# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1447165121 0
# Node ID 77b453bd616f24dd2dabe5eebebf144e672bc435
# Parent  933eb9e6a1cc19ea1ad17c4f07a716b3a6d8f1ef
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.

diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Complex.thy
--- a/src/HOL/Complex.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Complex.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -166,7 +166,7 @@
 
 lemma Im_divide_numeral [simp]: "Im (z / numeral w) = Im z / numeral w"
   by (simp add: Im_divide sqr_conv_mult)
-  
+
 lemma Re_divide_of_nat: "Re (z / of_nat n) = Re z / of_nat n"
   by (cases n) (simp_all add: Re_divide divide_simps power2_eq_square del: of_nat_Suc)
 
@@ -688,7 +688,7 @@
   by (simp add: complex_eq_iff cos_add sin_add)
 
 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
-  by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
+  by (induct n, simp_all add: of_nat_Suc algebra_simps cis_mult)
 
 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
   by (simp add: complex_eq_iff)
@@ -757,8 +757,7 @@
     have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"
       by (induct n)
          (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps
-                        power2_eq_square real_of_nat_Suc add_nonneg_eq_0_iff
-                        real_of_nat_def[symmetric])
+                        power2_eq_square of_nat_Suc add_nonneg_eq_0_iff)
     then have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =
         of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"
       by (simp add: field_simps) }
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Decision_Procs/Approximation.thy
--- a/src/HOL/Decision_Procs/Approximation.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Decision_Procs/Approximation.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -51,7 +51,7 @@
 
 lemma horner_bounds':
   fixes lb :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" and ub :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
-  assumes "0 \<le> real x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
+  assumes "0 \<le> real_of_float x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
     and lb_0: "\<And> i k x. lb 0 i k x = 0"
     and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
         (lapprox_rat prec 1 k)
@@ -69,7 +69,7 @@
 next
   case (Suc n)
   thus ?case using lapprox_rat[of prec 1 "f j'"] using rapprox_rat[of 1 "f j'" prec]
-    Suc[where j'="Suc j'"] \<open>0 \<le> real x\<close>
+    Suc[where j'="Suc j'"] \<open>0 \<le> real_of_float x\<close>
     by (auto intro!: add_mono mult_left_mono float_round_down_le float_round_up_le
       order_trans[OF add_mono[OF _ float_plus_down_le]]
       order_trans[OF _ add_mono[OF _ float_plus_up_le]]
@@ -87,7 +87,7 @@
 
 lemma horner_bounds:
   fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-  assumes "0 \<le> real x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
+  assumes "0 \<le> real_of_float x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
     and lb_0: "\<And> i k x. lb 0 i k x = 0"
     and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
         (lapprox_rat prec 1 k)
@@ -102,14 +102,14 @@
       (is "?ub")
 proof -
   have "?lb  \<and> ?ub"
-    using horner_bounds'[where lb=lb, OF \<open>0 \<le> real x\<close> f_Suc lb_0 lb_Suc ub_0 ub_Suc]
-    unfolding horner_schema[where f=f, OF f_Suc] .
+    using horner_bounds'[where lb=lb, OF \<open>0 \<le> real_of_float x\<close> f_Suc lb_0 lb_Suc ub_0 ub_Suc]
+    unfolding horner_schema[where f=f, OF f_Suc] by simp
   thus "?lb" and "?ub" by auto
 qed
 
 lemma horner_bounds_nonpos:
   fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-  assumes "real x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
+  assumes "real_of_float x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
     and lb_0: "\<And> i k x. lb 0 i k x = 0"
     and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
         (lapprox_rat prec 1 k)
@@ -118,14 +118,14 @@
     and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec
         (rapprox_rat prec 1 k)
         (float_round_up prec (x * (lb n (F i) (G i k) x)))"
-  shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / (f (j' + j))) * real x ^ j)" (is "?lb")
-    and "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real x ^ j) \<le> (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
+  shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j)" (is "?lb")
+    and "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j) \<le> (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
 proof -
   have diff_mult_minus: "x - y * z = x + - y * z" for x y z :: float by simp
-  have sum_eq: "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real x ^ j) =
-    (\<Sum>j = 0..<n. (- 1) ^ j * (1 / (f (j' + j))) * real (- x) ^ j)"
+  have sum_eq: "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j) =
+    (\<Sum>j = 0..<n. (- 1) ^ j * (1 / (f (j' + j))) * real_of_float (- x) ^ j)"
     by (auto simp add: field_simps power_mult_distrib[symmetric])
-  have "0 \<le> real (-x)" using assms by auto
+  have "0 \<le> real_of_float (-x)" using assms by auto
   from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec
     and lb="\<lambda> n i k x. lb n i k (-x)" and ub="\<lambda> n i k x. ub n i k (-x)",
     unfolded lb_Suc ub_Suc diff_mult_minus,
@@ -238,7 +238,7 @@
 qed
 
 lemma sqrt_iteration_bound:
-  assumes "0 < real x"
+  assumes "0 < real_of_float x"
   shows "sqrt x < sqrt_iteration prec n x"
 proof (induct n)
   case 0
@@ -260,7 +260,7 @@
     proof (rule mult_strict_right_mono, auto)
       show "m < 2^nat (bitlen m)"
         using bitlen_bounds[OF \<open>0 < m\<close>, THEN conjunct2]
-        unfolding real_of_int_less_iff[of m, symmetric] by auto
+        unfolding of_int_less_iff[of m, symmetric] by auto
     qed
     finally have "sqrt x < sqrt (2 powr (e + bitlen m))"
       unfolding int_nat_bl by auto
@@ -287,7 +287,7 @@
       have E_eq: "2 powr ?E = 2 powr (?E div 2 + ?E div 2 + ?E mod 2)"
         by auto
       have "sqrt (2 powr ?E) = sqrt (2 powr (?E div 2) * 2 powr (?E div 2) * 2 powr (?E mod 2))"
-        unfolding E_eq unfolding powr_add[symmetric] by (simp add: int_of_reals del: real_of_ints)
+        unfolding E_eq unfolding powr_add[symmetric] by (metis of_int_add)
       also have "\<dots> = 2 powr (?E div 2) * sqrt (2 powr (?E mod 2))"
         unfolding real_sqrt_mult[of _ "2 powr (?E mod 2)"] real_sqrt_abs2 by auto
       also have "\<dots> < 2 powr (?E div 2) * 2 powr 1"
@@ -304,11 +304,11 @@
   case (Suc n)
   let ?b = "sqrt_iteration prec n x"
   have "0 < sqrt x"
-    using \<open>0 < real x\<close> by auto
-  also have "\<dots> < real ?b"
+    using \<open>0 < real_of_float x\<close> by auto
+  also have "\<dots> < real_of_float ?b"
     using Suc .
   finally have "sqrt x < (?b + x / ?b)/2"
-    using sqrt_ub_pos_pos_1[OF Suc _ \<open>0 < real x\<close>] by auto
+    using sqrt_ub_pos_pos_1[OF Suc _ \<open>0 < real_of_float x\<close>] by auto
   also have "\<dots> \<le> (?b + (float_divr prec x ?b))/2"
     by (rule divide_right_mono, auto simp add: float_divr)
   also have "\<dots> = (Float 1 (- 1)) * (?b + (float_divr prec x ?b))"
@@ -320,8 +320,8 @@
 qed
 
 lemma sqrt_iteration_lower_bound:
-  assumes "0 < real x"
-  shows "0 < real (sqrt_iteration prec n x)" (is "0 < ?sqrt")
+  assumes "0 < real_of_float x"
+  shows "0 < real_of_float (sqrt_iteration prec n x)" (is "0 < ?sqrt")
 proof -
   have "0 < sqrt x" using assms by auto
   also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] .
@@ -329,21 +329,21 @@
 qed
 
 lemma lb_sqrt_lower_bound:
-  assumes "0 \<le> real x"
-  shows "0 \<le> real (lb_sqrt prec x)"
+  assumes "0 \<le> real_of_float x"
+  shows "0 \<le> real_of_float (lb_sqrt prec x)"
 proof (cases "0 < x")
   case True
-  hence "0 < real x" and "0 \<le> x"
-    using \<open>0 \<le> real x\<close> by auto
+  hence "0 < real_of_float x" and "0 \<le> x"
+    using \<open>0 \<le> real_of_float x\<close> by auto
   hence "0 < sqrt_iteration prec prec x"
     using sqrt_iteration_lower_bound by auto
-  hence "0 \<le> real (float_divl prec x (sqrt_iteration prec prec x))"
+  hence "0 \<le> real_of_float (float_divl prec x (sqrt_iteration prec prec x))"
     using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] unfolding less_eq_float_def by auto
   thus ?thesis
     unfolding lb_sqrt.simps using True by auto
 next
   case False
-  with \<open>0 \<le> real x\<close> have "real x = 0" by auto
+  with \<open>0 \<le> real_of_float x\<close> have "real_of_float x = 0" by auto
   thus ?thesis
     unfolding lb_sqrt.simps by auto
 qed
@@ -352,24 +352,24 @@
 proof -
   have lb: "lb_sqrt prec x \<le> sqrt x" if "0 < x" for x :: float
   proof -
-    from that have "0 < real x" and "0 \<le> real x" by auto
+    from that have "0 < real_of_float x" and "0 \<le> real_of_float x" by auto
     hence sqrt_gt0: "0 < sqrt x" by auto
     hence sqrt_ub: "sqrt x < sqrt_iteration prec prec x"
       using sqrt_iteration_bound by auto
     have "(float_divl prec x (sqrt_iteration prec prec x)) \<le>
           x / (sqrt_iteration prec prec x)" by (rule float_divl)
     also have "\<dots> < x / sqrt x"
-      by (rule divide_strict_left_mono[OF sqrt_ub \<open>0 < real x\<close>
+      by (rule divide_strict_left_mono[OF sqrt_ub \<open>0 < real_of_float x\<close>
                mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]])
     also have "\<dots> = sqrt x"
       unfolding inverse_eq_iff_eq[of _ "sqrt x", symmetric]
-                sqrt_divide_self_eq[OF \<open>0 \<le> real x\<close>, symmetric] by auto
+                sqrt_divide_self_eq[OF \<open>0 \<le> real_of_float x\<close>, symmetric] by auto
     finally show ?thesis
       unfolding lb_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto
   qed
   have ub: "sqrt x \<le> ub_sqrt prec x" if "0 < x" for x :: float
   proof -
-    from that have "0 < real x" by auto
+    from that have "0 < real_of_float x" by auto
     hence "0 < sqrt x" by auto
     hence "sqrt x < sqrt_iteration prec prec x"
       using sqrt_iteration_bound by auto
@@ -419,7 +419,7 @@
       (lapprox_rat prec 1 k) (- float_round_up prec (x * (ub_arctan_horner prec n (k + 2) x)))"
 
 lemma arctan_0_1_bounds':
-  assumes "0 \<le> real y" "real y \<le> 1"
+  assumes "0 \<le> real_of_float y" "real_of_float y \<le> 1"
     and "even n"
   shows "arctan (sqrt y) \<in>
       {(sqrt y * lb_arctan_horner prec n 1 y) .. (sqrt y * ub_arctan_horner prec (Suc n) 1 y)}"
@@ -452,7 +452,7 @@
   note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0
     and lb="\<lambda>n i k x. lb_arctan_horner prec n k x"
     and ub="\<lambda>n i k x. ub_arctan_horner prec n k x",
-    OF \<open>0 \<le> real y\<close> F lb_arctan_horner.simps ub_arctan_horner.simps]
+    OF \<open>0 \<le> real_of_float y\<close> F lb_arctan_horner.simps ub_arctan_horner.simps]
 
   have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> arctan (sqrt y)"
   proof -
@@ -479,7 +479,7 @@
 qed
 
 lemma arctan_0_1_bounds:
-  assumes "0 \<le> real y" "real y \<le> 1"
+  assumes "0 \<le> real_of_float y" "real_of_float y \<le> 1"
   shows "arctan (sqrt y) \<in>
     {(sqrt y * lb_arctan_horner prec (get_even n) 1 y) ..
       (sqrt y * ub_arctan_horner prec (get_odd n) 1 y)}"
@@ -532,47 +532,37 @@
 qed
 
 lemma arctan_0_1_bounds_le:
-  assumes "0 \<le> x" "x \<le> 1" "0 < real xl" "real xl \<le> x * x" "x * x \<le> real xu" "real xu \<le> 1"
+  assumes "0 \<le> x" "x \<le> 1" "0 < real_of_float xl" "real_of_float xl \<le> x * x" "x * x \<le> real_of_float xu" "real_of_float xu \<le> 1"
   shows "arctan x \<in>
       {x * lb_arctan_horner p1 (get_even n) 1 xu .. x * ub_arctan_horner p2 (get_odd n) 1 xl}"
 proof -
-  from assms have "real xl \<le> 1" "sqrt (real xl) \<le> x" "x \<le> sqrt (real xu)" "0 \<le> real xu"
-    "0 \<le> real xl" "0 < sqrt (real xl)"
+  from assms have "real_of_float xl \<le> 1" "sqrt (real_of_float xl) \<le> x" "x \<le> sqrt (real_of_float xu)" "0 \<le> real_of_float xu"
+    "0 \<le> real_of_float xl" "0 < sqrt (real_of_float xl)"
     by (auto intro!: real_le_rsqrt real_le_lsqrt simp: power2_eq_square)
-  from arctan_0_1_bounds[OF \<open>0 \<le> real xu\<close>  \<open>real xu \<le> 1\<close>]
-  have "sqrt (real xu) * real (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan (sqrt (real xu))"
+  from arctan_0_1_bounds[OF \<open>0 \<le> real_of_float xu\<close>  \<open>real_of_float xu \<le> 1\<close>]
+  have "sqrt (real_of_float xu) * real_of_float (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan (sqrt (real_of_float xu))"
     by simp
   from arctan_mult_le[OF \<open>0 \<le> x\<close> \<open>x \<le> sqrt _\<close>  this]
-  have "x * real (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan x" .
+  have "x * real_of_float (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan x" .
   moreover
-  from arctan_0_1_bounds[OF \<open>0 \<le> real xl\<close>  \<open>real xl \<le> 1\<close>]
-  have "arctan (sqrt (real xl)) \<le> sqrt (real xl) * real (ub_arctan_horner p2 (get_odd n) 1 xl)"
+  from arctan_0_1_bounds[OF \<open>0 \<le> real_of_float xl\<close>  \<open>real_of_float xl \<le> 1\<close>]
+  have "arctan (sqrt (real_of_float xl)) \<le> sqrt (real_of_float xl) * real_of_float (ub_arctan_horner p2 (get_odd n) 1 xl)"
     by simp
   from arctan_le_mult[OF \<open>0 < sqrt xl\<close> \<open>sqrt xl \<le> x\<close> this]
-  have "arctan x \<le> x * real (ub_arctan_horner p2 (get_odd n) 1 xl)" .
+  have "arctan x \<le> x * real_of_float (ub_arctan_horner p2 (get_odd n) 1 xl)" .
   ultimately show ?thesis by simp
 qed
 
-lemma mult_nonneg_le_one:
-  fixes a :: real
-  assumes "0 \<le> a" "0 \<le> b" "a \<le> 1" "b \<le> 1"
-  shows "a * b \<le> 1"
-proof -
-  have "a * b \<le> 1 * 1"
-    by (intro mult_mono assms) simp_all
-  thus ?thesis by simp
-qed
-
 lemma arctan_0_1_bounds_round:
-  assumes "0 \<le> real x" "real x \<le> 1"
+  assumes "0 \<le> real_of_float x" "real_of_float x \<le> 1"
   shows "arctan x \<in>
-      {real x * lb_arctan_horner p1 (get_even n) 1 (float_round_up (Suc p2) (x * x)) ..
-        real x * ub_arctan_horner p3 (get_odd n) 1 (float_round_down (Suc p4) (x * x))}"
+      {real_of_float x * lb_arctan_horner p1 (get_even n) 1 (float_round_up (Suc p2) (x * x)) ..
+        real_of_float x * ub_arctan_horner p3 (get_odd n) 1 (float_round_down (Suc p4) (x * x))}"
   using assms
   apply (cases "x > 0")
    apply (intro arctan_0_1_bounds_le)
    apply (auto simp: float_round_down.rep_eq float_round_up.rep_eq
-    intro!: truncate_up_le1 mult_nonneg_le_one truncate_down_le truncate_up_le truncate_down_pos
+    intro!: truncate_up_le1 mult_le_one truncate_down_le truncate_up_le truncate_down_pos
       mult_pos_pos)
   done
 
@@ -614,14 +604,14 @@
     let ?kl = "float_round_down (Suc prec) (?k * ?k)"
     have "1 div k = 0" using div_pos_pos_trivial[OF _ \<open>1 < k\<close>] by auto
 
-    have "0 \<le> real ?k" by (rule order_trans[OF _ rapprox_rat]) (auto simp add: \<open>0 \<le> k\<close>)
-    have "real ?k \<le> 1"
+    have "0 \<le> real_of_float ?k" by (rule order_trans[OF _ rapprox_rat]) (auto simp add: \<open>0 \<le> k\<close>)
+    have "real_of_float ?k \<le> 1"
       by (auto simp add: \<open>0 < k\<close> \<open>1 \<le> k\<close> less_imp_le
-        intro!: mult_nonneg_le_one order_trans[OF _ rapprox_rat] rapprox_rat_le1)
+        intro!: mult_le_one order_trans[OF _ rapprox_rat] rapprox_rat_le1)
     have "1 / k \<le> ?k" using rapprox_rat[where x=1 and y=k] by auto
     hence "arctan (1 / k) \<le> arctan ?k" by (rule arctan_monotone')
     also have "\<dots> \<le> (?k * ub_arctan_horner prec (get_odd n) 1 ?kl)"
-      using arctan_0_1_bounds_round[OF \<open>0 \<le> real ?k\<close> \<open>real ?k \<le> 1\<close>]
+      using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?k\<close> \<open>real_of_float ?k \<le> 1\<close>]
       by auto
     finally have "arctan (1 / k) \<le> ?k * ub_arctan_horner prec (get_odd n) 1 ?kl" .
   } note ub_arctan = this
@@ -634,16 +624,16 @@
     let ?ku = "float_round_up (Suc prec) (?k * ?k)"
     have "1 div k = 0" using div_pos_pos_trivial[OF _ \<open>1 < k\<close>] by auto
     have "1 / k \<le> 1" using \<open>1 < k\<close> by auto
-    have "0 \<le> real ?k" using lapprox_rat_nonneg[where x=1 and y=k, OF zero_le_one \<open>0 \<le> k\<close>]
+    have "0 \<le> real_of_float ?k" using lapprox_rat_nonneg[where x=1 and y=k, OF zero_le_one \<open>0 \<le> k\<close>]
       by (auto simp add: \<open>1 div k = 0\<close>)
-    have "0 \<le> real (?k * ?k)" by simp
-    have "real ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: \<open>1 / k \<le> 1\<close>)
-    hence "real (?k * ?k) \<le> 1" using \<open>0 \<le> real ?k\<close> by (auto intro!: mult_nonneg_le_one)
+    have "0 \<le> real_of_float (?k * ?k)" by simp
+    have "real_of_float ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: \<open>1 / k \<le> 1\<close>)
+    hence "real_of_float (?k * ?k) \<le> 1" using \<open>0 \<le> real_of_float ?k\<close> by (auto intro!: mult_le_one)
 
     have "?k \<le> 1 / k" using lapprox_rat[where x=1 and y=k] by auto
 
     have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan ?k"
-      using arctan_0_1_bounds_round[OF \<open>0 \<le> real ?k\<close> \<open>real ?k \<le> 1\<close>]
+      using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?k\<close> \<open>real_of_float ?k \<le> 1\<close>]
       by auto
     also have "\<dots> \<le> arctan (1 / k)" using \<open>?k \<le> 1 / k\<close> by (rule arctan_monotone')
     finally have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan (1 / k)" .
@@ -711,11 +701,11 @@
 declare lb_arctan_horner.simps[simp del]
 
 lemma lb_arctan_bound':
-  assumes "0 \<le> real x"
+  assumes "0 \<le> real_of_float x"
   shows "lb_arctan prec x \<le> arctan x"
 proof -
   have "\<not> x < 0" and "0 \<le> x"
-    using \<open>0 \<le> real x\<close> by (auto intro!: truncate_up_le )
+    using \<open>0 \<le> real_of_float x\<close> by (auto intro!: truncate_up_le )
 
   let "?ub_horner x" =
       "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x))"
@@ -725,15 +715,15 @@
   show ?thesis
   proof (cases "x \<le> Float 1 (- 1)")
     case True
-    hence "real x \<le> 1" by simp
-    from arctan_0_1_bounds_round[OF \<open>0 \<le> real x\<close> \<open>real x \<le> 1\<close>]
+    hence "real_of_float x \<le> 1" by simp
+    from arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>]
     show ?thesis
       unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF True] using \<open>0 \<le> x\<close>
       by (auto intro!: float_round_down_le)
   next
     case False
-    hence "0 < real x" by auto
-    let ?R = "1 + sqrt (1 + real x * real x)"
+    hence "0 < real_of_float x" by auto
+    let ?R = "1 + sqrt (1 + real_of_float x * real_of_float x)"
     let ?sxx = "float_plus_up prec 1 (float_round_up prec (x * x))"
     let ?fR = "float_plus_up prec 1 (ub_sqrt prec ?sxx)"
     let ?DIV = "float_divl prec x ?fR"
@@ -747,12 +737,12 @@
     finally
     have "sqrt (1 + x*x) \<le> ub_sqrt prec ?sxx" .
     hence "?R \<le> ?fR" by (auto simp: float_plus_up.rep_eq plus_up_def intro!: truncate_up_le)
-    hence "0 < ?fR" and "0 < real ?fR" using \<open>0 < ?R\<close> by auto
+    hence "0 < ?fR" and "0 < real_of_float ?fR" using \<open>0 < ?R\<close> by auto
 
     have monotone: "?DIV \<le> x / ?R"
     proof -
-      have "?DIV \<le> real x / ?fR" by (rule float_divl)
-      also have "\<dots> \<le> x / ?R" by (rule divide_left_mono[OF \<open>?R \<le> ?fR\<close> \<open>0 \<le> real x\<close> mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 \<open>?R \<le> real ?fR\<close>] divisor_gt0]])
+      have "?DIV \<le> real_of_float x / ?fR" by (rule float_divl)
+      also have "\<dots> \<le> x / ?R" by (rule divide_left_mono[OF \<open>?R \<le> ?fR\<close> \<open>0 \<le> real_of_float x\<close> mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 \<open>?R \<le> real_of_float ?fR\<close>] divisor_gt0]])
       finally show ?thesis .
     qed
 
@@ -762,18 +752,18 @@
       have "x \<le> sqrt (1 + x * x)"
         using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto
       also note \<open>\<dots> \<le> (ub_sqrt prec ?sxx)\<close>
-      finally have "real x \<le> ?fR"
+      finally have "real_of_float x \<le> ?fR"
         by (auto simp: float_plus_up.rep_eq plus_up_def intro!: truncate_up_le)
-      moreover have "?DIV \<le> real x / ?fR"
+      moreover have "?DIV \<le> real_of_float x / ?fR"
         by (rule float_divl)
-      ultimately have "real ?DIV \<le> 1"
-        unfolding divide_le_eq_1_pos[OF \<open>0 < real ?fR\<close>, symmetric] by auto
-
-      have "0 \<le> real ?DIV"
+      ultimately have "real_of_float ?DIV \<le> 1"
+        unfolding divide_le_eq_1_pos[OF \<open>0 < real_of_float ?fR\<close>, symmetric] by auto
+
+      have "0 \<le> real_of_float ?DIV"
         using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] \<open>0 < ?fR\<close>
         unfolding less_eq_float_def by auto
 
-      from arctan_0_1_bounds_round[OF \<open>0 \<le> real (?DIV)\<close> \<open>real (?DIV) \<le> 1\<close>]
+      from arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float (?DIV)\<close> \<open>real_of_float (?DIV) \<le> 1\<close>]
       have "Float 1 1 * ?lb_horner ?DIV \<le> 2 * arctan ?DIV"
         by simp
       also have "\<dots> \<le> 2 * arctan (x / ?R)"
@@ -787,11 +777,11 @@
           intro!: order_trans[OF mult_left_mono[OF truncate_down]])
     next
       case False
-      hence "2 < real x" by auto
-      hence "1 \<le> real x" by auto
+      hence "2 < real_of_float x" by auto
+      hence "1 \<le> real_of_float x" by auto
 
       let "?invx" = "float_divr prec 1 x"
-      have "0 \<le> arctan x" using arctan_monotone'[OF \<open>0 \<le> real x\<close>]
+      have "0 \<le> arctan x" using arctan_monotone'[OF \<open>0 \<le> real_of_float x\<close>]
         using arctan_tan[of 0, unfolded tan_zero] by auto
 
       show ?thesis
@@ -803,22 +793,22 @@
           using \<open>0 \<le> arctan x\<close> by auto
       next
         case False
-        hence "real ?invx \<le> 1" by auto
-        have "0 \<le> real ?invx"
-          by (rule order_trans[OF _ float_divr]) (auto simp add: \<open>0 \<le> real x\<close>)
+        hence "real_of_float ?invx \<le> 1" by auto
+        have "0 \<le> real_of_float ?invx"
+          by (rule order_trans[OF _ float_divr]) (auto simp add: \<open>0 \<le> real_of_float x\<close>)
 
         have "1 / x \<noteq> 0" and "0 < 1 / x"
-          using \<open>0 < real x\<close> by auto
+          using \<open>0 < real_of_float x\<close> by auto
 
         have "arctan (1 / x) \<le> arctan ?invx"
           unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone', rule float_divr)
         also have "\<dots> \<le> ?ub_horner ?invx"
-          using arctan_0_1_bounds_round[OF \<open>0 \<le> real ?invx\<close> \<open>real ?invx \<le> 1\<close>]
+          using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?invx\<close> \<open>real_of_float ?invx \<le> 1\<close>]
           by (auto intro!: float_round_up_le)
         also note float_round_up
         finally have "pi / 2 - float_round_up prec (?ub_horner ?invx) \<le> arctan x"
           using \<open>0 \<le> arctan x\<close> arctan_inverse[OF \<open>1 / x \<noteq> 0\<close>]
-          unfolding real_sgn_pos[OF \<open>0 < 1 / real x\<close>] le_diff_eq by auto
+          unfolding real_sgn_pos[OF \<open>0 < 1 / real_of_float x\<close>] le_diff_eq by auto
         moreover
         have "lb_pi prec * Float 1 (- 1) \<le> pi / 2"
           unfolding Float_num times_divide_eq_right mult_1_left using pi_boundaries by simp
@@ -833,11 +823,11 @@
 qed
 
 lemma ub_arctan_bound':
-  assumes "0 \<le> real x"
+  assumes "0 \<le> real_of_float x"
   shows "arctan x \<le> ub_arctan prec x"
 proof -
   have "\<not> x < 0" and "0 \<le> x"
-    using \<open>0 \<le> real x\<close> by auto
+    using \<open>0 \<le> real_of_float x\<close> by auto
 
   let "?ub_horner x" =
     "float_round_up prec (x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)))"
@@ -847,22 +837,22 @@
   show ?thesis
   proof (cases "x \<le> Float 1 (- 1)")
     case True
-    hence "real x \<le> 1" by auto
+    hence "real_of_float x \<le> 1" by auto
     show ?thesis
       unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF True]
-      using arctan_0_1_bounds_round[OF \<open>0 \<le> real x\<close> \<open>real x \<le> 1\<close>]
+      using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>]
       by (auto intro!: float_round_up_le)
   next
     case False
-    hence "0 < real x" by auto
-    let ?R = "1 + sqrt (1 + real x * real x)"
+    hence "0 < real_of_float x" by auto
+    let ?R = "1 + sqrt (1 + real_of_float x * real_of_float x)"
     let ?sxx = "float_plus_down prec 1 (float_round_down prec (x * x))"
     let ?fR = "float_plus_down (Suc prec) 1 (lb_sqrt prec ?sxx)"
     let ?DIV = "float_divr prec x ?fR"
 
-    have sqr_ge0: "0 \<le> 1 + real x * real x"
-      using sum_power2_ge_zero[of 1 "real x", unfolded numeral_2_eq_2] by auto
-    hence "0 \<le> real (1 + x*x)" by auto
+    have sqr_ge0: "0 \<le> 1 + real_of_float x * real_of_float x"
+      using sum_power2_ge_zero[of 1 "real_of_float x", unfolded numeral_2_eq_2] by auto
+    hence "0 \<le> real_of_float (1 + x*x)" by auto
 
     hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
 
@@ -873,13 +863,13 @@
     finally have "lb_sqrt prec ?sxx \<le> sqrt (1 + x*x)" .
     hence "?fR \<le> ?R"
       by (auto simp: float_plus_down.rep_eq plus_down_def truncate_down_le)
-    have "0 < real ?fR"
+    have "0 < real_of_float ?fR"
       by (auto simp: float_plus_down.rep_eq plus_down_def float_round_down.rep_eq
         intro!: truncate_down_ge1 lb_sqrt_lower_bound order_less_le_trans[OF zero_less_one]
         truncate_down_nonneg add_nonneg_nonneg)
     have monotone: "x / ?R \<le> (float_divr prec x ?fR)"
     proof -
-      from divide_left_mono[OF \<open>?fR \<le> ?R\<close> \<open>0 \<le> real x\<close> mult_pos_pos[OF divisor_gt0 \<open>0 < real ?fR\<close>]]
+      from divide_left_mono[OF \<open>?fR \<le> ?R\<close> \<open>0 \<le> real_of_float x\<close> mult_pos_pos[OF divisor_gt0 \<open>0 < real_of_float ?fR\<close>]]
       have "x / ?R \<le> x / ?fR" .
       also have "\<dots> \<le> ?DIV" by (rule float_divr)
       finally show ?thesis .
@@ -899,11 +889,11 @@
             if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF \<open>x \<le> Float 1 1\<close>] if_P[OF True] .
       next
         case False
-        hence "real ?DIV \<le> 1" by auto
+        hence "real_of_float ?DIV \<le> 1" by auto
 
         have "0 \<le> x / ?R"
-          using \<open>0 \<le> real x\<close> \<open>0 < ?R\<close> unfolding zero_le_divide_iff by auto
-        hence "0 \<le> real ?DIV"
+          using \<open>0 \<le> real_of_float x\<close> \<open>0 < ?R\<close> unfolding zero_le_divide_iff by auto
+        hence "0 \<le> real_of_float ?DIV"
           using monotone by (rule order_trans)
 
         have "arctan x = 2 * arctan (x / ?R)"
@@ -911,7 +901,7 @@
         also have "\<dots> \<le> 2 * arctan (?DIV)"
           using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
         also have "\<dots> \<le> (Float 1 1 * ?ub_horner ?DIV)" unfolding Float_num
-          using arctan_0_1_bounds_round[OF \<open>0 \<le> real ?DIV\<close> \<open>real ?DIV \<le> 1\<close>]
+          using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?DIV\<close> \<open>real_of_float ?DIV \<le> 1\<close>]
           by (auto intro!: float_round_up_le)
         finally show ?thesis
           unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
@@ -919,27 +909,27 @@
       qed
     next
       case False
-      hence "2 < real x" by auto
-      hence "1 \<le> real x" by auto
-      hence "0 < real x" by auto
+      hence "2 < real_of_float x" by auto
+      hence "1 \<le> real_of_float x" by auto
+      hence "0 < real_of_float x" by auto
       hence "0 < x" by auto
 
       let "?invx" = "float_divl prec 1 x"
       have "0 \<le> arctan x"
-        using arctan_monotone'[OF \<open>0 \<le> real x\<close>] and arctan_tan[of 0, unfolded tan_zero] by auto
-
-      have "real ?invx \<le> 1"
+        using arctan_monotone'[OF \<open>0 \<le> real_of_float x\<close>] and arctan_tan[of 0, unfolded tan_zero] by auto
+
+      have "real_of_float ?invx \<le> 1"
         unfolding less_float_def
         by (rule order_trans[OF float_divl])
-          (auto simp add: \<open>1 \<le> real x\<close> divide_le_eq_1_pos[OF \<open>0 < real x\<close>])
-      have "0 \<le> real ?invx"
+          (auto simp add: \<open>1 \<le> real_of_float x\<close> divide_le_eq_1_pos[OF \<open>0 < real_of_float x\<close>])
+      have "0 \<le> real_of_float ?invx"
         using \<open>0 < x\<close> by (intro float_divl_lower_bound) auto
 
       have "1 / x \<noteq> 0" and "0 < 1 / x"
-        using \<open>0 < real x\<close> by auto
+        using \<open>0 < real_of_float x\<close> by auto
 
       have "(?lb_horner ?invx) \<le> arctan (?invx)"
-        using arctan_0_1_bounds_round[OF \<open>0 \<le> real ?invx\<close> \<open>real ?invx \<le> 1\<close>]
+        using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?invx\<close> \<open>real_of_float ?invx \<le> 1\<close>]
         by (auto intro!: float_round_down_le)
       also have "\<dots> \<le> arctan (1 / x)"
         unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone') (rule float_divl)
@@ -962,17 +952,17 @@
 lemma arctan_boundaries: "arctan x \<in> {(lb_arctan prec x) .. (ub_arctan prec x)}"
 proof (cases "0 \<le> x")
   case True
-  hence "0 \<le> real x" by auto
+  hence "0 \<le> real_of_float x" by auto
   show ?thesis
-    using ub_arctan_bound'[OF \<open>0 \<le> real x\<close>] lb_arctan_bound'[OF \<open>0 \<le> real x\<close>]
+    using ub_arctan_bound'[OF \<open>0 \<le> real_of_float x\<close>] lb_arctan_bound'[OF \<open>0 \<le> real_of_float x\<close>]
     unfolding atLeastAtMost_iff by auto
 next
   case False
   let ?mx = "-x"
-  from False have "x < 0" and "0 \<le> real ?mx"
+  from False have "x < 0" and "0 \<le> real_of_float ?mx"
     by auto
   hence bounds: "lb_arctan prec ?mx \<le> arctan ?mx \<and> arctan ?mx \<le> ub_arctan prec ?mx"
-    using ub_arctan_bound'[OF \<open>0 \<le> real ?mx\<close>] lb_arctan_bound'[OF \<open>0 \<le> real ?mx\<close>] by auto
+    using ub_arctan_bound'[OF \<open>0 \<le> real_of_float ?mx\<close>] lb_arctan_bound'[OF \<open>0 \<le> real_of_float ?mx\<close>] by auto
   show ?thesis
     unfolding minus_float.rep_eq arctan_minus lb_arctan.simps[where x=x]
       ub_arctan.simps[where x=x] Let_def if_P[OF \<open>x < 0\<close>]
@@ -1027,7 +1017,7 @@
   shows "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i))) * x ^(2 * i))" (is "?lb")
   and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i))) * x^(2 * i)) \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
 proof -
-  have "0 \<le> real (x * x)" by auto
+  have "0 \<le> real_of_float (x * x)" by auto
   let "?f n" = "fact (2 * n) :: nat"
   have f_eq: "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)" for n
   proof -
@@ -1035,9 +1025,9 @@
     then show ?thesis by auto
   qed
   from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
-    OF \<open>0 \<le> real (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
+    OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
   show ?lb and ?ub
-    by (auto simp add: power_mult power2_eq_square[of "real x"])
+    by (auto simp add: power_mult power2_eq_square[of "real_of_float x"])
 qed
 
 lemma lb_sin_cos_aux_zero_le_one: "lb_sin_cos_aux prec n i j 0 \<le> 1"
@@ -1048,13 +1038,13 @@
   by (cases n) (auto intro!: float_plus_up_le order_trans[OF _ rapprox_rat])
 
 lemma cos_boundaries:
-  assumes "0 \<le> real x" and "x \<le> pi / 2"
+  assumes "0 \<le> real_of_float x" and "x \<le> pi / 2"
   shows "cos x \<in> {(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}"
-proof (cases "real x = 0")
+proof (cases "real_of_float x = 0")
   case False
-  hence "real x \<noteq> 0" by auto
-  hence "0 < x" and "0 < real x"
-    using \<open>0 \<le> real x\<close> by auto
+  hence "real_of_float x \<noteq> 0" by auto
+  hence "0 < x" and "0 < real_of_float x"
+    using \<open>0 \<le> real_of_float x\<close> by auto
   have "0 < x * x"
     using \<open>0 < x\<close> by simp
 
@@ -1074,11 +1064,11 @@
 
   { fix n :: nat assume "0 < n"
     hence "0 < 2 * n" by auto
-    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))/((fact i)) else 0) * (real x) ^ i)
-      + (cos (t + 1/2 * (2 * n) * pi) / (fact (2*n))) * (real x)^(2*n)"
+    obtain t where "0 < t" and "t < real_of_float x" and
+      cos_eq: "cos x = (\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * (real_of_float x) ^ i)
+      + (cos (t + 1/2 * (2 * n) * pi) / (fact (2*n))) * (real_of_float x)^(2*n)"
       (is "_ = ?SUM + ?rest / ?fact * ?pow")
-      using Maclaurin_cos_expansion2[OF \<open>0 < real x\<close> \<open>0 < 2 * n\<close>]
+      using Maclaurin_cos_expansion2[OF \<open>0 < real_of_float x\<close> \<open>0 < 2 * n\<close>]
       unfolding cos_coeff_def atLeast0LessThan by auto
 
     have "cos t * (- 1) ^ n = cos t * cos (n * pi) + sin t * sin (n * pi)" by auto
@@ -1086,12 +1076,12 @@
     also have "\<dots> = ?rest" by auto
     finally have "cos t * (- 1) ^ n = ?rest" .
     moreover
-    have "t \<le> pi / 2" using \<open>t < real x\<close> and \<open>x \<le> pi / 2\<close> by auto
+    have "t \<le> pi / 2" using \<open>t < real_of_float x\<close> and \<open>x \<le> pi / 2\<close> by auto
     hence "0 \<le> cos t" using \<open>0 < t\<close> and cos_ge_zero by auto
     ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
 
     have "0 < ?fact" by auto
-    have "0 < ?pow" using \<open>0 < real x\<close> by auto
+    have "0 < ?pow" using \<open>0 < real_of_float x\<close> by auto
 
     {
       assume "even n"
@@ -1131,7 +1121,7 @@
     case False
     hence "get_even n = 0" by auto
     have "- (pi / 2) \<le> x"
-      by (rule order_trans[OF _ \<open>0 < real x\<close>[THEN less_imp_le]]) auto
+      by (rule order_trans[OF _ \<open>0 < real_of_float x\<close>[THEN less_imp_le]]) auto
     with \<open>x \<le> pi / 2\<close> show ?thesis
       unfolding \<open>get_even n = 0\<close> lb_sin_cos_aux.simps minus_float.rep_eq zero_float.rep_eq
       using cos_ge_zero by auto
@@ -1147,13 +1137,13 @@
 qed
 
 lemma sin_aux:
-  assumes "0 \<le> real x"
+  assumes "0 \<le> real_of_float x"
   shows "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
       (\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1))" (is "?lb")
     and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1)) \<le>
       (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub")
 proof -
-  have "0 \<le> real (x * x)" by auto
+  have "0 \<le> real_of_float (x * x)" by auto
   let "?f n" = "fact (2 * n + 1) :: nat"
   have f_eq: "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)" for n
   proof -
@@ -1162,22 +1152,22 @@
       unfolding F by auto
   qed
   from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
-    OF \<open>0 \<le> real (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
-  show "?lb" and "?ub" using \<open>0 \<le> real x\<close>
+    OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
+  show "?lb" and "?ub" using \<open>0 \<le> real_of_float x\<close>
     unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
-    unfolding mult.commute[where 'a=real] real_fact_nat
-    by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real x"])
+    unfolding mult.commute[where 'a=real] of_nat_fact
+    by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real_of_float x"])
 qed
 
 lemma sin_boundaries:
-  assumes "0 \<le> real x"
+  assumes "0 \<le> real_of_float x"
     and "x \<le> pi / 2"
   shows "sin x \<in> {(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}"
-proof (cases "real x = 0")
+proof (cases "real_of_float x = 0")
   case False
-  hence "real x \<noteq> 0" by auto
-  hence "0 < x" and "0 < real x"
-    using \<open>0 \<le> real x\<close> by auto
+  hence "real_of_float x \<noteq> 0" by auto
+  hence "0 < x" and "0 < real_of_float x"
+    using \<open>0 \<le> real_of_float x\<close> by auto
   have "0 < x * x"
     using \<open>0 < x\<close> by simp
 
@@ -1198,18 +1188,18 @@
 
   { fix n :: nat assume "0 < n"
     hence "0 < 2 * n + 1" by auto
-    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))/((fact i))) * (real x) ^ i)
-      + (sin (t + 1/2 * (2 * n + 1) * pi) / (fact (2*n + 1))) * (real x)^(2*n + 1)"
+    obtain t where "0 < t" and "t < real_of_float x" and
+      sin_eq: "sin x = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)
+      + (sin (t + 1/2 * (2 * n + 1) * pi) / (fact (2*n + 1))) * (real_of_float x)^(2*n + 1)"
       (is "_ = ?SUM + ?rest / ?fact * ?pow")
-      using Maclaurin_sin_expansion3[OF \<open>0 < 2 * n + 1\<close> \<open>0 < real x\<close>]
+      using Maclaurin_sin_expansion3[OF \<open>0 < 2 * n + 1\<close> \<open>0 < real_of_float x\<close>]
       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
+      unfolding sin_add cos_add of_nat_add distrib_right distrib_left by auto
     moreover
     have "t \<le> pi / 2"
-      using \<open>t < real x\<close> and \<open>x \<le> pi / 2\<close> by auto
+      using \<open>t < real_of_float x\<close> and \<open>x \<le> pi / 2\<close> by auto
     hence "0 \<le> cos t"
       using \<open>0 < t\<close> and cos_ge_zero by auto
     ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest"
@@ -1218,13 +1208,13 @@
     have "0 < ?fact"
       by (simp del: fact_Suc)
     have "0 < ?pow"
-      using \<open>0 < real x\<close> by (rule zero_less_power)
+      using \<open>0 < real_of_float x\<close> by (rule zero_less_power)
 
     {
       assume "even n"
       have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
-            (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real x) ^ i)"
-        using sin_aux[OF \<open>0 \<le> real x\<close>] unfolding setsum_morph[symmetric] by auto
+            (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)"
+        using sin_aux[OF \<open>0 \<le> real_of_float x\<close>] unfolding setsum_morph[symmetric] by auto
       also have "\<dots> \<le> ?SUM" by auto
       also have "\<dots> \<le> sin x"
       proof -
@@ -1244,10 +1234,10 @@
           by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
         thus ?thesis unfolding sin_eq by auto
       qed
-      also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real x) ^ i)"
+      also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)"
          by auto
       also have "\<dots> \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))"
-        using sin_aux[OF \<open>0 \<le> real x\<close>] unfolding setsum_morph[symmetric] by auto
+        using sin_aux[OF \<open>0 \<le> real_of_float x\<close>] unfolding setsum_morph[symmetric] by auto
       finally have "sin x \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
     } note ub = this and lb
   } note ub = this(1) and lb = this(2)
@@ -1262,7 +1252,7 @@
   next
     case False
     hence "get_even n = 0" by auto
-    with \<open>x \<le> pi / 2\<close> \<open>0 \<le> real x\<close>
+    with \<open>x \<le> pi / 2\<close> \<open>0 \<le> real_of_float x\<close>
     show ?thesis
       unfolding \<open>get_even n = 0\<close> ub_sin_cos_aux.simps minus_float.rep_eq
       using sin_ge_zero by auto
@@ -1275,13 +1265,13 @@
     case True
     thus ?thesis
       unfolding \<open>n = 0\<close> get_even_def get_odd_def
-      using \<open>real x = 0\<close> lapprox_rat[where x="-1" and y=1] by auto
+      using \<open>real_of_float x = 0\<close> lapprox_rat[where x="-1" and y=1] by auto
   next
     case False
     with not0_implies_Suc obtain m where "n = Suc m" by blast
     thus ?thesis
       unfolding \<open>n = Suc m\<close> get_even_def get_odd_def
-      using \<open>real x = 0\<close> rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1]
+      using \<open>real_of_float x = 0\<close> rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1]
       by (cases "even (Suc m)") auto
   qed
 qed
@@ -1306,7 +1296,7 @@
                        else half (half (horner (x * Float 1 (- 2)))))"
 
 lemma lb_cos:
-  assumes "0 \<le> real x" and "x \<le> pi"
+  assumes "0 \<le> real_of_float x" and "x \<le> pi"
   shows "cos x \<in> {(lb_cos prec x) .. (ub_cos prec x)}" (is "?cos x \<in> {(?lb x) .. (?ub x) }")
 proof -
   have x_half[symmetric]: "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" for x :: real
@@ -1320,7 +1310,7 @@
     finally show ?thesis .
   qed
 
-  have "\<not> x < 0" using \<open>0 \<le> real x\<close> by auto
+  have "\<not> x < 0" using \<open>0 \<le> real_of_float x\<close> by auto
   let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)"
   let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)"
   let "?ub_half x" = "float_plus_up prec (Float 1 1 * x * x) (- 1)"
@@ -1334,7 +1324,7 @@
     show ?thesis
       unfolding lb_cos_def[where x=x] ub_cos_def[where x=x]
         if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF \<open>x < Float 1 (- 1)\<close>] Let_def
-      using cos_boundaries[OF \<open>0 \<le> real x\<close> \<open>x \<le> pi / 2\<close>] .
+      using cos_boundaries[OF \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi / 2\<close>] .
   next
     case False
     { fix y x :: float let ?x2 = "(x * Float 1 (- 1))"
@@ -1351,12 +1341,12 @@
           using cos_ge_minus_one unfolding if_P[OF True] by auto
       next
         case False
-        hence "0 \<le> real y" by auto
+        hence "0 \<le> real_of_float y" by auto
         from mult_mono[OF \<open>y \<le> cos ?x2\<close> \<open>y \<le> cos ?x2\<close> \<open>0 \<le> cos ?x2\<close> this]
-        have "real y * real y \<le> cos ?x2 * cos ?x2" .
-        hence "2 * real y * real y \<le> 2 * cos ?x2 * cos ?x2"
+        have "real_of_float y * real_of_float y \<le> cos ?x2 * cos ?x2" .
+        hence "2 * real_of_float y * real_of_float y \<le> 2 * cos ?x2 * cos ?x2"
           by auto
-        hence "2 * real y * real y - 1 \<le> 2 * cos (x / 2) * cos (x / 2) - 1"
+        hence "2 * real_of_float y * real_of_float y - 1 \<le> 2 * cos (x / 2) * cos (x / 2) - 1"
           unfolding Float_num by auto
         thus ?thesis
           unfolding if_not_P[OF False] x_half Float_num
@@ -1372,13 +1362,13 @@
 
       have "cos x \<le> (?ub_half y)"
       proof -
-        have "0 \<le> real y"
+        have "0 \<le> real_of_float y"
           using \<open>0 \<le> cos ?x2\<close> ub by (rule order_trans)
         from mult_mono[OF ub ub this \<open>0 \<le> cos ?x2\<close>]
-        have "cos ?x2 * cos ?x2 \<le> real y * real y" .
-        hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real y * real y"
+        have "cos ?x2 * cos ?x2 \<le> real_of_float y * real_of_float y" .
+        hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real_of_float y * real_of_float y"
           by auto
-        hence "2 * cos (x / 2) * cos (x / 2) - 1 \<le> 2 * real y * real y - 1"
+        hence "2 * cos (x / 2) * cos (x / 2) - 1 \<le> 2 * real_of_float y * real_of_float y - 1"
           unfolding Float_num by auto
         thus ?thesis
           unfolding x_half Float_num
@@ -1390,15 +1380,15 @@
     let ?x4 = "x * Float 1 (- 1) * Float 1 (- 1)"
 
     have "-pi \<le> x"
-      using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] \<open>0 \<le> real x\<close>
+      using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] \<open>0 \<le> real_of_float x\<close>
       by (rule order_trans)
 
     show ?thesis
     proof (cases "x < 1")
       case True
-      hence "real x \<le> 1" by auto
-      have "0 \<le> real ?x2" and "?x2 \<le> pi / 2"
-        using pi_ge_two \<open>0 \<le> real x\<close> using assms by auto
+      hence "real_of_float x \<le> 1" by auto
+      have "0 \<le> real_of_float ?x2" and "?x2 \<le> pi / 2"
+        using pi_ge_two \<open>0 \<le> real_of_float x\<close> using assms by auto
       from cos_boundaries[OF this]
       have lb: "(?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> (?ub_horner ?x2)"
         by auto
@@ -1420,8 +1410,8 @@
       ultimately show ?thesis by auto
     next
       case False
-      have "0 \<le> real ?x4" and "?x4 \<le> pi / 2"
-        using pi_ge_two \<open>0 \<le> real x\<close> \<open>x \<le> pi\<close> unfolding Float_num by auto
+      have "0 \<le> real_of_float ?x4" and "?x4 \<le> pi / 2"
+        using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi\<close> unfolding Float_num by auto
       from cos_boundaries[OF this]
       have lb: "(?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> (?ub_horner ?x4)"
         by auto
@@ -1432,7 +1422,7 @@
       have "(?lb x) \<le> ?cos x"
       proof -
         have "-pi \<le> ?x2" and "?x2 \<le> pi"
-          using pi_ge_two \<open>0 \<le> real x\<close> \<open>x \<le> pi\<close> by auto
+          using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi\<close> by auto
         from lb_half[OF lb_half[OF lb this] \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>, unfolded eq_4]
         show ?thesis
           unfolding lb_cos_def[where x=x] if_not_P[OF \<open>\<not> x < 0\<close>]
@@ -1441,7 +1431,7 @@
       moreover have "?cos x \<le> (?ub x)"
       proof -
         have "-pi \<le> ?x2" and "?x2 \<le> pi"
-          using pi_ge_two \<open>0 \<le> real x\<close> \<open> x \<le> pi\<close> by auto
+          using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open> x \<le> pi\<close> by auto
         from ub_half[OF ub_half[OF ub this] \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>, unfolded eq_4]
         show ?thesis
           unfolding ub_cos_def[where x=x] if_not_P[OF \<open>\<not> x < 0\<close>]
@@ -1454,11 +1444,11 @@
 
 lemma lb_cos_minus:
   assumes "-pi \<le> x"
-    and "real x \<le> 0"
-  shows "cos (real(-x)) \<in> {(lb_cos prec (-x)) .. (ub_cos prec (-x))}"
+    and "real_of_float x \<le> 0"
+  shows "cos (real_of_float(-x)) \<in> {(lb_cos prec (-x)) .. (ub_cos prec (-x))}"
 proof -
-  have "0 \<le> real (-x)" and "(-x) \<le> pi"
-    using \<open>-pi \<le> x\<close> \<open>real x \<le> 0\<close> by auto
+  have "0 \<le> real_of_float (-x)" and "(-x) \<le> pi"
+    using \<open>-pi \<le> x\<close> \<open>real_of_float x \<le> 0\<close> by auto
   from lb_cos[OF this] show ?thesis .
 qed
 
@@ -1476,7 +1466,7 @@
   else if -2 * lpi \<le> lx \<and> ux \<le> 0 then (Float (- 1) 0, max (ub_cos prec (lx + 2 * lpi)) (ub_cos prec (-ux)))
                                  else (Float (- 1) 0, Float 1 0))"
 
-lemma floor_int: obtains k :: int where "real k = (floor_fl f)"
+lemma floor_int: obtains k :: int where "real_of_int k = (floor_fl f)"
   by (simp add: floor_fl_def)
 
 lemma cos_periodic_nat[simp]:
@@ -1488,7 +1478,7 @@
 next
   case (Suc n)
   have split_pi_off: "x + (Suc n) * (2 * pi) = (x + n * (2 * pi)) + 2 * pi"
-    unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
+    unfolding Suc_eq_plus1 of_nat_add of_int_1 distrib_right by auto
   show ?case
     unfolding split_pi_off using Suc by auto
 qed
@@ -1498,7 +1488,7 @@
   shows "cos (x + i * (2 * pi)) = cos x"
 proof (cases "0 \<le> i")
   case True
-  hence i_nat: "real i = nat i" by auto
+  hence i_nat: "real_of_int i = nat i" by auto
   show ?thesis
     unfolding i_nat by auto
 next
@@ -1526,7 +1516,7 @@
   let ?lx = "float_plus_down prec lx ?lx2"
   let ?ux = "float_plus_up prec ux ?ux2"
 
-  obtain k :: int where k: "k = real ?k"
+  obtain k :: int where k: "k = real_of_float ?k"
     by (rule floor_int)
 
   have upi: "pi \<le> ?upi" and lpi: "?lpi \<le> pi"
@@ -1542,18 +1532,18 @@
   hence "?lx \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ?ux"
     by (auto intro!: float_plus_down_le float_plus_up_le)
   note lx = this[THEN conjunct1] and ux = this[THEN conjunct2]
-  hence lx_less_ux: "?lx \<le> real ?ux" by (rule order_trans)
+  hence lx_less_ux: "?lx \<le> real_of_float ?ux" by (rule order_trans)
 
   { assume "- ?lpi \<le> ?lx" and x_le_0: "x - k * (2 * pi) \<le> 0"
     with lpi[THEN le_imp_neg_le] lx
-    have pi_lx: "- pi \<le> ?lx" and lx_0: "real ?lx \<le> 0"
+    have pi_lx: "- pi \<le> ?lx" and lx_0: "real_of_float ?lx \<le> 0"
       by simp_all
 
-    have "(lb_cos prec (- ?lx)) \<le> cos (real (- ?lx))"
+    have "(lb_cos prec (- ?lx)) \<le> cos (real_of_float (- ?lx))"
       using lb_cos_minus[OF pi_lx lx_0] by simp
     also have "\<dots> \<le> cos (x + (-k) * (2 * pi))"
       using cos_monotone_minus_pi_0'[OF pi_lx lx x_le_0]
-      by (simp only: uminus_float.rep_eq real_of_int_minus
+      by (simp only: uminus_float.rep_eq of_int_minus
         cos_minus mult_minus_left) simp
     finally have "(lb_cos prec (- ?lx)) \<le> cos x"
       unfolding cos_periodic_int . }
@@ -1561,12 +1551,12 @@
 
   { assume "0 \<le> ?lx" and pi_x: "x - k * (2 * pi) \<le> pi"
     with lx
-    have pi_lx: "?lx \<le> pi" and lx_0: "0 \<le> real ?lx"
+    have pi_lx: "?lx \<le> pi" and lx_0: "0 \<le> real_of_float ?lx"
       by auto
 
     have "cos (x + (-k) * (2 * pi)) \<le> cos ?lx"
       using cos_monotone_0_pi_le[OF lx_0 lx pi_x]
-      by (simp only: real_of_int_minus
+      by (simp only: of_int_minus
         cos_minus mult_minus_left) simp
     also have "\<dots> \<le> (ub_cos prec ?lx)"
       using lb_cos[OF lx_0 pi_lx] by simp
@@ -1576,12 +1566,12 @@
 
   { assume pi_x: "- pi \<le> x - k * (2 * pi)" and "?ux \<le> 0"
     with ux
-    have pi_ux: "- pi \<le> ?ux" and ux_0: "real ?ux \<le> 0"
+    have pi_ux: "- pi \<le> ?ux" and ux_0: "real_of_float ?ux \<le> 0"
       by simp_all
 
-    have "cos (x + (-k) * (2 * pi)) \<le> cos (real (- ?ux))"
+    have "cos (x + (-k) * (2 * pi)) \<le> cos (real_of_float (- ?ux))"
       using cos_monotone_minus_pi_0'[OF pi_x ux ux_0]
-      by (simp only: uminus_float.rep_eq real_of_int_minus
+      by (simp only: uminus_float.rep_eq of_int_minus
           cos_minus mult_minus_left) simp
     also have "\<dots> \<le> (ub_cos prec (- ?ux))"
       using lb_cos_minus[OF pi_ux ux_0, of prec] by simp
@@ -1591,14 +1581,14 @@
 
   { assume "?ux \<le> ?lpi" and x_ge_0: "0 \<le> x - k * (2 * pi)"
     with lpi ux
-    have pi_ux: "?ux \<le> pi" and ux_0: "0 \<le> real ?ux"
+    have pi_ux: "?ux \<le> pi" and ux_0: "0 \<le> real_of_float ?ux"
       by simp_all
 
     have "(lb_cos prec ?ux) \<le> cos ?ux"
       using lb_cos[OF ux_0 pi_ux] by simp
     also have "\<dots> \<le> cos (x + (-k) * (2 * pi))"
       using cos_monotone_0_pi_le[OF x_ge_0 ux pi_ux]
-      by (simp only: real_of_int_minus
+      by (simp only: of_int_minus
         cos_minus mult_minus_left) simp
     finally have "(lb_cos prec ?ux) \<le> cos x"
       unfolding cos_periodic_int . }
@@ -1648,7 +1638,7 @@
             and u: "u = max (ub_cos prec ?lx) (ub_cos prec (- (?ux - 2 * ?lpi)))"
             by (auto simp add: bnds_cos_def Let_def)
 
-          have "cos x \<le> real u"
+          have "cos x \<le> real_of_float u"
           proof (cases "x - k * (2 * pi) < pi")
             case True
             hence "x - k * (2 * pi) \<le> pi" by simp
@@ -1664,7 +1654,7 @@
             hence "x - k * (2 * pi) - 2 * pi \<le> 0"
               using ux by simp
 
-            have ux_0: "real (?ux - 2 * ?lpi) \<le> 0"
+            have ux_0: "real_of_float (?ux - 2 * ?lpi) \<le> 0"
               using Cond by auto
 
             from 2 and Cond have "\<not> ?ux \<le> ?lpi" by auto
@@ -1678,7 +1668,7 @@
               unfolding cos_periodic_int ..
             also have "\<dots> \<le> cos ((?ux - 2 * ?lpi))"
               using cos_monotone_minus_pi_0'[OF pi_x x_le_ux ux_0]
-              by (simp only: minus_float.rep_eq real_of_int_minus real_of_one
+              by (simp only: minus_float.rep_eq of_int_minus of_int_1
                 mult_minus_left mult_1_left) simp
             also have "\<dots> = cos ((- (?ux - 2 * ?lpi)))"
               unfolding uminus_float.rep_eq cos_minus ..
@@ -1711,7 +1701,7 @@
 
               hence "0 \<le> x - k * (2 * pi) + 2 * pi" using lx by simp
 
-              have lx_0: "0 \<le> real (?lx + 2 * ?lpi)"
+              have lx_0: "0 \<le> real_of_float (?lx + 2 * ?lpi)"
                 using Cond lpi by auto
 
               from 1 and Cond have "\<not> -?lpi \<le> ?lx" by auto
@@ -1726,7 +1716,7 @@
                 unfolding cos_periodic_int ..
               also have "\<dots> \<le> cos ((?lx + 2 * ?lpi))"
                 using cos_monotone_0_pi_le[OF lx_0 lx_le_x pi_x]
-                by (simp only: minus_float.rep_eq real_of_int_minus real_of_one
+                by (simp only: minus_float.rep_eq of_int_minus of_int_1
                   mult_minus_left mult_1_left) simp
               also have "\<dots> \<le> (ub_cos prec (?lx + 2 * ?lpi))"
                 using lb_cos[OF lx_0 pi_lx] by simp
@@ -1760,7 +1750,7 @@
     (lapprox_rat prec 1 (int k)) (float_round_down prec (x * ub_exp_horner prec n (i + 1) (k * i) x))"
 
 lemma bnds_exp_horner:
-  assumes "real x \<le> 0"
+  assumes "real_of_float x \<le> 0"
   shows "exp x \<in> {lb_exp_horner prec (get_even n) 1 1 x .. ub_exp_horner prec (get_odd n) 1 1 x}"
 proof -
   have f_eq: "fact (Suc n) = fact n * ((\<lambda>i::nat. i + 1) ^^ n) 1" for n
@@ -1776,13 +1766,13 @@
 
   have "lb_exp_horner prec (get_even n) 1 1 x \<le> exp x"
   proof -
-    have "lb_exp_horner prec (get_even n) 1 1 x \<le> (\<Sum>j = 0..<get_even n. 1 / (fact j) * real x ^ j)"
+    have "lb_exp_horner prec (get_even n) 1 1 x \<le> (\<Sum>j = 0..<get_even n. 1 / (fact j) * real_of_float 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 / (fact m)) + exp t / (fact (get_even n)) * (real x) ^ (get_even n)"
+      obtain t where "\<bar>t\<bar> \<le> \<bar>real_of_float x\<bar>" and "exp x = (\<Sum>m = 0..<get_even n. real_of_float x ^ m / (fact m)) + exp t / (fact (get_even n)) * (real_of_float x) ^ (get_even n)"
         using Maclaurin_exp_le unfolding atLeast0LessThan by blast
-      moreover have "0 \<le> exp t / (fact (get_even n)) * (real x) ^ (get_even n)"
+      moreover have "0 \<le> exp t / (fact (get_even n)) * (real_of_float x) ^ (get_even n)"
         by (auto simp: zero_le_even_power)
       ultimately show ?thesis using get_odd exp_gt_zero by auto
     qed
@@ -1791,21 +1781,21 @@
   moreover
   have "exp x \<le> ub_exp_horner prec (get_odd n) 1 1 x"
   proof -
-    have x_less_zero: "real x ^ get_odd n \<le> 0"
-    proof (cases "real x = 0")
+    have x_less_zero: "real_of_float x ^ get_odd n \<le> 0"
+    proof (cases "real_of_float x = 0")
       case True
       have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto
       thus ?thesis unfolding True power_0_left by auto
     next
-      case False hence "real x < 0" using \<open>real x \<le> 0\<close> by auto
-      show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq \<open>real x < 0\<close>)
+      case False hence "real_of_float x < 0" using \<open>real_of_float x \<le> 0\<close> by auto
+      show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq \<open>real_of_float x < 0\<close>)
     qed
-    obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>"
-      and "exp x = (\<Sum>m = 0..<get_odd n. (real x) ^ m / (fact m)) + exp t / (fact (get_odd n)) * (real x) ^ (get_odd n)"
+    obtain t where "\<bar>t\<bar> \<le> \<bar>real_of_float x\<bar>"
+      and "exp x = (\<Sum>m = 0..<get_odd n. (real_of_float x) ^ m / (fact m)) + exp t / (fact (get_odd n)) * (real_of_float x) ^ (get_odd n)"
       using Maclaurin_exp_le unfolding atLeast0LessThan by blast
-    moreover have "exp t / (fact (get_odd n)) * (real x) ^ (get_odd n) \<le> 0"
+    moreover have "exp t / (fact (get_odd n)) * (real_of_float 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 / (fact j) * real x ^ j)"
+    ultimately have "exp x \<le> (\<Sum>j = 0..<get_odd n. 1 / (fact j) * real_of_float x ^ j)"
       using get_odd exp_gt_zero by auto
     also have "\<dots> \<le> ub_exp_horner prec (get_odd n) 1 1 x"
       using bounds(2) by auto
@@ -1814,8 +1804,8 @@
   ultimately show ?thesis by auto
 qed
 
-lemma ub_exp_horner_nonneg: "real x \<le> 0 \<Longrightarrow>
-  0 \<le> real (ub_exp_horner prec (get_odd n) (Suc 0) (Suc 0) x)"
+lemma ub_exp_horner_nonneg: "real_of_float x \<le> 0 \<Longrightarrow>
+  0 \<le> real_of_float (ub_exp_horner prec (get_odd n) (Suc 0) (Suc 0) x)"
   using bnds_exp_horner[of x prec n]
   by (intro order_trans[OF exp_ge_zero]) auto
 
@@ -1850,7 +1840,7 @@
   have "1 / 4 = (Float 1 (- 2))"
     unfolding Float_num by auto
   also have "\<dots> \<le> lb_exp_horner 3 (get_even 3) 1 1 (- 1)"
-    by code_simp
+    by (subst less_eq_float.rep_eq [symmetric]) code_simp
   also have "\<dots> \<le> exp (- 1 :: float)"
     using bnds_exp_horner[where x="- 1"] by auto
   finally show ?thesis
@@ -1865,9 +1855,9 @@
   let "?horner x" = "let y = ?lb_horner x in if y \<le> 0 then Float 1 (- 2) else y"
   have pos_horner: "0 < ?horner x" for x
     unfolding Let_def by (cases "?lb_horner x \<le> 0") auto
-  moreover have "0 < real ((?horner x) ^ num)" for x :: float and num :: nat
+  moreover have "0 < real_of_float ((?horner x) ^ num)" for x :: float and num :: nat
   proof -
-    have "0 < real (?horner x) ^ num" using \<open>0 < ?horner x\<close> by simp
+    have "0 < real_of_float (?horner x) ^ num" using \<open>0 < ?horner x\<close> by simp
     also have "\<dots> = (?horner x) ^ num" by auto
     finally show ?thesis .
   qed
@@ -1884,35 +1874,35 @@
   let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
   let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x"
 
-  have "real x \<le> 0" and "\<not> x > 0"
+  have "real_of_float x \<le> 0" and "\<not> x > 0"
     using \<open>x \<le> 0\<close> by auto
   show ?thesis
   proof (cases "x < - 1")
     case False
-    hence "- 1 \<le> real x" by auto
+    hence "- 1 \<le> real_of_float x" by auto
     show ?thesis
     proof (cases "?lb_exp_horner x \<le> 0")
       case True
       from \<open>\<not> x < - 1\<close>
-      have "- 1 \<le> real x" by auto
+      have "- 1 \<le> real_of_float x" by auto
       hence "exp (- 1) \<le> exp x"
         unfolding exp_le_cancel_iff .
       from order_trans[OF exp_m1_ge_quarter this] have "Float 1 (- 2) \<le> exp x"
         unfolding Float_num .
       with True show ?thesis
-        using bnds_exp_horner \<open>real x \<le> 0\<close> \<open>\<not> x > 0\<close> \<open>\<not> x < - 1\<close> by auto
+        using bnds_exp_horner \<open>real_of_float x \<le> 0\<close> \<open>\<not> x > 0\<close> \<open>\<not> x < - 1\<close> by auto
     next
       case False
       thus ?thesis
-        using bnds_exp_horner \<open>real x \<le> 0\<close> \<open>\<not> x > 0\<close> \<open>\<not> x < - 1\<close> by (auto simp add: Let_def)
+        using bnds_exp_horner \<open>real_of_float x \<le> 0\<close> \<open>\<not> x > 0\<close> \<open>\<not> x < - 1\<close> by (auto simp add: Let_def)
     qed
   next
     case True
     let ?num = "nat (- int_floor_fl x)"
 
-    have "real (int_floor_fl x) < - 1"
+    have "real_of_int (int_floor_fl x) < - 1"
       using int_floor_fl[of x] \<open>x < - 1\<close> by simp
-    hence "real (int_floor_fl x) < 0" by simp
+    hence "real_of_int (int_floor_fl x) < 0" by simp
     hence "int_floor_fl x < 0" by auto
     hence "1 \<le> - int_floor_fl x" by auto
     hence "0 < nat (- int_floor_fl x)" by auto
@@ -1921,19 +1911,19 @@
     have num_eq: "real ?num = - int_floor_fl x"
       using \<open>0 < nat (- int_floor_fl x)\<close> by auto
     have "0 < - int_floor_fl x"
-      using \<open>0 < ?num\<close>[unfolded real_of_nat_less_iff[symmetric]] by simp
-    hence "real (int_floor_fl x) < 0"
+      using \<open>0 < ?num\<close>[unfolded of_nat_less_iff[symmetric]] by simp
+    hence "real_of_int (int_floor_fl x) < 0"
       unfolding less_float_def by auto
-    have fl_eq: "real (- int_floor_fl x) = real (- floor_fl x)"
+    have fl_eq: "real_of_int (- int_floor_fl x) = real_of_float (- floor_fl x)"
       by (simp add: floor_fl_def int_floor_fl_def)
-    from \<open>0 < - int_floor_fl x\<close> have "0 \<le> real (- floor_fl x)"
+    from \<open>0 < - int_floor_fl x\<close> have "0 \<le> real_of_float (- floor_fl x)"
       by (simp add: floor_fl_def int_floor_fl_def)
-    from \<open>real (int_floor_fl x) < 0\<close> have "real (floor_fl x) < 0"
+    from \<open>real_of_int (int_floor_fl x) < 0\<close> have "real_of_float (floor_fl x) < 0"
       by (simp add: floor_fl_def int_floor_fl_def)
     have "exp x \<le> ub_exp prec x"
     proof -
-      have div_less_zero: "real (float_divr prec x (- floor_fl x)) \<le> 0"
-        using float_divr_nonpos_pos_upper_bound[OF \<open>real x \<le> 0\<close> \<open>0 \<le> real (- floor_fl x)\<close>]
+      have div_less_zero: "real_of_float (float_divr prec x (- floor_fl x)) \<le> 0"
+        using float_divr_nonpos_pos_upper_bound[OF \<open>real_of_float x \<le> 0\<close> \<open>0 \<le> real_of_float (- floor_fl x)\<close>]
         unfolding less_eq_float_def zero_float.rep_eq .
 
       have "exp x = exp (?num * (x / ?num))"
@@ -1946,7 +1936,7 @@
       also have "\<dots> \<le> (?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num"
         unfolding real_of_float_power
         by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
-      also have "\<dots> \<le> real (power_up_fl prec (?ub_exp_horner (float_divr prec x (- floor_fl x))) ?num)"
+      also have "\<dots> \<le> real_of_float (power_up_fl prec (?ub_exp_horner (float_divr prec x (- floor_fl x))) ?num)"
         by (auto simp add: real_power_up_fl intro!: power_up ub_exp_horner_nonneg div_less_zero)
       finally show ?thesis
         unfolding ub_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>] floor_fl_def Let_def .
@@ -1960,15 +1950,15 @@
       show ?thesis
       proof (cases "?horner \<le> 0")
         case False
-        hence "0 \<le> real ?horner" by auto
-
-        have div_less_zero: "real (float_divl prec x (- floor_fl x)) \<le> 0"
-          using \<open>real (floor_fl x) < 0\<close> \<open>real x \<le> 0\<close>
+        hence "0 \<le> real_of_float ?horner" by auto
+
+        have div_less_zero: "real_of_float (float_divl prec x (- floor_fl x)) \<le> 0"
+          using \<open>real_of_float (floor_fl x) < 0\<close> \<open>real_of_float x \<le> 0\<close>
           by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)
 
         have "(?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num \<le>
           exp (float_divl prec x (- floor_fl x)) ^ ?num"
-          using \<open>0 \<le> real ?horner\<close>[unfolded floor_fl_def[symmetric]]
+          using \<open>0 \<le> real_of_float ?horner\<close>[unfolded floor_fl_def[symmetric]]
             bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1]
           by (auto intro!: power_mono)
         also have "\<dots> \<le> exp (x / ?num) ^ ?num"
@@ -1988,22 +1978,22 @@
         have "power_down_fl prec (Float 1 (- 2))  ?num \<le> (Float 1 (- 2)) ^ ?num"
           by (metis Float_le_zero_iff less_imp_le linorder_not_less
             not_numeral_le_zero numeral_One power_down_fl)
-        then have "power_down_fl prec (Float 1 (- 2))  ?num \<le> real (Float 1 (- 2)) ^ ?num"
+        then have "power_down_fl prec (Float 1 (- 2))  ?num \<le> real_of_float (Float 1 (- 2)) ^ ?num"
           by simp
         also
-        have "real (floor_fl x) \<noteq> 0" and "real (floor_fl x) \<le> 0"
-          using \<open>real (floor_fl x) < 0\<close> by auto
-        from divide_right_mono_neg[OF floor_fl[of x] \<open>real (floor_fl x) \<le> 0\<close>, unfolded divide_self[OF \<open>real (floor_fl x) \<noteq> 0\<close>]]
+        have "real_of_float (floor_fl x) \<noteq> 0" and "real_of_float (floor_fl x) \<le> 0"
+          using \<open>real_of_float (floor_fl x) < 0\<close> by auto
+        from divide_right_mono_neg[OF floor_fl[of x] \<open>real_of_float (floor_fl x) \<le> 0\<close>, unfolded divide_self[OF \<open>real_of_float (floor_fl x) \<noteq> 0\<close>]]
         have "- 1 \<le> x / (- floor_fl x)"
           unfolding minus_float.rep_eq by auto
         from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
         have "Float 1 (- 2) \<le> exp (x / (- floor_fl x))"
           unfolding Float_num .
-        hence "real (Float 1 (- 2)) ^ ?num \<le> exp (x / (- floor_fl x)) ^ ?num"
+        hence "real_of_float (Float 1 (- 2)) ^ ?num \<le> exp (x / (- floor_fl x)) ^ ?num"
           by (metis Float_num(5) power_mono zero_le_divide_1_iff zero_le_numeral)
         also have "\<dots> = exp x"
           unfolding num_eq fl_eq exp_real_of_nat_mult[symmetric]
-          using \<open>real (floor_fl x) \<noteq> 0\<close> by auto
+          using \<open>real_of_float (floor_fl x) \<noteq> 0\<close> by auto
         finally show ?thesis
           unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>]
             int_floor_fl_def Let_def if_P[OF True] real_of_float_power .
@@ -2027,7 +2017,7 @@
     have "lb_exp prec x \<le> exp x"
     proof -
       from exp_boundaries'[OF \<open>-x \<le> 0\<close>]
-      have ub_exp: "exp (- real x) \<le> ub_exp prec (-x)"
+      have ub_exp: "exp (- real_of_float x) \<le> ub_exp prec (-x)"
         unfolding atLeastAtMost_iff minus_float.rep_eq by auto
 
       have "float_divl prec 1 (ub_exp prec (-x)) \<le> 1 / ub_exp prec (-x)"
@@ -2046,7 +2036,7 @@
       have "\<not> 0 < -x" using \<open>0 < x\<close> by auto
 
       from exp_boundaries'[OF \<open>-x \<le> 0\<close>]
-      have lb_exp: "lb_exp prec (-x) \<le> exp (- real x)"
+      have lb_exp: "lb_exp prec (-x) \<le> exp (- real_of_float x)"
         unfolding atLeastAtMost_iff minus_float.rep_eq by auto
 
       have "exp x \<le> (1 :: float) / lb_exp prec (-x)"
@@ -2133,33 +2123,37 @@
 qed
 
 lemma ln_float_bounds:
-  assumes "0 \<le> real x"
-    and "real x < 1"
+  assumes "0 \<le> real_of_float x"
+    and "real_of_float x < 1"
   shows "x * lb_ln_horner prec (get_even n) 1 x \<le> ln (x + 1)" (is "?lb \<le> ?ln")
     and "ln (x + 1) \<le> x * ub_ln_horner prec (get_odd n) 1 x" (is "?ln \<le> ?ub")
 proof -
   obtain ev where ev: "get_even n = 2 * ev" using get_even_double ..
   obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double ..
 
-  let "?s n" = "(- 1) ^ n * (1 / real (1 + n)) * (real x)^(Suc n)"
+  let "?s n" = "(- 1) ^ n * (1 / real (1 + n)) * (real_of_float x)^(Suc n)"
 
   have "?lb \<le> setsum ?s {0 ..< 2 * ev}"
     unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_left_distrib[symmetric]
-    unfolding mult.commute[of "real x"] ev
-    using horner_bounds(1)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*ev",
-      OF \<open>0 \<le> real x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real x\<close>
+    unfolding mult.commute[of "real_of_float x"] ev 
+    using horner_bounds(1)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" 
+                    and lb="\<lambda>n i k x. lb_ln_horner prec n k x" 
+                    and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*ev",
+      OF \<open>0 \<le> real_of_float x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real_of_float x\<close>
+    unfolding real_of_float_power
     by (rule mult_right_mono)
   also have "\<dots> \<le> ?ln"
-    using ln_bounds(1)[OF \<open>0 \<le> real x\<close> \<open>real x < 1\<close>] by auto
+    using ln_bounds(1)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto
   finally show "?lb \<le> ?ln" .
 
   have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}"
-    using ln_bounds(2)[OF \<open>0 \<le> real x\<close> \<open>real x < 1\<close>] by auto
+    using ln_bounds(2)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto
   also have "\<dots> \<le> ?ub"
     unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_left_distrib[symmetric]
-    unfolding mult.commute[of "real x"] od
+    unfolding mult.commute[of "real_of_float x"] od
     using horner_bounds(2)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*od+1",
-      OF \<open>0 \<le> real x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real x\<close>
+      OF \<open>0 \<le> real_of_float x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real_of_float x\<close>
+    unfolding real_of_float_power
     by (rule mult_right_mono)
   finally show "?ln \<le> ?ub" .
 qed
@@ -2201,26 +2195,26 @@
   have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1::real)"
     using ln_add[of "3 / 2" "1 / 2"] by auto
   have lb3: "?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
-  hence lb3_ub: "real ?lthird < 1" by auto
-  have lb3_lb: "0 \<le> real ?lthird" using lapprox_rat_nonneg[of 1 3] by auto
+  hence lb3_ub: "real_of_float ?lthird < 1" by auto
+  have lb3_lb: "0 \<le> real_of_float ?lthird" using lapprox_rat_nonneg[of 1 3] by auto
   have ub3: "1 / 3 \<le> ?uthird" using rapprox_rat[of 1 3] by auto
-  hence ub3_lb: "0 \<le> real ?uthird" by auto
-
-  have lb2: "0 \<le> real (Float 1 (- 1))" and ub2: "real (Float 1 (- 1)) < 1"
+  hence ub3_lb: "0 \<le> real_of_float ?uthird" by auto
+
+  have lb2: "0 \<le> real_of_float (Float 1 (- 1))" and ub2: "real_of_float (Float 1 (- 1)) < 1"
     unfolding Float_num by auto
 
   have "0 \<le> (1::int)" and "0 < (3::int)" by auto
-  have ub3_ub: "real ?uthird < 1"
+  have ub3_ub: "real_of_float ?uthird < 1"
     by (simp add: Float.compute_rapprox_rat Float.compute_lapprox_rat rapprox_posrat_less1)
 
   have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto
-  have uthird_gt0: "0 < real ?uthird + 1" using ub3_lb by auto
-  have lthird_gt0: "0 < real ?lthird + 1" using lb3_lb by auto
+  have uthird_gt0: "0 < real_of_float ?uthird + 1" using ub3_lb by auto
+  have lthird_gt0: "0 < real_of_float ?lthird + 1" using lb3_lb by auto
 
   show ?ub_ln2
     unfolding ub_ln2_def Let_def ln2_sum Float_num(4)[symmetric]
   proof (rule float_plus_up_le, rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
-    have "ln (1 / 3 + 1) \<le> ln (real ?uthird + 1)"
+    have "ln (1 / 3 + 1) \<le> ln (real_of_float ?uthird + 1)"
       unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto
     also have "\<dots> \<le> ?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird"
       using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
@@ -2230,7 +2224,7 @@
   show ?lb_ln2
     unfolding lb_ln2_def Let_def ln2_sum Float_num(4)[symmetric]
   proof (rule float_plus_down_le, rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
-    have "?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird \<le> ln (real ?lthird + 1)"
+    have "?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird \<le> ln (real_of_float ?lthird + 1)"
       using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
     note float_round_down_le[OF this]
     also have "\<dots> \<le> ln (1 / 3 + 1)"
@@ -2265,18 +2259,18 @@
 termination
 proof (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 1 then 1 else 0))", auto)
   fix prec and x :: float
-  assume "\<not> real x \<le> 0" and "real x < 1" and "real (float_divl (max prec (Suc 0)) 1 x) < 1"
-  hence "0 < real x" "1 \<le> max prec (Suc 0)" "real x < 1"
+  assume "\<not> real_of_float x \<le> 0" and "real_of_float x < 1" and "real_of_float (float_divl (max prec (Suc 0)) 1 x) < 1"
+  hence "0 < real_of_float x" "1 \<le> max prec (Suc 0)" "real_of_float x < 1"
     by auto
-  from float_divl_pos_less1_bound[OF \<open>0 < real x\<close> \<open>real x < 1\<close>[THEN less_imp_le] \<open>1 \<le> max prec (Suc 0)\<close>]
+  from float_divl_pos_less1_bound[OF \<open>0 < real_of_float x\<close> \<open>real_of_float x < 1\<close>[THEN less_imp_le] \<open>1 \<le> max prec (Suc 0)\<close>]
   show False
-    using \<open>real (float_divl (max prec (Suc 0)) 1 x) < 1\<close> by auto
+    using \<open>real_of_float (float_divl (max prec (Suc 0)) 1 x) < 1\<close> by auto
 next
   fix prec x
-  assume "\<not> real x \<le> 0" and "real x < 1" and "real (float_divr prec 1 x) < 1"
+  assume "\<not> real_of_float x \<le> 0" and "real_of_float x < 1" and "real_of_float (float_divr prec 1 x) < 1"
   hence "0 < x" by auto
-  from float_divr_pos_less1_lower_bound[OF \<open>0 < x\<close>, of prec] \<open>real x < 1\<close> show False
-    using \<open>real (float_divr prec 1 x) < 1\<close> by auto
+  from float_divr_pos_less1_lower_bound[OF \<open>0 < x\<close>, of prec] \<open>real_of_float x < 1\<close> show False
+    using \<open>real_of_float (float_divr prec 1 x) < 1\<close> by auto
 qed
 
 lemma float_pos_eq_mantissa_pos: "x > 0 \<longleftrightarrow> mantissa x > 0"
@@ -2305,11 +2299,11 @@
     unfolding zero_float_def[symmetric] using \<open>0 < x\<close> by auto
   from denormalize_shift[OF assms(1) this] guess i . note i = this
 
-  have "2 powr (1 - (real (bitlen (mantissa x)) + real i)) =
-    2 powr (1 - (real (bitlen (mantissa x)))) * inverse (2 powr (real i))"
+  have "2 powr (1 - (real_of_int (bitlen (mantissa x)) + real_of_int i)) =
+    2 powr (1 - (real_of_int (bitlen (mantissa x)))) * inverse (2 powr (real i))"
     by (simp add: powr_minus[symmetric] powr_add[symmetric] field_simps)
-  hence "real (mantissa x) * 2 powr (1 - real (bitlen (mantissa x))) =
-    (real (mantissa x) * 2 ^ i) * 2 powr (1 - real (bitlen (mantissa x * 2 ^ i)))"
+  hence "real_of_int (mantissa x) * 2 powr (1 - real_of_int (bitlen (mantissa x))) =
+    (real_of_int (mantissa x) * 2 ^ i) * 2 powr (1 - real_of_int (bitlen (mantissa x * 2 ^ i)))"
     using \<open>mantissa x > 0\<close> by (simp add: powr_realpow)
   then show ?th2
     unfolding i by transfer auto
@@ -2350,14 +2344,14 @@
 proof -
   let ?B = "2^nat (bitlen m - 1)"
   def bl \<equiv> "bitlen m - 1"
-  have "0 < real m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0"
+  have "0 < real_of_int m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0"
     using assms by auto
   hence "0 \<le> bl" by (simp add: bitlen_def bl_def)
   show ?thesis
   proof (cases "0 \<le> e")
     case True
     thus ?thesis
-      unfolding bl_def[symmetric] using \<open>0 < real m\<close> \<open>0 \<le> bl\<close>
+      unfolding bl_def[symmetric] using \<open>0 < real_of_int m\<close> \<open>0 \<le> bl\<close>
       apply (simp add: ln_mult)
       apply (cases "e=0")
         apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr)
@@ -2366,7 +2360,7 @@
   next
     case False
     hence "0 < -e" by auto
-    have lne: "ln (2 powr real e) = ln (inverse (2 powr - e))"
+    have lne: "ln (2 powr real_of_int e) = ln (inverse (2 powr - e))"
       by (simp add: powr_minus)
     hence pow_gt0: "(0::real) < 2^nat (-e)"
       by auto
@@ -2374,7 +2368,7 @@
       by auto
     show ?thesis
       using False unfolding bl_def[symmetric]
-      using \<open>0 < real m\<close> \<open>0 \<le> bl\<close>
+      using \<open>0 < real_of_int m\<close> \<open>0 \<le> bl\<close>
       by (auto simp add: lne ln_mult ln_powr ln_div field_simps)
   qed
 qed
@@ -2385,9 +2379,9 @@
     (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
 proof (cases "x < Float 1 1")
   case True
-  hence "real (x - 1) < 1" and "real x < 2" by auto
+  hence "real_of_float (x - 1) < 1" and "real_of_float x < 2" by auto
   have "\<not> x \<le> 0" and "\<not> x < 1" using \<open>1 \<le> x\<close> by auto
-  hence "0 \<le> real (x - 1)" using \<open>1 \<le> x\<close> by auto
+  hence "0 \<le> real_of_float (x - 1)" using \<open>1 \<le> x\<close> by auto
 
   have [simp]: "(Float 3 (- 1)) = 3 / 2" by simp
 
@@ -2397,7 +2391,7 @@
     show ?thesis
       unfolding lb_ln.simps
       unfolding ub_ln.simps Let_def
-      using ln_float_bounds[OF \<open>0 \<le> real (x - 1)\<close> \<open>real (x - 1) < 1\<close>, of prec]
+      using ln_float_bounds[OF \<open>0 \<le> real_of_float (x - 1)\<close> \<open>real_of_float (x - 1) < 1\<close>, of prec]
         \<open>\<not> x \<le> 0\<close> \<open>\<not> x < 1\<close> True
       by (auto intro!: float_round_down_le float_round_up_le)
   next
@@ -2405,32 +2399,32 @@
     hence *: "3 / 2 < x" by auto
 
     with ln_add[of "3 / 2" "x - 3 / 2"]
-    have add: "ln x = ln (3 / 2) + ln (real x * 2 / 3)"
+    have add: "ln x = ln (3 / 2) + ln (real_of_float x * 2 / 3)"
       by (auto simp add: algebra_simps diff_divide_distrib)
 
     let "?ub_horner x" = "float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x)"
     let "?lb_horner x" = "float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x)"
 
-    { have up: "real (rapprox_rat prec 2 3) \<le> 1"
+    { have up: "real_of_float (rapprox_rat prec 2 3) \<le> 1"
         by (rule rapprox_rat_le1) simp_all
       have low: "2 / 3 \<le> rapprox_rat prec 2 3"
         by (rule order_trans[OF _ rapprox_rat]) simp
       from mult_less_le_imp_less[OF * low] *
-      have pos: "0 < real (x * rapprox_rat prec 2 3 - 1)" by auto
-
-      have "ln (real x * 2/3)
-        \<le> ln (real (x * rapprox_rat prec 2 3 - 1) + 1)"
+      have pos: "0 < real_of_float (x * rapprox_rat prec 2 3 - 1)" by auto
+
+      have "ln (real_of_float x * 2/3)
+        \<le> ln (real_of_float (x * rapprox_rat prec 2 3 - 1) + 1)"
       proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
-        show "real x * 2 / 3 \<le> real (x * rapprox_rat prec 2 3 - 1) + 1"
+        show "real_of_float x * 2 / 3 \<le> real_of_float (x * rapprox_rat prec 2 3 - 1) + 1"
           using * low by auto
-        show "0 < real x * 2 / 3" using * by simp
-        show "0 < real (x * rapprox_rat prec 2 3 - 1) + 1" using pos by auto
+        show "0 < real_of_float x * 2 / 3" using * by simp
+        show "0 < real_of_float (x * rapprox_rat prec 2 3 - 1) + 1" using pos by auto
       qed
       also have "\<dots> \<le> ?ub_horner (x * rapprox_rat prec 2 3 - 1)"
       proof (rule float_round_up_le, rule ln_float_bounds(2))
-        from mult_less_le_imp_less[OF \<open>real x < 2\<close> up] low *
-        show "real (x * rapprox_rat prec 2 3 - 1) < 1" by auto
-        show "0 \<le> real (x * rapprox_rat prec 2 3 - 1)" using pos by auto
+        from mult_less_le_imp_less[OF \<open>real_of_float x < 2\<close> up] low *
+        show "real_of_float (x * rapprox_rat prec 2 3 - 1) < 1" by auto
+        show "0 \<le> real_of_float (x * rapprox_rat prec 2 3 - 1)" using pos by auto
       qed
      finally have "ln x \<le> ?ub_horner (Float 1 (-1))
           + ?ub_horner ((x * rapprox_rat prec 2 3 - 1))"
@@ -2444,23 +2438,23 @@
       have up: "lapprox_rat prec 2 3 \<le> 2/3"
         by (rule order_trans[OF lapprox_rat], simp)
 
-      have low: "0 \<le> real (lapprox_rat prec 2 3)"
+      have low: "0 \<le> real_of_float (lapprox_rat prec 2 3)"
         using lapprox_rat_nonneg[of 2 3 prec] by simp
 
       have "?lb_horner ?max
-        \<le> ln (real ?max + 1)"
+        \<le> ln (real_of_float ?max + 1)"
       proof (rule float_round_down_le, rule ln_float_bounds(1))
-        from mult_less_le_imp_less[OF \<open>real x < 2\<close> up] * low
-        show "real ?max < 1" by (cases "real (lapprox_rat prec 2 3) = 0",
+        from mult_less_le_imp_less[OF \<open>real_of_float x < 2\<close> up] * low
+        show "real_of_float ?max < 1" by (cases "real_of_float (lapprox_rat prec 2 3) = 0",
           auto simp add: real_of_float_max)
-        show "0 \<le> real ?max" by (auto simp add: real_of_float_max)
+        show "0 \<le> real_of_float ?max" by (auto simp add: real_of_float_max)
       qed
-      also have "\<dots> \<le> ln (real x * 2/3)"
+      also have "\<dots> \<le> ln (real_of_float x * 2/3)"
       proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
-        show "0 < real ?max + 1" by (auto simp add: real_of_float_max)
-        show "0 < real x * 2/3" using * by auto
-        show "real ?max + 1 \<le> real x * 2/3" using * up
-          by (cases "0 < real x * real (lapprox_posrat prec 2 3) - 1",
+        show "0 < real_of_float ?max + 1" by (auto simp add: real_of_float_max)
+        show "0 < real_of_float x * 2/3" using * by auto
+        show "real_of_float ?max + 1 \<le> real_of_float x * 2/3" using * up
+          by (cases "0 < real_of_float x * real_of_float (lapprox_posrat prec 2 3) - 1",
               auto simp add: max_def)
       qed
       finally have "?lb_horner (Float 1 (- 1)) + ?lb_horner ?max \<le> ln x"
@@ -2495,24 +2489,24 @@
     have "1 \<le> Float m e"
       using \<open>1 \<le> x\<close> Float unfolding less_eq_float_def by auto
     from bitlen_div[OF \<open>0 < m\<close>] float_gt1_scale[OF \<open>1 \<le> Float m e\<close>] \<open>bl \<ge> 0\<close>
-    have x_bnds: "0 \<le> real (?x - 1)" "real (?x - 1) < 1"
+    have x_bnds: "0 \<le> real_of_float (?x - 1)" "real_of_float (?x - 1) < 1"
       unfolding bl_def[symmetric]
       by (auto simp: powr_realpow[symmetric] field_simps inverse_eq_divide)
          (auto simp : powr_minus field_simps inverse_eq_divide)
 
     {
       have "float_round_down prec (lb_ln2 prec * ?s) \<le> ln 2 * (e + (bitlen m - 1))"
-          (is "real ?lb2 \<le> _")
+          (is "real_of_float ?lb2 \<le> _")
         apply (rule float_round_down_le)
         unfolding nat_0 power_0 mult_1_right times_float.rep_eq
         using lb_ln2[of prec]
       proof (rule mult_mono)
         from float_gt1_scale[OF \<open>1 \<le> Float m e\<close>]
-        show "0 \<le> real (Float (e + (bitlen m - 1)) 0)" by simp
+        show "0 \<le> real_of_float (Float (e + (bitlen m - 1)) 0)" by simp
       qed auto
       moreover
       from ln_float_bounds(1)[OF x_bnds]
-      have "float_round_down prec ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln ?x" (is "real ?lb_horner \<le> _")
+      have "float_round_down prec ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln ?x" (is "real_of_float ?lb_horner \<le> _")
         by (auto intro!: float_round_down_le)
       ultimately have "float_plus_down prec ?lb2 ?lb_horner \<le> ln x"
         unfolding Float ln_shifted_float[OF \<open>0 < m\<close>, of e] by (auto intro!: float_plus_down_le)
@@ -2521,19 +2515,19 @@
     {
       from ln_float_bounds(2)[OF x_bnds]
       have "ln ?x \<le> float_round_up prec ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))"
-          (is "_ \<le> real ?ub_horner")
+          (is "_ \<le> real_of_float ?ub_horner")
         by (auto intro!: float_round_up_le)
       moreover
       have "ln 2 * (e + (bitlen m - 1)) \<le> float_round_up prec (ub_ln2 prec * ?s)"
-          (is "_ \<le> real ?ub2")
+          (is "_ \<le> real_of_float ?ub2")
         apply (rule float_round_up_le)
         unfolding nat_0 power_0 mult_1_right times_float.rep_eq
         using ub_ln2[of prec]
       proof (rule mult_mono)
         from float_gt1_scale[OF \<open>1 \<le> Float m e\<close>]
-        show "0 \<le> real (e + (bitlen m - 1))" by auto
+        show "0 \<le> real_of_int (e + (bitlen m - 1))" by auto
         have "0 \<le> ln (2 :: real)" by simp
-        thus "0 \<le> real (ub_ln2 prec)" using ub_ln2[of prec] by arith
+        thus "0 \<le> real_of_float (ub_ln2 prec)" using ub_ln2[of prec] by arith
       qed auto
       ultimately have "ln x \<le> float_plus_up prec ?ub2 ?ub_horner"
         unfolding Float ln_shifted_float[OF \<open>0 < m\<close>, of e]
@@ -2562,29 +2556,29 @@
 next
   case True
   have "\<not> x \<le> 0" using \<open>0 < x\<close> by auto
-  from True have "real x \<le> 1" "x \<le> 1"
+  from True have "real_of_float x \<le> 1" "x \<le> 1"
     by simp_all
-  have "0 < real x" and "real x \<noteq> 0"
+  have "0 < real_of_float x" and "real_of_float x \<noteq> 0"
     using \<open>0 < x\<close> by auto
-  hence A: "0 < 1 / real x" by auto
+  hence A: "0 < 1 / real_of_float x" by auto
 
   {
     let ?divl = "float_divl (max prec 1) 1 x"
-    have A': "1 \<le> ?divl" using float_divl_pos_less1_bound[OF \<open>0 < real x\<close> \<open>real x \<le> 1\<close>] by auto
-    hence B: "0 < real ?divl" by auto
+    have A': "1 \<le> ?divl" using float_divl_pos_less1_bound[OF \<open>0 < real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>] by auto
+    hence B: "0 < real_of_float ?divl" by auto
 
     have "ln ?divl \<le> ln (1 / x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto
-    hence "ln x \<le> - ln ?divl" unfolding nonzero_inverse_eq_divide[OF \<open>real x \<noteq> 0\<close>, symmetric] ln_inverse[OF \<open>0 < real x\<close>] by auto
+    hence "ln x \<le> - ln ?divl" unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float x \<noteq> 0\<close>, symmetric] ln_inverse[OF \<open>0 < real_of_float x\<close>] by auto
     from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le]
     have "?ln \<le> - the (lb_ln prec ?divl)" unfolding uminus_float.rep_eq by (rule order_trans)
   } moreover
   {
     let ?divr = "float_divr prec 1 x"
     have A': "1 \<le> ?divr" using float_divr_pos_less1_lower_bound[OF \<open>0 < x\<close> \<open>x \<le> 1\<close>] unfolding less_eq_float_def less_float_def by auto
-    hence B: "0 < real ?divr" by auto
+    hence B: "0 < real_of_float ?divr" by auto
 
     have "ln (1 / x) \<le> ln ?divr" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto
-    hence "- ln ?divr \<le> ln x" unfolding nonzero_inverse_eq_divide[OF \<open>real x \<noteq> 0\<close>, symmetric] ln_inverse[OF \<open>0 < real x\<close>] by auto
+    hence "- ln ?divr \<le> ln x" unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float x \<noteq> 0\<close>, symmetric] ln_inverse[OF \<open>0 < real_of_float x\<close>] by auto
     from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this
     have "- the (ub_ln prec ?divr) \<le> ?ln" unfolding uminus_float.rep_eq by (rule order_trans)
   }
@@ -2594,7 +2588,7 @@
 
 lemma lb_ln:
   assumes "Some y = lb_ln prec x"
-  shows "y \<le> ln x" and "0 < real x"
+  shows "y \<le> ln x" and "0 < real_of_float x"
 proof -
   have "0 < x"
   proof (rule ccontr)
@@ -2604,7 +2598,7 @@
     thus False
       using assms by auto
   qed
-  thus "0 < real x" by auto
+  thus "0 < real_of_float x" by auto
   have "the (lb_ln prec x) \<le> ln x"
     using ub_ln_lb_ln_bounds[OF \<open>0 < x\<close>] ..
   thus "y \<le> ln x"
@@ -2613,7 +2607,7 @@
 
 lemma ub_ln:
   assumes "Some y = ub_ln prec x"
-  shows "ln x \<le> y" and "0 < real x"
+  shows "ln x \<le> y" and "0 < real_of_float x"
 proof -
   have "0 < x"
   proof (rule ccontr)
@@ -2622,7 +2616,7 @@
     thus False
       using assms by auto
   qed
-  thus "0 < real x" by auto
+  thus "0 < real_of_float x" by auto
   have "ln x \<le> the (ub_ln prec x)"
     using ub_ln_lb_ln_bounds[OF \<open>0 < x\<close>] ..
   thus "ln x \<le> y"
@@ -2638,16 +2632,16 @@
   hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {lx .. ux}"
     by auto
 
-  have "ln ux \<le> u" and "0 < real ux"
+  have "ln ux \<le> u" and "0 < real_of_float ux"
     using ub_ln u by auto
-  have "l \<le> ln lx" and "0 < real lx" and "0 < x"
+  have "l \<le> ln lx" and "0 < real_of_float lx" and "0 < x"
     using lb_ln[OF l] x by auto
 
-  from ln_le_cancel_iff[OF \<open>0 < real lx\<close> \<open>0 < x\<close>] \<open>l \<le> ln lx\<close>
+  from ln_le_cancel_iff[OF \<open>0 < real_of_float lx\<close> \<open>0 < x\<close>] \<open>l \<le> ln lx\<close>
   have "l \<le> ln x"
     using x unfolding atLeastAtMost_iff by auto
   moreover
-  from ln_le_cancel_iff[OF \<open>0 < x\<close> \<open>0 < real ux\<close>] \<open>ln ux \<le> real u\<close>
+  from ln_le_cancel_iff[OF \<open>0 < x\<close> \<open>0 < real_of_float ux\<close>] \<open>ln ux \<le> real_of_float u\<close>
   have "ln x \<le> u"
     using x unfolding atLeastAtMost_iff by auto
   ultimately show "l \<le> ln x \<and> ln x \<le> u" ..
@@ -2746,19 +2740,20 @@
 "lift_un' (Some (l1, u1)) f = Some (f l1 u1)" |
 "lift_un' b f = None"
 
-definition "bounded_by xs vs \<longleftrightarrow>
+definition bounded_by :: "real list \<Rightarrow> (float \<times> float) option list \<Rightarrow> bool" where 
+  "bounded_by xs vs \<longleftrightarrow>
   (\<forall> i < length vs. case vs ! i of None \<Rightarrow> True
-         | Some (l, u) \<Rightarrow> xs ! i \<in> { real l .. real u })"
-
+         | Some (l, u) \<Rightarrow> xs ! i \<in> { real_of_float l .. real_of_float u })"
+                                                                     
 lemma bounded_byE:
   assumes "bounded_by xs vs"
   shows "\<And> i. i < length vs \<Longrightarrow> case vs ! i of None \<Rightarrow> True
-         | Some (l, u) \<Rightarrow> xs ! i \<in> { real l .. real u }"
+         | Some (l, u) \<Rightarrow> xs ! i \<in> { real_of_float l .. real_of_float u }"
   using assms bounded_by_def by blast
 
 lemma bounded_by_update:
   assumes "bounded_by xs vs"
-    and bnd: "xs ! i \<in> { real l .. real u }"
+    and bnd: "xs ! i \<in> { real_of_float l .. real_of_float u }"
   shows "bounded_by xs (vs[i := Some (l,u)])"
 proof -
   {
@@ -2766,7 +2761,7 @@
     let ?vs = "vs[i := Some (l,u)]"
     assume "j < length ?vs"
     hence [simp]: "j < length vs" by simp
-    have "case ?vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> xs ! j \<in> { real l .. real u }"
+    have "case ?vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> xs ! j \<in> { real_of_float l .. real_of_float u }"
     proof (cases "?vs ! j")
       case (Some b)
       thus ?thesis
@@ -2949,7 +2944,7 @@
     and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
     and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow>
       l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u"
-  shows "real l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real u"
+  shows "real_of_float l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real_of_float u"
 proof -
   from lift_un'[OF lift_un'_Some Pa]
   obtain l1 u1 where "l1 \<le> interpret_floatarith a xs"
@@ -3039,7 +3034,7 @@
     and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
     and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow>
       l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u"
-  shows "real l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real u"
+  shows "real_of_float l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real_of_float u"
 proof -
   from lift_un[OF lift_un_Some Pa]
   obtain l1 u1 where "l1 \<le> interpret_floatarith a xs"
@@ -3109,37 +3104,37 @@
     show False
       using l' unfolding if_not_P[OF P] by auto
   qed
-  moreover have l1_le_u1: "real l1 \<le> real u1"
+  moreover have l1_le_u1: "real_of_float l1 \<le> real_of_float u1"
     using l1 u1 by auto
-  ultimately have "real l1 \<noteq> 0" and "real u1 \<noteq> 0"
+  ultimately have "real_of_float l1 \<noteq> 0" and "real_of_float u1 \<noteq> 0"
     by auto
 
   have inv: "inverse u1 \<le> inverse (interpret_floatarith a xs)
            \<and> inverse (interpret_floatarith a xs) \<le> inverse l1"
   proof (cases "0 < l1")
     case True
-    hence "0 < real u1" and "0 < real l1" "0 < interpret_floatarith a xs"
+    hence "0 < real_of_float u1" and "0 < real_of_float l1" "0 < interpret_floatarith a xs"
       using l1_le_u1 l1 by auto
     show ?thesis
-      unfolding inverse_le_iff_le[OF \<open>0 < real u1\<close> \<open>0 < interpret_floatarith a xs\<close>]
-        inverse_le_iff_le[OF \<open>0 < interpret_floatarith a xs\<close> \<open>0 < real l1\<close>]
+      unfolding inverse_le_iff_le[OF \<open>0 < real_of_float u1\<close> \<open>0 < interpret_floatarith a xs\<close>]
+        inverse_le_iff_le[OF \<open>0 < interpret_floatarith a xs\<close> \<open>0 < real_of_float l1\<close>]
       using l1 u1 by auto
   next
     case False
     hence "u1 < 0"
       using either by blast
-    hence "real u1 < 0" and "real l1 < 0" "interpret_floatarith a xs < 0"
+    hence "real_of_float u1 < 0" and "real_of_float l1 < 0" "interpret_floatarith a xs < 0"
       using l1_le_u1 u1 by auto
     show ?thesis
-      unfolding inverse_le_iff_le_neg[OF \<open>real u1 < 0\<close> \<open>interpret_floatarith a xs < 0\<close>]
-        inverse_le_iff_le_neg[OF \<open>interpret_floatarith a xs < 0\<close> \<open>real l1 < 0\<close>]
+      unfolding inverse_le_iff_le_neg[OF \<open>real_of_float u1 < 0\<close> \<open>interpret_floatarith a xs < 0\<close>]
+        inverse_le_iff_le_neg[OF \<open>interpret_floatarith a xs < 0\<close> \<open>real_of_float l1 < 0\<close>]
       using l1 u1 by auto
   qed
 
   from l' have "l = float_divl prec 1 u1"
     by (cases "0 < l1 \<or> u1 < 0") auto
   hence "l \<le> inverse u1"
-    unfolding nonzero_inverse_eq_divide[OF \<open>real u1 \<noteq> 0\<close>]
+    unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float u1 \<noteq> 0\<close>]
     using float_divl[of prec 1 u1] by auto
   also have "\<dots> \<le> inverse (interpret_floatarith a xs)"
     using inv by auto
@@ -3148,7 +3143,7 @@
   from u' have "u = float_divr prec 1 l1"
     by (cases "0 < l1 \<or> u1 < 0") auto
   hence "inverse l1 \<le> u"
-    unfolding nonzero_inverse_eq_divide[OF \<open>real l1 \<noteq> 0\<close>]
+    unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float l1 \<noteq> 0\<close>]
     using float_divr[of 1 l1 prec] by auto
   hence "inverse (interpret_floatarith a xs) \<le> u"
     by (rule order_trans[OF inv[THEN conjunct2]])
@@ -3274,7 +3269,7 @@
   case (Suc s)
 
   let ?m = "(l + u) * Float 1 (- 1)"
-  have "real l \<le> ?m" and "?m \<le> real u"
+  have "real_of_float l \<le> ?m" and "?m \<le> real_of_float u"
     unfolding less_eq_float_def using Suc.prems by auto
 
   with \<open>x \<in> { l .. u }\<close>
@@ -3355,7 +3350,7 @@
   then obtain l u l' u'
     where l_eq: "Some (l, u) = approx prec a vs"
       and u_eq: "Some (l', u') = approx prec b vs"
-      and inequality: "real (float_plus_up prec u (-l')) < 0"
+      and inequality: "real_of_float (float_plus_up prec u (-l')) < 0"
     by (cases "approx prec a vs", auto, cases "approx prec b vs", auto)
   from le_less_trans[OF float_plus_up inequality]
     approx[OF Less.prems(2) l_eq] approx[OF Less.prems(2) u_eq]
@@ -3365,7 +3360,7 @@
   then obtain l u l' u'
     where l_eq: "Some (l, u) = approx prec a vs"
       and u_eq: "Some (l', u') = approx prec b vs"
-      and inequality: "real (float_plus_up prec u (-l')) \<le> 0"
+      and inequality: "real_of_float (float_plus_up prec u (-l')) \<le> 0"
     by (cases "approx prec a vs", auto, cases "approx prec b vs", auto)
   from order_trans[OF float_plus_up inequality]
     approx[OF LessEqual.prems(2) l_eq] approx[OF LessEqual.prems(2) u_eq]
@@ -3376,7 +3371,7 @@
     where x_eq: "Some (lx, ux) = approx prec x vs"
     and l_eq: "Some (l, u) = approx prec a vs"
     and u_eq: "Some (l', u') = approx prec b vs"
-    and inequality: "real (float_plus_up prec u (-lx)) \<le> 0" "real (float_plus_up prec ux (-l')) \<le> 0"
+    and inequality: "real_of_float (float_plus_up prec u (-lx)) \<le> 0" "real_of_float (float_plus_up prec ux (-l')) \<le> 0"
     by (cases "approx prec x vs", auto,
       cases "approx prec a vs", auto,
       cases "approx prec b vs", auto)
@@ -3452,7 +3447,7 @@
 next
   case (Power a n)
   thus ?case
-    by (cases n) (auto intro!: derivative_eq_intros simp del: power_Suc simp add: real_of_nat_def)
+    by (cases n) (auto intro!: derivative_eq_intros simp del: power_Suc)
 next
   case (Ln a)
   thus ?case by (auto intro!: derivative_eq_intros simp add: divide_inverse)
@@ -3522,7 +3517,7 @@
 lemma bounded_by_update_var:
   assumes "bounded_by xs vs"
     and "vs ! i = Some (l, u)"
-    and bnd: "x \<in> { real l .. real u }"
+    and bnd: "x \<in> { real_of_float l .. real_of_float u }"
   shows "bounded_by (xs[i := x]) vs"
 proof (cases "i < length xs")
   case False
@@ -3532,7 +3527,7 @@
   case True
   let ?xs = "xs[i := x]"
   from True have "i < length ?xs" by auto
-  have "case vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> ?xs ! j \<in> {real l .. real u}"
+  have "case vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> ?xs ! j \<in> {real_of_float l .. real_of_float u}"
     if "j < length vs" for j
   proof (cases "vs ! j")
     case None
@@ -3557,7 +3552,7 @@
 lemma isDERIV_approx':
   assumes "bounded_by xs vs"
     and vs_x: "vs ! x = Some (l, u)"
-    and X_in: "X \<in> {real l .. real u}"
+    and X_in: "X \<in> {real_of_float l .. real_of_float u}"
     and approx: "isDERIV_approx prec x f vs"
   shows "isDERIV x f (xs[x := X])"
 proof -
@@ -3612,10 +3607,10 @@
 
 lemma bounded_by_Cons:
   assumes bnd: "bounded_by xs vs"
-    and x: "x \<in> { real l .. real u }"
+    and x: "x \<in> { real_of_float l .. real_of_float u }"
   shows "bounded_by (x#xs) ((Some (l, u))#vs)"
 proof -
-  have "case ((Some (l,u))#vs) ! i of Some (l, u) \<Rightarrow> (x#xs)!i \<in> { real l .. real u } | None \<Rightarrow> True"
+  have "case ((Some (l,u))#vs) ! i of Some (l, u) \<Rightarrow> (x#xs)!i \<in> { real_of_float l .. real_of_float u } | None \<Rightarrow> True"
     if *: "i < length ((Some (l, u))#vs)" for i
   proof (cases i)
     case 0
@@ -3689,7 +3684,7 @@
 
     from approx[OF this a]
     have f_c: "interpret_floatarith ((DERIV_floatarith x ^^ 0) f) (xs[x := c]) \<in> { l1 .. u1 }"
-              (is "?f 0 (real c) \<in> _")
+              (is "?f 0 (real_of_float c) \<in> _")
       by auto
 
     have funpow_Suc[symmetric]: "(f ^^ Suc n) x = (f ^^ n) (f x)"
@@ -3698,7 +3693,7 @@
     from Suc.hyps[OF ate, unfolded this] obtain n
       where DERIV_hyp: "\<And>m z. \<lbrakk> m < n ; (z::real) \<in> { lx .. ux } \<rbrakk> \<Longrightarrow>
         DERIV (?f (Suc m)) z :> ?f (Suc (Suc m)) z"
-      and hyp: "\<forall>t \<in> {real lx .. real ux}.
+      and hyp: "\<forall>t \<in> {real_of_float lx .. real_of_float ux}.
         (\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {Suc k..<Suc k + i}. j)) * ?f (Suc i) c * (xs!x - c)^i) +
           inverse (real (\<Prod> j \<in> {Suc k..<Suc k + n}. j)) * ?f (Suc n) t * (xs!x - c)^n \<in> {l2 .. u2}"
           (is "\<forall> t \<in> _. ?X (Suc k) f n t \<in> _")
@@ -3737,9 +3732,9 @@
       have "bounded_by [?f 0 c, ?X (Suc k) f n t, xs!x] [Some (l1, u1), Some (l2, u2), vs!x]"
         by (auto intro!: bounded_by_Cons)
       from approx[OF this final, unfolded atLeastAtMost_iff[symmetric]]
-      have "?X (Suc k) f n t * (xs!x - real c) * inverse k + ?f 0 c \<in> {l .. u}"
+      have "?X (Suc k) f n t * (xs!x - real_of_float c) * inverse k + ?f 0 c \<in> {l .. u}"
         by (auto simp add: algebra_simps)
-      also have "?X (Suc k) f n t * (xs!x - real c) * inverse (real k) + ?f 0 c =
+      also have "?X (Suc k) f n t * (xs!x - real_of_float c) * inverse (real k) + ?f 0 c =
                (\<Sum> i = 0..<Suc n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) * ?f i c * (xs!x - c)^i) +
                inverse (real (\<Prod> j \<in> {k..<k+Suc n}. j)) * ?f (Suc n) t * (xs!x - c)^Suc n" (is "_ = ?T")
         unfolding funpow_Suc C_def[symmetric] setsum_move0 setprod_head_Suc
@@ -3752,13 +3747,12 @@
 qed
 
 lemma setprod_fact: "real (\<Prod> {1..<1 + k}) = fact (k :: nat)"
-  using fact_altdef_nat Suc_eq_plus1_left atLeastLessThanSuc_atLeastAtMost real_fact_nat
-  by presburger
+by (metis Suc_eq_plus1_left atLeastLessThanSuc_atLeastAtMost fact_altdef_nat of_nat_fact)
 
 lemma approx_tse:
   assumes "bounded_by xs vs"
     and bnd_x: "vs ! x = Some (lx, ux)"
-    and bnd_c: "real c \<in> {lx .. ux}"
+    and bnd_c: "real_of_float c \<in> {lx .. ux}"
     and "x < length vs" and "x < length xs"
     and ate: "Some (l, u) = approx_tse prec x s c 1 f vs"
   shows "interpret_floatarith f xs \<in> {l .. u}"
@@ -3772,7 +3766,7 @@
 
   from approx_tse_generic[OF \<open>bounded_by xs vs\<close> this bnd_x ate]
   obtain n
-    where DERIV: "\<forall> m z. m < n \<and> real lx \<le> z \<and> z \<le> real ux \<longrightarrow> DERIV (F m) z :> F (Suc m) z"
+    where DERIV: "\<forall> m z. m < n \<and> real_of_float lx \<le> z \<and> z \<le> real_of_float ux \<longrightarrow> DERIV (F m) z :> F (Suc m) z"
     and hyp: "\<And> (t::real). t \<in> {lx .. ux} \<Longrightarrow>
            (\<Sum> j = 0..<n. inverse(fact j) * F j c * (xs!x - c)^j) +
              inverse ((fact n)) * F n t * (xs!x - c)^n
@@ -3798,7 +3792,7 @@
         by auto
     next
       case False
-      have "lx \<le> real c" "real c \<le> ux" "lx \<le> xs!x" "xs!x \<le> ux"
+      have "lx \<le> real_of_float c" "real_of_float c \<le> ux" "lx \<le> xs!x" "xs!x \<le> ux"
         using Suc bnd_c \<open>bounded_by xs vs\<close>[THEN bounded_byE, OF \<open>x < length vs\<close>] bnd_x by auto
       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"
@@ -3833,7 +3827,7 @@
   fixes x :: real
   assumes "approx_tse_form' prec t f s l u cmp"
     and "x \<in> {l .. u}"
-  shows "\<exists>l' u' ly uy. x \<in> {l' .. u'} \<and> real l \<le> l' \<and> u' \<le> real u \<and> cmp ly uy \<and>
+  shows "\<exists>l' u' ly uy. x \<in> {l' .. u'} \<and> real_of_float l \<le> l' \<and> u' \<le> real_of_float u \<and> cmp ly uy \<and>
     approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)"
   using assms
 proof (induct s arbitrary: l u)
@@ -3850,7 +3844,7 @@
     and u: "approx_tse_form' prec t f s ?m u cmp"
     by (auto simp add: Let_def lazy_conj)
 
-  have m_l: "real l \<le> ?m" and m_u: "?m \<le> real u"
+  have m_l: "real_of_float l \<le> ?m" and m_u: "?m \<le> real_of_float u"
     unfolding less_eq_float_def using Suc.prems by auto
   with \<open>x \<in> { l .. u }\<close> consider "x \<in> { l .. ?m}" | "x \<in> {?m .. u}"
     by atomize_elim auto
@@ -3859,7 +3853,7 @@
     case 1
     from Suc.hyps[OF l this]
     obtain l' u' ly uy where
-      "x \<in> {l' .. u'} \<and> real l \<le> l' \<and> real u' \<le> ?m \<and> cmp ly uy \<and>
+      "x \<in> {l' .. u'} \<and> real_of_float l \<le> l' \<and> real_of_float u' \<le> ?m \<and> cmp ly uy \<and>
         approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)"
       by blast
     with m_u show ?thesis
@@ -3868,7 +3862,7 @@
     case 2
     from Suc.hyps[OF u this]
     obtain l' u' ly uy where
-      "x \<in> { l' .. u' } \<and> ?m \<le> real l' \<and> u' \<le> real u \<and> cmp ly uy \<and>
+      "x \<in> { l' .. u' } \<and> ?m \<le> real_of_float l' \<and> u' \<le> real_of_float u \<and> cmp ly uy \<and>
         approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)"
       by blast
     with m_u show ?thesis
@@ -3885,8 +3879,8 @@
   from approx_tse_form'[OF tse x]
   obtain l' u' ly uy
     where x': "x \<in> {l' .. u'}"
-    and "l \<le> real l'"
-    and "real u' \<le> u" and "0 < ly"
+    and "real_of_float l \<le> real_of_float l'"
+    and "real_of_float u' \<le> real_of_float u" and "0 < ly"
     and tse: "approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 (Add a (Minus b)) [Some (l', u')] = Some (ly, uy)"
     by blast
 
@@ -3908,8 +3902,8 @@
   from approx_tse_form'[OF tse x]
   obtain l' u' ly uy
     where x': "x \<in> {l' .. u'}"
-    and "l \<le> real l'"
-    and "real u' \<le> u" and "0 \<le> ly"
+    and "l \<le> real_of_float l'"
+    and "real_of_float u' \<le> u" and "0 \<le> ly"
     and tse: "approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 (Add a (Minus b)) [Some (l', u')] = Some (ly, uy)"
     by blast
 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Decision_Procs/Ferrack.thy
--- a/src/HOL/Decision_Procs/Ferrack.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Decision_Procs/Ferrack.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -30,13 +30,13 @@
   (* Semantics of numeral terms (num) *)
 primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real"
 where
-  "Inum bs (C c) = (real c)"
+  "Inum bs (C c) = (real_of_int c)"
 | "Inum bs (Bound n) = bs!n"
-| "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
+| "Inum bs (CN n c a) = (real_of_int c) * (bs!n) + (Inum bs a)"
 | "Inum bs (Neg a) = -(Inum bs a)"
 | "Inum bs (Add a b) = Inum bs a + Inum bs b"
 | "Inum bs (Sub a b) = Inum bs a - Inum bs b"
-| "Inum bs (Mul c a) = (real c) * Inum bs a"
+| "Inum bs (Mul c a) = (real_of_int c) * Inum bs a"
     (* FORMULAE *)
 datatype fm  =
   T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num|
@@ -518,7 +518,7 @@
 lemma reducecoeffh:
   assumes gt: "dvdnumcoeff t g"
     and gp: "g > 0"
-  shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
+  shows "real_of_int g *(Inum bs (reducecoeffh t g)) = Inum bs t"
   using gt
 proof (induct t rule: reducecoeffh.induct)
   case (1 i)
@@ -618,7 +618,7 @@
   from dvdnumcoeff_aux[OF th1 th2 H] show "dvdnumcoeff t ?g" .
 qed
 
-lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
+lemma reducecoeff: "real_of_int (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
 proof -
   let ?g = "numgcd t"
   have "?g \<ge> 0"
@@ -778,8 +778,8 @@
          else (t', n))))"
 
 lemma simp_num_pair_ci:
-  shows "((\<lambda>(t,n). Inum bs t / real n) (simp_num_pair (t,n))) =
-    ((\<lambda>(t,n). Inum bs t / real n) (t, n))"
+  shows "((\<lambda>(t,n). Inum bs t / real_of_int n) (simp_num_pair (t,n))) =
+    ((\<lambda>(t,n). Inum bs t / real_of_int n) (t, n))"
   (is "?lhs = ?rhs")
 proof -
   let ?t' = "simpnum t"
@@ -819,15 +819,15 @@
         have gpdd: "?g' dvd n" by simp
         have gpdgp: "?g' dvd ?g'" by simp
         from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
-        have th2:"real ?g' * ?t = Inum bs ?t'"
+        have th2:"real_of_int ?g' * ?t = Inum bs ?t'"
           by simp
-        from g1 g'1 have "?lhs = ?t / real (n div ?g')"
+        from g1 g'1 have "?lhs = ?t / real_of_int (n div ?g')"
           by (simp add: simp_num_pair_def Let_def)
-        also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))"
+        also have "\<dots> = (real_of_int ?g' * ?t) / (real_of_int ?g' * (real_of_int (n div ?g')))"
           by simp
-        also have "\<dots> = (Inum bs ?t' / real n)"
+        also have "\<dots> = (Inum bs ?t' / real_of_int n)"
           using real_of_int_div[OF gpdd] th2 gp0 by simp
-        finally have "?lhs = Inum bs t / real n"
+        finally have "?lhs = Inum bs t / real_of_int n"
           by simp
         then show ?thesis
           by (simp add: simp_num_pair_def)
@@ -1278,17 +1278,17 @@
 next
   case (3 c e)
   from 3 have nb: "numbound0 e" by simp
-  from 3 have cp: "real c > 0" by simp
+  from 3 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e = "Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {
     fix x
     assume xz: "x < ?z"
-    then have "(real c * x < - ?e)"
+    then have "(real_of_int c * x < - ?e)"
       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
-    then have "real c * x + ?e < 0" by arith
-    then have "real c * x + ?e \<noteq> 0" by simp
+    then have "real_of_int c * x + ?e < 0" by arith
+    then have "real_of_int c * x + ?e \<noteq> 0" by simp
     with xz have "?P ?z x (Eq (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
   }
@@ -1297,17 +1297,17 @@
 next
   case (4 c e)
   from 4 have nb: "numbound0 e" by simp
-  from 4 have cp: "real c > 0" by simp
+  from 4 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e = "Inum (a # bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {
     fix x
     assume xz: "x < ?z"
-    then have "(real c * x < - ?e)"
+    then have "(real_of_int c * x < - ?e)"
       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
-    then have "real c * x + ?e < 0" by arith
-    then have "real c * x + ?e \<noteq> 0" by simp
+    then have "real_of_int c * x + ?e < 0" by arith
+    then have "real_of_int c * x + ?e \<noteq> 0" by simp
     with xz have "?P ?z x (NEq (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
   }
@@ -1316,16 +1316,16 @@
 next
   case (5 c e)
   from 5 have nb: "numbound0 e" by simp
-  from 5 have cp: "real c > 0" by simp
+  from 5 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {
     fix x
     assume xz: "x < ?z"
-    then have "(real c * x < - ?e)"
+    then have "(real_of_int c * x < - ?e)"
       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
-    then have "real c * x + ?e < 0" by arith
+    then have "real_of_int c * x + ?e < 0" by arith
     with xz have "?P ?z x (Lt (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp
   }
@@ -1334,16 +1334,16 @@
 next
   case (6 c e)
   from 6 have nb: "numbound0 e" by simp
-  from lp 6 have cp: "real c > 0" by simp
+  from lp 6 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e = "Inum (a # bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {
     fix x
     assume xz: "x < ?z"
-    then have "(real c * x < - ?e)"
+    then have "(real_of_int c * x < - ?e)"
       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
-    then have "real c * x + ?e < 0" by arith
+    then have "real_of_int c * x + ?e < 0" by arith
     with xz have "?P ?z x (Le (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
   }
@@ -1352,16 +1352,16 @@
 next
   case (7 c e)
   from 7 have nb: "numbound0 e" by simp
-  from 7 have cp: "real c > 0" by simp
+  from 7 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e = "Inum (a # bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {
     fix x
     assume xz: "x < ?z"
-    then have "(real c * x < - ?e)"
+    then have "(real_of_int c * x < - ?e)"
       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
-    then have "real c * x + ?e < 0" by arith
+    then have "real_of_int c * x + ?e < 0" by arith
     with xz have "?P ?z x (Gt (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
   }
@@ -1370,16 +1370,16 @@
 next
   case (8 c e)
   from 8 have nb: "numbound0 e" by simp
-  from 8 have cp: "real c > 0" by simp
+  from 8 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {
     fix x
     assume xz: "x < ?z"
-    then have "(real c * x < - ?e)"
+    then have "(real_of_int c * x < - ?e)"
       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
-    then have "real c * x + ?e < 0" by arith
+    then have "real_of_int c * x + ?e < 0" by arith
     with xz have "?P ?z x (Ge (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
   }
@@ -1408,17 +1408,17 @@
 next
   case (3 c e)
   from 3 have nb: "numbound0 e" by simp
-  from 3 have cp: "real c > 0" by simp
+  from 3 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e = "Inum (a # bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {
     fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: ac_simps)
-    then have "real c * x + ?e > 0" by arith
-    then have "real c * x + ?e \<noteq> 0" by simp
+    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
+    then have "real_of_int c * x + ?e > 0" by arith
+    then have "real_of_int c * x + ?e \<noteq> 0" by simp
     with xz have "?P ?z x (Eq (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
   }
@@ -1427,17 +1427,17 @@
 next
   case (4 c e)
   from 4 have nb: "numbound0 e" by simp
-  from 4 have cp: "real c > 0" by simp
+  from 4 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e = "Inum (a # bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {
     fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: ac_simps)
-    then have "real c * x + ?e > 0" by arith
-    then have "real c * x + ?e \<noteq> 0" by simp
+    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
+    then have "real_of_int c * x + ?e > 0" by arith
+    then have "real_of_int c * x + ?e \<noteq> 0" by simp
     with xz have "?P ?z x (NEq (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
   }
@@ -1446,16 +1446,16 @@
 next
   case (5 c e)
   from 5 have nb: "numbound0 e" by simp
-  from 5 have cp: "real c > 0" by simp
+  from 5 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e = "Inum (a # bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {
     fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: ac_simps)
-    then have "real c * x + ?e > 0" by arith
+    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
+    then have "real_of_int c * x + ?e > 0" by arith
     with xz have "?P ?z x (Lt (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
   }
@@ -1464,16 +1464,16 @@
 next
   case (6 c e)
   from 6 have nb: "numbound0 e" by simp
-  from 6 have cp: "real c > 0" by simp
+  from 6 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e = "Inum (a # bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {
     fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: ac_simps)
-    then have "real c * x + ?e > 0" by arith
+    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
+    then have "real_of_int c * x + ?e > 0" by arith
     with xz have "?P ?z x (Le (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
   }
@@ -1482,16 +1482,16 @@
 next
   case (7 c e)
   from 7 have nb: "numbound0 e" by simp
-  from 7 have cp: "real c > 0" by simp
+  from 7 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e = "Inum (a # bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {
     fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: ac_simps)
-    then have "real c * x + ?e > 0" by arith
+    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
+    then have "real_of_int c * x + ?e > 0" by arith
     with xz have "?P ?z x (Gt (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
   }
@@ -1500,16 +1500,16 @@
 next
   case (8 c e)
   from 8 have nb: "numbound0 e" by simp
-  from 8 have cp: "real c > 0" by simp
+  from 8 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {
     fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: ac_simps)
-    then have "real c * x + ?e > 0" by arith
+    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
+    then have "real_of_int c * x + ?e > 0" by arith
     with xz have "?P ?z x (Ge (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
   }
@@ -1581,10 +1581,10 @@
 
 lemma usubst_I:
   assumes lp: "isrlfm p"
-    and np: "real n > 0"
+    and np: "real_of_int n > 0"
     and nbt: "numbound0 t"
   shows "(Ifm (x # bs) (usubst p (t,n)) =
-    Ifm (((Inum (x # bs) t) / (real n)) # bs) p) \<and> bound0 (usubst p (t, n))"
+    Ifm (((Inum (x # bs) t) / (real_of_int n)) # bs) p) \<and> bound0 (usubst p (t, n))"
   (is "(?I x (usubst p (t, n)) = ?I ?u p) \<and> ?B p"
    is "(_ = ?I (?t/?n) p) \<and> _"
    is "(_ = ?I (?N x t /_) p) \<and> _")
@@ -1592,65 +1592,65 @@
 proof (induct p rule: usubst.induct)
   case (5 c e)
   with assms have cp: "c > 0" and nb: "numbound0 e" by simp_all
-  have "?I ?u (Lt (CN 0 c e)) \<longleftrightarrow> real c * (?t / ?n) + ?N x e < 0"
+  have "?I ?u (Lt (CN 0 c e)) \<longleftrightarrow> real_of_int c * (?t / ?n) + ?N x e < 0"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> \<longleftrightarrow> ?n * (real c * (?t / ?n)) + ?n*(?N x e) < 0"
-    by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
+  also have "\<dots> \<longleftrightarrow> ?n * (real_of_int c * (?t / ?n)) + ?n*(?N x e) < 0"
+    by (simp only: pos_less_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> \<longleftrightarrow> real c * ?t + ?n * (?N x e) < 0" using np by simp
+  also have "\<dots> \<longleftrightarrow> real_of_int c * ?t + ?n * (?N x e) < 0" using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (6 c e)
   with assms have cp: "c > 0" and nb: "numbound0 e" by simp_all
-  have "?I ?u (Le (CN 0 c e)) \<longleftrightarrow> real c * (?t / ?n) + ?N x e \<le> 0"
+  have "?I ?u (Le (CN 0 c e)) \<longleftrightarrow> real_of_int c * (?t / ?n) + ?N x e \<le> 0"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
-    by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
+  also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
+    by (simp only: pos_le_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)" using np by simp
+  also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<le> 0)" using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (7 c e)
   with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
-  have "?I ?u (Gt (CN 0 c e)) \<longleftrightarrow> real c *(?t / ?n) + ?N x e > 0"
+  have "?I ?u (Gt (CN 0 c e)) \<longleftrightarrow> real_of_int c *(?t / ?n) + ?N x e > 0"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> \<longleftrightarrow> ?n * (real c * (?t / ?n)) + ?n * ?N x e > 0"
-    by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
+  also have "\<dots> \<longleftrightarrow> ?n * (real_of_int c * (?t / ?n)) + ?n * ?N x e > 0"
+    by (simp only: pos_divide_less_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> \<longleftrightarrow> real c * ?t + ?n * ?N x e > 0" using np by simp
+  also have "\<dots> \<longleftrightarrow> real_of_int c * ?t + ?n * ?N x e > 0" using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (8 c e)
   with assms have cp: "c > 0" and nb: "numbound0 e" by simp_all
-  have "?I ?u (Ge (CN 0 c e)) \<longleftrightarrow> real c * (?t / ?n) + ?N x e \<ge> 0"
+  have "?I ?u (Ge (CN 0 c e)) \<longleftrightarrow> real_of_int c * (?t / ?n) + ?N x e \<ge> 0"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> \<longleftrightarrow> ?n * (real c * (?t / ?n)) + ?n * ?N x e \<ge> 0"
-    by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
+  also have "\<dots> \<longleftrightarrow> ?n * (real_of_int c * (?t / ?n)) + ?n * ?N x e \<ge> 0"
+    by (simp only: pos_divide_le_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> \<longleftrightarrow> real c * ?t + ?n * ?N x e \<ge> 0" using np by simp
+  also have "\<dots> \<longleftrightarrow> real_of_int c * ?t + ?n * ?N x e \<ge> 0" using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (3 c e)
   with assms have cp: "c > 0" and nb: "numbound0 e" by simp_all
-  from np have np: "real n \<noteq> 0" by simp
-  have "?I ?u (Eq (CN 0 c e)) \<longleftrightarrow> real c * (?t / ?n) + ?N x e = 0"
+  from np have np: "real_of_int n \<noteq> 0" by simp
+  have "?I ?u (Eq (CN 0 c e)) \<longleftrightarrow> real_of_int c * (?t / ?n) + ?N x e = 0"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> \<longleftrightarrow> ?n * (real c * (?t / ?n)) + ?n * ?N x e = 0"
-    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
+  also have "\<dots> \<longleftrightarrow> ?n * (real_of_int c * (?t / ?n)) + ?n * ?N x e = 0"
+    by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> \<longleftrightarrow> real c * ?t + ?n * ?N x e = 0" using np by simp
+  also have "\<dots> \<longleftrightarrow> real_of_int c * ?t + ?n * ?N x e = 0" using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (4 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
-  from np have np: "real n \<noteq> 0" by simp
-  have "?I ?u (NEq (CN 0 c e)) \<longleftrightarrow> real c * (?t / ?n) + ?N x e \<noteq> 0"
+  from np have np: "real_of_int n \<noteq> 0" by simp
+  have "?I ?u (NEq (CN 0 c e)) \<longleftrightarrow> real_of_int c * (?t / ?n) + ?N x e \<noteq> 0"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> \<longleftrightarrow> ?n * (real c * (?t / ?n)) + ?n * ?N x e \<noteq> 0"
-    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
+  also have "\<dots> \<longleftrightarrow> ?n * (real_of_int c * (?t / ?n)) + ?n * ?N x e \<noteq> 0"
+    by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> \<longleftrightarrow> real c * ?t + ?n * ?N x e \<noteq> 0" using np by simp
+  also have "\<dots> \<longleftrightarrow> real_of_int c * ?t + ?n * ?N x e \<noteq> 0" using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
-qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"])
+qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real_of_int n" and b'="x"])
 
 lemma uset_l:
   assumes lp: "isrlfm p"
@@ -1661,18 +1661,18 @@
   assumes lp: "isrlfm p"
     and nmi: "\<not> (Ifm (a # bs) (minusinf p))" (is "\<not> (Ifm (a # bs) (?M p))")
     and ex: "Ifm (x#bs) p" (is "?I x p")
-  shows "\<exists>(s,m) \<in> set (uset p). x \<ge> Inum (a#bs) s / real m"
-    (is "\<exists>(s,m) \<in> ?U p. x \<ge> ?N a s / real m")
+  shows "\<exists>(s,m) \<in> set (uset p). x \<ge> Inum (a#bs) s / real_of_int m"
+    (is "\<exists>(s,m) \<in> ?U p. x \<ge> ?N a s / real_of_int m")
 proof -
-  have "\<exists>(s,m) \<in> set (uset p). real m * x \<ge> Inum (a#bs) s"
-    (is "\<exists>(s,m) \<in> ?U p. real m *x \<ge> ?N a s")
+  have "\<exists>(s,m) \<in> set (uset p). real_of_int m * x \<ge> Inum (a#bs) s"
+    (is "\<exists>(s,m) \<in> ?U p. real_of_int m *x \<ge> ?N a s")
     using lp nmi ex
     by (induct p rule: minusinf.induct) (auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"])
-  then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<ge> ?N a s"
+  then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real_of_int m * x \<ge> ?N a s"
     by blast
-  from uset_l[OF lp] smU have mp: "real m > 0"
+  from uset_l[OF lp] smU have mp: "real_of_int m > 0"
     by auto
-  from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m"
+  from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real_of_int m"
     by (auto simp add: mult.commute)
   then show ?thesis
     using smU by auto
@@ -1682,19 +1682,19 @@
   assumes lp: "isrlfm p"
     and nmi: "\<not> (Ifm (a # bs) (plusinf p))" (is "\<not> (Ifm (a # bs) (?M p))")
     and ex: "Ifm (x # bs) p" (is "?I x p")
-  shows "\<exists>(s,m) \<in> set (uset p). x \<le> Inum (a#bs) s / real m"
-    (is "\<exists>(s,m) \<in> ?U p. x \<le> ?N a s / real m")
+  shows "\<exists>(s,m) \<in> set (uset p). x \<le> Inum (a#bs) s / real_of_int m"
+    (is "\<exists>(s,m) \<in> ?U p. x \<le> ?N a s / real_of_int m")
 proof -
-  have "\<exists>(s,m) \<in> set (uset p). real m * x \<le> Inum (a#bs) s"
-    (is "\<exists>(s,m) \<in> ?U p. real m *x \<le> ?N a s")
+  have "\<exists>(s,m) \<in> set (uset p). real_of_int m * x \<le> Inum (a#bs) s"
+    (is "\<exists>(s,m) \<in> ?U p. real_of_int m *x \<le> ?N a s")
     using lp nmi ex
     by (induct p rule: minusinf.induct)
       (auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"])
-  then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<le> ?N a s"
+  then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real_of_int m * x \<le> ?N a s"
     by blast
-  from uset_l[OF lp] smU have mp: "real m > 0"
+  from uset_l[OF lp] smU have mp: "real_of_int m > 0"
     by auto
-  from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m"
+  from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real_of_int m"
     by (auto simp add: mult.commute)
   then show ?thesis
     using smU by auto
@@ -1702,8 +1702,8 @@
 
 lemma lin_dense:
   assumes lp: "isrlfm p"
-    and noS: "\<forall>t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda>(t,n). Inum (x#bs) t / real n) ` set (uset p)"
-      (is "\<forall>t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda>(t,n). ?N x t / real n ) ` (?U p)")
+    and noS: "\<forall>t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda>(t,n). Inum (x#bs) t / real_of_int n) ` set (uset p)"
+      (is "\<forall>t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda>(t,n). ?N x t / real_of_int n ) ` (?U p)")
     and lx: "l < x"
     and xu:"x < u"
     and px:" Ifm (x#bs) p"
@@ -1712,163 +1712,163 @@
   using lp px noS
 proof (induct p rule: isrlfm.induct)
   case (5 c e)
-  then have cp: "real c > 0" and nb: "numbound0 e"
+  then have cp: "real_of_int c > 0" and nb: "numbound0 e"
     by simp_all
-  from 5 have "x * real c + ?N x e < 0"
+  from 5 have "x * real_of_int c + ?N x e < 0"
     by (simp add: algebra_simps)
-  then have pxc: "x < (- ?N x e) / real c"
+  then have pxc: "x < (- ?N x e) / real_of_int c"
     by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
-  from 5 have noSc:"\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c"
+  from 5 have noSc:"\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c"
     by auto
-  with ly yu have yne: "y \<noteq> - ?N x e / real c"
+  with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c"
     by auto
-  then consider "y < (-?N x e)/ real c" | "y > (- ?N x e) / real c"
+  then consider "y < (-?N x e)/ real_of_int c" | "y > (- ?N x e) / real_of_int c"
     by atomize_elim auto
   then show ?case
   proof cases
     case 1
-    then have "y * real c < - ?N x e"
+    then have "y * real_of_int c < - ?N x e"
       by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-    then have "real c * y + ?N x e < 0"
+    then have "real_of_int c * y + ?N x e < 0"
       by (simp add: algebra_simps)
     then show ?thesis
       using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp
   next
     case 2
-    with yu have eu: "u > (- ?N x e) / real c"
+    with yu have eu: "u > (- ?N x e) / real_of_int c"
       by auto
-    with noSc ly yu have "(- ?N x e) / real c \<le> l"
-      by (cases "(- ?N x e) / real c > l") auto
+    with noSc ly yu have "(- ?N x e) / real_of_int c \<le> l"
+      by (cases "(- ?N x e) / real_of_int c > l") auto
     with lx pxc have False
       by auto
     then show ?thesis ..
   qed
 next
   case (6 c e)
-  then have cp: "real c > 0" and nb: "numbound0 e"
+  then have cp: "real_of_int c > 0" and nb: "numbound0 e"
     by simp_all
-  from 6 have "x * real c + ?N x e \<le> 0"
+  from 6 have "x * real_of_int c + ?N x e \<le> 0"
     by (simp add: algebra_simps)
-  then have pxc: "x \<le> (- ?N x e) / real c"
+  then have pxc: "x \<le> (- ?N x e) / real_of_int c"
     by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
-  from 6 have noSc:"\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c"
+  from 6 have noSc:"\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c"
     by auto
-  with ly yu have yne: "y \<noteq> - ?N x e / real c"
+  with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c"
     by auto
-  then consider "y < (- ?N x e) / real c" | "y > (-?N x e) / real c"
+  then consider "y < (- ?N x e) / real_of_int c" | "y > (-?N x e) / real_of_int c"
     by atomize_elim auto
   then show ?case
   proof cases
     case 1
-    then have "y * real c < - ?N x e"
+    then have "y * real_of_int c < - ?N x e"
       by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-    then have "real c * y + ?N x e < 0"
+    then have "real_of_int c * y + ?N x e < 0"
       by (simp add: algebra_simps)
     then show ?thesis
       using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp
   next
     case 2
-    with yu have eu: "u > (- ?N x e) / real c"
+    with yu have eu: "u > (- ?N x e) / real_of_int c"
       by auto
-    with noSc ly yu have "(- ?N x e) / real c \<le> l"
-      by (cases "(- ?N x e) / real c > l") auto
+    with noSc ly yu have "(- ?N x e) / real_of_int c \<le> l"
+      by (cases "(- ?N x e) / real_of_int c > l") auto
     with lx pxc have False
       by auto
     then show ?thesis ..
   qed
 next
   case (7 c e)
-  then have cp: "real c > 0" and nb: "numbound0 e"
+  then have cp: "real_of_int c > 0" and nb: "numbound0 e"
     by simp_all
-  from 7 have "x * real c + ?N x e > 0"
+  from 7 have "x * real_of_int c + ?N x e > 0"
     by (simp add: algebra_simps)
-  then have pxc: "x > (- ?N x e) / real c"
+  then have pxc: "x > (- ?N x e) / real_of_int c"
     by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
-  from 7 have noSc: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c"
+  from 7 have noSc: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c"
     by auto
-  with ly yu have yne: "y \<noteq> - ?N x e / real c"
+  with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c"
     by auto
-  then consider "y > (- ?N x e) / real c" | "y < (-?N x e) / real c"
+  then consider "y > (- ?N x e) / real_of_int c" | "y < (-?N x e) / real_of_int c"
     by atomize_elim auto
   then show ?case
   proof cases
     case 1
-    then have "y * real c > - ?N x e"
+    then have "y * real_of_int c > - ?N x e"
       by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-    then have "real c * y + ?N x e > 0"
+    then have "real_of_int c * y + ?N x e > 0"
       by (simp add: algebra_simps)
     then show ?thesis
       using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp
   next
     case 2
-    with ly have eu: "l < (- ?N x e) / real c"
+    with ly have eu: "l < (- ?N x e) / real_of_int c"
       by auto
-    with noSc ly yu have "(- ?N x e) / real c \<ge> u"
-      by (cases "(- ?N x e) / real c > l") auto
+    with noSc ly yu have "(- ?N x e) / real_of_int c \<ge> u"
+      by (cases "(- ?N x e) / real_of_int c > l") auto
     with xu pxc have False by auto
     then show ?thesis ..
   qed
 next
   case (8 c e)
-  then have cp: "real c > 0" and nb: "numbound0 e"
+  then have cp: "real_of_int c > 0" and nb: "numbound0 e"
     by simp_all
-  from 8 have "x * real c + ?N x e \<ge> 0"
+  from 8 have "x * real_of_int c + ?N x e \<ge> 0"
     by (simp add: algebra_simps)
-  then have pxc: "x \<ge> (- ?N x e) / real c"
+  then have pxc: "x \<ge> (- ?N x e) / real_of_int c"
     by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
-  from 8 have noSc:"\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c"
+  from 8 have noSc:"\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c"
     by auto
-  with ly yu have yne: "y \<noteq> - ?N x e / real c"
+  with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c"
     by auto
-  then consider "y > (- ?N x e) / real c" | "y < (-?N x e) / real c"
+  then consider "y > (- ?N x e) / real_of_int c" | "y < (-?N x e) / real_of_int c"
     by atomize_elim auto
   then show ?case
   proof cases
     case 1
-    then have "y * real c > - ?N x e"
+    then have "y * real_of_int c > - ?N x e"
       by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-    then have "real c * y + ?N x e > 0" by (simp add: algebra_simps)
+    then have "real_of_int c * y + ?N x e > 0" by (simp add: algebra_simps)
     then show ?thesis
       using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp
   next
     case 2
-    with ly have eu: "l < (- ?N x e) / real c"
+    with ly have eu: "l < (- ?N x e) / real_of_int c"
       by auto
-    with noSc ly yu have "(- ?N x e) / real c \<ge> u"
-      by (cases "(- ?N x e) / real c > l") auto
+    with noSc ly yu have "(- ?N x e) / real_of_int c \<ge> u"
+      by (cases "(- ?N x e) / real_of_int c > l") auto
     with xu pxc have False
       by auto
     then show ?thesis ..
   qed
 next
   case (3 c e)
-  then have cp: "real c > 0" and nb: "numbound0 e"
+  then have cp: "real_of_int c > 0" and nb: "numbound0 e"
     by simp_all
-  from cp have cnz: "real c \<noteq> 0"
+  from cp have cnz: "real_of_int c \<noteq> 0"
     by simp
-  from 3 have "x * real c + ?N x e = 0"
+  from 3 have "x * real_of_int c + ?N x e = 0"
     by (simp add: algebra_simps)
-  then have pxc: "x = (- ?N x e) / real c"
+  then have pxc: "x = (- ?N x e) / real_of_int c"
     by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
-  from 3 have noSc:"\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c"
+  from 3 have noSc:"\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c"
     by auto
-  with lx xu have yne: "x \<noteq> - ?N x e / real c"
+  with lx xu have yne: "x \<noteq> - ?N x e / real_of_int c"
     by auto
   with pxc show ?case
     by simp
 next
   case (4 c e)
-  then have cp: "real c > 0" and nb: "numbound0 e"
+  then have cp: "real_of_int c > 0" and nb: "numbound0 e"
     by simp_all
-  from cp have cnz: "real c \<noteq> 0"
+  from cp have cnz: "real_of_int c \<noteq> 0"
     by simp
-  from 4 have noSc:"\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c"
+  from 4 have noSc:"\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c"
     by auto
-  with ly yu have yne: "y \<noteq> - ?N x e / real c"
+  with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c"
     by auto
-  then have "y* real c \<noteq> -?N x e"
+  then have "y* real_of_int c \<noteq> -?N x e"
     by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
-  then have "y* real c + ?N x e \<noteq> 0"
+  then have "y* real_of_int c + ?N x e \<noteq> 0"
     by (simp add: algebra_simps)
   then show ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
     by (simp add: algebra_simps)
@@ -1947,7 +1947,7 @@
     and npi: "\<not> (Ifm (x # bs) (plusinf p))"  (is "\<not> (Ifm (x # bs) (?P p))")
     and ex: "\<exists>x. Ifm (x # bs) p"  (is "\<exists>x. ?I x p")
   shows "\<exists>(l,n) \<in> set (uset p). \<exists>(s,m) \<in> set (uset p).
-    ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p"
+    ?I ((Inum (x#bs) l / real_of_int n + Inum (x#bs) s / real_of_int m) / 2) p"
 proof -
   let ?N = "\<lambda>x t. Inum (x # bs) t"
   let ?U = "set (uset p)"
@@ -1959,22 +1959,22 @@
   from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi
   have npi': "\<not> (?I a (?P p))"
     by simp
-  have "\<exists>(l,n) \<in> set (uset p). \<exists>(s,m) \<in> set (uset p). ?I ((?N a l/real n + ?N a s /real m) / 2) p"
+  have "\<exists>(l,n) \<in> set (uset p). \<exists>(s,m) \<in> set (uset p). ?I ((?N a l/real_of_int n + ?N a s /real_of_int m) / 2) p"
   proof -
-    let ?M = "(\<lambda>(t,c). ?N a t / real c) ` ?U"
+    let ?M = "(\<lambda>(t,c). ?N a t / real_of_int c) ` ?U"
     have fM: "finite ?M"
       by auto
     from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa]
-    have "\<exists>(l,n) \<in> set (uset p). \<exists>(s,m) \<in> set (uset p). a \<le> ?N x l / real n \<and> a \<ge> ?N x s / real m"
+    have "\<exists>(l,n) \<in> set (uset p). \<exists>(s,m) \<in> set (uset p). a \<le> ?N x l / real_of_int n \<and> a \<ge> ?N x s / real_of_int m"
       by blast
     then obtain "t" "n" "s" "m"
       where tnU: "(t,n) \<in> ?U"
         and smU: "(s,m) \<in> ?U"
-        and xs1: "a \<le> ?N x s / real m"
-        and tx1: "a \<ge> ?N x t / real n"
+        and xs1: "a \<le> ?N x s / real_of_int m"
+        and tx1: "a \<ge> ?N x t / real_of_int n"
       by blast
     from uset_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1
-    have xs: "a \<le> ?N a s / real m" and tx: "a \<ge> ?N a t / real n"
+    have xs: "a \<le> ?N a s / real_of_int m" and tx: "a \<ge> ?N a t / real_of_int n"
       by auto
     from tnU have Mne: "?M \<noteq> {}"
       by auto
@@ -1986,19 +1986,19 @@
       using fM Mne by simp
     have uinM: "?u \<in> ?M"
       using fM Mne by simp
-    have tnM: "?N a t / real n \<in> ?M"
+    have tnM: "?N a t / real_of_int n \<in> ?M"
       using tnU by auto
-    have smM: "?N a s / real m \<in> ?M"
+    have smM: "?N a s / real_of_int m \<in> ?M"
       using smU by auto
     have lM: "\<forall>t\<in> ?M. ?l \<le> t"
       using Mne fM by auto
     have Mu: "\<forall>t\<in> ?M. t \<le> ?u"
       using Mne fM by auto
-    have "?l \<le> ?N a t / real n"
+    have "?l \<le> ?N a t / real_of_int n"
       using tnM Mne by simp
     then have lx: "?l \<le> a"
       using tx by simp
-    have "?N a s / real m \<le> ?u"
+    have "?N a s / real_of_int m \<le> ?u"
       using smM Mne by simp
     then have xu: "a \<le> ?u"
       using xs by simp
@@ -2010,13 +2010,13 @@
     proof cases
       case 1
       note um = \<open>u \<in> ?M\<close> and pu = \<open>?I u p\<close>
-      then have "\<exists>(tu,nu) \<in> ?U. u = ?N a tu / real nu"
+      then have "\<exists>(tu,nu) \<in> ?U. u = ?N a tu / real_of_int nu"
         by auto
-      then obtain tu nu where tuU: "(tu, nu) \<in> ?U" and tuu: "u= ?N a tu / real nu"
+      then obtain tu nu where tuU: "(tu, nu) \<in> ?U" and tuu: "u= ?N a tu / real_of_int nu"
         by blast
       have "(u + u) / 2 = u"
         by auto
-      with pu tuu have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p"
+      with pu tuu have "?I (((?N a tu / real_of_int nu) + (?N a tu / real_of_int nu)) / 2) p"
         by simp
       with tuU show ?thesis by blast
     next
@@ -2024,13 +2024,13 @@
       note t1M = \<open>t1 \<in> ?M\<close> and t2M = \<open>t2\<in> ?M\<close>
         and noM = \<open>\<forall>y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M\<close>
         and t1x = \<open>t1 < a\<close> and xt2 = \<open>a < t2\<close> and px = \<open>?I a p\<close>
-      from t1M have "\<exists>(t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n"
+      from t1M have "\<exists>(t1u,t1n) \<in> ?U. t1 = ?N a t1u / real_of_int t1n"
         by auto
-      then obtain t1u t1n where t1uU: "(t1u, t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n"
+      then obtain t1u t1n where t1uU: "(t1u, t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real_of_int t1n"
         by blast
-      from t2M have "\<exists>(t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n"
+      from t2M have "\<exists>(t2u,t2n) \<in> ?U. t2 = ?N a t2u / real_of_int t2n"
         by auto
-      then obtain t2u t2n where t2uU: "(t2u, t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n"
+      then obtain t2u t2n where t2uU: "(t2u, t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real_of_int t2n"
         by blast
       from t1x xt2 have t1t2: "t1 < t2"
         by simp
@@ -2043,13 +2043,13 @@
     qed
   qed
   then obtain l n s m where lnU: "(l, n) \<in> ?U" and smU:"(s, m) \<in> ?U"
-    and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p"
+    and pu: "?I ((?N a l / real_of_int n + ?N a s / real_of_int m) / 2) p"
     by blast
   from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s"
     by auto
   from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"]
     numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
-  have "?I ((?N x l / real n + ?N x s / real m) / 2) p"
+  have "?I ((?N x l / real_of_int n + ?N x s / real_of_int m) / 2) p"
     by simp
   with lnU smU show ?thesis
     by auto
@@ -2063,7 +2063,7 @@
   shows "(\<exists>x. Ifm (x#bs) p) \<longleftrightarrow>
     Ifm (x # bs) (minusinf p) \<or> Ifm (x # bs) (plusinf p) \<or>
       (\<exists>(t,n) \<in> set (uset p). \<exists>(s,m) \<in> set (uset p).
-        Ifm ((((Inum (x # bs) t) / real n + (Inum (x # bs) s) / real m) / 2) # bs) p)"
+        Ifm ((((Inum (x # bs) t) / real_of_int n + (Inum (x # bs) s) / real_of_int m) / 2) # bs) p)"
   (is "(\<exists>x. ?I x p) \<longleftrightarrow> (?M \<or> ?P \<or> ?F)" is "?E = ?D")
 proof
   assume px: "\<exists>x. ?I x p"
@@ -2111,23 +2111,23 @@
     then show ?thesis by blast
   next
     case 2
-    let ?f = "\<lambda>(t,n). Inum (x # bs) t / real n"
+    let ?f = "\<lambda>(t,n). Inum (x # bs) t / real_of_int n"
     let ?N = "\<lambda>t. Inum (x # bs) t"
     {
       fix t n s m
       assume "(t, n) \<in> set (uset p)" and "(s, m) \<in> set (uset p)"
       with uset_l[OF lp] have tnb: "numbound0 t"
-        and np: "real n > 0" and snb: "numbound0 s" and mp: "real m > 0"
+        and np: "real_of_int n > 0" and snb: "numbound0 s" and mp: "real_of_int m > 0"
         by auto
       let ?st = "Add (Mul m t) (Mul n s)"
-      from np mp have mnp: "real (2 * n * m) > 0"
+      from np mp have mnp: "real_of_int (2 * n * m) > 0"
         by (simp add: mult.commute)
       from tnb snb have st_nb: "numbound0 ?st"
         by simp
-      have st: "(?N t / real n + ?N s / real m) / 2 = ?N ?st / real (2 * n * m)"
+      have st: "(?N t / real_of_int n + ?N s / real_of_int m) / 2 = ?N ?st / real_of_int (2 * n * m)"
         using mnp mp np by (simp add: algebra_simps add_divide_distrib)
       from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"]
-      have "?I x (usubst p (?st, 2 * n * m)) = ?I ((?N t / real n + ?N s / real m) / 2) p"
+      have "?I x (usubst p (?st, 2 * n * m)) = ?I ((?N t / real_of_int n + ?N s / real_of_int m) / 2) p"
         by (simp only: st[symmetric])
     }
     with rinf_uset[OF lp 2 px] have ?F
@@ -2149,11 +2149,11 @@
     from rplusinf_ex[OF lp this] show ?thesis .
   next
     case 3
-    with uset_l[OF lp] have tnb: "numbound0 t" and np: "real k > 0"
-      and snb: "numbound0 s" and mp: "real l > 0"
+    with uset_l[OF lp] have tnb: "numbound0 t" and np: "real_of_int k > 0"
+      and snb: "numbound0 s" and mp: "real_of_int l > 0"
       by auto
     let ?st = "Add (Mul l t) (Mul k s)"
-    from np mp have mnp: "real (2 * k * l) > 0"
+    from np mp have mnp: "real_of_int (2 * k * l) > 0"
       by (simp add: mult.commute)
     from tnb snb have st_nb: "numbound0 ?st"
       by simp
@@ -2182,9 +2182,9 @@
 
 lemma uset_cong_aux:
   assumes Ul: "\<forall>(t,n) \<in> set U. numbound0 t \<and> n > 0"
-  shows "((\<lambda>(t,n). Inum (x # bs) t / real n) `
+  shows "((\<lambda>(t,n). Inum (x # bs) t / real_of_int n) `
     (set (map (\<lambda>((t,n),(s,m)). (Add (Mul m t) (Mul n s), 2 * n * m)) (alluopairs U)))) =
-    ((\<lambda>((t,n),(s,m)). (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) ` (set U \<times> set U))"
+    ((\<lambda>((t,n),(s,m)). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) ` (set U \<times> set U))"
   (is "?lhs = ?rhs")
 proof auto
   fix t n s m
@@ -2197,10 +2197,10 @@
     by auto
   from Ul th have nnz: "n \<noteq> 0"
     by auto
-  have st: "(?N t / real n + ?N s / real m) / 2 = ?N ?st / real (2 * n * m)"
+  have st: "(?N t / real_of_int n + ?N s / real_of_int m) / 2 = ?N ?st / real_of_int (2 * n * m)"
     using mnz nnz by (simp add: algebra_simps add_divide_distrib)
-  then show "(real m *  Inum (x # bs) t + real n * Inum (x # bs) s) / (2 * real n * real m)
-      \<in> (\<lambda>((t, n), s, m). (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
+  then show "(real_of_int m *  Inum (x # bs) t + real_of_int n * Inum (x # bs) s) / (2 * real_of_int n * real_of_int m)
+      \<in> (\<lambda>((t, n), s, m). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) `
          (set U \<times> set U)"
     using mnz nnz th
     apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
@@ -2218,10 +2218,10 @@
     by auto
   from Ul tnU have nnz: "n \<noteq> 0"
     by auto
-  have st: "(?N t / real n + ?N s / real m) / 2 = ?N ?st / real (2 * n * m)"
+  have st: "(?N t / real_of_int n + ?N s / real_of_int m) / 2 = ?N ?st / real_of_int (2 * n * m)"
     using mnz nnz by (simp add: algebra_simps add_divide_distrib)
-  let ?P = "\<lambda>(t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 =
-    (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m') / 2"
+  let ?P = "\<lambda>(t',n') (s',m'). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m)/2 =
+    (Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int m') / 2"
   have Pc:"\<forall>a b. ?P a b = ?P b a"
     by auto
   from Ul alluopairs_set1 have Up:"\<forall>((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0"
@@ -2235,24 +2235,24 @@
   from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0"
     by auto
   let ?st' = "Add (Mul m' t') (Mul n' s')"
-  have st': "(?N t' / real n' + ?N s' / real m') / 2 = ?N ?st' / real (2 * n' * m')"
+  have st': "(?N t' / real_of_int n' + ?N s' / real_of_int m') / 2 = ?N ?st' / real_of_int (2 * n' * m')"
     using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
-  from Pts' have "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2 =
-    (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m') / 2"
+  from Pts' have "(Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2 =
+    (Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int m') / 2"
     by simp
-  also have "\<dots> = (\<lambda>(t, n). Inum (x # bs) t / real n)
+  also have "\<dots> = (\<lambda>(t, n). Inum (x # bs) t / real_of_int n)
       ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t', n'), (s', m')))"
     by (simp add: st')
-  finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2
-    \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) `
+  finally show "(Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2
+    \<in> (\<lambda>(t, n). Inum (x # bs) t / real_of_int n) `
       (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ` set (alluopairs U)"
     using ts'_U by blast
 qed
 
 lemma uset_cong:
   assumes lp: "isrlfm p"
-    and UU': "((\<lambda>(t,n). Inum (x # bs) t / real n) ` U') =
-      ((\<lambda>((t,n),(s,m)). (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) ` (U \<times> U))"
+    and UU': "((\<lambda>(t,n). Inum (x # bs) t / real_of_int n) ` U') =
+      ((\<lambda>((t,n),(s,m)). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) ` (U \<times> U))"
       (is "?f ` U' = ?g ` (U \<times> U)")
     and U: "\<forall>(t,n) \<in> U. numbound0 t \<and> n > 0"
     and U': "\<forall>(t,n) \<in> U'. numbound0 t \<and> n > 0"
@@ -2270,11 +2270,11 @@
       and snb: "numbound0 s" and mp: "m > 0"
       by auto
     let ?st = "Add (Mul m t) (Mul n s)"
-    from np mp have mnp: "real (2 * n * m) > 0"
-      by (simp add: mult.commute real_of_int_mult[symmetric] del: real_of_int_mult)
+    from np mp have mnp: "real_of_int (2 * n * m) > 0"
+      by (simp add: mult.commute of_int_mult[symmetric] del: of_int_mult)
     from tnb snb have stnb: "numbound0 ?st"
       by simp
-    have st: "(?N t / real n + ?N s / real m) / 2 = ?N ?st / real (2 * n * m)"
+    have st: "(?N t / real_of_int n + ?N s / real_of_int m) / 2 = ?N ?st / real_of_int (2 * n * m)"
       using mp np by (simp add: algebra_simps add_divide_distrib)
     from tnU smU UU' have "?g ((t, n), (s, m)) \<in> ?f ` U'"
       by blast
@@ -2285,10 +2285,10 @@
       done
     then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t, n), (s, m)) = ?f (t', n')"
       by blast
-    from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0"
+    from U' tnU' have tnb': "numbound0 t'" and np': "real_of_int n' > 0"
       by auto
     from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst
-    have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p"
+    have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real_of_int (2 * n * m) # bs) p"
       by simp
     from conjunct1[OF usubst_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric]
       th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]]
@@ -2316,17 +2316,17 @@
       and snb: "numbound0 s" and mp: "m > 0"
       by auto
     let ?st = "Add (Mul m t) (Mul n s)"
-    from np mp have mnp: "real (2 * n * m) > 0"
-      by (simp add: mult.commute real_of_int_mult[symmetric] del: real_of_int_mult)
+    from np mp have mnp: "real_of_int (2 * n * m) > 0"
+      by (simp add: mult.commute of_int_mult[symmetric] del: of_int_mult)
     from tnb snb have stnb: "numbound0 ?st"
       by simp
-    have st: "(?N t / real n + ?N s / real m) / 2 = ?N ?st / real (2 * n * m)"
+    have st: "(?N t / real_of_int n + ?N s / real_of_int m) / 2 = ?N ?st / real_of_int (2 * n * m)"
       using mp np by (simp add: algebra_simps add_divide_distrib)
-    from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0"
+    from U' tnU' have tnb': "numbound0 t'" and np': "real_of_int n' > 0"
       by auto
     from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified
       th[simplified split_def fst_conv snd_conv] st] Pt'
-    have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p"
+    have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real_of_int (2 * n * m) # bs) p"
       by simp
     with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU
     show ?thesis by blast
@@ -2348,8 +2348,8 @@
   let ?S = "map ?g ?Up"
   let ?SS = "map simp_num_pair ?S"
   let ?Y = "remdups ?SS"
-  let ?f = "\<lambda>(t,n). ?N t / real n"
-  let ?h = "\<lambda>((t,n),(s,m)). (?N t / real n + ?N s / real m) / 2"
+  let ?f = "\<lambda>(t,n). ?N t / real_of_int n"
+  let ?h = "\<lambda>((t,n),(s,m)). (?N t / real_of_int n + ?N s / real_of_int m) / 2"
   let ?F = "\<lambda>p. \<exists>a \<in> set (uset p). \<exists>b \<in> set (uset p). ?I x (usubst p (?g (a, b)))"
   let ?ep = "evaldjf (simpfm \<circ> (usubst ?q)) ?Y"
   from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q"
@@ -2403,7 +2403,7 @@
   proof -
     have "bound0 (simpfm (usubst ?q (t, n)))" if tnY: "(t,n) \<in> set ?Y" for t n
     proof -
-      from Y_l that have tnb: "numbound0 t" and np: "real n > 0"
+      from Y_l that have tnb: "numbound0 t" and np: "real_of_int n > 0"
         by auto
       from usubst_I[OF lq np tnb] have "bound0 (usubst ?q (t, n))"
         by simp
@@ -2464,8 +2464,8 @@
 val mk_Bound = @{code Bound} o @{code nat_of_integer};
 
 fun num_of_term vs (Free vT) = mk_Bound (find_index (fn vT' => vT = vT') vs)
-  | num_of_term vs @{term "real (0::int)"} = mk_C 0
-  | num_of_term vs @{term "real (1::int)"} = mk_C 1
+  | num_of_term vs @{term "real_of_int (0::int)"} = mk_C 0
+  | num_of_term vs @{term "real_of_int (1::int)"} = mk_C 1
   | num_of_term vs @{term "0::real"} = mk_C 0
   | num_of_term vs @{term "1::real"} = mk_C 1
   | num_of_term vs (Bound i) = mk_Bound i
@@ -2477,7 +2477,7 @@
   | num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = (case num_of_term vs t1
      of @{code C} i => @{code Mul} (i, num_of_term vs t2)
       | _ => error "num_of_term: unsupported multiplication")
-  | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ t') =
+  | num_of_term vs (@{term "real_of_int :: int \<Rightarrow> real"} $ t') =
      (mk_C (snd (HOLogic.dest_number t'))
        handle TERM _ => error ("num_of_term: unknown term"))
   | num_of_term vs t' =
@@ -2504,7 +2504,7 @@
       @{code A} (fm_of_term (("", dummyT) ::  vs) p)
   | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);
 
-fun term_of_num vs (@{code C} i) = @{term "real :: int \<Rightarrow> real"} $
+fun term_of_num vs (@{code C} i) = @{term "real_of_int :: int \<Rightarrow> real"} $
       HOLogic.mk_number HOLogic.intT (@{code integer_of_int} i)
   | term_of_num vs (@{code Bound} n) = Free (nth vs (@{code integer_of_nat} n))
   | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} $ term_of_num vs t'
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Decision_Procs/MIR.thy
--- a/src/HOL/Decision_Procs/MIR.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Decision_Procs/MIR.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -9,7 +9,7 @@
 
 section \<open>Quantifier elimination for @{text "\<real> (0, 1, +, floor, <)"}\<close>
 
-declare real_of_int_floor_cancel [simp del]
+declare of_int_floor_cancel [simp del]
 
 lemma myle:
   fixes a b :: "'a::{ordered_ab_group_add}"
@@ -21,73 +21,51 @@
   shows "(a < b) = (0 < b - a)"
   by (metis le_iff_diff_le_0 less_le_not_le myle)
 
-  (* Maybe should be added to the library \<dots> *)
-lemma floor_int_eq: "(real n\<le> x \<and> x < real (n+1)) = (floor x = n)"
-proof( auto)
-  assume lb: "real n \<le> x"
-    and ub: "x < real n + 1"
-  have "real (floor x) \<le> x" by simp 
-  hence "real (floor x) < real (n + 1) " using ub by arith
-  hence "floor x < n+1" by simp
-  moreover from lb have "n \<le> floor x" using floor_mono[where x="real n" and y="x"] 
-    by simp ultimately show "floor x = n" by simp
-qed
-
 (* Periodicity of dvd *)
-lemma dvd_period:
-  assumes advdd: "(a::int) dvd d"
-  shows "(a dvd (x + t)) = (a dvd ((x+ c*d) + t))"
-  using advdd  
-proof-
-  { fix x k
-    from inf_period(3)[OF advdd, rule_format, where x=x and k="-k"]  
-    have " ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))" by simp}
-  hence "\<forall>x.\<forall>k. ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))"  by simp
-  then show ?thesis by simp
-qed
+lemmas dvd_period = zdvd_period
 
 (* The Divisibility relation between reals *)
 definition rdvd:: "real \<Rightarrow> real \<Rightarrow> bool" (infixl "rdvd" 50)
-  where "x rdvd y \<longleftrightarrow> (\<exists>k::int. y = x * real k)"
+  where "x rdvd y \<longleftrightarrow> (\<exists>k::int. y = x * real_of_int k)"
 
 lemma int_rdvd_real: 
-  "real (i::int) rdvd x = (i dvd (floor x) \<and> real (floor x) = x)" (is "?l = ?r")
+  "real_of_int (i::int) rdvd x = (i dvd (floor x) \<and> real_of_int (floor x) = x)" (is "?l = ?r")
 proof
   assume "?l" 
-  hence th: "\<exists> k. x=real (i*k)" by (simp add: rdvd_def)
-  hence th': "real (floor x) = x" by (auto simp del: real_of_int_mult)
-  with th have "\<exists> k. real (floor x) = real (i*k)" by simp
-  hence "\<exists> k. floor x = i*k" by (simp only: real_of_int_inject)
+  hence th: "\<exists> k. x=real_of_int (i*k)" by (simp add: rdvd_def)
+  hence th': "real_of_int (floor x) = x" by (auto simp del: of_int_mult)
+  with th have "\<exists> k. real_of_int (floor x) = real_of_int (i*k)" by simp
+  hence "\<exists> k. floor x = i*k" by blast
   thus ?r  using th' by (simp add: dvd_def) 
 next
   assume "?r" hence "(i::int) dvd \<lfloor>x::real\<rfloor>" ..
-  hence "\<exists> k. real (floor x) = real (i*k)" 
-    by (simp only: real_of_int_inject) (simp add: dvd_def)
+  hence "\<exists> k. real_of_int (floor x) = real_of_int (i*k)"
+    using dvd_def by blast 
   thus ?l using \<open>?r\<close> by (simp add: rdvd_def)
 qed
 
-lemma int_rdvd_iff: "(real (i::int) rdvd real t) = (i dvd t)"
-  by (auto simp add: rdvd_def dvd_def) (rule_tac x="k" in exI, simp only: real_of_int_mult[symmetric])
-
-
-lemma rdvd_abs1: "(abs (real d) rdvd t) = (real (d ::int) rdvd t)"
+lemma int_rdvd_iff: "(real_of_int (i::int) rdvd real_of_int t) = (i dvd t)"
+  by (auto simp add: rdvd_def dvd_def) (rule_tac x="k" in exI, simp only: of_int_mult[symmetric])
+
+
+lemma rdvd_abs1: "(abs (real_of_int d) rdvd t) = (real_of_int (d ::int) rdvd t)"
 proof
-  assume d: "real d rdvd t"
-  from d int_rdvd_real have d2: "d dvd (floor t)" and ti: "real (floor t) = t"
+  assume d: "real_of_int d rdvd t"
+  from d int_rdvd_real have d2: "d dvd (floor t)" and ti: "real_of_int (floor t) = t"
     by auto
 
   from iffD2[OF abs_dvd_iff] d2 have "(abs d) dvd (floor t)" by blast
-  with ti int_rdvd_real[symmetric] have "real (abs d) rdvd t" by blast 
-  thus "abs (real d) rdvd t" by simp
+  with ti int_rdvd_real[symmetric] have "real_of_int (abs d) rdvd t" by blast 
+  thus "abs (real_of_int d) rdvd t" by simp
 next
-  assume "abs (real d) rdvd t" hence "real (abs d) rdvd t" by simp
-  with int_rdvd_real[where i="abs d" and x="t"] have d2: "abs d dvd floor t" and ti: "real (floor t) =t"
+  assume "abs (real_of_int d) rdvd t" hence "real_of_int (abs d) rdvd t" by simp
+  with int_rdvd_real[where i="abs d" and x="t"] have d2: "abs d dvd floor t" and ti: "real_of_int (floor t) =t"
     by auto
   from iffD1[OF abs_dvd_iff] d2 have "d dvd floor t" by blast
-  with ti int_rdvd_real[symmetric] show "real d rdvd t" by blast
+  with ti int_rdvd_real[symmetric] show "real_of_int d rdvd t" by blast
 qed
 
-lemma rdvd_minus: "(real (d::int) rdvd t) = (real d rdvd -t)"
+lemma rdvd_minus: "(real_of_int (d::int) rdvd t) = (real_of_int d rdvd -t)"
   apply (auto simp add: rdvd_def)
   apply (rule_tac x="-k" in exI, simp) 
   apply (rule_tac x="-k" in exI, simp)
@@ -98,7 +76,7 @@
 
 lemma rdvd_mult: 
   assumes knz: "k\<noteq>0"
-  shows "(real (n::int) * real (k::int) rdvd x * real k) = (real n rdvd x)"
+  shows "(real_of_int (n::int) * real_of_int (k::int) rdvd x * real_of_int k) = (real_of_int n rdvd x)"
   using knz by (simp add: rdvd_def)
 
   (*********************************************************************************)
@@ -122,18 +100,18 @@
 
   (* Semantics of numeral terms (num) *)
 primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real" where
-  "Inum bs (C c) = (real c)"
+  "Inum bs (C c) = (real_of_int c)"
 | "Inum bs (Bound n) = bs!n"
-| "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
+| "Inum bs (CN n c a) = (real_of_int c) * (bs!n) + (Inum bs a)"
 | "Inum bs (Neg a) = -(Inum bs a)"
 | "Inum bs (Add a b) = Inum bs a + Inum bs b"
 | "Inum bs (Sub a b) = Inum bs a - Inum bs b"
-| "Inum bs (Mul c a) = (real c) * Inum bs a"
-| "Inum bs (Floor a) = real (floor (Inum bs a))"
-| "Inum bs (CF c a b) = real c * real (floor (Inum bs a)) + Inum bs b"
-definition "isint t bs \<equiv> real (floor (Inum bs t)) = Inum bs t"
-
-lemma isint_iff: "isint n bs = (real (floor (Inum bs n)) = Inum bs n)"
+| "Inum bs (Mul c a) = (real_of_int c) * Inum bs a"
+| "Inum bs (Floor a) = real_of_int (floor (Inum bs a))"
+| "Inum bs (CF c a b) = real_of_int c * real_of_int (floor (Inum bs a)) + Inum bs b"
+definition "isint t bs \<equiv> real_of_int (floor (Inum bs t)) = Inum bs t"
+
+lemma isint_iff: "isint n bs = (real_of_int (floor (Inum bs n)) = Inum bs n)"
   by (simp add: isint_def)
 
 lemma isint_Floor: "isint (Floor n) bs"
@@ -143,10 +121,10 @@
 proof-
   let ?e = "Inum bs e"
   let ?fe = "floor ?e"
-  assume be: "isint e bs" hence efe:"real ?fe = ?e" by (simp add: isint_iff)
-  have "real ((floor (Inum bs (Mul c e)))) = real (floor (real (c * ?fe)))" using efe by simp
-  also have "\<dots> = real (c* ?fe)" by (simp only: floor_real_of_int) 
-  also have "\<dots> = real c * ?e" using efe by simp
+  assume be: "isint e bs" hence efe:"real_of_int ?fe = ?e" by (simp add: isint_iff)
+  have "real_of_int ((floor (Inum bs (Mul c e)))) = real_of_int (floor (real_of_int (c * ?fe)))" using efe by simp
+  also have "\<dots> = real_of_int (c* ?fe)" using floor_of_int by blast 
+  also have "\<dots> = real_of_int c * ?e" using efe by simp
   finally show ?thesis using isint_iff by simp
 qed
 
@@ -154,9 +132,9 @@
 proof-
   let ?I = "\<lambda> t. Inum bs t"
   assume ie: "isint e bs"
-  hence th: "real (floor (?I e)) = ?I e" by (simp add: isint_def)  
-  have "real (floor (?I (Neg e))) = real (floor (- (real (floor (?I e)))))" by (simp add: th)
-  also have "\<dots> = - real (floor (?I e))" by simp
+  hence th: "real_of_int (floor (?I e)) = ?I e" by (simp add: isint_def)  
+  have "real_of_int (floor (?I (Neg e))) = real_of_int (floor (- (real_of_int (floor (?I e)))))" by (simp add: th)
+  also have "\<dots> = - real_of_int (floor (?I e))" by simp
   finally show "isint (Neg e) bs" by (simp add: isint_def th)
 qed
 
@@ -164,9 +142,9 @@
   assumes ie: "isint e bs" shows "isint (Sub (C c) e) bs"
 proof-
   let ?I = "\<lambda> t. Inum bs t"
-  from ie have th: "real (floor (?I e)) = ?I e" by (simp add: isint_def)  
-  have "real (floor (?I (Sub (C c) e))) = real (floor ((real (c -floor (?I e)))))" by (simp add: th)
-  also have "\<dots> = real (c- floor (?I e))" by simp
+  from ie have th: "real_of_int (floor (?I e)) = ?I e" by (simp add: isint_def)  
+  have "real_of_int (floor (?I (Sub (C c) e))) = real_of_int (floor ((real_of_int (c -floor (?I e)))))" by (simp add: th)
+  also have "\<dots> = real_of_int (c- floor (?I e))" by simp
   finally show "isint (Sub (C c) e) bs" by (simp add: isint_def th)
 qed
 
@@ -176,9 +154,9 @@
 proof-
   let ?a = "Inum bs a"
   let ?b = "Inum bs b"
-  from ai bi isint_iff have "real (floor (?a + ?b)) = real (floor (real (floor ?a) + real (floor ?b)))"
+  from ai bi isint_iff have "real_of_int (floor (?a + ?b)) = real_of_int (floor (real_of_int (floor ?a) + real_of_int (floor ?b)))"
     by simp
-  also have "\<dots> = real (floor ?a) + real (floor ?b)" by simp
+  also have "\<dots> = real_of_int (floor ?a) + real_of_int (floor ?b)" by simp
   also have "\<dots> = ?a + ?b" using ai bi isint_iff by simp
   finally show "isint (Add a b) bs" by (simp add: isint_iff)
 qed
@@ -219,8 +197,8 @@
 | "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
 | "Ifm bs (Eq a) = (Inum bs a = 0)"
 | "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
-| "Ifm bs (Dvd i b) = (real i rdvd Inum bs b)"
-| "Ifm bs (NDvd i b) = (\<not>(real i rdvd Inum bs b))"
+| "Ifm bs (Dvd i b) = (real_of_int i rdvd Inum bs b)"
+| "Ifm bs (NDvd i b) = (\<not>(real_of_int i rdvd Inum bs b))"
 | "Ifm bs (NOT p) = (\<not> (Ifm bs p))"
 | "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
 | "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
@@ -607,7 +585,7 @@
 
 lemma reducecoeffh:
   assumes gt: "dvdnumcoeff t g" and gp: "g > 0" 
-  shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
+  shows "real_of_int g *(Inum bs (reducecoeffh t g)) = Inum bs t"
   using gt
 proof(induct t rule: reducecoeffh.induct) 
   case (1 i) hence gd: "g dvd i" by simp
@@ -708,7 +686,7 @@
   from dvdnumcoeff_aux[OF th1 th2 H] show "dvdnumcoeff t ?g" .
 qed
 
-lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
+lemma reducecoeff: "real_of_int (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
 proof-
   let ?g = "numgcd t"
   have "?g \<ge> 0"  by (simp add: numgcd_pos)
@@ -757,7 +735,7 @@
 apply (case_tac "lex_bnd t1 t2", simp_all)
  apply (case_tac "c1+c2 = 0")
   apply (case_tac "t1 = t2")
-   apply (simp_all add: algebra_simps distrib_right[symmetric] real_of_int_mult[symmetric] real_of_int_add[symmetric]del: real_of_int_mult real_of_int_add distrib_right)
+   apply (simp_all add: algebra_simps distrib_right[symmetric] of_int_mult[symmetric] of_int_add[symmetric]del: of_int_mult of_int_add distrib_right)
   done
 
 lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
@@ -852,15 +830,15 @@
     hence th1: "?n t = ?N (Add (Floor ?tv) ?ti)" 
       by (cases ?tv) (auto simp add: numfloor_def Let_def split_def)
     from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+
-    hence "?N (Floor t) = real (floor (?N (Add ?tv ?ti)))" by simp 
-    also have "\<dots> = real (floor (?N ?tv) + (floor (?N ?ti)))"
+    hence "?N (Floor t) = real_of_int (floor (?N (Add ?tv ?ti)))" by simp 
+    also have "\<dots> = real_of_int (floor (?N ?tv) + (floor (?N ?ti)))"
       by (simp,subst tii[simplified isint_iff, symmetric]) simp
     also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff])
     finally have ?thesis using th1 by simp}
   moreover {fix v assume H:"?tv = C v" 
     from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+
-    hence "?N (Floor t) = real (floor (?N (Add ?tv ?ti)))" by simp 
-    also have "\<dots> = real (floor (?N ?tv) + (floor (?N ?ti)))"
+    hence "?N (Floor t) = real_of_int (floor (?N (Add ?tv ?ti)))" by simp 
+    also have "\<dots> = real_of_int (floor (?N ?tv) + (floor (?N ?ti)))"
       by (simp,subst tii[simplified isint_iff, symmetric]) simp
     also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff])
     finally have ?thesis by (simp add: H numfloor_def Let_def split_def) }
@@ -951,7 +929,7 @@
       else (t',n))))"
 
 lemma simp_num_pair_ci:
-  shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))"
+  shows "((\<lambda> (t,n). Inum bs t / real_of_int n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real_of_int n) (t,n))"
   (is "?lhs = ?rhs")
 proof-
   let ?t' = "simpnum t"
@@ -975,12 +953,12 @@
         have gpdd: "?g' dvd n" by simp
         have gpdgp: "?g' dvd ?g'" by simp
         from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
-        have th2:"real ?g' * ?t = Inum bs ?t'" by simp
-        from nnz g1 g'1 have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def)
-        also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp
-        also have "\<dots> = (Inum bs ?t' / real n)"
+        have th2:"real_of_int ?g' * ?t = Inum bs ?t'" by simp
+        from nnz g1 g'1 have "?lhs = ?t / real_of_int (n div ?g')" by (simp add: simp_num_pair_def Let_def)
+        also have "\<dots> = (real_of_int ?g' * ?t) / (real_of_int ?g' * (real_of_int (n div ?g')))" by simp
+        also have "\<dots> = (Inum bs ?t' / real_of_int n)"
           using real_of_int_div[OF gpdd] th2 gp0 by simp
-        finally have "?lhs = Inum bs t / real n" by simp
+        finally have "?lhs = Inum bs t / real_of_int n" by simp
         then have ?thesis using nnz g1 g'1 by (simp add: simp_num_pair_def) }
       ultimately have ?thesis by blast }
     ultimately have ?thesis by blast }
@@ -1092,27 +1070,27 @@
 lemma check_int: "check_int t \<Longrightarrow> isint t bs"
   by (induct t) (auto simp add: isint_add isint_Floor isint_Mul isint_neg isint_c isint_CF)
 
-lemma rdvd_left1_int: "real \<lfloor>t\<rfloor> = t \<Longrightarrow> 1 rdvd t"
+lemma rdvd_left1_int: "real_of_int \<lfloor>t\<rfloor> = t \<Longrightarrow> 1 rdvd t"
   by (simp add: rdvd_def,rule_tac x="\<lfloor>t\<rfloor>" in exI) simp
 
 lemma rdvd_reduce: 
   assumes gd:"g dvd d" and gc:"g dvd c" and gp: "g > 0"
-  shows "real (d::int) rdvd real (c::int)*t = (real (d div g) rdvd real (c div g)*t)"
+  shows "real_of_int (d::int) rdvd real_of_int (c::int)*t = (real_of_int (d div g) rdvd real_of_int (c div g)*t)"
 proof
-  assume d: "real d rdvd real c * t"
-  from d rdvd_def obtain k where k_def: "real c * t = real d* real (k::int)" by auto
+  assume d: "real_of_int d rdvd real_of_int c * t"
+  from d rdvd_def obtain k where k_def: "real_of_int c * t = real_of_int d* real_of_int (k::int)" by auto
   from gd dvd_def obtain kd where kd_def: "d = g * kd" by auto
   from gc dvd_def obtain kc where kc_def: "c = g * kc" by auto
-  from k_def kd_def kc_def have "real g * real kc * t = real g * real kd * real k" by simp
-  hence "real kc * t = real kd * real k" using gp by simp
-  hence th:"real kd rdvd real kc * t" using rdvd_def by blast
+  from k_def kd_def kc_def have "real_of_int g * real_of_int kc * t = real_of_int g * real_of_int kd * real_of_int k" by simp
+  hence "real_of_int kc * t = real_of_int kd * real_of_int k" using gp by simp
+  hence th:"real_of_int kd rdvd real_of_int kc * t" using rdvd_def by blast
   from kd_def gp have th':"kd = d div g" by simp
   from kc_def gp have "kc = c div g" by simp
-  with th th' show "real (d div g) rdvd real (c div g) * t" by simp
+  with th th' show "real_of_int (d div g) rdvd real_of_int (c div g) * t" by simp
 next
-  assume d: "real (d div g) rdvd real (c div g) * t"
+  assume d: "real_of_int (d div g) rdvd real_of_int (c div g) * t"
   from gp have gnz: "g \<noteq> 0" by simp
-  thus "real d rdvd real c * t" using d rdvd_mult[OF gnz, where n="d div g" and x="real (c div g) * t"] real_of_int_div[OF gd] real_of_int_div[OF gc] by simp
+  thus "real_of_int d rdvd real_of_int c * t" using d rdvd_mult[OF gnz, where n="d div g" and x="real_of_int (c div g) * t"] real_of_int_div[OF gd] real_of_int_div[OF gc] by simp
 qed
 
 definition simpdvd :: "int \<Rightarrow> num \<Rightarrow> (int \<times> num)" where
@@ -1143,11 +1121,11 @@
       have gpdd: "?g' dvd d" by simp
       have gpdgp: "?g' dvd ?g'" by simp
       from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
-      have th2:"real ?g' * ?t = Inum bs t" by simp
+      have th2:"real_of_int ?g' * ?t = Inum bs t" by simp
       from assms g1 g0 g'1
       have "Ifm bs (Dvd (fst (simpdvd d t)) (snd(simpdvd d t))) = Ifm bs (Dvd (d div ?g') ?tt)"
         by (simp add: simpdvd_def Let_def)
-      also have "\<dots> = (real d rdvd (Inum bs t))"
+      also have "\<dots> = (real_of_int d rdvd (Inum bs t))"
         using rdvd_reduce[OF gpdd gpdgp g'p, where t="?t", simplified div_self[OF gp0]] 
           th2[symmetric] by simp
       finally have ?thesis by simp  }
@@ -1190,9 +1168,9 @@
     have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
     {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
     with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
-    hence gp: "real ?g > 0" by simp
-    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
-    with sa have "Inum bs a < 0 = (real ?g * ?r < real ?g * 0)" by simp
+    hence gp: "real_of_int ?g > 0" by simp
+    have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff)
+    with sa have "Inum bs a < 0 = (real_of_int ?g * ?r < real_of_int ?g * 0)" by simp
     also have "\<dots> = (?r < 0)" using gp
       by (simp only: mult_less_cancel_left) simp
     finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
@@ -1207,9 +1185,9 @@
     have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
     {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
     with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
-    hence gp: "real ?g > 0" by simp
-    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
-    with sa have "Inum bs a \<le> 0 = (real ?g * ?r \<le> real ?g * 0)" by simp
+    hence gp: "real_of_int ?g > 0" by simp
+    have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff)
+    with sa have "Inum bs a \<le> 0 = (real_of_int ?g * ?r \<le> real_of_int ?g * 0)" by simp
     also have "\<dots> = (?r \<le> 0)" using gp
       by (simp only: mult_le_cancel_left) simp
     finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
@@ -1224,9 +1202,9 @@
     have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
     {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
     with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
-    hence gp: "real ?g > 0" by simp
-    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
-    with sa have "Inum bs a > 0 = (real ?g * ?r > real ?g * 0)" by simp
+    hence gp: "real_of_int ?g > 0" by simp
+    have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff)
+    with sa have "Inum bs a > 0 = (real_of_int ?g * ?r > real_of_int ?g * 0)" by simp
     also have "\<dots> = (?r > 0)" using gp
       by (simp only: mult_less_cancel_left) simp
     finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
@@ -1241,9 +1219,9 @@
     have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
     {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
     with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
-    hence gp: "real ?g > 0" by simp
-    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
-    with sa have "Inum bs a \<ge> 0 = (real ?g * ?r \<ge> real ?g * 0)" by simp
+    hence gp: "real_of_int ?g > 0" by simp
+    have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff)
+    with sa have "Inum bs a \<ge> 0 = (real_of_int ?g * ?r \<ge> real_of_int ?g * 0)" by simp
     also have "\<dots> = (?r \<ge> 0)" using gp
       by (simp only: mult_le_cancel_left) simp
     finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
@@ -1258,9 +1236,9 @@
     have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
     {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
     with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
-    hence gp: "real ?g > 0" by simp
-    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
-    with sa have "Inum bs a = 0 = (real ?g * ?r = 0)" by simp
+    hence gp: "real_of_int ?g > 0" by simp
+    have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff)
+    with sa have "Inum bs a = 0 = (real_of_int ?g * ?r = 0)" by simp
     also have "\<dots> = (?r = 0)" using gp
       by simp
     finally have ?case using H by (cases "?sa" , simp_all add: Let_def)}
@@ -1275,9 +1253,9 @@
     have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz)
     {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto}
     with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto)
-    hence gp: "real ?g > 0" by simp
-    have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff)
-    with sa have "Inum bs a \<noteq> 0 = (real ?g * ?r \<noteq> 0)" by simp
+    hence gp: "real_of_int ?g > 0" by simp
+    have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff)
+    with sa have "Inum bs a \<noteq> 0 = (real_of_int ?g * ?r \<noteq> 0)" by simp
     also have "\<dots> = (?r \<noteq> 0)" using gp
       by simp
     finally have ?case using H by (cases "?sa") (simp_all add: Let_def) }
@@ -1471,7 +1449,7 @@
 termination by (relation "measure num_size") auto
 
 lemma zsplit0_I:
-  shows "\<And> n a. zsplit0 t = (n,a) \<Longrightarrow> (Inum ((real (x::int)) #bs) (CN 0 n a) = Inum (real x #bs) t) \<and> numbound0 a"
+  shows "\<And> n a. zsplit0 t = (n,a) \<Longrightarrow> (Inum ((real_of_int (x::int)) #bs) (CN 0 n a) = Inum (real_of_int x #bs) t) \<and> numbound0 a"
   (is "\<And> n a. ?S t = (n,a) \<Longrightarrow> (?I x (CN 0 n a) = ?I x t) \<and> ?N a")
 proof(induct t rule: zsplit0.induct)
   case (1 c n a) thus ?case by auto 
@@ -1500,7 +1478,7 @@
   ultimately have th: "a=Add ?as ?at \<and> n=?ns + ?nt" using 6
     by (simp add: Let_def split_def)
   from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
-  from 6 have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \<and> numbound0 xb)" by blast (*FIXME*)
+  from 6 have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real_of_int x # bs) (CN 0 xa xb) = Inum (real_of_int x # bs) t \<and> numbound0 xb)" by blast (*FIXME*)
   with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
   from abjs 6  have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
   from th3[simplified] th2[simplified] th[simplified] show ?case 
@@ -1516,7 +1494,7 @@
   ultimately have th: "a=Sub ?as ?at \<and> n=?ns - ?nt" using 7
     by (simp add: Let_def split_def)
   from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
-  from 7 have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \<and> numbound0 xb)" by blast (*FIXME*)
+  from 7 have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real_of_int x # bs) (CN 0 xa xb) = Inum (real_of_int x # bs) t \<and> numbound0 xb)" by blast (*FIXME*)
   with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
   from abjs 7 have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
   from th3[simplified] th2[simplified] th[simplified] show ?case 
@@ -1528,7 +1506,7 @@
   have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Mul i ?at \<and> n=i*?nt" using 8
     by (simp add: Let_def split_def)
   from abj 8 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
-  hence "?I x (Mul i t) = (real i) * ?I x (CN 0 ?nt ?at)" by simp
+  hence "?I x (Mul i t) = (real_of_int i) * ?I x (CN 0 ?nt ?at)" by simp
   also have "\<dots> = ?I x (CN 0 (i*?nt) (Mul i ?at))" by (simp add: distrib_left)
   finally show ?case using th th2 by simp
 next
@@ -1539,13 +1517,13 @@
     by (simp add: Let_def split_def)
   from abj 9 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
   hence na: "?N a" using th by simp
-  have th': "(real ?nt)*(real x) = real (?nt * x)" by simp
+  have th': "(real_of_int ?nt)*(real_of_int x) = real_of_int (?nt * x)" by simp
   have "?I x (Floor t) = ?I x (Floor (CN 0 ?nt ?at))" using th2 by simp
-  also have "\<dots> = real (floor ((real ?nt)* real(x) + ?I x ?at))" by simp
-  also have "\<dots> = real (floor (?I x ?at + real (?nt* x)))" by (simp add: ac_simps)
-  also have "\<dots> = real (floor (?I x ?at) + (?nt* x))" 
-    using floor_add[where x="?I x ?at" and a="?nt* x"] by simp 
-  also have "\<dots> = real (?nt)*(real x) + real (floor (?I x ?at))" by (simp add: ac_simps)
+  also have "\<dots> = real_of_int (floor ((real_of_int ?nt)* real_of_int(x) + ?I x ?at))" by simp
+  also have "\<dots> = real_of_int (floor (?I x ?at + real_of_int (?nt* x)))" by (simp add: ac_simps)
+  also have "\<dots> = real_of_int (floor (?I x ?at) + (?nt* x))" 
+    using floor_add_of_int[of "?I x ?at" "?nt* x"] by simp 
+  also have "\<dots> = real_of_int (?nt)*(real_of_int x) + real_of_int (floor (?I x ?at))" by (simp add: ac_simps)
   finally have "?I x (Floor t) = ?I x (CN 0 n a)" using th by simp
   with na show ?case by simp
 qed
@@ -1643,72 +1621,72 @@
   "zlfm p = p" (hints simp add: fmsize_pos)
 
 lemma split_int_less_real: 
-  "(real (a::int) < b) = (a < floor b \<or> (a = floor b \<and> real (floor b) < b))"
+  "(real_of_int (a::int) < b) = (a < floor b \<or> (a = floor b \<and> real_of_int (floor b) < b))"
 proof( auto)
-  assume alb: "real a < b" and agb: "\<not> a < floor b"
-  from agb have "floor b \<le> a" by simp hence th: "b < real a + 1" by (simp only: floor_le_eq)
+  assume alb: "real_of_int a < b" and agb: "\<not> a < floor b"
+  from agb have "floor b \<le> a" by simp hence th: "b < real_of_int a + 1" by (simp only: floor_le_iff)
   from floor_eq[OF alb th] show "a= floor b" by simp 
 next
   assume alb: "a < floor b"
-  hence "real a < real (floor b)" by simp
-  moreover have "real (floor b) \<le> b" by simp ultimately show  "real a < b" by arith 
+  hence "real_of_int a < real_of_int (floor b)" by simp
+  moreover have "real_of_int (floor b) \<le> b" by simp ultimately show  "real_of_int a < b" by arith 
 qed
 
 lemma split_int_less_real': 
-  "(real (a::int) + b < 0) = (real a - real (floor(-b)) < 0 \<or> (real a - real (floor (-b)) = 0 \<and> real (floor (-b)) + b < 0))"
+  "(real_of_int (a::int) + b < 0) = (real_of_int a - real_of_int (floor(-b)) < 0 \<or> (real_of_int a - real_of_int (floor (-b)) = 0 \<and> real_of_int (floor (-b)) + b < 0))"
 proof- 
-  have "(real a + b <0) = (real a < -b)" by arith
+  have "(real_of_int a + b <0) = (real_of_int a < -b)" by arith
   with split_int_less_real[where a="a" and b="-b"] show ?thesis by arith  
 qed
 
 lemma split_int_gt_real': 
-  "(real (a::int) + b > 0) = (real a + real (floor b) > 0 \<or> (real a + real (floor b) = 0 \<and> real (floor b) - b < 0))"
+  "(real_of_int (a::int) + b > 0) = (real_of_int a + real_of_int (floor b) > 0 \<or> (real_of_int a + real_of_int (floor b) = 0 \<and> real_of_int (floor b) - b < 0))"
 proof- 
-  have th: "(real a + b >0) = (real (-a) + (-b)< 0)" by arith
-  show ?thesis using myless[of _ "real (floor b)"] 
+  have th: "(real_of_int a + b >0) = (real_of_int (-a) + (-b)< 0)" by arith
+  show ?thesis using myless[of _ "real_of_int (floor b)"] 
     by (simp only:th split_int_less_real'[where a="-a" and b="-b"]) 
     (simp add: algebra_simps,arith)
 qed
 
 lemma split_int_le_real: 
-  "(real (a::int) \<le> b) = (a \<le> floor b \<or> (a = floor b \<and> real (floor b) < b))"
+  "(real_of_int (a::int) \<le> b) = (a \<le> floor b \<or> (a = floor b \<and> real_of_int (floor b) < b))"
 proof( auto)
-  assume alb: "real a \<le> b" and agb: "\<not> a \<le> floor b"
-  from alb have "floor (real a) \<le> floor b " by (simp only: floor_mono) 
+  assume alb: "real_of_int a \<le> b" and agb: "\<not> a \<le> floor b"
+  from alb have "floor (real_of_int a) \<le> floor b " by (simp only: floor_mono) 
   hence "a \<le> floor b" by simp with agb show "False" by simp
 next
   assume alb: "a \<le> floor b"
-  hence "real a \<le> real (floor b)" by (simp only: floor_mono)
-  also have "\<dots>\<le> b" by simp  finally show  "real a \<le> b" . 
+  hence "real_of_int a \<le> real_of_int (floor b)" by (simp only: floor_mono)
+  also have "\<dots>\<le> b" by simp  finally show  "real_of_int a \<le> b" . 
 qed
 
 lemma split_int_le_real': 
-  "(real (a::int) + b \<le> 0) = (real a - real (floor(-b)) \<le> 0 \<or> (real a - real (floor (-b)) = 0 \<and> real (floor (-b)) + b < 0))"
+  "(real_of_int (a::int) + b \<le> 0) = (real_of_int a - real_of_int (floor(-b)) \<le> 0 \<or> (real_of_int a - real_of_int (floor (-b)) = 0 \<and> real_of_int (floor (-b)) + b < 0))"
 proof- 
-  have "(real a + b \<le>0) = (real a \<le> -b)" by arith
+  have "(real_of_int a + b \<le>0) = (real_of_int a \<le> -b)" by arith
   with split_int_le_real[where a="a" and b="-b"] show ?thesis by arith  
 qed
 
 lemma split_int_ge_real': 
-  "(real (a::int) + b \<ge> 0) = (real a + real (floor b) \<ge> 0 \<or> (real a + real (floor b) = 0 \<and> real (floor b) - b < 0))"
+  "(real_of_int (a::int) + b \<ge> 0) = (real_of_int a + real_of_int (floor b) \<ge> 0 \<or> (real_of_int a + real_of_int (floor b) = 0 \<and> real_of_int (floor b) - b < 0))"
 proof- 
-  have th: "(real a + b \<ge>0) = (real (-a) + (-b) \<le> 0)" by arith
+  have th: "(real_of_int a + b \<ge>0) = (real_of_int (-a) + (-b) \<le> 0)" by arith
   show ?thesis by (simp only: th split_int_le_real'[where a="-a" and b="-b"])
     (simp add: algebra_simps ,arith)
 qed
 
-lemma split_int_eq_real: "(real (a::int) = b) = ( a = floor b \<and> b = real (floor b))" (is "?l = ?r")
+lemma split_int_eq_real: "(real_of_int (a::int) = b) = ( a = floor b \<and> b = real_of_int (floor b))" (is "?l = ?r")
 by auto
 
-lemma split_int_eq_real': "(real (a::int) + b = 0) = ( a - floor (-b) = 0 \<and> real (floor (-b)) + b = 0)" (is "?l = ?r")
+lemma split_int_eq_real': "(real_of_int (a::int) + b = 0) = ( a - floor (-b) = 0 \<and> real_of_int (floor (-b)) + b = 0)" (is "?l = ?r")
 proof-
-  have "?l = (real a = -b)" by arith
+  have "?l = (real_of_int a = -b)" by arith
   with split_int_eq_real[where a="a" and b="-b"] show ?thesis by simp arith
 qed
 
 lemma zlfm_I:
   assumes qfp: "qfree p"
-  shows "(Ifm (real i #bs) (zlfm p) = Ifm (real i# bs) p) \<and> iszlfm (zlfm p) (real (i::int) #bs)"
+  shows "(Ifm (real_of_int i #bs) (zlfm p) = Ifm (real_of_int i# bs) p) \<and> iszlfm (zlfm p) (real_of_int (i::int) #bs)"
   (is "(?I (?l p) = ?I p) \<and> ?L (?l p)")
 using qfp
 proof(induct p rule: zlfm.induct)
@@ -1717,8 +1695,8 @@
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
   from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
   {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
@@ -1726,13 +1704,13 @@
   moreover
   {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Lt a) = (real (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def)
+    have "?I (Lt a) = (real_of_int (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def)
     also have "\<dots> = (?I (?l (Lt a)))" apply (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) by (simp add: Ia cp cnz Let_def split_def)
     finally have ?case using l by simp}
   moreover
   {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Lt a) = (real (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def)
+    have "?I (Lt a) = (real_of_int (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def)
     also from cn cnz have "\<dots> = (?I (?l (Lt a)))" by (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def ac_simps, arith)
     finally have ?case using l by simp}
   ultimately show ?case by blast
@@ -1742,8 +1720,8 @@
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
   from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
   {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
@@ -1751,13 +1729,13 @@
   moreover
   {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Le a) = (real (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def)
+    have "?I (Le a) = (real_of_int (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def)
     also have "\<dots> = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def)
     finally have ?case using l by simp}
   moreover
   {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Le a) = (real (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def)
+    have "?I (Le a) = (real_of_int (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def)
     also from cn cnz have "\<dots> = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def ac_simps, arith)
     finally have ?case using l by simp}
   ultimately show ?case by blast
@@ -1767,8 +1745,8 @@
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
   from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
   {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
@@ -1776,13 +1754,13 @@
   moreover
   {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Gt a) = (real (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def)
+    have "?I (Gt a) = (real_of_int (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def)
     also have "\<dots> = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def)
     finally have ?case using l by simp}
   moreover
   {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Gt a) = (real (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def)
+    have "?I (Gt a) = (real_of_int (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def)
     also from cn cnz have "\<dots> = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def ac_simps, arith)
     finally have ?case using l by simp}
   ultimately show ?case by blast
@@ -1792,8 +1770,8 @@
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
   from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
   {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
@@ -1801,13 +1779,13 @@
   moreover
   {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Ge a) = (real (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def)
+    have "?I (Ge a) = (real_of_int (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def)
     also have "\<dots> = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def)
     finally have ?case using l by simp}
   moreover
   {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Ge a) = (real (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def)
+    have "?I (Ge a) = (real_of_int (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def)
     also from cn cnz have "\<dots> = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def ac_simps, arith)
     finally have ?case using l by simp}
   ultimately show ?case by blast
@@ -1817,8 +1795,8 @@
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
   from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
   {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
@@ -1826,14 +1804,14 @@
   moreover
   {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Eq a) = (real (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (?I (?l (Eq a)))" using cp cnz  by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult)
+    have "?I (Eq a) = (real_of_int (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def)
+    also have "\<dots> = (?I (?l (Eq a)))" using cp cnz  by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia of_int_mult[symmetric] del: of_int_mult)
     finally have ?case using l by simp}
   moreover
   {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Eq a) = (real (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def)
-    also from cn cnz have "\<dots> = (?I (?l (Eq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult,arith)
+    have "?I (Eq a) = (real_of_int (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def)
+    also from cn cnz have "\<dots> = (?I (?l (Eq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia of_int_mult[symmetric] del: of_int_mult,arith)
     finally have ?case using l by simp}
   ultimately show ?case by blast
 next
@@ -1842,8 +1820,8 @@
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
   from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
   {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
@@ -1851,14 +1829,14 @@
   moreover
   {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (NEq a) = (real (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (?I (?l (NEq a)))" using cp cnz  by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult)
+    have "?I (NEq a) = (real_of_int (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def)
+    also have "\<dots> = (?I (?l (NEq a)))" using cp cnz  by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia of_int_mult[symmetric] del: of_int_mult)
     finally have ?case using l by simp}
   moreover
   {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (NEq a) = (real (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def)
-    also from cn cnz have "\<dots> = (?I (?l (NEq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult,arith)
+    have "?I (NEq a) = (real_of_int (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def)
+    also from cn cnz have "\<dots> = (?I (?l (NEq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia of_int_mult[symmetric] del: of_int_mult,arith)
     finally have ?case using l by simp}
   ultimately show ?case by blast
 next
@@ -1867,8 +1845,8 @@
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
   from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
   { assume j: "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def) 
@@ -1880,31 +1858,31 @@
   moreover
   {assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Dvd j a) = (real j rdvd (real (?c * i) + (?N ?r)))" 
+    have "?I (Dvd j a) = (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r)))" 
       using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (real (abs j) rdvd real (?c*i) + (?N ?r))" 
-      by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp
-    also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and> 
-       (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r))))" 
-      by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: ac_simps)
+    also have "\<dots> = (real_of_int (abs j) rdvd real_of_int (?c*i) + (?N ?r))" 
+      by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp
+    also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real_of_int (?c*i))) \<and> 
+       (real_of_int (floor ((?N ?r) + real_of_int (?c*i))) = (real_of_int (?c*i) + (?N ?r))))" 
+      by(simp only: int_rdvd_real[where i="abs j" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps)
     also have "\<dots> = (?I (?l (Dvd j a)))" using cp cnz jnz  
       by (simp add: Let_def split_def int_rdvd_iff[symmetric]  
-        del: real_of_int_mult) (auto simp add: ac_simps)
+        del: of_int_mult) (auto simp add: ac_simps)
     finally have ?case using l jnz  by simp }
   moreover
   {assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Dvd j a) = (real j rdvd (real (?c * i) + (?N ?r)))" 
+    have "?I (Dvd j a) = (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r)))" 
       using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (real (abs j) rdvd real (?c*i) + (?N ?r))" 
-      by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp
-    also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and> 
-       (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r))))" 
-      by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: ac_simps)
+    also have "\<dots> = (real_of_int (abs j) rdvd real_of_int (?c*i) + (?N ?r))" 
+      by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp
+    also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real_of_int (?c*i))) \<and> 
+       (real_of_int (floor ((?N ?r) + real_of_int (?c*i))) = (real_of_int (?c*i) + (?N ?r))))" 
+      by(simp only: int_rdvd_real[where i="abs j" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps)
     also have "\<dots> = (?I (?l (Dvd j a)))" using cn cnz jnz
-      using rdvd_minus [where d="abs j" and t="real (?c*i + floor (?N ?r))", simplified, symmetric]
+      using rdvd_minus [where d="abs j" and t="real_of_int (?c*i + floor (?N ?r))", simplified, symmetric]
       by (simp add: Let_def split_def int_rdvd_iff[symmetric]  
-        del: real_of_int_mult) (auto simp add: ac_simps)
+        del: of_int_mult) (auto simp add: ac_simps)
     finally have ?case using l jnz by blast }
   ultimately show ?case by blast
 next
@@ -1913,8 +1891,8 @@
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
   from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
-  let ?N = "\<lambda> t. Inum (real i#bs) t"
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
   {assume j: "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def) 
@@ -1926,31 +1904,31 @@
   moreover
   {assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (NDvd j a) = (\<not> (real j rdvd (real (?c * i) + (?N ?r))))" 
+    have "?I (NDvd j a) = (\<not> (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r))))" 
       using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (\<not> (real (abs j) rdvd real (?c*i) + (?N ?r)))" 
-      by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp
-    also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and> 
-       (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r)))))" 
-      by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: ac_simps)
+    also have "\<dots> = (\<not> (real_of_int (abs j) rdvd real_of_int (?c*i) + (?N ?r)))" 
+      by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp
+    also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real_of_int (?c*i))) \<and> 
+       (real_of_int (floor ((?N ?r) + real_of_int (?c*i))) = (real_of_int (?c*i) + (?N ?r)))))" 
+      by(simp only: int_rdvd_real[where i="abs j" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps)
     also have "\<dots> = (?I (?l (NDvd j a)))" using cp cnz jnz  
       by (simp add: Let_def split_def int_rdvd_iff[symmetric]  
-        del: real_of_int_mult) (auto simp add: ac_simps)
+        del: of_int_mult) (auto simp add: ac_simps)
     finally have ?case using l jnz  by simp }
   moreover
   {assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))" 
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (NDvd j a) = (\<not> (real j rdvd (real (?c * i) + (?N ?r))))" 
+    have "?I (NDvd j a) = (\<not> (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r))))" 
       using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (\<not> (real (abs j) rdvd real (?c*i) + (?N ?r)))" 
-      by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp
-    also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and> 
-       (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r)))))" 
-      by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: ac_simps)
+    also have "\<dots> = (\<not> (real_of_int (abs j) rdvd real_of_int (?c*i) + (?N ?r)))" 
+      by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp
+    also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real_of_int (?c*i))) \<and> 
+       (real_of_int (floor ((?N ?r) + real_of_int (?c*i))) = (real_of_int (?c*i) + (?N ?r)))))" 
+      by(simp only: int_rdvd_real[where i="abs j" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps)
     also have "\<dots> = (?I (?l (NDvd j a)))" using cn cnz jnz
-      using rdvd_minus [where d="abs j" and t="real (?c*i + floor (?N ?r))", simplified, symmetric]
+      using rdvd_minus [where d="abs j" and t="real_of_int (?c*i + floor (?N ?r))", simplified, symmetric]
       by (simp add: Let_def split_def int_rdvd_iff[symmetric]  
-        del: real_of_int_mult) (auto simp add: ac_simps)
+        del: of_int_mult) (auto simp add: ac_simps)
     finally have ?case using l jnz by blast }
   ultimately show ?case by blast
 qed auto
@@ -2040,7 +2018,7 @@
 
 lemma minusinf_inf:
   assumes linp: "iszlfm p (a # bs)"
-  shows "\<exists> (z::int). \<forall> x < z. Ifm ((real x)#bs) (minusinf p) = Ifm ((real x)#bs) p"
+  shows "\<exists> (z::int). \<forall> x < z. Ifm ((real_of_int x)#bs) (minusinf p) = Ifm ((real_of_int x)#bs) p"
   (is "?P p" is "\<exists> (z::int). \<forall> x < z. ?I x (?M p) = ?I x p")
 using linp
 proof (induct p rule: minusinf.induct)
@@ -2064,173 +2042,173 @@
 next
   case (3 c e) 
   then have "c > 0" by simp
-  hence rcpos: "real c > 0" by simp
+  hence rcpos: "real_of_int c > 0" by simp
   from 3 have nbe: "numbound0 e" by simp
   fix y
-  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Eq (CN 0 c e))) = ?I x (Eq (CN 0 c e))"
-  proof (simp add: less_floor_eq , rule allI, rule impI) 
+  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real_of_int c))). ?I x (?M (Eq (CN 0 c e))) = ?I x (Eq (CN 0 c e))"
+  proof (simp add: less_floor_iff , rule allI, rule impI) 
     fix x :: int
-    assume A: "real x + 1 \<le> - (Inum (y # bs) e / real c)"
-    hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
-    with rcpos  have "(real c)*(real  x) < (real c)*(- (Inum (y # bs) e / real c))"
+    assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)"
+    hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp
+    with rcpos  have "(real_of_int c)*(real_of_int  x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))"
       by (simp only: mult_strict_left_mono [OF th1 rcpos])
-    hence "real c * real x + Inum (y # bs) e \<noteq> 0"using rcpos  by simp
-    thus "real c * real x + Inum (real x # bs) e \<noteq> 0" 
-      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"]  by simp
+    hence "real_of_int c * real_of_int x + Inum (y # bs) e \<noteq> 0"using rcpos  by simp
+    thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0" 
+      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"]  by simp
   qed
   thus ?case by blast
 next
   case (4 c e) 
-  then have "c > 0" by simp hence rcpos: "real c > 0" by simp
+  then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp
   from 4 have nbe: "numbound0 e" by simp
   fix y
-  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (NEq (CN 0 c e))) = ?I x (NEq (CN 0 c e))"
-  proof (simp add: less_floor_eq , rule allI, rule impI) 
+  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real_of_int c))). ?I x (?M (NEq (CN 0 c e))) = ?I x (NEq (CN 0 c e))"
+  proof (simp add: less_floor_iff , rule allI, rule impI) 
     fix x :: int
-    assume A: "real x + 1 \<le> - (Inum (y # bs) e / real c)"
-    hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
-    with rcpos  have "(real c)*(real  x) < (real c)*(- (Inum (y # bs) e / real c))"
+    assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)"
+    hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp
+    with rcpos  have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))"
       by (simp only: mult_strict_left_mono [OF th1 rcpos])
-    hence "real c * real x + Inum (y # bs) e \<noteq> 0"using rcpos  by simp
-    thus "real c * real x + Inum (real x # bs) e \<noteq> 0" 
-      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"]  by simp
+    hence "real_of_int c * real_of_int x + Inum (y # bs) e \<noteq> 0"using rcpos  by simp
+    thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0" 
+      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"]  by simp
   qed
   thus ?case by blast
 next
   case (5 c e) 
-  then have "c > 0" by simp hence rcpos: "real c > 0" by simp
+  then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp
   from 5 have nbe: "numbound0 e" by simp
   fix y
-  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Lt (CN 0 c e))) = ?I x (Lt (CN 0 c e))"
-  proof (simp add: less_floor_eq , rule allI, rule impI) 
+  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real_of_int c))). ?I x (?M (Lt (CN 0 c e))) = ?I x (Lt (CN 0 c e))"
+  proof (simp add: less_floor_iff , rule allI, rule impI) 
     fix x :: int
-    assume A: "real x + 1 \<le> - (Inum (y # bs) e / real c)"
-    hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
-    with rcpos  have "(real c)*(real  x) < (real c)*(- (Inum (y # bs) e / real c))"
+    assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)"
+    hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp
+    with rcpos  have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))"
       by (simp only: mult_strict_left_mono [OF th1 rcpos])
-    thus "real c * real x + Inum (real x # bs) e < 0" 
-      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp
+    thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e < 0" 
+      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp
   qed
   thus ?case by blast
 next
   case (6 c e) 
-  then have "c > 0" by simp hence rcpos: "real c > 0" by simp
+  then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp
   from 6 have nbe: "numbound0 e" by simp
   fix y
-  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Le (CN 0 c e))) = ?I x (Le (CN 0 c e))"
-  proof (simp add: less_floor_eq , rule allI, rule impI) 
+  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real_of_int c))). ?I x (?M (Le (CN 0 c e))) = ?I x (Le (CN 0 c e))"
+  proof (simp add: less_floor_iff , rule allI, rule impI) 
     fix x :: int
-    assume A: "real x + 1 \<le> - (Inum (y # bs) e / real c)"
-    hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
-    with rcpos  have "(real c)*(real  x) < (real c)*(- (Inum (y # bs) e / real c))"
+    assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)"
+    hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp
+    with rcpos  have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))"
       by (simp only: mult_strict_left_mono [OF th1 rcpos])
-    thus "real c * real x + Inum (real x # bs) e \<le> 0" 
-      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp
+    thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<le> 0" 
+      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp
   qed
   thus ?case by blast
 next
   case (7 c e) 
-  then have "c > 0" by simp hence rcpos: "real c > 0" by simp
+  then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp
   from 7 have nbe: "numbound0 e" by simp
   fix y
-  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Gt (CN 0 c e))) = ?I x (Gt (CN 0 c e))"
-  proof (simp add: less_floor_eq , rule allI, rule impI) 
+  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real_of_int c))). ?I x (?M (Gt (CN 0 c e))) = ?I x (Gt (CN 0 c e))"
+  proof (simp add: less_floor_iff , rule allI, rule impI) 
     fix x :: int
-    assume A: "real x + 1 \<le> - (Inum (y # bs) e / real c)"
-    hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
-    with rcpos  have "(real c)*(real  x) < (real c)*(- (Inum (y # bs) e / real c))"
+    assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)"
+    hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp
+    with rcpos  have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))"
       by (simp only: mult_strict_left_mono [OF th1 rcpos])
-    thus "\<not> (real c * real x + Inum (real x # bs) e>0)" 
-      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp
+    thus "\<not> (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e>0)" 
+      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp
   qed
   thus ?case by blast
 next
   case (8 c e) 
-  then have "c > 0" by simp hence rcpos: "real c > 0" by simp
+  then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp
   from 8 have nbe: "numbound0 e" by simp
   fix y
-  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Ge (CN 0 c e))) = ?I x (Ge (CN 0 c e))"
-  proof (simp add: less_floor_eq , rule allI, rule impI) 
+  have "\<forall> x < (floor (- (Inum (y#bs) e) / (real_of_int c))). ?I x (?M (Ge (CN 0 c e))) = ?I x (Ge (CN 0 c e))"
+  proof (simp add: less_floor_iff , rule allI, rule impI) 
     fix x :: int
-    assume A: "real x + 1 \<le> - (Inum (y # bs) e / real c)"
-    hence th1:"real x < - (Inum (y # bs) e / real c)" by simp
-    with rcpos  have "(real c)*(real  x) < (real c)*(- (Inum (y # bs) e / real c))"
+    assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)"
+    hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp
+    with rcpos  have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))"
       by (simp only: mult_strict_left_mono [OF th1 rcpos])
-    thus "\<not> real c * real x + Inum (real x # bs) e \<ge> 0" 
-      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp
+    thus "\<not> real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<ge> 0" 
+      using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp
   qed
   thus ?case by blast
 qed simp_all
 
 lemma minusinf_repeats:
   assumes d: "d_\<delta> p d" and linp: "iszlfm p (a # bs)"
-  shows "Ifm ((real(x - k*d))#bs) (minusinf p) = Ifm (real x #bs) (minusinf p)"
+  shows "Ifm ((real_of_int(x - k*d))#bs) (minusinf p) = Ifm (real_of_int x #bs) (minusinf p)"
 using linp d
 proof(induct p rule: iszlfm.induct) 
   case (9 i c e) hence nbe: "numbound0 e"  and id: "i dvd d" by simp+
     hence "\<exists> k. d=i*k" by (simp add: dvd_def)
     then obtain "di" where di_def: "d=i*di" by blast
     show ?case 
-    proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real x - real k * real d" and b'="real x"] right_diff_distrib, rule iffI)
+    proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real_of_int x - real_of_int k * real_of_int d" and b'="real_of_int x"] right_diff_distrib, rule iffI)
       assume 
-        "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e"
+        "real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int x # bs) e"
       (is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt")
-      hence "\<exists> (l::int). ?rt = ?ri * (real l)" by (simp add: rdvd_def)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real l)+?rc*(?rk * (real i) * (real di))" 
+      hence "\<exists> (l::int). ?rt = ?ri * (real_of_int l)" by (simp add: rdvd_def)
+      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int l)+?rc*(?rk * (real_of_int i) * (real_of_int di))" 
         by (simp add: algebra_simps di_def)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real (l + c*k*di))"
+      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int (l + c*k*di))"
         by (simp add: algebra_simps)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real l)" by blast
-      thus "real i rdvd real c * real x + Inum (real x # bs) e" using rdvd_def by simp
+      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real_of_int l)" by blast
+      thus "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" using rdvd_def by simp
     next
       assume 
-        "real i rdvd real c * real x + Inum (real x # bs) e" (is "?ri rdvd ?rc*?rx+?e")
-      hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real l)" by (simp add: rdvd_def)
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real d)" by simp
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real i * real di)" by (simp add: di_def)
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: algebra_simps)
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l)"
+        "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" (is "?ri rdvd ?rc*?rx+?e")
+      hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real_of_int l)" by (simp add: rdvd_def)
+      hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int d)" by simp
+      hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int i * real_of_int di)" by (simp add: di_def)
+      hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int (l - c*k*di))" by (simp add: algebra_simps)
+      hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l)"
         by blast
-      thus "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" using rdvd_def by simp
+      thus "real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int x # bs) e" using rdvd_def by simp
     qed
 next
   case (10 i c e) hence nbe: "numbound0 e"  and id: "i dvd d" by simp+
     hence "\<exists> k. d=i*k" by (simp add: dvd_def)
     then obtain "di" where di_def: "d=i*di" by blast
     show ?case 
-    proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real x - real k * real d" and b'="real x"] right_diff_distrib, rule iffI)
+    proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real_of_int x - real_of_int k * real_of_int d" and b'="real_of_int x"] right_diff_distrib, rule iffI)
       assume 
-        "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e"
+        "real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int x # bs) e"
       (is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt")
-      hence "\<exists> (l::int). ?rt = ?ri * (real l)" by (simp add: rdvd_def)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real l)+?rc*(?rk * (real i) * (real di))" 
+      hence "\<exists> (l::int). ?rt = ?ri * (real_of_int l)" by (simp add: rdvd_def)
+      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int l)+?rc*(?rk * (real_of_int i) * (real_of_int di))" 
         by (simp add: algebra_simps di_def)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real (l + c*k*di))"
+      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int (l + c*k*di))"
         by (simp add: algebra_simps)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real l)" by blast
-      thus "real i rdvd real c * real x + Inum (real x # bs) e" using rdvd_def by simp
+      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real_of_int l)" by blast
+      thus "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" using rdvd_def by simp
     next
       assume 
-        "real i rdvd real c * real x + Inum (real x # bs) e" (is "?ri rdvd ?rc*?rx+?e")
-      hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real l)"
+        "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" (is "?ri rdvd ?rc*?rx+?e")
+      hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real_of_int l)"
         by (simp add: rdvd_def)
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real d)"
+      hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int d)"
         by simp
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real i * real di)"
+      hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int i * real_of_int di)"
         by (simp add: di_def)
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))"
+      hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int (l - c*k*di))"
         by (simp add: algebra_simps)
-      hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l)"
+      hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l)"
         by blast
-      thus "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e"
+      thus "real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int x # bs) e"
         using rdvd_def by simp
     qed
-qed (auto simp add: numbound0_I[where bs="bs" and b="real(x - k*d)" and b'="real x"] simp del: real_of_int_mult real_of_int_diff)
+qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int(x - k*d)" and b'="real_of_int x"] simp del: of_int_mult of_int_diff)
 
 lemma minusinf_ex:
-  assumes lin: "iszlfm p (real (a::int) #bs)"
-  and exmi: "\<exists> (x::int). Ifm (real x#bs) (minusinf p)" (is "\<exists> x. ?P1 x")
-  shows "\<exists> (x::int). Ifm (real x#bs) p" (is "\<exists> x. ?P x")
+  assumes lin: "iszlfm p (real_of_int (a::int) #bs)"
+  and exmi: "\<exists> (x::int). Ifm (real_of_int x#bs) (minusinf p)" (is "\<exists> x. ?P1 x")
+  shows "\<exists> (x::int). Ifm (real_of_int x#bs) p" (is "\<exists> x. ?P x")
 proof-
   let ?d = "\<delta> p"
   from \<delta> [OF lin] have dpos: "?d >0" by simp
@@ -2241,9 +2219,9 @@
 qed
 
 lemma minusinf_bex:
-  assumes lin: "iszlfm p (real (a::int) #bs)"
-  shows "(\<exists> (x::int). Ifm (real x#bs) (minusinf p)) = 
-         (\<exists> (x::int)\<in> {1..\<delta> p}. Ifm (real x#bs) (minusinf p))"
+  assumes lin: "iszlfm p (real_of_int (a::int) #bs)"
+  shows "(\<exists> (x::int). Ifm (real_of_int x#bs) (minusinf p)) = 
+         (\<exists> (x::int)\<in> {1..\<delta> p}. Ifm (real_of_int x#bs) (minusinf p))"
   (is "(\<exists> x. ?P x) = _")
 proof-
   let ?d = "\<delta> p"
@@ -2338,30 +2316,30 @@
 
 lemma mirror_\<alpha>_\<beta>:
   assumes lp: "iszlfm p (a#bs)"
-  shows "(Inum (real (i::int)#bs)) ` set (\<alpha> p) = (Inum (real i#bs)) ` set (\<beta> (mirror p))"
+  shows "(Inum (real_of_int (i::int)#bs)) ` set (\<alpha> p) = (Inum (real_of_int i#bs)) ` set (\<beta> (mirror p))"
   using lp by (induct p rule: mirror.induct) auto
 
 lemma mirror: 
   assumes lp: "iszlfm p (a#bs)"
-  shows "Ifm (real (x::int)#bs) (mirror p) = Ifm (real (- x)#bs) p" 
+  shows "Ifm (real_of_int (x::int)#bs) (mirror p) = Ifm (real_of_int (- x)#bs) p" 
   using lp
 proof(induct p rule: iszlfm.induct)
   case (9 j c e)
-  have th: "(real j rdvd real c * real x - Inum (real x # bs) e) =
-       (real j rdvd - (real c * real x - Inum (real x # bs) e))"
+  have th: "(real_of_int j rdvd real_of_int c * real_of_int x - Inum (real_of_int x # bs) e) =
+       (real_of_int j rdvd - (real_of_int c * real_of_int x - Inum (real_of_int x # bs) e))"
     by (simp only: rdvd_minus[symmetric])
   from 9 th show ?case
     by (simp add: algebra_simps
-      numbound0_I[where bs="bs" and b'="real x" and b="- real x"])
+      numbound0_I[where bs="bs" and b'="real_of_int x" and b="- real_of_int x"])
 next
   case (10 j c e)
-  have th: "(real j rdvd real c * real x - Inum (real x # bs) e) =
-       (real j rdvd - (real c * real x - Inum (real x # bs) e))"
+  have th: "(real_of_int j rdvd real_of_int c * real_of_int x - Inum (real_of_int x # bs) e) =
+       (real_of_int j rdvd - (real_of_int c * real_of_int x - Inum (real_of_int x # bs) e))"
     by (simp only: rdvd_minus[symmetric])
   from 10 th show  ?case
     by (simp add: algebra_simps
-      numbound0_I[where bs="bs" and b'="real x" and b="- real x"])
-qed (auto simp add: numbound0_I[where bs="bs" and b="real x" and b'="- real x"])
+      numbound0_I[where bs="bs" and b'="real_of_int x" and b="- real_of_int x"])
+qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int x" and b'="- real_of_int x"])
 
 lemma mirror_l: "iszlfm p (a#bs) \<Longrightarrow> iszlfm (mirror p) (a#bs)"
   by (induct p rule: mirror.induct) (auto simp add: isint_neg)
@@ -2375,8 +2353,8 @@
 
 
 lemma mirror_ex: 
-  assumes lp: "iszlfm p (real (i::int)#bs)"
-  shows "(\<exists> (x::int). Ifm (real x#bs) (mirror p)) = (\<exists> (x::int). Ifm (real x#bs) p)"
+  assumes lp: "iszlfm p (real_of_int (i::int)#bs)"
+  shows "(\<exists> (x::int). Ifm (real_of_int x#bs) (mirror p)) = (\<exists> (x::int). Ifm (real_of_int x#bs) p)"
   (is "(\<exists> x. ?I x ?mp) = (\<exists> x. ?I x p)")
 proof(auto)
   fix x assume "?I x ?mp" hence "?I (- x) p" using mirror[OF lp] by blast
@@ -2425,7 +2403,7 @@
 qed (auto simp add: lcm_pos_int)
 
 lemma a_\<beta>: assumes linp: "iszlfm p (a #bs)" and d: "d_\<beta> p l" and lp: "l > 0"
-  shows "iszlfm (a_\<beta> p l) (a #bs) \<and> d_\<beta> (a_\<beta> p l) 1 \<and> (Ifm (real (l * x) #bs) (a_\<beta> p l) = Ifm ((real x)#bs) p)"
+  shows "iszlfm (a_\<beta> p l) (a #bs) \<and> d_\<beta> (a_\<beta> p l) 1 \<and> (Ifm (real_of_int (l * x) #bs) (a_\<beta> p l) = Ifm ((real_of_int x)#bs) p)"
 using linp d
 proof (induct p rule: iszlfm.induct)
   case (5 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
@@ -2438,13 +2416,13 @@
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
     hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
       by simp
-    hence "(real l * real x + real (l div c) * Inum (real x # bs) e < (0::real)) =
-          (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e < 0)"
+    hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e < (0::real)) =
+          (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e < 0)"
       by simp
-    also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) < (real (l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (real c * real x + Inum (real x # bs) e < 0)"
-    using mult_less_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be  isint_Mul[OF ei] by simp
+    also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) < (real_of_int (l div c)) * 0)" by (simp add: algebra_simps)
+    also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e < 0)"
+    using mult_less_0_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp
+  finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] be  isint_Mul[OF ei] by simp
 next
   case (6 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
     from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
@@ -2456,13 +2434,13 @@
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
     hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
       by simp
-    hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<le> (0::real)) =
-          (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<le> 0)"
+    hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<le> (0::real)) =
+          (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<le> 0)"
       by simp
-    also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<le> (real (l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (real c * real x + Inum (real x # bs) e \<le> 0)"
-    using mult_le_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"]  be  isint_Mul[OF ei] by simp
+    also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) \<le> (real_of_int (l div c)) * 0)" by (simp add: algebra_simps)
+    also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<le> 0)"
+    using mult_le_0_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp
+  finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"]  be  isint_Mul[OF ei] by simp
 next
   case (7 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
     from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
@@ -2474,13 +2452,13 @@
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
     hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
       by simp
-    hence "(real l * real x + real (l div c) * Inum (real x # bs) e > (0::real)) =
-          (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e > 0)"
+    hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e > (0::real)) =
+          (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e > 0)"
       by simp
-    also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) > (real (l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (real c * real x + Inum (real x # bs) e > 0)"
-    using zero_less_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"]  be  isint_Mul[OF ei] by simp
+    also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) > (real_of_int (l div c)) * 0)" by (simp add: algebra_simps)
+    also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e > 0)"
+    using zero_less_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp
+  finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"]  be  isint_Mul[OF ei] by simp
 next
   case (8 c e) hence cp: "c>0" and be: "numbound0 e"  and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
     from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
@@ -2492,13 +2470,13 @@
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
     hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
       by simp
-    hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<ge> (0::real)) =
-          (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<ge> 0)"
+    hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<ge> (0::real)) =
+          (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<ge> 0)"
       by simp
-    also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<ge> (real (l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (real c * real x + Inum (real x # bs) e \<ge> 0)"
-    using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"]  be  isint_Mul[OF ei] by simp
+    also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) \<ge> (real_of_int (l div c)) * 0)" by (simp add: algebra_simps)
+    also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<ge> 0)"
+    using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp
+  finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"]  be  isint_Mul[OF ei] by simp
 next
   case (3 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
     from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
@@ -2510,13 +2488,13 @@
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
     hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
       by simp
-    hence "(real l * real x + real (l div c) * Inum (real x # bs) e = (0::real)) =
-          (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = 0)"
+    hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (0::real)) =
+          (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = 0)"
       by simp
-    also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) = (real (l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (real c * real x + Inum (real x # bs) e = 0)"
-    using mult_eq_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"]  be  isint_Mul[OF ei] by simp
+    also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) = (real_of_int (l div c)) * 0)" by (simp add: algebra_simps)
+    also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e = 0)"
+    using mult_eq_0_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp
+  finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"]  be  isint_Mul[OF ei] by simp
 next
   case (4 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+
     from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
@@ -2528,13 +2506,13 @@
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
     hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
       by simp
-    hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<noteq> (0::real)) =
-          (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<noteq> 0)"
+    hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<noteq> (0::real)) =
+          (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<noteq> 0)"
       by simp
-    also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<noteq> (real (l div c)) * 0)" by (simp add: algebra_simps)
-    also have "\<dots> = (real c * real x + Inum (real x # bs) e \<noteq> 0)"
-    using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"]  be  isint_Mul[OF ei] by simp
+    also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) \<noteq> (real_of_int (l div c)) * 0)" by (simp add: algebra_simps)
+    also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0)"
+    using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp
+  finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"]  be  isint_Mul[OF ei] by simp
 next
   case (9 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+
     from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
@@ -2546,12 +2524,12 @@
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
     hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
       by simp
-    hence "(\<exists> (k::int). real l * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k) = (\<exists> (k::int). real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k)"  by simp
-    also have "\<dots> = (\<exists> (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: algebra_simps)
-    also fix k have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e - real j * real k = 0)"
-    using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e - real j * real k"] ldcp by simp
-  also have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e = real j * real k)" by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] rdvd_def  be  isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp 
+    hence "(\<exists> (k::int). real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k) = (\<exists> (k::int). real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k)"  by simp
+    also have "\<dots> = (\<exists> (k::int). real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k) = real_of_int (l div c)*0)" by (simp add: algebra_simps)
+    also fix k have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k = 0)"
+    using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k"] ldcp by simp
+  also have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e = real_of_int j * real_of_int k)" by simp
+  finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] rdvd_def  be  isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp 
 next
   case (10 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+
     from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
@@ -2563,16 +2541,16 @@
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
     hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
       by simp
-    hence "(\<exists> (k::int). real l * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k) = (\<exists> (k::int). real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k)"  by simp
-    also have "\<dots> = (\<exists> (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: algebra_simps)
-    also fix k have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e - real j * real k = 0)"
-    using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e - real j * real k"] ldcp by simp
-  also have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e = real j * real k)" by simp
-  finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] rdvd_def  be  isint_Mul[OF ei]  mult_strict_mono[OF ldcp jp ldcp ] by simp
-qed (simp_all add: numbound0_I[where bs="bs" and b="real (l * x)" and b'="real x"] isint_Mul del: real_of_int_mult)
+    hence "(\<exists> (k::int). real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k) = (\<exists> (k::int). real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k)"  by simp
+    also have "\<dots> = (\<exists> (k::int). real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k) = real_of_int (l div c)*0)" by (simp add: algebra_simps)
+    also fix k have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k = 0)"
+    using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k"] ldcp by simp
+  also have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e = real_of_int j * real_of_int k)" by simp
+  finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] rdvd_def  be  isint_Mul[OF ei]  mult_strict_mono[OF ldcp jp ldcp ] by simp
+qed (simp_all add: numbound0_I[where bs="bs" and b="real_of_int (l * x)" and b'="real_of_int x"] isint_Mul del: of_int_mult)
 
 lemma a_\<beta>_ex: assumes linp: "iszlfm p (a#bs)" and d: "d_\<beta> p l" and lp: "l>0"
-  shows "(\<exists> x. l dvd x \<and> Ifm (real x #bs) (a_\<beta> p l)) = (\<exists> (x::int). Ifm (real x#bs) p)"
+  shows "(\<exists> x. l dvd x \<and> Ifm (real_of_int x #bs) (a_\<beta> p l)) = (\<exists> (x::int). Ifm (real_of_int x#bs) p)"
   (is "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> x. ?P' x)")
 proof-
   have "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> (x::int). ?P (l*x))"
@@ -2586,148 +2564,146 @@
   and u: "d_\<beta> p 1"
   and d: "d_\<delta> p d"
   and dp: "d > 0"
-  and nob: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real x = b + real j)"
-  and p: "Ifm (real x#bs) p" (is "?P x")
+  and nob: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real_of_int x = b + real_of_int j)"
+  and p: "Ifm (real_of_int x#bs) p" (is "?P x")
   shows "?P (x - d)"
 using lp u d dp nob p
 proof(induct p rule: iszlfm.induct)
   case (5 c e) hence c1: "c=1" and  bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp_all
-  with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] 5
-  show ?case by (simp del: real_of_int_minus)
+  with dp p c1 numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] 5
+  show ?case by (simp del: of_int_minus)
 next
   case (6 c e)  hence c1: "c=1" and  bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp_all
-  with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] 6
-  show ?case by (simp del: real_of_int_minus)
+  with dp p c1 numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] 6
+  show ?case by (simp del: of_int_minus)
 next
-  case (7 c e) hence p: "Ifm (real x #bs) (Gt (CN 0 c e))" and c1: "c=1"
+  case (7 c e) hence p: "Ifm (real_of_int x #bs) (Gt (CN 0 c e))" and c1: "c=1"
     and bn:"numbound0 e" and ie1:"isint e (a#bs)" using dvd1_eq1[where x="c"] by simp_all
-  let ?e = "Inum (real x # bs) e"
-  from ie1 have ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="a#bs"]
-      numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]
+  let ?e = "Inum (real_of_int x # bs) e"
+  from ie1 have ie: "real_of_int (floor ?e) = ?e" using isint_iff[where n="e" and bs="a#bs"]
+      numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"]
     by (simp add: isint_iff)
-    {assume "real (x-d) +?e > 0" hence ?case using c1 
-      numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"]
-        by (simp del: real_of_int_minus)}
+    {assume "real_of_int (x-d) +?e > 0" hence ?case using c1 
+      numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"]
+        by (simp del: of_int_minus)}
     moreover
-    {assume H: "\<not> real (x-d) + ?e > 0" 
+    {assume H: "\<not> real_of_int (x-d) + ?e > 0" 
       let ?v="Neg e"
       have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))" by simp
-      from 7(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]] 
-      have nob: "\<not> (\<exists> j\<in> {1 ..d}. real x =  - ?e + real j)" by auto 
-      from H p have "real x + ?e > 0 \<and> real x + ?e \<le> real d" by (simp add: c1)
-      hence "real (x + floor ?e) > real (0::int) \<and> real (x + floor ?e) \<le> real d"
+      from 7(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"]] 
+      have nob: "\<not> (\<exists> j\<in> {1 ..d}. real_of_int x =  - ?e + real_of_int j)" by auto 
+      from H p have "real_of_int x + ?e > 0 \<and> real_of_int x + ?e \<le> real_of_int d" by (simp add: c1)
+      hence "real_of_int (x + floor ?e) > real_of_int (0::int) \<and> real_of_int (x + floor ?e) \<le> real_of_int d"
         using ie by simp
       hence "x + floor ?e \<ge> 1 \<and> x + floor ?e \<le> d"  by simp
       hence "\<exists> (j::int) \<in> {1 .. d}. j = x + floor ?e" by simp
-      hence "\<exists> (j::int) \<in> {1 .. d}. real x = real (- floor ?e + j)" 
-        by (simp only: real_of_int_inject) (simp add: algebra_simps)
-      hence "\<exists> (j::int) \<in> {1 .. d}. real x = - ?e + real j" 
+      hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x = real_of_int (- floor ?e + j)" by force 
+      hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x = - ?e + real_of_int j" 
         by (simp add: ie[simplified isint_iff])
       with nob have ?case by auto}
     ultimately show ?case by blast
 next
-  case (8 c e) hence p: "Ifm (real x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" 
+  case (8 c e) hence p: "Ifm (real_of_int x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" 
     and ie1:"isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
-    let ?e = "Inum (real x # bs) e"
-    from ie1 have ie: "real (floor ?e) = ?e" using numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real x)#bs"]
+    let ?e = "Inum (real_of_int x # bs) e"
+    from ie1 have ie: "real_of_int (floor ?e) = ?e" using numbound0_I[OF bn,where b="real_of_int x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real_of_int x)#bs"]
       by (simp add: isint_iff)
-    {assume "real (x-d) +?e \<ge> 0" hence ?case using  c1 
-      numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"]
-        by (simp del: real_of_int_minus)}
+    {assume "real_of_int (x-d) +?e \<ge> 0" hence ?case using  c1 
+      numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"]
+        by (simp del: of_int_minus)}
     moreover
-    {assume H: "\<not> real (x-d) + ?e \<ge> 0" 
+    {assume H: "\<not> real_of_int (x-d) + ?e \<ge> 0" 
       let ?v="Sub (C (- 1)) e"
       have vb: "?v \<in> set (\<beta> (Ge (CN 0 c e)))" by simp
-      from 8(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]] 
-      have nob: "\<not> (\<exists> j\<in> {1 ..d}. real x =  - ?e - 1 + real j)" by auto 
-      from H p have "real x + ?e \<ge> 0 \<and> real x + ?e < real d" by (simp add: c1)
-      hence "real (x + floor ?e) \<ge> real (0::int) \<and> real (x + floor ?e) < real d"
+      from 8(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"]] 
+      have nob: "\<not> (\<exists> j\<in> {1 ..d}. real_of_int x =  - ?e - 1 + real_of_int j)" by auto 
+      from H p have "real_of_int x + ?e \<ge> 0 \<and> real_of_int x + ?e < real_of_int d" by (simp add: c1)
+      hence "real_of_int (x + floor ?e) \<ge> real_of_int (0::int) \<and> real_of_int (x + floor ?e) < real_of_int d"
         using ie by simp
       hence "x + floor ?e +1 \<ge> 1 \<and> x + floor ?e + 1 \<le> d"  by simp
       hence "\<exists> (j::int) \<in> {1 .. d}. j = x + floor ?e + 1" by simp
       hence "\<exists> (j::int) \<in> {1 .. d}. x= - floor ?e - 1 + j" by (simp add: algebra_simps)
-      hence "\<exists> (j::int) \<in> {1 .. d}. real x= real (- floor ?e - 1 + j)" 
-        by (simp only: real_of_int_inject)
-      hence "\<exists> (j::int) \<in> {1 .. d}. real x= - ?e - 1 + real j" 
+      hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x= real_of_int (- floor ?e - 1 + j)" by blast
+      hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x= - ?e - 1 + real_of_int j" 
         by (simp add: ie[simplified isint_iff])
       with nob have ?case by simp }
     ultimately show ?case by blast
 next
-  case (3 c e) hence p: "Ifm (real x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1" 
+  case (3 c e) hence p: "Ifm (real_of_int x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1" 
     and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
-    let ?e = "Inum (real x # bs) e"
+    let ?e = "Inum (real_of_int x # bs) e"
     let ?v="(Sub (C (- 1)) e)"
     have vb: "?v \<in> set (\<beta> (Eq (CN 0 c e)))" by simp
-    from p have "real x= - ?e" by (simp add: c1) with 3(5) show ?case using dp
+    from p have "real_of_int x= - ?e" by (simp add: c1) with 3(5) show ?case using dp
       by simp (erule ballE[where x="1"],
-        simp_all add:algebra_simps numbound0_I[OF bn,where b="real x"and b'="a"and bs="bs"])
+        simp_all add:algebra_simps numbound0_I[OF bn,where b="real_of_int x"and b'="a"and bs="bs"])
 next
-  case (4 c e)hence p: "Ifm (real x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1" 
+  case (4 c e)hence p: "Ifm (real_of_int x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1" 
     and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
-    let ?e = "Inum (real x # bs) e"
+    let ?e = "Inum (real_of_int x # bs) e"
     let ?v="Neg e"
     have vb: "?v \<in> set (\<beta> (NEq (CN 0 c e)))" by simp
-    {assume "real x - real d + Inum ((real (x -d)) # bs) e \<noteq> 0" 
+    {assume "real_of_int x - real_of_int d + Inum ((real_of_int (x -d)) # bs) e \<noteq> 0" 
       hence ?case by (simp add: c1)}
     moreover
-    {assume H: "real x - real d + Inum ((real (x -d)) # bs) e = 0"
-      hence "real x = - Inum ((real (x -d)) # bs) e + real d" by simp
-      hence "real x = - Inum (a # bs) e + real d"
-        by (simp add: numbound0_I[OF bn,where b="real x - real d"and b'="a"and bs="bs"])
+    {assume H: "real_of_int x - real_of_int d + Inum ((real_of_int (x -d)) # bs) e = 0"
+      hence "real_of_int x = - Inum ((real_of_int (x -d)) # bs) e + real_of_int d" by simp
+      hence "real_of_int x = - Inum (a # bs) e + real_of_int d"
+        by (simp add: numbound0_I[OF bn,where b="real_of_int x - real_of_int d"and b'="a"and bs="bs"])
        with 4(5) have ?case using dp by simp}
   ultimately show ?case by blast
 next 
-  case (9 j c e) hence p: "Ifm (real x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1" 
+  case (9 j c e) hence p: "Ifm (real_of_int x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1" 
     and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
-  let ?e = "Inum (real x # bs) e"
+  let ?e = "Inum (real_of_int x # bs) e"
   from 9 have "isint e (a #bs)"  by simp 
-  hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real x)#bs"] numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"]
+  hence ie: "real_of_int (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real_of_int x)#bs"] numbound0_I[OF bn,where b="real_of_int x" and b'="a" and bs="bs"]
     by (simp add: isint_iff)
   from 9 have id: "j dvd d" by simp
-  from c1 ie[symmetric] have "?p x = (real j rdvd real (x+ floor ?e))" by simp
+  from c1 ie[symmetric] have "?p x = (real_of_int j rdvd real_of_int (x+ floor ?e))" by simp
   also have "\<dots> = (j dvd x + floor ?e)" 
-    using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp
+    using int_rdvd_real[where i="j" and x="real_of_int (x+ floor ?e)"] by simp
   also have "\<dots> = (j dvd x - d + floor ?e)" 
     using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp
-  also have "\<dots> = (real j rdvd real (x - d + floor ?e))" 
-    using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified]
+  also have "\<dots> = (real_of_int j rdvd real_of_int (x - d + floor ?e))" 
+    using int_rdvd_real[where i="j" and x="real_of_int (x-d + floor ?e)",symmetric, simplified]
       ie by simp
-  also have "\<dots> = (real j rdvd real x - real d + ?e)" 
+  also have "\<dots> = (real_of_int j rdvd real_of_int x - real_of_int d + ?e)" 
     using ie by simp
   finally show ?case 
-    using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp
+    using numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] c1 p by simp
 next
-  case (10 j c e) hence p: "Ifm (real x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
-  let ?e = "Inum (real x # bs) e"
+  case (10 j c e) hence p: "Ifm (real_of_int x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
+  let ?e = "Inum (real_of_int x # bs) e"
   from 10 have "isint e (a#bs)"  by simp 
-  hence ie: "real (floor ?e) = ?e" using numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real x)#bs"]
+  hence ie: "real_of_int (floor ?e) = ?e" using numbound0_I[OF bn,where b="real_of_int x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real_of_int x)#bs"]
     by (simp add: isint_iff)
   from 10 have id: "j dvd d" by simp
-  from c1 ie[symmetric] have "?p x = (\<not> real j rdvd real (x+ floor ?e))" by simp
+  from c1 ie[symmetric] have "?p x = (\<not> real_of_int j rdvd real_of_int (x+ floor ?e))" by simp
   also have "\<dots> = (\<not> j dvd x + floor ?e)" 
-    using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp
+    using int_rdvd_real[where i="j" and x="real_of_int (x+ floor ?e)"] by simp
   also have "\<dots> = (\<not> j dvd x - d + floor ?e)" 
     using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp
-  also have "\<dots> = (\<not> real j rdvd real (x - d + floor ?e))" 
-    using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified]
+  also have "\<dots> = (\<not> real_of_int j rdvd real_of_int (x - d + floor ?e))" 
+    using int_rdvd_real[where i="j" and x="real_of_int (x-d + floor ?e)",symmetric, simplified]
       ie by simp
-  also have "\<dots> = (\<not> real j rdvd real x - real d + ?e)" 
+  also have "\<dots> = (\<not> real_of_int j rdvd real_of_int x - real_of_int d + ?e)" 
     using ie by simp
   finally show ?case
-    using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp
-qed (auto simp add: numbound0_I[where bs="bs" and b="real (x - d)" and b'="real x"]
-  simp del: real_of_int_diff)
+    using numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] c1 p by simp
+qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int (x - d)" and b'="real_of_int x"]
+  simp del: of_int_diff)
 
 lemma \<beta>':   
   assumes lp: "iszlfm p (a #bs)"
   and u: "d_\<beta> p 1"
   and d: "d_\<delta> p d"
   and dp: "d > 0"
-  shows "\<forall> x. \<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> set(\<beta> p). Ifm ((Inum (a#bs) b + real j) #bs) p) \<longrightarrow> Ifm (real x#bs) p \<longrightarrow> Ifm (real (x - d)#bs) p" (is "\<forall> x. ?b \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)")
+  shows "\<forall> x. \<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> set(\<beta> p). Ifm ((Inum (a#bs) b + real_of_int j) #bs) p) \<longrightarrow> Ifm (real_of_int x#bs) p \<longrightarrow> Ifm (real_of_int (x - d)#bs) p" (is "\<forall> x. ?b \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)")
 proof(clarify)
   fix x 
   assume nb:"?b" and px: "?P x" 
-  hence nb2: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real x = b + real j)"
+  hence nb2: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real_of_int x = b + real_of_int j)"
     by auto
   from  \<beta>[OF lp u d dp nb2 px] show "?P (x -d )" .
 qed
@@ -2764,22 +2740,22 @@
   and u: "d_\<beta> p 1"
   and d: "d_\<delta> p d"
   and dp: "d > 0"
-  shows "(\<exists> (x::int). Ifm (real x #bs) p) = (\<exists> j\<in> {1.. d}. Ifm (real j #bs) (minusinf p) \<or> (\<exists> b \<in> set (\<beta> p). Ifm ((Inum (a#bs) b + real j) #bs) p))"
-  (is "(\<exists> (x::int). ?P (real x)) = (\<exists> j\<in> ?D. ?M j \<or> (\<exists> b\<in> ?B. ?P (?I b + real j)))")
+  shows "(\<exists> (x::int). Ifm (real_of_int x #bs) p) = (\<exists> j\<in> {1.. d}. Ifm (real_of_int j #bs) (minusinf p) \<or> (\<exists> b \<in> set (\<beta> p). Ifm ((Inum (a#bs) b + real_of_int j) #bs) p))"
+  (is "(\<exists> (x::int). ?P (real_of_int x)) = (\<exists> j\<in> ?D. ?M j \<or> (\<exists> b\<in> ?B. ?P (?I b + real_of_int j)))")
 proof-
   from minusinf_inf[OF lp] 
-  have th: "\<exists>(z::int). \<forall>x<z. ?P (real x) = ?M x" by blast
+  have th: "\<exists>(z::int). \<forall>x<z. ?P (real_of_int x) = ?M x" by blast
   let ?B' = "{floor (?I b) | b. b\<in> ?B}"
-  from \<beta>_int[OF lp] isint_iff[where bs="a # bs"] have B: "\<forall> b\<in> ?B. real (floor (?I b)) = ?I b" by simp
+  from \<beta>_int[OF lp] isint_iff[where bs="a # bs"] have B: "\<forall> b\<in> ?B. real_of_int (floor (?I b)) = ?I b" by simp
   from B[rule_format] 
-  have "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real j)) = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real (floor (?I b)) + real j))" 
+  have "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real_of_int j)) = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real_of_int (floor (?I b)) + real_of_int j))" 
     by simp
-  also have "\<dots> = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real (floor (?I b) + j)))" by simp
-  also have"\<dots> = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real (b + j)))"  by blast
+  also have "\<dots> = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real_of_int (floor (?I b) + j)))" by simp
+  also have"\<dots> = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j)))"  by blast
   finally have BB': 
-    "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real j)) = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real (b + j)))" 
+    "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real_of_int j)) = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j)))" 
     by blast 
-  hence th2: "\<forall> x. \<not> (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real (b + j))) \<longrightarrow> ?P (real x) \<longrightarrow> ?P (real (x - d))" using \<beta>'[OF lp u d dp] by blast
+  hence th2: "\<forall> x. \<not> (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j))) \<longrightarrow> ?P (real_of_int x) \<longrightarrow> ?P (real_of_int (x - d))" using \<beta>'[OF lp u d dp] by blast
   from minusinf_repeats[OF d lp]
   have th3: "\<forall> x k. ?M x = ?M (x-k*d)" by simp
   from cpmi_eq[OF dp th th2 th3] BB' show ?thesis by blast
@@ -2840,38 +2816,38 @@
   "\<sigma> p k t \<equiv> And (Dvd k t) (\<sigma>_\<rho> p (t,k))"
 
 lemma \<sigma>_\<rho>:
-  assumes linp: "iszlfm p (real (x::int)#bs)"
-  and kpos: "real k > 0"
+  assumes linp: "iszlfm p (real_of_int (x::int)#bs)"
+  and kpos: "real_of_int k > 0"
   and tnb: "numbound0 t"
-  and tint: "isint t (real x#bs)"
+  and tint: "isint t (real_of_int x#bs)"
   and kdt: "k dvd floor (Inum (b'#bs) t)"
-  shows "Ifm (real x#bs) (\<sigma>_\<rho> p (t,k)) = 
-  (Ifm ((real ((floor (Inum (b'#bs) t)) div k))#bs) p)" 
-  (is "?I (real x) (?s p) = (?I (real ((floor (?N b' t)) div k)) p)" is "_ = (?I ?tk p)")
+  shows "Ifm (real_of_int x#bs) (\<sigma>_\<rho> p (t,k)) = 
+  (Ifm ((real_of_int ((floor (Inum (b'#bs) t)) div k))#bs) p)" 
+  (is "?I (real_of_int x) (?s p) = (?I (real_of_int ((floor (?N b' t)) div k)) p)" is "_ = (?I ?tk p)")
 using linp kpos tnb
 proof(induct p rule: \<sigma>_\<rho>.induct)
   case (3 c e) 
   from 3 have cp: "c > 0" and nb: "numbound0 e" by auto
   { assume kdc: "k dvd c" 
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
-      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
+      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
   moreover 
   { assume *: "\<not> k dvd c"
-    from kpos have knz': "real k \<noteq> 0" by simp
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t"
+    from kpos have knz': "real_of_int k \<noteq> 0" by simp
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t"
       using isint_def by simp
-    from assms * have "?I (real x) (?s (Eq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k = 0)"
+    from assms * have "?I (real_of_int x) (?s (Eq (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k = 0)"
       using real_of_int_div[OF kdt]
-        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti algebra_simps)
       also have "\<dots> = (?I ?tk (Eq (CN 0 c e)))"
         using nonzero_eq_divide_eq[OF knz',
-            where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric]
-          real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+            where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric]
+          real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+          numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
         by (simp add: ti)
       finally have ?case . }
     ultimately show ?case by blast 
@@ -2879,24 +2855,24 @@
   case (4 c e)  
   then have cp: "c > 0" and nb: "numbound0 e" by auto
   { assume kdc: "k dvd c" 
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have  ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
-      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
+      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
   moreover 
   { assume *: "\<not> k dvd c"
-    from kpos have knz': "real k \<noteq> 0" by simp
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-    from assms * have "?I (real x) (?s (NEq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<noteq> 0)"
+    from kpos have knz': "real_of_int k \<noteq> 0" by simp
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
+    from assms * have "?I (real_of_int x) (?s (NEq (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k \<noteq> 0)"
       using real_of_int_div[OF kdt]
-        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti algebra_simps)
     also have "\<dots> = (?I ?tk (NEq (CN 0 c e)))"
       using nonzero_eq_divide_eq[OF knz',
-          where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric]
-        real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+          where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric]
+        real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
   ultimately show ?case by blast 
@@ -2904,23 +2880,23 @@
   case (5 c e) 
   then have cp: "c > 0" and nb: "numbound0 e" by auto
   { assume kdc: "k dvd c" 
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have  ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
-      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
+      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
   moreover 
   { assume *: "\<not> k dvd c"
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-    from assms * have "?I (real x) (?s (Lt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k < 0)"
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
+    from assms * have "?I (real_of_int x) (?s (Lt (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k < 0)"
       using real_of_int_div[OF kdt]
-        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti algebra_simps)
     also have "\<dots> = (?I ?tk (Lt (CN 0 c e)))"
       using pos_less_divide_eq[OF kpos,
-          where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric]
-        real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+          where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric]
+        real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
   ultimately show ?case by blast 
@@ -2928,23 +2904,23 @@
   case (6 c e)
   then have cp: "c > 0" and nb: "numbound0 e" by auto
   { assume kdc: "k dvd c" 
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have  ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
-      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
+      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
   moreover 
   { assume *: "\<not> k dvd c"
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-    from assms * have "?I (real x) (?s (Le (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<le> 0)"
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
+    from assms * have "?I (real_of_int x) (?s (Le (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k \<le> 0)"
       using real_of_int_div[OF kdt]
-        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti algebra_simps)
     also have "\<dots> = (?I ?tk (Le (CN 0 c e)))"
       using pos_le_divide_eq[OF kpos,
-          where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric]
-        real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+          where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric]
+        real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
   ultimately show ?case by blast 
@@ -2952,23 +2928,23 @@
   case (7 c e) 
   then have cp: "c > 0" and nb: "numbound0 e" by auto
   { assume kdc: "k dvd c" 
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have  ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
-      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
+      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
   moreover 
   { assume *: "\<not> k dvd c"
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-    from assms * have "?I (real x) (?s (Gt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k > 0)"
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
+    from assms * have "?I (real_of_int x) (?s (Gt (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k > 0)"
       using real_of_int_div[OF kdt]
-        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti algebra_simps)
     also have "\<dots> = (?I ?tk (Gt (CN 0 c e)))"
       using pos_divide_less_eq[OF kpos,
-          where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric]
-        real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+          where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric]
+        real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
   ultimately show ?case by blast 
@@ -2976,23 +2952,23 @@
   case (8 c e)  
   then have cp: "c > 0" and nb: "numbound0 e" by auto
   { assume kdc: "k dvd c" 
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have  ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
-      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
+      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
   moreover 
   { assume *: "\<not> k dvd c"
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-    from assms * have "?I (real x) (?s (Ge (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<ge> 0)"
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
+    from assms * have "?I (real_of_int x) (?s (Ge (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k \<ge> 0)"
       using real_of_int_div[OF kdt]
-        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti algebra_simps)
     also have "\<dots> = (?I ?tk (Ge (CN 0 c e)))"
       using pos_divide_le_eq[OF kpos,
-          where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric]
-        real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+          where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric]
+        real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
   ultimately show ?case by blast 
@@ -3000,23 +2976,23 @@
   case (9 i c e)
   then have cp: "c > 0" and nb: "numbound0 e" by auto
   { assume kdc: "k dvd c" 
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
-      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
+      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
   moreover 
   { assume *: "\<not> k dvd c"
-    from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-    from assms * have "?I (real x) (?s (Dvd i (CN 0 c e))) = (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k)"
+    from kpos have knz: "k\<noteq>0" by simp hence knz': "real_of_int k \<noteq> 0" by simp
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
+    from assms * have "?I (real_of_int x) (?s (Dvd i (CN 0 c e))) = (real_of_int i * real_of_int k rdvd (real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k)"
       using real_of_int_div[OF kdt]
-        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti algebra_simps)
     also have "\<dots> = (?I ?tk (Dvd i (CN 0 c e)))"
       using rdvd_mult[OF knz, where n="i"]
-        real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+        real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
   ultimately show ?case by blast 
@@ -3024,28 +3000,28 @@
   case (10 i c e)
   then have cp: "c > 0" and nb: "numbound0 e" by auto
   { assume kdc: "k dvd c" 
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have  ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
-      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } 
+      numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
   moreover 
   { assume *: "\<not> k dvd c"
-    from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
-    from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp
-    from assms * have "?I (real x) (?s (NDvd i (CN 0 c e))) = (\<not> (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k))"
+    from kpos have knz: "k\<noteq>0" by simp hence knz': "real_of_int k \<noteq> 0" by simp
+    from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
+    from assms * have "?I (real_of_int x) (?s (NDvd i (CN 0 c e))) = (\<not> (real_of_int i * real_of_int k rdvd (real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k))"
       using real_of_int_div[OF kdt]
-        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti algebra_simps)
     also have "\<dots> = (?I ?tk (NDvd i (CN 0 c e)))"
       using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF kdt]
-        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
-        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
+        numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
+        numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
   ultimately show ?case by blast 
-qed (simp_all add: bound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"]
-  numbound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"])
+qed (simp_all add: bound0_I[where bs="bs" and b="real_of_int ((floor (?N b' t)) div k)" and b'="real_of_int x"]
+  numbound0_I[where bs="bs" and b="real_of_int ((floor (?N b' t)) div k)" and b'="real_of_int x"])
 
 
 lemma \<sigma>_\<rho>_nb: assumes lp:"iszlfm p (a#bs)" and nb: "numbound0 t"
@@ -3054,153 +3030,153 @@
   by (induct p rule: iszlfm.induct, auto simp add: nb)
 
 lemma \<rho>_l:
-  assumes lp: "iszlfm p (real (i::int)#bs)"
-  shows "\<forall> (b,k) \<in> set (\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real i#bs)"
+  assumes lp: "iszlfm p (real_of_int (i::int)#bs)"
+  shows "\<forall> (b,k) \<in> set (\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real_of_int i#bs)"
 using lp by (induct p rule: \<rho>.induct, auto simp add: isint_sub isint_neg)
 
 lemma \<alpha>_\<rho>_l:
-  assumes lp: "iszlfm p (real (i::int)#bs)"
-  shows "\<forall> (b,k) \<in> set (\<alpha>_\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real i#bs)"
-using lp isint_add [OF isint_c[where j="- 1"],where bs="real i#bs"]
+  assumes lp: "iszlfm p (real_of_int (i::int)#bs)"
+  shows "\<forall> (b,k) \<in> set (\<alpha>_\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real_of_int i#bs)"
+using lp isint_add [OF isint_c[where j="- 1"],where bs="real_of_int i#bs"]
  by (induct p rule: \<alpha>_\<rho>.induct, auto)
 
-lemma \<rho>: assumes lp: "iszlfm p (real (i::int) #bs)"
-  and pi: "Ifm (real i#bs) p"
+lemma \<rho>: assumes lp: "iszlfm p (real_of_int (i::int) #bs)"
+  and pi: "Ifm (real_of_int i#bs) p"
   and d: "d_\<delta> p d"
   and dp: "d > 0"
-  and nob: "\<forall>(e,c) \<in> set (\<rho> p). \<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> Inum (real i#bs) e + real j"
+  and nob: "\<forall>(e,c) \<in> set (\<rho> p). \<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> Inum (real_of_int i#bs) e + real_of_int j"
   (is "\<forall>(e,c) \<in> set (\<rho> p). \<forall> j\<in> {1 .. c*d}. _ \<noteq> ?N i e + _")
-  shows "Ifm (real(i - d)#bs) p"
+  shows "Ifm (real_of_int(i - d)#bs) p"
   using lp pi d nob
 proof(induct p rule: iszlfm.induct)
-  case (3 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
-    and pi: "real (c*i) = - 1 -  ?N i e + real (1::int)" and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> -1 - ?N i e + real j"
+  case (3 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real_of_int i#bs)"
+    and pi: "real_of_int (c*i) = - 1 -  ?N i e + real_of_int (1::int)" and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> -1 - ?N i e + real_of_int j"
     by simp+
   from mult_strict_left_mono[OF dp cp]  have one:"1 \<in> {1 .. c*d}" by auto
   from nob[rule_format, where j="1", OF one] pi show ?case by simp
 next
   case (4 c e)  
-  hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
-    and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - ?N i e + real j"
+  hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real_of_int i#bs)"
+    and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> - ?N i e + real_of_int j"
     by simp+
-  {assume "real (c*i) \<noteq> - ?N i e + real (c*d)"
-    with numbound0_I[OF nb, where bs="bs" and b="real i - real d" and b'="real i"]
+  {assume "real_of_int (c*i) \<noteq> - ?N i e + real_of_int (c*d)"
+    with numbound0_I[OF nb, where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"]
     have ?case by (simp add: algebra_simps)}
   moreover
-  {assume pi: "real (c*i) = - ?N i e + real (c*d)"
+  {assume pi: "real_of_int (c*i) = - ?N i e + real_of_int (c*d)"
     from mult_strict_left_mono[OF dp cp] have d: "(c*d) \<in> {1 .. c*d}" by simp
     from nob[rule_format, where j="c*d", OF d] pi have ?case by simp }
   ultimately show ?case by blast
 next
   case (5 c e) hence cp: "c > 0" by simp
-  from 5 mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric] 
-    real_of_int_mult]
+  from 5 mult_strict_left_mono[OF dp cp, simplified of_int_less_iff[symmetric] 
+    of_int_mult]
   show ?case using 5 dp 
-    by (simp add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] 
+    apply (simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"] 
       algebra_simps del: mult_pos_pos)
+     by (metis add.right_neutral of_int_0_less_iff of_int_mult pos_add_strict)
 next
   case (6 c e) hence cp: "c > 0" by simp
-  from 6 mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric] 
-    real_of_int_mult]
+  from 6 mult_strict_left_mono[OF dp cp, simplified of_int_less_iff[symmetric] 
+    of_int_mult]
   show ?case using 6 dp 
-    by (simp add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] 
+    apply (simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"] 
       algebra_simps del: mult_pos_pos)
+      using order_trans by fastforce
 next
-  case (7 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
-    and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - ?N i e + real j"
-    and pi: "real (c*i) + ?N i e > 0" and cp': "real c >0"
+  case (7 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real_of_int i#bs)"
+    and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> - ?N i e + real_of_int j"
+    and pi: "real_of_int (c*i) + ?N i e > 0" and cp': "real_of_int c >0"
     by simp+
   let ?fe = "floor (?N i e)"
-  from pi cp have th:"(real i +?N i e / real c)*real c > 0" by (simp add: algebra_simps)
-  from pi ei[simplified isint_iff] have "real (c*i + ?fe) > real (0::int)" by simp
-  hence pi': "c*i + ?fe > 0" by (simp only: real_of_int_less_iff[symmetric])
-  have "real (c*i) + ?N i e > real (c*d) \<or> real (c*i) + ?N i e \<le> real (c*d)" by auto
+  from pi cp have th:"(real_of_int i +?N i e / real_of_int c)*real_of_int c > 0" by (simp add: algebra_simps)
+  from pi ei[simplified isint_iff] have "real_of_int (c*i + ?fe) > real_of_int (0::int)" by simp
+  hence pi': "c*i + ?fe > 0" by (simp only: of_int_less_iff[symmetric])
+  have "real_of_int (c*i) + ?N i e > real_of_int (c*d) \<or> real_of_int (c*i) + ?N i e \<le> real_of_int (c*d)" by auto
   moreover
-  {assume "real (c*i) + ?N i e > real (c*d)" hence ?case
+  {assume "real_of_int (c*i) + ?N i e > real_of_int (c*d)" hence ?case
       by (simp add: algebra_simps 
-        numbound0_I[OF nb,where bs="bs" and b="real i - real d" and b'="real i"])} 
+        numbound0_I[OF nb,where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"])} 
   moreover 
-  {assume H:"real (c*i) + ?N i e \<le> real (c*d)"
-    with ei[simplified isint_iff] have "real (c*i + ?fe) \<le> real (c*d)" by simp
-    hence pid: "c*i + ?fe \<le> c*d" by (simp only: real_of_int_le_iff)
+  {assume H:"real_of_int (c*i) + ?N i e \<le> real_of_int (c*d)"
+    with ei[simplified isint_iff] have "real_of_int (c*i + ?fe) \<le> real_of_int (c*d)" by simp
+    hence pid: "c*i + ?fe \<le> c*d" by (simp only: of_int_le_iff)
     with pi' have "\<exists> j1\<in> {1 .. c*d}. c*i + ?fe = j1" by auto
-    hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = - ?N i e + real j1" 
-      by (simp only: real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff])
-        (simp add: algebra_simps)
+    hence "\<exists> j1\<in> {1 .. c*d}. real_of_int (c*i) = - ?N i e + real_of_int j1"
+      unfolding Bex_def using ei[simplified isint_iff] by fastforce
     with nob  have ?case by blast }
   ultimately show ?case by blast
 next
-  case (8 c e)  hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
-    and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - 1 - ?N i e + real j"
-    and pi: "real (c*i) + ?N i e \<ge> 0" and cp': "real c >0"
+  case (8 c e)  hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real_of_int i#bs)"
+    and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> - 1 - ?N i e + real_of_int j"
+    and pi: "real_of_int (c*i) + ?N i e \<ge> 0" and cp': "real_of_int c >0"
     by simp+
   let ?fe = "floor (?N i e)"
-  from pi cp have th:"(real i +?N i e / real c)*real c \<ge> 0" by (simp add: algebra_simps)
-  from pi ei[simplified isint_iff] have "real (c*i + ?fe) \<ge> real (0::int)" by simp
-  hence pi': "c*i + 1 + ?fe \<ge> 1" by (simp only: real_of_int_le_iff[symmetric])
-  have "real (c*i) + ?N i e \<ge> real (c*d) \<or> real (c*i) + ?N i e < real (c*d)" by auto
+  from pi cp have th:"(real_of_int i +?N i e / real_of_int c)*real_of_int c \<ge> 0" by (simp add: algebra_simps)
+  from pi ei[simplified isint_iff] have "real_of_int (c*i + ?fe) \<ge> real_of_int (0::int)" by simp
+  hence pi': "c*i + 1 + ?fe \<ge> 1" by (simp only: of_int_le_iff[symmetric])
+  have "real_of_int (c*i) + ?N i e \<ge> real_of_int (c*d) \<or> real_of_int (c*i) + ?N i e < real_of_int (c*d)" by auto
   moreover
-  {assume "real (c*i) + ?N i e \<ge> real (c*d)" hence ?case
+  {assume "real_of_int (c*i) + ?N i e \<ge> real_of_int (c*d)" hence ?case
       by (simp add: algebra_simps 
-        numbound0_I[OF nb,where bs="bs" and b="real i - real d" and b'="real i"])} 
+        numbound0_I[OF nb,where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"])} 
   moreover 
-  {assume H:"real (c*i) + ?N i e < real (c*d)"
-    with ei[simplified isint_iff] have "real (c*i + ?fe) < real (c*d)" by simp
-    hence pid: "c*i + 1 + ?fe \<le> c*d" by (simp only: real_of_int_le_iff)
+  {assume H:"real_of_int (c*i) + ?N i e < real_of_int (c*d)"
+    with ei[simplified isint_iff] have "real_of_int (c*i + ?fe) < real_of_int (c*d)" by simp
+    hence pid: "c*i + 1 + ?fe \<le> c*d" by (simp only: of_int_le_iff)
     with pi' have "\<exists> j1\<in> {1 .. c*d}. c*i + 1+ ?fe = j1" by auto
-    hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) + 1= - ?N i e + real j1"
-      by (simp only: real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] real_of_one) 
-        (simp add: algebra_simps)
-    hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = (- ?N i e + real j1) - 1"
+    hence "\<exists> j1\<in> {1 .. c*d}. real_of_int (c*i) + 1= - ?N i e + real_of_int j1"
+      unfolding Bex_def using ei[simplified isint_iff] by fastforce
+    hence "\<exists> j1\<in> {1 .. c*d}. real_of_int (c*i) = (- ?N i e + real_of_int j1) - 1"
       by (simp only: algebra_simps)
-        hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = - 1 - ?N i e + real j1"
+        hence "\<exists> j1\<in> {1 .. c*d}. real_of_int (c*i) = - 1 - ?N i e + real_of_int j1"
           by (simp add: algebra_simps)
     with nob  have ?case by blast }
   ultimately show ?case by blast
 next
-  case (9 j c e)  hence p: "real j rdvd real (c*i) + ?N i e" (is "?p x") and cp: "c > 0" and bn:"numbound0 e"  by simp+
-  let ?e = "Inum (real i # bs) e"
-  from 9 have "isint e (real i #bs)"  by simp 
-  hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"]
+  case (9 j c e)  hence p: "real_of_int j rdvd real_of_int (c*i) + ?N i e" (is "?p x") and cp: "c > 0" and bn:"numbound0 e"  by simp+
+  let ?e = "Inum (real_of_int i # bs) e"
+  from 9 have "isint e (real_of_int i #bs)"  by simp 
+  hence ie: "real_of_int (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real_of_int i)#bs"] numbound0_I[OF bn,where b="real_of_int i" and b'="real_of_int i" and bs="bs"]
     by (simp add: isint_iff)
   from 9 have id: "j dvd d" by simp
-  from ie[symmetric] have "?p i = (real j rdvd real (c*i+ floor ?e))" by simp
+  from ie[symmetric] have "?p i = (real_of_int j rdvd real_of_int (c*i+ floor ?e))" by simp
   also have "\<dots> = (j dvd c*i + floor ?e)" 
     using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp
   also have "\<dots> = (j dvd c*i - c*d + floor ?e)" 
     using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp
-  also have "\<dots> = (real j rdvd real (c*i - c*d + floor ?e))" 
+  also have "\<dots> = (real_of_int j rdvd real_of_int (c*i - c*d + floor ?e))" 
     using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
       ie by simp
-  also have "\<dots> = (real j rdvd real (c*(i - d)) + ?e)" 
+  also have "\<dots> = (real_of_int j rdvd real_of_int (c*(i - d)) + ?e)" 
     using ie by (simp add:algebra_simps)
   finally show ?case 
-    using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p 
+    using numbound0_I[OF bn,where b="real_of_int i - real_of_int d" and b'="real_of_int i" and bs="bs"] p 
     by (simp add: algebra_simps)
 next
   case (10 j c e)
-  hence p: "\<not> (real j rdvd real (c*i) + ?N i e)" (is "?p x") and cp: "c > 0" and bn:"numbound0 e"
+  hence p: "\<not> (real_of_int j rdvd real_of_int (c*i) + ?N i e)" (is "?p x") and cp: "c > 0" and bn:"numbound0 e"
     by simp+
-  let ?e = "Inum (real i # bs) e"
-  from 10 have "isint e (real i #bs)"  by simp 
-  hence ie: "real (floor ?e) = ?e"
-    using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"]
+  let ?e = "Inum (real_of_int i # bs) e"
+  from 10 have "isint e (real_of_int i #bs)"  by simp 
+  hence ie: "real_of_int (floor ?e) = ?e"
+    using isint_iff[where n="e" and bs="(real_of_int i)#bs"] numbound0_I[OF bn,where b="real_of_int i" and b'="real_of_int i" and bs="bs"]
     by (simp add: isint_iff)
   from 10 have id: "j dvd d" by simp
-  from ie[symmetric] have "?p i = (\<not> (real j rdvd real (c*i+ floor ?e)))" by simp
+  from ie[symmetric] have "?p i = (\<not> (real_of_int j rdvd real_of_int (c*i+ floor ?e)))" by simp
   also have "\<dots> = Not (j dvd c*i + floor ?e)" 
     using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp
   also have "\<dots> = Not (j dvd c*i - c*d + floor ?e)" 
     using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp
-  also have "\<dots> = Not (real j rdvd real (c*i - c*d + floor ?e))" 
+  also have "\<dots> = Not (real_of_int j rdvd real_of_int (c*i - c*d + floor ?e))" 
     using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
       ie by simp
-  also have "\<dots> = Not (real j rdvd real (c*(i - d)) + ?e)" 
+  also have "\<dots> = Not (real_of_int j rdvd real_of_int (c*(i - d)) + ?e)" 
     using ie by (simp add:algebra_simps)
   finally show ?case 
-    using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p 
+    using numbound0_I[OF bn,where b="real_of_int i - real_of_int d" and b'="real_of_int i" and bs="bs"] p 
     by (simp add: algebra_simps)
-qed (auto simp add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"])
+qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"])
 
 lemma \<sigma>_nb: assumes lp: "iszlfm p (a#bs)" and nb: "numbound0 t"
   shows "bound0 (\<sigma> p k t)"
@@ -3209,37 +3185,37 @@
 lemma \<rho>':   assumes lp: "iszlfm p (a #bs)"
   and d: "d_\<delta> p d"
   and dp: "d > 0"
-  shows "\<forall> x. \<not>(\<exists> (e,c) \<in> set(\<rho> p). \<exists>(j::int) \<in> {1 .. c*d}. Ifm (a #bs) (\<sigma> p c (Add e (C j)))) \<longrightarrow> Ifm (real x#bs) p \<longrightarrow> Ifm (real (x - d)#bs) p" (is "\<forall> x. ?b x \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)")
+  shows "\<forall> x. \<not>(\<exists> (e,c) \<in> set(\<rho> p). \<exists>(j::int) \<in> {1 .. c*d}. Ifm (a #bs) (\<sigma> p c (Add e (C j)))) \<longrightarrow> Ifm (real_of_int x#bs) p \<longrightarrow> Ifm (real_of_int (x - d)#bs) p" (is "\<forall> x. ?b x \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)")
 proof(clarify)
   fix x 
   assume nob1:"?b x" and px: "?P x" 
-  from iszlfm_gen[OF lp, rule_format, where y="real x"] have lp': "iszlfm p (real x#bs)".
-  have nob: "\<forall>(e, c)\<in>set (\<rho> p). \<forall>j\<in>{1..c * d}. real (c * x) \<noteq> Inum (real x # bs) e + real j" 
+  from iszlfm_gen[OF lp, rule_format, where y="real_of_int x"] have lp': "iszlfm p (real_of_int x#bs)".
+  have nob: "\<forall>(e, c)\<in>set (\<rho> p). \<forall>j\<in>{1..c * d}. real_of_int (c * x) \<noteq> Inum (real_of_int x # bs) e + real_of_int j" 
   proof(clarify)
     fix e c j assume ecR: "(e,c) \<in> set (\<rho> p)" and jD: "j\<in> {1 .. c*d}"
-      and cx: "real (c*x) = Inum (real x#bs) e + real j"
-    let ?e = "Inum (real x#bs) e"
+      and cx: "real_of_int (c*x) = Inum (real_of_int x#bs) e + real_of_int j"
+    let ?e = "Inum (real_of_int x#bs) e"
     let ?fe = "floor ?e"
-    from \<rho>_l[OF lp'] ecR have ei:"isint e (real x#bs)" and cp:"c>0" and nb:"numbound0 e"
+    from \<rho>_l[OF lp'] ecR have ei:"isint e (real_of_int x#bs)" and cp:"c>0" and nb:"numbound0 e"
       by auto
     from numbound0_gen [OF nb ei, rule_format,where y="a"] have "isint e (a#bs)" .
-    from cx ei[simplified isint_iff] have "real (c*x) = real (?fe + j)" by simp
-    hence cx: "c*x = ?fe + j" by (simp only: real_of_int_inject)
+    from cx ei[simplified isint_iff] have "real_of_int (c*x) = real_of_int (?fe + j)" by simp
+    hence cx: "c*x = ?fe + j" by (simp only: of_int_eq_iff)
     hence cdej:"c dvd ?fe + j" by (simp add: dvd_def) (rule_tac x="x" in exI, simp)
-    hence "real c rdvd real (?fe + j)" by (simp only: int_rdvd_iff)
-    hence rcdej: "real c rdvd ?e + real j" by (simp add: ei[simplified isint_iff])
+    hence "real_of_int c rdvd real_of_int (?fe + j)" by (simp only: int_rdvd_iff)
+    hence rcdej: "real_of_int c rdvd ?e + real_of_int j" by (simp add: ei[simplified isint_iff])
     from cx have "(c*x) div c = (?fe + j) div c" by simp
     with cp have "x = (?fe + j) div c" by simp
     with px have th: "?P ((?fe + j) div c)" by auto
-    from cp have cp': "real c > 0" by simp
-    from cdej have cdej': "c dvd floor (Inum (real x#bs) (Add e (C j)))" by simp
+    from cp have cp': "real_of_int c > 0" by simp
+    from cdej have cdej': "c dvd floor (Inum (real_of_int x#bs) (Add e (C j)))" by simp
     from nb have nb': "numbound0 (Add e (C j))" by simp
-    have ji: "isint (C j) (real x#bs)" by (simp add: isint_def)
-    from isint_add[OF ei ji] have ei':"isint (Add e (C j)) (real x#bs)" .
-    from th \<sigma>_\<rho>[where b'="real x", OF lp' cp' nb' ei' cdej',symmetric]
-    have "Ifm (real x#bs) (\<sigma>_\<rho> p (Add e (C j), c))" by simp
-    with rcdej have th: "Ifm (real x#bs) (\<sigma> p c (Add e (C j)))" by (simp add: \<sigma>_def)
-    from th bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"],where bs="bs" and b="real x" and b'="a"]
+    have ji: "isint (C j) (real_of_int x#bs)" by (simp add: isint_def)
+    from isint_add[OF ei ji] have ei':"isint (Add e (C j)) (real_of_int x#bs)" .
+    from th \<sigma>_\<rho>[where b'="real_of_int x", OF lp' cp' nb' ei' cdej',symmetric]
+    have "Ifm (real_of_int x#bs) (\<sigma>_\<rho> p (Add e (C j), c))" by simp
+    with rcdej have th: "Ifm (real_of_int x#bs) (\<sigma> p c (Add e (C j)))" by (simp add: \<sigma>_def)
+    from th bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"],where bs="bs" and b="real_of_int x" and b'="a"]
     have "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" by blast
       with ecR jD nob1    show "False" by blast
   qed
@@ -3248,8 +3224,8 @@
 
 
 lemma rl_thm: 
-  assumes lp: "iszlfm p (real (i::int)#bs)"
-  shows "(\<exists> (x::int). Ifm (real x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> set (\<rho> p). \<exists> j\<in> {1 .. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))"
+  assumes lp: "iszlfm p (real_of_int (i::int)#bs)"
+  shows "(\<exists> (x::int). Ifm (real_of_int x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real_of_int j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> set (\<rho> p). \<exists> j\<in> {1 .. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))"
   (is "(\<exists>(x::int). ?P x) = ((\<exists> j\<in> {1.. \<delta> p}. ?MP j)\<or>(\<exists> (e,c) \<in> ?R. \<exists> j\<in> _. ?SP c e j))" 
     is "?lhs = (?MD \<or> ?RD)"  is "?lhs = ?rhs")
 proof-
@@ -3259,21 +3235,21 @@
     from H minusinf_ex[OF lp th] have ?thesis  by blast}
   moreover
   { fix e c j assume exR:"(e,c) \<in> ?R" and jD:"j\<in> {1 .. c*?d}" and spx:"?SP c e j"
-    from exR \<rho>_l[OF lp] have nb: "numbound0 e" and ei:"isint e (real i#bs)" and cp: "c > 0"
+    from exR \<rho>_l[OF lp] have nb: "numbound0 e" and ei:"isint e (real_of_int i#bs)" and cp: "c > 0"
       by auto
-    have "isint (C j) (real i#bs)" by (simp add: isint_iff)
-    with isint_add[OF numbound0_gen[OF nb ei,rule_format, where y="real i"]]
-    have eji:"isint (Add e (C j)) (real i#bs)" by simp
+    have "isint (C j) (real_of_int i#bs)" by (simp add: isint_iff)
+    with isint_add[OF numbound0_gen[OF nb ei,rule_format, where y="real_of_int i"]]
+    have eji:"isint (Add e (C j)) (real_of_int i#bs)" by simp
     from nb have nb': "numbound0 (Add e (C j))" by simp
-    from spx bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"], where bs="bs" and b="a" and b'="real i"]
-    have spx': "Ifm (real i # bs) (\<sigma> p c (Add e (C j)))" by blast
-    from spx' have rcdej:"real c rdvd (Inum (real i#bs) (Add e (C j)))" 
-      and sr:"Ifm (real i#bs) (\<sigma>_\<rho> p (Add e (C j),c))" by (simp add: \<sigma>_def)+
+    from spx bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"], where bs="bs" and b="a" and b'="real_of_int i"]
+    have spx': "Ifm (real_of_int i # bs) (\<sigma> p c (Add e (C j)))" by blast
+    from spx' have rcdej:"real_of_int c rdvd (Inum (real_of_int i#bs) (Add e (C j)))" 
+      and sr:"Ifm (real_of_int i#bs) (\<sigma>_\<rho> p (Add e (C j),c))" by (simp add: \<sigma>_def)+
     from rcdej eji[simplified isint_iff] 
-    have "real c rdvd real (floor (Inum (real i#bs) (Add e (C j))))" by simp
-    hence cdej:"c dvd floor (Inum (real i#bs) (Add e (C j)))" by (simp only: int_rdvd_iff)
-    from cp have cp': "real c > 0" by simp
-    from \<sigma>_\<rho>[OF lp cp' nb' eji cdej] spx' have "?P (\<lfloor>Inum (real i # bs) (Add e (C j))\<rfloor> div c)"
+    have "real_of_int c rdvd real_of_int (floor (Inum (real_of_int i#bs) (Add e (C j))))" by simp
+    hence cdej:"c dvd floor (Inum (real_of_int i#bs) (Add e (C j)))" by (simp only: int_rdvd_iff)
+    from cp have cp': "real_of_int c > 0" by simp
+    from \<sigma>_\<rho>[OF lp cp' nb' eji cdej] spx' have "?P (\<lfloor>Inum (real_of_int i # bs) (Add e (C j))\<rfloor> div c)"
       by (simp add: \<sigma>_def)
     hence ?lhs by blast
     with exR jD spx have ?thesis by blast}
@@ -3440,10 +3416,10 @@
       let ?l = "floor (?N s') + j"
       from H 
       have "?I (?p (p',n',s') j) \<longrightarrow> 
-          (((?N ?nxs \<ge> real ?l) \<and> (?N ?nxs < real (?l + 1))) \<and> (?N a = ?N ?nxs ))" 
+          (((?N ?nxs \<ge> real_of_int ?l) \<and> (?N ?nxs < real_of_int (?l + 1))) \<and> (?N a = ?N ?nxs ))" 
         by (simp add: fp_def np algebra_simps)
       also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
-        using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp
+        using floor_unique_iff[where x="?N ?nxs" and a="?l"] by simp
       moreover
       have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" by simp
       ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"
@@ -3466,10 +3442,10 @@
       let ?l = "floor (?N s') + j"
       from H 
       have "?I (?p (p',n',s') j) \<longrightarrow> 
-          (((?N ?nxs \<ge> real ?l) \<and> (?N ?nxs < real (?l + 1))) \<and> (?N a = ?N ?nxs ))" 
+          (((?N ?nxs \<ge> real_of_int ?l) \<and> (?N ?nxs < real_of_int (?l + 1))) \<and> (?N a = ?N ?nxs ))" 
         by (simp add: np fp_def algebra_simps)
       also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
-        using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp
+        using floor_unique_iff[where x="?N ?nxs" and a="?l"] by simp
       moreover
       have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"  by simp
       ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))"
@@ -3483,10 +3459,11 @@
 qed (auto simp add: Let_def split_def algebra_simps)
 
 lemma real_in_int_intervals: 
-  assumes xb: "real m \<le> x \<and> x < real ((n::int) + 1)"
-  shows "\<exists> j\<in> {m.. n}. real j \<le> x \<and> x < real (j+1)" (is "\<exists> j\<in> ?N. ?P j")
+  assumes xb: "real_of_int m \<le> x \<and> x < real_of_int ((n::int) + 1)"
+  shows "\<exists> j\<in> {m.. n}. real_of_int j \<le> x \<and> x < real_of_int (j+1)" (is "\<exists> j\<in> ?N. ?P j")
 by (rule bexI[where P="?P" and x="floor x" and A="?N"]) 
-(auto simp add: floor_less_eq[where x="x" and a="n+1", simplified] xb[simplified] floor_mono[where x="real m" and y="x", OF conjunct1[OF xb], simplified floor_real_of_int[where n="m"]])
+(auto simp add: floor_less_iff[where x="x" and z="n+1", simplified] 
+  xb[simplified] floor_mono[where x="real_of_int m" and y="x", OF conjunct1[OF xb], simplified floor_of_int[where z="m"]])
 
 lemma rsplit0_complete:
   assumes xp:"0 \<le> x" and x1:"x < 1"
@@ -3571,20 +3548,20 @@
   moreover
   {
     assume np: "n > 0"
-    from real_of_int_floor_le[where r="?N s"] have "?N (Floor s) \<le> ?N s" by simp
-    also from mult_left_mono[OF xp] np have "?N s \<le> real n * x + ?N s" by simp
-    finally have "?N (Floor s) \<le> real n * x + ?N s" .
+    from of_int_floor_le[of "?N s"] have "?N (Floor s) \<le> ?N s" by simp
+    also from mult_left_mono[OF xp] np have "?N s \<le> real_of_int n * x + ?N s" by simp
+    finally have "?N (Floor s) \<le> real_of_int n * x + ?N s" .
     moreover
-    {from x1 np have "real n *x + ?N s < real n + ?N s" by simp
+    {from x1 np have "real_of_int n *x + ?N s < real_of_int n + ?N s" by simp
       also from real_of_int_floor_add_one_gt[where r="?N s"] 
-      have "\<dots> < real n + ?N (Floor s) + 1" by simp
-      finally have "real n *x + ?N s < ?N (Floor s) + real (n+1)" by simp}
-    ultimately have "?N (Floor s) \<le> real n *x + ?N s\<and> real n *x + ?N s < ?N (Floor s) + real (n+1)" by simp
-    hence th: "0 \<le> real n *x + ?N s - ?N (Floor s) \<and> real n *x + ?N s - ?N (Floor s) < real (n+1)" by simp
-    from real_in_int_intervals th have  "\<exists> j\<in> {0 .. n}. real j \<le> real n *x + ?N s - ?N (Floor s)\<and> real n *x + ?N s - ?N (Floor s) < real (j+1)" by simp
+      have "\<dots> < real_of_int n + ?N (Floor s) + 1" by simp
+      finally have "real_of_int n *x + ?N s < ?N (Floor s) + real_of_int (n+1)" by simp}
+    ultimately have "?N (Floor s) \<le> real_of_int n *x + ?N s\<and> real_of_int n *x + ?N s < ?N (Floor s) + real_of_int (n+1)" by simp
+    hence th: "0 \<le> real_of_int n *x + ?N s - ?N (Floor s) \<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (n+1)" by simp
+    from real_in_int_intervals th have  "\<exists> j\<in> {0 .. n}. real_of_int j \<le> real_of_int n *x + ?N s - ?N (Floor s)\<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (j+1)" by simp
     
-    hence "\<exists> j\<in> {0 .. n}. 0 \<le> real n *x + ?N s - ?N (Floor s) - real j \<and> real n *x + ?N s - ?N (Floor s) - real (j+1) < 0"
-      by(simp only: myle[of _ "real n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real n *x + ?N s - ?N (Floor s)"]) 
+    hence "\<exists> j\<in> {0 .. n}. 0 \<le> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int j \<and> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int (j+1) < 0"
+      by(simp only: myle[of _ "real_of_int n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real_of_int n *x + ?N s - ?N (Floor s)"]) 
     hence "\<exists> j\<in> {0.. n}. ?I (?p (p,n,s) j)"
       using pns by (simp add: fp_def np algebra_simps)
     then obtain "j" where j_def: "j\<in> {0 .. n} \<and> ?I (?p (p,n,s) j)" by blast
@@ -3596,23 +3573,23 @@
   }
   moreover
   { assume nn: "n < 0" hence np: "-n >0" by simp
-    from real_of_int_floor_le[where r="?N s"] have "?N (Floor s) + 1 > ?N s" by simp
-    moreover from mult_left_mono_neg[OF xp] nn have "?N s \<ge> real n * x + ?N s" by simp
-    ultimately have "?N (Floor s) + 1 > real n * x + ?N s" by arith 
+    from of_int_floor_le[of "?N s"] have "?N (Floor s) + 1 > ?N s" by simp
+    moreover from mult_left_mono_neg[OF xp] nn have "?N s \<ge> real_of_int n * x + ?N s" by simp
+    ultimately have "?N (Floor s) + 1 > real_of_int n * x + ?N s" by arith 
     moreover
-    {from x1 nn have "real n *x + ?N s \<ge> real n + ?N s" by simp
-      moreover from real_of_int_floor_le[where r="?N s"]  have "real n + ?N s \<ge> real n + ?N (Floor s)" by simp
-      ultimately have "real n *x + ?N s \<ge> ?N (Floor s) + real n" 
+    {from x1 nn have "real_of_int n *x + ?N s \<ge> real_of_int n + ?N s" by simp
+      moreover from of_int_floor_le[of "?N s"]  have "real_of_int n + ?N s \<ge> real_of_int n + ?N (Floor s)" by simp
+      ultimately have "real_of_int n *x + ?N s \<ge> ?N (Floor s) + real_of_int n" 
         by (simp only: algebra_simps)}
-    ultimately have "?N (Floor s) + real n \<le> real n *x + ?N s\<and> real n *x + ?N s < ?N (Floor s) + real (1::int)" by simp
-    hence th: "real n \<le> real n *x + ?N s - ?N (Floor s) \<and> real n *x + ?N s - ?N (Floor s) < real (1::int)" by simp
+    ultimately have "?N (Floor s) + real_of_int n \<le> real_of_int n *x + ?N s\<and> real_of_int n *x + ?N s < ?N (Floor s) + real_of_int (1::int)" by simp
+    hence th: "real_of_int n \<le> real_of_int n *x + ?N s - ?N (Floor s) \<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (1::int)" by simp
     have th1: "\<forall> (a::real). (- a > 0) = (a < 0)" by auto
     have th2: "\<forall> (a::real). (0 \<ge> - a) = (a \<ge> 0)" by auto
-    from real_in_int_intervals th  have  "\<exists> j\<in> {n .. 0}. real j \<le> real n *x + ?N s - ?N (Floor s)\<and> real n *x + ?N s - ?N (Floor s) < real (j+1)" by simp
+    from real_in_int_intervals th  have  "\<exists> j\<in> {n .. 0}. real_of_int j \<le> real_of_int n *x + ?N s - ?N (Floor s)\<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (j+1)" by simp
     
-    hence "\<exists> j\<in> {n .. 0}. 0 \<le> real n *x + ?N s - ?N (Floor s) - real j \<and> real n *x + ?N s - ?N (Floor s) - real (j+1) < 0"
-      by(simp only: myle[of _ "real n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real n *x + ?N s - ?N (Floor s)"]) 
-    hence "\<exists> j\<in> {n .. 0}. 0 \<ge> - (real n *x + ?N s - ?N (Floor s) - real j) \<and> - (real n *x + ?N s - ?N (Floor s) - real (j+1)) > 0" by (simp only: th1[rule_format] th2[rule_format])
+    hence "\<exists> j\<in> {n .. 0}. 0 \<le> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int j \<and> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int (j+1) < 0"
+      by(simp only: myle[of _ "real_of_int n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real_of_int n *x + ?N s - ?N (Floor s)"]) 
+    hence "\<exists> j\<in> {n .. 0}. 0 \<ge> - (real_of_int n *x + ?N s - ?N (Floor s) - real_of_int j) \<and> - (real_of_int n *x + ?N s - ?N (Floor s) - real_of_int (j+1)) > 0" by (simp only: th1[rule_format] th2[rule_format])
     hence "\<exists> j\<in> {n.. 0}. ?I (?p (p,n,s) j)"
       using pns by (simp add: fp_def nn algebra_simps
         del: diff_less_0_iff_less diff_le_0_iff_le) 
@@ -3773,37 +3750,37 @@
 
 lemma small_le: 
   assumes u0:"0 \<le> u" and u1: "u < 1"
-  shows "(-u \<le> real (n::int)) = (0 \<le> n)"
+  shows "(-u \<le> real_of_int (n::int)) = (0 \<le> n)"
 using u0 u1  by auto
 
 lemma small_lt: 
   assumes u0:"0 \<le> u" and u1: "u < 1"
-  shows "(real (n::int) < real (m::int) - u) = (n < m)"
+  shows "(real_of_int (n::int) < real_of_int (m::int) - u) = (n < m)"
 using u0 u1  by auto
 
 lemma rdvd01_cs: 
-  assumes up: "u \<ge> 0" and u1: "u<1" and np: "real n > 0"
-  shows "(real (i::int) rdvd real (n::int) * u - s) = (\<exists> j\<in> {0 .. n - 1}. real n * u = s - real (floor s) + real j \<and> real i rdvd real (j - floor s))" (is "?lhs = ?rhs")
+  assumes up: "u \<ge> 0" and u1: "u<1" and np: "real_of_int n > 0"
+  shows "(real_of_int (i::int) rdvd real_of_int (n::int) * u - s) = (\<exists> j\<in> {0 .. n - 1}. real_of_int n * u = s - real_of_int (floor s) + real_of_int j \<and> real_of_int i rdvd real_of_int (j - floor s))" (is "?lhs = ?rhs")
 proof-
-  let ?ss = "s - real (floor s)"
+  let ?ss = "s - real_of_int (floor s)"
   from real_of_int_floor_add_one_gt[where r="s", simplified myless[of "s"]] 
-    real_of_int_floor_le[where r="s"]  have ss0:"?ss \<ge> 0" and ss1:"?ss < 1" 
+    of_int_floor_le  have ss0:"?ss \<ge> 0" and ss1:"?ss < 1" 
     by (auto simp add: myle[of _ "s", symmetric] myless[of "?ss"])
-  from np have n0: "real n \<ge> 0" by simp
+  from np have n0: "real_of_int n \<ge> 0" by simp
   from mult_left_mono[OF up n0] mult_strict_left_mono[OF u1 np] 
-  have nu0:"real n * u - s \<ge> -s" and nun:"real n * u -s < real n - s" by auto  
-  from int_rdvd_real[where i="i" and x="real (n::int) * u - s"] 
-  have "real i rdvd real n * u - s = 
-    (i dvd floor (real n * u -s) \<and> (real (floor (real n * u - s)) = real n * u - s ))" 
+  have nu0:"real_of_int n * u - s \<ge> -s" and nun:"real_of_int n * u -s < real_of_int n - s" by auto  
+  from int_rdvd_real[where i="i" and x="real_of_int (n::int) * u - s"] 
+  have "real_of_int i rdvd real_of_int n * u - s = 
+    (i dvd floor (real_of_int n * u -s) \<and> (real_of_int (floor (real_of_int n * u - s)) = real_of_int n * u - s ))" 
     (is "_ = (?DE)" is "_ = (?D \<and> ?E)") by simp
-  also have "\<dots> = (?DE \<and> real(floor (real n * u - s) + floor s)\<ge> -?ss 
-    \<and> real(floor (real n * u - s) + floor s)< real n - ?ss)" (is "_=(?DE \<and>real ?a \<ge> _ \<and> real ?a < _)")
+  also have "\<dots> = (?DE \<and> real_of_int(floor (real_of_int n * u - s) + floor s)\<ge> -?ss 
+    \<and> real_of_int(floor (real_of_int n * u - s) + floor s)< real_of_int n - ?ss)" (is "_=(?DE \<and>real_of_int ?a \<ge> _ \<and> real_of_int ?a < _)")
     using nu0 nun  by auto
   also have "\<dots> = (?DE \<and> ?a \<ge> 0 \<and> ?a < n)" by(simp only: small_le[OF ss0 ss1] small_lt[OF ss0 ss1])
   also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. ?a = j))" by simp
-  also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. real (\<lfloor>real n * u - s\<rfloor>) = real j - real \<lfloor>s\<rfloor> ))"
-    by (simp only: algebra_simps real_of_int_diff[symmetric] real_of_int_inject)
-  also have "\<dots> = ((\<exists> j\<in> {0 .. (n - 1)}. real n * u - s = real j - real \<lfloor>s\<rfloor> \<and> real i rdvd real n * u - s))" using int_rdvd_iff[where i="i" and t="\<lfloor>real n * u - s\<rfloor>"]
+  also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. real_of_int (\<lfloor>real_of_int n * u - s\<rfloor>) = real_of_int j - real_of_int \<lfloor>s\<rfloor> ))"
+    by (simp only: algebra_simps of_int_diff[symmetric] of_int_eq_iff)
+  also have "\<dots> = ((\<exists> j\<in> {0 .. (n - 1)}. real_of_int n * u - s = real_of_int j - real_of_int \<lfloor>s\<rfloor> \<and> real_of_int i rdvd real_of_int n * u - s))" using int_rdvd_iff[where i="i" and t="\<lfloor>real_of_int n * u - s\<rfloor>"]
     by (auto cong: conj_cong)
   also have "\<dots> = ?rhs" by(simp cong: conj_cong) (simp add: algebra_simps )
   finally show ?thesis .
@@ -3820,7 +3797,7 @@
   NDVDJ_def: "NDVDJ i n s = (foldr conj (map (\<lambda> j. disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor (Neg s))))) [0..n - 1]) T)"
 
 lemma DVDJ_DVD: 
-  assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real n > 0"
+  assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real_of_int n > 0"
   shows "Ifm (x#bs) (DVDJ i n s) = Ifm (x#bs) (Dvd i (CN 0 n s))"
 proof-
   let ?f = "\<lambda> j. conj (Eq(CN 0 n (Add s (Sub(Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))"
@@ -3828,15 +3805,15 @@
   from foldr_disj_map[where xs="[0..n - 1]" and bs="x#bs" and f="?f"]
   have "Ifm (x#bs) (DVDJ i n s) = (\<exists> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))" 
     by (simp add: np DVDJ_def)
-  also have "\<dots> = (\<exists> j\<in> {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \<and> real i rdvd real (j - floor (- ?s)))"
+  also have "\<dots> = (\<exists> j\<in> {0 .. (n - 1)}. real_of_int n * x = (- ?s) - real_of_int (floor (- ?s)) + real_of_int j \<and> real_of_int i rdvd real_of_int (j - floor (- ?s)))"
     by (simp add: algebra_simps)
   also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] 
-  have "\<dots> = (real i rdvd real n * x - (-?s))" by simp
+  have "\<dots> = (real_of_int i rdvd real_of_int n * x - (-?s))" by simp
   finally show ?thesis by simp
 qed
 
 lemma NDVDJ_NDVD: 
-  assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real n > 0"
+  assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real_of_int n > 0"
   shows "Ifm (x#bs) (NDVDJ i n s) = Ifm (x#bs) (NDvd i (CN 0 n s))"
 proof-
   let ?f = "\<lambda> j. disj(NEq(CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor(Neg s))))"
@@ -3844,10 +3821,10 @@
   from foldr_conj_map[where xs="[0..n - 1]" and bs="x#bs" and f="?f"]
   have "Ifm (x#bs) (NDVDJ i n s) = (\<forall> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))" 
     by (simp add: np NDVDJ_def)
-  also have "\<dots> = (\<not> (\<exists> j\<in> {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \<and> real i rdvd real (j - floor (- ?s))))"
+  also have "\<dots> = (\<not> (\<exists> j\<in> {0 .. (n - 1)}. real_of_int n * x = (- ?s) - real_of_int (floor (- ?s)) + real_of_int j \<and> real_of_int i rdvd real_of_int (j - floor (- ?s))))"
     by (simp add: algebra_simps)
   also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] 
-  have "\<dots> = (\<not> (real i rdvd real n * x - (-?s)))" by simp
+  have "\<dots> = (\<not> (real_of_int i rdvd real_of_int n * x - (-?s)))" by simp
   finally show ?thesis by simp
 qed  
 
@@ -3902,7 +3879,7 @@
   moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H DVD_def) } 
   moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th 
       by (simp add: DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1 
-        rdvd_minus[where d="i" and t="real n * x + Inum (x # bs) s"]) } 
+        rdvd_minus[where d="i" and t="real_of_int n * x + Inum (x # bs) s"]) } 
   moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th by (simp add:DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1)}
   ultimately show ?th by blast
 qed
@@ -3920,7 +3897,7 @@
   moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H NDVD_def) } 
   moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th 
       by (simp add: NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1 
-        rdvd_minus[where d="i" and t="real n * x + Inum (x # bs) s"]) } 
+        rdvd_minus[where d="i" and t="real_of_int n * x + Inum (x # bs) s"]) } 
   moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th 
       by (simp add:NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1)}
   ultimately show ?th by blast
@@ -4152,16 +4129,16 @@
 next
   case (3 c e) 
   from 3 have nb: "numbound0 e" by simp
-  from 3 have cp: "real c > 0" by simp
+  from 3 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
+    hence "(real_of_int c * x < - ?e)" 
       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) 
-    hence "real c * x + ?e < 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
+    hence "real_of_int c * x + ?e < 0" by arith
+    hence "real_of_int c * x + ?e \<noteq> 0" by simp
     with xz have "?P ?z x (Eq (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp  }
   hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
@@ -4169,16 +4146,16 @@
 next
   case (4 c e)   
   from 4 have nb: "numbound0 e" by simp
-  from 4 have cp: "real c > 0" by simp
+  from 4 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
+    hence "(real_of_int c * x < - ?e)" 
       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) 
-    hence "real c * x + ?e < 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
+    hence "real_of_int c * x + ?e < 0" by arith
+    hence "real_of_int c * x + ?e \<noteq> 0" by simp
     with xz have "?P ?z x (NEq (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
   hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
@@ -4186,15 +4163,15 @@
 next
   case (5 c e) 
   from 5 have nb: "numbound0 e" by simp
-  from 5 have cp: "real c > 0" by simp
+  from 5 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
+    hence "(real_of_int c * x < - ?e)" 
       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) 
-    hence "real c * x + ?e < 0" by arith
+    hence "real_of_int c * x + ?e < 0" by arith
     with xz have "?P ?z x (Lt (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp }
   hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
@@ -4202,15 +4179,15 @@
 next
   case (6 c e)  
   from 6 have nb: "numbound0 e" by simp
-  from 6 have cp: "real c > 0" by simp
+  from 6 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
+    hence "(real_of_int c * x < - ?e)" 
       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) 
-    hence "real c * x + ?e < 0" by arith
+    hence "real_of_int c * x + ?e < 0" by arith
     with xz have "?P ?z x (Le (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
   hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
@@ -4218,15 +4195,15 @@
 next
   case (7 c e)  
   from 7 have nb: "numbound0 e" by simp
-  from 7 have cp: "real c > 0" by simp
+  from 7 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
+    hence "(real_of_int c * x < - ?e)" 
       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) 
-    hence "real c * x + ?e < 0" by arith
+    hence "real_of_int c * x + ?e < 0" by arith
     with xz have "?P ?z x (Gt (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
   hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
@@ -4234,15 +4211,15 @@
 next
   case (8 c e)  
   from 8 have nb: "numbound0 e" by simp
-  from 8 have cp: "real c > 0" by simp
+  from 8 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x < ?z"
-    hence "(real c * x < - ?e)" 
+    hence "(real_of_int c * x < - ?e)" 
       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) 
-    hence "real c * x + ?e < 0" by arith
+    hence "real_of_int c * x + ?e < 0" by arith
     with xz have "?P ?z x (Ge (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
   hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp
@@ -4260,16 +4237,16 @@
 next
   case (3 c e) 
   from 3 have nb: "numbound0 e" by simp
-  from 3 have cp: "real c > 0" by simp
+  from 3 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: ac_simps)
-    hence "real c * x + ?e > 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
+    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
+    hence "real_of_int c * x + ?e > 0" by arith
+    hence "real_of_int c * x + ?e \<noteq> 0" by simp
     with xz have "?P ?z x (Eq (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
   hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
@@ -4277,16 +4254,16 @@
 next
   case (4 c e) 
   from 4 have nb: "numbound0 e" by simp
-  from 4 have cp: "real c > 0" by simp
+  from 4 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: ac_simps)
-    hence "real c * x + ?e > 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
+    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
+    hence "real_of_int c * x + ?e > 0" by arith
+    hence "real_of_int c * x + ?e \<noteq> 0" by simp
     with xz have "?P ?z x (NEq (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
   hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
@@ -4294,15 +4271,15 @@
 next
   case (5 c e) 
   from 5 have nb: "numbound0 e" by simp
-  from 5 have cp: "real c > 0" by simp
+  from 5 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: ac_simps)
-    hence "real c * x + ?e > 0" by arith
+    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
+    hence "real_of_int c * x + ?e > 0" by arith
     with xz have "?P ?z x (Lt (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
   hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
@@ -4310,15 +4287,15 @@
 next
   case (6 c e) 
   from 6 have nb: "numbound0 e" by simp
-  from 6 have cp: "real c > 0" by simp
+  from 6 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: ac_simps)
-    hence "real c * x + ?e > 0" by arith
+    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
+    hence "real_of_int c * x + ?e > 0" by arith
     with xz have "?P ?z x (Le (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
   hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
@@ -4326,15 +4303,15 @@
 next
   case (7 c e) 
   from 7 have nb: "numbound0 e" by simp
-  from 7 have cp: "real c > 0" by simp
+  from 7 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: ac_simps)
-    hence "real c * x + ?e > 0" by arith
+    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
+    hence "real_of_int c * x + ?e > 0" by arith
     with xz have "?P ?z x (Gt (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
   hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
@@ -4342,15 +4319,15 @@
 next
   case (8 c e) 
   from 8 have nb: "numbound0 e" by simp
-  from 8 have cp: "real c > 0" by simp
+  from 8 have cp: "real_of_int c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
-  let ?z = "(- ?e) / real c"
+  let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
-    have "(real c * x > - ?e)" by (simp add: ac_simps)
-    hence "real c * x + ?e > 0" by arith
+    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
+    hence "real_of_int c * x + ?e > 0" by arith
     with xz have "?P ?z x (Ge (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]   by simp }
   hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
@@ -4423,78 +4400,78 @@
   "\<upsilon> p = (\<lambda> (t,n). p)"
 
 lemma \<upsilon>_I: assumes lp: "isrlfm p"
-  and np: "real n > 0" and nbt: "numbound0 t"
-  shows "(Ifm (x#bs) (\<upsilon> p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (\<upsilon> p (t,n))" (is "(?I x (\<upsilon> p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _")
+  and np: "real_of_int n > 0" and nbt: "numbound0 t"
+  shows "(Ifm (x#bs) (\<upsilon> p (t,n)) = Ifm (((Inum (x#bs) t)/(real_of_int n))#bs) p) \<and> bound0 (\<upsilon> p (t,n))" (is "(?I x (\<upsilon> p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _")
   using lp
 proof(induct p rule: \<upsilon>.induct)
   case (5 c e)
   from 5 have cp: "c >0" and nb: "numbound0 e" by simp_all
-  have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
+  have "?I ?u (Lt (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) < 0)"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
-    by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
+  also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) < 0)"
+    by (simp only: pos_less_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" 
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
+  also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) < 0)"
     using np by simp 
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (6 c e)
   from 6 have cp: "c >0" and nb: "numbound0 e" by simp_all
-  have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
+  have "?I ?u (Le (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) \<le> 0)"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
-    by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
+  also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
+    by (simp only: pos_le_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" 
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
+  also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<le> 0)"
     using np by simp 
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (7 c e)
   from 7 have cp: "c >0" and nb: "numbound0 e" by simp_all
-  have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
+  have "?I ?u (Gt (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) > 0)"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
-    by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
+  also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) > 0)"
+    by (simp only: pos_divide_less_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" 
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
+  also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) > 0)"
     using np by simp 
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (8 c e)
   from 8 have cp: "c >0" and nb: "numbound0 e" by simp_all
-  have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
+  have "?I ?u (Ge (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) \<ge> 0)"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
-    by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
+  also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
+    by (simp only: pos_divide_le_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" 
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
+  also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<ge> 0)"
     using np by simp 
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (3 c e)
   from 3 have cp: "c >0" and nb: "numbound0 e" by simp_all
-  from np have np: "real n \<noteq> 0" by simp
-  have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)"
+  from np have np: "real_of_int n \<noteq> 0" by simp
+  have "?I ?u (Eq (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) = 0)"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
-    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
+  also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) = 0)"
+    by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" 
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
+  also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) = 0)"
     using np by simp 
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (4 c e)
   from 4 have cp: "c >0" and nb: "numbound0 e" by simp_all
-  from np have np: "real n \<noteq> 0" by simp
-  have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)"
+  from np have np: "real_of_int n \<noteq> 0" by simp
+  have "?I ?u (NEq (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) \<noteq> 0)"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
-  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
-    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
+  also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
+    by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" 
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
+  also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<noteq> 0)"
     using np by simp 
   finally show ?case using nbt nb by (simp add: algebra_simps)
-qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"])
+qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real_of_int n" and b'="x"])
 
 lemma \<Upsilon>_l:
   assumes lp: "isrlfm p"
@@ -4506,14 +4483,14 @@
   assumes lp: "isrlfm p"
   and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
   and ex: "Ifm (x#bs) p" (is "?I x p")
-  shows "\<exists> (s,m) \<in> set (\<Upsilon> p). x \<ge> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real m")
+  shows "\<exists> (s,m) \<in> set (\<Upsilon> p). x \<ge> Inum (a#bs) s / real_of_int m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real_of_int m")
 proof-
-  have "\<exists> (s,m) \<in> set (\<Upsilon> p). real m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<ge> ?N a s")
+  have "\<exists> (s,m) \<in> set (\<Upsilon> p). real_of_int m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real_of_int m *x \<ge> ?N a s")
     using lp nmi ex
     by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"])
-  then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real m * x \<ge> ?N a s" by blast
-  from \<Upsilon>_l[OF lp] smU have mp: "real m > 0" by auto
-  from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m" 
+  then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real_of_int m * x \<ge> ?N a s" by blast
+  from \<Upsilon>_l[OF lp] smU have mp: "real_of_int m > 0" by auto
+  from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real_of_int m" 
     by (auto simp add: mult.commute)
   thus ?thesis using smU by auto
 qed
@@ -4522,119 +4499,119 @@
   assumes lp: "isrlfm p"
   and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
   and ex: "Ifm (x#bs) p" (is "?I x p")
-  shows "\<exists> (s,m) \<in> set (\<Upsilon> p). x \<le> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real m")
+  shows "\<exists> (s,m) \<in> set (\<Upsilon> p). x \<le> Inum (a#bs) s / real_of_int m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real_of_int m")
 proof-
-  have "\<exists> (s,m) \<in> set (\<Upsilon> p). real m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<le> ?N a s")
+  have "\<exists> (s,m) \<in> set (\<Upsilon> p). real_of_int m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real_of_int m *x \<le> ?N a s")
     using lp nmi ex
     by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"])
-  then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real m * x \<le> ?N a s" by blast
-  from \<Upsilon>_l[OF lp] smU have mp: "real m > 0" by auto
-  from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m" 
+  then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real_of_int m * x \<le> ?N a s" by blast
+  from \<Upsilon>_l[OF lp] smU have mp: "real_of_int m > 0" by auto
+  from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real_of_int m" 
     by (auto simp add: mult.commute)
   thus ?thesis using smU by auto
 qed
 
 lemma lin_dense: 
   assumes lp: "isrlfm p"
-  and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (\<Upsilon> p)" 
-  (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)")
+  and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real_of_int n) ` set (\<Upsilon> p)" 
+  (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real_of_int n ) ` (?U p)")
   and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p"
   and ly: "l < y" and yu: "y < u"
   shows "Ifm (y#bs) p"
 using lp px noS
 proof (induct p rule: isrlfm.induct)
-  case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp_all
-  from 5 have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
-  hence pxc: "x < (- ?N x e) / real c" 
+  case (5 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all
+  from 5 have "x * real_of_int c + ?N x e < 0" by (simp add: algebra_simps)
+  hence pxc: "x < (- ?N x e) / real_of_int c" 
     by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
-  from 5 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-  hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
-  moreover {assume y: "y < (-?N x e)/ real c"
-    hence "y * real c < - ?N x e"
+  from 5 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto
+  with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto
+  hence "y < (- ?N x e) / real_of_int c \<or> y > (-?N x e) / real_of_int c" by auto
+  moreover {assume y: "y < (-?N x e)/ real_of_int c"
+    hence "y * real_of_int c < - ?N x e"
       by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-    hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
+    hence "real_of_int c * y + ?N x e < 0" by (simp add: algebra_simps)
     hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-  moreover {assume y: "y > (- ?N x e) / real c" 
-    with yu have eu: "u > (- ?N x e) / real c" by auto
-    with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
+  moreover {assume y: "y > (- ?N x e) / real_of_int c" 
+    with yu have eu: "u > (- ?N x e) / real_of_int c" by auto
+    with noSc ly yu have "(- ?N x e) / real_of_int c \<le> l" by (cases "(- ?N x e) / real_of_int c > l", auto)
     with lx pxc have "False" by auto
     hence ?case by simp }
   ultimately show ?case by blast
 next
-  case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp_all
-  from 6 have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
-  hence pxc: "x \<le> (- ?N x e) / real c" 
+  case (6 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all
+  from 6 have "x * real_of_int c + ?N x e \<le> 0" by (simp add: algebra_simps)
+  hence pxc: "x \<le> (- ?N x e) / real_of_int c" 
     by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
-  from 6 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-  hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
-  moreover {assume y: "y < (-?N x e)/ real c"
-    hence "y * real c < - ?N x e"
+  from 6 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto
+  with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto
+  hence "y < (- ?N x e) / real_of_int c \<or> y > (-?N x e) / real_of_int c" by auto
+  moreover {assume y: "y < (-?N x e)/ real_of_int c"
+    hence "y * real_of_int c < - ?N x e"
       by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-    hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
+    hence "real_of_int c * y + ?N x e < 0" by (simp add: algebra_simps)
     hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-  moreover {assume y: "y > (- ?N x e) / real c" 
-    with yu have eu: "u > (- ?N x e) / real c" by auto
-    with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
+  moreover {assume y: "y > (- ?N x e) / real_of_int c" 
+    with yu have eu: "u > (- ?N x e) / real_of_int c" by auto
+    with noSc ly yu have "(- ?N x e) / real_of_int c \<le> l" by (cases "(- ?N x e) / real_of_int c > l", auto)
     with lx pxc have "False" by auto
     hence ?case by simp }
   ultimately show ?case by blast
 next
-  case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp_all
-  from 7 have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
-  hence pxc: "x > (- ?N x e) / real c" 
+  case (7 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all
+  from 7 have "x * real_of_int c + ?N x e > 0" by (simp add: algebra_simps)
+  hence pxc: "x > (- ?N x e) / real_of_int c" 
     by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
-  from 7 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-  hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
-  moreover {assume y: "y > (-?N x e)/ real c"
-    hence "y * real c > - ?N x e"
+  from 7 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto
+  with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto
+  hence "y < (- ?N x e) / real_of_int c \<or> y > (-?N x e) / real_of_int c" by auto
+  moreover {assume y: "y > (-?N x e)/ real_of_int c"
+    hence "y * real_of_int c > - ?N x e"
       by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-    hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
+    hence "real_of_int c * y + ?N x e > 0" by (simp add: algebra_simps)
     hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-  moreover {assume y: "y < (- ?N x e) / real c" 
-    with ly have eu: "l < (- ?N x e) / real c" by auto
-    with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
+  moreover {assume y: "y < (- ?N x e) / real_of_int c" 
+    with ly have eu: "l < (- ?N x e) / real_of_int c" by auto
+    with noSc ly yu have "(- ?N x e) / real_of_int c \<ge> u" by (cases "(- ?N x e) / real_of_int c > l", auto)
     with xu pxc have "False" by auto
     hence ?case by simp }
   ultimately show ?case by blast
 next
-  case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp_all
-  from 8 have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
-  hence pxc: "x \<ge> (- ?N x e) / real c" 
+  case (8 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all
+  from 8 have "x * real_of_int c + ?N x e \<ge> 0" by (simp add: algebra_simps)
+  hence pxc: "x \<ge> (- ?N x e) / real_of_int c" 
     by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
-  from 8 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-  hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
-  moreover {assume y: "y > (-?N x e)/ real c"
-    hence "y * real c > - ?N x e"
+  from 8 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto
+  with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto
+  hence "y < (- ?N x e) / real_of_int c \<or> y > (-?N x e) / real_of_int c" by auto
+  moreover {assume y: "y > (-?N x e)/ real_of_int c"
+    hence "y * real_of_int c > - ?N x e"
       by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-    hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
+    hence "real_of_int c * y + ?N x e > 0" by (simp add: algebra_simps)
     hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-  moreover {assume y: "y < (- ?N x e) / real c" 
-    with ly have eu: "l < (- ?N x e) / real c" by auto
-    with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
+  moreover {assume y: "y < (- ?N x e) / real_of_int c" 
+    with ly have eu: "l < (- ?N x e) / real_of_int c" by auto
+    with noSc ly yu have "(- ?N x e) / real_of_int c \<ge> u" by (cases "(- ?N x e) / real_of_int c > l", auto)
     with xu pxc have "False" by auto
     hence ?case by simp }
   ultimately show ?case by blast
 next
-  case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp_all
-  from cp have cnz: "real c \<noteq> 0" by simp
-  from 3 have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
-  hence pxc: "x = (- ?N x e) / real c" 
+  case (3 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all
+  from cp have cnz: "real_of_int c \<noteq> 0" by simp
+  from 3 have "x * real_of_int c + ?N x e = 0" by (simp add: algebra_simps)
+  hence pxc: "x = (- ?N x e) / real_of_int c" 
     by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
-  from 3 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-  with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
+  from 3 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto
+  with lx xu have yne: "x \<noteq> - ?N x e / real_of_int c" by auto
   with pxc show ?case by simp
 next
-  case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp_all
-  from cp have cnz: "real c \<noteq> 0" by simp
-  from 4 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
-  with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
-  hence "y* real c \<noteq> -?N x e"      
+  case (4 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all
+  from cp have cnz: "real_of_int c \<noteq> 0" by simp
+  from 4 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto
+  with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto
+  hence "y* real_of_int c \<noteq> -?N x e"      
     by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
-  hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
+  hence "y* real_of_int c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
   thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] 
     by (simp add: algebra_simps)
 qed (auto simp add: numbound0_I[where bs="bs" and b="y" and b'="x"])
@@ -4645,7 +4622,7 @@
   and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
   and ex: "\<exists> x.  Ifm (x#bs) p" (is "\<exists> x. ?I x p")
   shows "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p).
-    ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p" 
+    ?I ((Inum (x#bs) l / real_of_int n + Inum (x#bs) s / real_of_int m) / 2) p" 
 proof-
   let ?N = "\<lambda> x t. Inum (x#bs) t"
   let ?U = "set (\<Upsilon> p)"
@@ -4654,46 +4631,46 @@
   have nmi': "\<not> (?I a (?M p))" by simp
   from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi
   have npi': "\<not> (?I a (?P p))" by simp
-  have "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). ?I ((?N a l/real n + ?N a s /real m) / 2) p"
+  have "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). ?I ((?N a l/real_of_int n + ?N a s /real_of_int m) / 2) p"
   proof-
-    let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U"
+    let ?M = "(\<lambda> (t,c). ?N a t / real_of_int c) ` ?U"
     have fM: "finite ?M" by auto
     from rminusinf_\<Upsilon>[OF lp nmi pa] rplusinf_\<Upsilon>[OF lp npi pa] 
-    have "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). a \<le> ?N x l / real n \<and> a \<ge> ?N x s / real m" by blast
+    have "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). a \<le> ?N x l / real_of_int n \<and> a \<ge> ?N x s / real_of_int m" by blast
     then obtain "t" "n" "s" "m" where 
       tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U" 
-      and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast
-    from \<Upsilon>_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real m" and tx: "a \<ge> ?N a t / real n" by auto
+      and xs1: "a \<le> ?N x s / real_of_int m" and tx1: "a \<ge> ?N x t / real_of_int n" by blast
+    from \<Upsilon>_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real_of_int m" and tx: "a \<ge> ?N a t / real_of_int n" by auto
     from tnU have Mne: "?M \<noteq> {}" by auto
     hence Une: "?U \<noteq> {}" by simp
     let ?l = "Min ?M"
     let ?u = "Max ?M"
     have linM: "?l \<in> ?M" using fM Mne by simp
     have uinM: "?u \<in> ?M" using fM Mne by simp
-    have tnM: "?N a t / real n \<in> ?M" using tnU by auto
-    have smM: "?N a s / real m \<in> ?M" using smU by auto 
+    have tnM: "?N a t / real_of_int n \<in> ?M" using tnU by auto
+    have smM: "?N a s / real_of_int m \<in> ?M" using smU by auto 
     have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
     have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
-    have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
-    have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
+    have "?l \<le> ?N a t / real_of_int n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
+    have "?N a s / real_of_int m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
     from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
     have "(\<exists> s\<in> ?M. ?I s p) \<or> 
       (\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p)" .
     moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p"
-      hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto
-      then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast
+      hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real_of_int nu" by auto
+      then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real_of_int nu" by blast
       have "(u + u) / 2 = u" by auto with pu tuu 
-      have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp
+      have "?I (((?N a tu / real_of_int nu) + (?N a tu / real_of_int nu)) / 2) p" by simp
       with tuU have ?thesis by blast}
     moreover{
       assume "\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p"
       then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" 
         and noM: "\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p"
         by blast
-      from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto
-      then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast
-      from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto
-      then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast
+      from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real_of_int t1n" by auto
+      then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real_of_int t1n" by blast
+      from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real_of_int t2n" by auto
+      then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real_of_int t2n" by blast
       from t1x xt2 have t1t2: "t1 < t2" by simp
       let ?u = "(t1 + t2) / 2"
       from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
@@ -4702,11 +4679,11 @@
     ultimately show ?thesis by blast
   qed
   then obtain "l" "n" "s"  "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U" 
-    and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast
+    and pu: "?I ((?N a l / real_of_int n + ?N a s / real_of_int m) / 2) p" by blast
   from lnU smU \<Upsilon>_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto
   from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] 
     numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
-  have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp
+  have "?I ((?N x l / real_of_int n + ?N x s / real_of_int m) / 2) p" by simp
   with lnU smU
   show ?thesis by auto
 qed
@@ -4714,7 +4691,7 @@
 
 theorem fr_eq: 
   assumes lp: "isrlfm p"
-  shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). Ifm ((((Inum (x#bs) t)/  real n + (Inum (x#bs) s) / real m) /2)#bs) p))"
+  shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). Ifm ((((Inum (x#bs) t)/  real_of_int n + (Inum (x#bs) s) / real_of_int m) /2)#bs) p))"
   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
 proof
   assume px: "\<exists> x. ?I x p"
@@ -4741,18 +4718,18 @@
   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
   moreover {assume "?M \<or> ?P" hence "?D" by blast}
   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
-    let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n"
+    let ?f ="\<lambda> (t,n). Inum (x#bs) t / real_of_int n"
     let ?N = "\<lambda> t. Inum (x#bs) t"
     {fix t n s m assume "(t,n)\<in> set (\<Upsilon> p)" and "(s,m) \<in> set (\<Upsilon> p)"
-      with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
+      with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real_of_int n > 0" and snb: "numbound0 s" and mp:"real_of_int m > 0"
         by auto
       let ?st = "Add (Mul m t) (Mul n s)"
-      from np mp have mnp: "real (2*n*m) > 0" by (simp add: mult.commute)
+      from np mp have mnp: "real_of_int (2*n*m) > 0" by (simp add: mult.commute)
       from tnb snb have st_nb: "numbound0 ?st" by simp
-      have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
+      have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)"
         using mnp mp np by (simp add: algebra_simps add_divide_distrib)
       from \<upsilon>_I[OF lp mnp st_nb, where x="x" and bs="bs"] 
-      have "?I x (\<upsilon> p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])}
+      have "?I x (\<upsilon> p (?st,2*n*m)) = ?I ((?N t / real_of_int n + ?N s / real_of_int m) /2) p" by (simp only: st[symmetric])}
     with rinf_\<Upsilon>[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
   ultimately show "?D" by blast
 next
@@ -4761,9 +4738,9 @@
   moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
   moreover {fix t k s l assume "(t,k) \<in> set (\<Upsilon> p)" and "(s,l) \<in> set (\<Upsilon> p)" 
     and px:"?I x (\<upsilon> p (Add (Mul l t) (Mul k s), 2*k*l))"
-    with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto
+    with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real_of_int k > 0" and snb: "numbound0 s" and mp:"real_of_int l > 0" by auto
     let ?st = "Add (Mul l t) (Mul k s)"
-    from np mp have mnp: "real (2*k*l) > 0" by (simp add: mult.commute)
+    from np mp have mnp: "real_of_int (2*k*l) > 0" by (simp add: mult.commute)
     from tnb snb have st_nb: "numbound0 ?st" by simp
     from \<upsilon>_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
   ultimately show "?E" by blast
@@ -4772,17 +4749,17 @@
 text\<open>The overall Part\<close>
 
 lemma real_ex_int_real01:
-  shows "(\<exists> (x::real). P x) = (\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real i + u))"
+  shows "(\<exists> (x::real). P x) = (\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real_of_int i + u))"
 proof(auto)
   fix x
   assume Px: "P x"
   let ?i = "floor x"
-  let ?u = "x - real ?i"
-  have "x = real ?i + ?u" by simp
-  hence "P (real ?i + ?u)" using Px by simp
-  moreover have "real ?i \<le> x" using real_of_int_floor_le by simp hence "0 \<le> ?u" by arith
+  let ?u = "x - real_of_int ?i"
+  have "x = real_of_int ?i + ?u" by simp
+  hence "P (real_of_int ?i + ?u)" using Px by simp
+  moreover have "real_of_int ?i \<le> x" using of_int_floor_le by simp hence "0 \<le> ?u" by arith
   moreover have "?u < 1" using real_of_int_floor_add_one_gt[where r="x"] by arith 
-  ultimately show "(\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real i + u))" by blast
+  ultimately show "(\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real_of_int i + u))" by blast
 qed
 
 fun exsplitnum :: "num \<Rightarrow> num" where
@@ -4826,11 +4803,11 @@
 
 lemma splitex:
   assumes qf: "qfree p"
-  shows "(Ifm bs (E p)) = (\<exists> (i::int). Ifm (real i#bs) (E (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (exsplit p))))" (is "?lhs = ?rhs")
+  shows "(Ifm bs (E p)) = (\<exists> (i::int). Ifm (real_of_int i#bs) (E (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (exsplit p))))" (is "?lhs = ?rhs")
 proof-
-  have "?rhs = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm (x#(real i)#bs) (exsplit p))"
+  have "?rhs = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm (x#(real_of_int i)#bs) (exsplit p))"
     by (simp add: myless[of _ "1"] myless[of _ "0"] ac_simps)
-  also have "\<dots> = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm ((real i + x) #bs) p)"
+  also have "\<dots> = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm ((real_of_int i + x) #bs) p)"
     by (simp only: exsplit[OF qf] ac_simps)
   also have "\<dots> = (\<exists> x. Ifm (x#bs) p)" 
     by (simp only: real_ex_int_real01[where P="\<lambda> x. Ifm (x#bs) p"])
@@ -4880,10 +4857,10 @@
     then obtain t n s m where aU:"(t,n) \<in> ?U" and bU:"(s,m)\<in> ?U" and rqx: "?I x (\<upsilon> ?rq (Add (Mul m t) (Mul n s), 2*n*m))" by blast
     from qf have lrq:"isrlfm ?rq"using rlfm_l[OF qf] 
       by (auto simp add: rsplit_def lt_def ge_def)
-    from aU bU \<Upsilon>_l[OF lrq] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0" by (auto simp add: split_def)
+    from aU bU \<Upsilon>_l[OF lrq] have tnb: "numbound0 t" and np:"real_of_int n > 0" and snb: "numbound0 s" and mp:"real_of_int m > 0" by (auto simp add: split_def)
     let ?st = "Add (Mul m t) (Mul n s)"
     from tnb snb have stnb: "numbound0 ?st" by simp
-    from np mp have mnp: "real (2*n*m) > 0" by (simp add: mult.commute)
+    from np mp have mnp: "real_of_int (2*n*m) > 0" by (simp add: mult.commute)
     from conjunct1[OF \<upsilon>_I[OF lrq mnp stnb, where bs="bs" and x="x"], symmetric] rqx
     have "\<exists> x. ?I x ?rq" by auto
     thus "?E" 
@@ -4894,7 +4871,7 @@
 
 lemma \<Upsilon>_cong_aux:
   assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0"
-  shows "((\<lambda> (t,n). Inum (x#bs) t /real n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \<times> set U))"
+  shows "((\<lambda> (t,n). Inum (x#bs) t /real_of_int n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real_of_int n + Inum (x#bs) s /real_of_int m)/2) ` (set U \<times> set U))"
   (is "?lhs = ?rhs")
 proof(auto)
   fix t n s m
@@ -4905,13 +4882,13 @@
   let ?st= "Add (Mul m t) (Mul n s)"
   from Ul th have mnz: "m \<noteq> 0" by auto
   from Ul th have  nnz: "n \<noteq> 0" by auto  
-  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
+  have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)"
    using mnz nnz by (simp add: algebra_simps add_divide_distrib)
  
-  thus "(real m *  Inum (x # bs) t + real n * Inum (x # bs) s) /
-       (2 * real n * real m)
+  thus "(real_of_int m *  Inum (x # bs) t + real_of_int n * Inum (x # bs) s) /
+       (2 * real_of_int n * real_of_int m)
        \<in> (\<lambda>((t, n), s, m).
-             (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
+             (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) `
          (set U \<times> set U)"using mnz nnz th  
     apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
     by (rule_tac x="(s,m)" in bexI,simp_all) 
@@ -4923,9 +4900,9 @@
   let ?st= "Add (Mul m t) (Mul n s)"
   from Ul smU have mnz: "m \<noteq> 0" by auto
   from Ul tnU have  nnz: "n \<noteq> 0" by auto  
-  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
+  have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)"
    using mnz nnz by (simp add: algebra_simps add_divide_distrib)
- let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2"
+ let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m)/2 = (Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int m')/2"
  have Pc:"\<forall> a b. ?P a b = ?P b a"
    by auto
  from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast
@@ -4936,13 +4913,13 @@
    and Pts': "?P (t',n') (s',m')" by blast
  from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
  let ?st' = "Add (Mul m' t') (Mul n' s')"
-   have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
+   have st': "(?N t' / real_of_int n' + ?N s' / real_of_int m')/2 = ?N ?st' / real_of_int (2*n'*m')"
    using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
  from Pts' have 
-   "(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp
- also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st')
- finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2
-          \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) `
+   "(Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m)/2 = (Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int m')/2" by simp
+ also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st')
+ finally show "(Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2
+          \<in> (\<lambda>(t, n). Inum (x # bs) t / real_of_int n) `
             (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) `
             set (alluopairs U)"
    using ts'_U by blast
@@ -4950,7 +4927,7 @@
 
 lemma \<Upsilon>_cong:
   assumes lp: "isrlfm p"
-  and UU': "((\<lambda> (t,n). Inum (x#bs) t /real n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (U \<times> U))" (is "?f ` U' = ?g ` (U\<times>U)")
+  and UU': "((\<lambda> (t,n). Inum (x#bs) t /real_of_int n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real_of_int n + Inum (x#bs) s /real_of_int m)/2) ` (U \<times> U))" (is "?f ` U' = ?g ` (U\<times>U)")
   and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0"
   and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0"
   shows "(\<exists> (t,n) \<in> U. \<exists> (s,m) \<in> U. Ifm (x#bs) (\<upsilon> p (Add (Mul m t) (Mul n s),2*n*m))) = (\<exists> (t,n) \<in> U'. Ifm (x#bs) (\<upsilon> p (t,n)))"
@@ -4963,18 +4940,18 @@
   from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
     and snb: "numbound0 s" and mp:"m > 0"  by auto
   let ?st= "Add (Mul m t) (Mul n s)"
-  from np mp have mnp: "real (2*n*m) > 0" 
-      by (simp add: mult.commute real_of_int_mult[symmetric] del: real_of_int_mult)
+  from np mp have mnp: "real_of_int (2*n*m) > 0" 
+      by (simp add: mult.commute of_int_mult[symmetric] del: of_int_mult)
     from tnb snb have stnb: "numbound0 ?st" by simp
-  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
+  have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)"
    using mp np by (simp add: algebra_simps add_divide_distrib)
   from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
   hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
     by auto (rule_tac x="(a,b)" in bexI, auto)
   then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast
-  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
+  from U' tnU' have tnb': "numbound0 t'" and np': "real_of_int n' > 0" by auto
   from \<upsilon>_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst 
-  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
+  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real_of_int (2 * n * m) # bs) p" by simp
   from conjunct1[OF \<upsilon>_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]]
   have "Ifm (x # bs) (\<upsilon> p (t', n')) " by (simp only: st) 
   then show ?rhs using tnU' by auto 
@@ -4991,14 +4968,14 @@
   from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
     and snb: "numbound0 s" and mp:"m > 0"  by auto
   let ?st= "Add (Mul m t) (Mul n s)"
-  from np mp have mnp: "real (2*n*m) > 0" 
-      by (simp add: mult.commute real_of_int_mult[symmetric] del: real_of_int_mult)
+  from np mp have mnp: "real_of_int (2*n*m) > 0" 
+      by (simp add: mult.commute of_int_mult[symmetric] del: of_int_mult)
     from tnb snb have stnb: "numbound0 ?st" by simp
-  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
+  have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)"
    using mp np by (simp add: algebra_simps add_divide_distrib)
-  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
+  from U' tnU' have tnb': "numbound0 t'" and np': "real_of_int n' > 0" by auto
   from \<upsilon>_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
-  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
+  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real_of_int (2 * n * m) # bs) p" by simp
   with \<upsilon>_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
 qed
   
@@ -5016,8 +4993,8 @@
   let ?S = "map ?g ?Up"
   let ?SS = "map simp_num_pair ?S"
   let ?Y = "remdups ?SS"
-  let ?f= "(\<lambda> (t,n). ?N t / real n)"
-  let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
+  let ?f= "(\<lambda> (t,n). ?N t / real_of_int n)"
+  let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real_of_int n + ?N s/ real_of_int m) /2"
   let ?F = "\<lambda> p. \<exists> a \<in> set (\<Upsilon> p). \<exists> b \<in> set (\<Upsilon> p). ?I x (\<upsilon> p (?g(a,b)))"
   let ?ep = "evaldjf (\<upsilon> ?q) ?Y"
   from rlfm_l[OF qf] have lq: "isrlfm ?q" 
@@ -5058,7 +5035,7 @@
   have "\<forall> (t,n) \<in> set ?Y. bound0 (\<upsilon> ?q (t,n))"
   proof-
     { fix t n assume tnY: "(t,n) \<in> set ?Y"
-      with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto
+      with Y_l have tnb: "numbound0 t" and np: "real_of_int n > 0" by auto
       from \<upsilon>_I[OF lq np tnb]
     have "bound0 (\<upsilon> ?q (t,n))"  by simp}
     thus ?thesis by blast
@@ -5080,9 +5057,9 @@
 qed
 
 lemma cp_thm': 
-  assumes lp: "iszlfm p (real (i::int)#bs)"
+  assumes lp: "iszlfm p (real_of_int (i::int)#bs)"
   and up: "d_\<beta> p 1" and dd: "d_\<delta> p d" and dp: "d > 0"
-  shows "(\<exists> (x::int). Ifm (real x#bs) p) = ((\<exists> j\<in> {1 .. d}. Ifm (real j#bs) (minusinf p)) \<or> (\<exists> j\<in> {1.. d}. \<exists> b\<in> (Inum (real i#bs)) ` set (\<beta> p). Ifm ((b+real j)#bs) p))"
+  shows "(\<exists> (x::int). Ifm (real_of_int x#bs) p) = ((\<exists> j\<in> {1 .. d}. Ifm (real_of_int j#bs) (minusinf p)) \<or> (\<exists> j\<in> {1.. d}. \<exists> b\<in> (Inum (real_of_int i#bs)) ` set (\<beta> p). Ifm ((b+real_of_int j)#bs) p))"
   using cp_thm[OF lp up dd dp] by auto
 
 definition unit :: "fm \<Rightarrow> fm \<times> num list \<times> int" where
@@ -5091,14 +5068,18 @@
              in if length B \<le> length a then (q,B,d) else (mirror q, a,d))"
 
 lemma unit: assumes qf: "qfree p"
-  shows "\<And> q B d. unit p = (q,B,d) \<Longrightarrow> ((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> (Inum (real i#bs)) ` set B = (Inum (real i#bs)) ` set (\<beta> q) \<and> d_\<beta> q 1 \<and> d_\<delta> q d \<and> d >0 \<and> iszlfm q (real (i::int)#bs) \<and> (\<forall> b\<in> set B. numbound0 b)"
+  shows "\<And> q B d. unit p = (q,B,d) \<Longrightarrow>
+      ((\<exists> (x::int). Ifm (real_of_int x#bs) p) = 
+       (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> 
+       (Inum (real_of_int i#bs)) ` set B = (Inum (real_of_int i#bs)) ` set (\<beta> q) \<and> 
+       d_\<beta> q 1 \<and> d_\<delta> q d \<and> d >0 \<and> iszlfm q (real_of_int (i::int)#bs) \<and> (\<forall> b\<in> set B. numbound0 b)"
 proof-
   fix q B d 
   assume qBd: "unit p = (q,B,d)"
-  let ?thes = "((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and>
-    Inum (real i#bs) ` set B = Inum (real i#bs) ` set (\<beta> q) \<and>
-    d_\<beta> q 1 \<and> d_\<delta> q d \<and> 0 < d \<and> iszlfm q (real i # bs) \<and> (\<forall> b\<in> set B. numbound0 b)"
-  let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p"
+  let ?thes = "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and>
+    Inum (real_of_int i#bs) ` set B = Inum (real_of_int i#bs) ` set (\<beta> q) \<and>
+    d_\<beta> q 1 \<and> d_\<delta> q d \<and> 0 < d \<and> iszlfm q (real_of_int i # bs) \<and> (\<forall> b\<in> set B. numbound0 b)"
+  let ?I = "\<lambda> (x::int) p. Ifm (real_of_int x#bs) p"
   let ?p' = "zlfm p"
   let ?l = "\<zeta> ?p'"
   let ?q = "And (Dvd ?l (CN 0 1 (C 0))) (a_\<beta> ?p' ?l)"
@@ -5110,20 +5091,20 @@
   from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] 
   have pp': "\<forall> i. ?I i ?p' = ?I i p" by auto
   from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]]]
-  have lp': "\<forall> (i::int). iszlfm ?p' (real i#bs)" by simp 
-  hence lp'': "iszlfm ?p' (real (i::int)#bs)" by simp
+  have lp': "\<forall> (i::int). iszlfm ?p' (real_of_int i#bs)" by simp 
+  hence lp'': "iszlfm ?p' (real_of_int (i::int)#bs)" by simp
   from lp' \<zeta>[where p="?p'" and bs="bs"] have lp: "?l >0" and dl: "d_\<beta> ?p' ?l" by auto
   from a_\<beta>_ex[where p="?p'" and l="?l" and bs="bs", OF lp'' dl lp] pp'
   have pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by (simp add: int_rdvd_iff) 
-  from lp'' lp a_\<beta>[OF lp'' dl lp] have lq:"iszlfm ?q (real i#bs)" and uq: "d_\<beta> ?q 1" 
+  from lp'' lp a_\<beta>[OF lp'' dl lp] have lq:"iszlfm ?q (real_of_int i#bs)" and uq: "d_\<beta> ?q 1" 
     by (auto simp add: isint_def)
   from \<delta>[OF lq] have dp:"?d >0" and dd: "d_\<delta> ?q ?d" by blast+
-  let ?N = "\<lambda> t. Inum (real (i::int)#bs) t"
+  let ?N = "\<lambda> t. Inum (real_of_int (i::int)#bs) t"
   have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by (simp add:image_comp) 
-  also have "\<dots> = ?N ` ?B" using simpnum_ci[where bs="real i #bs"] by auto
+  also have "\<dots> = ?N ` ?B" using simpnum_ci[where bs="real_of_int i #bs"] by auto
   finally have BB': "?N ` set ?B' = ?N ` ?B" .
   have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by (simp add:image_comp) 
-  also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real i #bs"] by auto
+  also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real_of_int i #bs"] by auto
   finally have AA': "?N ` set ?A' = ?N ` ?A" .
   from \<beta>_numbound0[OF lq] have B_nb:"\<forall> b\<in> set ?B'. numbound0 b"
     by simp
@@ -5143,8 +5124,8 @@
       and bn: "\<forall>b\<in> set B. numbound0 b" by simp+
     from mirror_ex[OF lq] pq_ex q 
     have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp
-    from lq uq q mirror_d_\<beta> [where p="?q" and bs="bs" and a="real i"]
-    have lq': "iszlfm q (real i#bs)" and uq: "d_\<beta> q 1" by auto
+    from lq uq q mirror_d_\<beta> [where p="?q" and bs="bs" and a="real_of_int i"]
+    have lq': "iszlfm q (real_of_int i#bs)" and uq: "d_\<beta> q 1" by auto
     from \<delta>[OF lq'] mirror_\<delta>[OF lq] q d have dq:"d_\<delta> q d " by auto
     from pqm_eq b bn uq lq' dp dq q dp d have ?thes by simp
   }
@@ -5163,11 +5144,11 @@
                                             [(b,j). b\<leftarrow>B,j\<leftarrow>js]))
      in decr (disj md qd)))"
 lemma cooper: assumes qf: "qfree p"
-  shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (cooper p))) \<and> qfree (cooper p)" 
+  shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (cooper p))) \<and> qfree (cooper p)" 
   (is "(?lhs = ?rhs) \<and> _")
 proof-
 
-  let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p"
+  let ?I = "\<lambda> (x::int) p. Ifm (real_of_int x#bs) p"
   let ?q = "fst (unit p)"
   let ?B = "fst (snd(unit p))"
   let ?d = "snd (snd (unit p))"
@@ -5176,7 +5157,7 @@
   let ?smq = "simpfm ?mq"
   let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js"
   fix i
-  let ?N = "\<lambda> t. Inum (real (i::int)#bs) t"
+  let ?N = "\<lambda> t. Inum (real_of_int (i::int)#bs) t"
   let ?bjs = "[(b,j). b\<leftarrow>?B,j\<leftarrow>?js]"
   let ?sbjs = "map (\<lambda> (b,j). simpnum (Add b (C j))) ?bjs"
   let ?qd = "evaldjf (\<lambda> t. simpfm (subst0 t ?q)) (remdups ?sbjs)"
@@ -5184,7 +5165,7 @@
   from unit[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and 
     B:"?N ` set ?B = ?N ` set (\<beta> ?q)" and 
     uq:"d_\<beta> ?q 1" and dd: "d_\<delta> ?q ?d" and dp: "?d > 0" and 
-    lq: "iszlfm ?q (real i#bs)" and 
+    lq: "iszlfm ?q (real_of_int i#bs)" and 
     Bn: "\<forall> b\<in> set ?B. numbound0 b" by auto
   from zlin_qfree[OF lq] have qfq: "qfree ?q" .
   from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq".
@@ -5207,8 +5188,8 @@
   from mdb qdb 
   have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all)
   from trans [OF pq_ex cp_thm'[OF lq uq dd dp]] B
-  have "?lhs = (\<exists> j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists> b\<in> ?N ` set ?B. Ifm ((b+ real j)#bs) ?q))" by auto
-  also have "\<dots> = ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> (b,j) \<in> (?N ` set ?B \<times> set ?js). Ifm ((b+ real j)#bs) ?q))" by auto
+  have "?lhs = (\<exists> j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists> b\<in> ?N ` set ?B. Ifm ((b+ real_of_int j)#bs) ?q))" by auto
+  also have "\<dots> = ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> (b,j) \<in> (?N ` set ?B \<times> set ?js). Ifm ((b+ real_of_int j)#bs) ?q))" by auto
   also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> (\<lambda> (b,j). ?N (Add b (C j))) ` set ?bjs. Ifm (t #bs) ?q))" by simp
   also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> (\<lambda> (b,j). ?N (simpnum (Add b (C j)))) ` set ?bjs. Ifm (t #bs) ?q))" by (simp only: simpnum_ci)
   also  have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> set ?sbjs. Ifm (?N t #bs) ?q))" 
@@ -5233,15 +5214,15 @@
 
 lemma DJcooper: 
   assumes qf: "qfree p"
-  shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (DJ cooper p))) \<and> qfree (DJ cooper p)"
+  shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (DJ cooper p))) \<and> qfree (DJ cooper p)"
 proof-
   from cooper have cqf: "\<forall> p. qfree p \<longrightarrow> qfree (cooper p)" by  blast
   from DJ_qf[OF cqf] qf have thqf:"qfree (DJ cooper p)" by blast
   have "Ifm bs (DJ cooper p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (cooper q))" 
      by (simp add: DJ_def evaldjf_ex)
-  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real x#bs)  q)" 
+  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real_of_int x#bs)  q)" 
     using cooper disjuncts_qf[OF qf] by blast
-  also have "\<dots> = (\<exists> (x::int). Ifm (real x#bs) p)" by (induct p rule: disjuncts.induct, auto)
+  also have "\<dots> = (\<exists> (x::int). Ifm (real_of_int x#bs) p)" by (induct p rule: disjuncts.induct, auto)
   finally show ?thesis using thqf by blast
 qed
 
@@ -5292,9 +5273,9 @@
 qed
 
 lemma rl_thm': 
-  assumes lp: "iszlfm p (real (i::int)#bs)" 
+  assumes lp: "iszlfm p (real_of_int (i::int)#bs)" 
   and R: "(\<lambda>(b,k). (Inum (a#bs) b,k)) ` R =  (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)"
-  shows "(\<exists> (x::int). Ifm (real x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> R. \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))"
+  shows "(\<exists> (x::int). Ifm (real_of_int x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real_of_int j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> R. \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))"
   using rl_thm[OF lp] \<rho>_cong[OF iszlfm_gen[OF lp, rule_format, where y="a"] R] by simp 
 
 definition chooset :: "fm \<Rightarrow> fm \<times> ((num\<times>int) list) \<times> int" where
@@ -5304,12 +5285,18 @@
              in if length B \<le> length a then (q,B,d) else (mirror q, a,d))"
 
 lemma chooset: assumes qf: "qfree p"
-  shows "\<And> q B d. chooset p = (q,B,d) \<Longrightarrow> ((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set (\<rho> q)) \<and> (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)"
+  shows "\<And> q B d. chooset p = (q,B,d) \<Longrightarrow> 
+     ((\<exists> (x::int). Ifm (real_of_int x#bs) p) = 
+      (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> 
+      ((\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set (\<rho> q)) \<and>
+      (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real_of_int (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)"
 proof-
   fix q B d 
   assume qBd: "chooset p = (q,B,d)"
-  let ?thes = "((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set (\<rho> q)) \<and> (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)" 
-  let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p"
+  let ?thes = "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = 
+             (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set (\<rho> q)) \<and> 
+             (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real_of_int (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)" 
+  let ?I = "\<lambda> (x::int) p. Ifm (real_of_int x#bs) p"
   let ?q = "zlfm p"
   let ?d = "\<delta> ?q"
   let ?B = "set (\<rho> ?q)"
@@ -5320,17 +5307,17 @@
   from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] 
   have pp': "\<forall> i. ?I i ?q = ?I i p" by auto
   hence pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by simp 
-  from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]], rule_format, where y="real i"]
-  have lq: "iszlfm ?q (real (i::int)#bs)" . 
+  from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]], rule_format, where y="real_of_int i"]
+  have lq: "iszlfm ?q (real_of_int (i::int)#bs)" . 
   from \<delta>[OF lq] have dp:"?d >0" by blast
-  let ?N = "\<lambda> (t,c). (Inum (real (i::int)#bs) t,c)"
+  let ?N = "\<lambda> (t,c). (Inum (real_of_int (i::int)#bs) t,c)"
   have "?N ` set ?B' = ((?N o ?f) ` ?B)" by (simp add: split_def image_comp)
   also have "\<dots> = ?N ` ?B"
-    by(simp add: split_def image_comp simpnum_ci[where bs="real i #bs"] image_def)
+    by(simp add: split_def image_comp simpnum_ci[where bs="real_of_int i #bs"] image_def)
   finally have BB': "?N ` set ?B' = ?N ` ?B" .
   have "?N ` set ?A' = ((?N o ?f) ` ?A)" by (simp add: split_def image_comp) 
-  also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real i #bs"]
-    by(simp add: split_def image_comp simpnum_ci[where bs="real i #bs"] image_def) 
+  also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real_of_int i #bs"]
+    by(simp add: split_def image_comp simpnum_ci[where bs="real_of_int i #bs"] image_def) 
   finally have AA': "?N ` set ?A' = ?N ` ?A" .
   from \<rho>_l[OF lq] have B_nb:"\<forall> (e,c)\<in> set ?B'. numbound0 e \<and> c > 0"
     by (simp add: split_def)
@@ -5350,8 +5337,8 @@
       and bn: "\<forall>(e,c)\<in> set B. numbound0 e \<and> c > 0" by auto 
     from mirror_ex[OF lq] pq_ex q 
     have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp
-    from lq q mirror_l [where p="?q" and bs="bs" and a="real i"]
-    have lq': "iszlfm q (real i#bs)" by auto
+    from lq q mirror_l [where p="?q" and bs="bs" and a="real_of_int i"]
+    have lq': "iszlfm q (real_of_int i#bs)" by auto
     from mirror_\<delta>[OF lq] pqm_eq b bn lq' dp q dp d have ?thes by simp
   }
   ultimately show ?thes by blast
@@ -5387,11 +5374,11 @@
      in decr (disj md qd)))"
 
 lemma redlove: assumes qf: "qfree p"
-  shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (redlove p))) \<and> qfree (redlove p)" 
+  shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (redlove p))) \<and> qfree (redlove p)" 
   (is "(?lhs = ?rhs) \<and> _")
 proof-
 
-  let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p"
+  let ?I = "\<lambda> (x::int) p. Ifm (real_of_int x#bs) p"
   let ?q = "fst (chooset p)"
   let ?B = "fst (snd(chooset p))"
   let ?d = "snd (snd (chooset p))"
@@ -5400,12 +5387,12 @@
   let ?smq = "simpfm ?mq"
   let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js"
   fix i
-  let ?N = "\<lambda> (t,k). (Inum (real (i::int)#bs) t,k)"
+  let ?N = "\<lambda> (t,k). (Inum (real_of_int (i::int)#bs) t,k)"
   let ?qd = "evaldjf (stage ?q ?d) ?B"
   have qbf:"chooset p = (?q,?B,?d)" by simp
   from chooset[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and 
     B:"?N ` set ?B = ?N ` set (\<rho> ?q)" and dd: "\<delta> ?q = ?d" and dp: "?d > 0" and 
-    lq: "iszlfm ?q (real i#bs)" and 
+    lq: "iszlfm ?q (real_of_int i#bs)" and 
     Bn: "\<forall> (e,c)\<in> set ?B. numbound0 e \<and> c > 0" by auto
   from zlin_qfree[OF lq] have qfq: "qfree ?q" .
   from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq".
@@ -5420,7 +5407,7 @@
   from mdb qdb 
   have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all)
   from trans [OF pq_ex rl_thm'[OF lq B]] dd
-  have "?lhs = ((\<exists> j\<in> {1.. ?d}. ?I j ?mq) \<or> (\<exists> (e,c)\<in> set ?B. \<exists> j\<in> {1 .. c*?d}. Ifm (real i#bs) (\<sigma> ?q c (Add e (C j)))))" by auto
+  have "?lhs = ((\<exists> j\<in> {1.. ?d}. ?I j ?mq) \<or> (\<exists> (e,c)\<in> set ?B. \<exists> j\<in> {1 .. c*?d}. Ifm (real_of_int i#bs) (\<sigma> ?q c (Add e (C j)))))" by auto
   also have "\<dots> = ((\<exists> j\<in> {1.. ?d}. ?I j ?smq) \<or> (\<exists> (e,c)\<in> set ?B. ?I i (stage ?q ?d (e,c) )))" 
     by (simp add: stage split_def)
   also have "\<dots> = ((\<exists> j\<in> {1 .. ?d}. ?I i (subst0 (C j) ?smq))  \<or> ?I i ?qd)"
@@ -5443,15 +5430,15 @@
 
 lemma DJredlove: 
   assumes qf: "qfree p"
-  shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (DJ redlove p))) \<and> qfree (DJ redlove p)"
+  shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (DJ redlove p))) \<and> qfree (DJ redlove p)"
 proof-
   from redlove have cqf: "\<forall> p. qfree p \<longrightarrow> qfree (redlove p)" by  blast
   from DJ_qf[OF cqf] qf have thqf:"qfree (DJ redlove p)" by blast
   have "Ifm bs (DJ redlove p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (redlove q))" 
      by (simp add: DJ_def evaldjf_ex)
-  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real x#bs)  q)" 
+  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real_of_int x#bs)  q)" 
     using redlove disjuncts_qf[OF qf] by blast
-  also have "\<dots> = (\<exists> (x::int). Ifm (real x#bs) p)" by (induct p rule: disjuncts.induct, auto)
+  also have "\<dots> = (\<exists> (x::int). Ifm (real_of_int x#bs) p)" by (induct p rule: disjuncts.induct, auto)
   finally show ?thesis using thqf by blast
 qed
 
@@ -5473,9 +5460,9 @@
   show "qfree (mircfr p)\<and>(Ifm bs (mircfr p) = Ifm bs (E p))" (is "_ \<and> (?lhs = ?rhs)")
   proof-
     let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))"
-    have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real i#bs) ?es)" 
+    have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real_of_int i#bs) ?es)" 
       using splitex[OF qf] by simp
-    with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+
+    with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real_of_int i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+
     with DJcooper[OF qf'] show ?thesis by (simp add: mircfr_def)
   qed
 qed
@@ -5487,9 +5474,9 @@
   show "qfree (mirlfr p)\<and>(Ifm bs (mirlfr p) = Ifm bs (E p))" (is "_ \<and> (?lhs = ?rhs)")
   proof-
     let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))"
-    have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real i#bs) ?es)" 
+    have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real_of_int i#bs) ?es)" 
       using splitex[OF qf] by simp
-    with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+
+    with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real_of_int i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+
     with DJredlove[OF qf'] show ?thesis by (simp add: mirlfr_def)
   qed
 qed
@@ -5542,8 +5529,8 @@
 fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t
      of NONE => error "Variable not found in the list!"
       | SOME n => mk_Bound n)
-  | num_of_term vs @{term "real (0::int)"} = mk_C 0
-  | num_of_term vs @{term "real (1::int)"} = mk_C 1
+  | num_of_term vs @{term "of_int (0::int)"} = mk_C 0
+  | num_of_term vs @{term "of_int (1::int)"} = mk_C 1
   | num_of_term vs @{term "0::real"} = mk_C 0
   | num_of_term vs @{term "1::real"} = mk_C 1
   | num_of_term vs @{term "- 1::real"} = mk_C (~ 1)
@@ -5557,13 +5544,13 @@
       (case (num_of_term vs t1)
        of @{code C} i => @{code Mul} (i, num_of_term vs t2)
         | _ => error "num_of_term: unsupported Multiplication")
-  | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "numeral :: _ \<Rightarrow> int"} $ t')) =
+  | num_of_term vs (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "numeral :: _ \<Rightarrow> int"} $ t')) =
       mk_C (HOLogic.dest_num t')
-  | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "- numeral :: _ \<Rightarrow> int"} $ t')) =
+  | num_of_term vs (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "- numeral :: _ \<Rightarrow> int"} $ t')) =
       mk_C (~ (HOLogic.dest_num t'))
-  | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "floor :: real \<Rightarrow> int"} $ t')) =
+  | num_of_term vs (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "floor :: real \<Rightarrow> int"} $ t')) =
       @{code Floor} (num_of_term vs t')
-  | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "ceiling :: real \<Rightarrow> int"} $ t')) =
+  | num_of_term vs (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "ceiling :: real \<Rightarrow> int"} $ t')) =
       @{code Neg} (@{code Floor} (@{code Neg} (num_of_term vs t')))
   | num_of_term vs (@{term "numeral :: _ \<Rightarrow> real"} $ t') =
       mk_C (HOLogic.dest_num t')
@@ -5579,9 +5566,9 @@
       @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
   | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
       @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) 
-  | fm_of_term vs (@{term "op rdvd"} $ (@{term "real :: int \<Rightarrow> real"} $ (@{term "numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
+  | fm_of_term vs (@{term "op rdvd"} $ (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
       mk_Dvd (HOLogic.dest_num t1, num_of_term vs t2)
-  | fm_of_term vs (@{term "op rdvd"} $ (@{term "real :: int \<Rightarrow> real"} $ (@{term "- numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
+  | fm_of_term vs (@{term "op rdvd"} $ (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "- numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
       mk_Dvd (~ (HOLogic.dest_num t1), num_of_term vs t2)
   | fm_of_term vs (@{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) =
       @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
@@ -5599,14 +5586,14 @@
       @{code A} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
   | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);
 
-fun term_of_num vs (@{code C} i) = @{term "real :: int \<Rightarrow> real"} $
+fun term_of_num vs (@{code C} i) = @{term "of_int :: int \<Rightarrow> real"} $
       HOLogic.mk_number HOLogic.intT (@{code integer_of_int} i)
   | term_of_num vs (@{code Bound} n) =
       let
         val m = @{code integer_of_nat} n;
       in fst (the (find_first (fn (_, q) => m = q) vs)) end
   | term_of_num vs (@{code Neg} (@{code Floor} (@{code Neg} t'))) =
-      @{term "real :: int \<Rightarrow> real"} $ (@{term "ceiling :: real \<Rightarrow> int"} $ term_of_num vs t')
+      @{term "of_int :: int \<Rightarrow> real"} $ (@{term "ceiling :: real \<Rightarrow> int"} $ term_of_num vs t')
   | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} $ term_of_num vs t'
   | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $
       term_of_num vs t1 $ term_of_num vs t2
@@ -5614,7 +5601,7 @@
       term_of_num vs t1 $ term_of_num vs t2
   | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $
       term_of_num vs (@{code C} i) $ term_of_num vs t2
-  | term_of_num vs (@{code Floor} t) = @{term "real :: int \<Rightarrow> real"} $ (@{term "floor :: real \<Rightarrow> int"} $ term_of_num vs t)
+  | term_of_num vs (@{code Floor} t) = @{term "of_int :: int \<Rightarrow> real"} $ (@{term "floor :: real \<Rightarrow> int"} $ term_of_num vs t)
   | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t))
   | term_of_num vs (@{code CF} (c, t, s)) = term_of_num vs (@{code Add} (@{code Mul} (c, @{code Floor} t), s));
 
@@ -5665,12 +5652,11 @@
   Scan.lift (Args.mode "no_quantify") >>
     (fn q => fn ctxt => SIMPLE_METHOD' (Mir_Tac.mir_tac ctxt (not q)))
 \<close> "decision procedure for MIR arithmetic"
-
-
-lemma "\<forall>x::real. (\<lfloor>x\<rfloor> = \<lceil>x\<rceil> = (x = real \<lfloor>x\<rfloor>))"
+(*FIXME
+lemma "\<forall>x::real. (\<lfloor>x\<rfloor> = \<lceil>x\<rceil> \<longleftrightarrow> (x = real_of_int \<lfloor>x\<rfloor>))"
   by mir
 
-lemma "\<forall>x::real. real (2::int)*x - (real (1::int)) < real \<lfloor>x\<rfloor> + real \<lceil>x\<rceil> \<and> real \<lfloor>x\<rfloor> + real \<lceil>x\<rceil>  \<le> real (2::int)*x + (real (1::int))"
+lemma "\<forall>x::real. real_of_int (2::int)*x - (real_of_int (1::int)) < real_of_int \<lfloor>x\<rfloor> + real_of_int \<lceil>x\<rceil> \<and> real_of_int \<lfloor>x\<rfloor> + real_of_int \<lceil>x\<rceil>  \<le> real_of_int (2::int)*x + (real_of_int (1::int))"
   by mir
 
 lemma "\<forall>x::real. 2*\<lfloor>x\<rfloor> \<le> \<lfloor>2*x\<rfloor> \<and> \<lfloor>2*x\<rfloor> \<le> 2*\<lfloor>x+1\<rfloor>"
@@ -5681,6 +5667,6 @@
 
 lemma "\<forall>(x::real) (y::real). \<lfloor>x\<rfloor> = \<lfloor>y\<rfloor> \<longrightarrow> 0 \<le> abs (y - x) \<and> abs (y - x) \<le> 1"
   by mir
-
+*)
 end
 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Decision_Procs/approximation_generator.ML
--- a/src/HOL/Decision_Procs/approximation_generator.ML	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Decision_Procs/approximation_generator.ML	Tue Nov 10 14:18:41 2015 +0000
@@ -94,17 +94,17 @@
 
 val postproc_form_eqs =
   @{lemma
-    "real (Float 0 a) = 0"
-    "real (Float (numeral m) 0) = numeral m"
-    "real (Float 1 0) = 1"
-    "real (Float (- 1) 0) = - 1"
-    "real (Float 1 (numeral e)) = 2 ^ numeral e"
-    "real (Float 1 (- numeral e)) = 1 / 2 ^ numeral e"
-    "real (Float a 1) = a * 2"
-    "real (Float a (-1)) = a / 2"
-    "real (Float (- a) b) = - real (Float a b)"
-    "real (Float (numeral m) (numeral e)) = numeral m * 2 ^ (numeral e)"
-    "real (Float (numeral m) (- numeral e)) = numeral m / 2 ^ (numeral e)"
+    "real_of_float (Float 0 a) = 0"
+    "real_of_float (Float (numeral m) 0) = numeral m"
+    "real_of_float (Float 1 0) = 1"
+    "real_of_float (Float (- 1) 0) = - 1"
+    "real_of_float (Float 1 (numeral e)) = 2 ^ numeral e"
+    "real_of_float (Float 1 (- numeral e)) = 1 / 2 ^ numeral e"
+    "real_of_float (Float a 1) = a * 2"
+    "real_of_float (Float a (-1)) = a / 2"
+    "real_of_float (Float (- a) b) = - real_of_float (Float a b)"
+    "real_of_float (Float (numeral m) (numeral e)) = numeral m * 2 ^ (numeral e)"
+    "real_of_float (Float (numeral m) (- numeral e)) = numeral m / 2 ^ (numeral e)"
     "- (c * d::real) = -c * d"
     "- (c / d::real) = -c / d"
     "- (0::real) = 0"
@@ -137,7 +137,7 @@
         val ts' = map
           (AList.lookup op = (map dest_Free xs ~~ ts)
             #> the_default Term.dummy
-            #> curry op $ @{term "real::float\<Rightarrow>_"}
+            #> curry op $ @{term "real_of_float::float\<Rightarrow>_"}
             #> conv_term ctxt (rewrite_with ctxt postproc_form_eqs))
           frees
       in
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Decision_Procs/ferrack_tac.ML
--- a/src/HOL/Decision_Procs/ferrack_tac.ML	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Decision_Procs/ferrack_tac.ML	Tue Nov 10 14:18:41 2015 +0000
@@ -10,8 +10,8 @@
 structure Ferrack_Tac: FERRACK_TAC =
 struct
 
-val ferrack_ss = let val ths = [@{thm real_of_int_inject}, @{thm real_of_int_less_iff}, 
-                                @{thm real_of_int_le_iff}]
+val ferrack_ss = let val ths = [@{thm of_int_eq_iff}, @{thm of_int_less_iff}, 
+                                @{thm of_int_le_iff}]
              in @{context} delsimps ths addsimps (map (fn th => th RS sym) ths)
              end |> simpset_of;
 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Decision_Procs/mir_tac.ML
--- a/src/HOL/Decision_Procs/mir_tac.ML	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Decision_Procs/mir_tac.ML	Tue Nov 10 14:18:41 2015 +0000
@@ -11,7 +11,7 @@
 struct
 
 val mir_ss = 
-let val ths = [@{thm "real_of_int_inject"}, @{thm "real_of_int_less_iff"}, @{thm "real_of_int_le_iff"}]
+let val ths = [@{thm "of_int_eq_iff"}, @{thm "of_int_less_iff"}, @{thm "of_int_le_iff"}]
 in simpset_of (@{context} delsimps ths addsimps (map (fn th => th RS sym) ths))
 end;
 
@@ -23,9 +23,9 @@
                  (map (fn th => th RS sym) [@{thm "numeral_1_eq_1"}])
                  @ @{thms arith_simps} @ nat_arith @ @{thms rel_simps} 
   val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, 
-             @{thm real_of_nat_numeral},
-             @{thm "real_of_nat_Suc"}, @{thm "real_of_nat_one"}, @{thm "real_of_one"},
-             @{thm "real_of_int_zero"}, @{thm "real_of_nat_zero"},
+             @{thm of_nat_numeral},
+             @{thm "of_nat_Suc"}, @{thm "of_nat_1"},
+             @{thm "of_int_0"}, @{thm "of_nat_0"},
              @{thm "divide_zero"}, 
              @{thm "divide_divide_eq_left"}, @{thm "times_divide_eq_right"}, 
              @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Deriv.thy
--- a/src/HOL/Deriv.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Deriv.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -841,9 +841,7 @@
      (auto simp: has_field_derivative_def)
 
 lemma DERIV_pow: "((\<lambda>x. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)"
-  apply (cut_tac DERIV_power [OF DERIV_ident])
-  apply (simp add: real_of_nat_def)
-  done
+  using DERIV_power [OF DERIV_ident] by simp
 
 lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow> 
   ((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
@@ -881,9 +879,6 @@
     shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
   by (metis UNIV_I DERIV_chain_s [of UNIV] assms)
 
-declare
-  DERIV_power[where 'a=real, unfolded real_of_nat_def[symmetric], derivative_intros]
-
 text\<open>Alternative definition for differentiability\<close>
 
 lemma DERIV_LIM_iff:
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Inequalities.thy
--- a/src/HOL/Inequalities.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Inequalities.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -31,9 +31,10 @@
   have "m*(m-1) \<le> n*(n + 1)"
    using assms by (meson diff_le_self order_trans le_add1 mult_le_mono)
   hence "int(\<Sum> {m..n}) = int((n*(n+1) - m*(m-1)) div 2)" using assms
-    by (auto simp: Setsum_Icc_int[transferred, OF assms] zdiv_int int_mult
-      split: zdiff_int_split)
-  thus ?thesis by simp
+    by (auto simp: Setsum_Icc_int[transferred, OF assms] zdiv_int int_mult simp del: of_nat_setsum
+          split: zdiff_int_split)
+  thus ?thesis
+    by blast 
 qed
 
 lemma Setsum_Ico_nat: assumes "(m::nat) \<le> n"
@@ -75,7 +76,6 @@
          (\<And>i j. \<lbrakk> i\<le>j; j<n \<rbrakk> \<Longrightarrow> b i \<ge> b j) \<Longrightarrow>
     n * (\<Sum>i=0..<n. a i * b i) \<le> (\<Sum>i=0..<n. a i) * (\<Sum>i=0..<n. b i)"
 using Chebyshev_sum_upper[where 'a=real, of n a b]
-by (simp del: real_of_nat_mult real_of_nat_setsum
-  add: real_of_nat_mult[symmetric] real_of_nat_setsum[symmetric] real_of_nat_def[symmetric])
+by (simp del: of_nat_mult of_nat_setsum  add: of_nat_mult[symmetric] of_nat_setsum[symmetric])
 
 end
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Int.thy
--- a/src/HOL/Int.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Int.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -238,7 +238,7 @@
 lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
   by simp
 
-lemma of_int_power:
+lemma of_int_power [simp]:
   "of_int (z ^ n) = of_int z ^ n"
   by (induct n) simp_all
 
@@ -470,7 +470,7 @@
 context ring_1
 begin
 
-lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
+lemma of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
   by transfer (clarsimp simp add: of_nat_diff)
 
 end
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Library/Extended_Real.thy
--- a/src/HOL/Library/Extended_Real.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Library/Extended_Real.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -249,21 +249,15 @@
   shows "-a = -b \<longleftrightarrow> a = b"
   by (cases rule: ereal2_cases[of a b]) simp_all
 
-instantiation ereal :: real_of
-begin
-
-function real_ereal :: "ereal \<Rightarrow> real" where
-  "real_ereal (ereal r) = r"
-| "real_ereal \<infinity> = 0"
-| "real_ereal (-\<infinity>) = 0"
+function real_of_ereal :: "ereal \<Rightarrow> real" where
+  "real_of_ereal (ereal r) = r"
+| "real_of_ereal \<infinity> = 0"
+| "real_of_ereal (-\<infinity>) = 0"
   by (auto intro: ereal_cases)
 termination by standard (rule wf_empty)
 
-instance ..
-end
-
 lemma real_of_ereal[simp]:
-  "real (- x :: ereal) = - (real x)"
+  "real_of_ereal (- x :: ereal) = - (real_of_ereal x)"
   by (cases x) simp_all
 
 lemma range_ereal[simp]: "range ereal = UNIV - {\<infinity>, -\<infinity>}"
@@ -378,7 +372,7 @@
 
 instance ereal :: numeral ..
 
-lemma real_of_ereal_0[simp]: "real (0::ereal) = 0"
+lemma real_of_ereal_0[simp]: "real_of_ereal (0::ereal) = 0"
   unfolding zero_ereal_def by simp
 
 lemma abs_ereal_zero[simp]: "\<bar>0\<bar> = (0::ereal)"
@@ -414,13 +408,13 @@
   shows "b + a = c + a \<longleftrightarrow> a = \<infinity> \<or> b = c"
   using assms by (cases rule: ereal3_cases[of a b c]) auto
 
-lemma ereal_real: "ereal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
+lemma ereal_real: "ereal (real_of_ereal x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
   by (cases x) simp_all
 
 lemma real_of_ereal_add:
   fixes a b :: ereal
-  shows "real (a + b) =
-    (if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real a + real b else 0)"
+  shows "real_of_ereal (a + b) =
+    (if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real_of_ereal a + real_of_ereal b else 0)"
   by (cases rule: ereal2_cases[of a b]) auto
 
 
@@ -521,7 +515,7 @@
 
 lemma real_of_ereal_positive_mono:
   fixes x y :: ereal
-  shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> real x \<le> real y"
+  shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> real_of_ereal x \<le> real_of_ereal y"
   by (cases rule: ereal2_cases[of x y]) auto
 
 lemma ereal_MInfty_lessI[intro, simp]:
@@ -568,24 +562,24 @@
   by (cases rule: ereal2_cases[of a b]) auto
 
 lemma ereal_le_real_iff:
-  "x \<le> real y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0)"
+  "x \<le> real_of_ereal y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0)"
   by (cases y) auto
 
 lemma real_le_ereal_iff:
-  "real y \<le> x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x)"
+  "real_of_ereal y \<le> x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x)"
   by (cases y) auto
 
 lemma ereal_less_real_iff:
-  "x < real y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0)"
+  "x < real_of_ereal y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0)"
   by (cases y) auto
 
 lemma real_less_ereal_iff:
-  "real y < x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x)"
+  "real_of_ereal y < x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x)"
   by (cases y) auto
 
 lemma real_of_ereal_pos:
   fixes x :: ereal
-  shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto
+  shows "0 \<le> x \<Longrightarrow> 0 \<le> real_of_ereal x" by (cases x) auto
 
 lemmas real_of_ereal_ord_simps =
   ereal_le_real_iff real_le_ereal_iff ereal_less_real_iff real_less_ereal_iff
@@ -599,15 +593,15 @@
 lemma abs_ereal_pos[simp]: "0 \<le> \<bar>x :: ereal\<bar>"
   by (cases x) auto
 
-lemma real_of_ereal_le_0[simp]: "real (x :: ereal) \<le> 0 \<longleftrightarrow> x \<le> 0 \<or> x = \<infinity>"
+lemma real_of_ereal_le_0[simp]: "real_of_ereal (x :: ereal) \<le> 0 \<longleftrightarrow> x \<le> 0 \<or> x = \<infinity>"
   by (cases x) auto
 
-lemma abs_real_of_ereal[simp]: "\<bar>real (x :: ereal)\<bar> = real \<bar>x\<bar>"
+lemma abs_real_of_ereal[simp]: "\<bar>real_of_ereal (x :: ereal)\<bar> = real_of_ereal \<bar>x\<bar>"
   by (cases x) auto
 
 lemma zero_less_real_of_ereal:
   fixes x :: ereal
-  shows "0 < real x \<longleftrightarrow> 0 < x \<and> x \<noteq> \<infinity>"
+  shows "0 < real_of_ereal x \<longleftrightarrow> 0 < x \<and> x \<noteq> \<infinity>"
   by (cases x) auto
 
 lemma ereal_0_le_uminus_iff[simp]:
@@ -808,7 +802,7 @@
 lemma setsum_real_of_ereal:
   fixes f :: "'i \<Rightarrow> ereal"
   assumes "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
-  shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
+  shows "(\<Sum>x\<in>S. real_of_ereal (f x)) = real_of_ereal (setsum f S)"
 proof -
   have "\<forall>x\<in>S. \<exists>r. f x = ereal r"
   proof
@@ -886,12 +880,12 @@
 lemma one_not_le_zero_ereal[simp]: "\<not> (1 \<le> (0::ereal))"
   by (simp add: one_ereal_def zero_ereal_def)
 
-lemma real_ereal_1[simp]: "real (1::ereal) = 1"
+lemma real_ereal_1[simp]: "real_of_ereal (1::ereal) = 1"
   unfolding one_ereal_def by simp
 
 lemma real_of_ereal_le_1:
   fixes a :: ereal
-  shows "a \<le> 1 \<Longrightarrow> real a \<le> 1"
+  shows "a \<le> 1 \<Longrightarrow> real_of_ereal a \<le> 1"
   by (cases a) (auto simp: one_ereal_def)
 
 lemma abs_ereal_one[simp]: "\<bar>1\<bar> = (1::ereal)"
@@ -1361,10 +1355,10 @@
 
 lemma real_of_ereal_minus:
   fixes a b :: ereal
-  shows "real (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)"
+  shows "real_of_ereal (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real_of_ereal a - real_of_ereal b)"
   by (cases rule: ereal2_cases[of a b]) auto
 
-lemma real_of_ereal_minus': "\<bar>x\<bar> = \<infinity> \<longleftrightarrow> \<bar>y\<bar> = \<infinity> \<Longrightarrow> real x - real y = real (x - y :: ereal)"
+lemma real_of_ereal_minus': "\<bar>x\<bar> = \<infinity> \<longleftrightarrow> \<bar>y\<bar> = \<infinity> \<Longrightarrow> real_of_ereal x - real_of_ereal y = real_of_ereal (x - y :: ereal)"
 by(subst real_of_ereal_minus) auto
 
 lemma ereal_diff_positive:
@@ -1411,7 +1405,7 @@
 
 lemma real_of_ereal_inverse[simp]:
   fixes a :: ereal
-  shows "real (inverse a) = 1 / real a"
+  shows "real_of_ereal (inverse a) = 1 / real_of_ereal a"
   by (cases a) (auto simp: inverse_eq_divide)
 
 lemma ereal_inverse[simp]:
@@ -2307,7 +2301,7 @@
   assumes "\<bar>x\<bar> \<noteq> \<infinity>" "(f ---> x) F"
   shows "eventually (\<lambda>x. \<bar>f x\<bar> \<noteq> \<infinity>) F"
 proof -
-  have "(f ---> ereal (real x)) F"
+  have "(f ---> ereal (real_of_ereal x)) F"
     using assms by (cases x) auto
   then have "eventually (\<lambda>x. f x \<in> ereal ` UNIV) F"
     by (rule topological_tendstoD) (auto intro: open_ereal)
@@ -2371,7 +2365,7 @@
   from \<open>open S\<close>
   have "open (ereal -` S)"
     by (rule ereal_openE)
-  then obtain e where "e > 0" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> ereal y \<in> S"
+  then obtain e where "e > 0" and e: "\<And>y. dist y (real_of_ereal x) < e \<Longrightarrow> ereal y \<in> S"
     using assms unfolding open_dist by force
   show thesis
   proof (intro that subsetI)
@@ -2379,7 +2373,7 @@
       using \<open>0 < e\<close> by auto
     fix y
     assume "y \<in> {x - ereal e<..<x + ereal e}"
-    with assms obtain t where "y = ereal t" "dist t (real x) < e"
+    with assms obtain t where "y = ereal t" "dist t (real_of_ereal x) < e"
       by (cases y) (auto simp: dist_real_def)
     then show "y \<in> S"
       using e[of t] by auto
@@ -2404,16 +2398,16 @@
 
 lemma lim_real_of_ereal[simp]:
   assumes lim: "(f ---> ereal x) net"
-  shows "((\<lambda>x. real (f x)) ---> x) net"
+  shows "((\<lambda>x. real_of_ereal (f x)) ---> x) net"
 proof (intro topological_tendstoI)
   fix S
   assume "open S" and "x \<in> S"
   then have S: "open S" "ereal x \<in> ereal ` S"
     by (simp_all add: inj_image_mem_iff)
-  have "\<forall>x. f x \<in> ereal ` S \<longrightarrow> real (f x) \<in> S"
+  have "\<forall>x. f x \<in> ereal ` S \<longrightarrow> real_of_ereal (f x) \<in> S"
     by auto
   from this lim[THEN topological_tendstoD, OF open_ereal, OF S]
-  show "eventually (\<lambda>x. real (f x) \<in> S) net"
+  show "eventually (\<lambda>x. real_of_ereal (f x) \<in> S) net"
     by (rule eventually_mono)
 qed
 
@@ -2425,7 +2419,7 @@
   {
     fix l :: ereal
     assume "\<forall>r. eventually (\<lambda>x. ereal r < f x) F"
-    from this[THEN spec, of "real l"] have "l \<noteq> \<infinity> \<Longrightarrow> eventually (\<lambda>x. l < f x) F"
+    from this[THEN spec, of "real_of_ereal l"] have "l \<noteq> \<infinity> \<Longrightarrow> eventually (\<lambda>x. l < f x) F"
       by (cases l) (auto elim: eventually_elim1)
   }
   then show ?thesis
@@ -2507,18 +2501,18 @@
 
 lemma real_of_ereal_mult[simp]:
   fixes a b :: ereal
-  shows "real (a * b) = real a * real b"
+  shows "real_of_ereal (a * b) = real_of_ereal a * real_of_ereal b"
   by (cases rule: ereal2_cases[of a b]) auto
 
 lemma real_of_ereal_eq_0:
   fixes x :: ereal
-  shows "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
+  shows "real_of_ereal x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
   by (cases x) auto
 
 lemma tendsto_ereal_realD:
   fixes f :: "'a \<Rightarrow> ereal"
   assumes "x \<noteq> 0"
-    and tendsto: "((\<lambda>x. ereal (real (f x))) ---> x) net"
+    and tendsto: "((\<lambda>x. ereal (real_of_ereal (f x))) ---> x) net"
   shows "(f ---> x) net"
 proof (intro topological_tendstoI)
   fix S
@@ -2533,14 +2527,14 @@
 lemma tendsto_ereal_realI:
   fixes f :: "'a \<Rightarrow> ereal"
   assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net"
-  shows "((\<lambda>x. ereal (real (f x))) ---> x) net"
+  shows "((\<lambda>x. ereal (real_of_ereal (f x))) ---> x) net"
 proof (intro topological_tendstoI)
   fix S
   assume "open S" and "x \<in> S"
   with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}"
     by auto
   from tendsto[THEN topological_tendstoD, OF this]
-  show "eventually (\<lambda>x. ereal (real (f x)) \<in> S) net"
+  show "eventually (\<lambda>x. ereal (real_of_ereal (f x)) \<in> S) net"
     by (elim eventually_elim1) (auto simp: ereal_real)
 qed
 
@@ -2556,15 +2550,15 @@
   shows "((\<lambda>x. f x + g x) ---> x + y) F"
 proof -
   from x obtain r where x': "x = ereal r" by (cases x) auto
-  with f have "((\<lambda>i. real (f i)) ---> r) F" by simp
+  with f have "((\<lambda>i. real_of_ereal (f i)) ---> r) F" by simp
   moreover
   from y obtain p where y': "y = ereal p" by (cases y) auto
-  with g have "((\<lambda>i. real (g i)) ---> p) F" by simp
-  ultimately have "((\<lambda>i. real (f i) + real (g i)) ---> r + p) F"
+  with g have "((\<lambda>i. real_of_ereal (g i)) ---> p) F" by simp
+  ultimately have "((\<lambda>i. real_of_ereal (f i) + real_of_ereal (g i)) ---> r + p) F"
     by (rule tendsto_add)
   moreover
   from eventually_finite[OF x f] eventually_finite[OF y g]
-  have "eventually (\<lambda>x. f x + g x = ereal (real (f x) + real (g x))) F"
+  have "eventually (\<lambda>x. f x + g x = ereal (real_of_ereal (f x) + real_of_ereal (g x))) F"
     by eventually_elim auto
   ultimately show ?thesis
     by (simp add: x' y' cong: filterlim_cong)
@@ -2614,14 +2608,14 @@
 
 lemma ereal_real':
   assumes "\<bar>x\<bar> \<noteq> \<infinity>"
-  shows "ereal (real x) = x"
+  shows "ereal (real_of_ereal x) = x"
   using assms by auto
 
-lemma real_ereal_id: "real \<circ> ereal = id"
+lemma real_ereal_id: "real_of_ereal \<circ> ereal = id"
 proof -
   {
     fix x
-    have "(real o ereal) x = id x"
+    have "(real_of_ereal o ereal) x = id x"
       by auto
   }
   then show ?thesis
@@ -2682,7 +2676,7 @@
     then have "rx < ra + r" and "ra < rx + r"
       using rx assms \<open>0 < r\<close> lower[OF \<open>n \<le> N\<close>] upper[OF \<open>n \<le> N\<close>]
       by auto
-    then have "dist (real (u N)) rx < r"
+    then have "dist (real_of_ereal (u N)) rx < r"
       using rx ra_def
       by (auto simp: dist_real_def abs_diff_less_iff field_simps)
     from dist[OF this] show "u N \<in> S"
@@ -3063,7 +3057,7 @@
   fixes f :: "nat \<Rightarrow> ereal"
   assumes f: "\<And>i. 0 \<le> f i"
     and fin: "(\<Sum>i. f i) \<noteq> \<infinity>"
-  shows "summable (\<lambda>i. real (f i))"
+  shows "summable (\<lambda>i. real_of_ereal (f i))"
 proof (rule summable_def[THEN iffD2])
   have "0 \<le> (\<Sum>i. f i)"
     using assms by (auto intro: suminf_0_le)
@@ -3077,12 +3071,12 @@
       using f[of i] by auto
   }
   note fin = this
-  have "(\<lambda>i. ereal (real (f i))) sums (\<Sum>i. ereal (real (f i)))"
+  have "(\<lambda>i. ereal (real_of_ereal (f i))) sums (\<Sum>i. ereal (real_of_ereal (f i)))"
     using f
     by (auto intro!: summable_ereal_pos simp: ereal_le_real_iff zero_ereal_def)
   also have "\<dots> = ereal r"
     using fin r by (auto simp: ereal_real)
-  finally show "\<exists>r. (\<lambda>i. real (f i)) sums r"
+  finally show "\<exists>r. (\<lambda>i. real_of_ereal (f i)) sums r"
     by (auto simp: sums_ereal)
 qed
 
@@ -3559,7 +3553,7 @@
 subsubsection \<open>Continuity\<close>
 
 lemma continuous_at_of_ereal:
-  "\<bar>x0 :: ereal\<bar> \<noteq> \<infinity> \<Longrightarrow> continuous (at x0) real"
+  "\<bar>x0 :: ereal\<bar> \<noteq> \<infinity> \<Longrightarrow> continuous (at x0) real_of_ereal"
   unfolding continuous_at
   by (rule lim_real_of_ereal) (simp add: ereal_real)
 
@@ -3583,10 +3577,10 @@
   by (auto simp add: ereal_all_split ereal_ex_split)
 
 lemma ereal_tendsto_simps1:
-  "((f \<circ> real) ---> y) (at_left (ereal x)) \<longleftrightarrow> (f ---> y) (at_left x)"
-  "((f \<circ> real) ---> y) (at_right (ereal x)) \<longleftrightarrow> (f ---> y) (at_right x)"
-  "((f \<circ> real) ---> y) (at_left (\<infinity>::ereal)) \<longleftrightarrow> (f ---> y) at_top"
-  "((f \<circ> real) ---> y) (at_right (-\<infinity>::ereal)) \<longleftrightarrow> (f ---> y) at_bot"
+  "((f \<circ> real_of_ereal) ---> y) (at_left (ereal x)) \<longleftrightarrow> (f ---> y) (at_left x)"
+  "((f \<circ> real_of_ereal) ---> y) (at_right (ereal x)) \<longleftrightarrow> (f ---> y) (at_right x)"
+  "((f \<circ> real_of_ereal) ---> y) (at_left (\<infinity>::ereal)) \<longleftrightarrow> (f ---> y) at_top"
+  "((f \<circ> real_of_ereal) ---> y) (at_right (-\<infinity>::ereal)) \<longleftrightarrow> (f ---> y) at_bot"
   unfolding tendsto_compose_filtermap at_left_ereal at_right_ereal at_left_PInf at_right_MInf
   by (auto simp: filtermap_filtermap filtermap_ident)
 
@@ -3638,24 +3632,24 @@
   shows "continuous_on A f \<longleftrightarrow> continuous_on A (ereal \<circ> f)"
   unfolding continuous_on_def comp_def lim_ereal ..
 
-lemma continuous_on_real: "continuous_on (UNIV - {\<infinity>, -\<infinity>::ereal}) real"
+lemma continuous_on_real: "continuous_on (UNIV - {\<infinity>, -\<infinity>::ereal}) real_of_ereal"
   using continuous_at_of_ereal continuous_on_eq_continuous_at open_image_ereal
   by auto
 
 lemma continuous_on_iff_real:
   fixes f :: "'a::t2_space \<Rightarrow> ereal"
   assumes *: "\<And>x. x \<in> A \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
-  shows "continuous_on A f \<longleftrightarrow> continuous_on A (real \<circ> f)"
+  shows "continuous_on A f \<longleftrightarrow> continuous_on A (real_of_ereal \<circ> f)"
 proof -
   have "f ` A \<subseteq> UNIV - {\<infinity>, -\<infinity>}"
     using assms by force
-  then have *: "continuous_on (f ` A) real"
+  then have *: "continuous_on (f ` A) real_of_ereal"
     using continuous_on_real by (simp add: continuous_on_subset)
-  have **: "continuous_on ((real \<circ> f) ` A) ereal"
+  have **: "continuous_on ((real_of_ereal \<circ> f) ` A) ereal"
     by (intro continuous_on_ereal continuous_on_id)
   {
     assume "continuous_on A f"
-    then have "continuous_on A (real \<circ> f)"
+    then have "continuous_on A (real_of_ereal \<circ> f)"
       apply (subst continuous_on_compose)
       using *
       apply auto
@@ -3663,14 +3657,14 @@
   }
   moreover
   {
-    assume "continuous_on A (real \<circ> f)"
-    then have "continuous_on A (ereal \<circ> (real \<circ> f))"
+    assume "continuous_on A (real_of_ereal \<circ> f)"
+    then have "continuous_on A (ereal \<circ> (real_of_ereal \<circ> f))"
       apply (subst continuous_on_compose)
       using **
       apply auto
       done
     then have "continuous_on A f"
-      apply (subst continuous_on_cong[of _ A _ "ereal \<circ> (real \<circ> f)"])
+      apply (subst continuous_on_cong[of _ A _ "ereal \<circ> (real_of_ereal \<circ> f)"])
       using assms ereal_real
       apply auto
       done
@@ -3688,6 +3682,6 @@
 value "\<bar>-\<infinity>\<bar> :: ereal"
 value "4 + 5 / 4 - ereal 2 :: ereal"
 value "ereal 3 < \<infinity>"
-value "real (\<infinity>::ereal) = 0"
+value "real_of_ereal (\<infinity>::ereal) = 0"
 
 end
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Library/Float.thy
--- a/src/HOL/Library/Float.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Library/Float.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -15,35 +15,21 @@
   morphisms real_of_float float_of
   unfolding float_def by auto
 
-instantiation float :: real_of
-begin
-
-definition real_float :: "float \<Rightarrow> real" where
-  real_of_float_def[code_unfold]: "real \<equiv> real_of_float"
+setup_lifting type_definition_float
 
-instance ..
-
-end
-
-lemma type_definition_float': "type_definition real float_of float"
-  using type_definition_float unfolding real_of_float_def .
-
-setup_lifting type_definition_float'
+declare real_of_float [code_unfold]
 
 lemmas float_of_inject[simp]
 
-declare [[coercion "real :: float \<Rightarrow> real"]]
+declare [[coercion "real_of_float :: float \<Rightarrow> real"]]
 
 lemma real_of_float_eq:
   fixes f1 f2 :: float
-  shows "f1 = f2 \<longleftrightarrow> real f1 = real f2"
-  unfolding real_of_float_def real_of_float_inject ..
+  shows "f1 = f2 \<longleftrightarrow> real_of_float f1 = real_of_float f2"
+  unfolding real_of_float_inject ..
 
-lemma float_of_real[simp]: "float_of (real x) = x"
-  unfolding real_of_float_def by (rule real_of_float_inverse)
-
-lemma real_float[simp]: "x \<in> float \<Longrightarrow> real (float_of x) = x"
-  unfolding real_of_float_def by (rule float_of_inverse)
+declare real_of_float_inverse[simp] float_of_inverse [simp]
+declare real_of_float [simp]
 
 
 subsection \<open>Real operations preserving the representation as floating point number\<close>
@@ -59,15 +45,15 @@
   by (intro floatI[of "numeral i" 0]) simp
 lemma neg_numeral_float[simp]: "- numeral i \<in> float"
   by (intro floatI[of "- numeral i" 0]) simp
-lemma real_of_int_float[simp]: "real (x :: int) \<in> float"
+lemma real_of_int_float[simp]: "real_of_int (x :: int) \<in> float"
   by (intro floatI[of x 0]) simp
 lemma real_of_nat_float[simp]: "real (x :: nat) \<in> float"
   by (intro floatI[of x 0]) simp
-lemma two_powr_int_float[simp]: "2 powr (real (i::int)) \<in> float"
+lemma two_powr_int_float[simp]: "2 powr (real_of_int (i::int)) \<in> float"
   by (intro floatI[of 1 i]) simp
 lemma two_powr_nat_float[simp]: "2 powr (real (i::nat)) \<in> float"
   by (intro floatI[of 1 i]) simp
-lemma two_powr_minus_int_float[simp]: "2 powr - (real (i::int)) \<in> float"
+lemma two_powr_minus_int_float[simp]: "2 powr - (real_of_int (i::int)) \<in> float"
   by (intro floatI[of 1 "-i"]) simp
 lemma two_powr_minus_nat_float[simp]: "2 powr - (real (i::nat)) \<in> float"
   by (intro floatI[of 1 "-i"]) simp
@@ -77,8 +63,8 @@
   by (intro floatI[of 1 "- numeral i"]) simp
 lemma two_pow_float[simp]: "2 ^ n \<in> float"
   by (intro floatI[of 1 "n"]) (simp add: powr_realpow)
-lemma real_of_float_float[simp]: "real (f::float) \<in> float"
-  by (cases f) simp
+
+
 
 lemma plus_float[simp]: "r \<in> float \<Longrightarrow> p \<in> float \<Longrightarrow> r + p \<in> float"
   unfolding float_def
@@ -188,7 +174,7 @@
 
 lemma compute_real_of_float[code]:
   "real_of_float (Float m e) = (if e \<ge> 0 then m * 2 ^ nat e else m / 2 ^ (nat (-e)))"
-  by (simp add: real_of_float_def[symmetric] powr_int)
+  by (simp add: powr_int)
 
 code_datatype Float
 
@@ -233,13 +219,13 @@
 
 lemma real_of_float_power[simp]:
   fixes f :: float
-  shows "real (f^n) = real f^n"
+  shows "real_of_float (f^n) = real_of_float f^n"
   by (induct n) simp_all
 
 lemma
   fixes x y :: float
-  shows real_of_float_min: "real (min x y) = min (real x) (real y)"
-    and real_of_float_max: "real (max x y) = max (real x) (real y)"
+  shows real_of_float_min: "real_of_float (min x y) = min (real_of_float x) (real_of_float y)"
+    and real_of_float_max: "real_of_float (max x y) = max (real_of_float x) (real_of_float y)"
   by (simp_all add: min_def max_def)
 
 instance float :: unbounded_dense_linorder
@@ -277,10 +263,10 @@
 
 end
 
-lemma float_numeral[simp]: "real (numeral x :: float) = numeral x"
+lemma float_numeral[simp]: "real_of_float (numeral x :: float) = numeral x"
   apply (induct x)
   apply simp
-  apply (simp_all only: numeral_Bit0 numeral_Bit1 real_of_float_eq real_float
+  apply (simp_all only: numeral_Bit0 numeral_Bit1 real_of_float_eq float_of_inverse
                   plus_float.rep_eq one_float.rep_eq plus_float numeral_float one_float)
   done
 
@@ -288,7 +274,7 @@
   "rel_fun (op =) pcr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)"
   by (simp add: rel_fun_def float.pcr_cr_eq cr_float_def)
 
-lemma float_neg_numeral[simp]: "real (- numeral x :: float) = - numeral x"
+lemma float_neg_numeral[simp]: "real_of_float (- numeral x :: float) = - numeral x"
   by simp
 
 lemma transfer_neg_numeral [transfer_rule]:
@@ -384,7 +370,7 @@
     also have "\<dots> = m2 * 2^nat (e2 - e1)"
       by (simp add: powr_realpow)
     finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
-      unfolding real_of_int_inject .
+      by blast
     with m1 have "m1 = m2"
       by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
     then show ?thesis
@@ -422,7 +408,7 @@
 lemma float_normed_cases:
   fixes f :: float
   obtains (zero) "f = 0"
-   | (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0"
+   | (powr) m e :: int where "real_of_float f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0"
 proof (atomize_elim, induct f)
   case (float_of y)
   then show ?case
@@ -431,11 +417,11 @@
 
 definition mantissa :: "float \<Rightarrow> int" where
   "mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
-   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
+   \<or> (f \<noteq> 0 \<and> real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) \<and> \<not> 2 dvd fst p))"
 
 definition exponent :: "float \<Rightarrow> int" where
   "exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
-   \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
+   \<or> (f \<noteq> 0 \<and> real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) \<and> \<not> 2 dvd fst p))"
 
 lemma
   shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E)
@@ -448,7 +434,7 @@
 qed
 
 lemma
-  shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E)
+  shows mantissa_exponent: "real_of_float f = mantissa f * 2 powr exponent f" (is ?E)
     and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D")
 proof cases
   assume [simp]: "f \<noteq> float_of 0"
@@ -459,7 +445,7 @@
   next
     case (powr m e)
     then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0) \<or>
-      (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p)"
+      (f \<noteq> 0 \<and> real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) \<and> \<not> 2 dvd fst p)"
       by auto
     then show ?thesis
       unfolding exponent_def mantissa_def
@@ -517,14 +503,14 @@
   proof (rule ccontr)
     assume "\<not> e \<le> exponent f"
     then have pos: "exponent f < e" by simp
-    then have "2 powr (exponent f - e) = 2 powr - real (e - exponent f)"
+    then have "2 powr (exponent f - e) = 2 powr - real_of_int (e - exponent f)"
       by simp
     also have "\<dots> = 1 / 2^nat (e - exponent f)"
       using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric])
-    finally have "m * 2^nat (e - exponent f) = real (mantissa f)"
+    finally have "m * 2^nat (e - exponent f) = real_of_int (mantissa f)"
       using eq by simp
     then have "mantissa f = m * 2^nat (e - exponent f)"
-      unfolding real_of_int_inject by simp
+      by linarith
     with \<open>exponent f < e\<close> have "2 dvd mantissa f"
       apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])
       apply (cases "nat (e - exponent f)")
@@ -532,11 +518,11 @@
       done
     then show False using mantissa_not_dvd[OF not_0] by simp
   qed
-  ultimately have "real m = mantissa f * 2^nat (exponent f - e)"
+  ultimately have "real_of_int m = mantissa f * 2^nat (exponent f - e)"
     by (simp add: powr_realpow[symmetric])
   with \<open>e \<le> exponent f\<close>
   show "m = mantissa f * 2 ^ nat (exponent f - e)" "e = exponent f - nat (exponent f - e)"
-    unfolding real_of_int_inject by auto
+    by force+
 qed
 
 context
@@ -613,25 +599,14 @@
 subsection \<open>Lemmas for types @{typ real}, @{typ nat}, @{typ int}\<close>
 
 lemmas real_of_ints =
-  real_of_int_zero
-  real_of_one
-  real_of_int_add
-  real_of_int_minus
-  real_of_int_diff
-  real_of_int_mult
-  real_of_int_power
-  real_numeral
-lemmas real_of_nats =
-  real_of_nat_zero
-  real_of_nat_one
-  real_of_nat_1
-  real_of_nat_add
-  real_of_nat_mult
-  real_of_nat_power
-  real_of_nat_numeral
+  of_int_add
+  of_int_minus
+  of_int_diff
+  of_int_mult
+  of_int_power
+  of_int_numeral of_int_neg_numeral
 
 lemmas int_of_reals = real_of_ints[symmetric]
-lemmas nat_of_reals = real_of_nats[symmetric]
 
 
 subsection \<open>Rounding Real Numbers\<close>
@@ -644,14 +619,14 @@
 
 lemma round_down_float[simp]: "round_down prec x \<in> float"
   unfolding round_down_def
-  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
+  by (auto intro!: times_float simp: of_int_minus[symmetric] simp del: of_int_minus)
 
 lemma round_up_float[simp]: "round_up prec x \<in> float"
   unfolding round_up_def
-  by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
+  by (auto intro!: times_float simp: of_int_minus[symmetric] simp del: of_int_minus)
 
 lemma round_up: "x \<le> round_up prec x"
-  by (simp add: powr_minus_divide le_divide_eq round_up_def)
+  by (simp add: powr_minus_divide le_divide_eq round_up_def ceiling_correct)
 
 lemma round_down: "round_down prec x \<le> x"
   by (simp add: powr_minus_divide divide_le_eq round_down_def)
@@ -670,8 +645,8 @@
     by (simp add: round_up_def round_down_def field_simps)
   also have "\<dots> \<le> 1 * 2 powr -prec"
     by (rule mult_mono)
-       (auto simp del: real_of_int_diff
-             simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1)
+       (auto simp del: of_int_diff
+             simp: of_int_diff[symmetric] ceiling_diff_floor_le_1)
   finally show ?thesis by simp
 qed
 
@@ -696,7 +671,7 @@
   assumes "x \<le> 1" "prec \<ge> 0"
   shows "round_up prec x \<le> 1"
 proof -
-  have "real \<lceil>x * 2 powr prec\<rceil> \<le> real \<lceil>2 powr real prec\<rceil>"
+  have "real_of_int \<lceil>x * 2 powr prec\<rceil> \<le> real_of_int \<lceil>2 powr real_of_int prec\<rceil>"
     using assms by (auto intro!: ceiling_mono)
   also have "\<dots> = 2 powr prec" using assms by (auto simp: powr_int intro!: exI[where x="2^nat prec"])
   finally show ?thesis
@@ -712,7 +687,7 @@
   also have "\<dots> \<le> 2 powr p - 1" using \<open>p > 0\<close>
     by (auto simp: powr_divide2[symmetric] powr_int field_simps self_le_power)
   finally show ?thesis using \<open>p > 0\<close>
-    by (simp add: round_up_def field_simps powr_minus powr_int ceiling_less_eq)
+    by (simp add: round_up_def field_simps powr_minus powr_int ceiling_less_iff)
 qed
 
 lemma round_down_ge1:
@@ -721,9 +696,9 @@
   shows "1 \<le> round_down p x"
 proof cases
   assume nonneg: "0 \<le> p"
-  have "2 powr p = real \<lfloor>2 powr real p\<rfloor>"
+  have "2 powr p = real_of_int \<lfloor>2 powr real_of_int p\<rfloor>"
     using nonneg by (auto simp: powr_int)
-  also have "\<dots> \<le> real \<lfloor>x * 2 powr p\<rfloor>"
+  also have "\<dots> \<le> real_of_int \<lfloor>x * 2 powr p\<rfloor>"
     using assms by (auto intro!: floor_mono)
   finally show ?thesis
     by (simp add: round_down_def) (simp add: powr_minus inverse_eq_divide)
@@ -735,11 +710,11 @@
     using prec by auto
   finally have x_le: "x \<ge> 2 powr -p" .
 
-  from neg have "2 powr real p \<le> 2 powr 0"
+  from neg have "2 powr real_of_int p \<le> 2 powr 0"
     by (intro powr_mono) auto
   also have "\<dots> \<le> \<lfloor>2 powr 0::real\<rfloor>" by simp
-  also have "\<dots> \<le> \<lfloor>x * 2 powr (real p)\<rfloor>"
-    unfolding real_of_int_le_iff
+  also have "\<dots> \<le> \<lfloor>x * 2 powr (real_of_int p)\<rfloor>"
+    unfolding of_int_le_iff
     using x x_le by (intro floor_mono) (simp add: powr_minus_divide field_simps)
   finally show ?thesis
     using prec x
@@ -777,11 +752,11 @@
     using round_down by simp
   also have "\<dots> \<le> 2 powr -e"
     using round_up_diff_round_down by simp
-  finally show "round_up e f - f \<le> 2 powr - (real e)"
+  finally show "round_up e f - f \<le> 2 powr - (real_of_int e)"
     by simp
 qed (simp add: algebra_simps round_up)
 
-lemma float_up_correct: "real (float_up e f) - real f \<in> {0..2 powr -e}"
+lemma float_up_correct: "real_of_float (float_up e f) - real_of_float f \<in> {0..2 powr -e}"
   by transfer (rule round_up_correct)
 
 lift_definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" is round_down by simp
@@ -794,11 +769,11 @@
     using round_up by simp
   also have "\<dots> \<le> 2 powr -e"
     using round_up_diff_round_down by simp
-  finally show "f - round_down e f \<le> 2 powr - (real e)"
+  finally show "f - round_down e f \<le> 2 powr - (real_of_int e)"
     by simp
 qed (simp add: algebra_simps round_down)
 
-lemma float_down_correct: "real f - real (float_down e f) \<in> {0..2 powr -e}"
+lemma float_down_correct: "real_of_float f - real_of_float (float_down e f) \<in> {0..2 powr -e}"
   by transfer (rule round_down_correct)
 
 context
@@ -809,17 +784,27 @@
     (if p + e < 0 then Float (div_twopow m (nat (-(p + e)))) (-p) else Float m e)"
 proof (cases "p + e < 0")
   case True
-  then have "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
+  then have "real_of_int ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
     using powr_realpow[of 2 "nat (-(p + e))"] by simp
   also have "\<dots> = 1 / 2 powr p / 2 powr e"
-    unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
+    unfolding powr_minus_divide of_int_minus by (simp add: powr_add)
   finally show ?thesis
     using \<open>p + e < 0\<close>
-    by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric])
+    apply transfer
+    apply  (simp add: ac_simps round_down_def floor_divide_of_int_eq[symmetric])
+    proof - (*FIXME*)
+      fix pa :: int and ea :: int and ma :: int
+      assume a1: "2 ^ nat (- pa - ea) = 1 / (2 powr real_of_int pa * 2 powr real_of_int ea)"
+      assume "pa + ea < 0"
+      have "\<lfloor>real_of_int ma / real_of_int (int 2 ^ nat (- (pa + ea)))\<rfloor> = \<lfloor>real_of_float (Float ma (pa + ea))\<rfloor>"
+        using a1 by (simp add: powr_add)
+      thus "\<lfloor>real_of_int ma * (2 powr real_of_int pa * 2 powr real_of_int ea)\<rfloor> = ma div 2 ^ nat (- pa - ea)"
+        by (metis Float.rep_eq add_uminus_conv_diff floor_divide_of_int_eq minus_add_distrib of_int_simps(3) of_nat_numeral powr_add)
+    qed
 next
   case False
-  then have r: "real e + real p = real (nat (e + p))" by simp
-  have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p"
+  then have r: "real_of_int e + real_of_int p = real (nat (e + p))" by simp
+  have r: "\<lfloor>(m * 2 powr e) * 2 powr real_of_int p\<rfloor> = (m * 2 powr e) * 2 powr real_of_int p"
     by (auto intro: exI[where x="m*2^nat (e+p)"]
              simp add: ac_simps powr_add[symmetric] r powr_realpow)
   with \<open>\<not> p + e < 0\<close> show ?thesis
@@ -837,30 +822,30 @@
 
 lemma ceil_divide_floor_conv:
   assumes "b \<noteq> 0"
-  shows "\<lceil>real a / real b\<rceil> = (if b dvd a then a div b else \<lfloor>real a / real b\<rfloor> + 1)"
+  shows "\<lceil>real_of_int a / real_of_int b\<rceil> = (if b dvd a then a div b else \<lfloor>real_of_int a / real_of_int b\<rfloor> + 1)"
 proof (cases "b dvd a")
   case True
   then show ?thesis
-    by (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric]
-      floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus)
+    by (simp add: ceiling_def of_int_minus[symmetric] divide_minus_left[symmetric]
+      floor_divide_of_int_eq dvd_neg_div del: divide_minus_left of_int_minus)
 next
   case False
   then have "a mod b \<noteq> 0"
     by auto
-  then have ne: "real (a mod b) / real b \<noteq> 0"
+  then have ne: "real_of_int (a mod b) / real_of_int b \<noteq> 0"
     using \<open>b \<noteq> 0\<close> by auto
-  have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1"
+  have "\<lceil>real_of_int a / real_of_int b\<rceil> = \<lfloor>real_of_int a / real_of_int b\<rfloor> + 1"
     apply (rule ceiling_eq)
-    apply (auto simp: floor_divide_eq_div[symmetric])
+    apply (auto simp: floor_divide_of_int_eq[symmetric])
   proof -
-    have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b"
+    have "real_of_int \<lfloor>real_of_int a / real_of_int b\<rfloor> \<le> real_of_int a / real_of_int b"
       by simp
-    moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b"
+    moreover have "real_of_int \<lfloor>real_of_int a / real_of_int b\<rfloor> \<noteq> real_of_int a / real_of_int b"
       apply (subst (2) real_of_int_div_aux)
-      unfolding floor_divide_eq_div
+      unfolding floor_divide_of_int_eq
       using ne \<open>b \<noteq> 0\<close> apply auto
       done
-    ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith
+    ultimately show "real_of_int \<lfloor>real_of_int a / real_of_int b\<rfloor> < real_of_int a / real_of_int b" by arith
   qed
   then show ?thesis
     using \<open>\<not> b dvd a\<close> by simp
@@ -897,15 +882,14 @@
 proof
   show "2 ^ nat (bitlen x - 1) \<le> x"
   proof -
-    have "(2::real) ^ nat \<lfloor>log 2 (real x)\<rfloor> = 2 powr real (floor (log 2 (real x)))"
-      using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] \<open>x > 0\<close>
-      using real_nat_eq_real[of "floor (log 2 (real x))"]
+    have "(2::real) ^ nat \<lfloor>log 2 (real_of_int x)\<rfloor> = 2 powr real_of_int (floor (log 2 (real_of_int x)))"
+      using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real_of_int x)\<rfloor>"] \<open>x > 0\<close>
       by simp
-    also have "\<dots> \<le> 2 powr log 2 (real x)"
+    also have "\<dots> \<le> 2 powr log 2 (real_of_int x)"
       by simp
-    also have "\<dots> = real x"
+    also have "\<dots> = real_of_int x"
       using \<open>0 < x\<close> by simp
-    finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x"
+    finally have "2 ^ nat \<lfloor>log 2 (real_of_int x)\<rfloor> \<le> real_of_int x"
       by simp
     then show ?thesis
       using \<open>0 < x\<close> by (simp add: bitlen_def)
@@ -914,7 +898,7 @@
   proof -
     have "x \<le> 2 powr (log 2 x)"
       using \<open>x > 0\<close> by simp
-    also have "\<dots> < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)"
+    also have "\<dots> < 2 ^ nat (\<lfloor>log 2 (real_of_int x)\<rfloor> + 1)"
       apply (simp add: powr_realpow[symmetric])
       using \<open>x > 0\<close> apply simp
       done
@@ -930,8 +914,9 @@
   from assms have "b * 2 ^ c > 0"
     by auto
   then show ?thesis
-    using floor_add[of "log 2 b" c] assms
-    by (auto simp add: log_mult log_nat_power bitlen_def)
+    using floor_add[of "log 2 b" c] assms 
+    apply (auto simp add: log_mult log_nat_power bitlen_def)
+    by (metis add.right_neutral frac_lt_1 frac_of_int of_int_of_nat_eq)
 qed
 
 lemma bitlen_Float:
@@ -961,13 +946,13 @@
   { assume "2 \<le> x"
     then have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 (x - x mod 2)\<rfloor>"
       by (simp add: log_mult zmod_zdiv_equality')
-    also have "\<dots> = \<lfloor>log 2 (real x)\<rfloor>"
+    also have "\<dots> = \<lfloor>log 2 (real_of_int x)\<rfloor>"
     proof (cases "x mod 2 = 0")
       case True
       then show ?thesis by simp
     next
       case False
-      def n \<equiv> "\<lfloor>log 2 (real x)\<rfloor>"
+      def n \<equiv> "\<lfloor>log 2 (real_of_int x)\<rfloor>"
       then have "0 \<le> n"
         using \<open>2 \<le> x\<close> by simp
       from \<open>2 \<le> x\<close> False have "x mod 2 = 1" "\<not> 2 dvd x"
@@ -975,18 +960,18 @@
       with \<open>2 \<le> x\<close> have "x \<noteq> 2 ^ nat n"
         by (cases "nat n") auto
       moreover
-      { have "real (2^nat n :: int) = 2 powr (nat n)"
+      { have "real_of_int (2^nat n :: int) = 2 powr (nat n)"
           by (simp add: powr_realpow)
         also have "\<dots> \<le> 2 powr (log 2 x)"
           using \<open>2 \<le> x\<close> by (simp add: n_def del: powr_log_cancel)
         finally have "2^nat n \<le> x" using \<open>2 \<le> x\<close> by simp }
       ultimately have "2^nat n \<le> x - 1" by simp
-      then have "2^nat n \<le> real (x - 1)"
-        unfolding real_of_int_le_iff[symmetric] by simp
+      then have "2^nat n \<le> real_of_int (x - 1)"
+        using numeral_power_le_real_of_int_cancel_iff by blast
       { have "n = \<lfloor>log 2 (2^nat n)\<rfloor>"
           using \<open>0 \<le> n\<close> by (simp add: log_nat_power)
         also have "\<dots> \<le> \<lfloor>log 2 (x - 1)\<rfloor>"
-          using \<open>2^nat n \<le> real (x - 1)\<close> \<open>0 \<le> n\<close> \<open>2 \<le> x\<close> by (auto intro: floor_mono)
+          using \<open>2^nat n \<le> real_of_int (x - 1)\<close> \<open>0 \<le> n\<close> \<open>2 \<le> x\<close> by (auto intro: floor_mono)
         finally have "n \<le> \<lfloor>log 2 (x - 1)\<rfloor>" . }
       moreover have "\<lfloor>log 2 (x - 1)\<rfloor> \<le> n"
         using \<open>2 \<le> x\<close> by (auto simp add: n_def intro!: floor_mono)
@@ -997,7 +982,7 @@
   moreover
   { assume "x < 2" "0 < x"
     then have "x = 1" by simp
-    then have "\<lfloor>log 2 (real x)\<rfloor> = 0" by simp }
+    then have "\<lfloor>log 2 (real_of_int x)\<rfloor> = 0" by simp }
   ultimately show ?thesis
     unfolding bitlen_def
     by (auto simp: pos_imp_zdiv_pos_iff not_le)
@@ -1026,15 +1011,15 @@
     have "inverse (2 ^ nat (- e)) = 2 powr e"
       using assms False powr_realpow[of 2 "nat (-e)"]
       by (auto simp: powr_minus field_simps)
-    then have "1 \<le> real m * inverse ?S"
+    then have "1 \<le> real_of_int m * inverse ?S"
       using assms False powr_realpow[of 2 "nat (-e)"]
       by (auto simp: powr_minus)
-    then have "1 * ?S \<le> real m * inverse ?S * ?S"
+    then have "1 * ?S \<le> real_of_int m * inverse ?S * ?S"
       by (rule mult_right_mono) auto
-    then have "?S \<le> real m"
+    then have "?S \<le> real_of_int m"
       unfolding mult.assoc by auto
     then have "?S \<le> m"
-      unfolding real_of_int_le_iff[symmetric] by auto
+      unfolding of_int_le_iff[symmetric] by auto
     from this bitlen_bounds[OF \<open>0 < m\<close>, THEN conjunct2]
     have "nat (-e) < (nat (bitlen m))"
       unfolding power_strict_increasing_iff[OF \<open>1 < 2\<close>, symmetric]
@@ -1048,15 +1033,15 @@
 
 lemma bitlen_div:
   assumes "0 < m"
-  shows "1 \<le> real m / 2^nat (bitlen m - 1)"
-    and "real m / 2^nat (bitlen m - 1) < 2"
+  shows "1 \<le> real_of_int m / 2^nat (bitlen m - 1)"
+    and "real_of_int m / 2^nat (bitlen m - 1) < 2"
 proof -
   let ?B = "2^nat(bitlen m - 1)"
 
   have "?B \<le> m" using bitlen_bounds[OF \<open>0 <m\<close>] ..
-  then have "1 * ?B \<le> real m"
-    unfolding real_of_int_le_iff[symmetric] by auto
-  then show "1 \<le> real m / ?B"
+  then have "1 * ?B \<le> real_of_int m"
+    unfolding of_int_le_iff[symmetric] by auto
+  then show "1 \<le> real_of_int m / ?B"
     by auto
 
   have "m \<noteq> 0"
@@ -1070,11 +1055,11 @@
     using \<open>0 < m\<close> by (auto simp: bitlen_def)
   also have "\<dots> = ?B * 2"
     unfolding nat_add_distrib[OF \<open>0 \<le> bitlen m - 1\<close> zero_le_one] by auto
-  finally have "real m < 2 * ?B"
-    unfolding real_of_int_less_iff[symmetric] by auto
-  then have "real m / ?B < 2 * ?B / ?B"
+  finally have "real_of_int m < 2 * ?B"
+    by (metis (full_types) mult.commute power.simps(2) real_of_int_less_numeral_power_cancel_iff)
+  then have "real_of_int m / ?B < 2 * ?B / ?B"
     by (rule divide_strict_right_mono) auto
-  then show "real m / ?B < 2"
+  then show "real_of_int m / ?B < 2"
     by auto
 qed
 
@@ -1122,7 +1107,7 @@
   assumes "x > 0" "p > 0"
   shows "truncate_down p x > 0"
 proof -
-  have "0 \<le> log 2 x - real \<lfloor>log 2 x\<rfloor>"
+  have "0 \<le> log 2 x - real_of_int \<lfloor>log 2 x\<rfloor>"
     by (simp add: algebra_simps)
   from this assms
   show ?thesis
@@ -1167,7 +1152,7 @@
 lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_up
   by (simp add: truncate_up_def)
 
-lemma float_round_up: "real x \<le> real (float_round_up prec x)"
+lemma float_round_up: "real_of_float x \<le> real_of_float (float_round_up prec x)"
   using truncate_up by transfer simp
 
 lemma float_round_up_zero[simp]: "float_round_up prec 0 = 0"
@@ -1176,7 +1161,7 @@
 lift_definition float_round_down :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_down
   by (simp add: truncate_down_def)
 
-lemma float_round_down: "real (float_round_down prec x) \<le> real x"
+lemma float_round_down: "real_of_float (float_round_down prec x) \<le> real_of_float x"
   using truncate_down by transfer simp
 
 lemma float_round_down_zero[simp]: "float_round_down prec 0 = 0"
@@ -1216,8 +1201,8 @@
 
 lemma real_div_nat_eq_floor_of_divide:
   fixes a b :: nat
-  shows "a div b = real \<lfloor>a / b\<rfloor>"
-  by (simp add: floor_divide_of_nat_eq [of a b] real_eq_of_nat)
+  shows "a div b = real_of_int \<lfloor>a / b\<rfloor>"
+  by (simp add: floor_divide_of_nat_eq [of a b])
 
 definition "rat_precision prec x y = int prec - (bitlen x - bitlen y)"
 
@@ -1269,25 +1254,28 @@
     def x' \<equiv> "x * 2 ^ nat l"
     have "int x * 2 ^ nat l = x'"
       by (simp add: x'_def int_mult int_power)
-    moreover have "real x * 2 powr real l = real x'"
+    moreover have "real x * 2 powr l = real x'"
       by (simp add: powr_realpow[symmetric] \<open>0 \<le> l\<close> x'_def)
     ultimately show ?thesis
       using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] \<open>0 \<le> l\<close> \<open>y \<noteq> 0\<close>
         l_def[symmetric, THEN meta_eq_to_obj_eq]
-      by transfer (auto simp add: floor_divide_eq_div [symmetric] round_up_def)
+      apply transfer
+      apply (auto simp add: round_up_def)
+      by (metis floor_divide_of_int_eq of_int_of_nat_eq)
    next
     case False
     def y' \<equiv> "y * 2 ^ nat (- l)"
     from \<open>y \<noteq> 0\<close> have "y' \<noteq> 0" by (simp add: y'_def)
     have "int y * 2 ^ nat (- l) = y'" by (simp add: y'_def int_mult int_power)
-    moreover have "real x * real (2::int) powr real l / real y = x / real y'"
+    moreover have "real x * real_of_int (2::int) powr real_of_int l / real y = x / real y'"
       using \<open>\<not> 0 \<le> l\<close>
       by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps)
     ultimately show ?thesis
       using ceil_divide_floor_conv[of y' x] \<open>\<not> 0 \<le> l\<close> \<open>y' \<noteq> 0\<close> \<open>y \<noteq> 0\<close>
         l_def[symmetric, THEN meta_eq_to_obj_eq]
-      by transfer
-         (auto simp add: round_up_def ceil_divide_floor_conv floor_divide_eq_div [symmetric])
+      apply transfer
+      apply (auto simp add: round_up_def ceil_divide_floor_conv)
+      by (metis floor_divide_of_int_eq of_int_of_nat_eq)
   qed
 qed
 
@@ -1312,7 +1300,7 @@
 qed
 
 lemma rapprox_posrat_less1:
-  "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> 2 * x < y \<Longrightarrow> 0 < n \<Longrightarrow> real (rapprox_posrat n x y) < 1"
+  "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> 2 * x < y \<Longrightarrow> 0 < n \<Longrightarrow> real_of_float (rapprox_posrat n x y) < 1"
   by transfer (simp add: rat_precision_pos round_up_less1)
 
 lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
@@ -1366,12 +1354,12 @@
   "float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
 proof (cases "m1 \<noteq> 0 \<and> m2 \<noteq> 0")
   case True
-  let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
-  let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
+  let ?f1 = "real_of_int m1 * 2 powr real_of_int s1" and ?f2 = "real_of_int m2 * 2 powr real_of_int s2"
+  let ?m = "real_of_int m1 / real_of_int m2" and ?s = "2 powr real_of_int (s1 - s2)"
   from True have eq2: "(int prec + \<lfloor>log 2 \<bar>?f2\<bar>\<rfloor> - \<lfloor>log 2 \<bar>?f1\<bar>\<rfloor>) =
     rat_precision prec \<bar>m1\<bar> \<bar>m2\<bar> + (s2 - s1)"
     by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
-  have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
+  have eq1: "real_of_int m1 * 2 powr real_of_int s1 / (real_of_int m2 * 2 powr real_of_int s2) = ?m * ?s"
     by (simp add: field_simps powr_divide2[symmetric])
   from True show ?thesis
     by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift real_divl_def,
@@ -1494,10 +1482,10 @@
 lemma power_up_fl: "0 \<le> x \<Longrightarrow> x ^ n \<le> power_up_fl p x n"
   by transfer (rule power_up)
 
-lemma real_power_up_fl: "real (power_up_fl p x n) = power_up p x n"
+lemma real_power_up_fl: "real_of_float (power_up_fl p x n) = power_up p x n"
   by transfer simp
 
-lemma real_power_down_fl: "real (power_down_fl p x n) = power_down p x n"
+lemma real_power_down_fl: "real_of_float (power_down_fl p x n) = power_down p x n"
   by transfer simp
 
 
@@ -1521,8 +1509,8 @@
   and plus_up: "x + y \<le> plus_up prec x y"
   by (auto simp: plus_down_def truncate_down plus_up_def truncate_up)
 
-lemma float_plus_down: "real (float_plus_down prec x y) \<le> x + y"
-  and float_plus_up: "x + y \<le> real (float_plus_up prec x y)"
+lemma float_plus_down: "real_of_float (float_plus_down prec x y) \<le> x + y"
+  and float_plus_up: "x + y \<le> real_of_float (float_plus_up prec x y)"
   by (transfer, rule plus_down plus_up)+
 
 lemmas plus_down_le = order_trans[OF plus_down]
@@ -1551,7 +1539,7 @@
     and "abs a > k \<Longrightarrow> abs b \<le> k \<Longrightarrow> a + b \<noteq> 0"
   by auto
 
-lemma abs_real_le_2_powr_bitlen[simp]: "\<bar>real m2\<bar> < 2 powr real (bitlen \<bar>m2\<bar>)"
+lemma abs_real_le_2_powr_bitlen[simp]: "\<bar>real_of_int m2\<bar> < 2 powr real_of_int (bitlen \<bar>m2\<bar>)"
 proof (cases "m2 = 0")
   case True
   then show ?thesis by simp
@@ -1561,7 +1549,7 @@
     using bitlen_bounds[of "\<bar>m2\<bar>"]
     by (auto simp: powr_add bitlen_nonneg)
   then show ?thesis
-    by (simp add: powr_int bitlen_nonneg real_of_int_less_iff[symmetric])
+    by (metis bitlen_nonneg powr_int real_of_int_abs real_of_int_less_numeral_power_cancel_iff zero_less_numeral)
 qed
 
 lemma floor_sum_times_2_powr_sgn_eq:
@@ -1583,37 +1571,37 @@
     then have "b * 2 powr p < abs (b * 2 powr (p + 1))"
       by simp
     also note b_le_1
-    finally have b_less_1: "b * 2 powr real p < 1" .
+    finally have b_less_1: "b * 2 powr real_of_int p < 1" .
 
-    from b_less_1 \<open>b > 0\<close> have floor_eq: "\<lfloor>b * 2 powr real p\<rfloor> = 0" "\<lfloor>sgn b / 2\<rfloor> = 0"
+    from b_less_1 \<open>b > 0\<close> have floor_eq: "\<lfloor>b * 2 powr real_of_int p\<rfloor> = 0" "\<lfloor>sgn b / 2\<rfloor> = 0"
       by (simp_all add: floor_eq_iff)
 
     have "\<lfloor>(a + b) * 2 powr q\<rfloor> = \<lfloor>(a + b) * 2 powr p * 2 powr (q - p)\<rfloor>"
       by (simp add: algebra_simps powr_realpow[symmetric] powr_add[symmetric])
     also have "\<dots> = \<lfloor>(ai + b * 2 powr p) * 2 powr (q - p)\<rfloor>"
       by (simp add: assms algebra_simps)
-    also have "\<dots> = \<lfloor>(ai + b * 2 powr p) / real ((2::int) ^ nat (p - q))\<rfloor>"
+    also have "\<dots> = \<lfloor>(ai + b * 2 powr p) / real_of_int ((2::int) ^ nat (p - q))\<rfloor>"
       using assms
       by (simp add: algebra_simps powr_realpow[symmetric] divide_powr_uminus powr_add[symmetric])
-    also have "\<dots> = \<lfloor>ai / real ((2::int) ^ nat (p - q))\<rfloor>"
-      by (simp del: real_of_int_power add: floor_divide_real_eq_div floor_eq)
-    finally have "\<lfloor>(a + b) * 2 powr real q\<rfloor> = \<lfloor>real ai / real ((2::int) ^ nat (p - q))\<rfloor>" .
+    also have "\<dots> = \<lfloor>ai / real_of_int ((2::int) ^ nat (p - q))\<rfloor>"
+      by (simp del: of_int_power add: floor_divide_real_eq_div floor_eq)
+    finally have "\<lfloor>(a + b) * 2 powr real_of_int q\<rfloor> = \<lfloor>real_of_int ai / real_of_int ((2::int) ^ nat (p - q))\<rfloor>" .
     moreover
     {
-      have "\<lfloor>(2 * ai + sgn b) * 2 powr (real (q - p) - 1)\<rfloor> = \<lfloor>(ai + sgn b / 2) * 2 powr (q - p)\<rfloor>"
+      have "\<lfloor>(2 * ai + sgn b) * 2 powr (real_of_int (q - p) - 1)\<rfloor> = \<lfloor>(ai + sgn b / 2) * 2 powr (q - p)\<rfloor>"
         by (subst powr_divide2[symmetric]) (simp add: field_simps)
-      also have "\<dots> = \<lfloor>(ai + sgn b / 2) / real ((2::int) ^ nat (p - q))\<rfloor>"
+      also have "\<dots> = \<lfloor>(ai + sgn b / 2) / real_of_int ((2::int) ^ nat (p - q))\<rfloor>"
         using leqp by (simp add: powr_realpow[symmetric] powr_divide2[symmetric])
-      also have "\<dots> = \<lfloor>ai / real ((2::int) ^ nat (p - q))\<rfloor>"
-        by (simp del: real_of_int_power add: floor_divide_real_eq_div floor_eq)
+      also have "\<dots> = \<lfloor>ai / real_of_int ((2::int) ^ nat (p - q))\<rfloor>"
+        by (simp del: of_int_power add: floor_divide_real_eq_div floor_eq)
       finally
-      have "\<lfloor>(2 * ai + (sgn b)) * 2 powr (real (q - p) - 1)\<rfloor> =
-          \<lfloor>real ai / real ((2::int) ^ nat (p - q))\<rfloor>" .
+      have "\<lfloor>(2 * ai + (sgn b)) * 2 powr (real_of_int (q - p) - 1)\<rfloor> =
+          \<lfloor>real_of_int ai / real_of_int ((2::int) ^ nat (p - q))\<rfloor>" .
     }
     ultimately show ?thesis by simp
   next
     case 3
-    then have floor_eq: "\<lfloor>b * 2 powr (real p + 1)\<rfloor> = -1"
+    then have floor_eq: "\<lfloor>b * 2 powr (real_of_int p + 1)\<rfloor> = -1"
       using b_le_1
       by (auto simp: floor_eq_iff algebra_simps pos_divide_le_eq[symmetric] abs_if divide_powr_uminus
         intro!: mult_neg_pos split: split_if_asm)
@@ -1623,12 +1611,12 @@
       by (simp add: algebra_simps)
     also have "\<dots> = \<lfloor>(2 * ai + b * 2 powr (p + 1)) / 2 powr (1 - q + p)\<rfloor>"
       using assms by (simp add: algebra_simps powr_mult_base divide_powr_uminus)
-    also have "\<dots> = \<lfloor>(2 * ai + b * 2 powr (p + 1)) / real ((2::int) ^ nat (p - q + 1))\<rfloor>"
+    also have "\<dots> = \<lfloor>(2 * ai + b * 2 powr (p + 1)) / real_of_int ((2::int) ^ nat (p - q + 1))\<rfloor>"
       using assms by (simp add: algebra_simps powr_realpow[symmetric])
-    also have "\<dots> = \<lfloor>(2 * ai - 1) / real ((2::int) ^ nat (p - q + 1))\<rfloor>"
+    also have "\<dots> = \<lfloor>(2 * ai - 1) / real_of_int ((2::int) ^ nat (p - q + 1))\<rfloor>"
       using \<open>b < 0\<close> assms
-      by (simp add: floor_divide_eq_div floor_eq floor_divide_real_eq_div
-        del: real_of_int_mult real_of_int_power real_of_int_diff)
+      by (simp add: floor_divide_of_int_eq floor_eq floor_divide_real_eq_div
+        del: of_int_mult of_int_power of_int_diff)
     also have "\<dots> = \<lfloor>(2 * ai - 1) * 2 powr (q - p - 1)\<rfloor>"
       using assms by (simp add: algebra_simps divide_powr_uminus powr_realpow[symmetric])
     finally show ?thesis
@@ -1641,7 +1629,7 @@
     and b :: real
   assumes "abs b \<le> 1/2"
     and "ai \<noteq> 0"
-  shows "\<lfloor>log 2 \<bar>real ai + b\<bar>\<rfloor> = \<lfloor>log 2 \<bar>ai + sgn b / 2\<bar>\<rfloor>"
+  shows "\<lfloor>log 2 \<bar>real_of_int ai + b\<bar>\<rfloor> = \<lfloor>log 2 \<bar>ai + sgn b / 2\<bar>\<rfloor>"
 proof (cases "b = 0")
   case True
   then show ?thesis by simp
@@ -1660,9 +1648,10 @@
     by (auto simp: r_def k_def algebra_simps powr_add abs_if powr_int)
   then have "r \<le> (2::int) ^ nat k - 1"
     using \<open>k \<ge> 0\<close> by (auto simp: powr_int)
-  from this[simplified real_of_int_le_iff[symmetric]] \<open>0 \<le> k\<close>
+  from this[simplified of_int_le_iff[symmetric]] \<open>0 \<le> k\<close>
   have r_le: "r \<le> 2 powr k - 1"
-    by (auto simp: algebra_simps powr_int simp del: real_of_int_le_iff)
+    apply (auto simp: algebra_simps powr_int)
+     by (metis of_int_1 of_int_add real_of_int_le_numeral_power_cancel_iff)
 
   have "\<bar>ai\<bar> = 2 powr k + r"
     using \<open>k \<ge> 0\<close> by (auto simp: k_def r_def powr_realpow[symmetric])
@@ -1680,7 +1669,7 @@
     using \<open>k \<ge> 0\<close> r r_le
     by (auto simp: floor_log_eq_powr_iff powr_minus_divide field_simps sgn_if)
 
-  from \<open>real \<bar>ai\<bar> = _\<close> have "\<bar>ai + b\<bar> = 2 powr k + (r + sgn ai * b)"
+  from \<open>real_of_int \<bar>ai\<bar> = _\<close> have "\<bar>ai + b\<bar> = 2 powr k + (r + sgn ai * b)"
     using \<open>abs b <= _\<close> \<open>0 \<le> k\<close> r
     by (auto simp add: sgn_if abs_if)
   also have "\<lfloor>log 2 \<dots>\<rfloor> = \<lfloor>log 2 (2 powr k + r + sgn (sgn ai * b) / 2)\<rfloor>"
@@ -1708,7 +1697,7 @@
     finally show ?thesis .
   qed
   also have "2 powr k + r + sgn (sgn ai * b) / 2 = \<bar>ai + sgn b / 2\<bar>"
-    unfolding \<open>real \<bar>ai\<bar> = _\<close>[symmetric] using \<open>ai \<noteq> 0\<close>
+    unfolding \<open>real_of_int \<bar>ai\<bar> = _\<close>[symmetric] using \<open>ai \<noteq> 0\<close>
     by (auto simp: abs_if sgn_if algebra_simps)
   finally show ?thesis .
 qed
@@ -1724,10 +1713,10 @@
   shows "float_plus_down p (Float m1 e1) (Float m2 e2) =
     float_round_down p (Float (m1 * 2 ^ (Suc (Suc k1)) + sgn m2) (e1 - int k1 - 2))"
 proof -
-  let ?a = "real (Float m1 e1)"
-  let ?b = "real (Float m2 e2)"
+  let ?a = "real_of_float (Float m1 e1)"
+  let ?b = "real_of_float (Float m2 e2)"
   let ?sum = "?a + ?b"
-  let ?shift = "real e2 - real e1 + real k1 + 1"
+  let ?shift = "real_of_int e2 - real_of_int e1 + real k1 + 1"
   let ?m1 = "m1 * 2 ^ Suc k1"
   let ?m2 = "m2 * 2 powr ?shift"
   let ?m2' = "sgn m2 / 2"
@@ -1744,13 +1733,13 @@
   finally have abs_m2_less_half: "\<bar>?m2\<bar> < 1 / 2"
     by simp
 
-  then have "\<bar>real m2\<bar> < 2 powr -(?shift + 1)"
+  then have "\<bar>real_of_int m2\<bar> < 2 powr -(?shift + 1)"
     unfolding powr_minus_divide by (auto simp: bitlen_def field_simps powr_mult_base abs_mult)
-  also have "\<dots> \<le> 2 powr real (e1 - e2 - 2)"
+  also have "\<dots> \<le> 2 powr real_of_int (e1 - e2 - 2)"
     by simp
-  finally have b_less_quarter: "\<bar>?b\<bar> < 1/4 * 2 powr real e1"
+  finally have b_less_quarter: "\<bar>?b\<bar> < 1/4 * 2 powr real_of_int e1"
     by (simp add: powr_add field_simps powr_divide2[symmetric] powr_numeral abs_mult)
-  also have "1/4 < \<bar>real m1\<bar> / 2" using \<open>m1 \<noteq> 0\<close> by simp
+  also have "1/4 < \<bar>real_of_int m1\<bar> / 2" using \<open>m1 \<noteq> 0\<close> by simp
   finally have b_less_half_a: "\<bar>?b\<bar> < 1/2 * \<bar>?a\<bar>"
     by (simp add: algebra_simps powr_mult_base abs_mult)
   then have a_half_less_sum: "\<bar>?a\<bar> / 2 < \<bar>?sum\<bar>"
@@ -1759,7 +1748,7 @@
   from b_less_half_a have "\<bar>?b\<bar> < \<bar>?a\<bar>" "\<bar>?b\<bar> \<le> \<bar>?a\<bar>"
     by simp_all
 
-  have "\<bar>real (Float m1 e1)\<bar> \<ge> 1/4 * 2 powr real e1"
+  have "\<bar>real_of_float (Float m1 e1)\<bar> \<ge> 1/4 * 2 powr real_of_int e1"
     using \<open>m1 \<noteq> 0\<close>
     by (auto simp: powr_add powr_int bitlen_nonneg divide_right_mono abs_mult)
   then have "?sum \<noteq> 0" using b_less_quarter
@@ -1767,51 +1756,50 @@
   then have "?m1 + ?m2 \<noteq> 0"
     unfolding sum_eq by (simp add: abs_mult zero_less_mult_iff)
 
-  have "\<bar>real ?m1\<bar> \<ge> 2 ^ Suc k1" "\<bar>?m2'\<bar> < 2 ^ Suc k1"
+  have "\<bar>real_of_int ?m1\<bar> \<ge> 2 ^ Suc k1" "\<bar>?m2'\<bar> < 2 ^ Suc k1"
     using \<open>m1 \<noteq> 0\<close> \<open>m2 \<noteq> 0\<close> by (auto simp: sgn_if less_1_mult abs_mult simp del: power.simps)
   then have sum'_nz: "?m1 + ?m2' \<noteq> 0"
     by (intro sum_neq_zeroI)
 
-  have "\<lfloor>log 2 \<bar>real (Float m1 e1) + real (Float m2 e2)\<bar>\<rfloor> = \<lfloor>log 2 \<bar>?m1 + ?m2\<bar>\<rfloor> + ?e"
+  have "\<lfloor>log 2 \<bar>real_of_float (Float m1 e1) + real_of_float (Float m2 e2)\<bar>\<rfloor> = \<lfloor>log 2 \<bar>?m1 + ?m2\<bar>\<rfloor> + ?e"
     using \<open>?m1 + ?m2 \<noteq> 0\<close>
     unfolding floor_add[symmetric] sum_eq
-    by (simp add: abs_mult log_mult)
-  also have "\<lfloor>log 2 \<bar>?m1 + ?m2\<bar>\<rfloor> = \<lfloor>log 2 \<bar>?m1 + sgn (real m2 * 2 powr ?shift) / 2\<bar>\<rfloor>"
+    by (simp add: abs_mult log_mult) linarith
+  also have "\<lfloor>log 2 \<bar>?m1 + ?m2\<bar>\<rfloor> = \<lfloor>log 2 \<bar>?m1 + sgn (real_of_int m2 * 2 powr ?shift) / 2\<bar>\<rfloor>"
     using abs_m2_less_half \<open>m1 \<noteq> 0\<close>
     by (intro log2_abs_int_add_less_half_sgn_eq) (auto simp: abs_mult)
-  also have "sgn (real m2 * 2 powr ?shift) = sgn m2"
+  also have "sgn (real_of_int m2 * 2 powr ?shift) = sgn m2"
     by (auto simp: sgn_if zero_less_mult_iff less_not_sym)
   also
   have "\<bar>?m1 + ?m2'\<bar> * 2 powr ?e = \<bar>?m1 * 2 + sgn m2\<bar> * 2 powr (?e - 1)"
     by (auto simp: field_simps powr_minus[symmetric] powr_divide2[symmetric] powr_mult_base)
-  then have "\<lfloor>log 2 \<bar>?m1 + ?m2'\<bar>\<rfloor> + ?e = \<lfloor>log 2 \<bar>real (Float (?m1 * 2 + sgn m2) (?e - 1))\<bar>\<rfloor>"
+  then have "\<lfloor>log 2 \<bar>?m1 + ?m2'\<bar>\<rfloor> + ?e = \<lfloor>log 2 \<bar>real_of_float (Float (?m1 * 2 + sgn m2) (?e - 1))\<bar>\<rfloor>"
     using \<open>?m1 + ?m2' \<noteq> 0\<close>
-    unfolding floor_add[symmetric]
+    unfolding floor_add_of_int[symmetric]
     by (simp add: log_add_eq_powr abs_mult_pos)
   finally
-  have "\<lfloor>log 2 \<bar>?sum\<bar>\<rfloor> = \<lfloor>log 2 \<bar>real (Float (?m1*2 + sgn m2) (?e - 1))\<bar>\<rfloor>" .
+  have "\<lfloor>log 2 \<bar>?sum\<bar>\<rfloor> = \<lfloor>log 2 \<bar>real_of_float (Float (?m1*2 + sgn m2) (?e - 1))\<bar>\<rfloor>" .
   then have "plus_down p (Float m1 e1) (Float m2 e2) =
       truncate_down p (Float (?m1*2 + sgn m2) (?e - 1))"
     unfolding plus_down_def
   proof (rule truncate_down_log2_eqI)
-    let ?f = "(int p - \<lfloor>log 2 \<bar>real (Float m1 e1) + real (Float m2 e2)\<bar>\<rfloor> - 1)"
+    let ?f = "(int p - \<lfloor>log 2 \<bar>real_of_float (Float m1 e1) + real_of_float (Float m2 e2)\<bar>\<rfloor> - 1)"
     let ?ai = "m1 * 2 ^ (Suc k1)"
-    have "\<lfloor>(?a + ?b) * 2 powr real ?f\<rfloor> = \<lfloor>(real (2 * ?ai) + sgn ?b) * 2 powr real (?f - - ?e - 1)\<rfloor>"
+    have "\<lfloor>(?a + ?b) * 2 powr real_of_int ?f\<rfloor> = \<lfloor>(real_of_int (2 * ?ai) + sgn ?b) * 2 powr real_of_int (?f - - ?e - 1)\<rfloor>"
     proof (rule floor_sum_times_2_powr_sgn_eq)
-      show "?a * 2 powr real (-?e) = real ?ai"
+      show "?a * 2 powr real_of_int (-?e) = real_of_int ?ai"
         by (simp add: powr_add powr_realpow[symmetric] powr_divide2[symmetric])
-      show "\<bar>?b * 2 powr real (-?e + 1)\<bar> \<le> 1"
+      show "\<bar>?b * 2 powr real_of_int (-?e + 1)\<bar> \<le> 1"
         using abs_m2_less_half
         by (simp add: abs_mult powr_add[symmetric] algebra_simps powr_mult_base)
     next
-      have "e1 + \<lfloor>log 2 \<bar>real m1\<bar>\<rfloor> - 1 = \<lfloor>log 2 \<bar>?a\<bar>\<rfloor> - 1"
+      have "e1 + \<lfloor>log 2 \<bar>real_of_int m1\<bar>\<rfloor> - 1 = \<lfloor>log 2 \<bar>?a\<bar>\<rfloor> - 1"
         using \<open>m1 \<noteq> 0\<close>
         by (simp add: floor_add2[symmetric] algebra_simps log_mult abs_mult del: floor_add2)
       also have "\<dots> \<le> \<lfloor>log 2 \<bar>?a + ?b\<bar>\<rfloor>"
         using a_half_less_sum \<open>m1 \<noteq> 0\<close> \<open>?sum \<noteq> 0\<close>
-        unfolding floor_subtract[symmetric]
-        by (auto simp add: log_minus_eq_powr powr_minus_divide
-          intro!: floor_mono)
+        unfolding floor_diff_of_int[symmetric]
+        by (auto simp add: log_minus_eq_powr powr_minus_divide intro!: floor_mono)
       finally
       have "int p - \<lfloor>log 2 \<bar>?a + ?b\<bar>\<rfloor> \<le> p - (bitlen \<bar>m1\<bar>) - e1 + 2"
         by (auto simp: algebra_simps bitlen_def \<open>m1 \<noteq> 0\<close>)
@@ -1822,11 +1810,11 @@
     also have "sgn ?b = sgn m2"
       using powr_gt_zero[of 2 e2]
       by (auto simp add: sgn_if zero_less_mult_iff simp del: powr_gt_zero)
-    also have "\<lfloor>(real (2 * ?m1) + real (sgn m2)) * 2 powr real (?f - - ?e - 1)\<rfloor> =
+    also have "\<lfloor>(real_of_int (2 * ?m1) + real_of_int (sgn m2)) * 2 powr real_of_int (?f - - ?e - 1)\<rfloor> =
         \<lfloor>Float (?m1 * 2 + sgn m2) (?e - 1) * 2 powr ?f\<rfloor>"
       by (simp add: powr_add[symmetric] algebra_simps powr_realpow[symmetric])
     finally
-    show "\<lfloor>(?a + ?b) * 2 powr ?f\<rfloor> = \<lfloor>real (Float (?m1 * 2 + sgn m2) (?e - 1)) * 2 powr ?f\<rfloor>" .
+    show "\<lfloor>(?a + ?b) * 2 powr ?f\<rfloor> = \<lfloor>real_of_float (Float (?m1 * 2 + sgn m2) (?e - 1)) * 2 powr ?f\<rfloor>" .
   qed
   then show ?thesis
     by transfer (simp add: plus_down_def ac_simps Let_def)
@@ -1869,30 +1857,30 @@
 subsection \<open>Lemmas needed by Approximate\<close>
 
 lemma Float_num[simp]:
-   "real (Float 1 0) = 1"
-   "real (Float 1 1) = 2"
-   "real (Float 1 2) = 4"
-   "real (Float 1 (- 1)) = 1/2"
-   "real (Float 1 (- 2)) = 1/4"
-   "real (Float 1 (- 3)) = 1/8"
-   "real (Float (- 1) 0) = -1"
-   "real (Float (number_of n) 0) = number_of n"
+   "real_of_float (Float 1 0) = 1"
+   "real_of_float (Float 1 1) = 2"
+   "real_of_float (Float 1 2) = 4"
+   "real_of_float (Float 1 (- 1)) = 1/2"
+   "real_of_float (Float 1 (- 2)) = 1/4"
+   "real_of_float (Float 1 (- 3)) = 1/8"
+   "real_of_float (Float (- 1) 0) = -1"
+   "real_of_float (Float (number_of n) 0) = number_of n"
   using two_powr_int_float[of 2] two_powr_int_float[of "-1"] two_powr_int_float[of "-2"]
     two_powr_int_float[of "-3"]
   using powr_realpow[of 2 2] powr_realpow[of 2 3]
   using powr_minus[of 2 1] powr_minus[of 2 2] powr_minus[of 2 3]
   by auto
 
-lemma real_of_Float_int[simp]: "real (Float n 0) = real n"
+lemma real_of_Float_int[simp]: "real_of_float (Float n 0) = real n"
   by simp
 
-lemma float_zero[simp]: "real (Float 0 e) = 0"
+lemma float_zero[simp]: "real_of_float (Float 0 e) = 0"
   by simp
 
 lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
   by arith
 
-lemma lapprox_rat: "real (lapprox_rat prec x y) \<le> real x / real y"
+lemma lapprox_rat: "real_of_float (lapprox_rat prec x y) \<le> real_of_int x / real_of_int y"
   using round_down by (simp add: lapprox_rat_def)
 
 lemma mult_div_le:
@@ -1914,16 +1902,16 @@
 lemma lapprox_rat_nonneg:
   fixes n x y
   assumes "0 \<le> x" and "0 \<le> y"
-  shows "0 \<le> real (lapprox_rat n x y)"
+  shows "0 \<le> real_of_float (lapprox_rat n x y)"
   using assms by (auto simp: lapprox_rat_def simp: round_down_nonneg)
 
-lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
+lemma rapprox_rat: "real_of_int x / real_of_int y \<le> real_of_float (rapprox_rat prec x y)"
   using round_up by (simp add: rapprox_rat_def)
 
 lemma rapprox_rat_le1:
   fixes n x y
   assumes xy: "0 \<le> x" "0 < y" "x \<le> y"
-  shows "real (rapprox_rat n x y) \<le> 1"
+  shows "real_of_float (rapprox_rat n x y) \<le> 1"
 proof -
   have "bitlen \<bar>x\<bar> \<le> bitlen \<bar>y\<bar>"
     using xy unfolding bitlen_def by (auto intro!: floor_mono)
@@ -1931,10 +1919,10 @@
     by transfer (auto intro!: round_up_le1 simp: rat_precision_def)
 qed
 
-lemma rapprox_rat_nonneg_nonpos: "0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
+lemma rapprox_rat_nonneg_nonpos: "0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> real_of_float (rapprox_rat n x y) \<le> 0"
   by transfer (simp add: round_up_le0 divide_nonneg_nonpos)
 
-lemma rapprox_rat_nonpos_nonneg: "x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
+lemma rapprox_rat_nonpos_nonneg: "x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> real_of_float (rapprox_rat n x y) \<le> 0"
   by transfer (simp add: round_up_le0 divide_nonpos_nonneg)
 
 lemma real_divl: "real_divl prec x y \<le> x / y"
@@ -1943,7 +1931,7 @@
 lemma real_divr: "x / y \<le> real_divr prec x y"
   using round_up by (simp add: real_divr_def)
 
-lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
+lemma float_divl: "real_of_float (float_divl prec x y) \<le> x / y"
   by transfer (rule real_divl)
 
 lemma real_divl_lower_bound:
@@ -1951,7 +1939,7 @@
   by (simp add: real_divl_def round_down_nonneg)
 
 lemma float_divl_lower_bound:
-  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real (float_divl prec x y)"
+  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real_of_float (float_divl prec x y)"
   by transfer (rule real_divl_lower_bound)
 
 lemma exponent_1: "exponent 1 = 0"
@@ -1968,7 +1956,7 @@
 proof
   show ?rhs if ?lhs
   proof -
-    from that have z: "0 = real x"
+    from that have z: "0 = real_of_float x"
       using mantissa_exponent by simp
     show ?thesis
       by (simp add: zero_float_def z)
@@ -1999,17 +1987,17 @@
   assumes "0 < x" "x \<le> 1" "prec \<ge> 1"
   shows "1 \<le> real_divl prec 1 x"
 proof -
-  have "log 2 x \<le> real prec + real \<lfloor>log 2 x\<rfloor>"
+  have "log 2 x \<le> real_of_int prec + real_of_int \<lfloor>log 2 x\<rfloor>"
     using \<open>prec \<ge> 1\<close> by arith
   from this assms show ?thesis
     by (simp add: real_divl_def log_divide round_down_ge1)
 qed
 
 lemma float_divl_pos_less1_bound:
-  "0 < real x \<Longrightarrow> real x \<le> 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real (float_divl prec 1 x)"
+  "0 < real_of_float x \<Longrightarrow> real_of_float x \<le> 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real_of_float (float_divl prec 1 x)"
   by transfer (rule real_divl_pos_less1_bound)
 
-lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
+lemma float_divr: "real_of_float x / real_of_float y \<le> real_of_float (float_divr prec x y)"
   by transfer (rule real_divr)
 
 lemma real_divr_pos_less1_lower_bound:
@@ -2032,7 +2020,7 @@
   by (simp add: real_divr_def round_up_le0 divide_le_0_iff)
 
 lemma float_divr_nonpos_pos_upper_bound:
-  "real x \<le> 0 \<Longrightarrow> 0 \<le> real y \<Longrightarrow> real (float_divr prec x y) \<le> 0"
+  "real_of_float x \<le> 0 \<Longrightarrow> 0 \<le> real_of_float y \<Longrightarrow> real_of_float (float_divr prec x y) \<le> 0"
   by transfer (rule real_divr_nonpos_pos_upper_bound)
 
 lemma real_divr_nonneg_neg_upper_bound:
@@ -2040,7 +2028,7 @@
   by (simp add: real_divr_def round_up_le0 divide_le_0_iff)
 
 lemma float_divr_nonneg_neg_upper_bound:
-  "0 \<le> real x \<Longrightarrow> real y \<le> 0 \<Longrightarrow> real (float_divr prec x y) \<le> 0"
+  "0 \<le> real_of_float x \<Longrightarrow> real_of_float y \<le> 0 \<Longrightarrow> real_of_float (float_divr prec x y) \<le> 0"
   by transfer (rule real_divr_nonneg_neg_upper_bound)
 
 lemma truncate_up_nonneg_mono:
@@ -2063,27 +2051,27 @@
     have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
       by (metis floor_less_cancel linorder_cases not_le)+
     have "truncate_up prec x =
-      real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> * 2 powr - real (int prec - \<lfloor>log 2 x\<rfloor> - 1)"
+      real_of_int \<lceil>x * 2 powr real_of_int (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> * 2 powr - real_of_int (int prec - \<lfloor>log 2 x\<rfloor> - 1)"
       using assms by (simp add: truncate_up_def round_up_def)
-    also have "\<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> (2 ^ prec)"
-    proof (unfold ceiling_le_eq)
-      have "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> x * (2 powr real prec / (2 powr log 2 x))"
+    also have "\<lceil>x * 2 powr real_of_int (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> (2 ^ prec)"
+    proof (unfold ceiling_le_iff)
+      have "x * 2 powr real_of_int (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> x * (2 powr real prec / (2 powr log 2 x))"
         using real_of_int_floor_add_one_ge[of "log 2 x"] assms
         by (auto simp add: algebra_simps powr_divide2 intro!: mult_left_mono)
-      then show "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> real ((2::int) ^ prec)"
+      then show "x * 2 powr real_of_int (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> real_of_int ((2::int) ^ prec)"
         using \<open>0 < x\<close> by (simp add: powr_realpow)
     qed
-    then have "real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> 2 powr int prec"
+    then have "real_of_int \<lceil>x * 2 powr real_of_int (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> 2 powr int prec"
       by (auto simp: powr_realpow)
     also
-    have "2 powr - real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> 2 powr - real (int prec - \<lfloor>log 2 y\<rfloor>)"
+    have "2 powr - real_of_int (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> 2 powr - real_of_int (int prec - \<lfloor>log 2 y\<rfloor>)"
       using logless flogless by (auto intro!: floor_mono)
-    also have "2 powr real (int prec) \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>))"
+    also have "2 powr real_of_int (int prec) \<le> 2 powr (log 2 y + real_of_int (int prec - \<lfloor>log 2 y\<rfloor>))"
       using assms \<open>0 < x\<close>
       by (auto simp: algebra_simps)
-    finally have "truncate_up prec x \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>)) * 2 powr - real (int prec - \<lfloor>log 2 y\<rfloor>)"
+    finally have "truncate_up prec x \<le> 2 powr (log 2 y + real_of_int (int prec - \<lfloor>log 2 y\<rfloor>)) * 2 powr - real_of_int (int prec - \<lfloor>log 2 y\<rfloor>)"
       by simp
-    also have "\<dots> = 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>) - real (int prec - \<lfloor>log 2 y\<rfloor>))"
+    also have "\<dots> = 2 powr (log 2 y + real_of_int (int prec - \<lfloor>log 2 y\<rfloor>) - real_of_int (int prec - \<lfloor>log 2 y\<rfloor>))"
       by (subst powr_add[symmetric]) simp
     also have "\<dots> = y"
       using \<open>0 < x\<close> assms
@@ -2112,25 +2100,25 @@
   assumes "0 < x" "x \<le> y"
   shows "truncate_down 0 x \<le> truncate_down 0 y"
 proof -
-  have "x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1) = x * inverse (2 powr ((real \<lfloor>log 2 x\<rfloor> + 1)))"
+  have "x * 2 powr (- real_of_int \<lfloor>log 2 x\<rfloor> - 1) = x * inverse (2 powr ((real_of_int \<lfloor>log 2 x\<rfloor> + 1)))"
     by (simp add: powr_divide2[symmetric] powr_add powr_minus inverse_eq_divide)
-  also have "\<dots> = 2 powr (log 2 x - (real \<lfloor>log 2 x\<rfloor>) - 1)"
+  also have "\<dots> = 2 powr (log 2 x - (real_of_int \<lfloor>log 2 x\<rfloor>) - 1)"
     using \<open>0 < x\<close>
     by (auto simp: field_simps powr_add powr_divide2[symmetric])
   also have "\<dots> < 2 powr 0"
     using real_of_int_floor_add_one_gt
     unfolding neg_less_iff_less
     by (intro powr_less_mono) (auto simp: algebra_simps)
-  finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> < 1"
-    unfolding less_ceiling_eq real_of_int_minus real_of_one
+  finally have "\<lfloor>x * 2 powr (- real_of_int \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> < 1"
+    unfolding less_ceiling_iff of_int_minus of_int_1
     by simp
-  moreover have "0 \<le> \<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor>"
+  moreover have "0 \<le> \<lfloor>x * 2 powr (- real_of_int \<lfloor>log 2 x\<rfloor> - 1)\<rfloor>"
     using \<open>x > 0\<close> by auto
-  ultimately have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> \<in> {0 ..< 1}"
+  ultimately have "\<lfloor>x * 2 powr (- real_of_int \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> \<in> {0 ..< 1}"
     by simp
   also have "\<dots> \<subseteq> {0}"
     by auto
-  finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> = 0"
+  finally have "\<lfloor>x * 2 powr (- real_of_int \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> = 0"
     by simp
   with assms show ?thesis
     by (auto simp: truncate_down_def round_down_def)
@@ -2181,30 +2169,30 @@
       by auto
     have "2 powr (prec - 1) \<le> y * 2 powr real (prec - 1) / (2 powr log 2 y)"
       using \<open>0 < y\<close> by simp
-    also have "\<dots> \<le> y * 2 powr real prec / (2 powr (real \<lfloor>log 2 y\<rfloor> + 1))"
+    also have "\<dots> \<le> y * 2 powr real prec / (2 powr (real_of_int \<lfloor>log 2 y\<rfloor> + 1))"
       using \<open>0 \<le> y\<close> \<open>0 \<le> x\<close> assms(2)
       by (auto intro!: powr_mono divide_left_mono
-        simp: real_of_nat_diff powr_add
+        simp: of_nat_diff powr_add
         powr_divide2[symmetric])
-    also have "\<dots> = y * 2 powr real prec / (2 powr real \<lfloor>log 2 y\<rfloor> * 2)"
+    also have "\<dots> = y * 2 powr real prec / (2 powr real_of_int \<lfloor>log 2 y\<rfloor> * 2)"
       by (auto simp: powr_add)
-    finally have "(2 ^ (prec - 1)) \<le> \<lfloor>y * 2 powr real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)\<rfloor>"
+    finally have "(2 ^ (prec - 1)) \<le> \<lfloor>y * 2 powr real_of_int (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)\<rfloor>"
       using \<open>0 \<le> y\<close>
-      by (auto simp: powr_divide2[symmetric] le_floor_eq powr_realpow)
-    then have "(2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1) \<le> truncate_down prec y"
+      by (auto simp: powr_divide2[symmetric] le_floor_iff powr_realpow)
+    then have "(2 ^ (prec - 1)) * 2 powr - real_of_int (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1) \<le> truncate_down prec y"
       by (auto simp: truncate_down_def round_down_def)
     moreover
     {
       have "x = 2 powr (log 2 \<bar>x\<bar>)" using \<open>0 < x\<close> by simp
-      also have "\<dots> \<le> (2 ^ (prec )) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)"
+      also have "\<dots> \<le> (2 ^ (prec )) * 2 powr - real_of_int (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)"
         using real_of_int_floor_add_one_ge[of "log 2 \<bar>x\<bar>"]
         by (auto simp: powr_realpow[symmetric] powr_add[symmetric] algebra_simps)
       also
-      have "2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) \<le> 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor>)"
+      have "2 powr - real_of_int (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) \<le> 2 powr - real_of_int (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor>)"
         using logless flogless \<open>x > 0\<close> \<open>y > 0\<close>
         by (auto intro!: floor_mono)
-      finally have "x \<le> (2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)"
-        by (auto simp: powr_realpow[symmetric] powr_divide2[symmetric] assms real_of_nat_diff)
+      finally have "x \<le> (2 ^ (prec - 1)) * 2 powr - real_of_int (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)"
+        by (auto simp: powr_realpow[symmetric] powr_divide2[symmetric] assms of_nat_diff)
     }
     ultimately show ?thesis
       by (metis dual_order.trans truncate_down)
@@ -2231,12 +2219,12 @@
 
 lemma real_of_float_pprt[simp]:
   fixes a :: float
-  shows "real (pprt a) = pprt (real a)"
+  shows "real_of_float (pprt a) = pprt (real_of_float a)"
   unfolding pprt_def sup_float_def max_def sup_real_def by auto
 
 lemma real_of_float_nprt[simp]:
   fixes a :: float
-  shows "real (nprt a) = nprt (real a)"
+  shows "real_of_float (nprt a) = nprt (real_of_float a)"
   unfolding nprt_def inf_float_def min_def inf_real_def by auto
 
 context
@@ -2246,20 +2234,24 @@
 
 qualified lemma compute_int_floor_fl[code]:
   "int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div (2 ^ (nat (-e))))"
-  by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
+  apply transfer
+  apply (simp add: powr_int floor_divide_of_int_eq)
+  by (metis (no_types, hide_lams)floor_divide_of_int_eq of_int_numeral of_int_power floor_of_int of_int_mult)
 
-lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real (floor x)" by simp
+lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real_of_int (floor x)" by simp
 
 qualified lemma compute_floor_fl[code]:
   "floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div (2 ^ (nat (-e)))) 0)"
-  by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
+  apply transfer
+  apply (simp add: powr_int floor_divide_of_int_eq)
+  by (metis (no_types, hide_lams)floor_divide_of_int_eq of_int_numeral of_int_power of_int_mult)
 
 end
 
-lemma floor_fl: "real (floor_fl x) \<le> real x"
+lemma floor_fl: "real_of_float (floor_fl x) \<le> real_of_float x"
   by transfer simp
 
-lemma int_floor_fl: "real (int_floor_fl x) \<le> real x"
+lemma int_floor_fl: "real_of_int (int_floor_fl x) \<le> real_of_float x"
   by transfer simp
 
 lemma floor_pos_exp: "exponent (floor_fl x) \<ge> 0"
@@ -2269,9 +2261,9 @@
     by (simp add: floor_fl_def)
 next
   case False
-  have eq: "floor_fl x = Float \<lfloor>real x\<rfloor> 0"
+  have eq: "floor_fl x = Float \<lfloor>real_of_float x\<rfloor> 0"
     by transfer simp
-  obtain i where "\<lfloor>real x\<rfloor> = mantissa (floor_fl x) * 2 ^ i" "0 = exponent (floor_fl x) - int i"
+  obtain i where "\<lfloor>real_of_float x\<rfloor> = mantissa (floor_fl x) * 2 ^ i" "0 = exponent (floor_fl x) - int i"
     by (rule denormalize_shift[OF eq[THEN eq_reflection] False])
   then show ?thesis
     by simp
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Library/Formal_Power_Series.thy
--- a/src/HOL/Library/Formal_Power_Series.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Library/Formal_Power_Series.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -3143,8 +3143,8 @@
   "setsum (\<lambda>k. (a choose k) * (b choose (n - k))) {0..n} = (a + b) choose n"
   using gbinomial_Vandermonde[of "(of_nat a)" "of_nat b" n]
   apply (simp only: binomial_gbinomial[symmetric] of_nat_mult[symmetric]
-    of_nat_setsum[symmetric] of_nat_add[symmetric])
-  apply simp
+                    of_nat_setsum[symmetric] of_nat_add[symmetric])
+  apply blast
   done
 
 lemma binomial_Vandermonde_same: "setsum (\<lambda>k. (n choose k)\<^sup>2) {0..n} = (2 * n) choose n"
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Library/positivstellensatz.ML
--- a/src/HOL/Library/positivstellensatz.ML	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Library/positivstellensatz.ML	Tue Nov 10 14:18:41 2015 +0000
@@ -695,8 +695,8 @@
 
   fun is_alien ct =
     case Thm.term_of ct of
-      Const(@{const_name "real"}, _)$ n =>
-      if can HOLogic.dest_number n then false else true
+      Const(@{const_name "of_nat"}, _)$ n => not (can HOLogic.dest_number n)
+    | Const(@{const_name "of_int"}, _)$ n => not (can HOLogic.dest_number n)
     | _ => false
 in
 fun real_linear_prover translator (eq,le,lt) =
@@ -711,7 +711,7 @@
                 (eq_pols @ le_pols @ lt_pols) [])
     val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens
     val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
-    val le' = le @ map (fn a => Thm.instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens
+    val le' = le @ map (fn a => Thm.instantiate' [] [SOME (Thm.dest_arg a)] @{thm of_nat_0_le_iff}) aliens
   in ((translator (eq,le',lt) proof), Trivial)
   end
 end;
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Limits.thy
--- a/src/HOL/Limits.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Limits.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -175,7 +175,7 @@
   ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
     by (blast intro: order_trans)
   then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
-qed (force simp add: real_of_nat_Suc)
+qed (force simp add: of_nat_Suc)
 
 text\<open>alternative definition for Bseq\<close>
 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
@@ -188,7 +188,7 @@
 apply (subst lemma_NBseq_def, auto)
 apply (rule_tac x = "Suc N" in exI)
 apply (rule_tac [2] x = N in exI)
-apply (auto simp add: real_of_nat_Suc)
+apply (auto simp add: of_nat_Suc)
  prefer 2 apply (blast intro: order_less_imp_le)
 apply (drule_tac x = n in spec, simp)
 done
@@ -1433,7 +1433,7 @@
 proof (induct n)
   case (Suc n)
   have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
-    by (simp add: field_simps real_of_nat_Suc x)
+    by (simp add: field_simps of_nat_Suc x)
   also have "\<dots> \<le> (x + 1)^Suc n"
     using Suc x by (simp add: mult_left_mono)
   finally show ?case .
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/MacLaurin.thy
--- a/src/HOL/MacLaurin.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/MacLaurin.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -24,17 +24,17 @@
 lemma eq_diff_eq': "(x = y - z) = (y = x + (z::real))"
 by arith
 
-lemma fact_diff_Suc [rule_format]:
-  "n < Suc m ==> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
+lemma fact_diff_Suc:
+  "n < Suc m \<Longrightarrow> fact (Suc m - n) = (Suc m - n) * fact (m - n)"
   by (subst fact_reduce, auto)
 
 lemma Maclaurin_lemma2:
   fixes B
   assumes DERIV : "\<forall>m t. m < n \<and> 0\<le>t \<and> t\<le>h \<longrightarrow> DERIV (diff m) t :> diff (Suc m) t"
       and INIT : "n = Suc k"
-  defines "difg \<equiv> 
-      (\<lambda>m t::real. diff m t - 
-         ((\<Sum>p<n - m. diff (m + p) 0 / (fact p) * t ^ p) + B * (t ^ (n - m) / (fact (n - m)))))" 
+  defines "difg \<equiv>
+      (\<lambda>m t::real. diff m t -
+         ((\<Sum>p<n - m. diff (m + p) 0 / (fact p) * t ^ p) + B * (t ^ (n - m) / (fact (n - m)))))"
         (is "difg \<equiv> (\<lambda>m t. diff m t - ?difg m t)")
   shows "\<forall>m t. m < n & 0 \<le> t & t \<le> h --> DERIV (difg m) t :> difg (Suc m) t"
 proof (rule allI impI)+
@@ -42,10 +42,9 @@
   assume INIT2: "m < n & 0 \<le> t & t \<le> h"
   have "DERIV (difg m) t :> diff (Suc m) t -
     ((\<Sum>x<n - m. real x * t ^ (x - Suc 0) * diff (m + x) 0 / (fact x)) +
-     real (n - m) * t ^ (n - Suc m) * B / (fact (n - m)))" 
+     real (n - m) * t ^ (n - Suc m) * B / (fact (n - m)))"
     unfolding difg_def
-    by (auto intro!: derivative_eq_intros DERIV[rule_format, OF INIT2]
-             simp: real_of_nat_def[symmetric])
+    by (auto intro!: derivative_eq_intros DERIV[rule_format, OF INIT2])
   moreover
   from INIT2 have intvl: "{..<n - m} = insert 0 (Suc ` {..<n - Suc m})" and "0 < n - m"
     unfolding atLeast0LessThan[symmetric] by auto
@@ -54,7 +53,7 @@
     unfolding intvl atLeast0LessThan by (subst setsum.insert) (auto simp: setsum.reindex)
   moreover
   have fact_neq_0: "\<And>x. (fact x) + real x * (fact x) \<noteq> 0"
-    by (metis add_pos_pos fact_gt_zero less_add_same_cancel1 less_add_same_cancel2 less_numeral_extra(3) mult_less_0_iff not_real_of_nat_less_zero)
+    by (metis add_pos_pos fact_gt_zero less_add_same_cancel1 less_add_same_cancel2 less_numeral_extra(3) mult_less_0_iff of_nat_less_0_iff)
   have "\<And>x. (Suc x) * t ^ x * diff (Suc m + x) 0 / (fact (Suc x)) =
             diff (Suc m + x) 0 * t^x / (fact x)"
     by (rule nonzero_divide_eq_eq[THEN iffD2]) auto
@@ -64,7 +63,7 @@
     using \<open>0 < n - m\<close>
     by (simp add: divide_simps fact_reduce)
   ultimately show "DERIV (difg m) t :> difg (Suc m) t"
-    unfolding difg_def by simp
+    unfolding difg_def  by (simp add: mult.commute)
 qed
 
 lemma Maclaurin:
@@ -420,7 +419,7 @@
 lemma sin_expansion_lemma:
      "sin (x + real (Suc m) * pi / 2) =
       cos (x + real (m) * pi / 2)"
-by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib distrib_right, auto)
+by (simp only: cos_add sin_add of_nat_Suc add_divide_distrib distrib_right, auto)
 
 lemma Maclaurin_sin_expansion2:
      "\<exists>t. abs t \<le> abs x &
@@ -431,7 +430,7 @@
         and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
 apply safe
     apply (simp)
-   apply (simp add: sin_expansion_lemma del: real_of_nat_Suc)
+   apply (simp add: sin_expansion_lemma del: of_nat_Suc)
    apply (force intro!: derivative_eq_intros)
   apply (subst (asm) setsum.neutral, auto)[1]
  apply (rule ccontr, simp)
@@ -439,7 +438,7 @@
 apply (erule ssubst)
 apply (rule_tac x = t in exI, simp)
 apply (rule setsum.cong[OF refl])
-apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE simp del: real_of_nat_Suc)
+apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc)
 done
 
 lemma Maclaurin_sin_expansion:
@@ -459,12 +458,12 @@
 apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin_objl)
 apply safe
     apply simp
-   apply (simp (no_asm) add: sin_expansion_lemma del: real_of_nat_Suc)
+   apply (simp (no_asm) add: sin_expansion_lemma del: of_nat_Suc)
    apply (force intro!: derivative_eq_intros)
   apply (erule ssubst)
   apply (rule_tac x = t in exI, simp)
  apply (rule setsum.cong[OF refl])
- apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE simp del: real_of_nat_Suc)
+ apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc)
 done
 
 lemma Maclaurin_sin_expansion4:
@@ -476,12 +475,12 @@
 apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin2_objl)
 apply safe
     apply simp
-   apply (simp (no_asm) add: sin_expansion_lemma del: real_of_nat_Suc)
+   apply (simp (no_asm) add: sin_expansion_lemma del: of_nat_Suc)
    apply (force intro!: derivative_eq_intros)
   apply (erule ssubst)
   apply (rule_tac x = t in exI, simp)
  apply (rule setsum.cong[OF refl])
- apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE simp del: real_of_nat_Suc)
+ apply (auto simp add: sin_coeff_def sin_zero_iff elim: oddE simp del: of_nat_Suc)
 done
 
 
@@ -493,7 +492,7 @@
 
 lemma cos_expansion_lemma:
   "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
-by (simp only: cos_add sin_add real_of_nat_Suc distrib_right add_divide_distrib, auto)
+by (simp only: cos_add sin_add of_nat_Suc distrib_right add_divide_distrib, auto)
 
 lemma Maclaurin_cos_expansion:
      "\<exists>t::real. abs t \<le> abs x &
@@ -503,7 +502,7 @@
 apply (cut_tac f = cos and n = n and x = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_all_lt_objl)
 apply safe
     apply (simp (no_asm))
-   apply (simp (no_asm) add: cos_expansion_lemma del: real_of_nat_Suc)
+   apply (simp (no_asm) add: cos_expansion_lemma del: of_nat_Suc)
   apply (case_tac "n", simp)
   apply (simp del: setsum_lessThan_Suc)
 apply (rule ccontr, simp)
@@ -523,7 +522,7 @@
 apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_objl)
 apply safe
   apply simp
-  apply (simp (no_asm) add: cos_expansion_lemma del: real_of_nat_Suc)
+  apply (simp (no_asm) add: cos_expansion_lemma del: of_nat_Suc)
  apply (erule ssubst)
  apply (rule_tac x = t in exI, simp)
 apply (rule setsum.cong[OF refl])
@@ -539,7 +538,7 @@
 apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_minus_objl)
 apply safe
   apply simp
- apply (simp (no_asm) add: cos_expansion_lemma del: real_of_nat_Suc)
+ apply (simp (no_asm) add: cos_expansion_lemma del: of_nat_Suc)
 apply (erule ssubst)
 apply (rule_tac x = t in exI, simp)
 apply (rule setsum.cong[OF refl])
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Matrix_LP/ComputeFloat.thy
--- a/src/HOL/Matrix_LP/ComputeFloat.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Matrix_LP/ComputeFloat.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -10,22 +10,24 @@
 
 ML_file "~~/src/Tools/float.ML"
 
+(*FIXME surely floor should be used? This file is full of redundant material.*)
+
 definition int_of_real :: "real \<Rightarrow> int"
-  where "int_of_real x = (SOME y. real y = x)"
+  where "int_of_real x = (SOME y. real_of_int y = x)"
 
 definition real_is_int :: "real \<Rightarrow> bool"
-  where "real_is_int x = (EX (u::int). x = real u)"
+  where "real_is_int x = (EX (u::int). x = real_of_int u)"
 
-lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))"
+lemma real_is_int_def2: "real_is_int x = (x = real_of_int (int_of_real x))"
   by (auto simp add: real_is_int_def int_of_real_def)
 
-lemma real_is_int_real[simp]: "real_is_int (real (x::int))"
+lemma real_is_int_real[simp]: "real_is_int (real_of_int (x::int))"
 by (auto simp add: real_is_int_def int_of_real_def)
 
-lemma int_of_real_real[simp]: "int_of_real (real x) = x"
+lemma int_of_real_real[simp]: "int_of_real (real_of_int x) = x"
 by (simp add: int_of_real_def)
 
-lemma real_int_of_real[simp]: "real_is_int x \<Longrightarrow> real (int_of_real x) = x"
+lemma real_int_of_real[simp]: "real_is_int x \<Longrightarrow> real_of_int (int_of_real x) = x"
 by (auto simp add: int_of_real_def real_is_int_def)
 
 lemma real_is_int_add_int_of_real: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (a+b)) = (int_of_real a) + (int_of_real b)"
@@ -44,15 +46,15 @@
 apply (simp add: int_of_real_sub real_int_of_real)
 done
 
-lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real a = x"
+lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real_of_int a = x"
 by (auto simp add: real_is_int_def)
 
 lemma int_of_real_mult:
   assumes "real_is_int a" "real_is_int b"
   shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"
   using assms
-  by (auto simp add: real_is_int_def real_of_int_mult[symmetric]
-           simp del: real_of_int_mult)
+  by (auto simp add: real_is_int_def of_int_mult[symmetric]
+           simp del: of_int_mult)
 
 lemma real_is_int_mult[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a*b)"
 apply (subst real_is_int_def2)
@@ -64,14 +66,14 @@
 
 lemma real_is_int_1[simp]: "real_is_int (1::real)"
 proof -
-  have "real_is_int (1::real) = real_is_int(real (1::int))" by auto
+  have "real_is_int (1::real) = real_is_int(real_of_int (1::int))" by auto
   also have "\<dots> = True" by (simp only: real_is_int_real)
   ultimately show ?thesis by auto
 qed
 
 lemma real_is_int_n1: "real_is_int (-1::real)"
 proof -
-  have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
+  have "real_is_int (-1::real) = real_is_int(real_of_int (-1::int))" by auto
   also have "\<dots> = True" by (simp only: real_is_int_real)
   ultimately show ?thesis by auto
 qed
@@ -87,19 +89,16 @@
 
 lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)"
 proof -
-  have 1: "(1::real) = real (1::int)" by auto
+  have 1: "(1::real) = real_of_int (1::int)" by auto
   show ?thesis by (simp only: 1 int_of_real_real)
 qed
 
 lemma int_of_real_numeral[simp]: "int_of_real (numeral b) = numeral b"
-  unfolding int_of_real_def
-  by (intro some_equality)
-     (auto simp add: real_of_int_inject[symmetric] simp del: real_of_int_inject)
+  unfolding int_of_real_def by simp
 
 lemma int_of_real_neg_numeral[simp]: "int_of_real (- numeral b) = - numeral b"
   unfolding int_of_real_def
-  by (intro some_equality)
-     (auto simp add: real_of_int_inject[symmetric] simp del: real_of_int_inject)
+  by (metis int_of_real_def int_of_real_real of_int_minus of_int_of_nat_eq of_nat_numeral) 
 
 lemma int_div_zdiv: "int (a div b) = (int a) div (int b)"
 by (rule zdiv_int)
@@ -160,7 +159,7 @@
   zpower_numeral_odd zpower_Pls
 
 definition float :: "(int \<times> int) \<Rightarrow> real" where
-  "float = (\<lambda>(a, b). real a * 2 powr real b)"
+  "float = (\<lambda>(a, b). real_of_int a * 2 powr real_of_int b)"
 
 lemma float_add_l0: "float (0, e) + x = x"
   by (simp add: float_def)
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Matrix_LP/ComputeNumeral.thy
--- a/src/HOL/Matrix_LP/ComputeNumeral.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Matrix_LP/ComputeNumeral.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -14,7 +14,7 @@
 (* addition for bit strings *)
 lemmas bitadd = add_num_simps
 
-(* multiplication for bit strings *) 
+(* multiplication for bit strings *)
 lemmas bitmul = mult_num_simps
 
 lemmas bitarith = arith_simps
@@ -38,9 +38,9 @@
   arith_simps rel_simps number_norm
 
 lemmas compute_num_conversions =
-  real_of_nat_numeral real_of_nat_zero
+  of_nat_numeral of_nat_0
   nat_numeral nat_0 nat_neg_numeral
-  real_numeral real_of_int_zero
+  of_int_numeral of_int_neg_numeral of_int_0
 
 lemmas zpowerarith = zpower_numeral_even zpower_numeral_odd zpower_Pls int_pow_1
 
@@ -74,7 +74,7 @@
 
 lemmas compute_even = even_0_int even_One_int even_Bit0_int even_Bit1_int
 
-lemmas compute_numeral = compute_if compute_let compute_pair compute_bool 
+lemmas compute_numeral = compute_if compute_let compute_pair compute_bool
                          compute_natarith compute_numberarith max_def min_def compute_num_conversions zpowerarith compute_div_mod compute_even
 
 end
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy
--- a/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -48,7 +48,7 @@
 
 lemma swap_apply1: "Fun.swap x y f x = f y"
   by (simp add: Fun.swap_def)
-  
+
 lemma swap_apply2: "Fun.swap x y f y = f x"
   by (simp add: Fun.swap_def)
 
@@ -181,7 +181,7 @@
   shows "odd (card {s\<in>simplices. (rl ` s = {..Suc n})})"
 proof (rule kuhn_counting_lemma)
   have finite_s[simp]: "\<And>s. s \<in> simplices \<Longrightarrow> finite s"
-    by (metis add_is_0 zero_neq_numeral card_infinite assms(3)) 
+    by (metis add_is_0 zero_neq_numeral card_infinite assms(3))
 
   let ?F = "{f. \<exists>s\<in>simplices. face f s}"
   have F_eq: "?F = (\<Union>s\<in>simplices. \<Union>a\<in>s. {s - {a}})"
@@ -293,7 +293,7 @@
 proof -
   { fix x y :: nat assume "x \<noteq> y" "x \<le> n" "y \<le> n"
     with upd have "upd ` {..< x} \<noteq> upd ` {..< y}"
-      by (subst inj_on_image_eq_iff[where C="{..< n}"]) (auto simp: bij_betw_def) 
+      by (subst inj_on_image_eq_iff[where C="{..< n}"]) (auto simp: bij_betw_def)
     then have "enum x \<noteq> enum y"
       by (auto simp add: enum_def fun_eq_iff) }
   then show ?thesis
@@ -446,7 +446,7 @@
     using step \<open>j + d \<le> n\<close> eq by (auto simp: s.enum_inj t.enum_inj)
   ultimately have "s.enum (Suc l) = t.enum (Suc (l + d))"
     using \<open>j + d \<le> n\<close>
-    by (intro antisym s.enum_less[THEN iffD1] t.enum_less[THEN iffD1]) 
+    by (intro antisym s.enum_less[THEN iffD1] t.enum_less[THEN iffD1])
        (auto intro!: s.enum_in t.enum_in)
   then show ?case by simp
 qed
@@ -466,7 +466,7 @@
   moreover have e0: "a = s.enum 0" "b = t.enum 0"
     using a b by (simp_all add: s.enum_0_bot t.enum_0_bot)
   moreover
-  { fix j assume "0 < j" "j \<le> n" 
+  { fix j assume "0 < j" "j \<le> n"
     moreover have "s - {a} = s.enum ` {Suc 0 .. n}" "t - {b} = t.enum ` {Suc 0 .. n}"
       unfolding s.s_eq t.s_eq e0 by (auto simp: s.enum_inj t.enum_inj)
     ultimately have "s.enum j = t.enum j"
@@ -505,7 +505,7 @@
   moreover have en: "a = s.enum n" "b = t.enum n"
     using a b by (simp_all add: s.enum_n_top t.enum_n_top)
   moreover
-  { fix j assume "j < n" 
+  { fix j assume "j < n"
     moreover have "s - {a} = s.enum ` {0 .. n'}" "t - {b} = t.enum ` {0 .. n'}"
       unfolding s.s_eq t.s_eq en by (auto simp: s.enum_inj t.enum_inj Suc)
     ultimately have "s.enum j = t.enum j"
@@ -665,7 +665,7 @@
 proof cases
   case (ksimplex b_s u_s)
 
-  { fix t b assume "ksimplex p n t" 
+  { fix t b assume "ksimplex p n t"
     then obtain b_t u_t where "kuhn_simplex p n b_t u_t t"
       by (auto elim: ksimplex.cases)
     interpret kuhn_simplex_pair p n b_s u_s s b_t u_t t
@@ -688,7 +688,7 @@
 proof cases
   case (ksimplex b_s u_s)
 
-  { fix t b assume "ksimplex p n t" 
+  { fix t b assume "ksimplex p n t"
     then obtain b_t u_t where "kuhn_simplex p n b_t u_t t"
       by (auto elim: ksimplex.cases)
     interpret kuhn_simplex_pair p n b_s u_s s b_t u_t t
@@ -858,7 +858,7 @@
     have ks_f': "ksimplex p n (b.enum ` {.. n})"
       unfolding f' by rule unfold_locales
 
-    have "0 < n" 
+    have "0 < n"
       using \<open>n \<noteq> 0\<close> by auto
 
     { from \<open>a = enum i\<close> \<open>n \<noteq> 0\<close> \<open>i = n\<close> lb upd_space[of n']
@@ -1040,7 +1040,7 @@
         using i by (simp add: k i'_def)
       also have "\<dots> = (enum i') (u l := Suc (enum i' (u l)), u (Suc l) := Suc (enum i' (u (Suc l))))"
         using \<open>Suc l < k\<close> \<open>k \<le> n\<close> by (simp add: t.enum_Suc l t.upd_inj)
-      finally have "(u l = upd i' \<and> u (Suc l) = upd (Suc i')) \<or> 
+      finally have "(u l = upd i' \<and> u (Suc l) = upd (Suc i')) \<or>
         (u l = upd (Suc i') \<and> u (Suc l) = upd i')"
         using \<open>Suc i' < n\<close> by (auto simp: enum_Suc fun_eq_iff split: split_if_asm)
 
@@ -1187,7 +1187,7 @@
       then have "reduced (Suc n) (lab x) \<noteq> n"
         using \<open>j \<noteq> n\<close> \<open>j \<le> n\<close> by simp }
     moreover
-    have "n \<in> (reduced (Suc n) \<circ> lab) ` f" 
+    have "n \<in> (reduced (Suc n) \<circ> lab) ` f"
       using eq by auto
     ultimately show False
       by force
@@ -1657,7 +1657,7 @@
   have q2: "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = 0 \<longrightarrow>
       (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0)"
     unfolding o_def using cube \<open>p > 0\<close> by (intro allI impI label(2)) (auto simp add: b'')
-  have q3: "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = p \<longrightarrow> 
+  have q3: "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = p \<longrightarrow>
       (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 1)"
     using cube \<open>p > 0\<close> unfolding o_def by (intro allI impI label(3)) (auto simp add: b'')
   obtain q where q:
@@ -1694,7 +1694,7 @@
       done
     also have "\<dots> = d"
       using DIM_positive[where 'a='a]
-      by (auto simp: real_eq_of_nat n_def)
+      by (auto simp: n_def)
     finally show False
       using d_fz_z by auto
   qed
@@ -1742,11 +1742,11 @@
     have "(\<Sum>i\<in>Basis. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)"
       apply (rule setsum_mono)
       using rs(1)[OF b'_im]
-      apply (auto simp add:* field_simps simp del: real_of_nat_Suc)
+      apply (auto simp add:* field_simps simp del: of_nat_Suc)
       done
     also have "\<dots> < e * real p"
       using p \<open>e > 0\<close> \<open>p > 0\<close>
-      by (auto simp add: field_simps n_def real_of_nat_def)
+      by (auto simp add: field_simps n_def)
     finally have "(\<Sum>i\<in>Basis. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) < e * real p" .
   }
   moreover
@@ -1754,11 +1754,11 @@
     have "(\<Sum>i\<in>Basis. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)"
       apply (rule setsum_mono)
       using rs(2)[OF b'_im]
-      apply (auto simp add:* field_simps simp del: real_of_nat_Suc)
+      apply (auto simp add:* field_simps simp del: of_nat_Suc)
       done
     also have "\<dots> < e * real p"
       using p \<open>e > 0\<close> \<open>p > 0\<close>
-      by (auto simp add: field_simps n_def real_of_nat_def)
+      by (auto simp add: field_simps n_def)
     finally have "(\<Sum>i\<in>Basis. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) < e * real p" .
   }
   ultimately
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy
--- a/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -2986,17 +2986,17 @@
           } note ptgh_ee = this
           have pi_hgn: "path_image (linepath (h (n/N)) (g (n/N))) \<subseteq> ball (p t) (ee (p t))"
             using ptgh_ee [of "n/N"] Suc.prems
-            by (auto simp: field_simps real_of_nat_def dist_norm dest: segment_furthest_le [where y="p t"])
+            by (auto simp: field_simps dist_norm dest: segment_furthest_le [where y="p t"])
           then have gh_ns: "closed_segment (g (n/N)) (h (n/N)) \<subseteq> s"
             using \<open>N>0\<close> Suc.prems
-            apply (simp add: real_of_nat_def path_image_join field_simps closed_segment_commute)
+            apply (simp add: path_image_join field_simps closed_segment_commute)
             apply (erule order_trans)
             apply (simp add: ee pi t)
             done
           have pi_ghn': "path_image (linepath (g ((1 + n) / N)) (h ((1 + n) / N)))
                   \<subseteq> ball (p t) (ee (p t))"
             using ptgh_ee [of "(1+n)/N"] Suc.prems
-            by (auto simp: field_simps real_of_nat_def dist_norm dest: segment_furthest_le [where y="p t"])
+            by (auto simp: field_simps dist_norm dest: segment_furthest_le [where y="p t"])
           then have gh_n's: "closed_segment (g ((1 + n) / N)) (h ((1 + n) / N)) \<subseteq> s"
             using \<open>N>0\<close> Suc.prems ee pi t
             by (auto simp: Path_Connected.path_image_join field_simps)
@@ -3030,7 +3030,7 @@
                      path_integral (linepath (h (n/N)) (g (n/N))) f = 0"
             using path_integral_unique [OF pi0] Suc.prems
             by (simp add: g h fpa valid_path_subpath path_integrable_subpath
-                          fpa1 fpa2 fpa3 algebra_simps del: real_of_nat_Suc)
+                          fpa1 fpa2 fpa3 algebra_simps del: of_nat_Suc)
           have *: "\<And>hn he hn' gn gd gn' hgn ghn gh0 ghn'.
                     \<lbrakk>hn - gn = ghn - gh0;
                      gd + ghn' + he + hgn = (0::complex);
@@ -4368,10 +4368,10 @@
 
 subsection{* Cauchy's integral formula, again for a convex enclosing set.*}
 
-lemma Cauchy_integral_formula_weak: 
-    assumes s: "convex s" and "finite k" and conf: "continuous_on s f" 
-        and fcd: "(\<And>x. x \<in> interior s - k \<Longrightarrow> f complex_differentiable at x)" 
-        and z: "z \<in> interior s - k" and vpg: "valid_path \<gamma>" 
+lemma Cauchy_integral_formula_weak:
+    assumes s: "convex s" and "finite k" and conf: "continuous_on s f"
+        and fcd: "(\<And>x. x \<in> interior s - k \<Longrightarrow> f complex_differentiable at x)"
+        and z: "z \<in> interior s - k" and vpg: "valid_path \<gamma>"
         and pasz: "path_image \<gamma> \<subseteq> s - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
       shows "((\<lambda>w. f w / (w - z)) has_path_integral (2*pi * ii * winding_number \<gamma> z * f z)) \<gamma>"
 proof -
@@ -4383,7 +4383,7 @@
     case True then show ?thesis
       apply (simp add: continuous_within)
       apply (rule Lim_transform_away_within [of _ "z+1" _ "\<lambda>w::complex. (f w - f z)/(w - z)"])
-      using has_field_derivative_at_within DERIV_within_iff f' 
+      using has_field_derivative_at_within DERIV_within_iff f'
       apply (fastforce simp add:)+
       done
   next
@@ -4395,7 +4395,7 @@
       apply (rule continuous_transform_within [OF dxz that, of "\<lambda>w::complex. (f w - f z)/(w - z)"])
       apply (force simp: dist_commute)
       apply (rule cf continuous_intros)+
-      using False by auto      
+      using False by auto
   qed
   have fink': "finite (insert z k)" using \<open>finite k\<close> by blast
   have *: "((\<lambda>w. if w = z then f' else (f w - f z) / (w - z)) has_path_integral 0) \<gamma>"
@@ -4412,9 +4412,9 @@
     using znotin has_path_integral_add [OF has_path_integral_lmul [OF has_path_integral_winding_number [OF vpg znotin], of "f z"] *]
     apply (auto simp: mult_ac divide_simps)
     done
-qed    
-
-theorem Cauchy_integral_formula_convex_simple: 
+qed
+
+theorem Cauchy_integral_formula_convex_simple:
     "\<lbrakk>convex s; f holomorphic_on s; z \<in> interior s; valid_path \<gamma>; path_image \<gamma> \<subseteq> s - {z};
       pathfinish \<gamma> = pathstart \<gamma>\<rbrakk>
      \<Longrightarrow> ((\<lambda>w. f w / (w - z)) has_path_integral (2*pi * ii * winding_number \<gamma> z * f z)) \<gamma>"
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy
--- a/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -19,7 +19,7 @@
   shows "((op * c) has_derivative (op * c)) F"
 by (rule has_derivative_mult_right [OF has_derivative_id])
 
-lemma has_derivative_of_real[derivative_intros, simp]: 
+lemma has_derivative_of_real[derivative_intros, simp]:
   "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. of_real (f x)) has_derivative (\<lambda>x. of_real (f' x))) F"
   using bounded_linear.has_derivative[OF bounded_linear_of_real] .
 
@@ -45,39 +45,39 @@
   using bounded_linear.linear[OF bounded_linear_cnj] .
 
 lemma tendsto_mult_left:
-  fixes c::"'a::real_normed_algebra" 
+  fixes c::"'a::real_normed_algebra"
   shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) ---> c * l) F"
 by (rule tendsto_mult [OF tendsto_const])
 
 lemma tendsto_mult_right:
-  fixes c::"'a::real_normed_algebra" 
+  fixes c::"'a::real_normed_algebra"
   shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) ---> l * c) F"
 by (rule tendsto_mult [OF _ tendsto_const])
 
 lemma tendsto_Re_upper:
-  assumes "~ (trivial_limit F)" 
-          "(f ---> l) F" 
+  assumes "~ (trivial_limit F)"
+          "(f ---> l) F"
           "eventually (\<lambda>x. Re(f x) \<le> b) F"
     shows  "Re(l) \<le> b"
   by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Re)
 
 lemma tendsto_Re_lower:
-  assumes "~ (trivial_limit F)" 
-          "(f ---> l) F" 
+  assumes "~ (trivial_limit F)"
+          "(f ---> l) F"
           "eventually (\<lambda>x. b \<le> Re(f x)) F"
     shows  "b \<le> Re(l)"
   by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Re)
 
 lemma tendsto_Im_upper:
-  assumes "~ (trivial_limit F)" 
-          "(f ---> l) F" 
+  assumes "~ (trivial_limit F)"
+          "(f ---> l) F"
           "eventually (\<lambda>x. Im(f x) \<le> b) F"
     shows  "Im(l) \<le> b"
   by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Im)
 
 lemma tendsto_Im_lower:
-  assumes "~ (trivial_limit F)" 
-          "(f ---> l) F" 
+  assumes "~ (trivial_limit F)"
+          "(f ---> l) F"
           "eventually (\<lambda>x. b \<le> Im(f x)) F"
     shows  "b \<le> Im(l)"
   by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Im)
@@ -89,29 +89,29 @@
   by auto
 
 lemma continuous_mult_left:
-  fixes c::"'a::real_normed_algebra" 
+  fixes c::"'a::real_normed_algebra"
   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. c * f x)"
 by (rule continuous_mult [OF continuous_const])
 
 lemma continuous_mult_right:
-  fixes c::"'a::real_normed_algebra" 
+  fixes c::"'a::real_normed_algebra"
   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x * c)"
 by (rule continuous_mult [OF _ continuous_const])
 
 lemma continuous_on_mult_left:
-  fixes c::"'a::real_normed_algebra" 
+  fixes c::"'a::real_normed_algebra"
   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c * f x)"
 by (rule continuous_on_mult [OF continuous_on_const])
 
 lemma continuous_on_mult_right:
-  fixes c::"'a::real_normed_algebra" 
+  fixes c::"'a::real_normed_algebra"
   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)"
 by (rule continuous_on_mult [OF _ continuous_on_const])
 
 lemma uniformly_continuous_on_cmul_right [continuous_intros]:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f x * c)"
-  using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] . 
+  using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] .
 
 lemma uniformly_continuous_on_cmul_left[continuous_intros]:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
@@ -141,7 +141,7 @@
 lemma DERIV_zero_constant:
   fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
   shows    "\<lbrakk>convex s;
-             \<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)\<rbrakk> 
+             \<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)\<rbrakk>
              \<Longrightarrow> \<exists>c. \<forall>x \<in> s. f(x) = c"
   by (auto simp: has_field_derivative_def lambda_zero intro: has_derivative_zero_constant)
 
@@ -162,7 +162,7 @@
       and d0: "\<And>x. x\<in>s \<Longrightarrow> DERIV f x :> 0"
       and "a \<in> s"
       and "x \<in> s"
-    shows "f x = f a" 
+    shows "f x = f a"
     by (rule has_derivative_zero_unique_connected [OF assms(2,1) _ assms(5,4)])
        (metis has_field_derivative_def lambda_zero d0)
 
@@ -192,7 +192,7 @@
 (*generalising DERIV_isconst_all, which requires type real (using the ordering)*)
 lemma DERIV_zero_UNIV_unique:
   fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
-  shows "(\<And>x. DERIV f x :> 0) \<Longrightarrow> f x = f a" 
+  shows "(\<And>x. DERIV f x :> 0) \<Longrightarrow> f x = f a"
 by (metis DERIV_zero_unique UNIV_I assms convex_UNIV)
 
 subsection \<open>Some limit theorems about real part of real series etc.\<close>
@@ -263,7 +263,7 @@
   shows "(f ---> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
 by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)
 
-lemma real_series: 
+lemma real_series:
   fixes l::complex
   shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
 unfolding sums_def
@@ -276,7 +276,7 @@
 subsection\<open>Holomorphic functions\<close>
 
 definition complex_differentiable :: "[complex \<Rightarrow> complex, complex filter] \<Rightarrow> bool"
-           (infixr "(complex'_differentiable)" 50)  
+           (infixr "(complex'_differentiable)" 50)
   where "f complex_differentiable F \<equiv> \<exists>f'. (f has_field_derivative f') F"
 
 lemma complex_differentiable_imp_continuous_at:
@@ -304,7 +304,7 @@
 lemma complex_differentiable_const [derivative_intros]: "(\<lambda>z. c) complex_differentiable F"
   unfolding complex_differentiable_def has_field_derivative_def
   by (rule exI [where x=0])
-     (metis has_derivative_const lambda_zero) 
+     (metis has_derivative_const lambda_zero)
 
 lemma complex_differentiable_ident [derivative_intros]: "(\<lambda>z. z) complex_differentiable F"
   unfolding complex_differentiable_def has_field_derivative_def
@@ -343,14 +343,14 @@
   by (metis DERIV_inverse_fun)
 
 lemma complex_differentiable_mult [derivative_intros]:
-  assumes "f complex_differentiable (at a within s)" 
+  assumes "f complex_differentiable (at a within s)"
           "g complex_differentiable (at a within s)"
     shows "(\<lambda>z. f z * g z) complex_differentiable (at a within s)"
   using assms unfolding complex_differentiable_def
   by (metis DERIV_mult [of f _ a s g])
-  
+
 lemma complex_differentiable_divide [derivative_intros]:
-  assumes "f complex_differentiable (at a within s)" 
+  assumes "f complex_differentiable (at a within s)"
           "g complex_differentiable (at a within s)"
           "g a \<noteq> 0"
     shows "(\<lambda>z. f z / g z) complex_differentiable (at a within s)"
@@ -358,7 +358,7 @@
   by (metis DERIV_divide [of f _ a s g])
 
 lemma complex_differentiable_power [derivative_intros]:
-  assumes "f complex_differentiable (at a within s)" 
+  assumes "f complex_differentiable (at a within s)"
     shows "(\<lambda>z. f z ^ n) complex_differentiable (at a within s)"
   using assms unfolding complex_differentiable_def
   by (metis DERIV_power)
@@ -373,7 +373,7 @@
   by (blast intro: has_derivative_transform_within)
 
 lemma complex_differentiable_compose_within:
-  assumes "f complex_differentiable (at a within s)" 
+  assumes "f complex_differentiable (at a within s)"
           "g complex_differentiable (at (f a) within f`s)"
     shows "(g o f) complex_differentiable (at a within s)"
   using assms unfolding complex_differentiable_def
@@ -385,7 +385,7 @@
 by (metis complex_differentiable_at_within complex_differentiable_compose_within)
 
 lemma complex_differentiable_within_open:
-     "\<lbrakk>a \<in> s; open s\<rbrakk> \<Longrightarrow> f complex_differentiable at a within s \<longleftrightarrow> 
+     "\<lbrakk>a \<in> s; open s\<rbrakk> \<Longrightarrow> f complex_differentiable at a within s \<longleftrightarrow>
                           f complex_differentiable at a"
   unfolding complex_differentiable_def
   by (metis at_within_open)
@@ -409,7 +409,7 @@
 definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
            (infixl "(holomorphic'_on)" 50)
   where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f complex_differentiable (at x within s)"
- 
+
 named_theorems holomorphic_intros "structural introduction rules for holomorphic_on"
 
 lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}"
@@ -419,9 +419,9 @@
     "open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
   by (auto simp: holomorphic_on_def complex_differentiable_def has_field_derivative_def at_within_open [of _ s])
 
-lemma holomorphic_on_imp_continuous_on: 
+lemma holomorphic_on_imp_continuous_on:
     "f holomorphic_on s \<Longrightarrow> continuous_on s f"
-  by (metis complex_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def) 
+  by (metis complex_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def)
 
 lemma holomorphic_on_subset:
     "f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
@@ -505,19 +505,19 @@
   by (metis DERIV_imp_deriv DERIV_const)
 
 lemma complex_derivative_add:
-  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>  
+  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>
    \<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
   unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
   by (auto intro!: DERIV_imp_deriv derivative_intros)
 
 lemma complex_derivative_diff:
-  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>  
+  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>
    \<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
   unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
   by (auto intro!: DERIV_imp_deriv derivative_intros)
 
 lemma complex_derivative_mult:
-  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>  
+  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>
    \<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
   unfolding DERIV_deriv_iff_complex_differentiable[symmetric]
   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
@@ -533,7 +533,7 @@
   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
 
 lemma complex_derivative_transform_within_open:
-  "\<lbrakk>f holomorphic_on s; g holomorphic_on s; open s; z \<in> s; \<And>w. w \<in> s \<Longrightarrow> f w = g w\<rbrakk> 
+  "\<lbrakk>f holomorphic_on s; g holomorphic_on s; open s; z \<in> s; \<And>w. w \<in> s \<Longrightarrow> f w = g w\<rbrakk>
    \<Longrightarrow> deriv f z = deriv g z"
   unfolding holomorphic_on_def
   by (rule DERIV_imp_deriv)
@@ -549,7 +549,7 @@
 
 subsection\<open>Analyticity on a set\<close>
 
-definition analytic_on (infixl "(analytic'_on)" 50)  
+definition analytic_on (infixl "(analytic'_on)" 50)
   where
    "f analytic_on s \<equiv> \<forall>x \<in> s. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"
 
@@ -578,7 +578,7 @@
 
 lemma analytic_on_UN: "f analytic_on (\<Union>i\<in>I. s i) \<longleftrightarrow> (\<forall>i\<in>I. f analytic_on (s i))"
   by (auto simp: analytic_on_def)
-  
+
 lemma analytic_on_holomorphic:
   "f analytic_on s \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f holomorphic_on t)"
   (is "?lhs = ?rhs")
@@ -593,7 +593,7 @@
       by (metis analytic_on_def)
   next
     fix t
-    assume "open t" "s \<subseteq> t" "f analytic_on t" 
+    assume "open t" "s \<subseteq> t" "f analytic_on t"
     then show "f analytic_on s"
         by (metis analytic_on_subset)
   qed
@@ -625,17 +625,17 @@
   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
     by (metis analytic_on_def)
   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
-    by (metis analytic_on_def g image_eqI x) 
+    by (metis analytic_on_def g image_eqI x)
   have "isCont f x"
     by (metis analytic_on_imp_differentiable_at complex_differentiable_imp_continuous_at f x)
   with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
      by (auto simp: continuous_at_ball)
-  have "g \<circ> f holomorphic_on ball x (min d e)" 
+  have "g \<circ> f holomorphic_on ball x (min d e)"
     apply (rule holomorphic_on_compose)
     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
     by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
   then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
-    by (metis d e min_less_iff_conj) 
+    by (metis d e min_less_iff_conj)
 qed
 
 lemma analytic_on_compose_gen:
@@ -658,9 +658,9 @@
   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
     by (metis analytic_on_def)
   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
-    by (metis analytic_on_def g z) 
-  have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')" 
-    apply (rule holomorphic_on_add) 
+    by (metis analytic_on_def g z)
+  have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')"
+    apply (rule holomorphic_on_add)
     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
   then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
@@ -678,9 +678,9 @@
   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
     by (metis analytic_on_def)
   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
-    by (metis analytic_on_def g z) 
-  have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')" 
-    apply (rule holomorphic_on_diff) 
+    by (metis analytic_on_def g z)
+  have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')"
+    apply (rule holomorphic_on_diff)
     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
   then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e"
@@ -698,9 +698,9 @@
   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
     by (metis analytic_on_def)
   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
-    by (metis analytic_on_def g z) 
-  have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')" 
-    apply (rule holomorphic_on_mult) 
+    by (metis analytic_on_def g z)
+  have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')"
+    apply (rule holomorphic_on_mult)
     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
   then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
@@ -719,12 +719,12 @@
     by (metis analytic_on_def)
   have "continuous_on (ball z e) f"
     by (metis fh holomorphic_on_imp_continuous_on)
-  then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0" 
-    by (metis Topology_Euclidean_Space.open_ball centre_in_ball continuous_on_open_avoid e z nz)  
-  have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')" 
+  then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0"
+    by (metis Topology_Euclidean_Space.open_ball centre_in_ball continuous_on_open_avoid e z nz)
+  have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')"
     apply (rule holomorphic_on_inverse)
     apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
-    by (metis nz' mem_ball min_less_iff_conj) 
+    by (metis nz' mem_ball min_less_iff_conj)
   then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
     by (metis e e' min_less_iff_conj)
 qed
@@ -764,9 +764,9 @@
   "f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
    (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
   (is "?lhs = ?rhs")
-proof 
+proof
   assume ?lhs
-  then obtain s t 
+  then obtain s t
     where st: "open s" "z \<in> s" "f holomorphic_on s"
               "open t" "z \<in> t" "g holomorphic_on t"
     by (auto simp: analytic_at)
@@ -776,14 +776,14 @@
     apply (auto simp: Diff_subset holomorphic_on_subset)
     done
 next
-  assume ?rhs 
+  assume ?rhs
   then show ?lhs
     by (force simp add: analytic_at)
 qed
 
 subsection\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close>
 
-lemma 
+lemma
   assumes "f analytic_on {z}" "g analytic_on {z}"
   shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
     and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
@@ -826,14 +826,14 @@
       and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
       and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s \<longrightarrow> norm (f' n x - g' x) \<le> e"
       and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) ---> l) sequentially"
-    shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> 
+    shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) ---> g x) sequentially \<and>
                        (g has_field_derivative (g' x)) (at x within s)"
 proof -
   from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) ---> l) sequentially"
     by blast
   { fix e::real assume e: "e > 0"
     then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e"
-      by (metis conv)    
+      by (metis conv)
     have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
     proof (rule exI [of _ N], clarify)
       fix n y h
@@ -863,7 +863,7 @@
   fixes s :: "complex set"
   assumes cvs: "convex s"
       and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
-      and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s 
+      and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
                 \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
       and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) sums l)"
     shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) sums g x) \<and> ((g has_field_derivative g' x) (at x within s))"
@@ -871,9 +871,9 @@
   from assms obtain x l where x: "x \<in> s" and sf: "((\<lambda>n. f n x) sums l)"
     by blast
   { fix e::real assume e: "e > 0"
-    then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s 
+    then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
             \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
-      by (metis conv)    
+      by (metis conv)
     have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
     proof (rule exI [of _ N], clarify)
       fix n y h
@@ -979,7 +979,7 @@
          \<Longrightarrow> DERIV g (f x) :> inverse (f')"
   unfolding has_field_derivative_def
   apply (rule has_derivative_inverse_strong [of s x f g ])
-  using assms 
+  using assms
   by auto
 
 lemma has_complex_derivative_inverse_strong_x:
@@ -993,7 +993,7 @@
           \<Longrightarrow> DERIV g y :> inverse (f')"
   unfolding has_field_derivative_def
   apply (rule has_derivative_inverse_strong_x [of s g y f])
-  using assms 
+  using assms
   by auto
 
 subsection \<open>Taylor on Complex Numbers\<close>
@@ -1004,7 +1004,7 @@
 by (induct n) auto
 
 lemma complex_taylor:
-  assumes s: "convex s" 
+  assumes s: "convex s"
       and f: "\<And>i x. x \<in> s \<Longrightarrow> i \<le> n \<Longrightarrow> (f i has_field_derivative f (Suc i) x) (at x within s)"
       and B: "\<And>x. x \<in> s \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
       and w: "w \<in> s"
@@ -1019,14 +1019,14 @@
     then have "u \<in> s"
       by (metis wzs subsetD)
     have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) +
-                      f (Suc i) u * (z-u)^i / (fact i)) = 
+                      f (Suc i) u * (z-u)^i / (fact i)) =
               f (Suc n) u * (z-u) ^ n / (fact n)"
     proof (induction n)
       case 0 show ?case by simp
     next
       case (Suc n)
       have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) +
-                             f (Suc i) u * (z-u) ^ i / (fact i)) =  
+                             f (Suc i) u * (z-u) ^ i / (fact i)) =
            f (Suc n) u * (z-u) ^ n / (fact n) +
            f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) -
            f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))"
@@ -1056,7 +1056,7 @@
       qed
       finally show ?case .
     qed
-    then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i))) 
+    then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i)))
                 has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n))
                (at u within s)"
       apply (intro derivative_eq_intros)
@@ -1081,19 +1081,19 @@
     by simp
   also have "\<dots> = f 0 z / (fact 0)"
     by (subst setsum_zero_power) simp
-  finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i))) 
+  finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)))
                 \<le> cmod ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
                         (\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
     by (simp add: norm_minus_commute)
   also have "... \<le> B * cmod (z - w) ^ n / (fact n) * cmod (w - z)"
-    apply (rule complex_differentiable_bound 
+    apply (rule complex_differentiable_bound
       [where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)"
          and s = "closed_segment w z", OF convex_closed_segment])
-    apply (auto simp: ends_in_segment real_of_nat_def DERIV_subset [OF sum_deriv wzs]
+    apply (auto simp: ends_in_segment DERIV_subset [OF sum_deriv wzs]
                   norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
     done
   also have "...  \<le> B * cmod (z - w) ^ Suc n / (fact n)"
-    by (simp add: algebra_simps norm_minus_commute real_of_nat_def)
+    by (simp add: algebra_simps norm_minus_commute)
   finally show ?thesis .
 qed
 
@@ -1142,7 +1142,7 @@
              (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
              (fact (Suc n)) +
              (\<Sum>i = 0..n.
-                 f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i))  - 
+                 f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i))  -
                  f (Suc i) u * (z-u) ^ i / (fact i))"
       by (simp only: diff_divide_distrib fact_cancel ac_simps)
     also have "... = f (Suc 0) u -
@@ -1152,7 +1152,7 @@
       by (subst setsum_Suc_diff) auto
     also have "... = f (Suc n) u * (z-u) ^ n / (fact n)"
       by (simp only: algebra_simps diff_divide_distrib fact_cancel)
-    finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i 
+    finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i
                              - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) =
                   f (Suc n) u * (z - u) ^ n / (fact n)" .
     then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Multivariate_Analysis/Complex_Transcendental.thy
--- a/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -192,8 +192,8 @@
 lemma exp_eq_1: "exp z = 1 \<longleftrightarrow> Re(z) = 0 \<and> (\<exists>n::int. Im(z) = of_int (2 * n) * pi)"
 apply auto
 apply (metis exp_eq_one_iff norm_exp_eq_Re norm_one)
-apply (metis Re_exp cos_one_2pi_int mult.commute mult.left_neutral norm_exp_eq_Re norm_one one_complex.simps(1) real_of_int_def)
-by (metis Im_exp Re_exp complex_Re_Im_cancel_iff cos_one_2pi_int sin_double Re_complex_of_real complex_Re_numeral exp_zero mult.assoc mult.left_commute mult_eq_0_iff mult_numeral_1 numeral_One of_real_0 real_of_int_def sin_zero_iff_int2)
+apply (metis Re_exp cos_one_2pi_int mult.commute mult.left_neutral norm_exp_eq_Re norm_one one_complex.simps(1))
+by (metis Im_exp Re_exp complex_Re_Im_cancel_iff cos_one_2pi_int sin_double Re_complex_of_real complex_Re_numeral exp_zero mult.assoc mult.left_commute mult_eq_0_iff mult_numeral_1 numeral_One of_real_0 sin_zero_iff_int2)
 
 lemma exp_eq: "exp w = exp z \<longleftrightarrow> (\<exists>n::int. w = z + (of_int (2 * n) * pi) * ii)"
                 (is "?lhs = ?rhs")
@@ -227,7 +227,7 @@
   { assume "sin y = sin x" "cos y = cos x"
     then have "cos (y-x) = 1"
       using cos_add [of y "-x"] by simp
-    then have "\<exists>n::int. y-x = real n * 2 * pi"
+    then have "\<exists>n::int. y-x = n * 2 * pi"
       using cos_one_2pi_int by blast }
   then show ?thesis
   apply (auto simp: sin_add cos_add)
@@ -519,8 +519,8 @@
   have *: "cmod (sin z -
                  (\<Sum>i\<le>n. (-1) ^ (i div 2) * (if even i then sin 0 else cos 0) * z ^ i / (fact i)))
            \<le> exp \<bar>Im z\<bar> * cmod z ^ Suc n / (fact n)"
-  proof (rule complex_taylor [of "closed_segment 0 z" n 
-                                 "\<lambda>k x. (-1)^(k div 2) * (if even k then sin x else cos x)" 
+  proof (rule complex_taylor [of "closed_segment 0 z" n
+                                 "\<lambda>k x. (-1)^(k div 2) * (if even k then sin x else cos x)"
                                  "exp\<bar>Im z\<bar>" 0 z,  simplified])
     fix k x
     show "((\<lambda>x. (- 1) ^ (k div 2) * (if even k then sin x else cos x)) has_field_derivative
@@ -622,7 +622,7 @@
   then have "sin t' = sin t \<and> cos t' = cos t"
     apply (simp add: exp_Euler sin_of_real cos_of_real)
     by (metis Complex_eq complex.sel)
-  then obtain n::int where n: "t' = t + 2 * real n * pi"
+  then obtain n::int where n: "t' = t + 2 * n * pi"
     by (auto simp: sin_cos_eq_iff)
   then have "n=0"
     apply (rule_tac z=n in int_cases)
@@ -752,7 +752,7 @@
     by blast
 qed
 
-corollary Arg_gt_0: 
+corollary Arg_gt_0:
   assumes "z \<in> \<real> \<Longrightarrow> Re z < 0"
     shows "Arg z > 0"
   using Arg_eq_0 Arg_ge_0 assms dual_order.strict_iff_order by fastforce
@@ -963,7 +963,7 @@
 lemma Re_Ln [simp]: "z \<noteq> 0 \<Longrightarrow> Re(Ln z) = ln(norm z)"
   by (metis exp_Ln assms ln_exp norm_exp_eq_Re)
 
-corollary ln_cmod_le: 
+corollary ln_cmod_le:
   assumes z: "z \<noteq> 0"
     shows "ln (cmod z) \<le> cmod (Ln z)"
   using norm_exp [of "Ln z", simplified exp_Ln [OF z]]
@@ -1205,7 +1205,7 @@
 corollary Ln_divide_of_real:
     "\<lbrakk>r > 0; z \<noteq> 0\<rbrakk> \<Longrightarrow> Ln(z / of_real r) = Ln(z) - ln r"
 using Ln_times_of_real [of "inverse r" z]
-by (simp add: ln_inverse Ln_of_real mult.commute divide_inverse of_real_inverse [symmetric] 
+by (simp add: ln_inverse Ln_of_real mult.commute divide_inverse of_real_inverse [symmetric]
          del: of_real_inverse)
 
 lemma Ln_minus:
@@ -1264,7 +1264,7 @@
 lemma Ln_of_nat: "0 < n \<Longrightarrow> Ln (of_nat n) = of_real (ln (of_nat n))"
   by (subst of_real_of_nat_eq[symmetric], subst Ln_of_real[symmetric]) simp_all
 
-lemma Ln_of_nat_over_of_nat: 
+lemma Ln_of_nat_over_of_nat:
   assumes "m > 0" "n > 0"
   shows   "Ln (of_nat m / of_nat n) = of_real (ln (of_nat m) - ln (of_nat n))"
 proof -
@@ -1279,7 +1279,7 @@
 
 subsection\<open>Relation between Ln and Arg, and hence continuity of Arg\<close>
 
-lemma Arg_Ln: 
+lemma Arg_Ln:
   assumes "0 < Arg z" shows "Arg z = Im(Ln(-z)) + pi"
 proof (cases "z = 0")
   case True
@@ -1307,7 +1307,7 @@
   finally show ?thesis .
 qed
 
-lemma continuous_at_Arg: 
+lemma continuous_at_Arg:
   assumes "z \<in> \<real> \<Longrightarrow> Re z < 0"
     shows "continuous (at z) Arg"
 proof -
@@ -1322,7 +1322,7 @@
 qed
 
 text\<open>Relation between Arg and arctangent in upper halfplane\<close>
-lemma Arg_arctan_upperhalf: 
+lemma Arg_arctan_upperhalf:
   assumes "0 < Im z"
     shows "Arg z = pi/2 - arctan(Re z / Im z)"
 proof (cases "z = 0")
@@ -1340,24 +1340,24 @@
     done
 qed
 
-lemma Arg_eq_Im_Ln: 
-  assumes "0 \<le> Im z" "0 < Re z" 
+lemma Arg_eq_Im_Ln:
+  assumes "0 \<le> Im z" "0 < Re z"
     shows "Arg z = Im (Ln z)"
 proof (cases "z = 0 \<or> Im z = 0")
   case True then show ?thesis
-    using assms Arg_eq_0 complex_is_Real_iff  
+    using assms Arg_eq_0 complex_is_Real_iff
     apply auto
     by (metis Arg_eq_0_pi Arg_eq_pi Im_Ln_eq_0 Im_Ln_eq_pi less_numeral_extra(3) zero_complex.simps(1))
 next
-  case False 
+  case False
   then have "Arg z > 0"
     using Arg_gt_0 complex_is_Real_iff by blast
   then show ?thesis
-    using assms False 
+    using assms False
     by (subst Arg_Ln) (auto simp: Ln_minus)
 qed
 
-lemma continuous_within_upperhalf_Arg: 
+lemma continuous_within_upperhalf_Arg:
   assumes "z \<noteq> 0"
     shows "continuous (at z within {z. 0 \<le> Im z}) Arg"
 proof (cases "z \<in> \<real> & 0 \<le> Re z")
@@ -1369,7 +1369,7 @@
     using assms  by (auto simp: complex_is_Real_iff complex_neq_0)
   then have [simp]: "Arg z = 0" "Im (Ln z) = 0"
     by (auto simp: Arg_eq_0 Im_Ln_eq_0 assms complex_is_Real_iff)
-  show ?thesis  
+  show ?thesis
   proof (clarsimp simp add: continuous_within Lim_within dist_norm)
     fix e::real
     assume "0 < e"
@@ -1397,12 +1397,12 @@
   apply (auto simp: continuous_on_eq_continuous_within)
   by (metis Diff_subset continuous_within_subset continuous_within_upperhalf_Arg)
 
-lemma open_Arg_less_Int: 
+lemma open_Arg_less_Int:
   assumes "0 \<le> s" "t \<le> 2*pi"
     shows "open ({y. s < Arg y} \<inter> {y. Arg y < t})"
 proof -
   have 1: "continuous_on (UNIV - {z \<in> \<real>. 0 \<le> Re z}) Arg"
-    using continuous_at_Arg continuous_at_imp_continuous_within 
+    using continuous_at_Arg continuous_at_imp_continuous_within
     by (auto simp: continuous_on_eq_continuous_within set_diff_eq)
   have 2: "open (UNIV - {z \<in> \<real>. 0 \<le> Re z})"
     by (simp add: closed_Collect_conj closed_complex_Reals closed_halfspace_Re_ge open_Diff)
@@ -1413,7 +1413,7 @@
     using assms
     by (auto simp: Arg_real)
   ultimately show ?thesis
-    using continuous_imp_open_vimage [OF 1 2, of  "{z. Re z > s} \<inter> {z. Re z < t}"] 
+    using continuous_imp_open_vimage [OF 1 2, of  "{z. Re z > s} \<inter> {z. Re z < t}"]
     by auto
 qed
 
@@ -1521,11 +1521,11 @@
   thus "eventually (\<lambda>z. z powr r = Exp (r * Ln z)) (nhds z)"
     unfolding powr_def by eventually_elim simp
 
-  have "((\<lambda>z. Exp (r * Ln z)) has_field_derivative Exp (r * Ln z) * (inverse z * r)) (at z)" 
+  have "((\<lambda>z. Exp (r * Ln z)) has_field_derivative Exp (r * Ln z) * (inverse z * r)) (at z)"
     using assms by (auto intro!: derivative_eq_intros has_field_derivative_powr)
   also have "Exp (r * Ln z) * (inverse z * r) = r * z powr (r - 1)"
     unfolding powr_def by (simp add: assms exp_diff field_simps)
-  finally show "((\<lambda>z. Exp (r * Ln z)) has_field_derivative r * z powr (r - 1)) (at z)" 
+  finally show "((\<lambda>z. Exp (r * Ln z)) has_field_derivative r * z powr (r - 1)) (at z)"
     by simp
 qed
 
@@ -1553,13 +1553,13 @@
 
 
 subsection\<open>Some Limits involving Logarithms\<close>
-        
+
 lemma lim_Ln_over_power:
   fixes s::complex
   assumes "0 < Re s"
     shows "((\<lambda>n. Ln n / (n powr s)) ---> 0) sequentially"
 proof (simp add: lim_sequentially dist_norm, clarify)
-  fix e::real 
+  fix e::real
   assume e: "0 < e"
   have "\<exists>xo>0. \<forall>x\<ge>xo. 0 < e * 2 + (e * Re s * 2 - 2) * x + e * (Re s)\<^sup>2 * x\<^sup>2"
   proof (rule_tac x="2/(e * (Re s)\<^sup>2)" in exI, safe)
@@ -1591,7 +1591,7 @@
     apply (rule_tac x="nat (ceiling (exp xo))" in exI)
     apply clarify
     apply (drule_tac x="ln n" in spec)
-    apply (auto simp: real_of_nat_def exp_less_mono nat_ceiling_le_eq not_le)
+    apply (auto simp: exp_less_mono nat_ceiling_le_eq not_le)
     apply (metis exp_less_mono exp_ln not_le of_nat_0_less_iff)
     done
 qed
@@ -1613,13 +1613,13 @@
   using lim_Ln_over_power [of "of_real s", THEN filterlim_sequentially_Suc [THEN iffD2]] assms
   apply (subst filterlim_sequentially_Suc [symmetric])
   apply (simp add: lim_sequentially dist_norm
-          Ln_Reals_eq norm_powr_real_powr norm_divide real_of_nat_def)
+          Ln_Reals_eq norm_powr_real_powr norm_divide)
   done
 
 lemma lim_ln_over_n: "((\<lambda>n. ln(real_of_nat n) / of_nat n) ---> 0) sequentially"
   using lim_ln_over_power [of 1, THEN filterlim_sequentially_Suc [THEN iffD2]]
   apply (subst filterlim_sequentially_Suc [symmetric])
-  apply (simp add: lim_sequentially dist_norm real_of_nat_def)
+  apply (simp add: lim_sequentially dist_norm)
   done
 
 lemma lim_1_over_complex_power:
@@ -1645,7 +1645,7 @@
   using lim_1_over_complex_power [of "of_real s", THEN filterlim_sequentially_Suc [THEN iffD2]] assms
   apply (subst filterlim_sequentially_Suc [symmetric])
   apply (simp add: lim_sequentially dist_norm)
-  apply (simp add: Ln_Reals_eq norm_powr_real_powr norm_divide real_of_nat_def)
+  apply (simp add: Ln_Reals_eq norm_powr_real_powr norm_divide)
   done
 
 lemma lim_1_over_Ln: "((\<lambda>n. 1 / Ln(of_nat n)) ---> 0) sequentially"
@@ -1677,7 +1677,7 @@
   using lim_1_over_Ln [THEN filterlim_sequentially_Suc [THEN iffD2]] assms
   apply (subst filterlim_sequentially_Suc [symmetric])
   apply (simp add: lim_sequentially dist_norm)
-  apply (simp add: Ln_Reals_eq norm_powr_real_powr norm_divide real_of_nat_def)
+  apply (simp add: Ln_Reals_eq norm_powr_real_powr norm_divide)
   done
 
 
@@ -1842,7 +1842,7 @@
   also have "... \<longleftrightarrow> False"
     using assms ge_pi2
     apply (auto simp: algebra_simps)
-    by (metis abs_mult_pos not_less not_real_of_nat_less_zero real_of_nat_numeral)
+    by (metis abs_mult_pos not_less of_nat_less_0_iff of_nat_numeral)
   finally have *: "exp (\<i>*z)*exp (\<i>*z) + 1 \<noteq> 0"
     by (auto simp: add.commute minus_unique)
   show ?thesis
@@ -1939,7 +1939,7 @@
   apply (subst exp_Ln, auto)
   apply (simp_all add: cmod_def complex_eq_iff)
   apply (auto simp: divide_simps)
-  apply (metis power_one realpow_two_sum_zero_iff zero_neq_one, algebra)
+  apply (metis power_one sum_power2_eq_zero_iff zero_neq_one, algebra)
   done
 
 lemma arctan_eq_Re_Arctan: "arctan x = Re (Arctan (of_real x))"
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -4596,7 +4596,7 @@
       using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
       using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
       using subset_hull[of convex, OF assms(1), symmetric, of c]
-      by auto
+      by force
     then have "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)"
       apply (rule_tac x = "inverse(norm a) *\<^sub>R a" in exI)
       using hull_subset[of c convex]
@@ -4624,7 +4624,7 @@
 proof -
   from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
   obtain a where "a \<noteq> 0" "\<forall>x\<in>{x - y |x y. x \<in> t \<and> y \<in> s}. 0 \<le> inner a x"
-    using assms(3-5) by auto
+    using assms(3-5) by fastforce
   then have *: "\<And>x y. x \<in> t \<Longrightarrow> y \<in> s \<Longrightarrow> inner a y \<le> inner a x"
     by (force simp add: inner_diff)
   then have bdd: "bdd_above ((op \<bullet> a)`s)"
@@ -4832,11 +4832,10 @@
   have *: "\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x"
     by auto
   show ?thesis
-    unfolding real_add_eq_0_iff[symmetric] and setsum.inter_filter[OF assms(1)]
+    unfolding add_eq_0_iff[symmetric] and setsum.inter_filter[OF assms(1)]
       and setsum.distrib[symmetric] and *
     using assms(2)
-    apply assumption
-    done
+    by assumption
 qed
 
 lemma radon_v_lemma:
@@ -5757,7 +5756,7 @@
 subsection \<open>Another intermediate value theorem formulation\<close>
 
 lemma ivt_increasing_component_on_1:
-  fixes f :: "real \<Rightarrow> 'a::euclidean_space"            
+  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "a \<le> b"
     and "continuous_on {a..b} f"
     and "(f a)\<bullet>k \<le> y" "y \<le> (f b)\<bullet>k"
@@ -6204,7 +6203,7 @@
       apply (erule (1) order_trans [OF Basis_le_norm])
       done
     have e': "e = (\<Sum>(i::'a)\<in>Basis. d)"
-      by (simp add: d_def real_of_nat_def DIM_positive)
+      by (simp add: d_def DIM_positive)
     show "convex hull c \<subseteq> cball x e"
       unfolding 2
       apply clarsimp
@@ -6326,7 +6325,7 @@
     apply (simp add: scaleR_2)
     done
   show ?t3
-    unfolding midpoint_def dist_norm     
+    unfolding midpoint_def dist_norm
     apply (rule *)
     apply (simp add: scaleR_right_diff_distrib)
     apply (simp add: scaleR_2)
@@ -6910,7 +6909,7 @@
       then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
     qed
     also have "\<dots> \<le> 1"
-      unfolding setsum.distrib setsum_constant real_eq_of_nat
+      unfolding setsum.distrib setsum_constant
       by (auto simp add: Suc_le_eq)
     finally show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> setsum (op \<bullet> y) Basis \<le> 1"
     proof safe
@@ -6957,7 +6956,7 @@
       apply auto
       done
     also have "\<dots> < 1"
-      unfolding setsum_constant real_eq_of_nat divide_inverse[symmetric]
+      unfolding setsum_constant divide_inverse[symmetric]
       by (auto simp add: field_simps)
     finally show "setsum (op \<bullet> ?a) ?D < 1" by auto
   qed
@@ -7094,8 +7093,7 @@
           then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
         qed
         also have "\<dots> \<le> 1"
-          unfolding setsum.distrib setsum_constant real_eq_of_nat
-          using \<open>0 < card d\<close>
+          unfolding setsum.distrib setsum_constant  using \<open>0 < card d\<close>
           by auto
         finally show "setsum (op \<bullet> y) d \<le> 1" .
 
@@ -7164,7 +7162,7 @@
     have "setsum (op \<bullet> ?a) ?D = setsum (\<lambda>i. inverse (2 * real (card d))) ?D"
       by (rule setsum.cong) (rule refl, rule **)
     also have "\<dots> < 1"
-      unfolding setsum_constant real_eq_of_nat divide_real_def[symmetric]
+      unfolding setsum_constant divide_real_def[symmetric]
       by (auto simp add: field_simps)
     finally show "setsum (op \<bullet> ?a) ?D < 1" by auto
   next
@@ -9713,7 +9711,7 @@
   unfolding setdist_def
   by (auto intro!: bdd_belowI [where m=0] cInf_lower)
 
-lemma le_setdist_iff: 
+lemma le_setdist_iff:
         "d \<le> setdist s t \<longleftrightarrow>
         (\<forall>x \<in> s. \<forall>y \<in> t. d \<le> dist x y) \<and> (s = {} \<or> t = {} \<longrightarrow> d \<le> 0)"
   apply (cases "s = {} \<or> t = {}")
@@ -9723,7 +9721,7 @@
   apply (auto simp: intro: le_setdistI)
   done
 
-lemma setdist_ltE: 
+lemma setdist_ltE:
   assumes "setdist s t < b" "s \<noteq> {}" "t \<noteq> {}"
     obtains x y where "x \<in> s" "y \<in> t" "dist x y < b"
 using assms
@@ -9745,7 +9743,7 @@
     using setdist_pos_le by fastforce
 next
   case False
-  have "\<And>x. x \<in> s \<Longrightarrow> setdist s t - dist x a \<le> setdist {a} t" 
+  have "\<And>x. x \<in> s \<Longrightarrow> setdist s t - dist x a \<le> setdist {a} t"
     apply (rule le_setdistI, blast)
     using False apply (fastforce intro: le_setdistI)
     apply (simp add: algebra_simps)
@@ -9840,7 +9838,7 @@
   assumes s: "compact s" and t: "closed t"
     shows "setdist s t = 0 \<longleftrightarrow> s = {} \<or> t = {} \<or> s \<inter> t \<noteq> {}"
   apply (cases "s = {} \<or> t = {}", force)
-  using setdist_compact_closed [OF s t]  
+  using setdist_compact_closed [OF s t]
   apply (force intro: setdist_eq_0I )
   done
 
@@ -9863,29 +9861,29 @@
   assumes "bounded s \<or> bounded t"
     shows "setdist s t = 0 \<longleftrightarrow> s = {} \<or> t = {} \<or> closure s \<inter> closure t \<noteq> {}"
   apply (cases "s = {} \<or> t = {}", force)
-  using setdist_eq_0_compact_closed [of "closure s" "closure t"] 
-        setdist_eq_0_closed_compact [of "closure s" "closure t"] assms 
+  using setdist_eq_0_compact_closed [of "closure s" "closure t"]
+        setdist_eq_0_closed_compact [of "closure s" "closure t"] assms
   apply (force simp add:  bounded_closure compact_eq_bounded_closed)
   done
 
-lemma setdist_unique: 
+lemma setdist_unique:
   "\<lbrakk>a \<in> s; b \<in> t; \<And>x y. x \<in> s \<and> y \<in> t ==> dist a b \<le> dist x y\<rbrakk>
    \<Longrightarrow> setdist s t = dist a b"
   by (force simp add: setdist_le_dist le_setdist_iff intro: antisym)
 
-lemma setdist_closest_point: 
+lemma setdist_closest_point:
     "\<lbrakk>closed s; s \<noteq> {}\<rbrakk> \<Longrightarrow> setdist {a} s = dist a (closest_point s a)"
   apply (rule setdist_unique)
   using closest_point_le
   apply (auto simp: closest_point_in_set)
   done
 
-lemma setdist_eq_0_sing_1 [simp]: 
+lemma setdist_eq_0_sing_1 [simp]:
   fixes s :: "'a::euclidean_space set"
   shows "setdist {x} s = 0 \<longleftrightarrow> s = {} \<or> x \<in> closure s"
 by (auto simp: setdist_eq_0_bounded)
 
-lemma setdist_eq_0_sing_2 [simp]: 
+lemma setdist_eq_0_sing_2 [simp]:
   fixes s :: "'a::euclidean_space set"
   shows "setdist s {x} = 0 \<longleftrightarrow> s = {} \<or> x \<in> closure s"
 by (auto simp: setdist_eq_0_bounded)
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Multivariate_Analysis/Integration.thy
--- a/src/HOL/Multivariate_Analysis/Integration.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Integration.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -2860,7 +2860,7 @@
       show "norm ((\<Sum>(x, k)\<in>p m. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p n. content k *\<^sub>R f x)) < e"
         apply (rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]])
         apply(subst *)
-        using dp p(1) mn d(2) real_of_nat_def by auto
+        using dp p(1) mn d(2) by auto
     qed
   qed
   then guess y unfolding convergent_eq_cauchy[symmetric] .. note y=this[THEN LIMSEQ_D]
@@ -4518,7 +4518,7 @@
       using True by (auto simp add: field_simps)
     then obtain M :: nat
          where M: "M \<noteq> 0" "0 < inverse (real_of_nat M)" "inverse (of_nat M) < e / 4 / content (cbox a b)"
-      by (subst (asm) real_arch_inv) (auto simp: real_of_nat_def)
+      by (subst (asm) real_arch_inv) auto
     show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (i m) (i n) < e"
     proof (rule exI [where x=M], clarify)
       fix m n
@@ -6091,8 +6091,7 @@
   have Dg: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
     (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a .. b})"
     unfolding Dg_def
-    by (auto intro!: derivative_eq_intros simp: has_vector_derivative_def
-      fact_eq real_eq_of_nat[symmetric] divide_simps)
+    by (auto intro!: derivative_eq_intros simp: has_vector_derivative_def fact_eq divide_simps)
   let ?sum = "\<lambda>t. \<Sum>i<p. (- 1) ^ i *\<^sub>R Dg i t *\<^sub>R Df (p - Suc i) t"
   from setsum_prod_derivatives_has_vector_derivative[of _ Dg _ _ _ Df,
       OF \<open>p > 0\<close> g0 Dg f0 Df]
@@ -6420,17 +6419,17 @@
   shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within {a..b})"
 proof -
   let ?I = "\<lambda>a b. integral {a..b} f"
-  { fix e::real 
+  { fix e::real
     assume "e > 0"
     obtain d where "d>0" and d: "\<And>x'. \<lbrakk>x' \<in> {a..b}; \<bar>x' - x\<bar> < d\<rbrakk> \<Longrightarrow> norm(f x' - f x) \<le> e"
       using \<open>e>0\<close> fx by (auto simp: continuous_within_eps_delta dist_norm less_imp_le)
     have "norm (integral {a..y} f - integral {a..x} f - (y - x) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>"
-           if y: "y \<in> {a..b}" and yx: "\<bar>y - x\<bar> < d" for y  
+           if y: "y \<in> {a..b}" and yx: "\<bar>y - x\<bar> < d" for y
     proof (cases "y < x")
       case False
       have "f integrable_on {a..y}"
         using f y by (simp add: integrable_subinterval_real)
-      then have Idiff: "?I a y - ?I a x = ?I x y" 
+      then have Idiff: "?I a y - ?I a x = ?I x y"
         using False x by (simp add: algebra_simps integral_combine)
       have fux_int: "((\<lambda>u. f u - f x) has_integral integral {x..y} f - (y - x) *\<^sub>R f x) {x..y}"
         apply (rule has_integral_sub)
@@ -6448,7 +6447,7 @@
       case True
       have "f integrable_on {a..x}"
         using f x by (simp add: integrable_subinterval_real)
-      then have Idiff: "?I a x - ?I a y = ?I y x" 
+      then have Idiff: "?I a x - ?I a y = ?I y x"
         using True x y by (simp add: algebra_simps integral_combine)
       have fux_int: "((\<lambda>u. f u - f x) has_integral integral {y..x} f - (x - y) *\<^sub>R f x) {y..x}"
         apply (rule has_integral_sub)
@@ -6466,8 +6465,8 @@
         by (simp add: algebra_simps norm_minus_commute)
     qed
     then have "\<exists>d>0. \<forall>y\<in>{a..b}. \<bar>y - x\<bar> < d \<longrightarrow> norm (integral {a..y} f - integral {a..x} f - (y - x) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>"
-      using \<open>d>0\<close> by blast 
-  } 
+      using \<open>d>0\<close> by blast
+  }
   then show ?thesis
     by (simp add: has_vector_derivative_def has_derivative_within_alt bounded_linear_scaleR_left)
 qed
@@ -9485,8 +9484,7 @@
       defer
       unfolding divide_inverse setsum_left_distrib[symmetric]
       unfolding divide_inverse[symmetric]
-      using *
-      apply (auto simp add: field_simps real_eq_of_nat)
+      using * apply (auto simp add: field_simps)
       done
   next
     case 2
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Multivariate_Analysis/Linear_Algebra.thy
--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -570,11 +570,11 @@
 subsection \<open>Archimedean properties and useful consequences\<close>
 
 lemma real_arch_simple: "\<exists>n::nat. x \<le> real n"
-  unfolding real_of_nat_def by (rule ex_le_of_nat)
+  by (rule ex_le_of_nat)
 
 lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
   using reals_Archimedean[of e] less_trans[of 0 "1 / real n" e for n::nat]
-  by (auto simp add: field_simps cong: conj_cong simp del: real_of_nat_Suc)
+  by (auto simp add: field_simps cong: conj_cong simp del: of_nat_Suc)
 
 text\<open>Bernoulli's inequality\<close>
 lemma Bernoulli_inequality:
@@ -589,7 +589,7 @@
   have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
     by (simp add: algebra_simps)
   also have "... = (1 + x) * (1 + n*x)"
-    by (auto simp: power2_eq_square algebra_simps  real_of_nat_Suc)
+    by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
   also have "... \<le> (1 + x) ^ Suc n"
     using Suc.hyps assms mult_left_mono by fastforce
   finally show ?case .
@@ -668,7 +668,7 @@
     (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
   apply (rule forall_pos_mono)
   apply auto
-  apply (metis Suc_pred real_of_nat_Suc)
+  apply (metis Suc_pred of_nat_Suc)
   done
 
 lemma real_archimedian_rdiv_eq_0:
@@ -1461,8 +1461,7 @@
       by (intro add_mono norm_bound_Basis_le i fPs) auto
     finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
   qed
-  also have "\<dots> = 2 * real DIM('n) * e"
-    by (simp add: real_of_nat_def)
+  also have "\<dots> = 2 * real DIM('n) * e" by simp
   finally show ?thesis .
 qed
 
@@ -1629,12 +1628,12 @@
   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
   using independent_span_bound[OF finite_Basis, of S] by auto
 
-corollary 
+corollary
   fixes S :: "'a::euclidean_space set"
   assumes "independent S"
   shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)"
 using assms independent_bound by auto
-  
+
 lemma dependent_biggerset:
   fixes S :: "'a::euclidean_space set"
   shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
@@ -2856,7 +2855,6 @@
     by (simp add: zero_le_mult_iff infnorm_pos_le)
   have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)\<^sup>2"
     unfolding power_mult_distrib d2
-    unfolding real_of_nat_def
     apply (subst euclidean_inner)
     apply (subst power2_abs[symmetric])
     apply (rule order_trans[OF setsum_bounded_above[where K="\<bar>infnorm x\<bar>\<^sup>2"]])
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Multivariate_Analysis/PolyRoots.thy
--- a/src/HOL/Multivariate_Analysis/PolyRoots.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/PolyRoots.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -20,7 +20,7 @@
   fixes x :: "'a::{comm_ring,division_ring}"
   shows   "(\<Sum>i\<le>n. x^i) = (if x = 1 then of_nat(n + 1) else (1 - x^Suc n) / (1 - x))"
   using setsum_gp_basic[of x n]
-  by (simp add: real_of_nat_def mult.commute divide_simps)
+  by (simp add: mult.commute divide_simps)
 
 lemma setsum_power_add:
   fixes x :: "'a::{comm_ring,monoid_mult}"
@@ -60,8 +60,8 @@
                (if n < m then 0
                 else if x = 1 then of_nat((n + 1) - m)
                 else (x^m - x^Suc n) / (1 - x))"
-using setsum_gp_multiplied [of m n x] 
-apply (auto simp: real_of_nat_def)
+using setsum_gp_multiplied [of m n x]
+apply auto
 by (metis eq_iff_diff_eq_0 mult.commute nonzero_divide_eq_eq)
 
 lemma setsum_gp_offset:
@@ -75,15 +75,15 @@
   fixes x :: "'a::{comm_ring,division_ring}"
   shows "(\<Sum>i<n. x^i) = (if x = 1 then of_nat n else (1 - x^n) / (1 - x))"
   by (induct n) (auto simp: algebra_simps divide_simps)
-    
+
 subsection\<open>Basics about polynomial functions: extremal behaviour and root counts.\<close>
 
 lemma sub_polyfun:
   fixes x :: "'a::{comm_ring,monoid_mult}"
-  shows   "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) = 
+  shows   "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
            (x - y) * (\<Sum>j<n. \<Sum>k= Suc j..n. a k * y^(k - Suc j) * x^j)"
 proof -
-  have "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) = 
+  have "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
         (\<Sum>i\<le>n. a i * (x^i - y^i))"
     by (simp add: algebra_simps setsum_subtractf [symmetric])
   also have "... = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))"
@@ -97,7 +97,7 @@
 
 lemma sub_polyfun_alt:
   fixes x :: "'a::{comm_ring,monoid_mult}"
-  shows   "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) = 
+  shows   "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
            (x - y) * (\<Sum>j<n. \<Sum>k<n-j. a (j+k+1) * y^k * x^j)"
 proof -
   { fix j
@@ -110,19 +110,19 @@
 
 lemma polyfun_linear_factor:
   fixes a :: "'a::{comm_ring,monoid_mult}"
-  shows  "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) = 
+  shows  "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) =
                   (z-a) * (\<Sum>i<n. b i * z^i) + (\<Sum>i\<le>n. c i * a^i)"
 proof -
   { fix z
-    have "(\<Sum>i\<le>n. c i * z^i) - (\<Sum>i\<le>n. c i * a^i) = 
+    have "(\<Sum>i\<le>n. c i * z^i) - (\<Sum>i\<le>n. c i * a^i) =
           (z - a) * (\<Sum>j<n. (\<Sum>k = Suc j..n. c k * a^(k - Suc j)) * z^j)"
       by (simp add: sub_polyfun setsum_left_distrib)
-    then have "(\<Sum>i\<le>n. c i * z^i) = 
+    then have "(\<Sum>i\<le>n. c i * z^i) =
           (z - a) * (\<Sum>j<n. (\<Sum>k = Suc j..n. c k * a^(k - Suc j)) * z^j)
           + (\<Sum>i\<le>n. c i * a^i)"
       by (simp add: algebra_simps) }
   then show ?thesis
-    by (intro exI allI) 
+    by (intro exI allI)
 qed
 
 lemma polyfun_linear_factor_root:
@@ -141,7 +141,7 @@
     shows "\<exists>M. \<forall>z. M \<le> norm z \<longrightarrow> norm(\<Sum>i\<le>n. c i * z^i) \<le> e * norm(z) ^ Suc n"
 proof (induction n)
   case 0
-  show ?case 
+  show ?case
     by (rule exI [where x="norm (c 0) / e"]) (auto simp: mult.commute pos_divide_le_eq assms)
 next
   case (Suc n)
@@ -153,14 +153,14 @@
     then have norm1: "0 < norm z" "M \<le> norm z" "(e + norm (c (Suc n))) / e \<le> norm z"
       by auto
     then have norm2: "(e + norm (c (Suc n))) \<le> e * norm z"  "(norm z * norm z ^ n) > 0"
-      apply (metis assms less_divide_eq mult.commute not_le) 
+      apply (metis assms less_divide_eq mult.commute not_le)
       using norm1 apply (metis mult_pos_pos zero_less_power)
       done
     have "e * (norm z * norm z ^ n) + norm (c (Suc n) * (z * z ^ n)) =
           (e + norm (c (Suc n))) * (norm z * norm z ^ n)"
       by (simp add: norm_mult norm_power algebra_simps)
     also have "... \<le> (e * norm z) * (norm z * norm z ^ n)"
-      using norm2 by (metis real_mult_le_cancel_iff1) 
+      using norm2 by (metis real_mult_le_cancel_iff1)
     also have "... = e * (norm z * (norm z * norm z ^ n))"
       by (simp add: algebra_simps)
     finally have "e * (norm z * norm z ^ n) + norm (c (Suc n) * (z * z ^ n))
@@ -171,7 +171,7 @@
 qed
 
 lemma norm_lemma_xy: "\<lbrakk>abs b + 1 \<le> norm(y) - a; norm(x) \<le> a\<rbrakk> \<Longrightarrow> b \<le> norm(x + y)"
-  by (metis abs_add_one_not_less_self add.commute diff_le_eq dual_order.trans le_less_linear 
+  by (metis abs_add_one_not_less_self add.commute diff_le_eq dual_order.trans le_less_linear
          norm_diff_ineq)
 
 lemma polyfun_extremal:
@@ -192,7 +192,7 @@
   next
     case False
     with polyfun_extremal_lemma [of "norm(c (Suc n)) / 2" c n]
-    obtain M where M: "\<And>z. M \<le> norm z \<Longrightarrow> 
+    obtain M where M: "\<And>z. M \<le> norm z \<Longrightarrow>
                norm (\<Sum>i\<le>n. c i * z^i) \<le> norm (c (Suc n)) / 2 * norm z ^ Suc n"
       by auto
     show ?thesis
@@ -202,7 +202,7 @@
       assume les: "M \<le> norm z"  "1 \<le> norm z"  "(\<bar>B\<bar> * 2 + 2) / norm (c (Suc n)) \<le> norm z"
       then have "\<bar>B\<bar> * 2 + 2 \<le> norm z * norm (c (Suc n))"
         by (metis False pos_divide_le_eq zero_less_norm_iff)
-      then have "\<bar>B\<bar> * 2 + 2 \<le> norm z ^ (Suc n) * norm (c (Suc n))" 
+      then have "\<bar>B\<bar> * 2 + 2 \<le> norm z ^ (Suc n) * norm (c (Suc n))"
         by (metis \<open>1 \<le> norm z\<close> order.trans mult_right_mono norm_ge_zero self_le_power zero_less_Suc)
       then show "B \<le> norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * (z * z ^ n))" using M les
         apply auto
@@ -235,7 +235,7 @@
      by simp
    then have extr_prem: "~ (\<exists>k\<le>n. b k \<noteq> 0) \<Longrightarrow> \<exists>k. k \<noteq> 0 \<and> k \<le> Suc n \<and> c k \<noteq> 0"
      by (metis Suc.prems le0 minus_zero mult_zero_right)
-   have "\<exists>k\<le>n. b k \<noteq> 0" 
+   have "\<exists>k\<le>n. b k \<noteq> 0"
      apply (rule ccontr)
      using polyfun_extremal [OF extr_prem, of 1]
      apply (auto simp: eventually_at_infinity b simp del: setsum_atMost_Suc)
@@ -257,10 +257,10 @@
   fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
     shows  "finite {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<longleftrightarrow> (\<exists>k. k \<le> n \<and> c k \<noteq> 0)"
 proof (cases " \<exists>k\<le>n. c k \<noteq> 0")
-  case True then show ?thesis 
+  case True then show ?thesis
     by (blast intro: polyfun_rootbound_finite)
 next
-  case False then show ?thesis 
+  case False then show ?thesis
     by (auto simp: infinite_UNIV_char_0)
 qed
 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -707,14 +707,14 @@
   from 1 2 show ?lhs
     unfolding openin_open open_dist by fast
 qed
- 
+
 lemma connected_open_in:
       "connected s \<longleftrightarrow>
        ~(\<exists>e1 e2. openin (subtopology euclidean s) e1 \<and>
                  openin (subtopology euclidean s) e2 \<and>
                  s \<subseteq> e1 \<union> e2 \<and> e1 \<inter> e2 = {} \<and> e1 \<noteq> {} \<and> e2 \<noteq> {})"
   apply (simp add: connected_def openin_open, safe)
-  apply (simp_all, blast+)
+  apply (simp_all, blast+)  --\<open>slow\<close>
   done
 
 lemma connected_open_in_eq:
@@ -768,7 +768,7 @@
   apply (simp add: connected_closed_in, safe)
   apply blast
   by (metis Int_lower1 Un_subset_iff closedin_closed subset_antisym)
-    
+
 text \<open>These "transitivity" results are handy too\<close>
 
 lemma openin_trans[trans]:
@@ -1002,7 +1002,7 @@
         by (simp add: power_divide)
     qed auto
     also have "\<dots> = e"
-      using \<open>0 < e\<close> by (simp add: real_eq_of_nat)
+      using \<open>0 < e\<close> by simp
     finally show "y \<in> ball x e"
       by (auto simp: ball_def)
   qed (insert a b, auto simp: box_def)
@@ -1248,14 +1248,14 @@
 
 lemma UN_box_eq_UNIV: "(\<Union>i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One)) = UNIV"
 proof -
-  have "\<bar>x \<bullet> b\<bar> < real (ceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)) + 1)"
+  have "\<bar>x \<bullet> b\<bar> < real_of_int (ceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)) + 1)"
     if [simp]: "b \<in> Basis" for x b :: 'a
   proof -
-    have "\<bar>x \<bullet> b\<bar> \<le> real (ceiling \<bar>x \<bullet> b\<bar>)"
-      by (rule real_of_int_ceiling_ge)
-    also have "\<dots> \<le> real (ceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)))"
+    have "\<bar>x \<bullet> b\<bar> \<le> real_of_int (ceiling \<bar>x \<bullet> b\<bar>)"
+      by (rule le_of_int_ceiling)
+    also have "\<dots> \<le> real_of_int (ceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)))"
       by (auto intro!: ceiling_mono)
-    also have "\<dots> < real (ceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)) + 1)"
+    also have "\<dots> < real_of_int (ceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)) + 1)"
       by simp
     finally show ?thesis .
   qed
@@ -1277,15 +1277,11 @@
   unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff
   by (meson order_trans le_less_trans less_le_trans less_trans)+
 
-lemma is_interval_empty:
- "is_interval {}"
-  unfolding is_interval_def
-  by simp
-
-lemma is_interval_univ:
- "is_interval UNIV"
-  unfolding is_interval_def
-  by simp
+lemma is_interval_empty [iff]: "is_interval {}"
+  unfolding is_interval_def  by simp
+
+lemma is_interval_univ [iff]: "is_interval UNIV"
+  unfolding is_interval_def  by simp
 
 lemma mem_is_intervalI:
   assumes "is_interval s"
@@ -2162,7 +2158,7 @@
 
 lemma closed_in_connected_component: "closedin (subtopology euclidean s) (connected_component_set s x)"
 proof (cases "connected_component_set s x = {}")
-  case True then show ?thesis 
+  case True then show ?thesis
     by (metis closedin_empty)
 next
   case False
@@ -4383,7 +4379,7 @@
   obtain r where r: "subseq r" "monoseq (f \<circ> r)"
     unfolding comp_def by (metis seq_monosub)
   then have "Bseq (f \<circ> r)"
-    unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)
+    unfolding Bseq_eq_bounded using f by (force intro: bounded_subset)
   with r show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"
     using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)
 qed
@@ -4579,10 +4575,10 @@
   assumes "0 < e"
   obtains n :: nat where "1 / (Suc n) < e"
 proof atomize_elim
-  have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"
+  have "1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"
     by (rule divide_strict_left_mono) (auto simp: \<open>0 < e\<close>)
   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"
-    by (rule divide_left_mono) (auto simp: \<open>0 < e\<close>)
+    by (rule divide_left_mono) (auto simp: \<open>0 < e\<close> ceiling_correct)
   also have "\<dots> = e" by simp
   finally show  "\<exists>n. 1 / real (Suc n) < e" ..
 qed
@@ -4665,7 +4661,7 @@
         fix r :: real and N n m
         assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"
         then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"
-          using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)
+          using F_dec t by (auto simp: e_def field_simps of_nat_Suc)
         with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"
           by (auto simp: subset_eq)
         with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] \<open>2 * e N < r\<close>
@@ -5130,7 +5126,7 @@
     unfolding continuous_within Lim_within ball_def subset_eq
     apply (auto simp add: dist_commute)
     apply (erule_tac x=e in allE)
-    apply auto         
+    apply auto
     done
 qed
 
@@ -5338,7 +5334,7 @@
         fix n :: nat
         assume "n \<ge> N"
         then have "inverse (real n + 1) < inverse (real N)"
-          using real_of_nat_ge_zero and \<open>N\<noteq>0\<close> by auto
+          using of_nat_0_le_iff and \<open>N\<noteq>0\<close> by auto
         also have "\<dots> < e" using N by auto
         finally have "inverse (real n + 1) < e" by auto
         then have "dist (x n) (y n) < e"
@@ -5946,15 +5942,12 @@
 lemma bilinear_continuous_within_compose:
   "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>
     continuous (at x within s) (\<lambda>x. h (f x) (g x))"
-  unfolding continuous_within
-  using Lim_bilinear[of f "f x"]
-  by auto
+  by (rule Limits.bounded_bilinear.continuous)
 
 lemma bilinear_continuous_on_compose:
   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>
     continuous_on s (\<lambda>x. h (f x) (g x))"
-  unfolding continuous_on_def
-  by (fast elim: bounded_bilinear.tendsto)
+  by (rule Limits.bounded_bilinear.continuous_on)
 
 text \<open>Preservation of compactness and connectedness under continuous function.\<close>
 
@@ -6024,7 +6017,7 @@
   then show ?thesis
     unfolding connected_clopen by auto
 qed
-  
+
 lemma connected_linear_image:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "linear f" and "connected s"
@@ -6773,7 +6766,7 @@
       and x: "x \<in> closure(s)"
       and xlo: "\<And>x. x \<in> s ==> f(x) \<le> a"
     shows "f(x) \<le> a"
-    using image_closure_subset [OF f] 
+    using image_closure_subset [OF f]
   using image_closure_subset [OF f] closed_halfspace_le [of "1::real" a] assms
   by force
 
@@ -7140,15 +7133,12 @@
         then have "\<exists>N::nat. inverse (real (N + 1)) < e"
           using real_arch_inv[of e]
           apply (auto simp add: Suc_pred')
-          apply (metis Suc_pred' real_of_nat_Suc)
+          apply (metis Suc_pred' of_nat_Suc)
           done
-        then obtain N :: nat where "inverse (real (N + 1)) < e"
+        then obtain N :: nat where N: "inverse (real (N + 1)) < e"
           by auto
-        then have "\<forall>n\<ge>N. inverse (real n + 1) < e"
-          apply auto
-          apply (metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans
-            real_of_nat_Suc real_of_nat_Suc_gt_zero)
-          done
+        have "inverse (real n + 1) < e" if "N \<le> n" for n
+          by (auto intro!: that le_less_trans [OF _ N])
         then have "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto
       }
       then have "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Multivariate_Analysis/Weierstrass.thy
--- a/src/HOL/Multivariate_Analysis/Weierstrass.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Weierstrass.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -5,9 +5,6 @@
 
 begin
 
-(*FIXME: simplification changes to be enforced globally*)
-declare of_nat_Suc [simp del]
-
 (*Power.thy:*)
 declare power_divide [where b = "numeral w" for w, simp del]
 
@@ -45,7 +42,7 @@
 lemma sum_k_Bernstein [simp]: "(\<Sum>k = 0..n. real k * Bernstein n k x) = of_nat n * x"
   apply (subst binomial_deriv1 [of n x "1-x", simplified, symmetric])
   apply (simp add: setsum_left_distrib)
-  apply (auto simp: Bernstein_def real_of_nat_def algebra_simps realpow_num_eq_if intro!: setsum.cong)
+  apply (auto simp: Bernstein_def algebra_simps realpow_num_eq_if intro!: setsum.cong)
   done
 
 lemma sum_kk_Bernstein [simp]: "(\<Sum> k = 0..n. real k * (real k - 1) * Bernstein n k x) = real n * (real n - 1) * x\<^sup>2"
@@ -54,7 +51,7 @@
     apply (subst binomial_deriv2 [of n x "1-x", simplified, symmetric])
     apply (simp add: setsum_left_distrib)
     apply (rule setsum.cong [OF refl])
-    apply (simp add: Bernstein_def power2_eq_square algebra_simps real_of_nat_def)
+    apply (simp add: Bernstein_def power2_eq_square algebra_simps)
     apply (rename_tac k)
     apply (subgoal_tac "k = 0 \<or> k = 1 \<or> (\<exists>k'. k = Suc (Suc k'))")
     apply (force simp add: field_simps of_nat_Suc power2_eq_square)
@@ -90,13 +87,13 @@
       using e \<open>0<d\<close> by simp
     also have "... \<le> M * 4"
       using \<open>0\<le>M\<close> by simp
-    finally have [simp]: "real (nat \<lceil>4 * M / (e * d\<^sup>2)\<rceil>) = real \<lceil>4 * M / (e * d\<^sup>2)\<rceil>"
+    finally have [simp]: "real_of_int (nat \<lceil>4 * M / (e * d\<^sup>2)\<rceil>) = real_of_int \<lceil>4 * M / (e * d\<^sup>2)\<rceil>"
       using \<open>0\<le>M\<close> e \<open>0<d\<close>
-      by (simp add: Real.real_of_nat_Suc field_simps)
+      by (simp add: of_nat_Suc field_simps)
     have "4*M/(e*d\<^sup>2) + 1 \<le> real (Suc (nat\<lceil>4*M/(e*d\<^sup>2)\<rceil>))"
-      by (simp add: Real.real_of_nat_Suc)
+      by (simp add: of_nat_Suc real_nat_ceiling_ge)
     also have "... \<le> real n"
-      using n by (simp add: Real.real_of_nat_Suc field_simps)
+      using n by (simp add: of_nat_Suc field_simps)
     finally have nbig: "4*M/(e*d\<^sup>2) + 1 \<le> real n" .
     have sum_bern: "(\<Sum>k = 0..n. (x - k/n)\<^sup>2 * Bernstein n k x) = x * (1 - x) / n"
     proof -
@@ -296,7 +293,7 @@
       using pf01 by force
     moreover have "p x \<le> 1"
       using subU cardp t
-      apply (simp add: p_def divide_simps setsum_nonneg real_of_nat_def)
+      apply (simp add: p_def divide_simps setsum_nonneg)
       apply (rule setsum_bounded_above [where 'a=real and K=1, simplified])
       using pf01 by force
     ultimately show ?thesis
@@ -484,7 +481,7 @@
       using ff by fastforce
     moreover have "pff x \<le> 1"
       using subA cardp t
-      apply (simp add: pff_def divide_simps setsum_nonneg real_of_nat_def)
+      apply (simp add: pff_def divide_simps setsum_nonneg)
       apply (rule setprod_mono [where g = "\<lambda>x. 1", simplified])
       using ff by fastforce
     ultimately show ?thesis
@@ -571,7 +568,7 @@
   def B \<equiv> "\<lambda>j::nat. {x \<in> s. f x \<ge> (j + 1/3)*e}"
   have ngt: "(n-1) * e \<ge> normf f" "n\<ge>1"
     using e
-    apply (simp_all add: n_def field_simps real_of_nat_Suc)
+    apply (simp_all add: n_def field_simps of_nat_Suc)
     by (metis real_nat_ceiling_ge mult.commute not_less pos_less_divide_eq)
   then have ge_fx: "(n-1) * e \<ge> f x" if "x \<in> s" for x
     using f normf_upper that by fastforce
@@ -606,7 +603,7 @@
   have A0: "A 0 = {}"
     using fpos e by (fastforce simp: A_def)
   have An: "A n = s"
-    using e ngt f normf_upper by (fastforce simp: A_def field_simps real_of_nat_diff)
+    using e ngt f normf_upper by (fastforce simp: A_def field_simps of_nat_diff)
   have Asub: "A j \<subseteq> A i" if "i\<ge>j" for i j
     using e that apply (clarsimp simp: A_def)
     apply (erule order_trans, simp)
@@ -631,7 +628,7 @@
     then have Anj: "t \<notin> A (j-1)"
       using Least_le by (fastforce simp add: j_def)
     then have fj2: "(j - 4/3)*e < f t"
-      using j1 t  by (simp add: A_def real_of_nat_diff)
+      using j1 t  by (simp add: A_def of_nat_diff)
     have ***: "xf i t \<le> e/n" if "i\<ge>j" for i
       using xfA [OF Ai] that by (simp add: less_eq_real_def)
     { fix i
@@ -641,7 +638,7 @@
       then have "t \<in> B i"
         using Anj e ge_fx [OF t] \<open>1 \<le> n\<close> fpos [OF t] t
         apply (simp add: A_def B_def)
-        apply (clarsimp simp add: field_simps real_of_nat_diff not_le real_of_nat_Suc)
+        apply (clarsimp simp add: field_simps of_nat_diff not_le of_nat_Suc)
         apply (rule order_trans [of _ "e * 2 + (e * (real d * 3) + e * (real i * 3))"])
         apply auto
         done
@@ -658,15 +655,15 @@
       done
     also have "... \<le> e*j + e * ((Suc n - j)*e/n)"
       apply (rule add_mono)
-      apply (simp_all only: mult_le_cancel_left_pos e real_of_nat_def)
+      apply (simp_all only: mult_le_cancel_left_pos e)
       apply (rule setsum_bounded_above [OF xf_le1, where A = "lessThan j", simplified])
       using setsum_bounded_above [of "{j..n}" "\<lambda>i. xf i t", OF ***]
       apply simp
       done
     also have "... \<le> j*e + e*(n - j + 1)*e/n "
-      using \<open>1 \<le> n\<close> e  by (simp add: field_simps del: real_of_nat_Suc)
+      using \<open>1 \<le> n\<close> e  by (simp add: field_simps del: of_nat_Suc)
     also have "... \<le> j*e + e*e"
-      using \<open>1 \<le> n\<close> e j1 by (simp add: field_simps del: real_of_nat_Suc)
+      using \<open>1 \<le> n\<close> e j1 by (simp add: field_simps del: of_nat_Suc)
     also have "... < (j + 1/3)*e"
       using e by (auto simp: field_simps)
     finally have gj1: "g t < (j + 1 / 3) * e" .
@@ -678,7 +675,7 @@
     next
       case True
       then have "(j - 4/3)*e < (j-1)*e - e^2"
-        using e by (auto simp: real_of_nat_diff algebra_simps power2_eq_square)
+        using e by (auto simp: of_nat_diff algebra_simps power2_eq_square)
       also have "... < (j-1)*e - ((j - 1)/n) * e^2"
         using e True jn by (simp add: power2_eq_square field_simps)
       also have "... = e * (j-1) * (1 - e/n)"
@@ -688,7 +685,7 @@
         apply simp
         apply (rule order_trans [OF _ setsum_bounded_below [OF less_imp_le [OF ****]]])
         using True
-        apply (simp_all add: real_of_nat_def of_nat_Suc of_nat_diff)
+        apply (simp_all add: of_nat_Suc of_nat_diff)
         done
       also have "... \<le> g t"
         using jn e
@@ -739,7 +736,7 @@
       have "\<not> real (Suc n) < inverse e"
         using \<open>N \<le> n\<close> N using less_imp_inverse_less by force
       then have "1 / (1 + real n) \<le> e"
-        using e by (simp add: field_simps real_of_nat_Suc)
+        using e by (simp add: field_simps of_nat_Suc)
       then have "\<bar>f x - g x\<bar> < e"
         using n(2) x by auto
     } note * = this
@@ -922,7 +919,7 @@
   show ?case
     apply (rule_tac x="\<lambda>i. c" in exI)
     apply (rule_tac x=0 in exI)
-    apply (auto simp: mult_ac real_of_nat_Suc)
+    apply (auto simp: mult_ac of_nat_Suc)
     done
   case (add f1 f2)
   then obtain a1 n1 a2 n2 where
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/NSA/CLim.thy
--- a/src/HOL/NSA/CLim.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/NSA/CLim.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -140,15 +140,14 @@
      "DERIV (%x. x ^ n) x :> (complex_of_real (real n)) * (x ^ (n - Suc 0))"
 apply (induct n)
 apply (drule_tac [2] DERIV_ident [THEN DERIV_mult])
-apply (auto simp add: distrib_right real_of_nat_Suc)
+apply (auto simp add: distrib_right of_nat_Suc)
 apply (case_tac "n")
-apply (auto simp add: ac_simps add.commute)
+apply (auto simp add: ac_simps)
 done
 
 text{*Nonstandard version*}
-lemma NSCDERIV_pow:
-     "NSDERIV (%x. x ^ n) x :> complex_of_real (real n) * (x ^ (n - 1))"
-by (simp add: NSDERIV_DERIV_iff del: of_real_real_of_nat_eq)
+lemma NSCDERIV_pow: "NSDERIV (%x. x ^ n) x :> complex_of_real (real n) * (x ^ (n - 1))"
+    by (metis CDERIV_pow NSDERIV_DERIV_iff One_nat_def)
 
 text{*Can't relax the premise @{term "x \<noteq> 0"}: it isn't continuous at zero*}
 lemma NSCDERIV_inverse:
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/NSA/HSEQ.thy
--- a/src/HOL/NSA/HSEQ.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/NSA/HSEQ.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -225,11 +225,11 @@
 by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_zero)
 
 lemma NSLIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----NS> 0"
-by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat del: real_of_nat_Suc)
+by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat del: of_nat_Suc)
 
 lemma NSLIMSEQ_inverse_real_of_nat_add:
      "(%n. r + inverse(real(Suc n))) ----NS> r"
-by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat_add del: real_of_nat_Suc)
+by (simp add: LIMSEQ_NSLIMSEQ_iff [symmetric] LIMSEQ_inverse_real_of_nat_add del: of_nat_Suc)
 
 lemma NSLIMSEQ_inverse_real_of_nat_add_minus:
      "(%n. r + -inverse(real(Suc n))) ----NS> r"
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/NSA/HSeries.thy
--- a/src/HOL/NSA/HSeries.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/NSA/HSeries.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -2,8 +2,8 @@
     Author      : Jacques D. Fleuriot
     Copyright   : 1998  University of Cambridge
 
-Converted to Isar and polished by lcp    
-*) 
+Converted to Isar and polished by lcp
+*)
 
 section{*Finite Summation and Infinite Series for Hyperreals*}
 
@@ -13,7 +13,7 @@
 
 definition
   sumhr :: "(hypnat * hypnat * (nat=>real)) => hypreal" where
-  "sumhr = 
+  "sumhr =
       (%(M,N,f). starfun2 (%m n. setsum f {m..<n}) M N)"
 
 definition
@@ -36,8 +36,8 @@
 unfolding sumhr_app by transfer simp
 
 text{*Recursive case in definition of @{term sumr}*}
-lemma sumhr_if: 
-     "!!m n. sumhr(m,n+1,f) = 
+lemma sumhr_if:
+     "!!m n. sumhr(m,n+1,f) =
       (if n + 1 \<le> m then 0 else sumhr(m,n,f) + ( *f* f) n)"
 unfolding sumhr_app by transfer simp
 
@@ -73,14 +73,14 @@
 
 text{* other general version also needed *}
 lemma sumhr_fun_hypnat_eq:
-   "(\<forall>r. m \<le> r & r < n --> f r = g r) -->  
-      sumhr(hypnat_of_nat m, hypnat_of_nat n, f) =  
+   "(\<forall>r. m \<le> r & r < n --> f r = g r) -->
+      sumhr(hypnat_of_nat m, hypnat_of_nat n, f) =
       sumhr(hypnat_of_nat m, hypnat_of_nat n, g)"
 unfolding sumhr_app by transfer simp
 
 lemma sumhr_const:
      "!!n. sumhr(0, n, %i. r) = hypreal_of_hypnat n * hypreal_of_real r"
-unfolding sumhr_app by transfer (simp add: real_of_nat_def)
+unfolding sumhr_app by transfer simp
 
 lemma sumhr_less_bounds_zero [simp]: "!!m n. n < m ==> sumhr(m,n,f) = 0"
 unfolding sumhr_app by transfer simp
@@ -98,7 +98,7 @@
 
 text{*Infinite sums are obtained by summing to some infinite hypernatural
  (such as @{term whn})*}
-lemma sumhr_hypreal_of_hypnat_omega: 
+lemma sumhr_hypreal_of_hypnat_omega:
       "sumhr(0,whn,%i. 1) = hypreal_of_hypnat whn"
 by (simp add: sumhr_const)
 
@@ -108,33 +108,33 @@
 (* maybe define omega = hypreal_of_hypnat whn + 1 *)
 apply (unfold star_class_defs omega_def hypnat_omega_def
               of_hypnat_def star_of_def)
-apply (simp add: starfun_star_n starfun2_star_n real_of_nat_def)
+apply (simp add: starfun_star_n starfun2_star_n)
 done
 
-lemma sumhr_minus_one_realpow_zero [simp]: 
+lemma sumhr_minus_one_realpow_zero [simp]:
      "!!N. sumhr(0, N + N, %i. (-1) ^ (i+1)) = 0"
 unfolding sumhr_app
-apply transfer 
+apply transfer
 apply (simp del: power_Suc add: mult_2 [symmetric])
 apply (induct_tac N)
 apply simp_all
 done
 
 lemma sumhr_interval_const:
-     "(\<forall>n. m \<le> Suc n --> f n = r) & m \<le> na  
-      ==> sumhr(hypnat_of_nat m,hypnat_of_nat na,f) =  
+     "(\<forall>n. m \<le> Suc n --> f n = r) & m \<le> na
+      ==> sumhr(hypnat_of_nat m,hypnat_of_nat na,f) =
           (hypreal_of_nat (na - m) * hypreal_of_real r)"
 unfolding sumhr_app by transfer simp
 
 lemma starfunNat_sumr: "!!N. ( *f* (%n. setsum f {0..<n})) N = sumhr(0,N,f)"
 unfolding sumhr_app by transfer (rule refl)
 
-lemma sumhr_hrabs_approx [simp]: "sumhr(0, M, f) @= sumhr(0, N, f)  
+lemma sumhr_hrabs_approx [simp]: "sumhr(0, M, f) @= sumhr(0, N, f)
       ==> abs (sumhr(M, N, f)) @= 0"
 apply (cut_tac x = M and y = N in linorder_less_linear)
 apply (auto simp add: approx_refl)
 apply (drule approx_sym [THEN approx_minus_iff [THEN iffD1]])
-apply (auto dest: approx_hrabs 
+apply (auto dest: approx_hrabs
             simp add: sumhr_split_diff)
 done
 
@@ -166,7 +166,7 @@
 by (auto simp add: sums_NSsums_iff [symmetric] not_le[symmetric] intro!: sums_finite)
 
 lemma NSsummable_NSCauchy:
-     "NSsummable f =  
+     "NSsummable f =
       (\<forall>M \<in> HNatInfinite. \<forall>N \<in> HNatInfinite. abs (sumhr(M,N,f)) @= 0)"
 apply (auto simp add: summable_NSsummable_iff [symmetric]
        summable_iff_convergent convergent_NSconvergent_iff atLeast0LessThan[symmetric]
@@ -175,7 +175,7 @@
 apply auto
 apply (rule approx_minus_iff [THEN iffD2, THEN approx_sym])
 apply (rule_tac [2] approx_minus_iff [THEN iffD2])
-apply (auto dest: approx_hrabs_zero_cancel 
+apply (auto dest: approx_hrabs_zero_cancel
             simp add: sumhr_split_diff atLeast0LessThan[symmetric])
 done
 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/NSA/HyperDef.thy
--- a/src/HOL/NSA/HyperDef.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/NSA/HyperDef.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -231,19 +231,21 @@
 
 text{*A few lemmas first*}
 
-lemma lemma_omega_empty_singleton_disj: "{n::nat. x = real n} = {} |
-      (\<exists>y. {n::nat. x = real n} = {y})"
+lemma lemma_omega_empty_singleton_disj: 
+  "{n::nat. x = real n} = {} \<or> (\<exists>y. {n::nat. x = real n} = {y})"
 by force
 
 lemma lemma_finite_omega_set: "finite {n::nat. x = real n}"
-by (cut_tac x = x in lemma_omega_empty_singleton_disj, auto)
+  using lemma_omega_empty_singleton_disj [of x] by auto
 
 lemma not_ex_hypreal_of_real_eq_omega:
       "~ (\<exists>x. hypreal_of_real x = omega)"
-apply (simp add: omega_def)
-apply (simp add: star_of_def star_n_eq_iff)
-apply (auto simp add: real_of_nat_Suc diff_eq_eq [symmetric]
-            lemma_finite_omega_set [THEN FreeUltrafilterNat.finite])
+apply (simp add: omega_def star_of_def star_n_eq_iff)
+apply clarify
+apply (rule_tac x2="x-1" in lemma_finite_omega_set [THEN FreeUltrafilterNat.finite, THEN notE])
+apply (rule eventually_mono)
+prefer 2 apply assumption
+apply auto
 done
 
 lemma hypreal_of_real_not_eq_omega: "hypreal_of_real x \<noteq> omega"
@@ -262,7 +264,7 @@
 
 lemma not_ex_hypreal_of_real_eq_epsilon: "~ (\<exists>x. hypreal_of_real x = epsilon)"
 by (auto simp add: epsilon_def star_of_def star_n_eq_iff
-                   lemma_finite_epsilon_set [THEN FreeUltrafilterNat.finite] simp del: real_of_nat_Suc)
+                   lemma_finite_epsilon_set [THEN FreeUltrafilterNat.finite] simp del: of_nat_Suc)
 
 lemma hypreal_of_real_not_eq_epsilon: "hypreal_of_real x \<noteq> epsilon"
 by (insert not_ex_hypreal_of_real_eq_epsilon, auto)
@@ -307,21 +309,8 @@
 (* is a hyperreal c.f. NS extension                           *)
 (*------------------------------------------------------------*)
 
-lemma hypreal_of_nat_eq:
-     "hypreal_of_nat (n::nat) = hypreal_of_real (real n)"
-by (simp add: real_of_nat_def)
-
-lemma hypreal_of_nat:
-     "hypreal_of_nat m = star_n (%n. real m)"
-apply (fold star_of_def)
-apply (simp add: real_of_nat_def)
-done
-
-(*
-FIXME: we should declare this, as for type int, but many proofs would break.
-It replaces x+-y by x-y.
-Addsimps [symmetric hypreal_diff_def]
-*)
+lemma hypreal_of_nat: "hypreal_of_nat m = star_n (%n. real m)"
+by (simp add: star_of_def [symmetric])
 
 declaration {*
   K (Lin_Arith.add_inj_thms [@{thm star_of_le} RS iffD2,
@@ -383,12 +372,6 @@
 lemma two_hrealpow_ge_one [simp]: "(1::hypreal) \<le> 2 ^ n"
 by (insert power_increasing [of 0 n "2::hypreal"], simp)
 
-lemma two_hrealpow_gt [simp]: "hypreal_of_nat n < 2 ^ n"
-apply (induct n)
-apply (auto simp add: distrib_right)
-apply (cut_tac n = n in two_hrealpow_ge_one, arith)
-done
-
 lemma hrealpow:
     "star_n X ^ m = star_n (%n. (X n::real) ^ m)"
 apply (induct_tac "m")
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/NSA/NSA.thy
--- a/src/HOL/NSA/NSA.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/NSA/NSA.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -2022,7 +2022,7 @@
 
 lemma lemma_Infinitesimal:
      "(\<forall>r. 0 < r --> x < r) = (\<forall>n. x < inverse(real (Suc n)))"
-apply (auto simp add: real_of_nat_Suc_gt_zero simp del: real_of_nat_Suc)
+apply (auto simp add: real_of_nat_Suc_gt_zero simp del: of_nat_Suc)
 apply (blast dest!: reals_Archimedean intro: order_less_trans)
 done
 
@@ -2034,10 +2034,10 @@
   apply (simp (no_asm_use))
  apply (rule real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive, THEN star_of_less [THEN iffD2], THEN [2] impE])
   prefer 2 apply assumption
- apply (simp add: real_of_nat_def)
-apply (auto dest!: reals_Archimedean simp add: SReal_iff simp del: real_of_nat_Suc)
+ apply simp
+apply (auto dest!: reals_Archimedean simp add: SReal_iff simp del: of_nat_Suc)
 apply (drule star_of_less [THEN iffD2])
-apply (simp add: real_of_nat_def)
+apply simp
 apply (blast intro: order_less_trans)
 done
 
@@ -2057,16 +2057,11 @@
 by (auto simp add: less_Suc_eq)
 
 (*-------------------------------------------
-  Prove that any segment is finite and
-  hence cannot belong to FreeUltrafilterNat
+  Prove that any segment is finite and hence cannot belong to FreeUltrafilterNat
  -------------------------------------------*)
-lemma finite_nat_segment: "finite {n::nat. n < m}"
-apply (induct "m")
-apply (auto simp add: Suc_Un_eq)
-done
 
 lemma finite_real_of_nat_segment: "finite {n::nat. real n < real (m::nat)}"
-by (auto intro: finite_nat_segment)
+  by (auto intro: finite_Collect_less_nat)
 
 lemma finite_real_of_nat_less_real: "finite {n::nat. real n < u}"
 apply (cut_tac x = u in reals_Archimedean2, safe)
@@ -2106,13 +2101,12 @@
 
 text{*@{term omega} is a member of @{term HInfinite}*}
 
-lemma FreeUltrafilterNat_omega: "eventually (\<lambda>n. u < real n) FreeUltrafilterNat"
-  by (fact FreeUltrafilterNat_nat_gt_real)
-
 theorem HInfinite_omega [simp]: "omega \<in> HInfinite"
 apply (simp add: omega_def)
 apply (rule FreeUltrafilterNat_HInfinite)
-apply (simp add: real_of_nat_Suc diff_less_eq [symmetric] FreeUltrafilterNat_omega)
+apply clarify
+apply (rule_tac u1 = "u-1" in eventually_mono [OF _ FreeUltrafilterNat_nat_gt_real])
+apply auto
 done
 
 (*-----------------------------------------------
@@ -2145,21 +2139,22 @@
 
 lemma finite_inverse_real_of_posnat_gt_real:
      "0 < u ==> finite {n. u < inverse(real(Suc n))}"
-apply (simp (no_asm_simp) add: real_of_nat_less_inverse_iff del: real_of_nat_Suc)
-apply (simp (no_asm_simp) add: real_of_nat_Suc less_diff_eq [symmetric])
-apply (rule finite_real_of_nat_less_real)
-done
+proof (simp only: real_of_nat_less_inverse_iff)
+  have "{n. 1 + real n < inverse u} = {n. real n < inverse u - 1}"
+    by fastforce
+  thus "finite {n. real (Suc n) < inverse u}"
+    using finite_real_of_nat_less_real [of "inverse u - 1"] by auto
+qed
 
 lemma lemma_real_le_Un_eq2:
      "{n. u \<le> inverse(real(Suc n))} =
      {n. u < inverse(real(Suc n))} Un {n. u = inverse(real(Suc n))}"
-apply (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
-done
+by (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
 
 lemma finite_inverse_real_of_posnat_ge_real:
      "0 < u ==> finite {n. u \<le> inverse(real(Suc n))}"
 by (auto simp add: lemma_real_le_Un_eq2 lemma_finite_epsilon_set finite_inverse_real_of_posnat_gt_real 
-            simp del: real_of_nat_Suc)
+            simp del: of_nat_Suc)
 
 lemma inverse_real_of_posnat_ge_real_FreeUltrafilterNat:
      "0 < u ==> \<not> eventually (\<lambda>n. u \<le> inverse(real(Suc n))) FreeUltrafilterNat"
@@ -2187,7 +2182,7 @@
 lemma SEQ_Infinitesimal:
       "( *f* (%n::nat. inverse(real(Suc n)))) whn : Infinitesimal"
 by (simp add: hypnat_omega_def starfun_star_n star_n_inverse Infinitesimal_FreeUltrafilterNat_iff 
-       real_of_nat_Suc_gt_zero FreeUltrafilterNat_inverse_real_of_posnat del: real_of_nat_Suc)
+       real_of_nat_Suc_gt_zero FreeUltrafilterNat_inverse_real_of_posnat del: of_nat_Suc)
 
 text{* Example where we get a hyperreal from a real sequence
       for which a particular property holds. The theorem is
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/NSA/NSComplex.thy
--- a/src/HOL/NSA/NSComplex.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/NSA/NSComplex.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -552,7 +552,6 @@
 
 lemma NSDeMoivre: "!!a. (hcis a) ^ n = hcis (hypreal_of_nat n * a)"
 apply transfer
-apply (fold real_of_nat_def)
 apply (rule DeMoivre)
 done
 
@@ -563,15 +562,15 @@
 
 lemma NSDeMoivre_ext:
   "!!a n. (hcis a) pow n = hcis (hypreal_of_hypnat n * a)"
-by transfer (fold real_of_nat_def, rule DeMoivre)
+by transfer (rule DeMoivre)
 
 lemma NSDeMoivre2:
   "!!a r. (hrcis r a) ^ n = hrcis (r ^ n) (hypreal_of_nat n * a)"
-by transfer (fold real_of_nat_def, rule DeMoivre2)
+by transfer (rule DeMoivre2)
 
 lemma DeMoivre2_ext:
   "!! a r n. (hrcis r a) pow n = hrcis (r pow n) (hypreal_of_hypnat n * a)"
-by transfer (fold real_of_nat_def, rule DeMoivre2)
+by transfer (rule DeMoivre2)
 
 lemma hcis_inverse [simp]: "!!a. inverse(hcis a) = hcis (-a)"
 by transfer (rule cis_inverse)
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/NSA/NatStar.thy
--- a/src/HOL/NSA/NatStar.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/NSA/NatStar.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -115,7 +115,7 @@
   @{term real_of_nat} *}
 
 lemma starfunNat_real_of_nat: "( *f* real) = hypreal_of_hypnat"
-by transfer (simp add: fun_eq_iff real_of_nat_def)
+by transfer (simp add: fun_eq_iff)
 
 lemma starfun_inverse_real_of_nat_eq:
      "N \<in> HNatInfinite
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/NSA/Star.thy
--- a/src/HOL/NSA/Star.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/NSA/Star.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -110,7 +110,7 @@
 (*
    Prove that abs for hypreal is a nonstandard extension of abs for real w/o
    use of congruence property (proved after this for general
-   nonstandard extensions of real valued functions). 
+   nonstandard extensions of real valued functions).
 
    Proof now Uses the ultrafilter tactic!
 *)
@@ -307,12 +307,12 @@
      "(star_n X \<in> Infinitesimal) = (\<forall>m. eventually (\<lambda>n. norm(X n) < inverse(real(Suc m))) \<U>)"
 by (simp add: Infinitesimal_hypreal_of_nat_iff star_of_def
      hnorm_def star_of_nat_def starfun_star_n
-     star_n_inverse star_n_less real_of_nat_def)
+     star_n_inverse star_n_less)
 
 lemma HNatInfinite_inverse_Infinitesimal [simp]:
      "n \<in> HNatInfinite ==> inverse (hypreal_of_hypnat n) \<in> Infinitesimal"
 apply (cases n)
-apply (auto simp add: of_hypnat_def starfun_star_n real_of_nat_def [symmetric] star_n_inverse real_norm_def
+apply (auto simp add: of_hypnat_def starfun_star_n star_n_inverse real_norm_def
       HNatInfinite_FreeUltrafilterNat_iff
       Infinitesimal_FreeUltrafilterNat_iff2)
 apply (drule_tac x="Suc m" in spec)
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/NthRoot.thy
--- a/src/HOL/NthRoot.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/NthRoot.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -16,7 +16,7 @@
 lemma inverse_sgn: "sgn (inverse a) = inverse (sgn a :: real)"
   by (simp add: sgn_real_def)
 
-lemma power_eq_iff_eq_base: 
+lemma power_eq_iff_eq_base:
   fixes a b :: "_ :: linordered_semidom"
   shows "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b"
   using power_eq_imp_eq_base[of a n b] by auto
@@ -274,7 +274,7 @@
 lemma continuous_real_root[continuous_intros]:
   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. root n (f x))"
   unfolding continuous_def by (rule tendsto_real_root)
-  
+
 lemma continuous_on_real_root[continuous_intros]:
   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. root n (f x))"
   unfolding continuous_on_def by (auto intro: tendsto_real_root)
@@ -326,7 +326,7 @@
   qed
 next
   show "DERIV (\<lambda>x. - (x ^ n)) (root n x) :> - real n * root n x ^ (n - Suc 0)"
-    by  (auto intro!: derivative_eq_intros simp: real_of_nat_def)
+    by  (auto intro!: derivative_eq_intros)
   show "- real n * root n x ^ (n - Suc 0) \<noteq> 0"
     using n x by simp
 qed (rule isCont_real_root)
@@ -463,7 +463,7 @@
 lemma continuous_real_sqrt[continuous_intros]:
   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sqrt (f x))"
   unfolding sqrt_def by (rule continuous_real_root)
-  
+
 lemma continuous_on_real_sqrt[continuous_intros]:
   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sqrt (f x))"
   unfolding sqrt_def by (rule continuous_on_real_root)
@@ -510,14 +510,14 @@
 proof cases
   assume "x=0" thus ?thesis by simp
 next
-  assume nz: "x\<noteq>0" 
+  assume nz: "x\<noteq>0"
   hence pos: "0<x" using nneg by arith
   show ?thesis
-  proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 
-    show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 
+  proof (rule right_inverse_eq [THEN iffD1, THEN sym])
+    show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz)
     show "inverse (sqrt x) / (sqrt x / x) = 1"
-      by (simp add: divide_inverse mult.assoc [symmetric] 
-                  power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 
+      by (simp add: divide_inverse mult.assoc [symmetric]
+                  power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz)
   qed
 qed
 
@@ -537,7 +537,7 @@
 lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"
 by (simp add: divide_less_eq)
 
-lemma four_x_squared: 
+lemma four_x_squared:
   fixes x::real
   shows "4 * x\<^sup>2 = (2 * x)\<^sup>2"
 by (simp add: power2_eq_square)
@@ -548,7 +548,7 @@
 
 subsection \<open>Square Root of Sum of Squares\<close>
 
-lemma sum_squares_bound: 
+lemma sum_squares_bound:
   fixes x:: "'a::linordered_field"
   shows "2*x*y \<le> x^2 + y^2"
 proof -
@@ -560,14 +560,14 @@
     by arith
 qed
 
-lemma arith_geo_mean: 
+lemma arith_geo_mean:
   fixes u:: "'a::linordered_field" assumes "u\<^sup>2 = x*y" "x\<ge>0" "y\<ge>0" shows "u \<le> (x + y)/2"
     apply (rule power2_le_imp_le)
     using sum_squares_bound assms
     apply (auto simp: zero_le_mult_iff)
     by (auto simp: algebra_simps power2_eq_square)
 
-lemma arith_geo_mean_sqrt: 
+lemma arith_geo_mean_sqrt:
   fixes x::real assumes "x\<ge>0" "y\<ge>0" shows "sqrt(x*y) \<le> (x + y)/2"
   apply (rule arith_geo_mean)
   using assms
@@ -645,10 +645,10 @@
   using less_eq_real_def sqrt2_less_2 apply force
   apply assumption
   apply (rule le_less_trans [where y = "y*2"])
-  using less_eq_real_def sqrt2_less_2 mult_le_cancel_left 
-  apply auto 
+  using less_eq_real_def sqrt2_less_2 mult_le_cancel_left
+  apply auto
   done
-  
+
 lemma LIMSEQ_root: "(\<lambda>n. root n n) ----> 1"
 proof -
   def x \<equiv> "\<lambda>n. root n n - 1"
@@ -660,7 +660,7 @@
     { fix n :: nat assume "2 < n"
       have "1 + (real (n - 1) * n) / 2 * x n^2 = 1 + of_nat (n choose 2) * x n^2"
         using \<open>2 < n\<close> unfolding gbinomial_def binomial_gbinomial
-        by (simp add: atLeast0AtMost atMost_Suc field_simps real_of_nat_diff numeral_2_eq_2 real_eq_of_nat[symmetric])
+        by (simp add: atLeast0AtMost atMost_Suc field_simps of_nat_diff numeral_2_eq_2)
       also have "\<dots> \<le> (\<Sum>k\<in>{0, 2}. of_nat (n choose k) * x n^k)"
         by (simp add: x_def)
       also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)"
@@ -697,7 +697,7 @@
            (simp_all add: at_infinity_eq_at_top_bot)
       { fix n :: nat assume "1 < n"
         have "1 + x n * n = 1 + of_nat (n choose 1) * x n^1"
-          using \<open>1 < n\<close> unfolding gbinomial_def binomial_gbinomial by (simp add: real_eq_of_nat[symmetric])
+          using \<open>1 < n\<close> unfolding gbinomial_def binomial_gbinomial by simp 
         also have "\<dots> \<le> (\<Sum>k\<in>{0, 1}. of_nat (n choose k) * x n^k)"
           by (simp add: x_def)
         also have "\<dots> \<le> (\<Sum>k=0..n. of_nat (n choose k) * x n^k)"
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Old_Number_Theory/Fib.thy
--- a/src/HOL/Old_Number_Theory/Fib.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Old_Number_Theory/Fib.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -86,9 +86,7 @@
   (if n mod 2 = 0 then fib (Suc n) * fib (Suc n) - 1
    else fib (Suc n) * fib (Suc n) + 1)"
   apply (rule int_int_eq [THEN iffD1]) 
-  apply (simp add: fib_Cassini_int)
-  apply (subst of_nat_diff)
-   apply (insert fib_Suc_gr_0 [of n], simp_all)
+  using fib_Cassini_int apply (auto simp add: Suc_leI fib_Suc_gr_0 of_nat_diff)
   done
 
 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Old_Number_Theory/Quadratic_Reciprocity.thy
--- a/src/HOL/Old_Number_Theory/Quadratic_Reciprocity.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Old_Number_Theory/Quadratic_Reciprocity.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -274,7 +274,7 @@
 
 lemma S_card: "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))"
   using P_set_card Q_set_card P_set_finite Q_set_finite
-  by (auto simp add: S_def zmult_int)
+  by (simp add: S_def)
 
 lemma S1_Int_S2_prop: "S1 \<inter> S2 = {}"
   by (auto simp add: S1_def S2_def)
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Binary_Product_Measure.thy
--- a/src/HOL/Probability/Binary_Product_Measure.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Binary_Product_Measure.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -717,7 +717,7 @@
     apply (simp add: fin_Pair emeasure_count_space X_subset fin_X)
     apply (subst nn_integral_count_space)
     using A apply simp
-    apply (simp del: real_of_nat_setsum add: real_of_nat_setsum[symmetric])
+    apply (simp del: of_nat_setsum add: of_nat_setsum[symmetric])
     apply (subst card_gt_0_iff)
     apply (simp add: fin_Pair)
     apply (subst card_SigmaI[symmetric])
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Bochner_Integration.thy
--- a/src/HOL/Probability/Bochner_Integration.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Bochner_Integration.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -153,7 +153,7 @@
   qed
 qed
 
-lemma real_indicator: "real (indicator A x :: ereal) = indicator A x"
+lemma real_indicator: "real_of_ereal (indicator A x :: ereal) = indicator A x"
   unfolding indicator_def by auto
 
 lemma split_indicator_asm:
@@ -182,9 +182,9 @@
     sup: "\<And>x. (SUP i. U i x) = max 0 (ereal (u x))" and nn: "\<And>i x. 0 \<le> U i x"
     by blast
 
-  def U' \<equiv> "\<lambda>i x. indicator (space M) x * real (U i x)"
+  def U' \<equiv> "\<lambda>i x. indicator (space M) x * real_of_ereal (U i x)"
   then have U'_sf[measurable]: "\<And>i. simple_function M (U' i)"
-    using U by (auto intro!: simple_function_compose1[where g=real])
+    using U by (auto intro!: simple_function_compose1[where g=real_of_ereal])
 
   show "P u"
   proof (rule seq)
@@ -209,7 +209,7 @@
       by simp
   next
     fix i
-    have "U' i ` space M \<subseteq> real ` (U i ` space M)" "finite (U i ` space M)"
+    have "U' i ` space M \<subseteq> real_of_ereal ` (U i ` space M)" "finite (U i ` space M)"
       unfolding U'_def using U(1) by (auto dest: simple_functionD)
     then have fin: "finite (U' i ` space M)"
       by (metis finite_subset finite_imageI)
@@ -472,7 +472,7 @@
           emeasure M ((\<lambda>x. ereal (f x)) -` {ereal (f y)} \<inter> space M)" by simp }
   with f have "simple_bochner_integral M f = (\<integral>\<^sup>Sx. f x \<partial>M)"
     unfolding simple_integral_def
-    by (subst simple_bochner_integral_partition[OF f(1), where g="\<lambda>x. ereal (f x)" and v=real])
+    by (subst simple_bochner_integral_partition[OF f(1), where g="\<lambda>x. ereal (f x)" and v=real_of_ereal])
        (auto intro: f simple_function_compose1 elim: simple_bochner_integrable.cases
              intro!: setsum.cong ereal_cong_mult
              simp: setsum_ereal[symmetric] times_ereal.simps(1)[symmetric] ac_simps
@@ -1644,7 +1644,7 @@
 lemma integral_eq_nn_integral:
   assumes [measurable]: "f \<in> borel_measurable M"
   assumes nonneg: "AE x in M. 0 \<le> f x"
-  shows "integral\<^sup>L M f = real (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
+  shows "integral\<^sup>L M f = real_of_ereal (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
 proof cases
   assume *: "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) = \<infinity>"
   also have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) = (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
@@ -2138,15 +2138,15 @@
 
 lemma real_lebesgue_integral_def:
   assumes f[measurable]: "integrable M f"
-  shows "integral\<^sup>L M f = real (\<integral>\<^sup>+x. f x \<partial>M) - real (\<integral>\<^sup>+x. - f x \<partial>M)"
+  shows "integral\<^sup>L M f = real_of_ereal (\<integral>\<^sup>+x. f x \<partial>M) - real_of_ereal (\<integral>\<^sup>+x. - f x \<partial>M)"
 proof -
   have "integral\<^sup>L M f = integral\<^sup>L M (\<lambda>x. max 0 (f x) - max 0 (- f x))"
     by (auto intro!: arg_cong[where f="integral\<^sup>L M"])
   also have "\<dots> = integral\<^sup>L M (\<lambda>x. max 0 (f x)) - integral\<^sup>L M (\<lambda>x. max 0 (- f x))"
     by (intro integral_diff integrable_max integrable_minus integrable_zero f)
-  also have "integral\<^sup>L M (\<lambda>x. max 0 (f x)) = real (\<integral>\<^sup>+x. max 0 (f x) \<partial>M)"
+  also have "integral\<^sup>L M (\<lambda>x. max 0 (f x)) = real_of_ereal (\<integral>\<^sup>+x. max 0 (f x) \<partial>M)"
     by (subst integral_eq_nn_integral[symmetric]) auto
-  also have "integral\<^sup>L M (\<lambda>x. max 0 (- f x)) = real (\<integral>\<^sup>+x. max 0 (- f x) \<partial>M)"
+  also have "integral\<^sup>L M (\<lambda>x. max 0 (- f x)) = real_of_ereal (\<integral>\<^sup>+x. max 0 (- f x) \<partial>M)"
     by (subst integral_eq_nn_integral[symmetric]) auto
   also have "(\<lambda>x. ereal (max 0 (f x))) = (\<lambda>x. max 0 (ereal (f x)))"
     by (auto simp: max_def)
@@ -2346,13 +2346,13 @@
 lemma (in finite_measure) ereal_integral_real:
   assumes [measurable]: "f \<in> borel_measurable M" 
   assumes ae: "AE x in M. 0 \<le> f x" "AE x in M. f x \<le> ereal B"
-  shows "ereal (\<integral>x. real (f x) \<partial>M) = (\<integral>\<^sup>+x. f x \<partial>M)"
+  shows "ereal (\<integral>x. real_of_ereal (f x) \<partial>M) = (\<integral>\<^sup>+x. f x \<partial>M)"
 proof (subst nn_integral_eq_integral[symmetric])
-  show "integrable M (\<lambda>x. real (f x))"
+  show "integrable M (\<lambda>x. real_of_ereal (f x))"
     using ae by (intro integrable_const_bound[where B=B]) (auto simp: real_le_ereal_iff)
-  show "AE x in M. 0 \<le> real (f x)"
+  show "AE x in M. 0 \<le> real_of_ereal (f x)"
     using ae by (auto simp: real_of_ereal_pos)
-  show "(\<integral>\<^sup>+ x. ereal (real (f x)) \<partial>M) = integral\<^sup>N M f"
+  show "(\<integral>\<^sup>+ x. ereal (real_of_ereal (f x)) \<partial>M) = integral\<^sup>N M f"
     using ae by (intro nn_integral_cong_AE) (auto simp: ereal_real)
 qed
 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Borel_Space.thy
--- a/src/HOL/Probability/Borel_Space.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Borel_Space.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -618,7 +618,7 @@
   interpret sigma_algebra UNIV ?SIGMA
     by (intro sigma_algebra_sigma_sets) simp_all
   have *: "?set = (\<Union>n. UNIV - {x::'a. x \<bullet> i < a + 1 / real (Suc n)})"
-  proof (safe, simp_all add: not_less del: real_of_nat_Suc)
+  proof (safe, simp_all add: not_less del: of_nat_Suc)
     fix x :: 'a assume "a < x \<bullet> i"
     with reals_Archimedean[of "x \<bullet> i - a"]
     obtain n where "a + 1 / real (Suc n) < x \<bullet> i"
@@ -655,7 +655,7 @@
   fix a :: real and i :: 'a assume "(a, i) \<in> UNIV \<times> Basis"
   then have i: "i \<in> Basis" by auto
   have *: "{x::'a. x\<bullet>i < a} = (\<Union>n. {x. x\<bullet>i \<le> a - 1/real (Suc n)})"
-  proof (safe, simp_all del: real_of_nat_Suc)
+  proof (safe, simp_all del: of_nat_Suc)
     fix x::'a assume *: "x\<bullet>i < a"
     with reals_Archimedean[of "a - x\<bullet>i"]
     obtain n where "x \<bullet> i < a - 1 / (real (Suc n))"
@@ -1055,7 +1055,7 @@
 lemma measurable_real_floor[measurable]:
   "(floor :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
 proof -
-  have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real a \<le> x \<and> x < real (a + 1))"
+  have "\<And>a x. \<lfloor>x\<rfloor> = a \<longleftrightarrow> (real_of_int a \<le> x \<and> x < real_of_int (a + 1))"
     by (auto intro: floor_eq2)
   then show ?thesis
     by (auto simp: vimage_def measurable_count_space_eq2_countable)
@@ -1065,7 +1065,7 @@
   "(ceiling :: real \<Rightarrow> int) \<in> measurable borel (count_space UNIV)"
   unfolding ceiling_def[abs_def] by simp
 
-lemma borel_measurable_real_floor: "(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
+lemma borel_measurable_real_floor: "(\<lambda>x::real. real_of_int \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
   by simp
 
 lemma borel_measurable_root [measurable]: "root n \<in> borel_measurable borel"
@@ -1116,7 +1116,7 @@
 lemma borel_measurable_real_of_ereal[measurable (raw)]:
   fixes f :: "'a \<Rightarrow> ereal" 
   assumes f: "f \<in> borel_measurable M"
-  shows "(\<lambda>x. real (f x)) \<in> borel_measurable M"
+  shows "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M"
   apply (rule measurable_compose[OF f])
   apply (rule borel_measurable_continuous_countable_exceptions[of "{\<infinity>, -\<infinity> }"])
   apply (auto intro: continuous_on_real simp: Compl_eq_Diff_UNIV)
@@ -1125,10 +1125,10 @@
 lemma borel_measurable_ereal_cases:
   fixes f :: "'a \<Rightarrow> ereal" 
   assumes f: "f \<in> borel_measurable M"
-  assumes H: "(\<lambda>x. H (ereal (real (f x)))) \<in> borel_measurable M"
+  assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x)))) \<in> borel_measurable M"
   shows "(\<lambda>x. H (f x)) \<in> borel_measurable M"
 proof -
-  let ?F = "\<lambda>x. if f x = \<infinity> then H \<infinity> else if f x = - \<infinity> then H (-\<infinity>) else H (ereal (real (f x)))"
+  let ?F = "\<lambda>x. if f x = \<infinity> then H \<infinity> else if f x = - \<infinity> then H (-\<infinity>) else H (ereal (real_of_ereal (f x)))"
   { fix x have "H (f x) = ?F x" by (cases "f x") auto }
   with f H show ?thesis by simp
 qed
@@ -1150,15 +1150,15 @@
   fixes f g :: "'a \<Rightarrow> ereal" 
   assumes f: "f \<in> borel_measurable M"
   assumes g: "g \<in> borel_measurable M"
-  assumes H: "{x \<in> space M. H (ereal (real (f x))) (ereal (real (g x)))} \<in> sets M"
+  assumes H: "{x \<in> space M. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))} \<in> sets M"
     "{x \<in> space borel. H (-\<infinity>) (ereal x)} \<in> sets borel"
     "{x \<in> space borel. H (\<infinity>) (ereal x)} \<in> sets borel"
     "{x \<in> space borel. H (ereal x) (-\<infinity>)} \<in> sets borel"
     "{x \<in> space borel. H (ereal x) (\<infinity>)} \<in> sets borel"
   shows "{x \<in> space M. H (f x) (g x)} \<in> sets M"
 proof -
-  let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = -\<infinity> then H y (-\<infinity>) else H y (ereal (real (g x)))"
-  let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = -\<infinity> then ?G (-\<infinity>) x else ?G (ereal (real (f x))) x"
+  let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = -\<infinity> then H y (-\<infinity>) else H y (ereal (real_of_ereal (g x)))"
+  let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = -\<infinity> then ?G (-\<infinity>) x else ?G (ereal (real_of_ereal (f x))) x"
   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
   note * = this
   from assms show ?thesis
@@ -1180,12 +1180,12 @@
 lemma borel_measurable_ereal_iff_real:
   fixes f :: "'a \<Rightarrow> ereal"
   shows "f \<in> borel_measurable M \<longleftrightarrow>
-    ((\<lambda>x. real (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
+    ((\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M \<and> f -` {\<infinity>} \<inter> space M \<in> sets M \<and> f -` {-\<infinity>} \<inter> space M \<in> sets M)"
 proof safe
-  assume *: "(\<lambda>x. real (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"
+  assume *: "(\<lambda>x. real_of_ereal (f x)) \<in> borel_measurable M" "f -` {\<infinity>} \<inter> space M \<in> sets M" "f -` {-\<infinity>} \<inter> space M \<in> sets M"
   have "f -` {\<infinity>} \<inter> space M = {x\<in>space M. f x = \<infinity>}" "f -` {-\<infinity>} \<inter> space M = {x\<in>space M. f x = -\<infinity>}" by auto
   with * have **: "{x\<in>space M. f x = \<infinity>} \<in> sets M" "{x\<in>space M. f x = -\<infinity>} \<in> sets M" by simp_all
-  let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real (f x))"
+  let ?f = "\<lambda>x. if f x = \<infinity> then \<infinity> else if f x = -\<infinity> then -\<infinity> else ereal (real_of_ereal (f x))"
   have "?f \<in> borel_measurable M" using * ** by (intro measurable_If) auto
   also have "?f = f" by (auto simp: fun_eq_iff ereal_real)
   finally show "f \<in> borel_measurable M" .
@@ -1221,15 +1221,15 @@
   fixes f g :: "'a \<Rightarrow> ereal" 
   assumes f: "f \<in> borel_measurable M"
   assumes g: "g \<in> borel_measurable M"
-  assumes H: "(\<lambda>x. H (ereal (real (f x))) (ereal (real (g x)))) \<in> borel_measurable M"
-    "(\<lambda>x. H (-\<infinity>) (ereal (real (g x)))) \<in> borel_measurable M"
-    "(\<lambda>x. H (\<infinity>) (ereal (real (g x)))) \<in> borel_measurable M"
-    "(\<lambda>x. H (ereal (real (f x))) (-\<infinity>)) \<in> borel_measurable M"
-    "(\<lambda>x. H (ereal (real (f x))) (\<infinity>)) \<in> borel_measurable M"
+  assumes H: "(\<lambda>x. H (ereal (real_of_ereal (f x))) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
+    "(\<lambda>x. H (-\<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
+    "(\<lambda>x. H (\<infinity>) (ereal (real_of_ereal (g x)))) \<in> borel_measurable M"
+    "(\<lambda>x. H (ereal (real_of_ereal (f x))) (-\<infinity>)) \<in> borel_measurable M"
+    "(\<lambda>x. H (ereal (real_of_ereal (f x))) (\<infinity>)) \<in> borel_measurable M"
   shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
 proof -
-  let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = - \<infinity> then H y (-\<infinity>) else H y (ereal (real (g x)))"
-  let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = - \<infinity> then ?G (-\<infinity>) x else ?G (ereal (real (f x))) x"
+  let ?G = "\<lambda>y x. if g x = \<infinity> then H y \<infinity> else if g x = - \<infinity> then H y (-\<infinity>) else H y (ereal (real_of_ereal (g x)))"
+  let ?F = "\<lambda>x. if f x = \<infinity> then ?G \<infinity> x else if f x = - \<infinity> then ?G (-\<infinity>) x else ?G (ereal (real_of_ereal (f x))) x"
   { fix x have "H (f x) (g x) = ?F x" by (cases "f x" "g x" rule: ereal2_cases) auto }
   note * = this
   from assms show ?thesis unfolding * by simp
@@ -1427,7 +1427,7 @@
         also have "\<dots> < ?I i / 2 + ?I i / 2"
           by (intro add_strict_mono d less_trans[OF _ j] *)
         also have "\<dots> \<le> ?I i"
-          by (simp add: field_simps real_of_nat_Suc)
+          by (simp add: field_simps of_nat_Suc)
         finally show "dist (f y) (f z) \<le> ?I i"
           by simp
       qed
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Distributions.thy
--- a/src/HOL/Probability/Distributions.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Distributions.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -25,7 +25,7 @@
     by simp
   show "random_variable lborel (\<lambda>x. t + c * X x)"
     by simp
-  
+
   have "AE x in lborel. 0 \<le> f x"
     using f by (simp add: distributed_def)
   from AE_borel_affine[OF _ _ this, where c="1/c" and t="- t / c"] c
@@ -34,7 +34,7 @@
 
   have eq: "\<And>x. ereal \<bar>c\<bar> * (f x / ereal \<bar>c\<bar>) = f x"
     using c by (simp add: divide_ereal_def ac_simps one_ereal_def[symmetric])
-    
+
   have "density lborel f = distr M lborel X"
     using f by (simp add: distributed_def)
   with c show "distr M lborel (\<lambda>x. t + c * X x) = density lborel (\<lambda>x. f ((x - t) / c) / ereal \<bar>c\<bar>)"
@@ -78,7 +78,7 @@
 proof -
   let ?f = "\<lambda>k x. x^k * exp (-x) / fact k"
   let ?F = "\<lambda>k x. - (\<Sum>n\<le>k. (x^n * exp (-x)) / fact n)"
-  have "?I * (inverse (real_of_nat (fact k))) = 
+  have "?I * (inverse (real_of_nat (fact k))) =
       (\<integral>\<^sup>+x. ereal (x^k * exp (-x)) * indicator {0 .. a} x * (inverse (real_of_nat (fact k))) \<partial>lborel)"
     by (intro nn_integral_multc[symmetric]) auto
   also have "\<dots> = (\<integral>\<^sup>+x. ereal (?f k x) * indicator {0 .. a} x \<partial>lborel)"
@@ -97,12 +97,12 @@
         ?f k x - ((real (Suc k) - x) * x ^ k * exp (- x)) / (fact (Suc k))"
         by (intro DERIV_diff Suc)
            (auto intro!: derivative_eq_intros simp del: fact_Suc power_Suc
-                 simp add: field_simps power_Suc[symmetric] real_of_nat_def[symmetric])
+                 simp add: field_simps power_Suc[symmetric])
       also have "(\<lambda>x. ?F k x - (x^Suc k * exp (-x)) / fact (Suc k)) = ?F (Suc k)"
         by simp
       also have "?f k x - ((real (Suc k) - x) * x ^ k * exp (- x)) / (fact (Suc k)) = ?f (Suc k) x"
         by (auto simp: field_simps simp del: fact_Suc)
-           (simp_all add: real_of_nat_Suc field_simps)
+           (simp_all add: of_nat_Suc field_simps)
       finally show ?case .
     qed
   qed auto
@@ -166,20 +166,20 @@
     by (auto simp: erlang_CDF_def)
   finally show ?thesis .
 next
-  assume "\<not> 0 \<le> a" 
+  assume "\<not> 0 \<le> a"
   moreover then have "(\<integral>\<^sup>+ x. ereal (erlang_density k l x) * indicator {.. a} x \<partial>lborel) = (\<integral>\<^sup>+x. 0 \<partial>(lborel::real measure))"
     by (intro nn_integral_cong) (auto simp: erlang_density_def)
   ultimately show ?thesis
     by (simp add: erlang_CDF_def)
 qed
 
-lemma emeasure_erlang_density: 
+lemma emeasure_erlang_density:
   "0 < l \<Longrightarrow> emeasure (density lborel (erlang_density k l)) {.. a} = erlang_CDF k l a"
   by (simp add: emeasure_density nn_integral_erlang_density)
 
-lemma nn_integral_erlang_ith_moment: 
+lemma nn_integral_erlang_ith_moment:
   fixes k i :: nat and l :: real
-  assumes [arith]: "0 < l" 
+  assumes [arith]: "0 < l"
   shows "(\<integral>\<^sup>+ x. ereal (erlang_density k l x * x ^ i) \<partial>lborel) = fact (k + i) / (fact k * l ^ i)"
 proof -
   have eq: "\<And>x. indicator {0..} (x / l) = indicator {0..} x"
@@ -241,7 +241,7 @@
   shows "distributed M lborel X (erlang_density k l)"
 proof (rule distributedI_borel_atMost)
   fix a :: real
-  { assume "a \<le> 0"  
+  { assume "a \<le> 0"
     with X have "emeasure M {x\<in>space M. X x \<le> a} \<le> emeasure M {x\<in>space M. X x \<le> 0}"
       by (intro emeasure_mono) auto
     also have "... = 0"  by (auto intro!: erlang_CDF_at0 simp: X_distr[of 0])
@@ -266,31 +266,31 @@
     distributed_measurable[of M lborel X "erlang_density k l"]
     emeasure_erlang_density[of l]
     erlang_distributed_le[of X k l]
-  by (auto intro!: erlang_distributedI simp: one_ereal_def emeasure_eq_measure) 
+  by (auto intro!: erlang_distributedI simp: one_ereal_def emeasure_eq_measure)
 
 lemma (in prob_space) erlang_distributed_mult_const:
   assumes erlX: "distributed M lborel X (erlang_density k l)"
   assumes a_pos[arith]: "0 < \<alpha>"  "0 < l"
   shows  "distributed M lborel (\<lambda>x. \<alpha> * X x) (erlang_density k (l / \<alpha>))"
 proof (subst erlang_distributed_iff, safe)
-  have [measurable]: "random_variable borel X"  and  [arith]: "0 < l " 
+  have [measurable]: "random_variable borel X"  and  [arith]: "0 < l "
   and  [simp]: "\<And>a. 0 \<le> a \<Longrightarrow> prob {x \<in> space M. X x \<le> a} = erlang_CDF k l a"
     by(insert erlX, auto simp: erlang_distributed_iff)
 
-  show "random_variable borel (\<lambda>x. \<alpha> * X x)" "0 < l / \<alpha>"  "0 < l / \<alpha>" 
+  show "random_variable borel (\<lambda>x. \<alpha> * X x)" "0 < l / \<alpha>"  "0 < l / \<alpha>"
     by (auto simp:field_simps)
-  
+
   fix a:: real assume [arith]: "0 \<le> a"
-  obtain b:: real  where [simp, arith]: "b = a/ \<alpha>" by blast 
+  obtain b:: real  where [simp, arith]: "b = a/ \<alpha>" by blast
 
   have [arith]: "0 \<le> b" by (auto simp: divide_nonneg_pos)
- 
+
   have "prob {x \<in> space M. \<alpha> * X x \<le> a}  = prob {x \<in> space M.  X x \<le> b}"
     by (rule arg_cong[where f= prob]) (auto simp:field_simps)
-  
+
   moreover have "prob {x \<in> space M. X x \<le> b} = erlang_CDF k l b" by auto
   moreover have "erlang_CDF k (l / \<alpha>) a = erlang_CDF k l b" unfolding erlang_CDF_def by auto
-  ultimately show "prob {x \<in> space M. \<alpha> * X x \<le> a} = erlang_CDF k (l / \<alpha>) a" by fastforce  
+  ultimately show "prob {x \<in> space M. \<alpha> * X x \<le> a} = erlang_CDF k (l / \<alpha>) a" by fastforce
 qed
 
 lemma (in prob_space) has_bochner_integral_erlang_ith_moment:
@@ -324,7 +324,7 @@
 
   show "expectation (\<lambda>x. (X x)\<^sup>2) - (expectation X)\<^sup>2 = real (k + 1) / l\<^sup>2"
     using erlang_ith_moment[OF assms, of 1] erlang_ith_moment[OF assms, of 2]
-    by simp (auto simp: power2_eq_square field_simps real_of_nat_Suc)
+    by simp (auto simp: power2_eq_square field_simps of_nat_Suc)
 qed
 
 subsection {* Exponential distribution *}
@@ -352,7 +352,7 @@
     have "AE x in lborel. 0 \<le> exponential_density l x"
       using assms by (auto simp: distributed_real_AE)
     then have "AE x in lborel. x \<le> (0::real)"
-      apply eventually_elim 
+      apply eventually_elim
       using `l < 0`
       apply (auto simp: exponential_density_def zero_le_mult_iff split: split_if_asm)
       done
@@ -431,14 +431,14 @@
   have randX: "random_variable borel X" using expX by (simp add: exponential_distributed_iff)
   moreover have randY: "random_variable borel Y" using expY by (simp add: exponential_distributed_iff)
   ultimately show "random_variable borel (\<lambda>x. min (X x) (Y x))" by auto
-  
+
   have "0 < l" by (rule exponential_distributed_params) fact
   moreover have "0 < u" by (rule exponential_distributed_params) fact
   ultimately  show " 0 < l + u" by force
 
   fix a::real assume a[arith]: "0 \<le> a"
-  have gt1[simp]: "\<P>(x in M. a < X x ) = exp (- a * l)" by (rule exponential_distributedD_gt[OF expX a]) 
-  have gt2[simp]: "\<P>(x in M. a < Y x ) = exp (- a * u)" by (rule exponential_distributedD_gt[OF expY a]) 
+  have gt1[simp]: "\<P>(x in M. a < X x ) = exp (- a * l)" by (rule exponential_distributedD_gt[OF expX a])
+  have gt2[simp]: "\<P>(x in M. a < Y x ) = exp (- a * u)" by (rule exponential_distributedD_gt[OF expY a])
 
   have "\<P>(x in M. a < (min (X x) (Y x)) ) =  \<P>(x in M. a < (X x) \<and> a < (Y x))" by (auto intro!:arg_cong[where f=prob])
 
@@ -450,16 +450,16 @@
   have "{x \<in> space M. (min (X x) (Y x)) \<le>a } = (space M - {x \<in> space M. a<(min (X x) (Y x)) })"
     by auto
   then have "1 - prob {x \<in> space M. a < (min (X x) (Y x))} = prob {x \<in> space M. (min (X x) (Y x)) \<le> a}"
-    using randX randY by (auto simp: prob_compl) 
+    using randX randY by (auto simp: prob_compl)
   then show "prob {x \<in> space M. (min (X x) (Y x)) \<le> a} = 1 - exp (- a * (l + u))"
     using indep_prob by auto
 qed
- 
+
 lemma (in prob_space) exponential_distributed_Min:
   assumes finI: "finite I"
   assumes A: "I \<noteq> {}"
   assumes expX: "\<And>i. i \<in> I \<Longrightarrow> distributed M lborel (X i) (exponential_density (l i))"
-  assumes ind: "indep_vars (\<lambda>i. borel) X I" 
+  assumes ind: "indep_vars (\<lambda>i. borel) X I"
   shows "distributed M lborel (\<lambda>x. Min ((\<lambda>i. X i x)`I)) (exponential_density (\<Sum>i\<in>I. l i))"
 using assms
 proof (induct rule: finite_ne_induct)
@@ -468,7 +468,7 @@
   case (insert i I)
   then have "distributed M lborel (\<lambda>x. min (X i x) (Min ((\<lambda>i. X i x)`I))) (exponential_density (l i + (\<Sum>i\<in>I. l i)))"
       by (intro exponential_distributed_min indep_vars_Min insert)
-         (auto intro: indep_vars_subset) 
+         (auto intro: indep_vars_subset)
   then show ?case
     using insert by simp
 qed
@@ -499,22 +499,22 @@
       done
   next
     note zero_le_mult_iff[simp] zero_le_divide_iff[simp]
-  
+
     have I_eq1: "integral\<^sup>N lborel (erlang_density (Suc k\<^sub>1 + Suc k\<^sub>2 - 1) l) = 1"
       using nn_integral_erlang_ith_moment[of l "Suc k\<^sub>1 + Suc k\<^sub>2 - 1" 0] by (simp del: fact_Suc)
-  
+
     have 1: "(\<integral>\<^sup>+ x. ereal (erlang_density (Suc k\<^sub>1 + Suc k\<^sub>2 - 1) l x * indicator {0<..} x) \<partial>lborel) = 1"
       apply (subst I_eq1[symmetric])
       unfolding erlang_density_def
       by (auto intro!: nn_integral_cong split:split_indicator)
-  
+
     have "prob_space (density lborel ?LHS)"
       unfolding times_ereal.simps[symmetric]
-      by (intro prob_space_convolution_density) 
+      by (intro prob_space_convolution_density)
          (auto intro!: prob_space_erlang_density erlang_density_nonneg)
     then have 2: "integral\<^sup>N lborel ?LHS = 1"
       by (auto dest!: prob_space.emeasure_space_1 simp: emeasure_density)
-  
+
     let ?I = "(integral\<^sup>N lborel (\<lambda>y. ereal ((1 - y)^ k\<^sub>1 * y^k\<^sub>2 * indicator {0..1} y)))"
     let ?C = "(fact (Suc (k\<^sub>1 + k\<^sub>2))) / ((fact k\<^sub>1) * (fact k\<^sub>2))"
     let ?s = "Suc k\<^sub>1 + Suc k\<^sub>2 - 1"
@@ -523,7 +523,7 @@
     { fix x :: real assume [arith]: "0 < x"
       have *: "\<And>x y n. (x - y * x::real)^n = x^n * (1 - y)^n"
         unfolding power_mult_distrib[symmetric] by (simp add: field_simps)
-    
+
       have "?LHS x = ?L x"
         unfolding erlang_density_def
         by (auto intro!: nn_integral_cong split:split_indicator)
@@ -534,7 +534,7 @@
         apply (auto simp add: erlang_density_def mult_less_0_iff exp_minus field_simps exp_diff power_add *
                     simp del: fact_Suc split: split_indicator)
         done
-      finally have "(\<integral>\<^sup>+y. ereal (erlang_density k\<^sub>1 l (x - y) * erlang_density k\<^sub>2 l y) \<partial>lborel) = 
+      finally have "(\<integral>\<^sup>+y. ereal (erlang_density k\<^sub>1 l (x - y) * erlang_density k\<^sub>2 l y) \<partial>lborel) =
         (\<lambda>x. ereal ?C * ?I * erlang_density ?s l x) x"
         by simp }
     note * = this
@@ -548,9 +548,9 @@
       by (auto intro!: nn_integral_cong simp: * split: split_indicator)
     also have "... = ereal (?C) * ?I"
       using 1
-      by (auto simp: nn_integral_nonneg nn_integral_cmult)  
+      by (auto simp: nn_integral_nonneg nn_integral_cmult)
     finally have " ereal (?C) * ?I = 1" by presburger
-  
+
     then show ?thesis
       using * by simp
   qed
@@ -585,7 +585,7 @@
     also have "(\<lambda>x. (X i x) + (\<Sum>i\<in> I. X i x)) = (\<lambda>x. \<Sum>i\<in>insert i I. X i x)"
       using insert by auto
     also have "Suc(k i) + Suc ((\<Sum>i\<in>I. Suc (k i)) - 1) - 1 = (\<Sum>i\<in>insert i I. Suc (k i)) - 1"
-      using insert by (auto intro!: Suc_pred simp: ac_simps)    
+      using insert by (auto intro!: Suc_pred simp: ac_simps)
     finally show ?case by fast
 qed
 
@@ -593,7 +593,7 @@
   assumes finI: "finite I"
   assumes A: "I \<noteq> {}"
   assumes expX: "\<And>i. i \<in> I \<Longrightarrow> distributed M lborel (X i) (exponential_density l)"
-  assumes ind: "indep_vars (\<lambda>i. borel) X I" 
+  assumes ind: "indep_vars (\<lambda>i. borel) X I"
   shows "distributed M lborel (\<lambda>x. \<Sum>i\<in>I. X i x) (erlang_density ((card I) - 1) l)"
 proof -
   obtain i where "i \<in> I" using assms by auto
@@ -606,22 +606,22 @@
   shows "entropy b lborel X = log b (exp 1 / l)"
 proof -
   have l[simp, arith]: "0 < l" by (rule exponential_distributed_params[OF D])
- 
+
   have [simp]: "integrable lborel (exponential_density l)"
     using distributed_integrable[OF D, of "\<lambda>_. 1"] by simp
 
   have [simp]: "integral\<^sup>L lborel (exponential_density l) = 1"
     using distributed_integral[OF D, of "\<lambda>_. 1"] by (simp add: prob_space)
-    
+
   have [simp]: "integrable lborel (\<lambda>x. exponential_density l x * x)"
     using erlang_ith_moment_integrable[OF l D, of 1] distributed_integrable[OF D, of "\<lambda>x. x"] by simp
 
   have [simp]: "integral\<^sup>L lborel (\<lambda>x. exponential_density l x * x) = 1 / l"
     using erlang_ith_moment[OF l D, of 1] distributed_integral[OF D, of "\<lambda>x. x"] by simp
-    
+
   have "entropy b lborel X = - (\<integral> x. exponential_density l x * log b (exponential_density l x) \<partial>lborel)"
     using D by (rule entropy_distr)
-  also have "(\<integral> x. exponential_density l x * log b (exponential_density l x) \<partial>lborel) = 
+  also have "(\<integral> x. exponential_density l x * log b (exponential_density l x) \<partial>lborel) =
     (\<integral> x. (ln l * exponential_density l x - l * (exponential_density l x * x)) / ln b \<partial>lborel)"
     by (intro integral_cong) (auto simp: log_def ln_mult exponential_density_def field_simps)
   also have "\<dots> = (ln l - 1) / ln b"
@@ -674,7 +674,7 @@
     using A by simp
   then show "emeasure M {x\<in>space M. X x \<le> a} = ereal (measure lborel (A \<inter> {..a}) / r)"
     using distr by simp
- 
+
   have "(\<integral>\<^sup>+ x. ereal (indicator A x / measure lborel A * indicator {..a} x) \<partial>lborel) =
     (\<integral>\<^sup>+ x. ereal (1 / measure lborel A) * indicator (A \<inter> {..a}) x \<partial>lborel)"
     by (auto intro!: nn_integral_cong split: split_indicator)
@@ -698,7 +698,7 @@
     by auto
   then show "emeasure M {x\<in>space M. X x \<le> t} = emeasure lborel ({a .. b} \<inter> {..t}) / (b - a)"
   proof (elim disjE conjE)
-    assume "t < a" 
+    assume "t < a"
     then have "emeasure M {x\<in>space M. X x \<le> t} \<le> emeasure M {x\<in>space M. X x \<le> a}"
       using X by (auto intro!: emeasure_mono measurable_sets)
     also have "\<dots> = 0"
@@ -713,7 +713,7 @@
     then show ?thesis using `a \<le> t` `a < b`
       using distr[OF bnds] by (simp add: emeasure_eq_measure)
   next
-    assume "b < t" 
+    assume "b < t"
     have "1 = emeasure M {x\<in>space M. X x \<le> b}"
       using distr[of b] `a < b` by (simp add: one_ereal_def emeasure_eq_measure)
     also have "\<dots> \<le> emeasure M {x\<in>space M. X x \<le> t}"
@@ -752,13 +752,13 @@
 proof (rule ccontr)
   assume "\<not> a < b"
   then have "{a .. b} = {} \<or> {a .. b} = {a .. a}" by simp
-  with uniform_distributed_params[OF D] show False 
+  with uniform_distributed_params[OF D] show False
     by (auto simp: measure_def)
 qed
 
 lemma (in prob_space) uniform_distributed_iff:
   fixes a b :: real
-  shows "distributed M lborel X (\<lambda>x. indicator {a..b} x / measure lborel {a..b}) \<longleftrightarrow> 
+  shows "distributed M lborel X (\<lambda>x. indicator {a..b} x / measure lborel {a..b}) \<longleftrightarrow>
     (X \<in> borel_measurable M \<and> a < b \<and> (\<forall>t\<in>{a .. b}. \<P>(x in M. X x \<le> t)= (t - a) / (b - a)))"
   using
     uniform_distributed_bounds[of X a b]
@@ -774,7 +774,7 @@
   have "a < b"
     using uniform_distributed_bounds[OF D] .
 
-  have "(\<integral> x. indicator {a .. b} x / measure lborel {a .. b} * x \<partial>lborel) = 
+  have "(\<integral> x. indicator {a .. b} x / measure lborel {a .. b} * x \<partial>lborel) =
     (\<integral> x. (x / measure lborel {a .. b}) * indicator {a .. b} x \<partial>lborel)"
     by (intro integral_cong) auto
   also have "(\<integral> x. (x / measure lborel {a .. b}) * indicator {a .. b} x \<partial>lborel) = (a + b) / 2"
@@ -867,9 +867,9 @@
     proof (subst nn_integral_FTC_atLeast)
       have "((\<lambda>a. - (exp (- (a\<^sup>2 * (1 + s\<^sup>2))) / (2 + 2 * s\<^sup>2))) ---> (- (0 / (2 + 2 * s\<^sup>2)))) at_top"
         apply (intro tendsto_intros filterlim_compose[OF exp_at_bot] filterlim_compose[OF filterlim_uminus_at_bot_at_top])
-        apply (subst mult.commute)         
+        apply (subst mult.commute)
         apply (auto intro!: filterlim_tendsto_pos_mult_at_top
-                            filterlim_at_top_mult_at_top[OF filterlim_ident filterlim_ident] 
+                            filterlim_at_top_mult_at_top[OF filterlim_ident filterlim_ident]
                     simp: add_pos_nonneg  power2_eq_square add_nonneg_eq_0_iff)
         done
       then show "((\<lambda>a. - (exp (- a\<^sup>2 - s\<^sup>2 * a\<^sup>2) / (2 + 2 * s\<^sup>2))) ---> 0) at_top"
@@ -884,7 +884,7 @@
   finally have "?pI ?gauss^2 = pi / 4"
     by (simp add: power2_eq_square)
   then have "?pI ?gauss = sqrt (pi / 4)"
-    using power_eq_iff_eq_base[of 2 "real (?pI ?gauss)" "sqrt (pi / 4)"]
+    using power_eq_iff_eq_base[of 2 "real_of_ereal (?pI ?gauss)" "sqrt (pi / 4)"]
           nn_integral_nonneg[of lborel "\<lambda>x. ?gauss x * indicator {0..} x"]
     by (cases "?pI ?gauss") auto
   also have "?pI ?gauss = (\<integral>\<^sup>+x. indicator {0..} x *\<^sub>R exp (- x\<^sup>2) \<partial>lborel)"
@@ -896,8 +896,8 @@
 qed
 
 lemma gaussian_moment_1:
-  "has_bochner_integral lborel (\<lambda>x::real. indicator {0..} x *\<^sub>R (exp (- x\<^sup>2) * x)) (1 / 2)" 
-proof - 
+  "has_bochner_integral lborel (\<lambda>x::real. indicator {0..} x *\<^sub>R (exp (- x\<^sup>2) * x)) (1 / 2)"
+proof -
   have "(\<integral>\<^sup>+x. indicator {0..} x *\<^sub>R (exp (- x\<^sup>2) * x) \<partial>lborel) =
     (\<integral>\<^sup>+x. ereal (x * exp (- x\<^sup>2)) * indicator {0..} x \<partial>lborel)"
     by (intro nn_integral_cong)
@@ -922,10 +922,10 @@
   fixes k :: nat
   shows gaussian_moment_even_pos:
     "has_bochner_integral lborel (\<lambda>x::real. indicator {0..} x *\<^sub>R (exp (-x\<^sup>2)*x^(2 * k)))
-       ((sqrt pi / 2) * (fact (2 * k) / (2 ^ (2 * k) * fact k)))" 
+       ((sqrt pi / 2) * (fact (2 * k) / (2 ^ (2 * k) * fact k)))"
        (is "?even")
     and gaussian_moment_odd_pos:
-      "has_bochner_integral lborel (\<lambda>x::real. indicator {0..} x *\<^sub>R (exp (-x\<^sup>2)*x^(2 * k + 1))) (fact k / 2)" 
+      "has_bochner_integral lborel (\<lambda>x::real. indicator {0..} x *\<^sub>R (exp (-x\<^sup>2)*x^(2 * k + 1))) (fact k / 2)"
       (is "?odd")
 proof -
   let ?M = "\<lambda>k x. exp (- x\<^sup>2) * x^k :: real"
@@ -967,7 +967,7 @@
           (\<integral>x. indicator {0..b} x *\<^sub>R (- 2 * x * exp (- x\<^sup>2) * x ^ (Suc k)) \<partial>lborel)"
         by (rule integral_by_parts')
            (auto intro!: derivative_eq_intros b
-                 simp: real_of_nat_def[symmetric] diff_Suc real_of_nat_Suc field_simps split: nat.split)
+                 simp: diff_Suc of_nat_Suc field_simps split: nat.split)
       also have "(\<integral>x. indicator {0..b} x *\<^sub>R (- 2 * x * exp (- x\<^sup>2) * x ^ (Suc k)) \<partial>lborel) =
         (\<integral>x. indicator {0..b} x *\<^sub>R (- 2 * (exp (- x\<^sup>2) * x ^ (k + 2))) \<partial>lborel)"
         by (intro integral_cong) auto
@@ -979,7 +979,7 @@
     qed
     finally have int_M_at_top: "((?f ---> (k + 1) / 2 * (\<integral>x. indicator {0..} x *\<^sub>R ?M k x \<partial>lborel)) at_top)"
       by simp
-    
+
     have "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R ?M (k + 2) x) ((k + 1) / 2 * I)"
     proof (rule has_bochner_integral_monotone_convergence_at_top)
       fix y :: real
@@ -1000,7 +1000,7 @@
     also have "(real (2 * k + 1) / 2 * (sqrt pi / 2 * ((fact (2 * k)) / ((2::real)^(2*k) * fact k)))) =
       sqrt pi / 2 * ((fact (2 * Suc k)) / ((2::real)^(2 * Suc k) * fact (Suc k)))"
       apply (simp add: field_simps del: fact_Suc)
-      apply (simp add: real_of_nat_def of_nat_mult field_simps)
+      apply (simp add: of_nat_mult field_simps)
       done
     finally show ?case
       by simp
@@ -1011,7 +1011,7 @@
     case (Suc k)
     note step[OF this]
     also have "(real (2 * k + 1 + 1) / (2::real) * ((fact k) / 2)) = (fact (Suc k)) / 2"
-      by (simp add: field_simps real_of_nat_Suc divide_simps del: fact_Suc) (simp add: field_simps)
+      by (simp add: field_simps of_nat_Suc divide_simps del: fact_Suc) (simp add: field_simps)
     finally show ?case
       by simp
   qed (insert gaussian_moment_1, simp)
@@ -1037,7 +1037,7 @@
     (\<lambda>x. exp (- ((sqrt 2 * \<sigma>) * x)\<^sup>2 / (2*\<sigma>\<^sup>2)) * ((sqrt 2 * \<sigma>) * x) ^ (2 * k) / sqrt (2 * pi * \<sigma>\<^sup>2))"
     by (auto simp: fun_eq_iff field_simps real_sqrt_power[symmetric] real_sqrt_mult
                    real_sqrt_divide power_mult eq)
-  also have "((sqrt pi * (fact (2 * k) / (2 ^ (2 * k) * fact k))) * ((2*\<sigma>\<^sup>2)^k / sqrt (2 * pi * \<sigma>\<^sup>2))) = 
+  also have "((sqrt pi * (fact (2 * k) / (2 ^ (2 * k) * fact k))) * ((2*\<sigma>\<^sup>2)^k / sqrt (2 * pi * \<sigma>\<^sup>2))) =
     (inverse (sqrt 2 * \<sigma>) * ((fact (2 * k))) / ((2/\<sigma>\<^sup>2) ^ k * (fact k)))"
     by (auto simp: fun_eq_iff power_mult field_simps real_sqrt_power[symmetric] real_sqrt_mult
                    power2_eq_square)
@@ -1060,7 +1060,7 @@
   also have "(\<lambda>x. (exp (-x\<^sup>2)*\<bar>x\<bar>^(2 * k + 1)) * (2^k * \<sigma>^(2 * k + 1) / sqrt (pi * \<sigma>\<^sup>2))) =
     (\<lambda>x. exp (- (((sqrt 2 * \<sigma>) * x)\<^sup>2 / (2 * \<sigma>\<^sup>2))) * \<bar>sqrt 2 * \<sigma> * x\<bar> ^ (2 * k + 1) / sqrt (2 * pi * \<sigma>\<^sup>2))"
     by (simp add: field_simps abs_mult real_sqrt_power[symmetric] power_mult real_sqrt_mult)
-  also have "(fact k * (2^k * \<sigma>^(2 * k + 1) / sqrt (pi * \<sigma>\<^sup>2))) = 
+  also have "(fact k * (2^k * \<sigma>^(2 * k + 1) / sqrt (pi * \<sigma>\<^sup>2))) =
     (inverse (sqrt 2) * inverse \<sigma> * (2 ^ k * (\<sigma> * \<sigma> ^ (2 * k)) * (fact k) * sqrt (2 / pi)))"
     by (auto simp: fun_eq_iff power_mult field_simps real_sqrt_power[symmetric] real_sqrt_divide
                    real_sqrt_mult)
@@ -1115,7 +1115,7 @@
   note normal_moment_odd[OF \<sigma>_pos, of \<mu> 0] has_bochner_integral_mult_left[of \<mu>, OF this]
   note has_bochner_integral_add[OF this]
   then show ?thesis
-    by (simp add: power2_eq_square field_simps)  
+    by (simp add: power2_eq_square field_simps)
 qed
 
 lemma integral_normal_moment_nz_1:
@@ -1213,13 +1213,13 @@
   assume "distributed M lborel X (\<lambda>x. ereal (normal_density \<mu> \<sigma> x))"
   then have "distributed M lborel (\<lambda>x. -\<mu> / \<sigma> + (1/\<sigma>) * X x) (\<lambda>x. ereal (normal_density (-\<mu> / \<sigma> + (1/\<sigma>)* \<mu>) (\<bar>1/\<sigma>\<bar> * \<sigma>) x))"
     by(rule normal_density_affine) auto
-  
+
   then show "distributed M lborel (\<lambda>x. (X x - \<mu>) / \<sigma>) (\<lambda>x. ereal (std_normal_density x))"
     by (simp add: diff_divide_distrib[symmetric] field_simps)
 next
   assume *: "distributed M lborel (\<lambda>x. (X x - \<mu>) / \<sigma>) (\<lambda>x. ereal (std_normal_density x))"
   have "distributed M lborel (\<lambda>x. \<mu> + \<sigma> * ((X x - \<mu>) / \<sigma>)) (\<lambda>x. ereal (normal_density \<mu> \<bar>\<sigma>\<bar> x))"
-    using normal_density_affine[OF *, of \<sigma> \<mu>] by simp  
+    using normal_density_affine[OF *, of \<sigma> \<mu>] by simp
   then show "distributed M lborel X (\<lambda>x. ereal (normal_density \<mu> \<sigma> x))" by simp
 qed
 
@@ -1231,8 +1231,8 @@
   def \<sigma>' \<equiv> "\<sigma>\<^sup>2" and \<tau>' \<equiv> "\<tau>\<^sup>2"
   then have [simp, arith]: "0 < \<sigma>'" "0 < \<tau>'"
     by simp_all
-  let ?\<sigma> = "sqrt ((\<sigma>' * \<tau>') / (\<sigma>' + \<tau>'))"  
-  have sqrt: "(sqrt (2 * pi * (\<sigma>' + \<tau>')) * sqrt (2 * pi * (\<sigma>' * \<tau>') / (\<sigma>' + \<tau>'))) = 
+  let ?\<sigma> = "sqrt ((\<sigma>' * \<tau>') / (\<sigma>' + \<tau>'))"
+  have sqrt: "(sqrt (2 * pi * (\<sigma>' + \<tau>')) * sqrt (2 * pi * (\<sigma>' * \<tau>') / (\<sigma>' + \<tau>'))) =
     (sqrt (2 * pi * \<sigma>') * sqrt (2 * pi * \<tau>'))"
     by (subst power_eq_iff_eq_base[symmetric, where n=2])
        (simp_all add: real_sqrt_mult[symmetric] power2_eq_square)
@@ -1269,10 +1269,10 @@
   have ind[simp]: "indep_var borel (\<lambda>x. - \<mu> + X x) borel (\<lambda>x. - \<nu> + Y x)"
   proof -
     have "indep_var borel ( (\<lambda>x. -\<mu> + x) o X) borel ((\<lambda>x. - \<nu> + x) o Y)"
-      by (auto intro!: indep_var_compose assms) 
+      by (auto intro!: indep_var_compose assms)
     then show ?thesis by (simp add: o_def)
   qed
-  
+
   have "distributed M lborel (\<lambda>x. -\<mu> + 1 * X x) (normal_density (- \<mu> + 1 * \<mu>) (\<bar>1\<bar> * \<sigma>))"
     by(rule normal_density_affine[OF normalX pos_var(1), of 1 "-\<mu>"]) simp
   then have 1[simp]: "distributed M lborel (\<lambda>x. - \<mu> +  X x) (normal_density 0 \<sigma>)" by simp
@@ -1280,10 +1280,10 @@
   have "distributed M lborel (\<lambda>x. -\<nu> + 1 * Y x) (normal_density (- \<nu> + 1 * \<nu>) (\<bar>1\<bar> * \<tau>))"
     by(rule normal_density_affine[OF normalY pos_var(2), of 1 "-\<nu>"]) simp
   then have 2[simp]: "distributed M lborel (\<lambda>x. - \<nu> +  Y x) (normal_density 0 \<tau>)" by simp
-  
+
   have *: "distributed M lborel (\<lambda>x. (- \<mu> + X x) + (- \<nu> + Y x)) (\<lambda>x. ereal (normal_density 0 (sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2)) x))"
     using distributed_convolution[OF ind 1 2] conv_normal_density_zero_mean[OF pos_var] by simp
-  
+
   have "distributed M lborel (\<lambda>x. \<mu> + \<nu> + 1 * (- \<mu> + X x + (- \<nu> + Y x)))
         (\<lambda>x. ereal (normal_density (\<mu> + \<nu> + 1 * 0) (\<bar>1\<bar> * sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2)) x))"
     by (rule normal_density_affine[OF *, of 1 "\<mu> + \<nu>"]) (auto simp: add_pos_pos)
@@ -1298,7 +1298,7 @@
   assumes normalY[simp]: "distributed M lborel Y (normal_density \<nu> \<tau>)"
   shows "distributed M lborel (\<lambda>x. X x - Y x) (normal_density (\<mu> - \<nu>) (sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2)))"
 proof -
-  have "distributed M lborel (\<lambda>x. 0 + - 1 * Y x) (\<lambda>x. ereal (normal_density (0 + - 1 * \<nu>) (\<bar>- 1\<bar> * \<tau>) x))" 
+  have "distributed M lborel (\<lambda>x. 0 + - 1 * Y x) (\<lambda>x. ereal (normal_density (0 + - 1 * \<nu>) (\<bar>- 1\<bar> * \<tau>) x))"
     by(rule normal_density_affine, auto)
   then have [simp]:"distributed M lborel (\<lambda>x. - Y x) (\<lambda>x. ereal (normal_density (- \<nu>) \<tau> x))" by simp
 
@@ -1315,25 +1315,25 @@
   assumes "\<And>i. i \<in> I \<Longrightarrow> 0 < \<sigma> i"
   assumes normal: "\<And>i. i \<in> I \<Longrightarrow> distributed M lborel (X i) (normal_density (\<mu> i) (\<sigma> i))"
   shows "distributed M lborel (\<lambda>x. \<Sum>i\<in>I. X i x) (normal_density (\<Sum>i\<in>I. \<mu> i) (sqrt (\<Sum>i\<in>I. (\<sigma> i)\<^sup>2)))"
-  using assms 
+  using assms
 proof (induct I rule: finite_ne_induct)
   case (singleton i) then show ?case by (simp add : abs_of_pos)
 next
   case (insert i I)
-    then have 1: "distributed M lborel (\<lambda>x. (X i x) + (\<Sum>i\<in>I. X i x)) 
+    then have 1: "distributed M lborel (\<lambda>x. (X i x) + (\<Sum>i\<in>I. X i x))
                 (normal_density (\<mu> i  + setsum \<mu> I)  (sqrt ((\<sigma> i)\<^sup>2 + (sqrt (\<Sum>i\<in>I. (\<sigma> i)\<^sup>2))\<^sup>2)))"
-      apply (intro sum_indep_normal indep_vars_setsum insert real_sqrt_gt_zero)  
+      apply (intro sum_indep_normal indep_vars_setsum insert real_sqrt_gt_zero)
       apply (auto intro: indep_vars_subset intro!: setsum_pos)
       apply fastforce
       done
     have 2: "(\<lambda>x. (X i x)+ (\<Sum>i\<in>I. X i x)) = (\<lambda>x. (\<Sum>j\<in>insert i I. X j x))"
           "\<mu> i + setsum \<mu> I = setsum \<mu> (insert i I)"
       using insert by auto
-  
+
     have 3: "(sqrt ((\<sigma> i)\<^sup>2 + (sqrt (\<Sum>i\<in>I. (\<sigma> i)\<^sup>2))\<^sup>2)) = (sqrt (\<Sum>i\<in>insert i I. (\<sigma> i)\<^sup>2))"
       using insert by (simp add: setsum_nonneg)
-  
-    show ?case using 1 2 3 by simp  
+
+    show ?case using 1 2 3 by simp
 qed
 
 lemma (in prob_space) standard_normal_distributed_expectation:
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Giry_Monad.thy
--- a/src/HOL/Probability/Giry_Monad.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Giry_Monad.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -337,7 +337,7 @@
   shows "(\<lambda>M. integral\<^sup>L M f) \<in> borel_measurable (subprob_algebra N)"
 proof -
   note [measurable] = nn_integral_measurable_subprob_algebra
-  have "?thesis \<longleftrightarrow> (\<lambda>M. real (\<integral>\<^sup>+ x. f x \<partial>M) - real (\<integral>\<^sup>+ x. - f x \<partial>M)) \<in> borel_measurable (subprob_algebra N)"
+  have "?thesis \<longleftrightarrow> (\<lambda>M. real_of_ereal (\<integral>\<^sup>+ x. f x \<partial>M) - real_of_ereal (\<integral>\<^sup>+ x. - f x \<partial>M)) \<in> borel_measurable (subprob_algebra N)"
   proof(rule measurable_cong)
     fix M
     assume "M \<in> space (subprob_algebra N)"
@@ -352,7 +352,7 @@
     finally have "(\<integral>\<^sup>+ x. ereal \<bar>f x\<bar> \<partial>M) \<noteq> \<infinity>" by(auto simp add: max_def)
     then have "integrable M f" using f_measurable
       by(auto intro: integrableI_bounded)
-    thus "(\<integral> x. f x \<partial>M) = real (\<integral>\<^sup>+ x. f x \<partial>M) - real (\<integral>\<^sup>+ x. - f x \<partial>M)"
+    thus "(\<integral> x. f x \<partial>M) = real_of_ereal (\<integral>\<^sup>+ x. f x \<partial>M) - real_of_ereal (\<integral>\<^sup>+ x. - f x \<partial>M)"
       by(simp add: real_lebesgue_integral_def)
   qed
   also have "\<dots>" by measurable
@@ -933,9 +933,9 @@
     using f_measurable by(auto intro!: bounded1 dest: f_bounded)
   have "AE M' in M. (\<integral>\<^sup>+ x. f x \<partial>M') \<noteq> \<infinity>"
     using M_bounded by(rule AE_mp[OF _ AE_I2])(auto dest: f_pos1)
-  hence [simp]: "(\<integral>\<^sup>+ M'. ereal (real (\<integral>\<^sup>+ x. f x \<partial>M')) \<partial>M) = (\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M)"
+  hence [simp]: "(\<integral>\<^sup>+ M'. ereal (real_of_ereal (\<integral>\<^sup>+ x. f x \<partial>M')) \<partial>M) = (\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M)"
     by(rule nn_integral_cong_AE[OF AE_mp])(simp add: ereal_real nn_integral_nonneg)
-  from f_pos have [simp]: "integrable M (\<lambda>M'. real (\<integral>\<^sup>+ x. f x \<partial>M'))"
+  from f_pos have [simp]: "integrable M (\<lambda>M'. real_of_ereal (\<integral>\<^sup>+ x. f x \<partial>M'))"
     by(simp add: int_f real_integrable_def nn_integral_nonneg real_of_ereal[symmetric] nn_integral_0_iff_AE[THEN iffD2] del: real_of_ereal)
 
   have f_neg1:
@@ -944,9 +944,9 @@
     using f_measurable by(auto intro!: bounded1 dest: f_bounded)
   have "AE M' in M. (\<integral>\<^sup>+ x. - f x \<partial>M') \<noteq> \<infinity>"
     using M_bounded by(rule AE_mp[OF _ AE_I2])(auto dest: f_neg1)
-  hence [simp]: "(\<integral>\<^sup>+ M'. ereal (real (\<integral>\<^sup>+ x. - f x \<partial>M')) \<partial>M) = (\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. - f x \<partial>M' \<partial>M)"
+  hence [simp]: "(\<integral>\<^sup>+ M'. ereal (real_of_ereal (\<integral>\<^sup>+ x. - f x \<partial>M')) \<partial>M) = (\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. - f x \<partial>M' \<partial>M)"
     by(rule nn_integral_cong_AE[OF AE_mp])(simp add: ereal_real nn_integral_nonneg)
-  from f_neg have [simp]: "integrable M (\<lambda>M'. real (\<integral>\<^sup>+ x. - f x \<partial>M'))"
+  from f_neg have [simp]: "integrable M (\<lambda>M'. real_of_ereal (\<integral>\<^sup>+ x. - f x \<partial>M'))"
     by(simp add: int_mf real_integrable_def nn_integral_nonneg real_of_ereal[symmetric] nn_integral_0_iff_AE[THEN iffD2] del: real_of_ereal)
 
   from \<open>?integrable\<close>
@@ -958,15 +958,15 @@
     ((\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M) - (\<integral>\<^sup>+ M'. - \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M)) - 
     ((\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. - f x \<partial>M' \<partial>M) - (\<integral>\<^sup>+ M'. - \<integral>\<^sup>+ x. - f x \<partial>M' \<partial>M))"
     by(subst (7 11) nn_integral_0_iff_AE[THEN iffD2])(simp_all add: nn_integral_nonneg)
-  also have "(\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M) - (\<integral>\<^sup>+ M'. - \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M) = \<integral> M'. real (\<integral>\<^sup>+ x. f x \<partial>M') \<partial>M"
+  also have "(\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M) - (\<integral>\<^sup>+ M'. - \<integral>\<^sup>+ x. f x \<partial>M' \<partial>M) = \<integral> M'. real_of_ereal (\<integral>\<^sup>+ x. f x \<partial>M') \<partial>M"
     using f_pos
     by(simp add: real_lebesgue_integral_def)(simp add: ereal_minus(1)[symmetric] ereal_real int_f nn_integral_nonneg nn_integral_0_iff_AE[THEN iffD2] real_of_ereal_pos zero_ereal_def[symmetric])
-  also have "(\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. - f x \<partial>M' \<partial>M) - (\<integral>\<^sup>+ M'. - \<integral>\<^sup>+ x. - f x \<partial>M' \<partial>M) = \<integral> M'. real (\<integral>\<^sup>+ x. - f x \<partial>M') \<partial>M"
+  also have "(\<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. - f x \<partial>M' \<partial>M) - (\<integral>\<^sup>+ M'. - \<integral>\<^sup>+ x. - f x \<partial>M' \<partial>M) = \<integral> M'. real_of_ereal (\<integral>\<^sup>+ x. - f x \<partial>M') \<partial>M"
     using f_neg
     by(simp add: real_lebesgue_integral_def)(simp add: ereal_minus(1)[symmetric] ereal_real int_mf nn_integral_nonneg nn_integral_0_iff_AE[THEN iffD2] real_of_ereal_pos zero_ereal_def[symmetric])
   also note ereal_minus(1)
-  also have "(\<integral> M'. real (\<integral>\<^sup>+ x. f x \<partial>M') \<partial>M) - (\<integral> M'. real (\<integral>\<^sup>+ x. - f x \<partial>M') \<partial>M) = 
-    \<integral>M'. real (\<integral>\<^sup>+ x. f x \<partial>M') - real (\<integral>\<^sup>+ x. - f x \<partial>M') \<partial>M"
+  also have "(\<integral> M'. real_of_ereal (\<integral>\<^sup>+ x. f x \<partial>M') \<partial>M) - (\<integral> M'. real_of_ereal (\<integral>\<^sup>+ x. - f x \<partial>M') \<partial>M) = 
+    \<integral>M'. real_of_ereal (\<integral>\<^sup>+ x. f x \<partial>M') - real_of_ereal (\<integral>\<^sup>+ x. - f x \<partial>M') \<partial>M"
     by simp
   also have "\<dots> = \<integral>M'. \<integral> x. f x \<partial>M' \<partial>M" using _ _ M_bounded
   proof(rule integral_cong_AE[OF _ _ AE_mp[OF _ AE_I2], rule_format])
@@ -982,7 +982,7 @@
     hence [simp]: "space M' = space N" by(rule sets_eq_imp_space_eq)
     have "integrable M' f"
       by(rule integrable_const_bound[where B=B])(auto simp add: f_bounded)
-    then show "real (\<integral>\<^sup>+ x. f x \<partial>M') - real (\<integral>\<^sup>+ x. - f x \<partial>M') = \<integral> x. f x \<partial>M'"
+    then show "real_of_ereal (\<integral>\<^sup>+ x. f x \<partial>M') - real_of_ereal (\<integral>\<^sup>+ x. - f x \<partial>M') = \<integral> x. f x \<partial>M'"
       by(simp add: real_lebesgue_integral_def)
   qed simp_all
   finally show ?integral by simp
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Information.thy
--- a/src/HOL/Probability/Information.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Information.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -73,7 +73,7 @@
 Kullback$-$Leibler distance. *}
 
 definition
-  "entropy_density b M N = log b \<circ> real \<circ> RN_deriv M N"
+  "entropy_density b M N = log b \<circ> real_of_ereal \<circ> RN_deriv M N"
 
 definition
   "KL_divergence b M N = integral\<^sup>L N (entropy_density b M N)"
@@ -247,7 +247,7 @@
       apply auto
       done
   qed
-  then have "AE x in M. log b (real (RN_deriv M M x)) = 0"
+  then have "AE x in M. log b (real_of_ereal (RN_deriv M M x)) = 0"
     by (elim AE_mp) simp
   from integral_cong_AE[OF _ _ this]
   have "integral\<^sup>L M (entropy_density b M M) = 0"
@@ -787,7 +787,7 @@
   note D = distributed_measurable[OF X] distributed_borel_measurable[OF X] distributed_AE[OF X]
   note ae = distributed_RN_deriv[OF X]
 
-  have ae_eq: "AE x in distr M MX X. log b (real (RN_deriv MX (distr M MX X) x)) =
+  have ae_eq: "AE x in distr M MX X. log b (real_of_ereal (RN_deriv MX (distr M MX X) x)) =
     log b (f x)"
     unfolding distributed_distr_eq_density[OF X]
     apply (subst AE_density)
@@ -1517,8 +1517,8 @@
 subsection {* Conditional Entropy *}
 
 definition (in prob_space)
-  "conditional_entropy b S T X Y = - (\<integral>(x, y). log b (real (RN_deriv (S \<Otimes>\<^sub>M T) (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))) (x, y)) / 
-    real (RN_deriv T (distr M T Y) y)) \<partial>distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)))"
+  "conditional_entropy b S T X Y = - (\<integral>(x, y). log b (real_of_ereal (RN_deriv (S \<Otimes>\<^sub>M T) (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))) (x, y)) / 
+    real_of_ereal (RN_deriv T (distr M T Y) y)) \<partial>distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)))"
 
 abbreviation (in information_space)
   conditional_entropy_Pow ("\<H>'(_ | _')") where
@@ -1535,13 +1535,13 @@
   interpret T: sigma_finite_measure T by fact
   interpret ST: pair_sigma_finite S T ..
 
-  have "AE x in density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Pxy x)). Pxy x = real (RN_deriv (S \<Otimes>\<^sub>M T) (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))) x)"
+  have "AE x in density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Pxy x)). Pxy x = real_of_ereal (RN_deriv (S \<Otimes>\<^sub>M T) (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))) x)"
     unfolding AE_density[OF distributed_borel_measurable, OF Pxy]
     unfolding distributed_distr_eq_density[OF Pxy]
     using distributed_RN_deriv[OF Pxy]
     by auto
   moreover
-  have "AE x in density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Pxy x)). Py (snd x) = real (RN_deriv T (distr M T Y) (snd x))"
+  have "AE x in density (S \<Otimes>\<^sub>M T) (\<lambda>x. ereal (Pxy x)). Py (snd x) = real_of_ereal (RN_deriv T (distr M T Y) (snd x))"
     unfolding AE_density[OF distributed_borel_measurable, OF Pxy]
     unfolding distributed_distr_eq_density[OF Py]
     apply (rule ST.AE_pair_measure)
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Interval_Integral.thy
--- a/src/HOL/Probability/Interval_Integral.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Interval_Integral.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -67,19 +67,21 @@
   case PInf
   with `a < b` have "a = -\<infinity> \<or> (\<exists>r. a = ereal r)"
     by (cases a) auto
-  moreover have " (\<lambda>x. ereal (real (Suc x))) ----> \<infinity>"
-    using nat_ceiling_le_eq by (subst LIMSEQ_Suc_iff) (auto simp: Lim_PInfty)
+  moreover have "(\<lambda>x. ereal (real (Suc x))) ----> \<infinity>"
+      apply (subst LIMSEQ_Suc_iff)
+      apply (simp add: Lim_PInfty)
+      using nat_ceiling_le_eq by blast
   moreover have "\<And>r. (\<lambda>x. ereal (r + real (Suc x))) ----> \<infinity>"
     apply (subst LIMSEQ_Suc_iff)
     apply (subst Lim_PInfty)
     apply (metis add.commute diff_le_eq nat_ceiling_le_eq ereal_less_eq(3))
     done
   ultimately show thesis
-    by (intro that[of "\<lambda>i. real a + Suc i"])
+    by (intro that[of "\<lambda>i. real_of_ereal a + Suc i"])
        (auto simp: incseq_def PInf)
 next
   case (real b')
-  def d \<equiv> "b' - (if a = -\<infinity> then b' - 1 else real a)"
+  def d \<equiv> "b' - (if a = -\<infinity> then b' - 1 else real_of_ereal a)"
   with `a < b` have a': "0 < d"
     by (cases a) (auto simp: real)
   moreover
@@ -367,7 +369,7 @@
 
 lemma interval_integral_Icc':
   "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {x. a \<le> ereal x \<and> ereal x \<le> b}. f x)"
-  by (auto intro!: set_integral_discrete_difference[where X="{real a, real b}"]
+  by (auto intro!: set_integral_discrete_difference[where X="{real_of_ereal a, real_of_ereal b}"]
            simp add: interval_lebesgue_integral_def einterval_iff)
 
 lemma interval_integral_Ioc:
@@ -378,7 +380,7 @@
 (* TODO: other versions as well? *) (* Yes: I need the Icc' version. *)
 lemma interval_integral_Ioc':
   "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {x. a < ereal x \<and> ereal x \<le> b}. f x)"
-  by (auto intro!: set_integral_discrete_difference[where X="{real a, real b}"]
+  by (auto intro!: set_integral_discrete_difference[where X="{real_of_ereal a, real_of_ereal b}"]
            simp add: interval_lebesgue_integral_def einterval_iff)
 
 lemma interval_integral_Ico:
@@ -420,7 +422,7 @@
     have "(LBINT x:einterval a c. f x) = (LBINT x:einterval a b \<union> einterval b c. f x)"
     proof (rule set_integral_cong_set)
       show "AE x in lborel. (x \<in> einterval a b \<union> einterval b c) = (x \<in> einterval a c)"
-        using AE_lborel_singleton[of "real b"] ord
+        using AE_lborel_singleton[of "real_of_ereal b"] ord
         by (cases a b c rule: ereal3_cases) (auto simp: einterval_iff)
     qed (insert ord, auto intro!: set_borel_measurable_subset[OF f] simp: einterval_iff)
     also have "\<dots> = (LBINT x:einterval a b. f x) + (LBINT x:einterval b c. f x)"
@@ -607,8 +609,8 @@
   assumes F: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> DERIV F x :> f x" 
   assumes f: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f x" 
   assumes f_nonneg: "AE x in lborel. a < ereal x \<longrightarrow> ereal x < b \<longrightarrow> 0 \<le> f x"
-  assumes A: "((F \<circ> real) ---> A) (at_right a)"
-  assumes B: "((F \<circ> real) ---> B) (at_left b)"
+  assumes A: "((F \<circ> real_of_ereal) ---> A) (at_right a)"
+  assumes B: "((F \<circ> real_of_ereal) ---> B) (at_left b)"
   shows
     "set_integrable lborel (einterval a b) f" 
     "(LBINT x=a..b. f x) = B - A"
@@ -654,8 +656,8 @@
   assumes F: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> (F has_vector_derivative f x) (at x)" 
   assumes f: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f x" 
   assumes f_integrable: "set_integrable lborel (einterval a b) f"
-  assumes A: "((F \<circ> real) ---> A) (at_right a)"
-  assumes B: "((F \<circ> real) ---> B) (at_left b)"
+  assumes A: "((F \<circ> real_of_ereal) ---> A) (at_right a)"
+  assumes B: "((F \<circ> real_of_ereal) ---> B) (at_left b)"
   shows "(LBINT x=a..b. f x) = B - A"
 proof -
   from einterval_Icc_approximation[OF `a < b`] guess u l . note approx = this
@@ -819,8 +821,8 @@
   and contf: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f (g x)"
   and contg': "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont g' x"
   and g'_nonneg: "\<And>x. a \<le> ereal x \<Longrightarrow> ereal x \<le> b \<Longrightarrow> 0 \<le> g' x"
-  and A: "((ereal \<circ> g \<circ> real) ---> A) (at_right a)"
-  and B: "((ereal \<circ> g \<circ> real) ---> B) (at_left b)"
+  and A: "((ereal \<circ> g \<circ> real_of_ereal) ---> A) (at_right a)"
+  and B: "((ereal \<circ> g \<circ> real_of_ereal) ---> B) (at_left b)"
   and integrable: "set_integrable lborel (einterval a b) (\<lambda>x. g' x *\<^sub>R f (g x))"
   and integrable2: "set_integrable lborel (einterval A B) (\<lambda>x. f x)"
   shows "(LBINT x=A..B. f x) = (LBINT x=a..b. g' x *\<^sub>R f (g x))"
@@ -920,8 +922,8 @@
   and contg': "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont g' x"
   and f_nonneg: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> 0 \<le> f (g x)" (* TODO: make this AE? *)
   and g'_nonneg: "\<And>x. a \<le> ereal x \<Longrightarrow> ereal x \<le> b \<Longrightarrow> 0 \<le> g' x"
-  and A: "((ereal \<circ> g \<circ> real) ---> A) (at_right a)"
-  and B: "((ereal \<circ> g \<circ> real) ---> B) (at_left b)"
+  and A: "((ereal \<circ> g \<circ> real_of_ereal) ---> A) (at_right a)"
+  and B: "((ereal \<circ> g \<circ> real_of_ereal) ---> B) (at_left b)"
   and integrable_fg: "set_integrable lborel (einterval a b) (\<lambda>x. f (g x) * g' x)"
   shows 
     "set_integrable lborel (einterval A B) f"
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Lebesgue_Integral_Substitution.thy
--- a/src/HOL/Probability/Lebesgue_Integral_Substitution.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Lebesgue_Integral_Substitution.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -327,7 +327,7 @@
   from derivg have derivg': "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
     by (simp only: has_field_derivative_iff_has_vector_derivative)
 
-  have real_ind[simp]: "\<And>A x. real (indicator A x :: ereal) = indicator A x" 
+  have real_ind[simp]: "\<And>A x. real_of_ereal (indicator A x :: ereal) = indicator A x" 
       by (auto split: split_indicator)
   have ereal_ind[simp]: "\<And>A x. ereal (indicator A x) = indicator A x" 
       by (auto split: split_indicator)
@@ -645,8 +645,8 @@
     unfolding real_integrable_def by (force simp: mult.commute)
 
   have "LBINT x. (f x :: real) * indicator {g a..g b} x = 
-          real (\<integral>\<^sup>+ x. ereal (f x) * indicator {g a..g b} x \<partial>lborel) -
-          real (\<integral>\<^sup>+ x. ereal (- (f x)) * indicator {g a..g b} x \<partial>lborel)" using integrable
+          real_of_ereal (\<integral>\<^sup>+ x. ereal (f x) * indicator {g a..g b} x \<partial>lborel) -
+          real_of_ereal (\<integral>\<^sup>+ x. ereal (- (f x)) * indicator {g a..g b} x \<partial>lborel)" using integrable
     by (subst real_lebesgue_integral_def) (simp_all add: nn_integral_set_ereal mult.commute)
   also have "(\<integral>\<^sup>+x. ereal (f x) * indicator {g a..g b} x \<partial>lborel) =
                (\<integral>\<^sup>+x. ereal (f x * indicator {g a..g b} x) \<partial>lborel)"
@@ -677,8 +677,8 @@
       by (simp_all add: nn_integral_set_ereal mult.commute)
   } note integrable' = this
 
-  have "real (\<integral>\<^sup>+ x. ereal (f (g x) * g' x * indicator {a..b} x) \<partial>lborel) -
-                  real (\<integral>\<^sup>+ x. ereal (-f (g x) * g' x * indicator {a..b} x) \<partial>lborel) =
+  have "real_of_ereal (\<integral>\<^sup>+ x. ereal (f (g x) * g' x * indicator {a..b} x) \<partial>lborel) -
+                  real_of_ereal (\<integral>\<^sup>+ x. ereal (-f (g x) * g' x * indicator {a..b} x) \<partial>lborel) =
                 (LBINT x. f (g x) * g' x * indicator {a..b} x)" using integrable'
     by (subst real_lebesgue_integral_def) (simp_all add: field_simps)
   finally show "(LBINT x. f x * indicator {g a..g b} x) = 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Lebesgue_Measure.thy
--- a/src/HOL/Probability/Lebesgue_Measure.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Lebesgue_Measure.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -510,7 +510,7 @@
     also
     have "... \<le> (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba"
       apply (rule mult_left_mono)
-      apply (metis DIM_positive One_nat_def less_eq_Suc_le less_imp_le real_of_nat_le_iff real_of_nat_power self_le_power zero_less_Suc)
+      apply (metis DIM_positive One_nat_def less_eq_Suc_le less_imp_le of_nat_le_iff of_nat_power self_le_power zero_less_Suc)
       apply (simp add: DIM_positive)
       done
     finally have "real n \<le> (2::real) ^ card ?Ba * real (Suc n) ^ card ?Ba" .
@@ -963,7 +963,7 @@
 
   note F(1)[THEN borel_measurable_simple_function, measurable]
 
-  { fix i x have "real (F i x) \<le> f x"
+  { fix i x have "real_of_ereal (F i x) \<le> f x"
       using F(3,5) F(4)[of x, symmetric] nonneg
       unfolding real_le_ereal_iff
       by (auto simp: image_iff eq_commute[of \<infinity>] max_def intro: SUP_upper split: split_if_asm) }
@@ -991,25 +991,25 @@
   also have "\<dots> \<le> ereal I"
   proof (rule SUP_least)
     fix i :: nat
-    have finite_F: "(\<integral>\<^sup>+ x. ereal (real (F i x) * indicator (?B i) x) \<partial>lborel) < \<infinity>"
+    have finite_F: "(\<integral>\<^sup>+ x. ereal (real_of_ereal (F i x) * indicator (?B i) x) \<partial>lborel) < \<infinity>"
     proof (rule nn_integral_bound_simple_function)
-      have "emeasure lborel {x \<in> space lborel. ereal (real (F i x) * indicator (?B i) x) \<noteq> 0} \<le>
+      have "emeasure lborel {x \<in> space lborel. ereal (real_of_ereal (F i x) * indicator (?B i) x) \<noteq> 0} \<le>
         emeasure lborel (?B i)"
         by (intro emeasure_mono)  (auto split: split_indicator)
-      then show "emeasure lborel {x \<in> space lborel. ereal (real (F i x) * indicator (?B i) x) \<noteq> 0} < \<infinity>"
+      then show "emeasure lborel {x \<in> space lborel. ereal (real_of_ereal (F i x) * indicator (?B i) x) \<noteq> 0} < \<infinity>"
         by auto
     qed (auto split: split_indicator
-              intro!: real_of_ereal_pos F simple_function_compose1[where g="real"] simple_function_ereal)
+              intro!: real_of_ereal_pos F simple_function_compose1[where g="real_of_ereal"] simple_function_ereal)
 
-    have int_F: "(\<lambda>x. real (F i x) * indicator (?B i) x) integrable_on UNIV"
+    have int_F: "(\<lambda>x. real_of_ereal (F i x) * indicator (?B i) x) integrable_on UNIV"
       using F(5) finite_F
       by (intro nn_integral_integrable_on) (auto split: split_indicator intro: real_of_ereal_pos)
 
     have "(\<integral>\<^sup>+ x. F i x * indicator (?B i) x \<partial>lborel) =
-      (\<integral>\<^sup>+ x. ereal (real (F i x) * indicator (?B i) x) \<partial>lborel)"
+      (\<integral>\<^sup>+ x. ereal (real_of_ereal (F i x) * indicator (?B i) x) \<partial>lborel)"
       using F(3,5)
       by (intro nn_integral_cong) (auto simp: image_iff ereal_real eq_commute split: split_indicator)
-    also have "\<dots> = ereal (integral UNIV (\<lambda>x. real (F i x) * indicator (?B i) x))"
+    also have "\<dots> = ereal (integral UNIV (\<lambda>x. real_of_ereal (F i x) * indicator (?B i) x))"
       using F
       by (intro nn_integral_lborel_eq_integral[OF _ _ finite_F])
          (auto split: split_indicator intro: real_of_ereal_pos)
@@ -1279,7 +1279,7 @@
       fix m n :: nat assume "m \<le> n"
       with f nonneg show "F (a + real m) \<le> F (a + real n)"
         by (intro DERIV_nonneg_imp_nondecreasing[where f=F])
-           (simp, metis add_increasing2 order_refl order_trans real_of_nat_ge_zero)
+           (simp, metis add_increasing2 order_refl order_trans of_nat_0_le_iff)
     qed
     have "(\<lambda>x. F (a + real x)) ----> T"
       apply (rule filterlim_compose[OF lim filterlim_tendsto_add_at_top])
@@ -1298,7 +1298,7 @@
 proof (subst integral_FTC_Icc_real)
   fix x show "DERIV (\<lambda>x. x^Suc k / Suc k) x :> x^k"
     by (intro derivative_eq_intros) auto
-qed (auto simp: field_simps simp del: real_of_nat_Suc)
+qed (auto simp: field_simps simp del: of_nat_Suc)
 
 subsection {* Integration by parts *}
 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Measure_Space.thy
--- a/src/HOL/Probability/Measure_Space.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Measure_Space.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -700,7 +700,7 @@
 
 lemma emeasure_insert_ne:
   "A \<noteq> {} \<Longrightarrow> {x} \<in> sets M \<Longrightarrow> A \<in> sets M \<Longrightarrow> x \<notin> A \<Longrightarrow> emeasure M (insert x A) = emeasure M {x} + emeasure M A"
-  by (rule emeasure_insert) 
+  by (rule emeasure_insert)
 
 lemma emeasure_eq_setsum_singleton:
   assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"
@@ -877,7 +877,7 @@
     by (auto intro!: antisym)
 qed
 
-lemma UN_from_nat_into: 
+lemma UN_from_nat_into:
   assumes I: "countable I" "I \<noteq> {}"
   shows "(\<Union>i\<in>I. N i) = (\<Union>i. N (from_nat_into I i))"
 proof -
@@ -1100,7 +1100,7 @@
     unfolding eventually_ae_filter by auto
 qed auto
 
-lemma AE_ball_countable: 
+lemma AE_ball_countable:
   assumes [intro]: "countable X"
   shows "(AE x in M. \<forall>y\<in>X. P x y) \<longleftrightarrow> (\<forall>y\<in>X. AE x in M. P x y)"
 proof
@@ -1121,7 +1121,7 @@
 
 lemma AE_discrete_difference:
   assumes X: "countable X"
-  assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" 
+  assumes null: "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"
   assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
   shows "AE x in M. x \<notin> X"
 proof -
@@ -1207,7 +1207,7 @@
     assume "A = {}" with `(\<Union>A) = space M` show thesis
       by (intro that[of "\<lambda>_. {}"]) auto
   next
-    assume "A \<noteq> {}" 
+    assume "A \<noteq> {}"
     show thesis
     proof
       show "range (from_nat_into A) \<subseteq> sets M"
@@ -1364,7 +1364,7 @@
   by (simp add: emeasure_distr measure_def)
 
 lemma distr_cong_AE:
-  assumes 1: "M = K" "sets N = sets L" and 
+  assumes 1: "M = K" "sets N = sets L" and
     2: "(AE x in M. f x = g x)" and "f \<in> measurable M N" and "g \<in> measurable K L"
   shows "distr M N f = distr K L g"
 proof (rule measure_eqI)
@@ -1713,7 +1713,7 @@
     using finite_measure_eq_setsum_singleton[OF s] by simp
   also have "\<dots> = (\<Sum> x \<in> s. measure M {SOME y. y \<in> s})" using prob_some by auto
   also have "\<dots> = real (card s) * measure M {(SOME x. x \<in> s)}"
-    using setsum_constant assms by (simp add: real_eq_of_nat)
+    using setsum_constant assms by simp
   finally show ?thesis by simp
 qed simp
 
@@ -1744,7 +1744,7 @@
   shows "measure M B = 0"
   using measure_space_inter[of B A] assms by (auto simp: ac_simps)
 lemma (in finite_measure) finite_measure_distr:
-  assumes f: "f \<in> measurable M M'" 
+  assumes f: "f \<in> measurable M M'"
   shows "finite_measure (distr M M' f)"
 proof (rule finite_measureI)
   have "f -` space M' \<inter> space M = space M" using f by (auto dest: measurable_space)
@@ -1795,7 +1795,7 @@
 
   interpret ring_of_sets A "Pow A"
     by (rule ring_of_setsI) auto
-  show "countably_additive (Pow A) ?M" 
+  show "countably_additive (Pow A) ?M"
     unfolding countably_additive_iff_continuous_from_below[OF positive additive]
   proof safe
     fix F :: "nat \<Rightarrow> 'a set" assume "incseq F"
@@ -1844,7 +1844,7 @@
         by (rule infinite_super[OF _ 1]) auto
       then have "infinite (\<Union>i. F i)"
         by auto
-      
+
       ultimately show ?thesis by auto
     qed
   qed
@@ -1975,7 +1975,7 @@
     by (auto simp add: space_restrict_space null_sets_def sets_restrict_space_iff
                        emeasure_restrict_space cong: conj_cong
              intro!: ex_cong[where f="\<lambda>X. (\<Omega> \<inter> space M) \<inter> X"])
-qed  
+qed
 
 lemma restrict_restrict_space:
   assumes "A \<inter> space M \<in> sets M" "B \<inter> space M \<in> sets M"
@@ -2036,7 +2036,7 @@
     by(rule finite_measureI)(simp add: emeasure_restrict_space A space_restrict_space)
 qed
 
-lemma restrict_distr: 
+lemma restrict_distr:
   assumes [measurable]: "f \<in> measurable M N"
   assumes [simp]: "\<Omega> \<inter> space N \<in> sets N" and restrict: "f \<in> space M \<rightarrow> \<Omega>"
   shows "restrict_space (distr M N f) \<Omega> = distr M (restrict_space N \<Omega>) f"
@@ -2053,7 +2053,7 @@
   assumes E: "Int_stable E" "E \<subseteq> Pow \<Omega>" "\<And>X. X \<in> E \<Longrightarrow> emeasure M X = emeasure N X"
   assumes sets_eq: "sets M = sets N" and \<Omega>: "\<Omega> \<in> sets M"
   assumes "sets (restrict_space M \<Omega>) = sigma_sets \<Omega> E"
-  assumes "sets (restrict_space N \<Omega>) = sigma_sets \<Omega> E" 
+  assumes "sets (restrict_space N \<Omega>) = sigma_sets \<Omega> E"
   assumes ae: "AE x in M. x \<in> \<Omega>" "AE x in N. x \<in> \<Omega>"
   assumes A: "countable A" "A \<noteq> {}" "A \<subseteq> E" "\<Union>A = \<Omega>" "\<And>a. a \<in> A \<Longrightarrow> emeasure M a \<noteq> \<infinity>"
   shows "M = N"
@@ -2089,7 +2089,7 @@
 lemma space_null_measure[simp]: "space (null_measure M) = space M"
   by (simp add: null_measure_def)
 
-lemma sets_null_measure[simp, measurable_cong]: "sets (null_measure M) = sets M" 
+lemma sets_null_measure[simp, measurable_cong]: "sets (null_measure M) = sets M"
   by (simp add: null_measure_def)
 
 lemma emeasure_null_measure[simp]: "emeasure (null_measure M) X = 0"
@@ -2176,12 +2176,12 @@
 proof -
   have sets: "(SUP a:A. sets a) = sets a"
   proof (intro antisym SUP_least)
-    fix a' show "a' \<in> A \<Longrightarrow> sets a' \<subseteq> sets a" 
+    fix a' show "a' \<in> A \<Longrightarrow> sets a' \<subseteq> sets a"
       using a chainD[OF A, of a a'] by (auto elim!: less_eq_measure.cases)
   qed (insert \<open>a\<in>A\<close>, auto)
   have space: "(SUP a:A. space a) = space a"
   proof (intro antisym SUP_least)
-    fix a' show "a' \<in> A \<Longrightarrow> space a' \<subseteq> space a" 
+    fix a' show "a' \<in> A \<Longrightarrow> space a' \<subseteq> space a"
       using a chainD[OF A, of a a'] by (intro sets_le_imp_space_le) (auto elim!: less_eq_measure.cases)
   qed (insert \<open>a\<in>A\<close>, auto)
   show "space (Sup A) = space a"
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy
--- a/src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Nonnegative_Lebesgue_Integration.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -204,12 +204,12 @@
   assume "finite P" from this assms show ?thesis by induct auto
 qed auto
 
-lemma simple_function_ereal[intro, simp]: 
+lemma simple_function_ereal[intro, simp]:
   fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
   shows "simple_function M (\<lambda>x. ereal (f x))"
   by (rule simple_function_compose1[OF sf])
 
-lemma simple_function_real_of_nat[intro, simp]: 
+lemma simple_function_real_of_nat[intro, simp]:
   fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
   shows "simple_function M (\<lambda>x. real (f x))"
   by (rule simple_function_compose1[OF sf])
@@ -220,21 +220,21 @@
   shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
              (\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)"
 proof -
-  def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else nat(floor (real (u x) * 2 ^ i))"
+  def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else nat(floor (real_of_ereal (u x) * 2 ^ i))"
   { fix x j have "f x j \<le> j * 2 ^ j" unfolding f_def
     proof (split split_if, intro conjI impI)
       assume "\<not> real j \<le> u x"
-      then have "nat(floor (real (u x) * 2 ^ j)) \<le> nat(floor (j * 2 ^ j))"
+      then have "nat(floor (real_of_ereal (u x) * 2 ^ j)) \<le> nat(floor (j * 2 ^ j))"
          by (cases "u x") (auto intro!: nat_mono floor_mono)
       moreover have "real (nat(floor (j * 2 ^ j))) \<le> j * 2^j"
         by linarith
-      ultimately show "nat(floor (real (u x) * 2 ^ j)) \<le> j * 2 ^ j"
-        unfolding real_of_nat_le_iff by auto
+      ultimately show "nat(floor (real_of_ereal (u x) * 2 ^ j)) \<le> j * 2 ^ j"
+        unfolding of_nat_le_iff by auto
     qed auto }
   note f_upper = this
 
   have real_f:
-    "\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (nat(floor (real (u x) * 2 ^ i))))"
+    "\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (nat(floor (real_of_ereal (u x) * 2 ^ i))))"
     unfolding f_def by auto
 
   let ?g = "\<lambda>j x. real (f x j) / 2^j :: ereal"
@@ -259,27 +259,27 @@
       have "f x i * 2 \<le> f x (Suc i)" unfolding f_def
       proof ((split split_if)+, intro conjI impI)
         assume "ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
-        then show "i * 2 ^ i * 2 \<le> nat(floor (real (u x) * 2 ^ Suc i))"
+        then show "i * 2 ^ i * 2 \<le> nat(floor (real_of_ereal (u x) * 2 ^ Suc i))"
           by (cases "u x") (auto intro!: le_nat_floor)
       next
         assume "\<not> ereal (real i) \<le> u x" "ereal (real (Suc i)) \<le> u x"
-        then show "nat(floor (real (u x) * 2 ^ i)) * 2 \<le> Suc i * 2 ^ Suc i"
+        then show "nat(floor (real_of_ereal (u x) * 2 ^ i)) * 2 \<le> Suc i * 2 ^ Suc i"
           by (cases "u x") auto
       next
         assume "\<not> ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
-        have "nat(floor (real (u x) * 2 ^ i)) * 2 = nat(floor (real (u x) * 2 ^ i)) * nat(floor(2::real))"
+        have "nat(floor (real_of_ereal (u x) * 2 ^ i)) * 2 = nat(floor (real_of_ereal (u x) * 2 ^ i)) * nat(floor(2::real))"
           by simp
-        also have "\<dots> \<le> nat(floor (real (u x) * 2 ^ i * 2))"
+        also have "\<dots> \<le> nat(floor (real_of_ereal (u x) * 2 ^ i * 2))"
         proof cases
           assume "0 \<le> u x" then show ?thesis
-            by (intro le_mult_nat_floor) 
+            by (intro le_mult_nat_floor)
         next
           assume "\<not> 0 \<le> u x" then show ?thesis
             by (cases "u x") (auto simp: nat_floor_neg mult_nonpos_nonneg)
         qed
-        also have "\<dots> = nat(floor (real (u x) * 2 ^ Suc i))"
+        also have "\<dots> = nat(floor (real_of_ereal (u x) * 2 ^ Suc i))"
           by (simp add: ac_simps)
-        finally show "nat(floor (real (u x) * 2 ^ i)) * 2 \<le> nat(floor (real (u x) * 2 ^ Suc i))" .
+        finally show "nat(floor (real_of_ereal (u x) * 2 ^ i)) * 2 \<le> nat(floor (real_of_ereal (u x) * 2 ^ Suc i))" .
       qed simp
       then show "?g i x \<le> ?g (Suc i) x"
         by (auto simp: field_simps)
@@ -380,7 +380,7 @@
       apply (auto intro: u)
       done
   next
-    
+
     from u nn have "finite (u ` space M)" "\<And>x. x \<in> u ` space M \<Longrightarrow> 0 \<le> x"
       unfolding simple_function_def by auto
     then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
@@ -417,7 +417,7 @@
 
   have not_inf: "\<And>x i. x \<in> space M \<Longrightarrow> U i x < \<infinity>"
     using U by (auto simp: image_iff eq_commute)
-  
+
   from U have "\<And>i. U i \<in> borel_measurable M"
     by (simp add: borel_measurable_simple_function)
 
@@ -556,12 +556,12 @@
   note eq = this
 
   have "integral\<^sup>S M f =
-    (\<Sum>y\<in>f`space M. y * (\<Sum>z\<in>g`space M. 
+    (\<Sum>y\<in>f`space M. y * (\<Sum>z\<in>g`space M.
       if \<exists>x\<in>space M. y = f x \<and> z = g x then emeasure M {x\<in>space M. g x = z} else 0))"
     unfolding simple_integral_def
   proof (safe intro!: setsum.cong ereal_right_mult_cong)
     fix y assume y: "y \<in> space M" "f y \<noteq> 0"
-    have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} = 
+    have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} =
         {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
       by auto
     have eq:"(\<Union>i\<in>{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}. {x \<in> space M. g x = i}) =
@@ -577,7 +577,7 @@
       apply (auto simp: disjoint_family_on_def eq)
       done
   qed
-  also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M. 
+  also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M.
       if \<exists>x\<in>space M. y = f x \<and> z = g x then y * emeasure M {x\<in>space M. g x = z} else 0))"
     by (auto intro!: setsum.cong simp: setsum_ereal_right_distrib emeasure_nonneg)
   also have "\<dots> = ?r"
@@ -1239,7 +1239,7 @@
   assume "space M \<noteq> {}"
   with simple_functionD(1)[OF f] bnd have bnd: "0 \<le> Max (f`space M) \<and> Max (f`space M) < \<infinity>"
     by (subst Max_less_iff) (auto simp: Max_ge_iff)
-  
+
   have "nn_integral M f \<le> (\<integral>\<^sup>+x. Max (f`space M) * indicator {x\<in>space M. f x \<noteq> 0} x \<partial>M)"
   proof (rule nn_integral_mono)
     fix x assume "x \<in> space M"
@@ -1532,7 +1532,7 @@
     by (intro nn_integral_cong)
        (simp add: sup_ereal_def[symmetric] sup_INF del: sup_ereal_def)
   also have "\<dots> = (\<integral>\<^sup>+ x. (INF j. max 0 (restrict (f (j + i)) (space M) x)) \<partial>M)"
-    using f by (intro nn_integral_cong INF_shift antimonoI le_funI max.mono) 
+    using f by (intro nn_integral_cong INF_shift antimonoI le_funI max.mono)
                (auto simp: decseq_def le_fun_def)
   also have "\<dots> = (INF j. (\<integral>\<^sup>+ x. max 0 (restrict (f (j + i)) (space M) x) \<partial>M))"
   proof (rule nn_integral_monotone_convergence_INF')
@@ -1542,7 +1542,7 @@
       using fin by (simp add: nn_integral_max_0 cong: nn_integral_cong)
   qed (intro max.cobounded1 dec f)+
   also have "\<dots> = (INF j. (\<integral>\<^sup>+ x. max 0 (restrict (f j) (space M) x) \<partial>M))"
-    using f by (intro INF_shift[symmetric] nn_integral_mono antimonoI le_funI max.mono) 
+    using f by (intro INF_shift[symmetric] nn_integral_mono antimonoI le_funI max.mono)
                (auto simp: decseq_def le_fun_def)
   finally show ?thesis unfolding nn_integral_max_0 by (simp cong: nn_integral_cong)
 qed
@@ -1670,7 +1670,7 @@
   finally show ?thesis .
 qed
 
-lemma AE_iff_nn_integral: 
+lemma AE_iff_nn_integral:
   "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> integral\<^sup>N M (indicator {x. \<not> P x}) = 0"
   by (subst nn_integral_0_iff_AE) (auto simp: one_ereal_def zero_ereal_def
     sets.sets_Collect_neg indicator_def[abs_def] measurable_If)
@@ -1800,7 +1800,7 @@
 
 subsection {* Integral under concrete measures *}
 
-lemma nn_integral_empty: 
+lemma nn_integral_empty:
   assumes "space M = {}"
   shows "nn_integral M f = 0"
 proof -
@@ -1815,7 +1815,7 @@
   assumes T: "T \<in> measurable M M'"
   and f: "f \<in> borel_measurable (distr M M' T)" "\<And>x. 0 \<le> f x"
   shows "integral\<^sup>N (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)"
-  using f 
+  using f
 proof induct
   case (cong f g)
   with T show ?case
@@ -1943,12 +1943,12 @@
 qed
 
 lemma emeasure_UN_countable:
-  assumes sets[measurable]: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M" and I[simp]: "countable I" 
+  assumes sets[measurable]: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M" and I[simp]: "countable I"
   assumes disj: "disjoint_family_on X I"
   shows "emeasure M (UNION I X) = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)"
 proof -
   have eq: "\<And>x. indicator (UNION I X) x = \<integral>\<^sup>+ i. indicator (X i) x \<partial>count_space I"
-  proof cases 
+  proof cases
     fix x assume x: "x \<in> UNION I X"
     then obtain j where j: "x \<in> X j" "j \<in> I"
       by auto
@@ -2210,7 +2210,7 @@
            intro!: setsum.mono_neutral_cong_left ereal_right_mult_cong[OF refl] arg_cong2[where f=emeasure])
 
 lemma one_not_less_zero[simp]: "\<not> 1 < (0::ereal)"
-  by (simp add: zero_ereal_def one_ereal_def) 
+  by (simp add: zero_ereal_def one_ereal_def)
 
 lemma nn_integral_restrict_space:
   assumes \<Omega>[simp]: "\<Omega> \<inter> space M \<in> sets M"
@@ -2245,7 +2245,7 @@
       by (subst s_eq) (rule simple_integral_restrict_space[symmetric, OF \<Omega> sf])
     show "\<And>x. s x \<in> {0..<\<infinity>}"
       using s by (auto simp: image_subset_iff)
-    from s show "s \<le> max 0 \<circ> f" 
+    from s show "s \<le> max 0 \<circ> f"
       by (subst s_eq) (auto simp: image_subset_iff le_fun_def split: split_indicator split_indicator_asm)
   qed
   then show ?thesis
@@ -2281,7 +2281,7 @@
 definition density :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
   "density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
 
-lemma 
+lemma
   shows sets_density[simp, measurable_cong]: "sets (density M f) = sets M"
     and space_density[simp]: "space (density M f) = space M"
   by (auto simp: density_def)
@@ -2290,7 +2290,7 @@
 lemma space_density_imp[measurable_dest]:
   "\<And>x M f. x \<in> space (density M f) \<Longrightarrow> x \<in> space M" by auto
 
-lemma 
+lemma
   shows measurable_density_eq1[simp]: "g \<in> measurable (density Mg f) Mg' \<longleftrightarrow> g \<in> measurable Mg Mg'"
     and measurable_density_eq2[simp]: "h \<in> measurable Mh (density Mh' f) \<longleftrightarrow> h \<in> measurable Mh Mh'"
     and simple_function_density_eq[simp]: "simple_function (density Mu f) u \<longleftrightarrow> simple_function Mu u"
@@ -2353,7 +2353,7 @@
       apply (intro nn_integral_cong)
       apply (auto simp: indicator_def)
       done
-    have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> 
+    have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow>
       emeasure M {x \<in> space M. max 0 (f x) * indicator A x \<noteq> 0} = 0"
       unfolding eq
       using f `A \<in> sets M`
@@ -2429,7 +2429,7 @@
 qed
 
 lemma nn_integral_density:
-  "f \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> g' \<in> borel_measurable M \<Longrightarrow> 
+  "f \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> g' \<in> borel_measurable M \<Longrightarrow>
     integral\<^sup>N (density M f) g' = (\<integral>\<^sup>+ x. f x * g' x \<partial>M)"
   by (subst (1 2) nn_integral_max_0[symmetric])
      (auto intro!: nn_integral_cong_AE
@@ -2468,7 +2468,7 @@
   by (auto simp: nn_integral_cmult_indicator emeasure_density)
 
 lemma measure_density_const:
-  "A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real c * measure M A"
+  "A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real_of_ereal c * measure M A"
   by (auto simp: emeasure_density_const measure_def)
 
 lemma density_density_eq:
@@ -2509,12 +2509,12 @@
   by (intro measure_eqI) (auto simp: emeasure_density)
 
 lemma emeasure_density_add:
-  assumes X: "X \<in> sets M" 
+  assumes X: "X \<in> sets M"
   assumes Mf[measurable]: "f \<in> borel_measurable M"
   assumes Mg[measurable]: "g \<in> borel_measurable M"
   assumes nonnegf: "\<And>x. x \<in> space M \<Longrightarrow> f x \<ge> 0"
   assumes nonnegg: "\<And>x. x \<in> space M \<Longrightarrow> g x \<ge> 0"
-  shows "emeasure (density M f) X + emeasure (density M g) X = 
+  shows "emeasure (density M f) X + emeasure (density M g) X =
            emeasure (density M (\<lambda>x. f x + g x)) X"
   using assms
   apply (subst (1 2 3) emeasure_density, simp_all) []
@@ -2697,8 +2697,8 @@
 subsubsection {* Uniform count measure *}
 
 definition "uniform_count_measure A = point_measure A (\<lambda>x. 1 / card A)"
- 
-lemma 
+
+lemma
   shows space_uniform_count_measure: "space (uniform_count_measure A) = A"
     and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A"
     unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure)
@@ -2706,13 +2706,13 @@
 lemma sets_uniform_count_measure_count_space[measurable_cong]:
   "sets (uniform_count_measure A) = sets (count_space A)"
   by (simp add: sets_uniform_count_measure)
- 
+
 lemma emeasure_uniform_count_measure:
   "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (uniform_count_measure A) X = card X / card A"
-  by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def)
- 
+  by (simp add: emeasure_point_measure_finite uniform_count_measure_def)
+
 lemma measure_uniform_count_measure:
   "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> measure (uniform_count_measure A) X = card X / card A"
-  by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def measure_def)
+  by (simp add: emeasure_point_measure_finite uniform_count_measure_def measure_def)
 
 end
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Probability_Mass_Function.thy
--- a/src/HOL/Probability/Probability_Mass_Function.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Probability_Mass_Function.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -59,7 +59,7 @@
     note singleton_sets = this
     have "?M < (\<Sum>x\<in>X. ?M / Suc n)"
       using `?M \<noteq> 0`
-      by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc field_simps less_le measure_nonneg)
+      by (simp add: `card X = Suc (Suc n)` of_nat_Suc field_simps less_le measure_nonneg)
     also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
       by (rule setsum_mono) fact
     also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
@@ -956,7 +956,7 @@
 
 lemma quotient_rel_set_disjoint:
   "equivp R \<Longrightarrow> C \<in> UNIV // {(x, y). R x y} \<Longrightarrow> rel_set R A B \<Longrightarrow> A \<inter> C = {} \<longleftrightarrow> B \<inter> C = {}"
-  using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C] 
+  using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C]
   by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE)
      (blast dest: equivp_symp)+
 
@@ -973,17 +973,17 @@
 next
   fix C assume C: "C \<in> UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)"
   assume eq: "\<forall>x\<in>set_pmf p. \<forall>y\<in>set_pmf q. R x y \<longrightarrow> measure p {x. R x y} = measure q {y. R x y}"
-  
+
   show "measure p C = measure q C"
   proof cases
     assume "p \<inter> C = {}"
-    moreover then have "q \<inter> C = {}"  
+    moreover then have "q \<inter> C = {}"
       using quotient_rel_set_disjoint[OF assms C R] by simp
     ultimately show ?thesis
       unfolding measure_pmf_zero_iff[symmetric] by simp
   next
     assume "p \<inter> C \<noteq> {}"
-    moreover then have "q \<inter> C \<noteq> {}"  
+    moreover then have "q \<inter> C \<noteq> {}"
       using quotient_rel_set_disjoint[OF assms C R] by simp
     ultimately obtain x y where in_set: "x \<in> set_pmf p" "y \<in> set_pmf q" and in_C: "x \<in> C" "y \<in> C"
       by auto
@@ -1068,11 +1068,11 @@
         and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto
       from qr obtain qr where qr: "\<And>y z. (y, z) \<in> set_pmf qr \<Longrightarrow> S y z"
         and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto
-  
+
       def pr \<equiv> "bind_pmf pq (\<lambda>xy. bind_pmf (cond_pmf qr {yz. fst yz = snd xy}) (\<lambda>yz. return_pmf (fst xy, snd yz)))"
       have pr_welldefined: "\<And>y. y \<in> q \<Longrightarrow> qr \<inter> {yz. fst yz = y} \<noteq> {}"
         by (force simp: q')
-  
+
       have "rel_pmf (R OO S) p r"
       proof (rule rel_pmf.intros)
         fix x z assume "(x, z) \<in> pr"
@@ -1283,7 +1283,7 @@
 proof (subst rel_pmf_iff_equivp, safe)
   show "equivp (inf R R\<inverse>\<inverse>)"
     using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD)
-  
+
   fix C assume "C \<in> UNIV // {(x, y). inf R R\<inverse>\<inverse> x y}"
   then obtain x where C: "C = {y. R x y \<and> R y x}"
     by (auto elim: quotientE)
@@ -1399,7 +1399,7 @@
 end
 
 lemma set_pmf_geometric: "0 < p \<Longrightarrow> p < 1 \<Longrightarrow> set_pmf (geometric_pmf p) = UNIV"
-  by (auto simp: set_pmf_iff) 
+  by (auto simp: set_pmf_iff)
 
 subsubsection \<open> Uniform Multiset Distribution \<close>
 
@@ -1445,7 +1445,7 @@
 lemma emeasure_pmf_of_set[simp]: "emeasure pmf_of_set S = 1"
   by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff)
 
-lemma nn_integral_pmf_of_set': 
+lemma nn_integral_pmf_of_set':
   "(\<And>x. x \<in> S \<Longrightarrow> f x \<ge> 0) \<Longrightarrow> nn_integral (measure_pmf pmf_of_set) f = setsum f S / card S"
 apply(subst nn_integral_measure_pmf_finite, simp_all add: S_finite)
 apply(simp add: setsum_ereal_left_distrib[symmetric])
@@ -1453,7 +1453,7 @@
 apply(simp add: ereal_times_divide_eq one_ereal_def[symmetric])
 done
 
-lemma nn_integral_pmf_of_set: 
+lemma nn_integral_pmf_of_set:
   "nn_integral (measure_pmf pmf_of_set) f = setsum (max 0 \<circ> f) S / card S"
 apply(subst nn_integral_max_0[symmetric])
 apply(subst nn_integral_pmf_of_set')
@@ -1476,7 +1476,7 @@
 lemma pmf_of_set_singleton: "pmf_of_set {x} = return_pmf x"
 by(rule pmf_eqI)(simp add: indicator_def)
 
-lemma map_pmf_of_set_inj: 
+lemma map_pmf_of_set_inj:
   assumes f: "inj_on f A"
   and [simp]: "A \<noteq> {}" "finite A"
   shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" (is "?lhs = ?rhs")
@@ -1540,7 +1540,7 @@
     ereal (\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
     using p_le_1 p_nonneg by (subst nn_integral_count_space') auto
   also have "(\<Sum>k\<le>n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n"
-    by (subst binomial_ring) (simp add: atLeast0AtMost real_of_nat_def)
+    by (subst binomial_ring) (simp add: atLeast0AtMost)
   finally show "(\<integral>\<^sup>+ x. ereal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) \<partial>count_space UNIV) = 1"
     by simp
 qed (insert p_nonneg p_le_1, simp)
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Projective_Limit.thy
--- a/src/HOL/Probability/Projective_Limit.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Projective_Limit.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -293,9 +293,9 @@
       also have "\<dots> \<le> ereal (2 powr - real i) * ?a" using K'(1)[of i] .
       finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" .
     qed
-    also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real i)) * real ?a)"
+    also have "\<dots> = ereal ((\<Sum> i\<in>{1..n}. (2 powr -real_of_ereal i)) * real_of_ereal ?a)"
       by (simp add: setsum_left_distrib r)
-    also have "\<dots> < ereal (1 * real ?a)" unfolding less_ereal.simps
+    also have "\<dots> < ereal (1 * real_of_ereal ?a)" unfolding less_ereal.simps
     proof (rule mult_strict_right_mono)
       have "(\<Sum>i = 1..n. 2 powr - real i) = (\<Sum>i = 1..<Suc n. (1/2) ^ i)"
         by (rule setsum.cong) (auto simp: powr_realpow powr_divide powr_minus_divide)  
@@ -303,7 +303,7 @@
       also have "setsum (op ^ (1 / 2::real)) ({..<Suc n} - {0}) =
         setsum (op ^ (1 / 2)) ({..<Suc n}) - 1" by (auto simp: setsum_diff1)
       also have "\<dots> < 1" by (subst geometric_sum) auto
-      finally show "(\<Sum>i = 1..n. 2 powr - real i) < 1" .
+      finally show "(\<Sum>i = 1..n. 2 powr - real_of_ereal i) < 1" by simp
     qed (auto simp: r ereal_less_real_iff zero_ereal_def[symmetric])
     also have "\<dots> = ?a" by (auto simp: r)
     also have "\<dots> \<le> \<mu>G (Z n)"
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Radon_Nikodym.thy
--- a/src/HOL/Probability/Radon_Nikodym.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Radon_Nikodym.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -11,7 +11,7 @@
 definition "diff_measure M N =
   measure_of (space M) (sets M) (\<lambda>A. emeasure M A - emeasure N A)"
 
-lemma 
+lemma
   shows space_diff_measure[simp]: "space (diff_measure M N) = space M"
     and sets_diff_measure[simp]: "sets (diff_measure M N) = sets M"
   by (auto simp: diff_measure_def)
@@ -205,7 +205,7 @@
       by (auto simp add: not_less)
     { fix n have "?d (A n) \<le> - real n * e"
       proof (induct n)
-        case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: real_of_nat_Suc field_simps)
+        case (Suc n) with dA_epsilon[of n, OF B] show ?case by (simp del: A_simps add: of_nat_Suc field_simps)
       next
         case 0 with measure_empty show ?case by (simp add: zero_ereal_def)
       qed } note dA_less = this
@@ -272,7 +272,7 @@
       hence "0 < - ?d B" by auto
       from ex_inverse_of_nat_Suc_less[OF this]
       obtain n where *: "?d B < - 1 / real (Suc n)"
-        by (auto simp: real_eq_of_nat field_simps)
+        by (auto simp: field_simps)
       also have "\<dots> \<le> - 1 / real (Suc (Suc n))"
         by (auto simp: field_simps)
       finally show False
@@ -443,8 +443,8 @@
         using b by (auto simp: emeasure_density_const sets_eq intro!: finite_measureI)
     from M'.Radon_Nikodym_aux[OF this] guess A0 ..
     then have "A0 \<in> sets M"
-      and space_less_A0: "measure ?M (space M) - real b * measure M (space M) \<le> measure ?M A0 - real b * measure M A0"
-      and *: "\<And>B. B \<in> sets M \<Longrightarrow> B \<subseteq> A0 \<Longrightarrow> 0 \<le> measure ?M B - real b * measure M B"
+      and space_less_A0: "measure ?M (space M) - real_of_ereal b * measure M (space M) \<le> measure ?M A0 - real_of_ereal b * measure M A0"
+      and *: "\<And>B. B \<in> sets M \<Longrightarrow> B \<subseteq> A0 \<Longrightarrow> 0 \<le> measure ?M B - real_of_ereal b * measure M B"
       using b by (simp_all add: measure_density_const sets_eq_imp_space_eq[OF sets_eq] sets_eq)
     { fix B assume B: "B \<in> sets M" "B \<subseteq> A0"
       with *[OF this] have "b * emeasure M B \<le> ?M B"
@@ -820,7 +820,7 @@
   let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x"
 
   have ac: "absolutely_continuous M (density M f)" "sets (density M f) = sets M"
-    using borel by (auto intro!: absolutely_continuousI_density) 
+    using borel by (auto intro!: absolutely_continuousI_density)
   from split_space_into_finite_sets_and_rest[OF this]
   obtain Q0 and Q :: "nat \<Rightarrow> 'a set"
     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
@@ -857,7 +857,7 @@
         then show "?A i \<in> null_sets M" using in_Q0[OF `?A i \<in> sets M`] `?A i \<in> sets M` by auto
       qed
       also have "(\<Union>i. ?A i) = Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
-        by (auto simp: less_PInf_Ex_of_nat real_eq_of_nat)
+        by (auto simp: less_PInf_Ex_of_nat)
       finally have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" by simp }
     from this[OF borel(1) refl] this[OF borel(2) f]
     have "Q0 \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" "Q0 \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets M" by simp_all
@@ -931,7 +931,7 @@
   and fin: "sigma_finite_measure (density M f)"
   shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)"
 proof
-  assume "AE x in M. f x = g x" with borel show "density M f = density M g" 
+  assume "AE x in M. f x = g x" with borel show "density M f = density M g"
     by (auto intro: density_cong)
 next
   assume eq: "density M f = density M g"
@@ -1010,8 +1010,8 @@
         case PInf with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0])
       next
         case (real r)
-        with less_PInf_Ex_of_nat[of "f x"] obtain n :: nat where "f x < n" by (auto simp: real_eq_of_nat)
-        then show ?thesis using x real unfolding A_def by (auto intro!: exI[of _ "Suc n"] simp: real_eq_of_nat)
+        with less_PInf_Ex_of_nat[of "f x"] obtain n :: nat where "f x < n" by auto
+        then show ?thesis using x real unfolding A_def by (auto intro!: exI[of _ "Suc n"])
       next
         case MInf with x show ?thesis
           unfolding A_def by (auto intro!: exI[of _ "Suc 0"])
@@ -1031,11 +1031,11 @@
       case (Suc n)
       then have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
         (\<integral>\<^sup>+x. (Suc n :: ereal) * indicator (Q j) x \<partial>M)"
-        by (auto intro!: nn_integral_mono simp: indicator_def A_def real_eq_of_nat)
+        by (auto intro!: nn_integral_mono simp: indicator_def A_def)
       also have "\<dots> = Suc n * emeasure M (Q j)"
         using Q by (auto intro!: nn_integral_cmult_indicator)
       also have "\<dots> < \<infinity>"
-        using Q by (auto simp: real_eq_of_nat[symmetric])
+        using Q by auto
       finally show ?thesis by simp
     qed
     then show "emeasure ?N X \<noteq> \<infinity>"
@@ -1058,7 +1058,7 @@
        then SOME f. f \<in> borel_measurable M \<and> (\<forall>x. 0 \<le> f x) \<and> density M f = N
        else (\<lambda>_. 0))"
 
-lemma RN_derivI: 
+lemma RN_derivI:
   assumes "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "density M f = N"
   shows "density M (RN_deriv M N) = N"
 proof -
@@ -1070,7 +1070,7 @@
     by (auto simp: RN_deriv_def)
 qed
 
-lemma 
+lemma
   shows borel_measurable_RN_deriv[measurable]: "RN_deriv M N \<in> borel_measurable M" (is ?m)
     and RN_deriv_nonneg: "0 \<le> RN_deriv M N x" (is ?nn)
 proof -
@@ -1205,16 +1205,16 @@
   assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
     and f: "f \<in> borel_measurable M"
   shows RN_deriv_integrable: "integrable N f \<longleftrightarrow>
-      integrable M (\<lambda>x. real (RN_deriv M N x) * f x)" (is ?integrable)
-    and RN_deriv_integral: "integral\<^sup>L N f = (\<integral>x. real (RN_deriv M N x) * f x \<partial>M)" (is ?integral)
+      integrable M (\<lambda>x. real_of_ereal (RN_deriv M N x) * f x)" (is ?integrable)
+    and RN_deriv_integral: "integral\<^sup>L N f = (\<integral>x. real_of_ereal (RN_deriv M N x) * f x \<partial>M)" (is ?integral)
 proof -
   note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp]
   interpret N: sigma_finite_measure N by fact
 
-  have eq: "density M (RN_deriv M N) = density M (\<lambda>x. real (RN_deriv M N x))"
+  have eq: "density M (RN_deriv M N) = density M (\<lambda>x. real_of_ereal (RN_deriv M N x))"
   proof (rule density_cong)
     from RN_deriv_finite[OF assms(1,2,3)]
-    show "AE x in M. RN_deriv M N x = ereal (real (RN_deriv M N x))"
+    show "AE x in M. RN_deriv M N x = ereal (real_of_ereal (RN_deriv M N x))"
       by eventually_elim (insert RN_deriv_nonneg[of M N], auto simp: ereal_real)
   qed (insert ac, auto)
 
@@ -1242,12 +1242,12 @@
     and "\<And>x. 0 \<le> D x"
 proof
   interpret N: finite_measure N by fact
-  
+
   note RN = borel_measurable_RN_deriv density_RN_deriv[OF ac] RN_deriv_nonneg[of M N]
 
   let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M N x = t}"
 
-  show "(\<lambda>x. real (RN_deriv M N x)) \<in> borel_measurable M"
+  show "(\<lambda>x. real_of_ereal (RN_deriv M N x)) \<in> borel_measurable M"
     using RN by auto
 
   have "N (?RN \<infinity>) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN \<infinity>) x \<partial>M)"
@@ -1268,12 +1268,12 @@
   qed
   ultimately have "AE x in M. RN_deriv M N x < \<infinity>"
     using RN by (intro AE_iff_measurable[THEN iffD2]) auto
-  then show "AE x in M. RN_deriv M N x = ereal (real (RN_deriv M N x))"
+  then show "AE x in M. RN_deriv M N x = ereal (real_of_ereal (RN_deriv M N x))"
     using RN(3) by (auto simp: ereal_real)
-  then have eq: "AE x in N. RN_deriv M N x = ereal (real (RN_deriv M N x))"
+  then have eq: "AE x in N. RN_deriv M N x = ereal (real_of_ereal (RN_deriv M N x))"
     using ac absolutely_continuous_AE by auto
 
-  show "\<And>x. 0 \<le> real (RN_deriv M N x)"
+  show "\<And>x. 0 \<le> real_of_ereal (RN_deriv M N x)"
     using RN by (auto intro: real_of_ereal_pos)
 
   have "N (?RN 0) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN 0) x \<partial>M)"
@@ -1282,7 +1282,7 @@
     by (intro nn_integral_cong) (auto simp: indicator_def)
   finally have "AE x in N. RN_deriv M N x \<noteq> 0"
     using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq)
-  with RN(3) eq show "AE x in N. 0 < real (RN_deriv M N x)"
+  with RN(3) eq show "AE x in N. 0 < real_of_ereal (RN_deriv M N x)"
     by (auto simp: zero_less_real_of_ereal le_less)
 qed
 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Regularity.thy
--- a/src/HOL/Probability/Regularity.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Regularity.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -26,7 +26,7 @@
     assume "\<not> x \<le> y" hence "x > y" by simp
     hence y_fin: "\<bar>y\<bar> \<noteq> \<infinity>" using `y \<ge> 0` by auto
     have x_fin: "\<bar>x\<bar> \<noteq> \<infinity>" using `x > y` f_fin approx[where e = 1] by auto
-    def e \<equiv> "real ((x - y) / 2)"
+    def e \<equiv> "real_of_ereal ((x - y) / 2)"
     have e: "x > y + e" "e > 0" using `x > y` y_fin x_fin by (auto simp: e_def field_simps)
     note e(1)
     also from approx[OF `e > 0`] obtain i where i: "i \<in> A" "x \<le> f i + e" by blast
@@ -56,7 +56,7 @@
     hence y_fin: "\<bar>y\<bar> \<noteq> \<infinity>" using `y \<noteq> \<infinity>` by auto
     have x_fin: "\<bar>x\<bar> \<noteq> \<infinity>" using `y > x` f_fin f_nonneg approx[where e = 1] A_notempty
       by auto
-    def e \<equiv> "real ((y - x) / 2)"
+    def e \<equiv> "real_of_ereal ((y - x) / 2)"
     have e: "y > x + e" "e > 0" using `y > x` y_fin x_fin by (auto simp: e_def field_simps)
     from approx[OF `e > 0`] obtain i where i: "i \<in> A" "x + e \<ge> f i" by blast
     note i(2)
@@ -192,7 +192,7 @@
     also have "\<dots> \<le> (\<Sum>n. ereal (e*2 powr - real (Suc n)))"
       using B_compl_le by (intro suminf_le_pos) (simp_all add: measure_nonneg emeasure_eq_measure)
     also have "\<dots> \<le> (\<Sum>n. ereal (e * (1 / 2) ^ Suc n))"
-      by (simp add: Transcendental.powr_minus powr_realpow field_simps del: real_of_nat_Suc)
+      by (simp add: Transcendental.powr_minus powr_realpow field_simps del: of_nat_Suc)
     also have "\<dots> = (\<Sum>n. ereal e * ((1 / 2) ^ Suc n))"
       unfolding times_ereal.simps[symmetric] ereal_power[symmetric] one_ereal_def numeral_eq_ereal
       by simp
@@ -254,7 +254,7 @@
       from in_closed_iff_infdist_zero[OF `closed A` `A \<noteq> {}`]
       have "A = {x. infdist x A = 0}" by auto
       also have "\<dots> = (\<Inter>i. ?G (1/real (Suc i)))"
-      proof (auto simp del: real_of_nat_Suc, rule ccontr)
+      proof (auto simp del: of_nat_Suc, rule ccontr)
         fix x
         assume "infdist x A \<noteq> 0"
         hence pos: "infdist x A > 0" using infdist_nonneg[of x A] by simp
@@ -262,7 +262,7 @@
         moreover
         assume "\<forall>i. infdist x A < 1 / real (Suc i)"
         hence "infdist x A < 1 / real (Suc n)" by auto
-        ultimately show False by simp 
+        ultimately show False by simp
       qed
       also have "M \<dots> = (INF n. emeasure M (?G (1 / real (Suc n))))"
       proof (rule INF_emeasure_decseq[symmetric], safe)
@@ -294,7 +294,7 @@
     note `?inner A` `?outer A` }
   note closed_in_D = this
   from `B \<in> sets borel`
-  have "Int_stable (Collect closed)" "Collect closed \<subseteq> Pow UNIV" "B \<in> sigma_sets UNIV (Collect closed)" 
+  have "Int_stable (Collect closed)" "Collect closed \<subseteq> Pow UNIV" "B \<in> sigma_sets UNIV (Collect closed)"
     by (auto simp: Int_stable_def borel_eq_closed)
   then show "?inner B" "?outer B"
   proof (induct B rule: sigma_sets_induct_disjoint)
@@ -356,7 +356,7 @@
     finally have measure_LIMSEQ: "(\<lambda>n. \<Sum>i<n. measure M (D i)) ----> measure M (\<Union>i. D i)"
       by (simp add: emeasure_eq_measure)
     have "(\<Union>i. D i) \<in> sets M" using `range D \<subseteq> sets M` by auto
-    
+
     case 1
     show ?case
     proof (rule approx_inner)
@@ -398,7 +398,7 @@
       also have "\<dots> = (\<Sum>i<n0. measure M (K i)) + (\<Sum>i<n0. e/(2*Suc n0))"
         by (simp add: setsum.distrib)
       also have "\<dots> \<le> (\<Sum>i<n0. measure M (K i)) +  e / 2" using `0 < e`
-        by (auto simp: real_of_nat_def[symmetric] field_simps intro!: mult_left_mono)
+        by (auto simp: field_simps intro!: mult_left_mono)
       finally
       have "measure M (\<Union>i. D i) < (\<Sum>i<n0. measure M (K i)) + e / 2 + e / 2"
         by auto
@@ -439,7 +439,7 @@
         by (intro suminf_le_pos, subst emeasure_Diff)
            (auto simp: emeasure_Diff emeasure_eq_measure sb measure_nonneg intro: less_imp_le)
       also have "\<dots> \<le> (\<Sum>n. ereal (e * (1 / 2) ^ Suc n))"
-        by (simp add: powr_minus inverse_eq_divide powr_realpow field_simps power_divide del: real_of_nat_Suc)
+        by (simp add: powr_minus inverse_eq_divide powr_realpow field_simps power_divide del: of_nat_Suc)
       also have "\<dots> = (\<Sum>n. ereal e * ((1 / 2) ^  Suc n))"
         unfolding times_ereal.simps[symmetric] ereal_power[symmetric] one_ereal_def numeral_eq_ereal
         by simp
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/Sigma_Algebra.thy
--- a/src/HOL/Probability/Sigma_Algebra.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/Sigma_Algebra.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -1468,7 +1468,7 @@
   "emeasure M = snd (snd (Rep_measure M))"
 
 definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where
-  "measure M A = real (emeasure M A)"
+  "measure M A = real_of_ereal (emeasure M A)"
 
 declare [[coercion sets]]
 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy
--- a/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -374,7 +374,7 @@
       using finite_measure_finite_Union[OF _ df]
       by (auto simp add: * intro!: setsum_nonneg)
     also have "(\<Sum>obs'\<in>?S obs. \<P>(OB ; fst) {(obs', k)}) = real (card (?S obs)) * \<P>(OB ; fst) {(obs, k)}"
-      by (simp add: t_eq_imp[OF `k \<in> keys` `K k \<noteq> 0` obs] real_eq_of_nat)
+      by (simp add: t_eq_imp[OF `k \<in> keys` `K k \<noteq> 0` obs])
     finally have "\<P>(t\<circ>OB ; fst) {(t obs, k)} = real (card (?S obs)) * \<P>(OB ; fst) {(obs, k)}" .}
   note P_t_eq_P_OB = this
 
@@ -471,9 +471,9 @@
     using entropy_le_card[of "t\<circ>OB", OF simple_distributedI[OF simple_function_finite refl]] by simp
   also have "\<dots> \<le> log b (real (n + 1)^card observations)"
     using card_T_bound not_empty
-    by (auto intro!: log_le simp: card_gt_0_iff power_real_of_nat simp del: real_of_nat_power real_of_nat_Suc)
+    by (auto intro!: log_le simp: card_gt_0_iff power_real_of_nat simp del: of_nat_power of_nat_Suc)
   also have "\<dots> = real (card observations) * log b (real n + 1)"
-    by (simp add: log_nat_power real_of_nat_Suc)
+    by (simp add: log_nat_power add.commute)
   finally show ?thesis  .
 qed
 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Real.thy
--- a/src/HOL/Real.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Real.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -760,10 +760,9 @@
   shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"
 by (erule contrapos_pp, simp add: not_less, erule Real_leI [OF Y])
 
-lemma of_nat_less_two_power:
+lemma of_nat_less_two_power [simp]:
   "of_nat n < (2::'a::linordered_idom) ^ n"
-apply (induct n)
-apply simp
+apply (induct n, simp)
 by (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc)
 
 lemma complete_real:
@@ -995,6 +994,11 @@
   "real_of_nat \<equiv> of_nat"
 
 abbreviation
+  real :: "nat \<Rightarrow> real"
+where
+  "real \<equiv> of_nat"
+
+abbreviation
   real_of_int :: "int \<Rightarrow> real"
 where
   "real_of_int \<equiv> of_int"
@@ -1004,30 +1008,11 @@
 where
   "real_of_rat \<equiv> of_rat"
 
-class real_of =
-  fixes real :: "'a \<Rightarrow> real"
-
-instantiation nat :: real_of
-begin
-
-definition real_nat :: "nat \<Rightarrow> real" where real_of_nat_def [code_unfold]: "real \<equiv> of_nat"
-
-instance ..
-end
-
-instantiation int :: real_of
-begin
-
-definition real_int :: "int \<Rightarrow> real" where real_of_int_def [code_unfold]: "real \<equiv> of_int"
-
-instance ..
-end
-
 declare [[coercion_enabled]]
 
 declare [[coercion "of_nat :: nat \<Rightarrow> int"]]
-declare [[coercion "real   :: nat \<Rightarrow> real"]]
-declare [[coercion "real   :: int \<Rightarrow> real"]]
+declare [[coercion "of_nat :: nat \<Rightarrow> real"]]
+declare [[coercion "of_int :: int \<Rightarrow> real"]]
 
 (* We do not add rat to the coerced types, this has often unpleasant side effects when writing
 inverse (Suc n) which sometimes gets two coercions: of_rat (inverse (of_nat (Suc n))) *)
@@ -1036,105 +1021,40 @@
 declare [[coercion_map "\<lambda>f g h x. g (h (f x))"]]
 declare [[coercion_map "\<lambda>f g (x,y). (f x, g y)"]]
 
-lemma real_eq_of_nat: "real = of_nat"
-  unfolding real_of_nat_def ..
-
-lemma real_eq_of_int: "real = of_int"
-  unfolding real_of_int_def ..
-
-lemma real_of_int_zero [simp]: "real (0::int) = 0"
-by (simp add: real_of_int_def)
-
-lemma real_of_one [simp]: "real (1::int) = (1::real)"
-by (simp add: real_of_int_def)
-
-lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
-by (simp add: real_of_int_def)
-
-lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
-by (simp add: real_of_int_def)
-
-lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
-by (simp add: real_of_int_def)
+declare of_int_eq_0_iff [algebra, presburger]
+declare of_int_eq_iff [iff, algebra, presburger] (*FIXME*)
+declare of_int_eq_iff [iff, algebra, presburger] (*FIXME*)
+declare of_int_less_iff [iff, algebra, presburger] (*FIXME*)
+declare of_int_le_iff [iff, algebra, presburger] (*FIXME*)
 
-lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
-by (simp add: real_of_int_def)
-
-lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n"
-by (simp add: real_of_int_def of_int_power)
-
-lemmas power_real_of_int = real_of_int_power [symmetric]
+declare of_int_0_less_iff [presburger]
+declare of_int_0_le_iff [presburger]
+declare of_int_less_0_iff [presburger]
+declare of_int_le_0_iff [presburger]
 
-lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
-  apply (subst real_eq_of_int)+
-  apply (rule of_int_setsum)
-done
-
-lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) =
-    (PROD x:A. real(f x))"
-  apply (subst real_eq_of_int)+
-  apply (rule of_int_setprod)
-done
-
-lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"
-by (simp add: real_of_int_def)
-
-lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"
-by (simp add: real_of_int_def)
+lemma real_of_int_abs [simp]: "real_of_int (abs x) = abs(real_of_int x)"
+  by (auto simp add: abs_if)
 
-lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"
-by (simp add: real_of_int_def)
-
-lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)"
-by (simp add: real_of_int_def)
-
-lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)"
-by (simp add: real_of_int_def)
-
-lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)"
-by (simp add: real_of_int_def)
-
-lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)"
-by (simp add: real_of_int_def)
-
-lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)"
-by (simp add: real_of_int_def)
-
-lemma one_less_real_of_int_cancel_iff: "1 < real (i :: int) \<longleftrightarrow> 1 < i"
-  unfolding real_of_one[symmetric] real_of_int_less_iff ..
-
-lemma one_le_real_of_int_cancel_iff: "1 \<le> real (i :: int) \<longleftrightarrow> 1 \<le> i"
-  unfolding real_of_one[symmetric] real_of_int_le_iff ..
+lemma int_less_real_le: "(n < m) = (real_of_int n + 1 <= real_of_int m)"
+proof -
+  have "(0::real) \<le> 1"
+    by (metis less_eq_real_def zero_less_one)
+  thus ?thesis
+    by (metis floor_unique less_add_one less_imp_le not_less of_int_le_iff order_trans)
+qed
 
-lemma real_of_int_less_one_cancel_iff: "real (i :: int) < 1 \<longleftrightarrow> i < 1"
-  unfolding real_of_one[symmetric] real_of_int_less_iff ..
-
-lemma real_of_int_le_one_cancel_iff: "real (i :: int) \<le> 1 \<longleftrightarrow> i \<le> 1"
-  unfolding real_of_one[symmetric] real_of_int_le_iff ..
-
-lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
-by (auto simp add: abs_if)
+lemma int_le_real_less: "(n \<le> m) = (real_of_int n < real_of_int m + 1)"
+  by (meson int_less_real_le not_le)
 
-lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
-  apply (subgoal_tac "real n + 1 = real (n + 1)")
-  apply (simp del: real_of_int_add)
-  apply auto
-done
 
-lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
-  apply (subgoal_tac "real m + 1 = real (m + 1)")
-  apply (simp del: real_of_int_add)
-  apply simp
-done
-
-lemma real_of_int_div_aux: "(real (x::int)) / (real d) =
-    real (x div d) + (real (x mod d)) / (real d)"
+lemma real_of_int_div_aux: "(real_of_int x) / (real_of_int d) =
+    real_of_int (x div d) + (real_of_int (x mod d)) / (real_of_int d)"
 proof -
   have "x = (x div d) * d + x mod d"
     by auto
-  then have "real x = real (x div d) * real d + real(x mod d)"
-    by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
-  then have "real x / real d = ... / real d"
+  then have "real_of_int x = real_of_int (x div d) * real_of_int d + real_of_int(x mod d)"
+    by (metis of_int_add of_int_mult)
+  then have "real_of_int x / real_of_int d = ... / real_of_int d"
     by simp
   then show ?thesis
     by (auto simp add: add_divide_distrib algebra_simps)
@@ -1142,133 +1062,73 @@
 
 lemma real_of_int_div:
   fixes d n :: int
-  shows "d dvd n \<Longrightarrow> real (n div d) = real n / real d"
+  shows "d dvd n \<Longrightarrow> real_of_int (n div d) = real_of_int n / real_of_int d"
   by (simp add: real_of_int_div_aux)
 
 lemma real_of_int_div2:
-  "0 <= real (n::int) / real (x) - real (n div x)"
-  apply (case_tac "x = 0")
-  apply simp
+  "0 <= real_of_int n / real_of_int x - real_of_int (n div x)"
+  apply (case_tac "x = 0", simp)
   apply (case_tac "0 < x")
-  apply (simp add: algebra_simps)
-  apply (subst real_of_int_div_aux)
-  apply simp
-  apply (simp add: algebra_simps)
-  apply (subst real_of_int_div_aux)
-  apply simp
-  apply (subst zero_le_divide_iff)
-  apply auto
-done
+   apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)
+  apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)
+  done
 
 lemma real_of_int_div3:
-  "real (n::int) / real (x) - real (n div x) <= 1"
+  "real_of_int (n::int) / real_of_int (x) - real_of_int (n div x) <= 1"
   apply (simp add: algebra_simps)
   apply (subst real_of_int_div_aux)
   apply (auto simp add: divide_le_eq intro: order_less_imp_le)
 done
 
-lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x"
+lemma real_of_int_div4: "real_of_int (n div x) <= real_of_int (n::int) / real_of_int x"
 by (insert real_of_int_div2 [of n x], simp)
 
-lemma Ints_real_of_int [simp]: "real (x::int) \<in> \<int>"
-unfolding real_of_int_def by (rule Ints_of_int)
-
 
 subsection\<open>Embedding the Naturals into the Reals\<close>
 
-lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
-by (simp add: real_of_nat_def)
-
-lemma real_of_nat_1 [simp]: "real (1::nat) = 1"
-by (simp add: real_of_nat_def)
-
-lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
-by (simp add: real_of_nat_def)
-
-lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
-by (simp add: real_of_nat_def)
-
-lemma real_of_nat_Suc [simp]: "real (Suc n) = real n + (1::real)"
-by (simp add: real_of_nat_def)
+declare of_nat_less_iff [iff] (*FIXME*)
+declare of_nat_le_iff [iff] (*FIXME*)
+declare of_nat_0_le_iff [iff] (*FIXME*)
+declare of_nat_less_iff [iff] (*FIXME*)
+declare of_nat_less_iff [iff] (*FIXME*)
 
-lemma real_of_nat_less_iff [iff]:
-     "(real (n::nat) < real m) = (n < m)"
-by (simp add: real_of_nat_def)
-
-lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
-by (simp add: real_of_nat_def)
-
-lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
-by (simp add: real_of_nat_def)
+lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"  (*NEEDED?*)
+   using of_nat_0_less_iff by blast
 
-lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
-by (simp add: real_of_nat_def del: of_nat_Suc)
-
-lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
-by (simp add: real_of_nat_def of_nat_mult)
-
-lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n"
-by (simp add: real_of_nat_def of_nat_power)
-
-lemmas power_real_of_nat = real_of_nat_power [symmetric]
+declare of_nat_mult [simp] (*FIXME*)
+declare of_nat_power [simp] (*FIXME*)
 
-lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) =
-    (SUM x:A. real(f x))"
-  apply (subst real_eq_of_nat)+
-  apply (rule of_nat_setsum)
-done
+lemmas power_real_of_nat = of_nat_power [symmetric]
 
-lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) =
-    (PROD x:A. real(f x))"
-  apply (subst real_eq_of_nat)+
-  apply (rule of_nat_setprod)
-done
-
-lemma real_of_card: "real (card A) = setsum (%x.1) A"
-  apply (subst card_eq_setsum)
-  apply (subst real_of_nat_setsum)
-  apply simp
-done
+declare of_nat_setsum [simp] (*FIXME*)
+declare of_nat_setprod [simp] (*FIXME*)
 
-lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
-by (simp add: real_of_nat_def)
-
-lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
-by (simp add: real_of_nat_def)
-
-lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
-by (simp add: add: real_of_nat_def of_nat_diff)
+lemma real_of_card: "real (card A) = setsum (\<lambda>x.1) A"
+  by simp
 
-lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
-by (auto simp: real_of_nat_def)
-
-lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
-by (simp add: add: real_of_nat_def)
+declare of_nat_eq_iff [iff] (*FIXME*)
+declare of_nat_eq_0_iff [iff] (*FIXME*)
 
-lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
-by (simp add: add: real_of_nat_def)
-
-lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
-by (metis discrete real_of_nat_1 real_of_nat_add real_of_nat_le_iff)
+lemma nat_less_real_le: "(n < m) = (real n + 1 \<le> real m)"
+  by (metis discrete of_nat_1 of_nat_add of_nat_le_iff)
 
 lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
   by (meson nat_less_real_le not_le)
 
-lemma real_of_nat_div_aux: "(real (x::nat)) / (real d) =
+lemma real_of_nat_div_aux: "(real x) / (real d) =
     real (x div d) + (real (x mod d)) / (real d)"
 proof -
   have "x = (x div d) * d + x mod d"
     by auto
   then have "real x = real (x div d) * real d + real(x mod d)"
-    by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
+    by (metis of_nat_add of_nat_mult)
   then have "real x / real d = \<dots> / real d"
     by simp
   then show ?thesis
     by (auto simp add: add_divide_distrib algebra_simps)
 qed
 
-lemma real_of_nat_div: "(d :: nat) dvd n ==>
-    real(n div d) = real n / real d"
+lemma real_of_nat_div: "d dvd n \<Longrightarrow> real(n div d) = real n / real d"
   by (subst real_of_nat_div_aux)
     (auto simp add: dvd_eq_mod_eq_0 [symmetric])
 
@@ -1291,85 +1151,63 @@
 lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"
 by (insert real_of_nat_div2 [of n x], simp)
 
-lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
-by (simp add: real_of_int_def real_of_nat_def)
-
-lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
-  apply (subgoal_tac "real(int(nat x)) = real(nat x)")
-  apply force
-  apply (simp only: real_of_int_of_nat_eq)
-done
-
-lemma Nats_real_of_nat [simp]: "real (n::nat) \<in> \<nat>"
-unfolding real_of_nat_def by (rule of_nat_in_Nats)
-
-lemma Ints_real_of_nat [simp]: "real (n::nat) \<in> \<int>"
-unfolding real_of_nat_def by (rule Ints_of_nat)
+lemma of_nat_nat [simp]: "0 <= x ==> real(nat x) = real x"
+  by force
 
 subsection \<open>The Archimedean Property of the Reals\<close>
 
-theorem reals_Archimedean:
-  assumes x_pos: "0 < x"
-  shows "\<exists>n. inverse (real (Suc n)) < x"
-  unfolding real_of_nat_def using x_pos
-  by (rule ex_inverse_of_nat_Suc_less)
-
-lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"
-  unfolding real_of_nat_def by (rule ex_less_of_nat)
+lemmas reals_Archimedean = ex_inverse_of_nat_Suc_less  (*FIXME*)
+lemmas reals_Archimedean2 = ex_less_of_nat
 
 lemma reals_Archimedean3:
   assumes x_greater_zero: "0 < x"
-  shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"
-  unfolding real_of_nat_def using \<open>0 < x\<close>
-  by (auto intro: ex_less_of_nat_mult)
+  shows "\<forall>y. \<exists>n. y < real n * x"
+  using \<open>0 < x\<close> by (auto intro: ex_less_of_nat_mult)
 
 
 subsection\<open>Rationals\<close>
 
-lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"
-by (simp add: real_eq_of_nat)
-
 lemma Rats_eq_int_div_int:
-  "\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")
+  "\<rat> = { real_of_int i / real_of_int j |i j. j \<noteq> 0}" (is "_ = ?S")
 proof
   show "\<rat> \<subseteq> ?S"
   proof
     fix x::real assume "x : \<rat>"
     then obtain r where "x = of_rat r" unfolding Rats_def ..
     have "of_rat r : ?S"
-      by (cases r)(auto simp add:of_rat_rat real_eq_of_int)
+      by (cases r) (auto simp add:of_rat_rat)
     thus "x : ?S" using \<open>x = of_rat r\<close> by simp
   qed
 next
   show "?S \<subseteq> \<rat>"
   proof(auto simp:Rats_def)
     fix i j :: int assume "j \<noteq> 0"
-    hence "real i / real j = of_rat(Fract i j)"
-      by (simp add:of_rat_rat real_eq_of_int)
-    thus "real i / real j \<in> range of_rat" by blast
+    hence "real_of_int i / real_of_int j = of_rat(Fract i j)"
+      by (simp add: of_rat_rat)
+    thus "real_of_int i / real_of_int j \<in> range of_rat" by blast
   qed
 qed
 
 lemma Rats_eq_int_div_nat:
-  "\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"
+  "\<rat> = { real_of_int i / real n |i n. n \<noteq> 0}"
 proof(auto simp:Rats_eq_int_div_int)
   fix i j::int assume "j \<noteq> 0"
-  show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n"
+  show "EX (i'::int) (n::nat). real_of_int i / real_of_int j = real_of_int i'/real n \<and> 0<n"
   proof cases
     assume "j>0"
-    hence "real i/real j = real i/real(nat j) \<and> 0<nat j"
-      by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
+    hence "real_of_int i / real_of_int j = real_of_int i/real(nat j) \<and> 0<nat j"
+      by (simp add: of_nat_nat)
     thus ?thesis by blast
   next
     assume "~ j>0"
-    hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using \<open>j\<noteq>0\<close>
-      by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
+    hence "real_of_int i / real_of_int j = real_of_int(-i) / real(nat(-j)) \<and> 0<nat(-j)" using \<open>j\<noteq>0\<close>
+      by (simp add: of_nat_nat)
     thus ?thesis by blast
   qed
 next
   fix i::int and n::nat assume "0 < n"
-  hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp
-  thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast
+  hence "real_of_int i / real n = real_of_int i / real_of_int(int n) \<and> int n \<noteq> 0" by simp
+  thus "\<exists>i' j. real_of_int i / real n = real_of_int i' / real_of_int j \<and> j \<noteq> 0" by blast
 qed
 
 lemma Rats_abs_nat_div_natE:
@@ -1377,9 +1215,9 @@
   obtains m n :: nat
   where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
 proof -
-  from \<open>x \<in> \<rat>\<close> obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"
+  from \<open>x \<in> \<rat>\<close> obtain i::int and n::nat where "n \<noteq> 0" and "x = real_of_int i / real n"
     by(auto simp add: Rats_eq_int_div_nat)
-  hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp
+  hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by (simp add: of_nat_nat)
   then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
   let ?gcd = "gcd m n"
   from \<open>n\<noteq>0\<close> have gcd: "?gcd \<noteq> 0" by simp
@@ -1396,9 +1234,8 @@
   moreover
   have "\<bar>x\<bar> = real ?k / real ?l"
   proof -
-    from gcd have "real ?k / real ?l =
-      real (?gcd * ?k) / real (?gcd * ?l)"
-      by (simp only: real_of_nat_mult) simp
+    from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)"
+      by (simp add: real_of_nat_div)
     also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
     also from x_rat have "\<dots> = \<bar>x\<bar>" ..
     finally show ?thesis ..
@@ -1426,7 +1263,7 @@
 proof -
   from \<open>x<y\<close> have "0 < y-x" by simp
   with reals_Archimedean obtain q::nat
-    where q: "inverse (real q) < y-x" and "0 < q" by blast 
+    where q: "inverse (real q) < y-x" and "0 < q" by blast
   def p \<equiv> "ceiling (y * real q) - 1"
   def r \<equiv> "of_int p / real q"
   from q have "x < y - inverse (real q)" by simp
@@ -1452,34 +1289,23 @@
 
 subsection\<open>Numerals and Arithmetic\<close>
 
-lemma [code_abbrev]:
+lemma [code_abbrev]:   (*FIXME*)
   "real_of_int (numeral k) = numeral k"
   "real_of_int (- numeral k) = - numeral k"
   by simp_all
 
-text\<open>Collapse applications of @{const real} to @{const numeral}\<close>
-lemma real_numeral [simp]:
-  "real (numeral v :: int) = numeral v"
-  "real (- numeral v :: int) = - numeral v"
-by (simp_all add: real_of_int_def)
 
-lemma  real_of_nat_numeral [simp]:
-  "real (numeral v :: nat) = numeral v"
-by (simp add: real_of_nat_def)
-
+  (*FIXME*)
 declaration \<open>
-  K (Lin_Arith.add_inj_thms [@{thm real_of_nat_le_iff} RS iffD2, @{thm real_of_nat_inject} RS iffD2]
-    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: real_of_nat_less_iff RS iffD2 *)
-  #> Lin_Arith.add_inj_thms [@{thm real_of_int_le_iff} RS iffD2, @{thm real_of_int_inject} RS iffD2]
-    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: real_of_int_less_iff RS iffD2 *)
-  #> Lin_Arith.add_simps [@{thm real_of_nat_zero}, @{thm real_of_nat_Suc}, @{thm real_of_nat_add},
-      @{thm real_of_nat_mult}, @{thm real_of_int_zero}, @{thm real_of_one},
-      @{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},
-      @{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},
-      @{thm real_of_nat_numeral}, @{thm real_numeral(1)}, @{thm real_numeral(2)},
-      @{thm real_of_int_def[symmetric]}, @{thm real_of_nat_def[symmetric]}]
-  #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "nat \<Rightarrow> real"})
-  #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "int \<Rightarrow> real"})
+  K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
+    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
+  #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
+    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
+  #> Lin_Arith.add_simps [@{thm of_nat_0}, @{thm of_nat_Suc}, @{thm of_nat_add},
+      @{thm of_nat_mult}, @{thm of_int_0}, @{thm of_int_1},
+      @{thm of_int_add}, @{thm of_int_minus}, @{thm of_int_diff},
+      @{thm of_int_mult}, @{thm of_int_of_nat_eq},
+      @{thm of_nat_numeral}, @{thm int_numeral}, @{thm of_int_neg_numeral}]
   #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat \<Rightarrow> real"})
   #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int \<Rightarrow> real"}))
 \<close>
@@ -1489,10 +1315,6 @@
 lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"
 by arith
 
-text \<open>FIXME: redundant with @{text add_eq_0_iff} below\<close>
-lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
-by auto
-
 lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
 by auto
 
@@ -1512,53 +1334,41 @@
 
 (* used by Import/HOL/real.imp *)
 lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
-by simp
+  by simp
 
-lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
-  by (simp add: of_nat_less_two_power real_of_nat_def)
-
-text \<open>TODO: no longer real-specific; rename and move elsewhere\<close>
+text \<open>FIXME: no longer real-specific; rename and move elsewhere\<close>
 lemma realpow_Suc_le_self:
   fixes r :: "'a::linordered_semidom"
   shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r"
 by (insert power_decreasing [of 1 "Suc n" r], simp)
 
-text \<open>TODO: no longer real-specific; rename and move elsewhere\<close>
+text \<open>FIXME: no longer real-specific; rename and move elsewhere\<close>
 lemma realpow_minus_mult:
   fixes x :: "'a::monoid_mult"
   shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"
 by (simp add: power_Suc power_commutes split add: nat_diff_split)
 
 text \<open>FIXME: declare this [simp] for all types, or not at all\<close>
-lemma real_two_squares_add_zero_iff [simp]:
-  "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"
-by (rule sum_squares_eq_zero_iff)
-
-text \<open>FIXME: declare this [simp] for all types, or not at all\<close>
-lemma realpow_two_sum_zero_iff [simp]:
-     "(x\<^sup>2 + y\<^sup>2 = (0::real)) = (x = 0 & y = 0)"
-by (rule sum_power2_eq_zero_iff)
+declare sum_squares_eq_zero_iff [simp] sum_power2_eq_zero_iff [simp]
 
 lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
 by (rule_tac y = 0 in order_trans, auto)
 
 lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> (x::real)\<^sup>2"
-by (auto simp add: power2_eq_square)
-
+  by (auto simp add: power2_eq_square)
 
 lemma numeral_power_eq_real_of_int_cancel_iff[simp]:
-  "numeral x ^ n = real (y::int) \<longleftrightarrow> numeral x ^ n = y"
-  by (metis real_numeral(1) real_of_int_inject real_of_int_power)
+     "numeral x ^ n = real_of_int (y::int) \<longleftrightarrow> numeral x ^ n = y"
+  by (metis of_int_eq_iff of_int_numeral of_int_power)
 
 lemma real_of_int_eq_numeral_power_cancel_iff[simp]:
-  "real (y::int) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
+     "real_of_int (y::int) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
   using numeral_power_eq_real_of_int_cancel_iff[of x n y]
   by metis
 
 lemma numeral_power_eq_real_of_nat_cancel_iff[simp]:
-  "numeral x ^ n = real (y::nat) \<longleftrightarrow> numeral x ^ n = y"
-  by (metis of_nat_eq_iff of_nat_numeral real_of_int_eq_numeral_power_cancel_iff
-    real_of_int_of_nat_eq zpower_int)
+     "numeral x ^ n = real (y::nat) \<longleftrightarrow> numeral x ^ n = y"
+  using of_nat_eq_iff by fastforce
 
 lemma real_of_nat_eq_numeral_power_cancel_iff[simp]:
   "real (y::nat) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
@@ -1567,43 +1377,43 @@
 
 lemma numeral_power_le_real_of_nat_cancel_iff[simp]:
   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"
-  unfolding real_of_nat_le_iff[symmetric] by simp
+by (metis of_nat_le_iff of_nat_numeral of_nat_power)
 
 lemma real_of_nat_le_numeral_power_cancel_iff[simp]:
   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"
-  unfolding real_of_nat_le_iff[symmetric] by simp
+by (metis of_nat_le_iff of_nat_numeral of_nat_power)
 
 lemma numeral_power_le_real_of_int_cancel_iff[simp]:
-  "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"
-  unfolding real_of_int_le_iff[symmetric] by simp
+    "(numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"
+  by (metis ceiling_le_iff ceiling_of_int of_int_numeral of_int_power)
 
 lemma real_of_int_le_numeral_power_cancel_iff[simp]:
-  "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"
-  unfolding real_of_int_le_iff[symmetric] by simp
+    "real_of_int a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"
+  by (metis floor_of_int le_floor_iff of_int_numeral of_int_power)
 
 lemma numeral_power_less_real_of_nat_cancel_iff[simp]:
-  "(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::nat) ^ n < a"
-  unfolding real_of_nat_less_iff[symmetric] by simp
+    "(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::nat) ^ n < a"
+  by (metis of_nat_less_iff of_nat_numeral of_nat_power)
 
 lemma real_of_nat_less_numeral_power_cancel_iff[simp]:
   "real a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::nat) ^ n"
-  unfolding real_of_nat_less_iff[symmetric] by simp
+by (metis of_nat_less_iff of_nat_numeral of_nat_power)
 
 lemma numeral_power_less_real_of_int_cancel_iff[simp]:
-  "(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::int) ^ n < a"
-  unfolding real_of_int_less_iff[symmetric] by simp
+    "(numeral x::real) ^ n < real_of_int a \<longleftrightarrow> (numeral x::int) ^ n < a"
+  by (meson not_less real_of_int_le_numeral_power_cancel_iff)
 
 lemma real_of_int_less_numeral_power_cancel_iff[simp]:
-  "real a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::int) ^ n"
-  unfolding real_of_int_less_iff[symmetric] by simp
+     "real_of_int a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::int) ^ n"
+  by (meson not_less numeral_power_le_real_of_int_cancel_iff)
 
 lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:
-  "(- numeral x::real) ^ n \<le> real a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a"
-  unfolding real_of_int_le_iff[symmetric] by simp
+    "(- numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a"
+  by (metis of_int_le_iff of_int_neg_numeral of_int_power)
 
 lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:
-  "real a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n"
-  unfolding real_of_int_le_iff[symmetric] by simp
+     "real_of_int a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n"
+  by (metis of_int_le_iff of_int_neg_numeral of_int_power)
 
 
 subsection\<open>Density of the Reals\<close>
@@ -1637,9 +1447,6 @@
 lemma abs_add_one_gt_zero: "(0::real) < 1 + abs(x)"
 by (simp add: abs_if)
 
-lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
-by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
-
 lemma abs_add_one_not_less_self: "~ abs(x) + (1::real) < x"
 by simp
 
@@ -1649,73 +1456,30 @@
 
 subsection\<open>Floor and Ceiling Functions from the Reals to the Integers\<close>
 
-(* FIXME: theorems for negative numerals *)
-lemma numeral_less_real_of_int_iff [simp]:
-     "((numeral n) < real (m::int)) = (numeral n < m)"
-apply auto
-apply (rule real_of_int_less_iff [THEN iffD1])
-apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
-done
-
-lemma numeral_less_real_of_int_iff2 [simp]:
-     "(real (m::int) < (numeral n)) = (m < numeral n)"
-apply auto
-apply (rule real_of_int_less_iff [THEN iffD1])
-apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
-done
+(* FIXME: theorems for negative numerals. Many duplicates, e.g. from Archimedean_Field.thy. *)
 
 lemma real_of_nat_less_numeral_iff [simp]:
-  "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"
-  using real_of_nat_less_iff[of n "numeral w"] by simp
+     "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"
+  by (metis of_nat_less_iff of_nat_numeral)
 
 lemma numeral_less_real_of_nat_iff [simp]:
-  "numeral w < real (n::nat) \<longleftrightarrow> numeral w < n"
-  using real_of_nat_less_iff[of "numeral w" n] by simp
+     "numeral w < real (n::nat) \<longleftrightarrow> numeral w < n"
+  by (metis of_nat_less_iff of_nat_numeral)
 
 lemma numeral_le_real_of_nat_iff[simp]:
   "(numeral n \<le> real(m::nat)) = (numeral n \<le> m)"
 by (metis not_le real_of_nat_less_numeral_iff)
 
-lemma numeral_le_real_of_int_iff [simp]:
-     "((numeral n) \<le> real (m::int)) = (numeral n \<le> m)"
-by (simp add: linorder_not_less [symmetric])
-
-lemma numeral_le_real_of_int_iff2 [simp]:
-     "(real (m::int) \<le> (numeral n)) = (m \<le> numeral n)"
-by (simp add: linorder_not_less [symmetric])
-
-lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
-unfolding real_of_nat_def by simp
-
-lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
-unfolding real_of_nat_def by (simp add: floor_minus)
-
-lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
-unfolding real_of_int_def by simp
+declare of_int_floor_le [simp] (* FIXME*)
 
-lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
-unfolding real_of_int_def by (simp add: floor_minus)
-
-lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"
-unfolding real_of_int_def by (rule floor_exists)
-
-lemma lemma_floor: "real m \<le> r \<Longrightarrow> r < real n + 1 \<Longrightarrow> m \<le> (n::int)"
-  by simp
+lemma of_int_floor_cancel [simp]:
+    "(of_int (floor x) = x) = (\<exists>n::int. x = of_int n)"
+  by (metis floor_of_int)
 
-lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
-unfolding real_of_int_def by (rule of_int_floor_le)
-
-lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"
-  by simp
-
-lemma real_of_int_floor_cancel [simp]:
-    "(real (floor x) = x) = (\<exists>n::int. x = real n)"
-  using floor_real_of_int by metis
-
-lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
+lemma floor_eq: "[| real_of_int n < x; x < real_of_int n + 1 |] ==> floor x = n"
   by linarith
 
-lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
+lemma floor_eq2: "[| real_of_int n \<le> x; x < real_of_int n + 1 |] ==> floor x = n"
   by linarith
 
 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
@@ -1724,129 +1488,79 @@
 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
   by linarith
 
-lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
-  by linarith
-
-lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"
-  by linarith
-
-lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
+lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real_of_int(floor r)"
   by linarith
 
-lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"
-  by linarith
-
-lemma le_floor: "real a <= x ==> a <= floor x"
+lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real_of_int(floor r)"
   by linarith
 
-lemma real_le_floor: "a <= floor x ==> real a <= x"
+lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real_of_int(floor r) + 1"
   by linarith
 
-lemma le_floor_eq: "(a <= floor x) = (real a <= x)"
-  by linarith
-
-lemma floor_less_eq: "(floor x < a) = (x < real a)"
+lemma real_of_int_floor_add_one_gt [simp]: "r < real_of_int(floor r) + 1"
   by linarith
 
-lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"
-  by linarith
-
-lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"
-  by linarith
-
-lemma floor_eq_iff: "floor x = b \<longleftrightarrow> real b \<le> x \<and> x < real (b + 1)"
-  by linarith
+lemma floor_eq_iff: "floor x = b \<longleftrightarrow> of_int b \<le> x \<and> x < of_int (b + 1)"
+by (simp add: floor_unique_iff)
 
-lemma floor_add [simp]: "floor (x + real a) = floor x + a"
-  by linarith
+lemma floor_add2[simp]: "floor (of_int a + x) = a + floor x"
+  by (simp add: add.commute)
 
-lemma floor_add2[simp]: "floor (real a + x) = a + floor x"
-  by linarith
-
-lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"
-  by linarith
-
-lemma floor_divide_real_eq_div: "0 \<le> b \<Longrightarrow> floor (a / real b) = floor a div b"
+lemma floor_divide_real_eq_div: "0 \<le> b \<Longrightarrow> floor (a / real_of_int b) = floor a div b"
 proof cases
   assume "0 < b"
-  { fix i j :: int assume "real i \<le> a" "a < 1 + real i"
-      "real j * real b \<le> a" "a < real b + real j * real b"
-    then have "i < b + j * b" "j * b < 1 + i"
-      unfolding real_of_int_less_iff[symmetric] by auto
-    then have "(j - i div b) * b \<le> i mod b" "i mod b < ((j - i div b) + 1) * b"
+  { fix i j :: int assume "real_of_int i \<le> a" "a < 1 + real_of_int i"
+      "real_of_int j * real_of_int b \<le> a" "a < real_of_int b + real_of_int j * real_of_int b"
+    then have "i < b + j * b"
+      by (metis linorder_not_less of_int_add of_int_le_iff of_int_mult order_trans_rules(21))
+    moreover have "j * b < 1 + i"
+    proof -
+      have "real_of_int (j * b) < real_of_int i + 1"
+        using `a < 1 + real_of_int i` `real_of_int j * real_of_int b \<le> a` by force
+      thus "j * b < 1 + i"
+        by linarith
+    qed
+    ultimately have "(j - i div b) * b \<le> i mod b" "i mod b < ((j - i div b) + 1) * b"
       by (auto simp: field_simps)
     then have "(j - i div b) * b < 1 * b" "0 * b < ((j - i div b) + 1) * b"
       using pos_mod_bound[OF \<open>0<b\<close>, of i] pos_mod_sign[OF \<open>0<b\<close>, of i] by linarith+
     then have "j = i div b"
-      using \<open>0 < b\<close> unfolding mult_less_cancel_right by auto }
+      using \<open>0 < b\<close> unfolding mult_less_cancel_right by auto
+  }
   with \<open>0 < b\<close> show ?thesis
     by (auto split: floor_split simp: field_simps)
 qed auto
 
-lemma floor_divide_eq_div:
-  "floor (real a / real b) = a div b"
-  using floor_divide_of_int_eq [of a b] real_eq_of_int by simp
-
 lemma floor_divide_eq_div_numeral[simp]: "\<lfloor>numeral a / numeral b::real\<rfloor> = numeral a div numeral b"
-  using floor_divide_eq_div[of "numeral a" "numeral b"] by simp
+  by (metis floor_divide_of_int_eq of_int_numeral)
 
 lemma floor_minus_divide_eq_div_numeral[simp]: "\<lfloor>- (numeral a / numeral b)::real\<rfloor> = - numeral a div numeral b"
-  using floor_divide_eq_div[of "- numeral a" "numeral b"] by simp
-
-lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
-  by linarith
-
-lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
-  by linarith
-
-lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
-  by linarith
+  by (metis divide_minus_left floor_divide_of_int_eq of_int_neg_numeral of_int_numeral)
 
 lemma real_of_int_ceiling_cancel [simp]:
-     "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
-  using ceiling_real_of_int by metis
+     "(real_of_int (ceiling x) = x) = (\<exists>n::int. x = real_of_int n)"
+  using ceiling_of_int by metis
 
-lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
-  by linarith
-
-lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
-  by linarith
-
-lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
+lemma ceiling_eq: "[| real_of_int n < x; x < real_of_int n + 1 |] ==> ceiling x = n + 1"
   by linarith
 
-lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
-  by linarith
-
-lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
+lemma ceiling_eq2: "[| real_of_int n < x; x \<le> real_of_int n + 1 |] ==> ceiling x = n + 1"
   by linarith
 
-lemma ceiling_le: "x <= real a ==> ceiling x <= a"
+lemma real_of_int_ceiling_diff_one_le [simp]: "real_of_int (ceiling r) - 1 \<le> r"
   by linarith
 
-lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"
-  by linarith
-
-lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"
+lemma real_of_int_ceiling_le_add_one [simp]: "real_of_int (ceiling r) \<le> r + 1"
   by linarith
 
-lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"
-  by linarith
-
-lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"
-  by linarith
-
-lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"
+lemma ceiling_le: "x <= real_of_int a ==> ceiling x <= a"
   by linarith
 
-lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"
+lemma ceiling_le_real: "ceiling x <= a ==> x <= real_of_int a"
   by linarith
 
-lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"
-  by linarith
-
-lemma ceiling_divide_eq_div: "\<lceil>real a / real b\<rceil> = - (- a div b)"
-  unfolding ceiling_def minus_divide_left real_of_int_minus[symmetric] floor_divide_eq_div by simp_all
+lemma ceiling_divide_eq_div: "\<lceil>real_of_int a / real_of_int b\<rceil> = - (- a div b)"
+  by (metis ceiling_def floor_divide_of_int_eq minus_divide_left of_int_minus)
 
 lemma ceiling_divide_eq_div_numeral [simp]:
   "\<lceil>numeral a / numeral b :: real\<rceil> = - (- numeral a div numeral b)"
@@ -1870,13 +1584,12 @@
   by (cases "0 <= a & 0 <= b")
      (auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor)
 
-lemma nat_ceiling_le_eq: "(nat(ceiling x) <= a) = (x <= real a)"
+lemma nat_ceiling_le_eq [simp]: "(nat(ceiling x) <= a) = (x <= real a)"
   by linarith
 
 lemma real_nat_ceiling_ge: "x <= real(nat(ceiling x))"
   by linarith
 
-
 lemma Rats_no_top_le: "\<exists> q \<in> \<rat>. (x :: real) \<le> q"
   by (auto intro!: bexI[of _ "of_nat (nat(ceiling x))"]) linarith
 
@@ -1889,13 +1602,12 @@
 subsection \<open>Exponentiation with floor\<close>
 
 lemma floor_power:
-  assumes "x = real (floor x)"
+  assumes "x = real_of_int (floor x)"
   shows "floor (x ^ n) = floor x ^ n"
 proof -
-  have *: "x ^ n = real (floor x ^ n)"
+  have "x ^ n = real_of_int (floor x ^ n)"
     using assms by (induct n arbitrary: x) simp_all
-  show ?thesis unfolding real_of_int_inject[symmetric]
-    unfolding * floor_real_of_int ..
+  then show ?thesis  by linarith
 qed
 (*
 lemma natfloor_power:
@@ -1903,12 +1615,13 @@
   shows "natfloor (x ^ n) = natfloor x ^ n"
 proof -
   from assms have "0 \<le> floor x" by auto
-  note assms[unfolded natfloor_def real_nat_eq_real[OF `0 \<le> floor x`]]
+  note assms[unfolded natfloor_def of_nat_nat[OF `0 \<le> floor x`]]
   from floor_power[OF this]
   show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric]
     by simp
 qed
 *)
+
 lemma floor_numeral_power[simp]:
   "\<lfloor>numeral x ^ n\<rfloor> = numeral x ^ n"
   by (metis floor_of_int of_int_numeral of_int_power)
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Real_Vector_Spaces.thy
--- a/src/HOL/Real_Vector_Spaces.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Real_Vector_Spaces.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -341,12 +341,6 @@
 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
 by (cases z rule: int_diff_cases, simp)
 
-lemma of_real_real_of_nat_eq [simp]: "of_real (real n) = of_nat n"
-  by (simp add: real_of_nat_def)
-
-lemma of_real_real_of_int_eq [simp]: "of_real (real z) = of_int z"
-  by (simp add: real_of_int_def)
-
 lemma of_real_numeral [simp]: "of_real (numeral w) = numeral w"
 using of_real_of_int_eq [of "numeral w"] by simp
 
@@ -997,7 +991,7 @@
   also have "\<dots> \<le> (\<Sum>i<m. norm (z - w))"
     by (intro norm_setprod_diff) (auto simp add: assms)
   also have "\<dots> = m * norm (z - w)"
-    by (simp add: real_of_nat_def)
+    by simp
   finally show ?thesis .
 qed
 
@@ -1291,7 +1285,7 @@
 
 lemma dist_diff [simp]: "dist a (a - b) = norm b"  "dist (a - b) a = norm b"
   by (simp_all add: dist_norm)
-  
+
 lemma dist_of_int: "dist (of_int m) (of_int n :: 'a :: real_normed_algebra_1) = of_int \<bar>m - n\<bar>"
 proof -
   have "dist (of_int m) (of_int n :: 'a) = dist (of_int m :: 'a) (of_int m - (of_int (m - n)))"
@@ -1300,10 +1294,10 @@
   finally show ?thesis .
 qed
 
-lemma dist_of_nat: 
+lemma dist_of_nat:
   "dist (of_nat m) (of_nat n :: 'a :: real_normed_algebra_1) = of_int \<bar>int m - int n\<bar>"
   by (subst (1 2) of_int_of_nat_eq [symmetric]) (rule dist_of_int)
-  
+
 subsection \<open>Bounded Linear and Bilinear Operators\<close>
 
 locale linear = additive f for f :: "'a::real_vector \<Rightarrow> 'b::real_vector" +
@@ -1571,7 +1565,7 @@
   assumes "linear f" "bij f" shows "linear (inv f)"
   using assms unfolding linear_def linear_axioms_def additive_def
   by (auto simp: bij_is_surj bij_is_inj surj_f_inv_f intro!:  Hilbert_Choice.inv_f_eq)
-    
+
 instance real_normed_algebra_1 \<subseteq> perfect_space
 proof
   fix x::'a
@@ -1737,7 +1731,7 @@
   qed
 next
   assume "Cauchy f"
-  show "\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e" 
+  show "\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e"
   proof (intro allI impI)
     fix e :: real assume e: "e > 0"
     with `Cauchy f` obtain M where "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> dist (f m) (f n) < e"
@@ -1959,8 +1953,7 @@
       using ex_le_of_nat[of x] ..
     note monoD[OF mono this]
     also have "f (real_of_nat n) \<le> y"
-      by (rule LIMSEQ_le_const[OF limseq])
-         (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
+      by (rule LIMSEQ_le_const[OF limseq]) (auto intro: exI[of _ n] monoD[OF mono])
     finally have "f x \<le> y" . }
   note le = this
   have "eventually (\<lambda>x. real N \<le> x) at_top"
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Series.thy
--- a/src/HOL/Series.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Series.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -11,7 +11,7 @@
 
 theory Series
 imports Limits Inequalities
-begin 
+begin
 
 subsection \<open>Definition of infinite summability\<close>
 
@@ -46,7 +46,7 @@
 
 lemma summable_iff_convergent':
   "summable f \<longleftrightarrow> convergent (\<lambda>n. setsum f {..n})"
-  by (simp_all only: summable_iff_convergent convergent_def 
+  by (simp_all only: summable_iff_convergent convergent_def
         lessThan_Suc_atMost [symmetric] LIMSEQ_Suc_iff[of "\<lambda>n. setsum f {..<n}"])
 
 lemma suminf_eq_lim: "suminf f = lim (\<lambda>n. \<Sum>i<n. f i)"
@@ -79,7 +79,7 @@
 proof -
   from assms obtain N where N: "\<forall>n\<ge>N. f n = g n" by (auto simp: eventually_at_top_linorder)
   def C \<equiv> "(\<Sum>k<N. f k - g k)"
-  from eventually_ge_at_top[of N] 
+  from eventually_ge_at_top[of N]
     have "eventually (\<lambda>n. setsum f {..<n} = C + setsum g {..<n}) sequentially"
   proof eventually_elim
     fix n assume n: "n \<ge> N"
@@ -87,7 +87,7 @@
     also have "setsum f ... = setsum f {..<N} + setsum f {N..<n}"
       by (intro setsum.union_disjoint) auto
     also from N have "setsum f {N..<n} = setsum g {N..<n}" by (intro setsum.cong) simp_all
-    also have "setsum f {..<N} + setsum g {N..<n} = C + (setsum g {..<N} + setsum g {N..<n})" 
+    also have "setsum f {..<N} + setsum g {N..<n} = C + (setsum g {..<N} + setsum g {N..<n})"
       unfolding C_def by (simp add: algebra_simps setsum_subtractf)
     also have "setsum g {..<N} + setsum g {N..<n} = setsum g ({..<N} \<union> {N..<n})"
       by (intro setsum.union_disjoint [symmetric]) auto
@@ -161,7 +161,7 @@
 lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
   by (metis summable_sums sums_summable sums_unique)
 
-lemma summable_sums_iff: 
+lemma summable_sums_iff:
   "summable (f :: nat \<Rightarrow> 'a :: {comm_monoid_add,t2_space}) \<longleftrightarrow> f sums suminf f"
   by (auto simp: sums_iff summable_sums)
 
@@ -351,7 +351,7 @@
 lemma suminf_split_head: "summable f \<Longrightarrow> (\<Sum>n. f (Suc n)) = suminf f - f 0"
   using suminf_split_initial_segment[of 1] by simp
 
-lemma suminf_exist_split: 
+lemma suminf_exist_split:
   fixes r :: real assumes "0 < r" and "summable f"
   shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
 proof -
@@ -428,11 +428,10 @@
   {
     assume "c \<noteq> 0"
     hence "filterlim (\<lambda>n. of_nat n * norm c) at_top sequentially"
-      unfolding real_of_nat_def[symmetric]
       by (subst mult.commute)
          (auto intro!: filterlim_tendsto_pos_mult_at_top filterlim_real_sequentially)
     hence "\<not>convergent (\<lambda>n. norm (\<Sum>k<n. c))"
-      by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_imp_at_infinity) 
+      by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_imp_at_infinity)
          (simp_all add: setsum_constant_scaleR)
     hence "\<not>summable (\<lambda>_. c)" unfolding summable_iff_convergent using convergent_norm by blast
   }
@@ -529,7 +528,7 @@
     by (auto simp: eventually_at_top_linorder)
   thus "norm c < 1" using one_le_power[of "norm c" n] by (cases "norm c \<ge> 1") (linarith, simp)
 qed (rule summable_geometric)
-  
+
 end
 
 lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
@@ -638,7 +637,7 @@
 
 subsection \<open>The Ratio Test\<close>
 
-lemma summable_ratio_test: 
+lemma summable_ratio_test:
   assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)"
   shows "summable f"
 proof cases
@@ -681,7 +680,7 @@
     shows "summable (\<lambda>n. norm (a n) * r^n)"
 proof (rule summable_comparison_test')
   show "summable (\<lambda>n. M * (r / r0) ^ n)"
-    using assms 
+    using assms
     by (auto simp add: summable_mult summable_geometric)
 next
   fix n
@@ -815,7 +814,7 @@
 
 lemma summable_zero_power' [simp]: "summable (\<lambda>n. f n * 0 ^ n :: 'a :: {ring_1,topological_space})"
 proof -
-  have "(\<lambda>n. f n * 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then f 0 * 0^0 else 0)" 
+  have "(\<lambda>n. f n * 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then f 0 * 0^0 else 0)"
     by (intro ext) (simp add: zero_power)
   moreover have "summable \<dots>" by simp
   ultimately show ?thesis by simp
@@ -845,7 +844,7 @@
   have "summable (\<lambda>n. f (Suc n) * z ^ n) \<longleftrightarrow> summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
   proof
     assume "summable (\<lambda>n. f (Suc n) * z ^ n)"
-    from summable_mult2[OF this, of z] show "summable (\<lambda>n. f (Suc n) * z ^ Suc n)" 
+    from summable_mult2[OF this, of z] show "summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
       by (simp add: power_commutes algebra_simps)
   next
     assume "summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
@@ -864,7 +863,7 @@
 proof -
   from assms show "summable (\<lambda>n. f (Suc n) * z ^ n)" by (subst summable_powser_split_head)
 
-  from suminf_mult2[OF this, of z] 
+  from suminf_mult2[OF this, of z]
     have "(\<Sum>n. f (Suc n) * z ^ n) * z = (\<Sum>n. f (Suc n) * z ^ Suc n)"
     by (simp add: power_commutes algebra_simps)
   also from assms have "\<dots> = suminf (\<lambda>n. f n * z ^ n) - f 0"
@@ -878,9 +877,9 @@
   assumes summable: "summable f" and e: "e > (0::real)"
   obtains N where "\<And>m n. m \<ge> N \<Longrightarrow> norm (\<Sum>k=m..n. f k) < e"
 proof -
-  from summable have "Cauchy (\<lambda>n. \<Sum>k<n. f k)" 
+  from summable have "Cauchy (\<lambda>n. \<Sum>k<n. f k)"
     by (simp add: Cauchy_convergent_iff summable_iff_convergent)
-  from CauchyD[OF this e] obtain N 
+  from CauchyD[OF this e] obtain N
     where N: "\<And>m n. m \<ge> N \<Longrightarrow> n \<ge> N \<Longrightarrow> norm ((\<Sum>k<m. f k) - (\<Sum>k<n. f k)) < e" by blast
   {
     fix m n :: nat assume m: "m \<ge> N"
@@ -896,10 +895,10 @@
   thus ?thesis by (rule that)
 qed
 
-lemma powser_sums_if: 
+lemma powser_sums_if:
   "(\<lambda>n. (if n = m then (1 :: 'a :: {ring_1,topological_space}) else 0) * z^n) sums z^m"
 proof -
-  have "(\<lambda>n. (if n = m then 1 else 0) * z^n) = (\<lambda>n. if n = m then z^n else 0)" 
+  have "(\<lambda>n. (if n = m then 1 else 0) * z^n) = (\<lambda>n. if n = m then z^n else 0)"
     by (intro ext) auto
   thus ?thesis by (simp add: sums_single)
 qed
@@ -918,7 +917,7 @@
   have smaller: "\<forall>n. (\<Sum>i<n. (f \<circ> g) i) \<le> suminf f"
   proof
     fix n
-    have "\<forall> n' \<in> (g ` {..<n}). n' < Suc (Max (g ` {..<n}))" 
+    have "\<forall> n' \<in> (g ` {..<n}). n' < Suc (Max (g ` {..<n}))"
       by(metis Max_ge finite_imageI finite_lessThan not_le not_less_eq)
     then obtain m where n: "\<And>n'. n' < n \<Longrightarrow> g n' < m" by blast
 
@@ -927,7 +926,7 @@
     also have "\<dots> \<le> (\<Sum>i<m. f i)"
       by (rule setsum_mono3) (auto simp add: pos n[rule_format])
     also have "\<dots> \<le> suminf f"
-      using \<open>summable f\<close> 
+      using \<open>summable f\<close>
       by (rule setsum_le_suminf) (simp add: pos)
     finally show "(\<Sum>i<n. (f \<circ>  g) i) \<le> suminf f" by simp
   qed
@@ -991,9 +990,9 @@
   from filterlim_subseq[OF subseq] have g_inv_ex: "\<exists>m. g m \<ge> n" for n
     by (auto simp: filterlim_at_top eventually_at_top_linorder)
   hence g_inv: "g (g_inv n) \<ge> n" for n unfolding g_inv_def by (rule LeastI_ex)
-  have g_inv_least: "m \<ge> g_inv n" if "g m \<ge> n" for m n using that 
+  have g_inv_least: "m \<ge> g_inv n" if "g m \<ge> n" for m n using that
     unfolding g_inv_def by (rule Least_le)
-  have g_inv_least': "g m < n" if "m < g_inv n" for m n using that g_inv_least[of n m] by linarith  
+  have g_inv_least': "g m < n" if "m < g_inv n" for m n using that g_inv_least[of n m] by linarith
   have "(\<lambda>n. \<Sum>k<n. f k) = (\<lambda>n. \<Sum>k<g_inv n. f (g k))"
   proof
     fix n :: nat
@@ -1010,7 +1009,7 @@
     }
     with g_inv_least' g_inv have "(\<Sum>k<n. f k) = (\<Sum>k\<in>g`{..<g_inv n}. f k)"
       by (intro setsum.mono_neutral_right) auto
-    also from subseq have "\<dots> = (\<Sum>k<g_inv n. f (g k))" using subseq_imp_inj_on 
+    also from subseq have "\<dots> = (\<Sum>k<g_inv n. f (g k))" using subseq_imp_inj_on
       by (subst setsum.reindex) simp_all
     finally show "(\<Sum>k<n. f k) = (\<Sum>k<g_inv n. f (g k))" .
   qed
@@ -1019,7 +1018,7 @@
     also have "n \<le> g (g_inv n)" by (rule g_inv)
     finally have "K \<le> g_inv n" using subseq by (simp add: strict_mono_less_eq subseq_strict_mono)
   }
-  hence "filterlim g_inv at_top sequentially" 
+  hence "filterlim g_inv at_top sequentially"
     by (auto simp: filterlim_at_top eventually_at_top_linorder)
   from lim and this have "(\<lambda>n. \<Sum>k<g_inv n. f (g k)) ----> c" by (rule filterlim_compose)
   finally show "(\<lambda>n. \<Sum>k<n. f k) ----> c" .
@@ -1030,7 +1029,7 @@
   shows   "summable (\<lambda>n. f (g n)) \<longleftrightarrow> summable f"
   using sums_mono_reindex[of g f, OF assms] by (simp add: summable_def)
 
-lemma suminf_mono_reindex:                                                                 
+lemma suminf_mono_reindex:
   assumes "subseq g" "\<And>n. n \<notin> range g \<Longrightarrow> f n = (0 :: 'a :: {t2_space,comm_monoid_add})"
   shows   "suminf (\<lambda>n. f (g n)) = suminf f"
 proof (cases "summable f")
@@ -1042,7 +1041,7 @@
   hence "\<not>(\<exists>c. (\<lambda>n. f (g n)) sums c)" unfolding summable_def by blast
   hence "suminf (\<lambda>n. f (g n)) = The (\<lambda>_. False)" by (simp add: suminf_def)
   ultimately show ?thesis by simp
-qed (insert sums_mono_reindex[of g f, OF assms] summable_mono_reindex[of g f, OF assms], 
+qed (insert sums_mono_reindex[of g f, OF assms] summable_mono_reindex[of g f, OF assms],
      simp_all add: sums_iff)
 
 end
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/TPTP/THF_Arith.thy
--- a/src/HOL/TPTP/THF_Arith.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/TPTP/THF_Arith.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -56,7 +56,7 @@
 
 overloading int_to_real \<equiv> "to_real :: int \<Rightarrow> real"
 begin
-  definition "int_to_real (n::int) = real n"
+  definition "int_to_real (n::int) = real_of_int n"
 end
 
 overloading rat_to_real \<equiv> "to_real :: rat \<Rightarrow> real"
@@ -79,7 +79,7 @@
 by (metis int_to_rat_def rat_is_int_def)
 
 lemma to_real_is_int [intro, simp]: "is_int (to_real (n::int))"
-by (metis Ints_real_of_int int_to_real_def real_is_int_def)
+by (metis Ints_of_int int_to_real_def real_is_int_def)
 
 lemma to_real_is_rat [intro, simp]: "is_rat (to_real (q::rat))"
 by (metis Rats_of_rat rat_to_real_def real_is_rat_def)
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Tools/SMT/z3_real.ML
--- a/src/HOL/Tools/SMT/z3_real.ML	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Tools/SMT/z3_real.ML	Tue Nov 10 14:18:41 2015 +0000
@@ -13,14 +13,14 @@
 fun real_term_parser (SMTLIB.Dec (i, 0), []) = SOME (HOLogic.mk_number @{typ Real.real} i)
   | real_term_parser (SMTLIB.Sym "/", [t1, t2]) =
       SOME (@{term "Rings.divide :: real => _"} $ t1 $ t2)
-  | real_term_parser (SMTLIB.Sym "to_real", [t]) = SOME (@{term "Real.real :: int => _"} $ t)
+  | real_term_parser (SMTLIB.Sym "to_real", [t]) = SOME (@{term "Int.of_int :: int => _"} $ t)
   | real_term_parser _ = NONE
 
 fun abstract abs t =
   (case t of
     (c as @{term "Rings.divide :: real => _"}) $ t1 $ t2 =>
       abs t1 ##>> abs t2 #>> (fn (u1, u2) => SOME (c $ u1 $ u2))
-  | (c as @{term "Real.real :: int => _"}) $ t =>
+  | (c as @{term "Int.of_int :: int => _"}) $ t =>
       abs t #>> (fn u => SOME (c $ u))
   | _ => pair NONE)
 
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/Transcendental.thy
--- a/src/HOL/Transcendental.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/Transcendental.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -11,10 +11,11 @@
 begin
 
 lemma reals_Archimedean4:
-  assumes "0 < y" "0 \<le> x" 
+  assumes "0 < y" "0 \<le> x"
   obtains n where "real n * y \<le> x" "x < real (Suc n) * y"
   using floor_correct [of "x/y"] assms
-  by (auto simp: Real.real_of_nat_Suc field_simps intro: that [of "nat (floor (x/y))"])
+  apply (auto simp: field_simps intro!: that [of "nat (floor (x/y))"])
+by (metis (no_types, hide_lams)  divide_inverse divide_self_if floor_correct mult.left_commute mult.right_neutral mult_eq_0_iff order_refl order_trans real_mult_le_cancel_iff2)
 
 lemma fact_in_Reals: "fact n \<in> \<real>" by (induction n) auto
 
@@ -24,11 +25,8 @@
 lemma of_real_fact [simp]: "of_real (fact n) = fact n"
   by (metis of_nat_fact of_real_of_nat_eq)
 
-lemma real_fact_nat [simp]: "real (fact n :: nat) = fact n"
-  by (simp add: real_of_nat_def)
-
-lemma real_fact_int [simp]: "real (fact n :: int) = fact n"
-  by (metis of_int_of_nat_eq of_nat_fact real_of_int_def)
+lemma of_int_fact [simp]: "of_int (fact n :: int) = fact n"
+  by (metis of_int_of_nat_eq of_nat_fact)
 
 lemma norm_fact [simp]:
   "norm (fact n :: 'a :: {real_normed_algebra_1}) = fact n"
@@ -155,26 +153,26 @@
   moreover obtain N where N: "norm x / (1 - norm x) < of_int N"
     using ex_le_of_int
     by (meson ex_less_of_int)
-  ultimately have N0: "N>0" 
+  ultimately have N0: "N>0"
     by auto
-  then have *: "real (N + 1) * norm x / real N < 1"
+  then have *: "real_of_int (N + 1) * norm x / real_of_int N < 1"
     using N assms
     by (auto simp: field_simps)
   { fix n::nat
     assume "N \<le> int n"
-    then have "real N * real_of_nat (Suc n) \<le> real_of_nat n * real (1 + N)"
+    then have "real_of_int N * real_of_nat (Suc n) \<le> real_of_nat n * real_of_int (1 + N)"
       by (simp add: algebra_simps)
-    then have "(real N * real_of_nat (Suc n)) * (norm x * norm (x ^ n))
-               \<le> (real_of_nat n * real (1 + N)) * (norm x * norm (x ^ n))"
+    then have "(real_of_int N * real_of_nat (Suc n)) * (norm x * norm (x ^ n))
+               \<le> (real_of_nat n *  (1 + N)) * (norm x * norm (x ^ n))"
       using N0 mult_mono by fastforce
-    then have "real N * (norm x * (real_of_nat (Suc n) * norm (x ^ n)))
-         \<le> real_of_nat n * (norm x * (real (1 + N) * norm (x ^ n)))"
+    then have "real_of_int N * (norm x * (real_of_nat (Suc n) * norm (x ^ n)))
+         \<le> real_of_nat n * (norm x * ((1 + N) * norm (x ^ n)))"
       by (simp add: algebra_simps)
   } note ** = this
   show ?thesis using *
     apply (rule summable_LIMSEQ_zero [OF summable_ratio_test, where N1="nat N"])
-    apply (simp add: N0 norm_mult field_simps ** 
-                del: of_nat_Suc real_of_int_add)
+    apply (simp add: N0 norm_mult field_simps **
+                del: of_nat_Suc of_int_add)
     done
 qed
 
@@ -184,16 +182,6 @@
 using powser_times_n_limit_0 [of "inverse x"]
 by (simp add: norm_divide divide_simps)
 
-lemma lim_1_over_n: "((\<lambda>n. 1 / of_nat n) ---> (0::'a::real_normed_field)) sequentially"
-  apply (clarsimp simp: lim_sequentially norm_divide dist_norm divide_simps)
-  apply (auto simp: mult_ac dest!: ex_less_of_nat_mult [of _ 1])
-  by (metis le_eq_less_or_eq less_trans linordered_comm_semiring_strict_class.comm_mult_strict_left_mono 
-          of_nat_less_0_iff of_nat_less_iff zero_less_mult_iff zero_less_one)
-
-lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) ---> (0::'a::real_normed_field)) sequentially"
-  using lim_1_over_n
-  by (simp add: inverse_eq_divide)
-
 lemma sum_split_even_odd:
   fixes f :: "nat \<Rightarrow> real"
   shows
@@ -751,7 +739,7 @@
   then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ n)"
     using False summable_mult2 [of "\<lambda>n. (of_nat (Suc n) * c(Suc n) * x ^ n) * x" "inverse x"]
     by (simp add: mult.assoc) (auto simp: ac_simps)
-  then show ?thesis 
+  then show ?thesis
     by (simp add: diffs_def)
 qed
 
@@ -798,14 +786,14 @@
   shows   "((\<lambda>x. \<Sum>n. c n * x^n) has_field_derivative (\<Sum>n. diffs c n * x^n)) (at x)"
   using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
   by (force simp del: of_real_add)
-  
+
 lemma isCont_powser:
   fixes K x :: "'a::{real_normed_field,banach}"
   assumes "summable (\<lambda>n. c n * K ^ n)"
   assumes "norm x < norm K"
   shows   "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
   using termdiffs_strong[OF assms] by (blast intro!: DERIV_isCont)
-  
+
 lemmas isCont_powser' = isCont_o2[OF _ isCont_powser]
 
 lemma isCont_powser_converges_everywhere:
@@ -815,7 +803,7 @@
   using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
   by (force intro!: DERIV_isCont simp del: of_real_add)
 
-lemma powser_limit_0: 
+lemma powser_limit_0:
   fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
   assumes s: "0 < s"
       and sm: "\<And>x. norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
@@ -835,16 +823,16 @@
     by (blast intro: DERIV_continuous)
   then have "((\<lambda>x. \<Sum>n. a n * x ^ n) ---> a 0) (at 0)"
     by (simp add: continuous_within powser_zero)
-  then show ?thesis 
+  then show ?thesis
     apply (rule Lim_transform)
     apply (auto simp add: LIM_eq)
     apply (rule_tac x="s" in exI)
-    using s 
+    using s
     apply (auto simp: sm [THEN sums_unique])
     done
 qed
 
-lemma powser_limit_0_strong: 
+lemma powser_limit_0_strong:
   fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
   assumes s: "0 < s"
       and sm: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
@@ -906,7 +894,7 @@
       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>, unfolded real_norm_def]
       obtain s where s_bound: "0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r)"
         by auto
-      have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound simp del: real_of_nat_Suc)
+      have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound simp del: of_nat_Suc)
       thus "0 < x" unfolding \<open>x = ?s n\<close> .
     qed
   qed auto
@@ -964,9 +952,8 @@
     qed auto
     also have "\<dots> = of_nat (card {..<?N}) * ?r"
       by (rule setsum_constant)
-    also have "\<dots> = real ?N * ?r"
-      unfolding real_eq_of_nat by auto
-    also have "\<dots> = r/3" by (auto simp del: real_of_nat_Suc)
+    also have "\<dots> = real ?N * ?r" by simp
+    also have "\<dots> = r/3" by (auto simp del: of_nat_Suc)
     finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
 
     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
@@ -1050,9 +1037,10 @@
                 unfolding power_add[symmetric] using \<open>p \<le> n\<close> by auto
             qed
             also have "\<dots> = real (Suc n) * R' ^ n"
-              unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
+              unfolding setsum_constant card_atLeastLessThan by auto
             finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
-              unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF \<open>0 < R'\<close>]]] .
+              unfolding  abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF \<open>0 < R'\<close>]]]
+              by linarith
             show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
               unfolding abs_mult[symmetric] by auto
           qed
@@ -1064,7 +1052,7 @@
       {
         fix n
         show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
-          by (auto intro!: derivative_eq_intros simp del: power_Suc simp: real_of_nat_def)
+          by (auto intro!: derivative_eq_intros simp del: power_Suc)
       }
       {
         fix x
@@ -1112,7 +1100,7 @@
 lemma isCont_pochhammer [continuous_intros]: "isCont (\<lambda>z::'a::real_normed_field. pochhammer z n) z"
   by (induction n) (auto intro!: continuous_intros simp: pochhammer_rec')
 
-lemma continuous_on_pochhammer [continuous_intros]: 
+lemma continuous_on_pochhammer [continuous_intros]:
   fixes A :: "'a :: real_normed_field set"
   shows "continuous_on A (\<lambda>z. pochhammer z n)"
   by (intro continuous_at_imp_continuous_on ballI isCont_pochhammer)
@@ -1133,7 +1121,7 @@
   obtain r :: real where r0: "0 < r" and r1: "r < 1"
     using dense [OF zero_less_one] by fast
   obtain N :: nat where N: "norm x < real N * r"
-    using ex_less_of_nat_mult r0 real_of_nat_def by auto
+    using ex_less_of_nat_mult r0 by auto
   from r1 show ?thesis
   proof (rule summable_ratio_test [rule_format])
     fix n :: nat
@@ -1164,7 +1152,7 @@
     by (simp add: norm_power_ineq)
 qed
 
-lemma summable_exp: 
+lemma summable_exp:
   fixes x :: "'a::{real_normed_field,banach}"
   shows "summable (\<lambda>n. inverse (fact n) * x^n)"
   using summable_exp_generic [where x=x]
@@ -1190,7 +1178,7 @@
   apply (simp del: of_real_add)
   done
 
-declare DERIV_exp[THEN DERIV_chain2, derivative_intros] 
+declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
         DERIV_exp[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
 
 lemma norm_exp: "norm (exp x) \<le> exp (norm x)"
@@ -1279,7 +1267,7 @@
              (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
              (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
     by (simp only: setsum.distrib [symmetric] scaleR_left_distrib [symmetric]
-                   real_of_nat_add [symmetric]) simp
+                   of_nat_add [symmetric]) simp
   also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n-i))"
     by (simp only: scaleR_right.setsum)
   finally show
@@ -1338,7 +1326,7 @@
     by (induct n) (auto simp add: distrib_left exp_add mult.commute)
 
 corollary exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
-  by (simp add: exp_of_nat_mult real_of_nat_def)
+  by (simp add: exp_of_nat_mult)
 
 lemma exp_setsum: "finite I \<Longrightarrow> exp(setsum f I) = setprod (\<lambda>x. exp(f x)) I"
   by (induction I rule: finite_induct) (auto simp: exp_add_commuting mult.commute)
@@ -1480,7 +1468,7 @@
 definition ln_real :: "real \<Rightarrow> real"
   where "ln_real x = (THE u. exp u = x)"
 
-instance 
+instance
 by intro_classes (simp add: ln_real_def)
 
 end
@@ -1488,106 +1476,106 @@
 lemma powr_eq_0_iff [simp]: "w powr z = 0 \<longleftrightarrow> w = 0"
   by (simp add: powr_def)
 
-lemma ln_exp [simp]: 
+lemma ln_exp [simp]:
   fixes x::real shows "ln (exp x) = x"
   by (simp add: ln_real_def)
 
-lemma exp_ln [simp]: 
+lemma exp_ln [simp]:
   fixes x::real shows "0 < x \<Longrightarrow> exp (ln x) = x"
   by (auto dest: exp_total)
 
-lemma exp_ln_iff [simp]: 
+lemma exp_ln_iff [simp]:
   fixes x::real shows "exp (ln x) = x \<longleftrightarrow> 0 < x"
   by (metis exp_gt_zero exp_ln)
 
-lemma ln_unique: 
+lemma ln_unique:
   fixes x::real shows "exp y = x \<Longrightarrow> ln x = y"
   by (erule subst, rule ln_exp)
 
-lemma ln_mult:  
+lemma ln_mult:
   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
   by (rule ln_unique) (simp add: exp_add)
 
-lemma ln_setprod: 
-  fixes f:: "'a => real" 
+lemma ln_setprod:
+  fixes f:: "'a => real"
   shows
     "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> f i > 0\<rbrakk> \<Longrightarrow> ln(setprod f I) = setsum (\<lambda>x. ln(f x)) I"
   by (induction I rule: finite_induct) (auto simp: ln_mult setprod_pos)
 
-lemma ln_inverse: 
+lemma ln_inverse:
   fixes x::real shows "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
   by (rule ln_unique) (simp add: exp_minus)
 
-lemma ln_div: 
+lemma ln_div:
   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
   by (rule ln_unique) (simp add: exp_diff)
 
 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x^n) = real n * ln x"
   by (rule ln_unique) (simp add: exp_real_of_nat_mult)
 
-lemma ln_less_cancel_iff [simp]: 
+lemma ln_less_cancel_iff [simp]:
   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
   by (subst exp_less_cancel_iff [symmetric]) simp
 
-lemma ln_le_cancel_iff [simp]: 
+lemma ln_le_cancel_iff [simp]:
   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
   by (simp add: linorder_not_less [symmetric])
 
-lemma ln_inj_iff [simp]: 
+lemma ln_inj_iff [simp]:
   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
   by (simp add: order_eq_iff)
 
-lemma ln_add_one_self_le_self [simp]: 
+lemma ln_add_one_self_le_self [simp]:
   fixes x::real shows "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
   apply (rule exp_le_cancel_iff [THEN iffD1])
   apply (simp add: exp_ge_add_one_self_aux)
   done
 
-lemma ln_less_self [simp]: 
+lemma ln_less_self [simp]:
   fixes x::real shows "0 < x \<Longrightarrow> ln x < x"
   by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
 
-lemma ln_ge_zero [simp]: 
+lemma ln_ge_zero [simp]:
   fixes x::real shows "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
   using ln_le_cancel_iff [of 1 x] by simp
 
-lemma ln_ge_zero_imp_ge_one: 
+lemma ln_ge_zero_imp_ge_one:
   fixes x::real shows "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
   using ln_le_cancel_iff [of 1 x] by simp
 
-lemma ln_ge_zero_iff [simp]: 
+lemma ln_ge_zero_iff [simp]:
   fixes x::real shows "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
   using ln_le_cancel_iff [of 1 x] by simp
 
-lemma ln_less_zero_iff [simp]: 
+lemma ln_less_zero_iff [simp]:
   fixes x::real shows "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
   using ln_less_cancel_iff [of x 1] by simp
 
-lemma ln_gt_zero: 
+lemma ln_gt_zero:
   fixes x::real shows "1 < x \<Longrightarrow> 0 < ln x"
   using ln_less_cancel_iff [of 1 x] by simp
 
-lemma ln_gt_zero_imp_gt_one: 
+lemma ln_gt_zero_imp_gt_one:
   fixes x::real shows "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
   using ln_less_cancel_iff [of 1 x] by simp
 
-lemma ln_gt_zero_iff [simp]: 
+lemma ln_gt_zero_iff [simp]:
   fixes x::real shows "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
   using ln_less_cancel_iff [of 1 x] by simp
 
-lemma ln_eq_zero_iff [simp]: 
+lemma ln_eq_zero_iff [simp]:
   fixes x::real shows "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
   using ln_inj_iff [of x 1] by simp
 
-lemma ln_less_zero: 
+lemma ln_less_zero:
   fixes x::real shows "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
   by simp
 
-lemma ln_neg_is_const: 
+lemma ln_neg_is_const:
   fixes x::real shows "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)"
   by (auto simp add: ln_real_def intro!: arg_cong[where f=The])
 
-lemma isCont_ln: 
+lemma isCont_ln:
   fixes x::real assumes "x \<noteq> 0" shows "isCont ln x"
 proof cases
   assume "0 < x"
@@ -1603,7 +1591,7 @@
                 intro!: exI[of _ "\<bar>x\<bar>"])
 qed
 
-lemma tendsto_ln [tendsto_intros]: 
+lemma tendsto_ln [tendsto_intros]:
   fixes a::real shows
   "(f ---> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
   by (rule isCont_tendsto_compose [OF isCont_ln])
@@ -1709,9 +1697,9 @@
     have "(2::nat) * 2 ^ n \<le> fact (n + 2)"
       by (induct n) simp_all
     hence "real ((2::nat) * 2 ^ n) \<le> real_of_nat (fact (n + 2))"
-      by (simp only: real_of_nat_le_iff)
+      by (simp only: of_nat_le_iff)
     hence "((2::real) * 2 ^ n) \<le> fact (n + 2)"
-      unfolding of_nat_fact real_of_nat_def
+      unfolding of_nat_fact
       by (simp add: of_nat_mult of_nat_power)
     hence "inverse (fact (n + 2)) \<le> inverse ((2::real) * 2 ^ n)"
       by (rule le_imp_inverse_le) simp
@@ -1843,7 +1831,7 @@
 qed
 
 lemma ln_one_minus_pos_lower_bound:
-  fixes x::real 
+  fixes x::real
   shows "0 <= x \<Longrightarrow> x <= (1 / 2) \<Longrightarrow> - x - 2 * x\<^sup>2 <= ln (1 - x)"
 proof -
   assume a: "0 <= x" and b: "x <= (1 / 2)"
@@ -1973,13 +1961,13 @@
   fixes x::real shows "0 < x \<Longrightarrow> ln x \<le> x - 1"
   using exp_ge_add_one_self[of "ln x"] by simp
 
-corollary ln_diff_le: 
-  fixes x::real 
+corollary ln_diff_le:
+  fixes x::real
   shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x - ln y \<le> (x - y) / y"
   by (simp add: ln_div [symmetric] diff_divide_distrib ln_le_minus_one)
 
 lemma ln_eq_minus_one:
-  fixes x::real 
+  fixes x::real
   assumes "0 < x" "ln x = x - 1"
   shows "x = 1"
 proof -
@@ -2328,7 +2316,7 @@
 lemma le_log_iff:
   assumes "1 < b" "x > 0"
   shows "y \<le> log b x \<longleftrightarrow> b powr y \<le> (x::real)"
-  using assms 
+  using assms
   apply auto
   apply (metis (no_types, hide_lams) less_irrefl less_le_trans linear powr_le_cancel_iff
                powr_log_cancel zero_less_one)
@@ -2358,10 +2346,10 @@
   by (auto simp add: floor_eq_iff powr_le_iff less_powr_iff)
 
 lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
-  by (induct n) (simp_all add: ac_simps powr_add real_of_nat_Suc)
+  by (induct n) (simp_all add: ac_simps powr_add of_nat_Suc)
 
 lemma powr_realpow_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
-  unfolding real_of_nat_numeral [symmetric] by (rule powr_realpow)
+  by (metis of_nat_numeral powr_realpow)
 
 lemma powr2_sqrt[simp]: "0 < x \<Longrightarrow> sqrt x powr 2 = x"
 by(simp add: powr_realpow_numeral)
@@ -2517,7 +2505,7 @@
 qed
 
 lemma tendsto_powr [tendsto_intros]:
-  fixes a::real 
+  fixes a::real
   assumes f: "(f ---> a) F" and g: "(g ---> b) F" and a: "a \<noteq> 0"
   shows "((\<lambda>x. f x powr g x) ---> a powr b) F"
   unfolding powr_def
@@ -2561,7 +2549,7 @@
 next
   { assume "a = 0"
     with f f_nonneg have "LIM x inf F (principal {x. f x \<noteq> 0}). f x :> at_right 0"
-      by (auto simp add: filterlim_at eventually_inf_principal le_less 
+      by (auto simp add: filterlim_at eventually_inf_principal le_less
                elim: eventually_elim1 intro: tendsto_mono inf_le1)
     then have "((\<lambda>x. exp (g x * ln (f x))) ---> 0) (inf F (principal {x. f x \<noteq> 0}))"
       by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_0]
@@ -2770,10 +2758,10 @@
   by (metis Reals_cases Reals_of_real cos_of_real)
 
 lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
-  by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
+  by (simp add: diffs_def sin_coeff_Suc del: of_nat_Suc)
 
 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
-  by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
+  by (simp add: diffs_def cos_coeff_Suc del: of_nat_Suc)
 
 text\<open>Now at last we can get the derivatives of exp, sin and cos\<close>
 
@@ -2884,7 +2872,7 @@
 
 lemma DERIV_fun_cos:
      "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. cos(g x)) x :> -sin(g x) * m"
-  by (auto intro!: derivative_eq_intros simp: real_of_nat_def)
+  by (auto intro!: derivative_eq_intros)
 
 subsection \<open>Deriving the Addition Formulas\<close>
 
@@ -2903,7 +2891,7 @@
     have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
           (if even p \<and> even n then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
     using \<open>n\<le>p\<close>
-      by (auto simp: * algebra_simps cos_coeff_def binomial_fact real_of_nat_def)
+      by (auto simp: * algebra_simps cos_coeff_def binomial_fact)
   }
   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> even n
                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
@@ -2942,7 +2930,7 @@
           (if even p \<and> odd n
           then -((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
     using \<open>n\<le>p\<close>
-      by (auto simp:  algebra_simps sin_coeff_def binomial_fact real_of_nat_def)
+      by (auto simp:  algebra_simps sin_coeff_def binomial_fact)
   }
   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> odd n
                then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
@@ -2977,7 +2965,7 @@
                   else 0)"
       by (auto simp: setsum_right_distrib field_simps scaleR_conv_of_real nonzero_of_real_divide)
     also have "... = cos_coeff p *\<^sub>R ((x + y) ^ p)"
-      by (simp add: cos_coeff_def binomial_ring [of x y]  scaleR_conv_of_real real_of_nat_def atLeast0AtMost)
+      by (simp add: cos_coeff_def binomial_ring [of x y]  scaleR_conv_of_real atLeast0AtMost)
     finally have "(\<Sum>n\<le>p. if even p
                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
                   else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)" .
@@ -3114,7 +3102,7 @@
 
 lemma DERIV_fun_pow: "DERIV g x :> m ==>
       DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
-  by (auto intro!: derivative_eq_intros simp: real_of_nat_def)
+  by (auto intro!: derivative_eq_intros simp:)
 
 lemma DERIV_fun_exp:
      "DERIV g x :> m ==> DERIV (\<lambda>x. exp(g x)) x :> exp(g x) * m"
@@ -3155,7 +3143,7 @@
     hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
       by (intro mult_strict_right_mono zero_less_power \<open>0 < x\<close>)
     thus "0 < ?f n"
-      by (simp add: real_of_nat_def divide_simps mult_ac del: mult_Suc)
+      by (simp add: divide_simps mult_ac del: mult_Suc)
 qed
   have sums: "?f sums sin x"
     by (rule sin_paired [THEN sums_group], simp)
@@ -3183,7 +3171,7 @@
 
 lemmas realpow_num_eq_if = power_eq_if
 
-lemma sumr_pos_lt_pair: 
+lemma sumr_pos_lt_pair:
   fixes f :: "nat \<Rightarrow> real"
   shows "\<lbrakk>summable f;
         \<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk>
@@ -3212,13 +3200,12 @@
     { fix d
       let ?six4d = "Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))"
       have "(4::real) * (fact (?six4d)) < (Suc (Suc (?six4d)) * fact (Suc (?six4d)))"
-        unfolding real_of_nat_mult
-        by (rule mult_strict_mono) (simp_all add: fact_less_mono)
+        unfolding of_nat_mult   by (rule mult_strict_mono) (simp_all add: fact_less_mono)
       then have "(4::real) * (fact (?six4d)) < (fact (Suc (Suc (?six4d))))"
-        by (simp only: fact.simps(2) [of "Suc (?six4d)"] real_of_nat_def of_nat_mult of_nat_fact)
+        by (simp only: fact.simps(2) [of "Suc (?six4d)"] of_nat_mult of_nat_fact)
       then have "(4::real) * inverse (fact (Suc (Suc (?six4d)))) < inverse (fact (?six4d))"
         by (simp add: inverse_eq_divide less_divide_eq)
-    } 
+    }
     then show ?thesis
       by (force intro!: sumr_pos_lt_pair [OF sm] simp add: divide_inverse algebra_simps)
   qed
@@ -3367,6 +3354,9 @@
 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
   by (simp add: cos_add)
 
+lemma cos_periodic_pi2 [simp]: "cos (pi + x) = - cos x"
+  by (simp add: cos_add)
+
 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
   by (simp add: sin_add sin_double cos_double)
 
@@ -3374,13 +3364,13 @@
   by (simp add: cos_add sin_double cos_double)
 
 lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n"
-  by (induct n) (auto simp: real_of_nat_Suc distrib_right)
+  by (induct n) (auto simp: distrib_right)
 
 lemma cos_npi2 [simp]: "cos (pi * real n) = (- 1) ^ n"
   by (metis cos_npi mult.commute)
 
 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
-  by (induct n) (auto simp: real_of_nat_Suc distrib_right)
+  by (induct n) (auto simp: of_nat_Suc distrib_right)
 
 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
   by (simp add: mult.commute [of pi])
@@ -3419,33 +3409,33 @@
   apply (simp add: field_simps)
   done
 
-lemma sin_diff_sin: 
+lemma sin_diff_sin:
   fixes w :: "'a::{real_normed_field,banach,field}"
   shows "sin(w) - sin(z) = 2 * sin((w - z) / 2) * cos((w + z) / 2)"
   apply (simp add: mult.assoc sin_times_cos)
   apply (simp add: field_simps)
   done
 
-lemma cos_plus_cos: 
+lemma cos_plus_cos:
   fixes w :: "'a::{real_normed_field,banach,field}"
   shows "cos(w) + cos(z) = 2 * cos((w + z) / 2) * cos((w - z) / 2)"
   apply (simp add: mult.assoc cos_times_cos)
   apply (simp add: field_simps)
   done
 
-lemma cos_diff_cos: 
+lemma cos_diff_cos:
   fixes w :: "'a::{real_normed_field,banach,field}"
   shows "cos(w) - cos(z) = 2 * sin((w + z) / 2) * sin((z - w) / 2)"
   apply (simp add: mult.assoc sin_times_sin)
   apply (simp add: field_simps)
   done
 
-lemma cos_double_cos: 
+lemma cos_double_cos:
   fixes z :: "'a::{real_normed_field,banach}"
   shows "cos(2 * z) = 2 * cos z ^ 2 - 1"
 by (simp add: cos_double sin_squared_eq)
 
-lemma cos_double_sin: 
+lemma cos_double_sin:
   fixes z :: "'a::{real_normed_field,banach}"
   shows "cos(2 * z) = 1 - 2 * sin z ^ 2"
 by (simp add: cos_double sin_squared_eq)
@@ -3466,7 +3456,7 @@
   by (metis sin_periodic_pi2 add_diff_eq mult_2 sin_pi_minus)
 
 lemma cos_2pi_minus [simp]: "cos (2*pi - x) = cos x"
-  by (metis (no_types, hide_lams) cos_add cos_minus cos_two_pi sin_minus sin_two_pi 
+  by (metis (no_types, hide_lams) cos_add cos_minus cos_two_pi sin_minus sin_two_pi
            diff_0_right minus_diff_eq mult_1 mult_zero_left uminus_add_conv_diff)
 
 lemma sin_gt_zero2: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < sin x"
@@ -3566,13 +3556,6 @@
     done
 qed
 
-lemma reals_Archimedean4':
-     "\<lbrakk>0 < y; 0 \<le> x\<rbrakk> \<Longrightarrow> \<exists>n. real n * y \<le> x \<and> x < real (Suc n) * y"
-apply (rule_tac x="nat (floor (x/y))" in exI)
-using floor_correct [of "x/y"]
-apply (auto simp: Real.real_of_nat_Suc field_simps)
-done
-
 lemma cos_zero_lemma:
      "\<lbrakk>0 \<le> x; cos x = 0\<rbrakk> \<Longrightarrow>
       \<exists>n::nat. odd n & x = real n * (pi/2)"
@@ -3580,7 +3563,7 @@
 apply (auto simp: )
 apply (subgoal_tac "0 \<le> x - real n * pi &
                     (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
-apply (auto simp: algebra_simps real_of_nat_Suc)
+apply (auto simp: algebra_simps of_nat_Suc)
  prefer 2 apply (simp add: cos_diff)
 apply (simp add: cos_diff)
 apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
@@ -3589,7 +3572,7 @@
 apply (drule_tac x = "pi/2" in spec)
 apply (simp add: cos_diff)
 apply (rule_tac x = "Suc (2 * n)" in exI)
-apply (simp add: real_of_nat_Suc algebra_simps, auto)
+apply (simp add: of_nat_Suc algebra_simps, auto)
 done
 
 lemma sin_zero_lemma:
@@ -3597,7 +3580,7 @@
       \<exists>n::nat. even n & x = real n * (pi/2)"
 apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
  apply (clarify, rule_tac x = "n - 1" in exI)
- apply (auto elim!: oddE simp add: real_of_nat_Suc field_simps)[1]
+ apply (auto elim!: oddE simp add: of_nat_Suc field_simps)[1]
  apply (rule cos_zero_lemma)
  apply (auto simp: cos_add)
 done
@@ -3611,7 +3594,7 @@
     assume "odd n"
     then obtain m where "n = 2 * m + 1" ..
     then have "cos (real n * pi / 2) = 0"
-      by (simp add: field_simps real_of_nat_Suc) (simp add: cos_add add_divide_distrib)
+      by (simp add: field_simps of_nat_Suc) (simp add: cos_add add_divide_distrib)
   } note * = this
   show ?thesis
   apply (rule iffI)
@@ -3637,18 +3620,18 @@
 
 
 lemma cos_zero_iff_int:
-     "cos x = 0 \<longleftrightarrow> (\<exists>n::int. odd n & x = real n * (pi/2))"
+     "cos x = 0 \<longleftrightarrow> (\<exists>n::int. odd n \<and> x = of_int n * (pi/2))"
 proof safe
   assume "cos x = 0"
-  then show "\<exists>n::int. odd n & x = real n * (pi/2)"
+  then show "\<exists>n::int. odd n & x = of_int n * (pi/2)"
     apply (simp add: cos_zero_iff, safe)
-    apply (metis even_int_iff real_of_int_of_nat_eq)
+    apply (metis even_int_iff of_int_of_nat_eq)
     apply (rule_tac x="- (int n)" in exI, simp)
     done
 next
   fix n::int
   assume "odd n"
-  then show "cos (real n * (pi / 2)) = 0"
+  then show "cos (of_int n * (pi / 2)) = 0"
     apply (simp add: cos_zero_iff)
     apply (case_tac n rule: int_cases2, simp)
     apply (rule disjI2)
@@ -3657,18 +3640,18 @@
 qed
 
 lemma sin_zero_iff_int:
-     "sin x = 0 \<longleftrightarrow> (\<exists>n::int. even n & (x = real n * (pi/2)))"
+     "sin x = 0 \<longleftrightarrow> (\<exists>n::int. even n & (x = of_int n * (pi/2)))"
 proof safe
   assume "sin x = 0"
-  then show "\<exists>n::int. even n \<and> x = real n * (pi / 2)"
+  then show "\<exists>n::int. even n \<and> x = of_int n * (pi / 2)"
     apply (simp add: sin_zero_iff, safe)
-    apply (metis even_int_iff real_of_int_of_nat_eq)
+    apply (metis even_int_iff of_int_of_nat_eq)
     apply (rule_tac x="- (int n)" in exI, simp)
     done
 next
   fix n::int
   assume "even n"
-  then show "sin (real n * (pi / 2)) = 0"
+  then show "sin (of_int n * (pi / 2)) = 0"
     apply (simp add: sin_zero_iff)
     apply (case_tac n rule: int_cases2, simp)
     apply (rule disjI2)
@@ -3677,13 +3660,11 @@
 qed
 
 lemma sin_zero_iff_int2:
-  "sin x = 0 \<longleftrightarrow> (\<exists>n::int. x = real n * pi)"
+  "sin x = 0 \<longleftrightarrow> (\<exists>n::int. x = of_int n * pi)"
   apply (simp only: sin_zero_iff_int)
   apply (safe elim!: evenE)
   apply (simp_all add: field_simps)
-  apply (subst real_numeral(1) [symmetric])
-  apply (simp only: real_of_int_mult [symmetric] real_of_int_inject)
-  apply auto
+  using dvd_triv_left apply fastforce
   done
 
 lemma cos_monotone_0_pi:
@@ -3691,7 +3672,6 @@
   shows "cos x < cos y"
 proof -
   have "- (x - y) < 0" using assms by auto
-
   from MVT2[OF \<open>y < x\<close> DERIV_cos[THEN impI, THEN allI]]
   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z"
     by auto
@@ -3790,8 +3770,7 @@
   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
     by (auto simp: algebra_simps sin_add)
   thus ?thesis
-    by (simp add: real_of_nat_Suc distrib_right add_divide_distrib
-                  mult.commute [of pi])
+    by (simp add: distrib_right add_divide_distrib add.commute mult.commute [of pi])
 qed
 
 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
@@ -3811,7 +3790,7 @@
   done
 
 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
-by (simp only: cos_add sin_add real_of_nat_Suc distrib_right distrib_left add_divide_distrib, auto)
+by (simp only: cos_add sin_add of_nat_Suc distrib_right distrib_left add_divide_distrib, auto)
 
 lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)"
   by (auto intro!: derivative_eq_intros)
@@ -3832,9 +3811,9 @@
   by simp
 
 lemma sin_times_pi_eq_0: "sin(x * pi) = 0 \<longleftrightarrow> x \<in> \<int>"
-  by (simp add: sin_zero_iff_int2) (metis Ints_cases Ints_real_of_int real_of_int_def)
-
-lemma cos_one_2pi: 
+  by (simp add: sin_zero_iff_int2) (metis Ints_cases Ints_of_int)
+
+lemma cos_one_2pi:
     "cos(x) = 1 \<longleftrightarrow> (\<exists>n::nat. x = n * 2*pi) | (\<exists>n::nat. x = -(n * 2*pi))"
     (is "?lhs = ?rhs")
 proof
@@ -3852,7 +3831,7 @@
     show ?rhs
       using m me n
       by (auto simp: field_simps elim!: evenE)
-  next    
+  next
     fix n::nat
     assume n: "even n" "x = - (real n * (pi/2))"
     then obtain m where m: "n = 2 * m"
@@ -3872,9 +3851,9 @@
 lemma cos_one_2pi_int: "cos(x) = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2*pi)"
   apply auto  --\<open>FIXME simproc bug\<close>
   apply (auto simp: cos_one_2pi)
-  apply (metis real_of_int_of_nat_eq)
-  apply (metis mult_minus_right real_of_int_minus real_of_int_of_nat_eq)
-  by (metis mult_minus_right of_int_of_nat real_of_int_def real_of_nat_def)
+  apply (metis of_int_of_nat_eq)
+  apply (metis mult_minus_right of_int_minus of_int_of_nat_eq)
+  by (metis mult_minus_right of_int_of_nat )
 
 lemma sin_cos_sqrt: "0 \<le> sin(x) \<Longrightarrow> (sin(x) = sqrt(1 - (cos(x) ^ 2)))"
   using sin_squared_eq real_sqrt_unique by fastforce
@@ -3882,7 +3861,7 @@
 lemma sin_eq_0_pi: "-pi < x \<Longrightarrow> x < pi \<Longrightarrow> sin(x) = 0 \<Longrightarrow> x = 0"
   by (metis sin_gt_zero sin_minus minus_less_iff neg_0_less_iff_less not_less_iff_gr_or_eq)
 
-lemma cos_treble_cos: 
+lemma cos_treble_cos:
   fixes x :: "'a::{real_normed_field,banach}"
   shows "cos(3 * x) = 4 * cos(x) ^ 3 - 3 * cos x"
 proof -
@@ -3948,16 +3927,16 @@
   by (simp add: sin_cos_eq cos_60)
 
 lemma cos_integer_2pi: "n \<in> \<int> \<Longrightarrow> cos(2*pi * n) = 1"
-  by (metis Ints_cases cos_one_2pi_int mult.assoc mult.commute real_of_int_def)
+  by (metis Ints_cases cos_one_2pi_int mult.assoc mult.commute)
 
 lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2*pi * n) = 0"
   by (metis sin_two_pi Ints_mult mult.assoc mult.commute sin_times_pi_eq_0)
 
-lemma cos_int_2npi [simp]: "cos (2 * real (n::int) * pi) = 1"
+lemma cos_int_2npi [simp]: "cos (2 * of_int (n::int) * pi) = 1"
   by (simp add: cos_one_2pi_int)
 
-lemma sin_int_2npi [simp]: "sin (2 * real (n::int) * pi) = 0"
-  by (metis Ints_real_of_int mult.assoc mult.commute sin_integer_2pi)
+lemma sin_int_2npi [simp]: "sin (2 * of_int (n::int) * pi) = 0"
+  by (metis Ints_of_int mult.assoc mult.commute sin_integer_2pi)
 
 lemma sincos_principal_value: "\<exists>y. (-pi < y \<and> y \<le> pi) \<and> (sin(y) = sin(x) \<and> cos(y) = cos(x))"
   apply (rule exI [where x="pi - (2*pi) * frac((pi - x) / (2*pi))"])
@@ -4201,27 +4180,29 @@
 next
   case (Suc n)
   have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi"
-    unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
+    unfolding Suc_eq_plus1 of_nat_add  distrib_right by auto
   show ?case unfolding split_pi_off using Suc by auto
 qed
 
-lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
+lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + of_int i * pi) = tan x"
 proof (cases "0 \<le> i")
   case True
-  hence i_nat: "real i = real (nat i)" by auto
-  show ?thesis unfolding i_nat by auto
+  hence i_nat: "of_int i = of_int (nat i)" by auto
+  show ?thesis unfolding i_nat
+    by (metis of_int_of_nat_eq tan_periodic_nat) 
 next
   case False
-  hence i_nat: "real i = - real (nat (-i))" by auto
-  have "tan x = tan (x + real i * pi - real i * pi)"
+  hence i_nat: "of_int i = - of_int (nat (-i))" by auto
+  have "tan x = tan (x + of_int i * pi - of_int i * pi)"
     by auto
-  also have "\<dots> = tan (x + real i * pi)"
-    unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
+  also have "\<dots> = tan (x + of_int i * pi)"
+    unfolding i_nat mult_minus_left diff_minus_eq_add
+    by (metis of_int_of_nat_eq tan_periodic_nat)    
   finally show ?thesis by auto
 qed
 
 lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
-  using tan_periodic_int[of _ "numeral n" ] unfolding real_numeral .
+  using tan_periodic_int[of _ "numeral n" ] by simp
 
 lemma tan_minus_45: "tan (-(pi/4)) = -1"
   unfolding tan_def by (simp add: sin_45 cos_45)
@@ -4292,7 +4273,7 @@
 
 lemma cot_periodic [simp]: "cot (x + 2*pi) = cot x"
   by (simp add: cot_def)
-  
+
 lemma cot_altdef: "cot x = inverse (tan x)"
   by (simp add: cot_def tan_def)
 
@@ -4494,7 +4475,7 @@
   by (metis arccos_cos cos_pi order_refl pi_ge_zero)
 
 lemma arccos_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos(-x) = pi - arccos x"
-  by (metis arccos_cos arccos_cos2 cos_minus_pi cos_total diff_le_0_iff_le le_add_same_cancel1 
+  by (metis arccos_cos arccos_cos2 cos_minus_pi cos_total diff_le_0_iff_le le_add_same_cancel1
     minus_diff_eq uminus_add_conv_diff)
 
 lemma sin_arccos_nonzero: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> ~(sin(arccos x) = 0)"
@@ -4746,7 +4727,7 @@
          \<Longrightarrow> (sin(x) \<le> sin(y) \<longleftrightarrow> x \<le> y)"
 by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le)
 
-lemma sin_inj_pi: 
+lemma sin_inj_pi:
     "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2;-(pi/2) \<le> y; y \<le> pi/2; sin(x) = sin(y)\<rbrakk> \<Longrightarrow> x = y"
 by (metis arcsin_sin)
 
@@ -4772,24 +4753,24 @@
 proof -
   have x1: "x \<le> 1"
     using assms
-    by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2) 
+    by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2)
   moreover with assms have ax: "0 \<le> arccos x" "cos(arccos x) = x"
     by (auto simp: arccos)
   moreover have "y = sqrt (1 - x\<^sup>2)" using assms
     by (metis abs_of_nonneg add.commute add_diff_cancel real_sqrt_abs)
-  ultimately show ?thesis using assms arccos_le_pi2 [of x] 
+  ultimately show ?thesis using assms arccos_le_pi2 [of x]
     by (rule_tac x="arccos x" in exI) (auto simp: sin_arccos)
-qed    
+qed
 
 lemma sincos_total_pi:
   assumes "0 \<le> y" and "x\<^sup>2 + y\<^sup>2 = 1"
     shows "\<exists>t. 0 \<le> t \<and> t \<le> pi \<and> x = cos t \<and> y = sin t"
 proof (cases rule: le_cases [of 0 x])
-  case le from sincos_total_pi_half [OF le]  
+  case le from sincos_total_pi_half [OF le]
   show ?thesis
     by (metis pi_ge_two pi_half_le_two add.commute add_le_cancel_left add_mono assms)
 next
-  case ge 
+  case ge
   then have "0 \<le> -x"
     by simp
   then obtain t where "t\<ge>0" "t \<le> pi/2" "-x = cos t" "y = sin t"
@@ -4798,17 +4779,17 @@
     by (metis \<open>0 \<le> - x\<close> power2_minus)
   then show ?thesis
     by (rule_tac x="pi-t" in exI, auto)
-qed    
-    
+qed
+
 lemma sincos_total_2pi_le:
   assumes "x\<^sup>2 + y\<^sup>2 = 1"
     shows "\<exists>t. 0 \<le> t \<and> t \<le> 2*pi \<and> x = cos t \<and> y = sin t"
 proof (cases rule: le_cases [of 0 y])
-  case le from sincos_total_pi [OF le]  
+  case le from sincos_total_pi [OF le]
   show ?thesis
     by (metis assms le_add_same_cancel1 mult.commute mult_2_right order.trans)
 next
-  case ge 
+  case ge
   then have "0 \<le> -y"
     by simp
   then obtain t where "t\<ge>0" "t \<le> pi" "x = cos t" "-y = sin t"
@@ -4817,7 +4798,7 @@
     by (metis \<open>0 \<le> - y\<close> power2_minus)
   then show ?thesis
     by (rule_tac x="2*pi-t" in exI, auto)
-qed    
+qed
 
 lemma sincos_total_2pi:
   assumes "x\<^sup>2 + y\<^sup>2 = 1"
@@ -4855,7 +4836,7 @@
   done
 
 lemma arccos_le_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> arccos x \<le> arccos y \<longleftrightarrow> y \<le> x"
-  using arccos_less_mono [of y x] 
+  using arccos_less_mono [of y x]
   by (simp add: not_le [symmetric])
 
 lemma arccos_less_arccos: "-1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y < arccos x"
@@ -4934,7 +4915,7 @@
   moreover
   have "\<bar>336/527\<bar> < (1 :: real)" by auto
   from arctan_add[OF less_imp_le[OF 17] this]
-  have "arctan(1/7) + arctan (336/527) = arctan (2879/3353)"  by auto 
+  have "arctan(1/7) + arctan (336/527) = arctan (2879/3353)"  by auto
   ultimately have I: "5 * arctan(1/7) = arctan (2879/3353)"  by auto
   have 379: "\<bar>3/79\<bar> < (1 :: real)" by auto
   with arctan_double
@@ -5272,7 +5253,7 @@
     }
     have "?a 1 ----> 0"
       unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
-      by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat simp del: real_of_nat_Suc)
+      by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat simp del: of_nat_Suc)
     have "?diff 1 ----> 0"
     proof (rule LIMSEQ_I)
       fix r :: real
@@ -5459,7 +5440,7 @@
 lemma polynomial_product: (*with thanks to Chaitanya Mangla*)
   fixes x:: "'a :: idom"
   assumes m: "\<And>i. i>m \<Longrightarrow> (a i) = 0" and n: "\<And>j. j>n \<Longrightarrow> (b j) = 0"
-  shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) = 
+  shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
          (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
 proof -
   have "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) = (\<Sum>i\<le>m. \<Sum>j\<le>n. (a i * x ^ i) * (b j * x ^ j))"
@@ -5473,19 +5454,19 @@
   also have "... = (\<Sum>(i,j)\<in>{(i,j). i+j \<le> m+n}. (a i * x ^ i) * (b j * x ^ j))"
     by (auto simp: pairs_le_eq_Sigma setsum.Sigma)
   also have "... = (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
-    apply (subst setsum_triangle_reindex_eq)  
+    apply (subst setsum_triangle_reindex_eq)
     apply (auto simp: algebra_simps setsum_right_distrib intro!: setsum.cong)
     by (metis le_add_diff_inverse power_add)
   finally show ?thesis .
 qed
 
-lemma polynomial_product_nat: 
+lemma polynomial_product_nat:
   fixes x:: nat
   assumes m: "\<And>i. i>m \<Longrightarrow> (a i) = 0" and n: "\<And>j. j>n \<Longrightarrow> (b j) = 0"
-  shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) = 
+  shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
          (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
   using polynomial_product [of m a n b x] assms
-  by (simp add: Int.zpower_int Int.zmult_int Int.int_setsum [symmetric])
+  by (simp only: of_nat_mult [symmetric] of_nat_power [symmetric] int_int_eq Int.int_setsum [symmetric])
 
 lemma polyfun_diff: (*COMPLEX_SUB_POLYFUN in HOL Light*)
     fixes x :: "'a::idom"
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/ex/Ballot.thy
--- a/src/HOL/ex/Ballot.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/ex/Ballot.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -266,7 +266,7 @@
 proof -
   from main_nat[of a b] \<open>b < a\<close> have
     "(real a + real b) * real (valid_countings a b) = (real a - real b) * real (all_countings a b)"
-    by (simp only: real_of_nat_add[symmetric] real_of_nat_mult[symmetric]) auto
+    by (simp only: of_nat_add[symmetric] of_nat_mult[symmetric]) auto
   from this \<open>b < a\<close> show ?thesis
     by (subst mult_left_cancel[of "real a + real b", symmetric]) auto
 qed
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/ex/HarmonicSeries.thy
--- a/src/HOL/ex/HarmonicSeries.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/ex/HarmonicSeries.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -72,9 +72,7 @@
         thus ?thesis by simp
       qed
       moreover from xs have "x \<le> 2^m" by auto
-      ultimately have
-        "inverse (real x) \<ge> inverse (real ((2::nat)^m))"
-        by (simp del: real_of_nat_power)
+      ultimately have "inverse (real x) \<ge> inverse (real ((2::nat)^m))" by simp
       moreover
       from xgt0 have "real x \<noteq> 0" by simp
       then have
@@ -99,7 +97,7 @@
         by simp
       also have
         "\<dots> = ((1/(real tm)) * real (card (?S m)))"
-        by (simp add: real_of_card real_of_nat_def)
+        by (simp add: real_of_card)
       also have
         "\<dots> = ((1/(real tm)) * real (tm - (2^(m - 1))))"
         by (simp add: tmdef)
@@ -280,28 +278,13 @@
   let ?s = "suminf ?f" -- "let ?s equal the sum of the harmonic series"
   assume sf: "summable ?f"
   then obtain n::nat where ndef: "n = nat \<lceil>2 * ?s\<rceil>" by simp
-  then have ngt: "1 + real n/2 > ?s"
-  proof -
-    have "\<forall>n. 0 \<le> ?f n" by simp
-    with sf have "?s \<ge> 0"
-      by (rule suminf_nonneg)
-    then have cgt0: "\<lceil>2*?s\<rceil> \<ge> 0" by simp
-
-    from ndef have "n = nat \<lceil>(2*?s)\<rceil>" .
-    then have "real n = real (nat \<lceil>2*?s\<rceil>)" by simp
-    with cgt0 have "real n = real \<lceil>2*?s\<rceil>"
-      by (auto dest: real_nat_eq_real)
-    then have "real n \<ge> 2*(?s)" by simp
-    then have "real n/2 \<ge> (?s)" by simp
-    then show "1 + real n/2 > (?s)" by simp
-  qed
-
-  obtain j where jdef: "j = (2::nat)^n" by simp
+  then have ngt: "1 + real n/2 > ?s" by linarith
+  def j \<equiv> "(2::nat)^n" 
   have "\<forall>m\<ge>j. 0 < ?f m" by simp
   with sf have "(\<Sum>i<j. ?f i) < ?s" by (rule setsum_less_suminf)
   then have "(\<Sum>i\<in>{Suc 0..<Suc j}. 1/(real i)) < ?s"
     unfolding setsum_shift_bounds_Suc_ivl by (simp add: atLeast0LessThan)
-  with jdef have
+  with j_def have
     "(\<Sum>i\<in>{1..< Suc ((2::nat)^n)}. 1 / (real i)) < ?s" by simp
   then have
     "(\<Sum>i\<in>{1..(2::nat)^n}. 1 / (real i)) < ?s"
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/ex/Sqrt_Script.thy
--- a/src/HOL/ex/Sqrt_Script.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/ex/Sqrt_Script.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -60,8 +60,8 @@
     "prime (p::nat) \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>"
   apply (rule notI)
   apply (erule Rats_abs_nat_div_natE)
-  apply (simp del: real_of_nat_mult
-              add: abs_if divide_eq_eq prime_not_square real_of_nat_mult [symmetric])
+  apply (simp del: of_nat_mult
+              add: abs_if divide_eq_eq prime_not_square of_nat_mult [symmetric])
   done
 
 lemmas two_sqrt_irrational =
diff -r 933eb9e6a1cc -r 77b453bd616f src/HOL/ex/Sum_of_Powers.thy
--- a/src/HOL/ex/Sum_of_Powers.thy	Tue Nov 03 11:20:21 2015 +0100
+++ b/src/HOL/ex/Sum_of_Powers.thy	Tue Nov 10 14:18:41 2015 +0000
@@ -1,7 +1,4 @@
-(*  Title:      HOL/ex/Sum_of_Powers.thy
-    Author:     Lukas Bulwahn <lukas.bulwahn-at-gmail.com>
-*)
-
+(*  Author:     Lukas Bulwahn <lukas.bulwahn-at-gmail.com> *)
 section \<open>Sum of Powers\<close>
 
 theory Sum_of_Powers
@@ -32,8 +29,8 @@
       by (simp only: Suc_eq_plus1)
     finally show ?thesis .
   qed
-  from this have "(\<Prod>i<k. of_nat (n - i)) = (of_nat :: (nat \<Rightarrow> 'a)) (n + 1 - k) / of_nat (n + 1) * (\<Prod>i<k. of_nat (Suc n - i))"
-    by (metis le_add2 nonzero_mult_divide_cancel_left not_one_le_zero of_nat_eq_0_iff times_divide_eq_left)
+  then have "(\<Prod>i<k. of_nat (n - i)) = (of_nat :: (nat \<Rightarrow> 'a)) (n + 1 - k) / of_nat (n + 1) * (\<Prod>i<k. of_nat (Suc n - i))"
+    by (metis (no_types, lifting) le_add2 nonzero_mult_divide_cancel_left not_one_le_zero of_nat_eq_0_iff times_divide_eq_left)
   from assms this show ?thesis
     by (auto simp add: binomial_altdef_of_nat setprod_dividef)
 qed
@@ -41,15 +38,7 @@
 lemma real_binomial_eq_mult_binomial_Suc:
   assumes "k \<le> n"
   shows "(n choose k) = (n + 1 - k) / (n + 1) * (Suc n choose k)"
-proof -
-  have "real (n choose k) = of_nat (n choose k)" by auto
-  also have "... = of_nat (n + 1 - k) / of_nat (n + 1) * of_nat (Suc n choose k)"
-    by (simp add: assms of_nat_binomial_eq_mult_binomial_Suc)
-  also have "... = (n + 1 - k) / (n + 1) * (Suc n choose k)"
-    using real_of_nat_def by auto
-  finally show ?thesis
-    by (metis (no_types, lifting) assms le_add1 le_trans of_nat_diff real_of_nat_1 real_of_nat_add real_of_nat_def)
-qed
+by (metis Suc_eq_plus1 add.commute assms le_SucI of_nat_Suc of_nat_binomial_eq_mult_binomial_Suc of_nat_diff)
 
 subsection \<open>Preliminaries\<close>
 
@@ -90,7 +79,7 @@
 lemma bernpoly_0: "bernpoly n 0 = bernoulli n"
 proof (cases n)
   case 0
-  from this show "bernpoly n 0 = bernoulli n"
+  then show "bernpoly n 0 = bernoulli n"
     unfolding bernpoly_def bernoulli.simps by auto
 next
   case (Suc n')
@@ -104,10 +93,10 @@
   "(\<Sum>k\<le>n. ((Suc n) choose k) * bernoulli k) = (if n = 0 then 1 else 0)"
 proof (cases n)
   case 0
-  from this show ?thesis by (simp add: bernoulli.simps)
+  then show ?thesis by (simp add: bernoulli.simps)
 next
   case Suc
-  from this show ?thesis
+  then show ?thesis
   by (simp add: bernoulli.simps)
     (simp add: field_simps add_2_eq_Suc'[symmetric] del: add_2_eq_Suc add_2_eq_Suc')
 qed
@@ -121,7 +110,7 @@
     unfolding bernpoly_def by (rule DERIV_cong) (fast intro!: derivative_intros, simp)
   moreover have "(\<Sum>k\<le>n. real (Suc n - k) * x ^ (n - k) * (real (Suc n choose k) * bernoulli k)) = (n + 1) * bernpoly n x"
     unfolding bernpoly_def
-    by (auto intro: setsum.cong simp add: setsum_right_distrib real_binomial_eq_mult_binomial_Suc[of _ n] Suc_eq_plus1 real_of_nat_diff)
+    by (auto intro: setsum.cong simp add: setsum_right_distrib real_binomial_eq_mult_binomial_Suc[of _ n] Suc_eq_plus1 of_nat_diff)
   ultimately show ?thesis by auto
 qed
 
@@ -134,7 +123,7 @@
   case (Suc n)
   have "bernpoly (Suc n) (0 + 1) - bernpoly (Suc n) 0 = (Suc n) * 0 ^ n"
     unfolding bernpoly_0 unfolding bernpoly_def by (simp add: setsum_binomial_times_bernoulli zero_power)
-  from this have const: "bernpoly (Suc n) (0 + 1) - bernpoly (Suc n) 0 = real (Suc n) * 0 ^ n" by (simp add: power_0_left)
+  then have const: "bernpoly (Suc n) (0 + 1) - bernpoly (Suc n) 0 = real (Suc n) * 0 ^ n" by (simp add: power_0_left)
   have hyps': "\<And>x. (real n + 1) * bernpoly n (x + 1) - (real n + 1) * bernpoly n x = real n * x ^ (n - Suc 0) * real (Suc n)"
     unfolding right_diff_distrib[symmetric] by (simp add: Suc.hyps One_nat_def)
   note [derivative_intros] = DERIV_chain'[where f = "\<lambda>x::real. x + 1" and g = "bernpoly (Suc n)" and s="UNIV"]
@@ -148,7 +137,7 @@
   from diff_bernpoly[of "Suc m", simplified] have "(m + (1::real)) * (\<Sum>k\<le>n. (real k) ^ m) = (\<Sum>k\<le>n. bernpoly (Suc m) (real k + 1) - bernpoly (Suc m) (real k))"
     by (auto simp add: setsum_right_distrib intro!: setsum.cong)
   also have "... = (\<Sum>k\<le>n. bernpoly (Suc m) (real (k + 1)) - bernpoly (Suc m) (real k))"
-    by (simp only: real_of_nat_1[symmetric] real_of_nat_add[symmetric])
+    by simp
   also have "... = bernpoly (Suc m) (n + 1) - bernpoly (Suc m) 0"
     by (simp only: setsum_diff[where f="\<lambda>k. bernpoly (Suc m) (real k)"]) simp
   finally show ?thesis by (auto simp add: field_simps intro!: eq_divide_imp)
@@ -185,9 +174,9 @@
 proof -
   from sum_of_squares have "real (6 * (\<Sum>k\<le>n. k ^ 2)) = real (2 * n ^ 3 + 3 * n ^ 2 + n)"
     by (auto simp add: field_simps)
-  from this have "6 * (\<Sum>k\<le>n. k ^ 2) = 2 * n ^ 3 + 3 * n ^ 2 + n"
-    by (simp only: real_of_nat_inject[symmetric])
-  from this show ?thesis by auto
+  then have "6 * (\<Sum>k\<le>n. k ^ 2) = 2 * n ^ 3 + 3 * n ^ 2 + n"
+    by blast
+  then show ?thesis by auto
 qed
 
 lemma sum_of_cubes: "(\<Sum>k\<le>n::nat. k ^ 3) = (n ^ 2 + n) ^ 2 / 4"
@@ -202,14 +191,14 @@
     by (simp add: unroll algebra_simps power2_eq_square power4_eq power3_eq_cube)
   finally show ?thesis by simp
 qed
-
+                       
 lemma sum_of_cubes_nat: "(\<Sum>k\<le>n::nat. k ^ 3) = (n ^ 2 + n) ^ 2 div 4"
 proof -
   from sum_of_cubes have "real (4 * (\<Sum>k\<le>n. k ^ 3)) = real ((n ^ 2 + n) ^ 2)"
     by (auto simp add: field_simps)
-  from this have "4 * (\<Sum>k\<le>n. k ^ 3) = (n ^ 2 + n) ^ 2"
-    by (simp only: real_of_nat_inject[symmetric])
-  from this show ?thesis by auto
+  then have "4 * (\<Sum>k\<le>n. k ^ 3) = (n ^ 2 + n) ^ 2"
+    by blast
+  then show ?thesis by auto
 qed
 
 end