src/HOL/Decision_Procs/Approximation.thy

   and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) + x * (lb n (F i) (G i k) x)"
   shows "Ifloat (lb n ((F o^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / real (f (j' + j))) * Ifloat x ^ j)" (is "?lb") and 
     "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x ^ j)) \<le> Ifloat (ub n ((F o^ j') s) (f j') x)" (is "?ub")
 proof -
   { fix x y z :: float have "x - y * z = x + - y * z"
-      by (cases x, cases y, cases z, simp add: plus_float.simps minus_float.simps uminus_float.simps times_float.simps algebra_simps)
+      by (cases x, cases y, cases z, simp add: plus_float.simps minus_float_def uminus_float.simps times_float.simps algebra_simps)
   } note diff_mult_minus = this
 
   { fix x :: float have "- (- x) = x" by (cases x, auto simp add: uminus_float.simps) } note minus_minus = this
 
   have move_minus: "Ifloat (-x) = -1 * Ifloat x" by auto

     have "0 < Ifloat (?horner x) ^ num" using `0 < ?horner x`[unfolded less_float_def Ifloat_0] by (rule zero_less_power)
     also have "\<dots> = Ifloat ((?horner x) ^ num)" using float_power by auto
     finally have "0 < Ifloat ((?horner x) ^ num)" .
   }
   ultimately show ?thesis
-    unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def by (cases "floor_fl x", cases "x < - 1", auto simp add: le_float_def less_float_def normfloat) 
+    unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def
+    by (cases "floor_fl x", cases "x < - 1", auto simp add: float_power le_float_def less_float_def)
 qed
 
 lemma exp_boundaries': assumes "x \<le> 0"
   shows "exp (Ifloat x) \<in> { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}"
 proof -