author wenzelm Thu, 09 Jul 2015 00:40:57 +0200 changeset 60698 29e8bdc41f90 parent 60697 e266d5463e9d child 60699 7bf560b196a3
tuned proofs;
```--- a/src/HOL/Decision_Procs/Polynomial_List.thy	Thu Jul 09 00:39:49 2015 +0200
+++ b/src/HOL/Decision_Procs/Polynomial_List.thy	Thu Jul 09 00:40:57 2015 +0200
@@ -129,29 +129,38 @@
qed

-lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
-  apply (induct a)
-  apply simp
-  apply clarify
-  apply (case_tac b)
-  done
+lemma (in comm_semiring_0) padd_assoc: "(a +++ b) +++ c = a +++ (b +++ c)"
+proof (induct a arbitrary: b c)
+  case Nil
+  then show ?case
+    by simp
+next
+  case Cons
+  then show ?case
+    by (cases b) (simp_all add: ac_simps)
+qed

lemma (in semiring_0) poly_cmult_distr: "a %* (p +++ q) = a %* p +++ a %* q"
-  apply (induct p arbitrary: q)
-  apply simp
-  apply (case_tac q)
-  done
+proof (induct p arbitrary: q)
+  case Nil
+  then show ?case
+    by simp
+next
+  case Cons
+  then show ?case
+    by (cases q) (simp_all add: distrib_left)
+qed

lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = 0 # t"
-  apply (induct t)
-  apply simp
-  apply (case_tac t)
-  apply auto
-  done
-
+proof (induct t)
+  case Nil
+  then show ?case
+    by simp
+next
+  case (Cons a t)
+  then show ?case
+qed

text \<open>Properties of evaluation of polynomials.\<close>

@@ -167,18 +176,21 @@
qed

lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
-  apply (induct p)
-  apply (case_tac [2] "x = zero")
-  apply (auto simp add: distrib_left ac_simps)
-  done
+proof (induct p)
+  case Nil
+  then show ?case
+    by simp
+next
+  case Cons
+  then show ?case
+    by (cases "x = zero") (auto simp add: distrib_left ac_simps)
+qed

lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c * poly p x"
by (induct p) (auto simp add: distrib_left ac_simps)

lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
-  apply (auto simp add: poly_cmult)
-  done

lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
proof (induct p1 arbitrary: p2)
@@ -186,7 +198,7 @@
then show ?case
by simp
next
-  case (Cons a as p2)
+  case (Cons a as)
then show ?case
qed
@@ -216,16 +228,16 @@

subsection \<open>Key Property: if @{term "f a = 0"} then @{term "(x - a)"} divides @{term "p(x)"}.\<close>

-lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
-proof (induct t)
+lemma (in comm_ring_1) lemma_poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
+proof (induct t arbitrary: h)
case Nil
-  have "[h] = [h] +++ [- a, 1] *** []" for h by simp
+  have "[h] = [h] +++ [- a, 1] *** []" by simp
then show ?case by blast
next
case (Cons  x xs)
-  have "\<exists>q r. h # x # xs = [r] +++ [-a, 1] *** q" for h
+  have "\<exists>q r. h # x # xs = [r] +++ [-a, 1] *** q"
proof -
-    from Cons.hyps obtain q r where qr: "x # xs = [r] +++ [- a, 1] *** q"
+    from Cons obtain q r where qr: "x # xs = [r] +++ [- a, 1] *** q"
by blast
have "h # x # xs = [a * r + h] +++ [-a, 1] *** (r # q)"
using qr by (cases q) (simp_all add: algebra_simps)
@@ -237,7 +249,7 @@
lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
using lemma_poly_linear_rem [where t = t and a = a] by auto

-lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) \<longleftrightarrow> p = [] \<or> (\<exists>q. p = [-a, 1] *** q)"
+lemma (in comm_ring_1) poly_linear_divides: "poly p a = 0 \<longleftrightarrow> p = [] \<or> (\<exists>q. p = [-a, 1] *** q)"
proof (cases p)
case Nil
then show ?thesis by simp
@@ -264,12 +276,12 @@
qed

lemma (in semiring_0) lemma_poly_length_mult[simp]:
-  "\<forall>h k a. length (k %* p +++  (h # (a %* p))) = Suc (length p)"
-  by (induct p) auto
+  "length (k %* p +++  (h # (a %* p))) = Suc (length p)"
+  by (induct p arbitrary: h k a) auto

lemma (in semiring_0) lemma_poly_length_mult2[simp]:
-  "\<forall>h k. length (k %* p +++  (h # p)) = Suc (length p)"
-  by (induct p) auto
+  "length (k %* p +++  (h # p)) = Suc (length p)"
+  by (induct p arbitrary: h k) auto

lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
by auto
@@ -281,9 +293,9 @@
by (induct p) auto

lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"
-  by (induct p1 arbitrary: p2) (simp_all, arith)
+  by (induct p1 arbitrary: p2) auto

-lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)"
+lemma (in semiring_0) poly_root_mult_length[simp]: "length ([a, b] *** p) = Suc (length p)"

lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:
@@ -301,15 +313,15 @@

text \<open>A nontrivial polynomial of degree n has no more than n roots.\<close>
lemma (in idom) poly_roots_index_lemma:
-  assumes p: "poly p x \<noteq> poly [] x"
-    and n: "length p = n"
+  assumes "poly p x \<noteq> poly [] x"
+    and "length p = n"
shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
-  using p n
+  using assms
proof (induct n arbitrary: p x)
case 0
then show ?case by simp
next
-  case (Suc n p x)
+  case (Suc n)
have False if C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
proof -
from Suc.prems have p0: "poly p x \<noteq> 0" "p \<noteq> []"
@@ -325,14 +337,14 @@
using q Suc.prems(2) by simp
from q p0 have qx: "poly q x \<noteq> poly [] x"
-    from Suc.hyps[OF qx lg] obtain i where i: "\<forall>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
+    from Suc.hyps[OF qx lg] obtain i where i: "\<And>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
by blast
let ?i = "\<lambda>m. if m = Suc n then a else i m"
from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m"
by blast
from y have "y = a \<or> poly q y = 0"
-    with i[rule_format, of y] y(1) y(2) show ?thesis
+    with i[of y] y(1) y(2) show ?thesis
apply auto
apply (erule_tac x = "m" in allE)
apply auto
@@ -343,12 +355,13 @@

lemma (in idom) poly_roots_index_length:
-  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. n \<le> length p \<and> x = i n)"
+  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>n. n \<le> length p \<and> x = i n)"
by (blast intro: poly_roots_index_lemma)

lemma (in idom) poly_roots_finite_lemma1:
-  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>N i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n::nat. n < N \<and> x = i n)"
-  apply (drule poly_roots_index_length, safe)
+  "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>N i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>n::nat. n < N \<and> x = i n)"
+  apply (drule poly_roots_index_length)
+  apply safe
apply (rule_tac x = "Suc (length p)" in exI)
apply (rule_tac x = i in exI)
@@ -358,8 +371,10 @@
assumes "\<forall>x. P x \<longrightarrow> (\<exists>n. n < length j \<and> x = j!n)"
shows "finite {x. P x}"
proof -
-  from assms have "{x. P x} \<subseteq> set j" by auto
-  then show ?thesis using finite_subset by auto
+  from assms have "{x. P x} \<subseteq> set j"
+    by auto
+  then show ?thesis
+    using finite_subset by auto
qed

lemma (in idom) poly_roots_finite_lemma2:
@@ -379,7 +394,8 @@
assume F: "finite (UNIV :: 'a set)"
have "finite (UNIV :: nat set)"
proof (rule finite_imageD)
-    have "of_nat ` UNIV \<subseteq> UNIV" by simp
+    have "of_nat ` UNIV \<subseteq> UNIV"
+      by simp
then show "finite (of_nat ` UNIV :: 'a set)"
using F by (rule finite_subset)
show "inj (of_nat :: nat \<Rightarrow> 'a)"
@@ -499,12 +515,18 @@
qed

lemma (in idom_char_0) poly_zero: "poly p = poly [] \<longleftrightarrow> (\<forall>c \<in> set p. c = 0)"
-  apply (induct p)
-  apply simp
-  apply (rule iffI)
-  apply (drule poly_zero_lemma')
-  apply auto
-  done
+proof (induct p)
+  case Nil
+  then show ?case by simp
+next
+  case Cons
+  show ?case
+    apply (rule iffI)
+    apply (drule poly_zero_lemma')
+    using Cons
+    apply auto
+    done
+qed

lemma (in idom_char_0) poly_0: "\<forall>c \<in> set p. c = 0 \<Longrightarrow> poly p x = 0"
unfolding poly_zero[symmetric] by simp
@@ -594,10 +616,10 @@
text \<open>At last, we can consider the order of a root.\<close>

lemma (in idom_char_0) poly_order_exists_lemma:
-  assumes lp: "length p = d"
-    and p: "poly p \<noteq> poly []"
+  assumes "length p = d"
+    and "poly p \<noteq> poly []"
shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0"
-  using lp p
+  using assms
proof (induct d arbitrary: p)
case 0
then show ?case by simp
@@ -613,15 +635,15 @@
from True[unfolded poly_linear_divides] pN obtain q where q: "p = [-a, 1] *** q"
by blast
from q h True have qh: "length q = n" "poly q \<noteq> poly []"
-      apply -
-      apply simp
+      apply simp_all
apply (simp only: fun_eq_iff)
apply (rule ccontr)
done
from Suc.hyps[OF qh] obtain m r where mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0"
by blast
-    from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0" by simp
+    from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0"
+      by simp
then show ?thesis by blast
next
case False
@@ -680,7 +702,7 @@
case 0
show ?case
proof (rule ccontr)
-        assume "\<not> poly (mulexp 0 [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc 0 *** m)"
+        assume "\<not> ?thesis"
then have "poly q a = 0"
with \<open>poly q a \<noteq> 0\<close> show False
@@ -689,7 +711,7 @@
next
case (Suc n)
show ?case
-        by (rule pexp_Suc [THEN ssubst], rule ccontr)
+        by (rule pexp_Suc [THEN ssubst])
(simp add: poly_mult_left_cancel poly_mult_assoc Suc del: pmult_Cons pexp_Suc)
qed
ultimately show False by simp
@@ -906,8 +928,7 @@
lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> 0 < length p \<and> last p \<noteq> 0"
using pnormal_last_length pnormal_length pnormal_last_nonzero by blast

-lemma (in idom_char_0) poly_Cons_eq:
-  "poly (c # cs) = poly (d # ds) \<longleftrightarrow> c = d \<and> poly cs = poly ds"
+lemma (in idom_char_0) poly_Cons_eq: "poly (c # cs) = poly (d # ds) \<longleftrightarrow> c = d \<and> poly cs = poly ds"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
show ?rhs if ?lhs
@@ -974,11 +995,12 @@
"last ((a %* p) +++ (x # (b %* p))) = (if p = [] then x else b * last p)"
apply (induct p arbitrary: a x b)
apply auto
-  apply (rename_tac a p aa x b)
-  apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []")
-  apply simp
-  apply (induct_tac p)
-  apply auto
+  subgoal for a p c x b
+    apply (subgoal_tac "padd (cmult c p) (times b a # cmult b p) \<noteq> []")
+    apply simp
+    apply (induct p)
+    apply auto
+    done
done

lemma (in semiring_1) last_linear_mul:
@@ -1097,13 +1119,18 @@
lemma poly_mono:
fixes x :: "'a::linordered_idom"
shows "abs x \<le> k \<Longrightarrow> abs (poly p x) \<le> poly (map abs p) k"
-  apply (induct p)
-  apply auto
-  apply (rename_tac a p)
-  apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
-  apply (rule abs_triangle_ineq)
-  apply (auto intro!: mult_mono simp add: abs_mult)
-  done
+proof (induct p)
+  case Nil
+  then show ?case by simp
+next
+  case (Cons a p)
+  then show ?case
+    apply auto
+    apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
+    apply (rule abs_triangle_ineq)
+    apply (auto intro!: mult_mono simp add: abs_mult)
+    done
+qed

lemma (in semiring_0) poly_Sing: "poly [c] x = c"
by simp```
```--- a/src/HOL/Decision_Procs/Rat_Pair.thy	Thu Jul 09 00:39:49 2015 +0200
+++ b/src/HOL/Decision_Procs/Rat_Pair.thy	Thu Jul 09 00:40:57 2015 +0200
@@ -328,13 +328,7 @@
proof cases
case 1
then show ?thesis
-      apply (cases "a = 0")
-      apply (simp_all add: x y Nmul_def INum_def Let_def)
-      apply (cases "b = 0")
-      apply simp_all
-      apply (cases "a' = 0")
-      apply simp_all
-      done
+      by (auto simp add: x y Nmul_def INum_def)
next
case neq: 2
let ?g = "gcd (a * a') (b * b')"
@@ -347,21 +341,21 @@
qed
qed

-lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a::field)"
+lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x :: 'a::field)"
by (simp add: Nneg_def split_def INum_def)

-lemma Nsub[simp]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0,field})"
+lemma Nsub[simp]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a::{field_char_0,field})"

-lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field) / (INum x)"
+lemma Ninv[simp]: "INum (Ninv x) = (1 :: 'a::field) / INum x"
by (simp add: Ninv_def INum_def split_def)

-lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {field_char_0,field})"
+lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y :: 'a::{field_char_0,field})"

lemma Nlt0_iff[simp]:
assumes nx: "isnormNum x"
-  shows "((INum x :: 'a::{field_char_0,linordered_field})< 0) = 0>\<^sub>N x"
+  shows "((INum x :: 'a::{field_char_0,linordered_field}) < 0) = 0>\<^sub>N x"
proof -
obtain a b where x: "x = (a, b)" by (cases x)
show ?thesis
@@ -401,7 +395,7 @@

lemma Ngt0_iff[simp]:
assumes nx: "isnormNum x"
-  shows "((INum x :: 'a::{field_char_0,linordered_field})> 0) = 0<\<^sub>N x"
+  shows "((INum x :: 'a::{field_char_0,linordered_field}) > 0) = 0<\<^sub>N x"
proof -
obtain a b where x: "x = (a, b)" by (cases x)
show ?thesis
@@ -421,7 +415,7 @@

lemma Nge0_iff[simp]:
assumes nx: "isnormNum x"
-  shows "((INum x :: 'a::{field_char_0,linordered_field}) \<ge> 0) = 0\<le>\<^sub>N x"
+  shows "(INum x :: 'a::{field_char_0,linordered_field}) \<ge> 0 \<longleftrightarrow> 0\<le>\<^sub>N x"
proof -
obtain a b where x: "x = (a, b)" by (cases x)
show ?thesis
@@ -442,12 +436,12 @@
lemma Nlt_iff[simp]:
assumes nx: "isnormNum x"
and ny: "isnormNum y"
-  shows "((INum x :: 'a::{field_char_0,linordered_field}) < INum y) = (x <\<^sub>N y)"
+  shows "((INum x :: 'a::{field_char_0,linordered_field}) < INum y) \<longleftrightarrow> x <\<^sub>N y"
proof -
let ?z = "0::'a"
-  have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)"
+  have "((INum x ::'a) < INum y) \<longleftrightarrow> INum (x -\<^sub>N y) < ?z"
using nx ny by simp
-  also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))"
+  also have "\<dots> \<longleftrightarrow> 0>\<^sub>N (x -\<^sub>N y)"
using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
finally show ?thesis
@@ -456,11 +450,11 @@
lemma Nle_iff[simp]:
assumes nx: "isnormNum x"
and ny: "isnormNum y"
-  shows "((INum x :: 'a::{field_char_0,linordered_field})\<le> INum y) = (x \<le>\<^sub>N y)"
+  shows "((INum x :: 'a::{field_char_0,linordered_field}) \<le> INum y) \<longleftrightarrow> x \<le>\<^sub>N y"
proof -
-  have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))"
+  have "((INum x ::'a) \<le> INum y) \<longleftrightarrow> INum (x -\<^sub>N y) \<le> (0::'a)"
using nx ny by simp
-  also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))"
+  also have "\<dots> \<longleftrightarrow> 0\<ge>\<^sub>N (x -\<^sub>N y)"
using Nle0_iff[OF Nsub_normN[OF ny]] by simp
finally show ?thesis
@@ -508,7 +502,7 @@
shows "normNum (normNum x) = normNum x"
by simp

-lemma normNum0[simp]: "normNum (0,b) = 0\<^sub>N" "normNum (a,0) = 0\<^sub>N"
+lemma normNum0[simp]: "normNum (0, b) = 0\<^sub>N" "normNum (a, 0) = 0\<^sub>N"

@@ -520,29 +514,40 @@
assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "normNum x +\<^sub>N y = x +\<^sub>N y"
proof -
-  have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all
-  have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)" by simp
-  also have "\<dots> = INum (x +\<^sub>N y)" by simp
-  finally show ?thesis using isnormNum_unique[OF n] by simp
+  have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)"
+    by simp_all
+  have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)"
+    by simp
+  also have "\<dots> = INum (x +\<^sub>N y)"
+    by simp
+  finally show ?thesis
+    using isnormNum_unique[OF n] by simp
qed

assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "x +\<^sub>N normNum y = x +\<^sub>N y"
proof -
-  have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all
-  have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)" by simp
-  also have "\<dots> = INum (x +\<^sub>N y)" by simp
-  finally show ?thesis using isnormNum_unique[OF n] by simp
+  have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)"
+    by simp_all
+  have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)"
+    by simp
+  also have "\<dots> = INum (x +\<^sub>N y)"
+    by simp
+  finally show ?thesis
+    using isnormNum_unique[OF n] by simp
qed

assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
proof -
-  have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all
-  have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
-  with isnormNum_unique[OF n] show ?thesis by simp
+  have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))"
+    by simp_all
+  have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)"
+    by simp
+  with isnormNum_unique[OF n] show ?thesis
+    by simp
qed

lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
@@ -557,8 +562,10 @@
proof -
from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))"
by simp_all
-  have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
-  with isnormNum_unique[OF n] show ?thesis by simp
+  have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)"
+    by simp
+  with isnormNum_unique[OF n] show ?thesis
+    by simp
qed

lemma Nsub0:
@@ -568,9 +575,12 @@
shows "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = y"
proof -
from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]
-  have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp
-  also have "\<dots> = (INum x = (INum y :: 'a))" by simp
-  also have "\<dots> = (x = y)" using x y by simp
+  have "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)"
+    by simp
+  also have "\<dots> \<longleftrightarrow> INum x = (INum y :: 'a)"
+    by simp
+  also have "\<dots> \<longleftrightarrow> x = y"
+    using x y by simp
finally show ?thesis .
qed
```
```--- a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Thu Jul 09 00:39:49 2015 +0200
+++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Thu Jul 09 00:40:57 2015 +0200
@@ -8,7 +8,7 @@
imports Complex_Main Rat_Pair Polynomial_List
begin

-subsection\<open>Datatype of polynomial expressions\<close>
+subsection \<open>Datatype of polynomial expressions\<close>

datatype poly = C Num | Bound nat | Add poly poly | Sub poly poly
| Mul poly poly| Neg poly| Pw poly nat| CN poly nat poly
@@ -66,7 +66,7 @@
| "decrpoly a = a"

+subsection \<open>Degrees and heads and coefficients\<close>

fun degree :: "poly \<Rightarrow> nat"
where
@@ -78,7 +78,8 @@

-(* More general notions of degree and head *)
+text \<open>More general notions of degree and head.\<close>
+
fun degreen :: "poly \<Rightarrow> nat \<Rightarrow> nat"
where
"degreen (CN c n p) = (\<lambda>m. if n = m then 1 + degreen p n else 0)"
@@ -110,7 +111,7 @@
| "headconst (C n) = n"

-subsection\<open>Operations for normalization\<close>
+subsection \<open>Operations for normalization\<close>

declare if_cong[fundef_cong del]
declare let_cong[fundef_cong del]
@@ -179,7 +180,7 @@
| "polynate (Pw p n) = polynate p ^\<^sub>p n"
| "polynate (CN c n p) = polynate (Add c (Mul (Bound n) p))"
| "polynate (C c) = C (normNum c)"
-by pat_completeness auto
+  by pat_completeness auto
termination by (relation "measure polysize") auto

fun poly_cmul :: "Num \<Rightarrow> poly \<Rightarrow> poly"
@@ -233,7 +234,7 @@
| "poly_deriv p = 0\<^sub>p"

-subsection\<open>Semantics of the polynomial representation\<close>
+subsection \<open>Semantics of the polynomial representation\<close>

primrec Ipoly :: "'a list \<Rightarrow> poly \<Rightarrow> 'a::{field_char_0,field,power}"
where
@@ -246,8 +247,7 @@
| "Ipoly bs (Pw t n) = Ipoly bs t ^ n"
| "Ipoly bs (CN c n p) = Ipoly bs c + (bs!n) * Ipoly bs p"

-abbreviation Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0,field,power}"
-    ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
+abbreviation Ipoly_syntax :: "poly \<Rightarrow> 'a list \<Rightarrow>'a::{field_char_0,field,power}"  ("\<lparr>_\<rparr>\<^sub>p\<^bsup>_\<^esup>")
where "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> \<equiv> Ipoly bs p"

lemma Ipoly_CInt: "Ipoly bs (C (i, 1)) = of_int i"
@@ -273,14 +273,14 @@
definition isnpoly :: "poly \<Rightarrow> bool"
where "isnpoly p = isnpolyh p 0"

lemma polyadd_normh: "isnpolyh p n0 \<Longrightarrow> isnpolyh q n1 \<Longrightarrow> isnpolyh (polyadd p q) (min n0 n1)"
proof (induct p q arbitrary: n0 n1 rule: polyadd.induct)
case (2 ab c' n' p' n0 n1)
from 2 have  th1: "isnpolyh (C ab) (Suc n')"
by simp
-  from 2(3) have th2: "isnpolyh c' (Suc n')"  and nplen1: "n' \<ge> n1"
+  from 2(3) have th2: "isnpolyh c' (Suc n')" and nplen1: "n' \<ge> n1"
by simp_all
with isnpolyh_mono have cp: "isnpolyh c' (Suc n')"
by simp
@@ -314,11 +314,11 @@
by simp
from 4 have n'gen1: "n' \<ge> n1"
by simp
-  have "n < n' \<or> n' < n \<or> n = n'"
-    by auto
-  moreover
-  {
-    assume eq: "n = n'"
+  consider (eq) "n = n'" | (lt) "n < n'" | (gt) "n > n'"
+    by arith
+  then show ?case
+  proof cases
+    case eq
with "4.hyps"(3)[OF nc nc']
have ncc':"isnpolyh (c +\<^sub>p c') (Suc n)"
by auto
@@ -329,12 +329,10 @@
by simp
have minle: "min n0 n1 \<le> n'"
using ngen0 n'gen1 eq by simp
-    from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' have ?case
+    from minle npp' ncc'n01 4 eq ngen0 n'gen1 ncc' show ?thesis
-  }
-  moreover
-  {
-    assume lt: "n < n'"
+  next
+    case lt
have "min n0 n1 \<le> n0"
by simp
with 4 lt have th1:"min n0 n1 \<le> n"
@@ -347,12 +345,10 @@
by arith
from "4.hyps"(1)[OF th21 th22] have "isnpolyh (polyadd c (CN c' n' p')) (Suc n)"
using th23 by simp
-    with 4 lt th1 have ?case
+    with 4 lt th1 show ?thesis
by simp
-  }
-  moreover
-  {
-    assume gt: "n' < n"
+  next
+    case gt
then have gt': "n' < n \<and> \<not> n < n'"
by simp
have "min n0 n1 \<le> n1"
@@ -367,10 +363,9 @@
by arith
from "4.hyps"(2)[OF th22 th21] have "isnpolyh (polyadd (CN c n p) c') (Suc n')"
using th23 by simp
-    with 4 gt th1 have ?case
+    with 4 gt th1 show ?thesis
by simp
-  }
-  ultimately show ?case by blast
+  qed
qed auto

lemma polyadd[simp]: "Ipoly bs (polyadd p q) = Ipoly bs p + Ipoly bs q"
@@ -378,9 +373,9 @@
(auto simp add: Let_def field_simps distrib_left[symmetric] simp del: distrib_left_NO_MATCH)

lemma polyadd_norm: "isnpoly p \<Longrightarrow> isnpoly q \<Longrightarrow> isnpoly (polyadd p q)"
-  using polyadd_normh[of "p" "0" "q" "0"] isnpoly_def by simp
+  using polyadd_normh[of p 0 q 0] isnpoly_def by simp

-text\<open>The degree of addition and other general lemmas needed for the normal form of polymul\<close>
+text \<open>The degree of addition and other general lemmas needed for the normal form of polymul.\<close>

assumes "isnpolyh p n0"
@@ -391,17 +386,13 @@
using assms
proof (induct p q arbitrary: m n0 n1 rule: polyadd.induct)
case (4 c n p c' n' p' m n0 n1)
-  have "n' = n \<or> n < n' \<or> n' < n" by arith
-  then show ?case
-  proof (elim disjE)
-    assume [simp]: "n' = n"
-    from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
+  show ?case
+  proof (cases "n = n'")
+    case True
+    with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
show ?thesis by (auto simp: Let_def)
next
-    assume "n < n'"
-    with 4 show ?thesis by auto
-  next
-    assume "n' < n"
+    case False
with 4 show ?thesis by auto
qed
qed auto
@@ -441,13 +432,15 @@
by (cases n) auto
next
case (4 c n p c' n' p' n0 n1 m)
-  have "n' = n \<or> n < n' \<or> n' < n" by arith
-  then show ?case
-  proof (elim disjE)
-    assume [simp]: "n' = n"
-    from 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
+  show ?case
+  proof (cases "n = n'")
+    case True
+    with 4(4)[of n n m] 4(3)[of "Suc n" "Suc n" m] 4(5-7)
show ?thesis by (auto simp: Let_def)
-  qed simp_all
+  next
+    case False
+    then show ?thesis by simp
+  qed
qed auto

@@ -458,26 +451,16 @@
using assms
proof (induct p q arbitrary: m n0 n1 c rule: polyadd.induct)
case (4 c n p c' n' p' m n0 n1 x)
-  {
-    assume nn': "n' < n"
-    then have ?case using 4 by simp
-  }
-  moreover
-  {
-    assume nn': "\<not> n' < n"
-    then have "n < n' \<or> n = n'" by arith
-    moreover { assume "n < n'" with 4 have ?case by simp }
-    moreover
-    {
-      assume eq: "n = n'"
-      then have ?case using 4
-        apply (cases "p +\<^sub>p p' = 0\<^sub>p")
-        apply (auto simp add: Let_def)
-        done
-    }
-    ultimately have ?case by blast
-  }
-  ultimately show ?case by blast
+  consider "n = n'" | "n > n' \<or> n < n'" by arith
+  then show ?case
+  proof cases
+    case 1
+    with 4 show ?thesis
+      by (cases "p +\<^sub>p p' = 0\<^sub>p") (auto simp add: Let_def)
+  next
+    case 2
+    with 4 show ?thesis by auto
+  qed
qed simp_all

lemma polymul_properties:
@@ -500,7 +483,7 @@
then show ?case by auto
next
case (3 n0 n1)
-    then show ?case  using "2.hyps" by auto
+    then show ?case using "2.hyps" by auto
}
next
case (3 c n p c')
@@ -720,7 +703,7 @@
qed

-text\<open>polyneg is a negation and preserves normal forms\<close>
+text \<open>polyneg is a negation and preserves normal forms\<close>

lemma polyneg[simp]: "Ipoly bs (polyneg p) = - Ipoly bs p"
by (induct p rule: polyneg.induct) auto
@@ -738,7 +721,7 @@
using isnpoly_def polyneg_normh by simp

-text\<open>polysub is a substraction and preserves normal forms\<close>
+text \<open>polysub is a substraction and preserves normal forms\<close>

lemma polysub[simp]: "Ipoly bs (polysub p q) = Ipoly bs p - Ipoly bs q"
@@ -762,7 +745,7 @@
by (induct p q arbitrary: n0 n1 rule:polyadd.induct)
(auto simp: Nsub0[simplified Nsub_def] Let_def)

-text\<open>polypow is a power function and preserves normal forms\<close>
+text \<open>polypow is a power function and preserves normal forms\<close>

lemma polypow[simp]:
"Ipoly bs (polypow n p) = (Ipoly bs p :: 'a::{field_char_0,field}) ^ n"
@@ -830,7 +813,7 @@
shows "isnpoly p \<Longrightarrow> isnpoly (polypow k p)"

-text\<open>Finally the whole normalization\<close>
+text \<open>Finally the whole normalization\<close>

lemma polynate [simp]:
"Ipoly bs (polynate p) = (Ipoly bs p :: 'a ::{field_char_0,field})"
@@ -843,7 +826,7 @@

-text\<open>shift1\<close>
+text \<open>shift1\<close>

lemma shift1: "Ipoly bs (shift1 p) = Ipoly bs (Mul (Bound 0) p)"
@@ -1254,7 +1237,7 @@
qed

-text\<open>consequences of unicity on the algorithms for polynomial normalization\<close>
+text \<open>consequences of unicity on the algorithms for polynomial normalization\<close>

assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
@@ -1328,7 +1311,7 @@
unfolding poly_nate_polypoly' by auto

+subsection \<open>heads, degrees and all that\<close>

lemma degree_eq_degreen0: "degree p = degreen p 0"
by (induct p rule: degree.induct) simp_all```
```--- a/src/HOL/Library/Float.thy	Thu Jul 09 00:39:49 2015 +0200
+++ b/src/HOL/Library/Float.thy	Thu Jul 09 00:40:57 2015 +0200
@@ -22,6 +22,7 @@
real_of_float_def[code_unfold]: "real \<equiv> real_of_float"

instance ..
+
end

lemma type_definition_float': "type_definition real float_of float"
@@ -34,7 +35,8 @@
declare [[coercion "real :: float \<Rightarrow> real"]]

lemma real_of_float_eq:
-  fixes f1 f2 :: float shows "f1 = f2 \<longleftrightarrow> real f1 = real f2"
+  fixes f1 f2 :: float
+  shows "f1 = f2 \<longleftrightarrow> real f1 = real f2"
unfolding real_of_float_def real_of_float_inject ..

lemma float_of_real[simp]: "float_of (real x) = x"
@@ -43,40 +45,63 @@
lemma real_float[simp]: "x \<in> float \<Longrightarrow> real (float_of x) = x"
unfolding real_of_float_def by (rule float_of_inverse)

+
subsection \<open>Real operations preserving the representation as floating point number\<close>

lemma floatI: fixes m e :: int shows "m * 2 powr e = x \<Longrightarrow> x \<in> float"
by (auto simp: float_def)

-lemma zero_float[simp]: "0 \<in> float" by (auto simp: float_def)
-lemma one_float[simp]: "1 \<in> float" by (intro floatI[of 1 0]) simp
-lemma numeral_float[simp]: "numeral i \<in> float" 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" 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" 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" 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
-lemma two_powr_numeral_float[simp]: "2 powr numeral i \<in> float" by (intro floatI[of 1 "numeral i"]) simp
-lemma two_powr_neg_numeral_float[simp]: "2 powr - numeral i \<in> float" 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 zero_float[simp]: "0 \<in> float"
+  by (auto simp: float_def)
+lemma one_float[simp]: "1 \<in> float"
+  by (intro floatI[of 1 0]) simp
+lemma numeral_float[simp]: "numeral i \<in> float"
+  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"
+  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"
+  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"
+  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
+lemma two_powr_numeral_float[simp]: "2 powr numeral i \<in> float"
+  by (intro floatI[of 1 "numeral i"]) simp
+lemma two_powr_neg_numeral_float[simp]: "2 powr - numeral i \<in> float"
+  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
proof (safe, simp)
-  fix e1 m1 e2 m2 :: int
-  { fix e1 m1 e2 m2 :: int assume "e1 \<le> e2"
-    then have "m1 * 2 powr e1 + m2 * 2 powr e2 = (m1 + m2 * 2 ^ nat (e2 - e1)) * 2 powr e1"
+  have *: "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
+    if "e1 \<le> e2" for e1 m1 e2 m2 :: int
+  proof -
+    from that have "m1 * 2 powr e1 + m2 * 2 powr e2 = (m1 + m2 * 2 ^ nat (e2 - e1)) * 2 powr e1"
by (simp add: powr_realpow[symmetric] powr_divide2[symmetric] field_simps)
-    then have "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
-      by blast }
-  note * = this
-  show "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
-  proof (cases e1 e2 rule: linorder_le_cases)
-    assume "e2 \<le> e1" from *[OF this, of m2 m1] show ?thesis by (simp add: ac_simps)
-  qed (rule *)
+    then show ?thesis
+      by blast
+  qed
+  fix e1 m1 e2 m2 :: int
+  consider "e2 \<le> e1" | "e1 \<le> e2" by (rule linorder_le_cases)
+  then show "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
+  proof cases
+    case 1
+    from *[OF this, of m2 m1] show ?thesis
+  next
+    case 2
+    then show ?thesis by (rule *)
+  qed
qed

lemma uminus_float[simp]: "x \<in> float \<Longrightarrow> -x \<in> float"
@@ -125,7 +150,8 @@
done

lemma div_numeral_Bit0_float[simp]:
-  assumes x: "x / numeral n \<in> float" shows "x / (numeral (Num.Bit0 n)) \<in> float"
+  assumes x: "x / numeral n \<in> float"
+  shows "x / (numeral (Num.Bit0 n)) \<in> float"
proof -
have "(x / numeral n) / 2^1 \<in> float"
by (intro x div_power_2_float)
@@ -135,32 +161,38 @@
qed

lemma div_neg_numeral_Bit0_float[simp]:
-  assumes x: "x / numeral n \<in> float" shows "x / (- numeral (Num.Bit0 n)) \<in> float"
+  assumes x: "x / numeral n \<in> float"
+  shows "x / (- numeral (Num.Bit0 n)) \<in> float"
proof -
-  have "- (x / numeral (Num.Bit0 n)) \<in> float" using x by simp
+  have "- (x / numeral (Num.Bit0 n)) \<in> float"
+    using x by simp
also have "- (x / numeral (Num.Bit0 n)) = x / - numeral (Num.Bit0 n)"
by simp
finally show ?thesis .
qed

-lemma power_float[simp]: assumes "a \<in> float" shows "a ^ b \<in> float"
+lemma power_float[simp]:
+  assumes "a \<in> float"
+  shows "a ^ b \<in> float"
proof -
-  from assms obtain m e::int where "a = m * 2 powr e"
+  from assms obtain m e :: int where "a = m * 2 powr e"
by (auto simp: float_def)
-  thus ?thesis
+  then show ?thesis
by (auto intro!: floatI[where m="m^b" and e = "e*b"]
simp: power_mult_distrib powr_realpow[symmetric] powr_powr)
qed

-lift_definition Float :: "int \<Rightarrow> int \<Rightarrow> float" is "\<lambda>(m::int) (e::int). m * 2 powr e" by simp
+lift_definition Float :: "int \<Rightarrow> int \<Rightarrow> float" is "\<lambda>(m::int) (e::int). m * 2 powr e"
+  by simp
declare Float.rep_eq[simp]

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)

code_datatype Float

+
subsection \<open>Arithmetic operations on floating point numbers\<close>

instantiation float :: "{ring_1, linorder, linordered_ring, linordered_idom, numeral, equal}"
@@ -192,16 +224,20 @@
declare less_float.rep_eq[simp]

instance
-  proof qed (transfer, fastforce simp add: field_simps intro: mult_left_mono mult_right_mono)+
+  by (standard; transfer; fastforce simp add: field_simps intro: mult_left_mono mult_right_mono)+
+
end

lemma Float_0_eq_0[simp]: "Float 0 e = 0"
by transfer simp

-lemma real_of_float_power[simp]: fixes f::float shows "real (f^n) = real f^n"
+lemma real_of_float_power[simp]:
+  fixes f :: float
+  shows "real (f^n) = real f^n"
by (induct n) simp_all

-lemma fixes x y::float
+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)"
@@ -219,9 +255,9 @@
apply transfer
apply simp
done
-  assume "a < b"
-  then show "\<exists>c. a < c \<and> c < b"
-    apply (intro exI[of _ "(a + b) * Float 1 (- 1)"])
+  show "\<exists>c. a < c \<and> c < b" if "a < b"
+    apply (rule exI[of _ "(a + b) * Float 1 (- 1)"])
+    using that
apply transfer
done
@@ -230,11 +266,11 @@
begin

-definition inf_float::"float\<Rightarrow>float\<Rightarrow>float"
-where "inf_float a b = min a b"
+definition inf_float :: "float \<Rightarrow> float \<Rightarrow> float"
+  where "inf_float a b = min a b"

-definition sup_float::"float\<Rightarrow>float\<Rightarrow>float"
-where "sup_float a b = max a b"
+definition sup_float :: "float \<Rightarrow> float \<Rightarrow> float"
+  where "sup_float a b = max a b"

instance
by (standard; transfer; simp add: inf_float_def sup_float_def real_of_float_min real_of_float_max)
@@ -250,20 +286,21 @@

lemma transfer_numeral [transfer_rule]:
"rel_fun (op =) pcr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)"
-  unfolding rel_fun_def float.pcr_cr_eq  cr_float_def by simp
+  by (simp add: rel_fun_def float.pcr_cr_eq cr_float_def)

lemma float_neg_numeral[simp]: "real (- numeral x :: float) = - numeral x"
by simp

lemma transfer_neg_numeral [transfer_rule]:
"rel_fun (op =) pcr_float (- numeral :: _ \<Rightarrow> real) (- numeral :: _ \<Rightarrow> float)"
-  unfolding rel_fun_def float.pcr_cr_eq cr_float_def by simp
+  by (simp add: rel_fun_def float.pcr_cr_eq cr_float_def)

lemma
shows float_of_numeral[simp]: "numeral k = float_of (numeral k)"
and float_of_neg_numeral[simp]: "- numeral k = float_of (- numeral k)"
unfolding real_of_float_eq by simp_all

+
subsection \<open>Quickcheck\<close>

instantiation float :: exhaustive
@@ -311,39 +348,51 @@
assumes H: "\<And>n. (\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i) \<Longrightarrow> P n"
shows "P j"
proof (induct "nat \<bar>j\<bar>" arbitrary: j rule: less_induct)
-  case less show ?case by (rule H[OF less]) simp
+  case less
+  show ?case by (rule H[OF less]) simp
qed

lemma int_cancel_factors:
-  fixes n :: int assumes "1 < r" shows "n = 0 \<or> (\<exists>k i. n = k * r ^ i \<and> \<not> r dvd k)"
+  fixes n :: int
+  assumes "1 < r"
+  shows "n = 0 \<or> (\<exists>k i. n = k * r ^ i \<and> \<not> r dvd k)"
proof (induct n rule: int_induct_abs)
case (less n)
-  { fix m assume n: "n \<noteq> 0" "n = m * r"
-    then have "\<bar>m \<bar> < \<bar>n\<bar>"
+  have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" if "n \<noteq> 0" "n = m * r" for m
+  proof -
+    from that have "\<bar>m \<bar> < \<bar>n\<bar>"
using \<open>1 < r\<close> by (simp add: abs_mult)
-    from less[OF this] n have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" by auto }
+    from less[OF this] that show ?thesis by auto
+  qed
then show ?case
by (metis dvd_def monoid_mult_class.mult.right_neutral mult.commute power_0)
qed

lemma mult_powr_eq_mult_powr_iff_asym:
fixes m1 m2 e1 e2 :: int
-  assumes m1: "\<not> 2 dvd m1" and "e1 \<le> e2"
+  assumes m1: "\<not> 2 dvd m1"
+    and "e1 \<le> e2"
shows "m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
+  (is "?lhs \<longleftrightarrow> ?rhs")
proof
-  have "m1 \<noteq> 0" using m1 unfolding dvd_def by auto
-  assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2"
-  with \<open>e1 \<le> e2\<close> have "m1 = m2 * 2 powr nat (e2 - e1)"
-    by (simp add: powr_divide2[symmetric] field_simps)
-  also have "\<dots> = m2 * 2^nat (e2 - e1)"
-  finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
-    unfolding real_of_int_inject .
-  with m1 have "m1 = m2"
-    by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
-  then show "m1 = m2 \<and> e1 = e2"
-    using eq \<open>m1 \<noteq> 0\<close> by (simp add: powr_inj)
-qed simp
+  show ?rhs if eq: ?lhs
+  proof -
+    have "m1 \<noteq> 0"
+      using m1 unfolding dvd_def by auto
+    from \<open>e1 \<le> e2\<close> eq have "m1 = m2 * 2 powr nat (e2 - e1)"
+      by (simp add: powr_divide2[symmetric] field_simps)
+    also have "\<dots> = m2 * 2^nat (e2 - e1)"
+    finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
+      unfolding real_of_int_inject .
+    with m1 have "m1 = m2"
+      by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
+    then show ?thesis
+      using eq \<open>m1 \<noteq> 0\<close> by (simp add: powr_inj)
+  qed
+  show ?lhs if ?rhs
+    using that by simp
+qed

lemma mult_powr_eq_mult_powr_iff:
fixes m1 m2 e1 e2 :: int
@@ -356,16 +405,18 @@
assumes x: "x \<in> float"
obtains (zero) "x = 0"
| (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0"
-proof atomize_elim
-  { assume "x \<noteq> 0"
-    from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def)
+proof -
+  {
+    assume "x \<noteq> 0"
+    from x obtain m e :: int where x: "x = m * 2 powr e"
+      by (auto simp: float_def)
with \<open>x \<noteq> 0\<close> int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k"
by auto
with \<open>\<not> 2 dvd k\<close> x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m"
by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"])
-  then show "x = 0 \<or> (\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m \<and> x \<noteq> 0)"
-    by blast
+  }
+  with that show thesis by blast
qed

lemma float_normed_cases:
@@ -373,7 +424,8 @@
obtains (zero) "f = 0"
| (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0"
proof (atomize_elim, induct f)
-  case (float_of y) then show ?case
+  case (float_of y)
+  then show ?case
by (cases rule: floatE_normed) (auto simp: zero_float_def)
qed

@@ -389,7 +441,8 @@
shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E)
and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M)
proof -
-  have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto
+  have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)"
+    by auto
then show ?E ?M
by (auto simp add: mantissa_def exponent_def zero_float_def)
qed
@@ -398,17 +451,20 @@
shows mantissa_exponent: "real 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)"
+  assume [simp]: "f \<noteq> float_of 0"
have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f"
proof (cases f rule: float_normed_cases)
+    case zero
+    then show ?thesis by  (simp add: zero_float_def)
+  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)"
+    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)"
by auto
then show ?thesis
unfolding exponent_def mantissa_def
by (rule someI2_ex) (simp add: zero_float_def)
+  qed
then show ?E ?D by auto
qed simp

@@ -422,31 +478,33 @@
shows mantissa_float: "mantissa f = m" (is "?M")
and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E")
proof cases
-  assume "m = 0" with dvd show "mantissa f = m" by auto
+  assume "m = 0"
+  with dvd show "mantissa f = m" by auto
next
assume "m \<noteq> 0"
then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def)
-  from mantissa_exponent[of f]
-  have "m * 2 powr e = mantissa f * 2 powr exponent f"
+  from mantissa_exponent[of f] have "m * 2 powr e = mantissa f * 2 powr exponent f"
then show "?M" "?E"
using mantissa_not_dvd[OF f_not_0] dvd
by (auto simp: mult_powr_eq_mult_powr_iff)
qed

+
subsection \<open>Compute arithmetic operations\<close>

lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"
unfolding real_of_float_eq mantissa_exponent[of f] by simp

-lemma Float_cases[case_names Float, cases type: float]:
+lemma Float_cases [cases type: float]:
fixes f :: float
obtains (Float) m e :: int where "f = Float m e"
using Float_mantissa_exponent[symmetric]
by (atomize_elim) auto

lemma denormalize_shift:
-  assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0"
+  assumes f_def: "f \<equiv> Float m e"
+    and not_0: "f \<noteq> float_of 0"
obtains i where "m = mantissa f * 2 ^ i" "e = exponent f - i"
proof
from mantissa_exponent[of f] f_def
@@ -481,87 +539,75 @@
unfolding real_of_int_inject by auto
qed

-lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
-  by transfer simp
-hide_fact (open) compute_float_zero
+context
+begin

-lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
+qualified lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
by transfer simp
-hide_fact (open) compute_float_one
+
+qualified lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
+  by transfer simp

lift_definition normfloat :: "float \<Rightarrow> float" is "\<lambda>x. x" .
lemma normloat_id[simp]: "normfloat x = x" by transfer rule

-lemma compute_normfloat[code]: "normfloat (Float m e) =
+qualified lemma compute_normfloat[code]: "normfloat (Float m e) =
(if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1))
else if m = 0 then 0 else Float m e)"
-hide_fact (open) compute_normfloat

-lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
+qualified lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
by transfer simp
-hide_fact (open) compute_float_numeral

-lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k"
+qualified lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k"
by transfer simp
-hide_fact (open) compute_float_neg_numeral

-lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
+qualified lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
by transfer simp
-hide_fact (open) compute_float_uminus

-lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"
+qualified lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"
-hide_fact (open) compute_float_times

-lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 =
+qualified lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 =
(if m1 = 0 then Float m2 e2 else if m2 = 0 then Float m1 e1 else
if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1
else Float (m2 + m1 * 2^nat (e1 - e2)) e2)"
by transfer (simp add: field_simps powr_realpow[symmetric] powr_divide2[symmetric])
-hide_fact (open) compute_float_plus

-lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)"
+qualified lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)"
by simp
-hide_fact (open) compute_float_minus

-lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"
+qualified lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"
-hide_fact (open) compute_float_sgn

lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" .

-lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"
+qualified lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"
by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])
-hide_fact (open) compute_is_float_pos

-lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)"
+qualified lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)"
-hide_fact (open) compute_float_less

lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" .

-lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"
+qualified lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"
by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])
-hide_fact (open) compute_is_float_nonneg

-lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)"
+qualified lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)"
-hide_fact (open) compute_float_le

lift_definition is_float_zero :: "float \<Rightarrow> bool"  is "op = 0 :: real \<Rightarrow> bool" .

-lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"
+qualified lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"
by transfer (auto simp add: is_float_zero_def)
-hide_fact (open) compute_is_float_zero

-lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e"
+qualified lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e"
-hide_fact (open) compute_float_abs

-lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)"
+qualified lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)"
by transfer simp
-hide_fact (open) compute_float_eq
+
+end

subsection \<open>Lemmas for types @{typ real}, @{typ nat}, @{typ int}\<close>
@@ -590,11 +636,11 @@

subsection \<open>Rounding Real Numbers\<close>

-definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where
-  "round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"
+definition round_down :: "int \<Rightarrow> real \<Rightarrow> real"
+  where "round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"

-definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where
-  "round_up prec x = ceiling (x * 2 powr prec) * 2 powr -prec"
+definition round_up :: "int \<Rightarrow> real \<Rightarrow> real"
+  where "round_up prec x = ceiling (x * 2 powr prec) * 2 powr -prec"

lemma round_down_float[simp]: "round_down prec x \<in> float"
unfolding round_down_def
@@ -692,7 +738,7 @@
from neg have "2 powr real p \<le> 2 powr 0"
by (intro powr_mono) auto
also have "\<dots> \<le> \<lfloor>2 powr 0::real\<rfloor>" by simp
-  also have "... \<le> \<lfloor>x * 2 powr (real p)\<rfloor>"
+  also have "\<dots> \<le> \<lfloor>x * 2 powr (real p)\<rfloor>"
unfolding real_of_int_le_iff
using x x_le by (intro floor_mono) (simp add: powr_minus_divide field_simps)
finally show ?thesis
@@ -707,9 +753,11 @@

subsection \<open>Rounding Floats\<close>

-definition div_twopow::"int \<Rightarrow> nat \<Rightarrow> int" where [simp]: "div_twopow x n = x div (2 ^ n)"
+definition div_twopow :: "int \<Rightarrow> nat \<Rightarrow> int"
+  where [simp]: "div_twopow x n = x div (2 ^ n)"

-definition mod_twopow::"int \<Rightarrow> nat \<Rightarrow> int" where [simp]: "mod_twopow x n = x mod (2 ^ n)"
+definition mod_twopow :: "int \<Rightarrow> nat \<Rightarrow> int"
+  where [simp]: "mod_twopow x n = x mod (2 ^ n)"

lemma compute_div_twopow[code]:
"div_twopow x n = (if x = 0 \<or> x = -1 \<or> n = 0 then x else div_twopow (x div 2) (n - 1))"
@@ -722,51 +770,54 @@
lift_definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" is round_up by simp
declare float_up.rep_eq[simp]

-lemma round_up_correct:
-  shows "round_up e f - f \<in> {0..2 powr -e}"
-unfolding atLeastAtMost_iff
+lemma round_up_correct: "round_up e f - f \<in> {0..2 powr -e}"
+  unfolding atLeastAtMost_iff
proof
-  have "round_up e f - f \<le> round_up e f - round_down e f" using round_down by simp
-  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
+  have "round_up e f - f \<le> round_up e f - round_down e f"
+    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)"
by simp

-lemma float_up_correct:
-  shows "real (float_up e f) - real f \<in> {0..2 powr -e}"
+lemma float_up_correct: "real (float_up e f) - real 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
declare float_down.rep_eq[simp]

-lemma round_down_correct:
-  shows "f - (round_down e f) \<in> {0..2 powr -e}"
-unfolding atLeastAtMost_iff
+lemma round_down_correct: "f - (round_down e f) \<in> {0..2 powr -e}"
+  unfolding atLeastAtMost_iff
proof
-  have "f - round_down e f \<le> round_up e f - round_down e f" using round_up by simp
-  also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
+  have "f - round_down e f \<le> round_up e f - round_down e f"
+    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)"
by simp

-lemma float_down_correct:
-  shows "real f - real (float_down e f) \<in> {0..2 powr -e}"
+lemma float_down_correct: "real f - real (float_down e f) \<in> {0..2 powr -e}"
by transfer (rule round_down_correct)

-lemma compute_float_down[code]:
+context
+begin
+
+qualified lemma compute_float_down[code]:
"float_down p (Float m e) =
(if p + e < 0 then Float (div_twopow m (nat (-(p + e)))) (-p) else Float m e)"
-proof cases
-  assume "p + e < 0"
-  hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
+proof (cases "p + e < 0")
+  case True
+  then have "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
using powr_realpow[of 2 "nat (-(p + e))"] by simp
-  also have "... = 1 / 2 powr p / 2 powr e"
+  also have "\<dots> = 1 / 2 powr p / 2 powr e"
finally show ?thesis
using \<open>p + e < 0\<close>
by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric])
next
-  assume "\<not> p + e < 0"
+  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"
by (auto intro: exI[where x="m*2^nat (e+p)"]
@@ -774,7 +825,6 @@
with \<open>\<not> p + e < 0\<close> show ?thesis
qed
-hide_fact (open) compute_float_down

lemma abs_round_down_le: "\<bar>f - (round_down e f)\<bar> \<le> 2 powr -e"
using round_down_correct[of f e] by simp
@@ -786,75 +836,100 @@
by (auto simp: round_down_def)

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)"
-proof cases
-  assume "\<not> b dvd a"
-  hence "a mod b \<noteq> 0" by auto
-  hence ne: "real (a mod b) / real b \<noteq> 0" using \<open>b \<noteq> 0\<close> by auto
+  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)"
+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)
+next
+  case False
+  then have "a mod b \<noteq> 0"
+    by auto
+  then have ne: "real (a mod b) / real 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"
-  apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric])
+    apply (rule ceiling_eq)
+    apply (auto simp: floor_divide_eq_div[symmetric])
proof -
-    have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp
+    have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b"
+      by simp
moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b"
-    apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne \<open>b \<noteq> 0\<close> by auto
+      apply (subst (2) real_of_int_div_aux)
+      unfolding floor_divide_eq_div
+      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
qed
-  thus ?thesis using \<open>\<not> b dvd a\<close> by simp
-qed (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)
+  then show ?thesis
+    using \<open>\<not> b dvd a\<close> by simp
+qed

-lemma compute_float_up[code]:
-  "float_up p x = - float_down p (-x)"
+qualified lemma compute_float_up[code]: "float_up p x = - float_down p (-x)"
-hide_fact (open) compute_float_up
+
+end

subsection \<open>Compute bitlen of integers\<close>

-definition bitlen :: "int \<Rightarrow> int" where
-  "bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"
+definition bitlen :: "int \<Rightarrow> int"
+  where "bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"

lemma bitlen_nonneg: "0 \<le> bitlen x"
proof -
-  {
-    assume "0 > x"
-    have "-1 = log 2 (inverse 2)" by (subst log_inverse) simp_all
-    also have "... < log 2 (-x)" using \<open>0 > x\<close> by auto
-    finally have "-1 < log 2 (-x)" .
-  } thus "0 \<le> bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)
+  have "-1 < log 2 (-x)" if "0 > x"
+  proof -
+    have "-1 = log 2 (inverse 2)"
+      by (subst log_inverse) simp_all
+    also have "\<dots> < log 2 (-x)"
+      using \<open>0 > x\<close> by auto
+    finally show ?thesis .
+  qed
+  then show ?thesis
+    unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)
qed

lemma bitlen_bounds:
assumes "x > 0"
shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen 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))"]
-    by simp
-  also have "... \<le> 2 powr log 2 (real x)"
-    by simp
-  also have "... = real x"
-    using \<open>0 < x\<close> by simp
-  finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x" by simp
-  thus "2 ^ nat (bitlen x - 1) \<le> x" using \<open>x > 0\<close>
-next
-  have "x \<le> 2 powr (log 2 x)" using \<open>x > 0\<close> by simp
-  also have "... < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)"
-    using \<open>x > 0\<close> by simp
-  finally show "x < 2 ^ nat (bitlen x)" using \<open>x > 0\<close>
-    by (simp add: bitlen_def ac_simps)
+  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))"]
+      by simp
+    also have "\<dots> \<le> 2 powr log 2 (real x)"
+      by simp
+    also have "\<dots> = real x"
+      using \<open>0 < x\<close> by simp
+    finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x"
+      by simp
+    then show ?thesis
+      using \<open>0 < x\<close> by (simp add: bitlen_def)
+  qed
+  show "x < 2 ^ nat (bitlen x)"
+  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)"
+      using \<open>x > 0\<close> apply simp
+      done
+    finally show ?thesis
+      using \<open>x > 0\<close> by (simp add: bitlen_def ac_simps)
+  qed
qed

lemma bitlen_pow2[simp]:
assumes "b > 0"
shows "bitlen (b * 2 ^ c) = bitlen b + c"
proof -
-  from assms have "b * 2 ^ c > 0" by auto
-  thus ?thesis
+  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)
qed
@@ -868,9 +943,9 @@
then show ?thesis by (simp add: f_def bitlen_def Float_def)
next
case False
-  hence "f \<noteq> float_of 0"
+  then have "f \<noteq> float_of 0"
unfolding real_of_float_eq by (simp add: f_def)
-  hence "mantissa f \<noteq> 0"
+  then have "mantissa f \<noteq> 0"
moreover
obtain i where "m = mantissa f * 2 ^ i" "e = exponent f - int i"
@@ -878,22 +953,27 @@
ultimately show ?thesis by (simp add: abs_mult)
qed

-lemma compute_bitlen[code]:
-  shows "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"
+context
+begin
+
+qualified lemma compute_bitlen[code]: "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"
proof -
{ assume "2 \<le> x"
then have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 (x - x mod 2)\<rfloor>"
also have "\<dots> = \<lfloor>log 2 (real x)\<rfloor>"
-    proof cases
-      assume "x mod 2 = 0" then show ?thesis by simp
+    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>"
then have "0 \<le> n"
using \<open>2 \<le> x\<close> by simp
-      assume "x mod 2 \<noteq> 0"
-      with \<open>2 \<le> x\<close> have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)
-      with \<open>2 \<le> x\<close> have "x \<noteq> 2^nat n" by (cases "nat n") auto
+      from \<open>2 \<le> x\<close> False have "x mod 2 = 1" "\<not> 2 dvd x"
+        by (auto simp add: dvd_eq_mod_eq_0)
+      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)"
@@ -922,62 +1002,87 @@
unfolding bitlen_def
by (auto simp: pos_imp_zdiv_pos_iff not_le)
qed
-hide_fact (open) compute_bitlen
+
+end

lemma float_gt1_scale: assumes "1 \<le> Float m e"
shows "0 \<le> e + (bitlen m - 1)"
proof -
have "0 < Float m e" using assms by auto
-  hence "0 < m" using powr_gt_zero[of 2 e]
+  then have "0 < m" using powr_gt_zero[of 2 e]
apply (auto simp: zero_less_mult_iff)
-    using not_le powr_ge_pzero by blast
-  hence "m \<noteq> 0" by auto
+    using not_le powr_ge_pzero apply blast
+    done
+  then have "m \<noteq> 0" by auto
show ?thesis
proof (cases "0 \<le> e")
-    case True thus ?thesis using \<open>0 < m\<close>  by (simp add: bitlen_def)
+    case True
+    then show ?thesis
+      using \<open>0 < m\<close> by (simp add: bitlen_def)
next
+    case False
have "(1::int) < 2" by simp
-    case False let ?S = "2^(nat (-e))"
-    have "inverse (2 ^ nat (- e)) = 2 powr e" using assms False powr_realpow[of 2 "nat (-e)"]
+    let ?S = "2^(nat (-e))"
+    have "inverse (2 ^ nat (- e)) = 2 powr e"
+      using assms False powr_realpow[of 2 "nat (-e)"]
by (auto simp: powr_minus field_simps)
-    hence "1 \<le> real m * inverse ?S" using assms False powr_realpow[of 2 "nat (-e)"]
+    then have "1 \<le> real m * inverse ?S"
+      using assms False powr_realpow[of 2 "nat (-e)"]
by (auto simp: powr_minus)
-    hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
-    hence "?S \<le> real m" unfolding mult.assoc by auto
-    hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
+    then have "1 * ?S \<le> real m * inverse ?S * ?S"
+      by (rule mult_right_mono) auto
+    then have "?S \<le> real m"
+      unfolding mult.assoc by auto
+    then have "?S \<le> m"
+      unfolding real_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]
+    have "nat (-e) < (nat (bitlen m))"
+      unfolding power_strict_increasing_iff[OF \<open>1 < 2\<close>, symmetric]
by (rule order_le_less_trans)
-    hence "-e < bitlen m" using False by auto
-    thus ?thesis by auto
+    then have "-e < bitlen m"
+      using False by auto
+    then show ?thesis
+      by auto
qed
qed

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 m / 2^nat (bitlen m - 1)"
+    and "real 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>] ..
-  hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
-  thus "1 \<le> real m / ?B" by auto
+  then have "1 * ?B \<le> real m"
+    unfolding real_of_int_le_iff[symmetric] by auto
+  then show "1 \<le> real m / ?B"
+    by auto

-  have "m \<noteq> 0" using assms by auto
-  have "0 \<le> bitlen m - 1" using \<open>0 < m\<close> by (auto simp: bitlen_def)
+  have "m \<noteq> 0"
+    using assms by auto
+  have "0 \<le> bitlen m - 1"
+    using \<open>0 < m\<close> by (auto simp: bitlen_def)

-  have "m < 2^nat(bitlen m)" using bitlen_bounds[OF \<open>0 <m\<close>] ..
-  also have "\<dots> = 2^nat(bitlen m - 1 + 1)" 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
-  hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
-  thus "real m / ?B < 2" by auto
+  have "m < 2^nat(bitlen m)"
+    using bitlen_bounds[OF \<open>0 <m\<close>] ..
+  also have "\<dots> = 2^nat(bitlen m - 1 + 1)"
+    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"
+    by (rule divide_strict_right_mono) auto
+  then show "real m / ?B < 2"
+    by auto
qed

+
subsection \<open>Truncating Real Numbers\<close>

-definition truncate_down::"nat \<Rightarrow> real \<Rightarrow> real" where
-  "truncate_down prec x = round_down (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"
+definition truncate_down::"nat \<Rightarrow> real \<Rightarrow> real"
+  where "truncate_down prec x = round_down (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"

lemma truncate_down: "truncate_down prec x \<le> x"
using round_down by (simp add: truncate_down_def)
@@ -991,8 +1096,8 @@
lemma truncate_down_float[simp]: "truncate_down p x \<in> float"
by (auto simp: truncate_down_def)

-definition truncate_up::"nat \<Rightarrow> real \<Rightarrow> real" where
-  "truncate_up prec x = round_up (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"
+definition truncate_up::"nat \<Rightarrow> real \<Rightarrow> real"
+  where "truncate_up prec x = round_up (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"

lemma truncate_up: "x \<le> truncate_up prec x"
using round_up by (simp add: truncate_up_def)
@@ -1035,22 +1140,28 @@
by (auto simp: truncate_up_def round_up_def intro!: mult_nonpos_nonneg)

lemma truncate_up_le1:
-  assumes "x \<le> 1" "1 \<le> p" shows "truncate_up p x \<le> 1"
+  assumes "x \<le> 1" "1 \<le> p"
+  shows "truncate_up p x \<le> 1"
proof -
-  {
-    assume "x \<le> 0"
-    with truncate_up_nonpos[OF this, of p] have ?thesis by simp
-  } moreover {
-    assume "x > 0"
-    hence le: "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<le> 0"
+  consider "x \<le> 0" | "x > 0"
+    by arith
+  then show ?thesis
+  proof cases
+    case 1
+    with truncate_up_nonpos[OF this, of p] show ?thesis
+      by simp
+  next
+    case 2
+    then have le: "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<le> 0"
using assms by (auto simp: log_less_iff)
from assms have "1 \<le> int p" by simp
-    have ?thesis using assms
-  } ultimately show ?thesis by arith
+    show ?thesis
+  qed
qed

+
subsection \<open>Truncating Floats\<close>

lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_up
@@ -1078,25 +1189,30 @@
and minus_float_round_down_eq: "- float_round_down prec x = float_round_up prec (- x)"
by (transfer, simp add: truncate_down_uminus_eq truncate_up_uminus_eq)+

-lemma compute_float_round_down[code]:
+context
+begin
+
+qualified lemma compute_float_round_down[code]:
"float_round_down prec (Float m e) = (let d = bitlen (abs m) - int prec in
if 0 < d then Float (div_twopow m (nat d)) (e + d)
else Float m e)"
using Float.compute_float_down[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
by transfer (simp add: field_simps abs_mult log_mult bitlen_def truncate_down_def
cong del: if_weak_cong)
-hide_fact (open) compute_float_round_down

-lemma compute_float_round_up[code]:
+qualified lemma compute_float_round_up[code]:
"float_round_up prec x = - float_round_down prec (-x)"
-hide_fact (open) compute_float_round_up
+
+end

subsection \<open>Approximation of positive rationals\<close>

-lemma div_mult_twopow_eq: fixes a b::nat shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)"
-  by (cases "b=0") (simp_all add: div_mult2_eq[symmetric] ac_simps)
+lemma div_mult_twopow_eq:
+  fixes a b :: nat
+  shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)"
+  by (cases "b = 0") (simp_all add: div_mult2_eq[symmetric] ac_simps)

lemma real_div_nat_eq_floor_of_divide:
fixes a b :: nat
@@ -1106,23 +1222,29 @@
definition "rat_precision prec x y = int prec - (bitlen x - bitlen y)"

lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
-  is "\<lambda>prec (x::nat) (y::nat). round_down (rat_precision prec x y) (x / y)" by simp
+  is "\<lambda>prec (x::nat) (y::nat). round_down (rat_precision prec x y) (x / y)"
+  by simp

-lemma compute_lapprox_posrat[code]:
+context
+begin
+
+qualified lemma compute_lapprox_posrat[code]:
fixes prec x y
shows "lapprox_posrat prec x y =
(let
-       l = rat_precision prec x y;
-       d = if 0 \<le> l then x * 2^nat l div y else x div 2^nat (- l) div y
+      l = rat_precision prec x y;
+      d = if 0 \<le> l then x * 2^nat l div y else x div 2^nat (- l) div y
in normfloat (Float d (- l)))"
unfolding div_mult_twopow_eq
by transfer
(simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def
del: two_powr_minus_int_float)
-hide_fact (open) compute_lapprox_posrat
+
+end

lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
-  is "\<lambda>prec (x::nat) (y::nat). round_up (rat_precision prec x y) (x / y)" by simp
+  is "\<lambda>prec (x::nat) (y::nat). round_up (rat_precision prec x y) (x / y)" by
+  simp

context
notes divmod_int_mod_div[simp]
@@ -1137,14 +1259,16 @@
(d, m) = divmod_int (fst X) (snd X)
in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) (- l)))"
proof (cases "y = 0")
-  assume "y = 0" thus ?thesis by transfer simp
+  assume "y = 0"
+  then show ?thesis by transfer simp
next
assume "y \<noteq> 0"
show ?thesis
proof (cases "0 \<le> l")
-    assume "0 \<le> l"
+    case True
def x' \<equiv> "x * 2 ^ nat l"
-    have "int x * 2 ^ nat l = x'" by (simp add: x'_def int_mult int_power)
+    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'"
by (simp add: powr_realpow[symmetric] \<open>0 \<le> l\<close> x'_def)
ultimately show ?thesis
@@ -1152,7 +1276,7 @@
l_def[symmetric, THEN meta_eq_to_obj_eq]
by transfer (auto simp add: floor_divide_eq_div [symmetric] round_up_def)
next
-    assume "\<not> 0 \<le> l"
+    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)
@@ -1170,38 +1294,48 @@
end

lemma rat_precision_pos:
-  assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
+  assumes "0 \<le> x"
+    and "0 < y"
+    and "2 * x < y"
+    and "0 < n"
shows "rat_precision n (int x) (int y) > 0"
proof -
-  { assume "0 < x" hence "log 2 x + 1 = log 2 (2 * x)" by (simp add: log_mult) }
-  hence "bitlen (int x) < bitlen (int y)" using assms
+  have "0 < x \<Longrightarrow> log 2 x + 1 = log 2 (2 * x)"
+  then have "bitlen (int x) < bitlen (int y)"
+    using assms
-  thus ?thesis
+  then show ?thesis
+    using assms
qed

lemma rapprox_posrat_less1:
-  shows "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 (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
-  "\<lambda>prec (x::int) (y::int). round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
+  "\<lambda>prec (x::int) (y::int). round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)"
+  by simp

-lemma compute_lapprox_rat[code]:
+context
+begin
+
+qualified lemma compute_lapprox_rat[code]:
"lapprox_rat prec x y =
-    (if y = 0 then 0
+   (if y = 0 then 0
else if 0 \<le> x then
-      (if 0 < y then lapprox_posrat prec (nat x) (nat y)
+     (if 0 < y then lapprox_posrat prec (nat x) (nat y)
else - (rapprox_posrat prec (nat x) (nat (-y))))
else (if 0 < y
then - (rapprox_posrat prec (nat (-x)) (nat y))
else lapprox_posrat prec (nat (-x)) (nat (-y))))"
by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
-hide_fact (open) compute_lapprox_rat

lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
-  "\<lambda>prec (x::int) (y::int). round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
+  "\<lambda>prec (x::int) (y::int). round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)"
+  by simp

lemma "rapprox_rat = rapprox_posrat"
by transfer auto
@@ -1209,10 +1343,12 @@
lemma "lapprox_rat = lapprox_posrat"
by transfer auto

-lemma compute_rapprox_rat[code]:
+qualified lemma compute_rapprox_rat[code]:
"rapprox_rat prec x y = - lapprox_rat prec (-x) y"
-hide_fact (open) compute_rapprox_rat
+
+end
+

subsection \<open>Division\<close>

@@ -1223,47 +1359,58 @@
lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divl

-lemma compute_float_divl[code]:
+context
+begin
+
+qualified lemma compute_float_divl[code]:
"float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
-proof cases
+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)"
-  assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
-  then 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)"
+  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"
-
-  show ?thesis
-    using not_0
+  from True show ?thesis
by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift real_divl_def,
-qed (transfer, auto simp: real_divl_def)
-hide_fact (open) compute_float_divl
+next
+  case False
+  then show ?thesis by transfer (auto simp: real_divl_def)
+qed

lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divr

-lemma compute_float_divr[code]:
+qualified lemma compute_float_divr[code]:
"float_divr prec x y = - float_divl prec (-x) y"
by transfer (simp add: real_divr_def real_divl_def round_down_uminus_eq)
-hide_fact (open) compute_float_divr
+
+end

subsection \<open>Approximate Power\<close>

-lemma div2_less_self[termination_simp]: fixes n::nat shows "odd n \<Longrightarrow> n div 2 < n"
+lemma div2_less_self[termination_simp]:
+  fixes n :: nat
+  shows "odd n \<Longrightarrow> n div 2 < n"

-fun power_down :: "nat \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real" where
+fun power_down :: "nat \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real"
+where
"power_down p x 0 = 1"
| "power_down p x (Suc n) =
-    (if odd n then truncate_down (Suc p) ((power_down p x (Suc n div 2))\<^sup>2) else truncate_down (Suc p) (x * power_down p x n))"
+    (if odd n then truncate_down (Suc p) ((power_down p x (Suc n div 2))\<^sup>2)
+     else truncate_down (Suc p) (x * power_down p x n))"

-fun power_up :: "nat \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real" where
+fun power_up :: "nat \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real"
+where
"power_up p x 0 = 1"
| "power_up p x (Suc n) =
-    (if odd n then truncate_up p ((power_up p x (Suc n div 2))\<^sup>2) else truncate_up p (x * power_up p x n))"
+    (if odd n then truncate_up p ((power_up p x (Suc n div 2))\<^sup>2)
+     else truncate_up p (x * power_up p x n))"

lift_definition power_up_fl :: "nat \<Rightarrow> float \<Rightarrow> nat \<Rightarrow> float" is power_up
by (induct_tac rule: power_up.induct) simp_all
@@ -1279,11 +1426,13 @@
lemma compute_power_up_fl[code]:
"power_up_fl p x 0 = 1"
"power_up_fl p x (Suc n) =
-    (if odd n then float_round_up p ((power_up_fl p x (Suc n div 2))\<^sup>2) else float_round_up p (x * power_up_fl p x n))"
+    (if odd n then float_round_up p ((power_up_fl p x (Suc n div 2))\<^sup>2)
+     else float_round_up p (x * power_up_fl p x n))"
and compute_power_down_fl[code]:
"power_down_fl p x 0 = 1"
"power_down_fl p x (Suc n) =
-    (if odd n then float_round_down (Suc p) ((power_down_fl p x (Suc n div 2))\<^sup>2) else float_round_down (Suc p) (x * power_down_fl p x n))"
+    (if odd n then float_round_down (Suc p) ((power_down_fl p x (Suc n div 2))\<^sup>2)
+     else float_round_down (Suc p) (x * power_down_fl p x n))"
unfolding atomize_conj
by transfer simp

@@ -1300,7 +1449,7 @@
case (2 p x n)
{
assume "odd n"
-    hence "(power_down p x (Suc n div 2)) ^ 2 \<le> (x ^ (Suc n div 2)) ^ 2"
+    then have "(power_down p x (Suc n div 2)) ^ 2 \<le> (x ^ (Suc n div 2)) ^ 2"
using 2
by (auto intro: power_mono power_down_nonneg simp del: odd_Suc_div_two)
also have "\<dots> = x ^ (Suc n div 2 * 2)"
@@ -1310,7 +1459,8 @@
finally have ?case
using \<open>odd n\<close>
by (auto intro!: truncate_down_le simp del: odd_Suc_div_two)
-  } thus ?case
+  }
+  then show ?case
by (auto intro!: truncate_down_le mult_left_mono 2 mult_nonneg_nonneg power_down_nonneg)
qed simp

@@ -1319,9 +1469,9 @@
case (2 p x n)
{
assume "odd n"
-    hence "Suc n = Suc n div 2 * 2"
+    then have "Suc n = Suc n div 2 * 2"
using \<open>odd n\<close> even_Suc by presburger
-    hence "x ^ Suc n \<le> (x ^ (Suc n div 2))\<^sup>2"
+    then have "x ^ Suc n \<le> (x ^ (Suc n div 2))\<^sup>2"
also have "\<dots> \<le> (power_up p x (Suc n div 2))\<^sup>2"
using 2 \<open>odd n\<close>
@@ -1329,7 +1479,8 @@
finally have ?case
using \<open>odd n\<close>
by (auto intro!: truncate_up_le simp del: odd_Suc_div_two )
-  } thus ?case
+  }
+  then show ?case
by (auto intro!: truncate_up_le mult_left_mono 2)
qed simp

@@ -1383,8 +1534,7 @@
using truncate_down_uminus_eq[of p "x + y"]
by (auto simp: plus_down_def plus_up_def)

-lemma
-  truncate_down_log2_eqI:
+lemma truncate_down_log2_eqI:
assumes "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
assumes "\<lfloor>x * 2 powr (p - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)\<rfloor> = \<lfloor>y * 2 powr (p - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)\<rfloor>"
shows "truncate_down p x = truncate_down p y"
@@ -1395,40 +1545,43 @@
zero_less_one)

-lemma
-  sum_neq_zeroI:
-  fixes a k::real
+lemma sum_neq_zeroI:
+  fixes a k :: real
shows "abs a \<ge> k \<Longrightarrow> abs b < k \<Longrightarrow> a + b \<noteq> 0"
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>)"
-proof cases
-  assume "m2 \<noteq> 0"
-  hence "\<bar>m2\<bar> < 2 ^ nat (bitlen \<bar>m2\<bar>)"
+lemma abs_real_le_2_powr_bitlen[simp]: "\<bar>real m2\<bar> < 2 powr real (bitlen \<bar>m2\<bar>)"
+proof (cases "m2 = 0")
+  case True
+  then show ?thesis by simp
+next
+  case False
+  then have "\<bar>m2\<bar> < 2 ^ nat (bitlen \<bar>m2\<bar>)"
using bitlen_bounds[of "\<bar>m2\<bar>"]
-  thus ?thesis
+  then show ?thesis
by (simp add: powr_int bitlen_nonneg real_of_int_less_iff[symmetric])
-qed simp
+qed

lemma floor_sum_times_2_powr_sgn_eq:
-  fixes ai p q::int
-  and a b::real
+  fixes ai p q :: int
+    and a b :: real
assumes "a * 2 powr p = ai"
-  assumes b_le_1: "abs (b * 2 powr (p + 1)) \<le> 1"
-  assumes leqp: "q \<le> p"
+    and b_le_1: "abs (b * 2 powr (p + 1)) \<le> 1"
+    and leqp: "q \<le> p"
shows "\<lfloor>(a + b) * 2 powr q\<rfloor> = \<lfloor>(2 * ai + sgn b) * 2 powr (q - p - 1)\<rfloor>"
proof -
-  {
-    assume "b = 0"
-    hence ?thesis
+  consider "b = 0" | "b > 0" | "b < 0" by arith
+  then show ?thesis
+  proof cases
+    case 1
+    then show ?thesis
-  } moreover {
-    assume "b > 0"
-    hence "b * 2 powr p < abs (b * 2 powr (p + 1))" by simp
+  next
+    case 2
+    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" .

@@ -1455,13 +1608,12 @@
by (simp del: real_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>"
-        .
-    } ultimately have ?thesis by simp
-  } moreover {
-    assume "\<not> 0 \<le> b"
-    hence "0 > b" by simp
-    hence floor_eq: "\<lfloor>b * 2 powr (real p + 1)\<rfloor> = -1"
+          \<lfloor>real ai / real ((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"
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)
@@ -1479,22 +1631,26 @@
del: real_of_int_mult real_of_int_power real_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 have ?thesis using \<open>b < 0\<close> by simp
-  } ultimately show ?thesis by arith
+    finally show ?thesis
+      using \<open>b < 0\<close> by simp
+  qed
qed

-lemma
-  fixes ai::int and b::real
-  assumes "abs b \<le> 1/2" "ai \<noteq> 0"
+  fixes ai :: int
+    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>"
-proof cases
-  assume "b = 0" thus ?thesis by simp
+proof (cases "b = 0")
+  case True
+  then show ?thesis by simp
next
-  assume "b \<noteq> 0"
+  case False
def k \<equiv> "\<lfloor>log 2 \<bar>ai\<bar>\<rfloor>"
-  hence "\<lfloor>log 2 \<bar>ai\<bar>\<rfloor> = k" by simp
-  hence k: "2 powr k \<le> \<bar>ai\<bar>" "\<bar>ai\<bar> < 2 powr (k + 1)"
+  then have "\<lfloor>log 2 \<bar>ai\<bar>\<rfloor> = k"
+    by simp
+  then have k: "2 powr k \<le> \<bar>ai\<bar>" "\<bar>ai\<bar> < 2 powr (k + 1)"
by (simp_all add: floor_log_eq_powr_iff \<open>ai \<noteq> 0\<close>)
have "k \<ge> 0"
using assms by (auto simp: k_def)
@@ -1502,7 +1658,7 @@
have r: "0 \<le> r" "r < 2 powr k"
using \<open>k \<ge> 0\<close> k
by (auto simp: r_def k_def algebra_simps powr_add abs_if powr_int)
-  hence "r \<le> (2::int) ^ nat k - 1"
+  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>
have r_le: "r \<le> 2 powr k - 1"
@@ -1511,7 +1667,7 @@
have "\<bar>ai\<bar> = 2 powr k + r"
using \<open>k \<ge> 0\<close> by (auto simp: k_def r_def powr_realpow[symmetric])

-  have pos: "\<And>b::real. abs b < 1 \<Longrightarrow> 0 < 2 powr k + (r + b)"
+  have pos: "abs b < 1 \<Longrightarrow> 0 < 2 powr k + (r + b)" for b :: real
using \<open>0 \<le> k\<close> \<open>ai \<noteq> 0\<close>
by (auto simp add: r_def powr_realpow[symmetric] abs_if sgn_if algebra_simps
split: split_if_asm)
@@ -1557,8 +1713,12 @@
finally show ?thesis .
qed

-lemma compute_far_float_plus_down:
-  fixes m1 e1 m2 e2::int and p::nat
+context
+begin
+
+qualified lemma compute_far_float_plus_down:
+  fixes m1 e1 m2 e2 :: int
+    and p :: nat
defines "k1 \<equiv> p - nat (bitlen \<bar>m1\<bar>)"
assumes H: "bitlen \<bar>m2\<bar> \<le> e1 - e2 - k1 - 2" "m1 \<noteq> 0" "m2 \<noteq> 0" "e1 \<ge> e2"
shows "float_plus_down p (Float m1 e1) (Float m2 e2) =
@@ -1584,7 +1744,7 @@
finally have abs_m2_less_half: "\<bar>?m2\<bar> < 1 / 2"
by simp

-  hence "\<bar>real m2\<bar> < 2 powr -(?shift + 1)"
+  then have "\<bar>real 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)"
by simp
@@ -1593,7 +1753,7 @@
also have "1/4 < \<bar>real 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)
-  hence a_half_less_sum: "\<bar>?a\<bar> / 2 < \<bar>?sum\<bar>"
+  then have a_half_less_sum: "\<bar>?a\<bar> / 2 < \<bar>?sum\<bar>"
by (auto simp: field_simps abs_if split: split_if_asm)

from b_less_half_a have "\<bar>?b\<bar> < \<bar>?a\<bar>" "\<bar>?b\<bar> \<le> \<bar>?a\<bar>"
@@ -1602,14 +1762,14 @@
have "\<bar>real (Float m1 e1)\<bar> \<ge> 1/4 * 2 powr real e1"
using \<open>m1 \<noteq> 0\<close>
by (auto simp: powr_add powr_int bitlen_nonneg divide_right_mono abs_mult)
-  hence "?sum \<noteq> 0" using b_less_quarter
+  then have "?sum \<noteq> 0" using b_less_quarter
by (rule sum_neq_zeroI)
-  hence "?m1 + ?m2 \<noteq> 0"
+  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"
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)
-  hence sum'_nz: "?m1 + ?m2' \<noteq> 0"
+  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"
@@ -1624,13 +1784,13 @@
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)
-  hence "\<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 (Float (?m1 * 2 + sgn m2) (?e - 1))\<bar>\<rfloor>"
using \<open>?m1 + ?m2' \<noteq> 0\<close>
finally
have "\<lfloor>log 2 \<bar>?sum\<bar>\<rfloor> = \<lfloor>log 2 \<bar>real (Float (?m1*2 + sgn m2) (?e - 1))\<bar>\<rfloor>" .
-  hence "plus_down p (Float m1 e1) (Float m2 e2) =
+  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)
@@ -1668,14 +1828,14 @@
finally
show "\<lfloor>(?a + ?b) * 2 powr ?f\<rfloor> = \<lfloor>real (Float (?m1 * 2 + sgn m2) (?e - 1)) * 2 powr ?f\<rfloor>" .
qed
-  thus ?thesis
+  then show ?thesis
by transfer (simp add: plus_down_def ac_simps Let_def)
qed

lemma compute_float_plus_down_naive[code]: "float_plus_down p x y = float_round_down p (x + y)"
by transfer (auto simp: plus_down_def)

-lemma compute_float_plus_down[code]:
+qualified lemma compute_float_plus_down[code]:
fixes p::nat and m1 e1 m2 e2::int
shows "float_plus_down p (Float m1 e1) (Float m2 e2) =
(if m1 = 0 then float_round_down p (Float m2 e2)
@@ -1689,53 +1849,66 @@
else float_plus_down p (Float m2 e2) (Float m1 e1)))"
proof -
{
-    assume H: "bitlen \<bar>m2\<bar> \<le> e1 - e2 - (p - nat (bitlen \<bar>m1\<bar>)) - 2" "m1 \<noteq> 0" "m2 \<noteq> 0" "e1 \<ge> e2"
-    note compute_far_float_plus_down[OF H]
+    assume "bitlen \<bar>m2\<bar> \<le> e1 - e2 - (p - nat (bitlen \<bar>m1\<bar>)) - 2" "m1 \<noteq> 0" "m2 \<noteq> 0" "e1 \<ge> e2"
+    note compute_far_float_plus_down[OF this]
}
-  thus ?thesis
+  then show ?thesis
by transfer (simp add: Let_def plus_down_def ac_simps)
qed
-hide_fact (open) compute_far_float_plus_down
-hide_fact (open) compute_float_plus_down

-lemma compute_float_plus_up[code]: "float_plus_up p x y = - float_plus_down p (-x) (-y)"
+qualified lemma compute_float_plus_up[code]: "float_plus_up p x y = - float_plus_down p (-x) (-y)"
using truncate_down_uminus_eq[of p "x + y"]
by transfer (simp add: plus_down_def plus_up_def ac_simps)
-hide_fact (open) compute_float_plus_up

lemma mantissa_zero[simp]: "mantissa 0 = 0"
-by (metis mantissa_0 zero_float.abs_eq)
+  by (metis mantissa_0 zero_float.abs_eq)
+
+end

subsection \<open>Lemmas needed by Approximate\<close>

-lemma Float_num[simp]: shows
-   "real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
-   "real (Float 1 (- 1)) = 1/2" and "real (Float 1 (- 2)) = 1/4" and "real (Float 1 (- 3)) = 1/8" and
-   "real (Float (- 1) 0) = -1" and "real (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 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"
+  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" by simp
+lemma real_of_Float_int[simp]: "real (Float n 0) = real n"
+  by simp

-lemma float_zero[simp]: "real (Float 0 e) = 0" by simp
+lemma float_zero[simp]: "real (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
+  by arith

-lemma lapprox_rat:
-  shows "real (lapprox_rat prec x y) \<le> real x / real y"
+lemma lapprox_rat: "real (lapprox_rat prec x y) \<le> real x / real y"
using round_down by (simp add: lapprox_rat_def)

-lemma mult_div_le: fixes a b:: int assumes "b > 0" shows "a \<ge> b * (a div b)"
+lemma mult_div_le:
+  fixes a b :: int
+  assumes "b > 0"
+  shows "a \<ge> b * (a div b)"
proof -
-  from zmod_zdiv_equality'[of a b]
-  have "a = b * (a div b) + a mod b" by simp
-  also have "... \<ge> b * (a div b) + 0" apply (rule add_left_mono) apply (rule pos_mod_sign)
-  using assms by simp
-  finally show ?thesis by simp
+  from zmod_zdiv_equality'[of a b] have "a = b * (a div b) + a mod b"
+    by simp
+  also have "\<dots> \<ge> b * (a div b) + 0"
+    apply (rule pos_mod_sign)
+    using assms apply simp
+    done
+  finally show ?thesis
+    by simp
qed

lemma lapprox_rat_nonneg:
@@ -1758,12 +1931,10 @@
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 (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 (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"
@@ -1793,17 +1964,32 @@

lemma mantissa_eq_zero_iff: "mantissa x = 0 \<longleftrightarrow> x = 0"
+  (is "?lhs \<longleftrightarrow> ?rhs")
proof
-  assume "mantissa x = 0" hence z: "0 = real x" using mantissa_exponent by simp
-  show "x = 0" by (simp add: zero_float_def z)
+  show ?rhs if ?lhs
+  proof -
+    from that have z: "0 = real x"
+      using mantissa_exponent by simp
+    show ?thesis
+      by (simp add: zero_float_def z)
+  qed
+  show ?lhs if ?rhs
+    using that by (simp add: zero_float_def)
+qed

lemma float_upper_bound: "x \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x)"
-proof (cases "x = 0", simp)
-  assume "x \<noteq> 0" hence "mantissa x \<noteq> 0" using mantissa_eq_zero_iff by auto
-  have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent)
-  also have "mantissa x \<le> \<bar>mantissa x\<bar>" by simp
-  also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"
+proof (cases "x = 0")
+  case True
+  then show ?thesis by simp
+next
+  case False
+  then have "mantissa x \<noteq> 0"
+    using mantissa_eq_zero_iff by auto
+  have "x = mantissa x * 2 powr (exponent x)"
+    by (rule mantissa_exponent)
+  also have "mantissa x \<le> \<bar>mantissa x\<bar>"
+    by simp
+  also have "\<dots> \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"
using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg \<open>mantissa x \<noteq> 0\<close>
by (auto simp del: real_of_int_abs simp add: powr_int)
@@ -1813,22 +1999,28 @@
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>" using \<open>prec \<ge> 1\<close> by arith
+  have "log 2 x \<le> real prec + real \<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)"
-  by (transfer, rule real_divl_pos_less1_bound)
+  by transfer (rule real_divl_pos_less1_bound)

lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
by transfer (rule real_divr)

-lemma real_divr_pos_less1_lower_bound: assumes "0 < x" and "x \<le> 1" shows "1 \<le> real_divr prec 1 x"
+lemma real_divr_pos_less1_lower_bound:
+  assumes "0 < x"
+    and "x \<le> 1"
+  shows "1 \<le> real_divr prec 1 x"
proof -
-  have "1 \<le> 1 / x" using \<open>0 < x\<close> and \<open>x <= 1\<close> by auto
-  also have "\<dots> \<le> real_divr prec 1 x" using real_divr[where x=1 and y=x] by auto
+  have "1 \<le> 1 / x"
+    using \<open>0 < x\<close> and \<open>x <= 1\<close> by auto
+  also have "\<dots> \<le> real_divr prec 1 x"
+    using real_divr[where x=1 and y=x] by auto
finally show ?thesis by auto
qed

@@ -1855,19 +2047,21 @@
assumes "0 \<le> x" "x \<le> y"
shows "truncate_up prec x \<le> truncate_up prec y"
proof -
-  {
-    assume "\<lfloor>log 2 x\<rfloor> = \<lfloor>log 2 y\<rfloor>"
-    hence ?thesis
+  consider "\<lfloor>log 2 x\<rfloor> = \<lfloor>log 2 y\<rfloor>" | "\<lfloor>log 2 x\<rfloor> \<noteq> \<lfloor>log 2 y\<rfloor>" "0 < x" | "x \<le> 0"
+    by arith
+  then show ?thesis
+  proof cases
+    case 1
+    then show ?thesis
using assms
by (auto simp: truncate_up_def round_up_def intro!: ceiling_mono)
-  } moreover {
-    assume "0 < x"
-    hence "log 2 x \<le> log 2 y" using assms by auto
-    moreover
-    assume "\<lfloor>log 2 x\<rfloor> \<noteq> \<lfloor>log 2 y\<rfloor>"
-    ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
-      unfolding atomize_conj
-      by (metis floor_less_cancel linorder_cases not_le)
+  next
+    case 2
+    from assms \<open>0 < x\<close> have "log 2 x \<le> log 2 y"
+      by auto
+    with \<open>\<lfloor>log 2 x\<rfloor> \<noteq> \<lfloor>log 2 y\<rfloor>\<close>
+    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)"
using assms by (simp add: truncate_up_def round_up_def)
@@ -1876,10 +2070,10 @@
have "x * 2 powr real (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)
-      thus "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> real ((2::int) ^ prec)"
+      then show "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> real ((2::int) ^ prec)"
using \<open>0 < x\<close> by (simp add: powr_realpow)
qed
-    hence "real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> 2 powr int prec"
+    then have "real \<lceil>x * 2 powr real (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>)"
@@ -1896,14 +2090,13 @@
also have "\<dots> \<le> truncate_up prec y"
by (rule truncate_up)
-    finally have ?thesis .
-  } moreover {
-    assume "~ 0 < x"
-    hence ?thesis
+    finally show ?thesis .
+  next
+    case 3
+    then show ?thesis
using assms
by (auto intro!: truncate_up_le)
-  } ultimately show ?thesis
-    by blast
+  qed
qed

lemma truncate_up_switch_sign_mono:
@@ -1931,20 +2124,22 @@
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
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 \<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}"
by simp
-  also have "\<dots> \<subseteq> {0}" by auto
-  finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> = 0" by simp
+  also have "\<dots> \<subseteq> {0}"
+    by auto
+  finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> = 0"
+    by simp
with assms show ?thesis
by (auto simp: truncate_down_def round_down_def)
qed

lemma truncate_down_switch_sign_mono:
-  assumes "x \<le> 0" "0 \<le> y"
-  assumes "x \<le> y"
+  assumes "x \<le> 0"
+    and "0 \<le> y"
+    and "x \<le> y"
shows "truncate_down prec x \<le> truncate_down prec y"
proof -
note truncate_down_le[OF \<open>x \<le> 0\<close>]
@@ -1956,32 +2151,36 @@
assumes "0 \<le> x" "x \<le> y"
shows "truncate_down prec x \<le> truncate_down prec y"
proof -
-  {
-    assume "0 < x" "prec = 0"
-    with assms have ?thesis
+  consider "0 < x" "prec = 0" | "x \<le> 0" | "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>" |
+    "0 < x" "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<noteq> \<lfloor>log 2 \<bar>y\<bar>\<rfloor>" "prec \<noteq> 0"
+    by arith
+  then show ?thesis
+  proof cases
+    case 1
+    with assms show ?thesis
-  } moreover {
-    assume "~ 0 < x"
+  next
+    case 2
with assms have "x = 0" "0 \<le> y" by simp_all
-    hence ?thesis
+    then show ?thesis
by (auto intro!: truncate_down_nonneg)
-  } moreover {
-    assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
-    hence ?thesis
+  next
+    case 3
+    then show ?thesis
using assms
by (auto simp: truncate_down_def round_down_def intro!: floor_mono)
-  } moreover {
-    assume "0 < x"
-    hence "log 2 x \<le> log 2 y" "0 < y" "0 \<le> y" using assms by auto
-    moreover
-    assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<noteq> \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
-    ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
+  next
+    case 4
+    from \<open>0 < x\<close> have "log 2 x \<le> log 2 y" "0 < y" "0 \<le> y"
+      using assms by auto
+    with \<open>\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<noteq> \<lfloor>log 2 \<bar>y\<bar>\<rfloor>\<close>
+    have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
unfolding atomize_conj abs_of_pos[OF \<open>0 < x\<close>] abs_of_pos[OF \<open>0 < y\<close>]
by (metis floor_less_cancel linorder_cases not_le)
-    assume "prec \<noteq> 0" hence [simp]: "prec \<ge> Suc 0" by auto
+    from \<open>prec \<noteq> 0\<close> have [simp]: "prec \<ge> Suc 0"
+      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
+      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))"
using \<open>0 \<le> y\<close> \<open>0 \<le> x\<close> assms(2)
by (auto intro!: powr_mono divide_left_mono
@@ -1992,7 +2191,7 @@
finally have "(2 ^ (prec - 1)) \<le> \<lfloor>y * 2 powr real (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)
-    hence "(2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1) \<le> truncate_down prec y"
+    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: truncate_down_def round_down_def)
moreover
{
@@ -2006,9 +2205,10 @@
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)
-    } ultimately have ?thesis
+    }
+    ultimately show ?thesis
by (metis dual_order.trans truncate_down)
-  } ultimately show ?thesis by blast
+  qed
qed

lemma truncate_down_eq_truncate_up: "truncate_down p x = - truncate_up p (-x)"
@@ -2029,49 +2229,62 @@
lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0"
by (auto simp: zero_float_def mult_le_0_iff) (simp add: not_less [symmetric])

-lemma real_of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real a)"
+lemma real_of_float_pprt[simp]:
+  fixes a :: float
+  shows "real (pprt a) = pprt (real 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)"
+lemma real_of_float_nprt[simp]:
+  fixes a :: float
+  shows "real (nprt a) = nprt (real a)"
unfolding nprt_def inf_float_def min_def inf_real_def by auto

+context
+begin
+
lift_definition int_floor_fl :: "float \<Rightarrow> int" is floor .

-lemma compute_int_floor_fl[code]:
+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)
-hide_fact (open) compute_int_floor_fl

lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real (floor x)" by simp

-lemma compute_floor_fl[code]:
+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)
-hide_fact (open) compute_floor_fl
+
+end

-lemma floor_fl: "real (floor_fl x) \<le> real x" by transfer simp
+lemma floor_fl: "real (floor_fl x) \<le> real x"
+  by transfer simp

-lemma int_floor_fl: "real (int_floor_fl x) \<le> real x" by transfer simp
+lemma int_floor_fl: "real (int_floor_fl x) \<le> real x"
+  by transfer simp

lemma floor_pos_exp: "exponent (floor_fl x) \<ge> 0"
proof (cases "floor_fl x = float_of 0")
case True
-  then show ?thesis by (simp add: floor_fl_def)
+  then show ?thesis
next
case False
-  have eq: "floor_fl x = Float \<lfloor>real x\<rfloor> 0" by transfer simp
+  have eq: "floor_fl x = Float \<lfloor>real 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"
by (rule denormalize_shift[OF eq[THEN eq_reflection] False])
-  then show ?thesis by simp
+  then show ?thesis
+    by simp
qed

lemma compute_mantissa[code]:
-  "mantissa (Float m e) = (if m = 0 then 0 else if 2 dvd m then mantissa (normfloat (Float m e)) else m)"
+  "mantissa (Float m e) =
+    (if m = 0 then 0 else if 2 dvd m then mantissa (normfloat (Float m e)) else m)"
by (auto simp: mantissa_float Float.abs_eq)

lemma compute_exponent[code]:
-  "exponent (Float m e) = (if m = 0 then 0 else if 2 dvd m then exponent (normfloat (Float m e)) else e)"
+  "exponent (Float m e) =
+    (if m = 0 then 0 else if 2 dvd m then exponent (normfloat (Float m e)) else e)"
by (auto simp: exponent_float Float.abs_eq)

end
-```
```--- a/src/HOL/Library/Lattice_Algebras.thy	Thu Jul 09 00:39:49 2015 +0200
+++ b/src/HOL/Library/Lattice_Algebras.thy	Thu Jul 09 00:40:57 2015 +0200
@@ -145,7 +145,8 @@
lemma nprt_le_zero[simp]: "nprt a \<le> 0"

-lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r")
+lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0"
+  (is "?l = ?r")
proof
assume ?l
then show ?r
@@ -177,25 +178,24 @@
lemma nprt_eq_0 [simp, no_atp]: "0 \<le> x \<Longrightarrow> nprt x = 0"

-lemma sup_0_imp_0: "sup a (- a) = 0 \<Longrightarrow> a = 0"
+lemma sup_0_imp_0:
+  assumes "sup a (- a) = 0"
+  shows "a = 0"
proof -
-  {
-    fix a :: 'a
-    assume hyp: "sup a (- a) = 0"
-    then have "sup a (- a) + a = a"
+  have p: "0 \<le> a" if "sup a (- a) = 0" for a :: 'a
+  proof -
+    from that have "sup a (- a) + a = a"
by simp
then have "sup (a + a) 0 = a"
then have "sup (a + a) 0 \<le> a"
by simp
-    then have "0 \<le> a"
+    then show ?thesis
by (blast intro: order_trans inf_sup_ord)
-  }
-  note p = this
-  assume hyp:"sup a (-a) = 0"
-  then have hyp2:"sup (-a) (-(-a)) = 0"
+  qed
+  from assms have **: "sup (-a) (-(-a)) = 0"
-  from p[OF hyp] p[OF hyp2] show "a = 0"
+  from p[OF assms] p[OF **] show "a = 0"
by simp
qed

@@ -217,49 +217,50 @@
apply simp
done

-  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
+lemma zero_le_double_add_iff_zero_le_single_add [simp]: "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
+  (is "?lhs \<longleftrightarrow> ?rhs")
proof
-  assume "0 \<le> a + a"
-  then have a: "inf (a + a) 0 = 0"
-    by (simp add: inf_commute inf_absorb1)
-  have "inf a 0 + inf a 0 = inf (inf (a + a) 0) a"  (is "?l=_")
-  then have "?l = 0 + inf a 0"
-  then have "inf a 0 = 0"
-  then show "0 \<le> a"
-    unfolding le_iff_inf by (simp add: inf_commute)
-next
-  assume a: "0 \<le> a"
-  show "0 \<le> a + a"
+  show ?rhs if ?lhs
+  proof -
+    from that have a: "inf (a + a) 0 = 0"
+      by (simp add: inf_commute inf_absorb1)
+    have "inf a 0 + inf a 0 = inf (inf (a + a) 0) a"  (is "?l=_")
+    then have "?l = 0 + inf a 0"
+    then have "inf a 0 = 0"
+    then show ?thesis
+      unfolding le_iff_inf by (simp add: inf_commute)
+  qed
+  show ?lhs if ?rhs
qed

lemma double_zero [simp]: "a + a = 0 \<longleftrightarrow> a = 0"
+  (is "?lhs \<longleftrightarrow> ?rhs")
proof
-  assume assm: "a + a = 0"
-  then have "a + a + - a = - a"
-    by simp
-  then have "a + (a + - a) = - a"
-  then have a: "- a = a"
-    by simp
-  show "a = 0"
-    apply (rule antisym)
-    apply (unfold neg_le_iff_le [symmetric, of a])
-    unfolding a
-    apply simp
-    unfolding assm
-    unfolding le_less
-    apply simp_all
-    done
-next
-  assume "a = 0"
-  then show "a + a = 0"
-    by simp
+  show ?rhs if ?lhs
+  proof -
+    from that have "a + a + - a = - a"
+      by simp
+    then have "a + (a + - a) = - a"
+    then have a: "- a = a"
+      by simp
+    show ?thesis
+      apply (rule antisym)
+      apply (unfold neg_le_iff_le [symmetric, of a])
+      unfolding a
+      apply simp
+      unfolding that
+      unfolding le_less
+      apply simp_all
+      done
+  qed
+  show ?lhs if ?rhs
+    using that by simp
qed

lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a"
@@ -281,19 +282,17 @@
done
qed

-  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
+lemma double_add_le_zero_iff_single_add_le_zero [simp]: "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
proof -
have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)"
-    by (subst le_minus_iff, simp)
+    by (subst le_minus_iff) simp
moreover have "\<dots> \<longleftrightarrow> a \<le> 0"
ultimately show ?thesis
by blast
qed

-  "a + a < 0 \<longleftrightarrow> a < 0"
+lemma double_add_less_zero_iff_single_less_zero [simp]: "a + a < 0 \<longleftrightarrow> a < 0"
proof -
have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)"
by (subst less_minus_iff) simp
@@ -316,8 +315,8 @@

lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"
proof -
-  from add_le_cancel_left [of "uminus a" zero "plus a a"]
have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a"
+    using add_le_cancel_left [of "uminus a" zero "plus a a"]
then show ?thesis
by simp
@@ -370,15 +369,14 @@

proof
-  have abs_ge_zero [simp]: "\<And>a. 0 \<le> \<bar>a\<bar>"
+  have abs_ge_zero [simp]: "0 \<le> \<bar>a\<bar>" for a
proof -
-    fix a b
have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"
show "0 \<le> \<bar>a\<bar>"
by (rule add_mono [OF a b, simplified])
qed
-  have abs_leI: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
+  have abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" for a b
fix a b
show "0 \<le> \<bar>a\<bar>"
@@ -387,15 +385,12 @@
show "\<bar>-a\<bar> = \<bar>a\<bar>"
-  {
-    assume "a \<le> b"
-    then show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
-      by (rule abs_leI)
-  }
+  show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" if "a \<le> b"
+    using that by (rule abs_leI)
show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
proof -
have g: "\<bar>a\<bar> + \<bar>b\<bar> = sup (a + b) (sup (- a - b) (sup (- a + b) (a + (- b))))"
-      (is "_=sup ?m ?n")
+      (is "_ = sup ?m ?n")
have a: "a + b \<le> sup ?m ?n"
by simp
@@ -420,25 +415,23 @@
end

lemma sup_eq_if:
-  fixes a :: "'a::{lattice_ab_group_add, linorder}"
shows "sup a (- a) = (if a < 0 then - a else a)"
-proof -
-  note add_le_cancel_right [of a a "- a", symmetric, simplified]
-  moreover note add_le_cancel_right [of "-a" a a, symmetric, simplified]
-  then show ?thesis by (auto simp: sup_max max.absorb1 max.absorb2)
-qed
+  using add_le_cancel_right [of a a "- a", symmetric, simplified]
+    and add_le_cancel_right [of "-a" a a, symmetric, simplified]
+  by (auto simp: sup_max max.absorb1 max.absorb2)

lemma abs_if_lattice:
-  fixes a :: "'a::{lattice_ab_group_add_abs, linorder}"
shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
by auto

lemma estimate_by_abs:
fixes a b c :: "'a::lattice_ab_group_add_abs"
-  shows "a + b \<le> c \<Longrightarrow> a \<le> c + \<bar>b\<bar>"
+  assumes "a + b \<le> c"
+  shows "a \<le> c + \<bar>b\<bar>"
proof -
-  assume "a + b \<le> c"
-  then have "a \<le> c + (- b)"
+  from assms have "a \<le> c + (- b)"
have "- b \<le> \<bar>b\<bar>"
by (rule abs_ge_minus_self)
@@ -464,15 +457,12 @@
let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
have a: "\<bar>a\<bar> * \<bar>b\<bar> = ?x"
by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
-  {
-    fix u v :: 'a
-    have bh: "u = a \<Longrightarrow> v = b \<Longrightarrow>
-              u * v = pprt a * pprt b + pprt a * nprt b +
-                      nprt a * pprt b + nprt a * nprt b"
-      apply (subst prts[of u], subst prts[of v])
-      done
-  }
+  have bh: "u = a \<Longrightarrow> v = b \<Longrightarrow>
+            u * v = pprt a * pprt b + pprt a * nprt b +
+                    nprt a * pprt b + nprt a * nprt b" for u v :: 'a
+    apply (subst prts[of u], subst prts[of v])
+    done
note b = this[OF refl[of a] refl[of b]]
have xy: "- ?x \<le> ?y"
apply simp
@@ -552,9 +542,7 @@
pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"
proof -
have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"
-    apply (subst prts[symmetric])+
-    apply simp
-    done
+    by (subst prts[symmetric])+ simp
then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
moreover have "pprt a * pprt b \<le> pprt a2 * pprt b2"
@@ -587,9 +575,7 @@
by simp
qed
ultimately show ?thesis
-    apply -
-    apply (rule add_mono | simp)+
-    done
+    by - (rule add_mono | simp)+
qed

lemma mult_ge_prts:
@@ -607,7 +593,8 @@
by auto
from mult_le_prts[of "- a2" "- a" "- a1" "b1" b "b2",
OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]
-  have le: "- (a * b) \<le> - nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
+  have le: "- (a * b) \<le>
+    - nprt a1 * pprt b2 + - nprt a2 * nprt b2 +
- pprt a1 * pprt b1 + - pprt a2 * nprt b1"
by simp
then have "- (- nprt a1 * pprt b2 + - nprt a2 * nprt b2 +```