and
       cos_eq: "cos x = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * (real x) ^ i)
       + (cos (t + 1/2 * (2 * n) * pi) / real (fact (2*n))) * (real x)^(2*n)"
       (is "_ = ?SUM + ?rest / ?fact * ?pow")
       using Maclaurin_cos_expansion2[OF `0 < real x` `0 < 2 * n`]
-      unfolding cos_coeff_def by auto
+      unfolding cos_coeff_def atLeast0LessThan by auto
 
     have "cos t * -1^n = cos t * cos (n * pi) + sin t * sin (n * pi)" by auto
     also have "\<dots> = cos (t + n * pi)"  using cos_add by auto
     also have "\<dots> = ?rest" by auto
     finally have "cos t * -1^n = ?rest" .

     = (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * x ^ i)" (is "?SUM = _")
     proof -
       have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" by auto
       have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM" by auto
       also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then 0 else -1 ^ ((i - Suc 0) div 2) / (real (fact i)) * x ^ i)"
-        unfolding sum_split_even_odd ..
+        unfolding sum_split_even_odd atLeast0LessThan ..
       also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then 0 else -1 ^ ((i - Suc 0) div 2) / (real (fact i))) * x ^ i)"
         by (rule setsum_cong2) auto
       finally show ?thesis by assumption
     qed } note setsum_morph = this
 

     obtain t where "0 < t" and "t < real x" and
       sin_eq: "sin x = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)
       + (sin (t + 1/2 * (2 * n + 1) * pi) / real (fact (2*n + 1))) * (real x)^(2*n + 1)"
       (is "_ = ?SUM + ?rest / ?fact * ?pow")
       using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < real x`]
-      unfolding sin_coeff_def by auto
+      unfolding sin_coeff_def atLeast0LessThan by auto
 
     have "?rest = cos t * -1^n" unfolding sin_add cos_add real_of_nat_add distrib_right distrib_left by auto
     moreover
     have "t \<le> pi / 2" using `t < real x` and `x \<le> pi / 2` by auto
     hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto

   { have "lb_exp_horner prec (get_even n) 1 1 x \<le> (\<Sum>j = 0..<get_even n. 1 / real (fact j) * real x ^ j)"
       using bounds(1) by auto
     also have "\<dots> \<le> exp x"
     proof -
       obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp x = (\<Sum>m = 0..<get_even n. real x ^ m / real (fact m)) + exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
-        using Maclaurin_exp_le by blast
+        using Maclaurin_exp_le unfolding atLeast0LessThan by blast
       moreover have "0 \<le> exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
         by (auto intro!: mult_nonneg_nonneg divide_nonneg_pos simp add: zero_le_even_power)
       ultimately show ?thesis
         using get_odd exp_gt_zero by (auto intro!: mult_nonneg_nonneg)
     qed

       case False hence "real x < 0" using `real x \<le> 0` by auto
       show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq `real x < 0`)
     qed
 
     obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp x = (\<Sum>m = 0..<get_odd n. (real x) ^ m / real (fact m)) + exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n)"
-      using Maclaurin_exp_le by blast
+      using Maclaurin_exp_le unfolding atLeast0LessThan by blast
     moreover have "exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n) \<le> 0"
       by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero)
     ultimately have "exp x \<le> (\<Sum>j = 0..<get_odd n. 1 / real (fact j) * real x ^ j)"
       using get_odd exp_gt_zero by (auto intro!: mult_nonneg_nonneg)
     also have "\<dots> \<le> ub_exp_horner prec (get_odd n) 1 1 x"

       have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" unfolding power_Suc2 mult_assoc[symmetric]
         by (rule mult_left_mono, fact less_imp_le[OF `x < 1`], auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
       thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
     qed auto }
   from summable_Leibniz'(2,4)[OF `?a ----> 0` `\<And>n. 0 \<le> ?a n`, OF `\<And>n. ?a (Suc n) \<le> ?a n`, unfolded ln_eq]
-  show "?lb" and "?ub" by auto
+  show "?lb" and "?ub" unfolding atLeast0LessThan by auto
 qed
 
 lemma ln_float_bounds:
   assumes "0 \<le> real x" and "real x < 1"
   shows "x * lb_ln_horner prec (get_even n) 1 x \<le> ln (x + 1)" (is "?lb \<le> ?ln")

       from Taylor.taylor[OF zero_less_Suc, of F, OF F0 DERIV[unfolded Suc] this False]
       obtain t::real where t_bnd: "if xs ! x < c then xs ! x < t \<and> t < c else c < t \<and> t < xs ! x"
         and fl_eq: "interpret_floatarith f (xs[x := xs ! x]) =
            (\<Sum>m = 0..<Suc n'. F m c / real (fact m) * (xs ! x - c) ^ m) +
            F (Suc n') t / real (fact (Suc n')) * (xs ! x - c) ^ Suc n'"
-        by blast
+        unfolding atLeast0LessThan by blast
 
       from t_bnd bnd_xs bnd_c have *: "t \<in> {lx .. ux}"
         by (cases "xs ! x < c", auto)
 
       have "interpret_floatarith f (xs[x := xs ! x]) = ?taylor t"