author wenzelm Fri, 22 Jul 2016 21:43:56 +0200 changeset 63545 c2f69dac0353 parent 63544 e0cd6469a6b8 child 63546 5f097087fa1e
misc tuning and modernization;
```--- a/src/HOL/Real_Vector_Spaces.thy	Fri Jul 22 19:04:30 2016 +0200
+++ b/src/HOL/Real_Vector_Spaces.thy	Fri Jul 22 21:43:56 2016 +0200
@@ -34,39 +34,36 @@

lemma setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))"
-apply (cases "finite A")
-apply (induct set: finite)
-done
+  apply (cases "finite A")
+   apply (induct set: finite)
+  done

end

+
subsection \<open>Vector spaces\<close>

locale vector_space =
fixes scale :: "'a::field \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b"
-  assumes scale_right_distrib [algebra_simps]:
-    "scale a (x + y) = scale a x + scale a y"
-  and scale_left_distrib [algebra_simps]:
-    "scale (a + b) x = scale a x + scale b x"
-  and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x"
-  and scale_one [simp]: "scale 1 x = x"
+  assumes scale_right_distrib [algebra_simps]: "scale a (x + y) = scale a x + scale a y"
+    and scale_left_distrib [algebra_simps]: "scale (a + b) x = scale a x + scale b x"
+    and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x"
+    and scale_one [simp]: "scale 1 x = x"
begin

-lemma scale_left_commute:
-  "scale a (scale b x) = scale b (scale a x)"
+lemma scale_left_commute: "scale a (scale b x) = scale b (scale a x)"

lemma scale_zero_left [simp]: "scale 0 x = 0"
and scale_minus_left [simp]: "scale (- a) x = - (scale a x)"
-  and scale_left_diff_distrib [algebra_simps]:
-        "scale (a - b) x = scale a x - scale b x"
+  and scale_left_diff_distrib [algebra_simps]: "scale (a - b) x = scale a x - scale b x"
and scale_setsum_left: "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)"
proof -
interpret s: additive "\<lambda>a. scale a x"
-    proof qed (rule scale_left_distrib)
+    by standard (rule scale_left_distrib)
show "scale 0 x = 0" by (rule s.zero)
show "scale (- a) x = - (scale a x)" by (rule s.minus)
show "scale (a - b) x = scale a x - scale b x" by (rule s.diff)
@@ -75,72 +72,70 @@

lemma scale_zero_right [simp]: "scale a 0 = 0"
and scale_minus_right [simp]: "scale a (- x) = - (scale a x)"
-  and scale_right_diff_distrib [algebra_simps]:
-        "scale a (x - y) = scale a x - scale a y"
+  and scale_right_diff_distrib [algebra_simps]: "scale a (x - y) = scale a x - scale a y"
and scale_setsum_right: "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))"
proof -
interpret s: additive "\<lambda>x. scale a x"
-    proof qed (rule scale_right_distrib)
+    by standard (rule scale_right_distrib)
show "scale a 0 = 0" by (rule s.zero)
show "scale a (- x) = - (scale a x)" by (rule s.minus)
show "scale a (x - y) = scale a x - scale a y" by (rule s.diff)
show "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))" by (rule s.setsum)
qed

-lemma scale_eq_0_iff [simp]:
-  "scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0"
-proof cases
-  assume "a = 0" thus ?thesis by simp
+lemma scale_eq_0_iff [simp]: "scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0"
+proof (cases "a = 0")
+  case True
+  then show ?thesis by simp
next
-  assume anz [simp]: "a \<noteq> 0"
-  { assume "scale a x = 0"
-    hence "scale (inverse a) (scale a x) = 0" by simp
-    hence "x = 0" by simp }
-  thus ?thesis by force
+  case False
+  have "x = 0" if "scale a x = 0"
+  proof -
+    from False that have "scale (inverse a) (scale a x) = 0" by simp
+    with False show ?thesis by simp
+  qed
+  then show ?thesis by force
qed

lemma scale_left_imp_eq:
-  "\<lbrakk>a \<noteq> 0; scale a x = scale a y\<rbrakk> \<Longrightarrow> x = y"
+  assumes nonzero: "a \<noteq> 0"
+    and scale: "scale a x = scale a y"
+  shows "x = y"
proof -
-  assume nonzero: "a \<noteq> 0"
-  assume "scale a x = scale a y"
-  hence "scale a (x - y) = 0"
+  from scale have "scale a (x - y) = 0"
-  hence "x - y = 0" by (simp add: nonzero)
-  thus "x = y" by (simp only: right_minus_eq)
+  with nonzero have "x - y = 0" by simp
+  then show "x = y" by (simp only: right_minus_eq)
qed

lemma scale_right_imp_eq:
-  "\<lbrakk>x \<noteq> 0; scale a x = scale b x\<rbrakk> \<Longrightarrow> a = b"
+  assumes nonzero: "x \<noteq> 0"
+    and scale: "scale a x = scale b x"
+  shows "a = b"
proof -
-  assume nonzero: "x \<noteq> 0"
-  assume "scale a x = scale b x"
-  hence "scale (a - b) x = 0"
+  from scale have "scale (a - b) x = 0"
-  hence "a - b = 0" by (simp add: nonzero)
-  thus "a = b" by (simp only: right_minus_eq)
+  with nonzero have "a - b = 0" by simp
+  then show "a = b" by (simp only: right_minus_eq)
qed

-lemma scale_cancel_left [simp]:
-  "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0"
-by (auto intro: scale_left_imp_eq)
+lemma scale_cancel_left [simp]: "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0"
+  by (auto intro: scale_left_imp_eq)

-lemma scale_cancel_right [simp]:
-  "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0"
-by (auto intro: scale_right_imp_eq)
+lemma scale_cancel_right [simp]: "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0"
+  by (auto intro: scale_right_imp_eq)

end

+
subsection \<open>Real vector spaces\<close>

class scaleR =
fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)
begin

-abbreviation
-  divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)
-where
-  "x /\<^sub>R r == scaleR (inverse r) x"
+abbreviation divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a"  (infixl "'/\<^sub>R" 70)
+  where "x /\<^sub>R r \<equiv> scaleR (inverse r) x"

end

@@ -150,14 +145,13 @@
and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x"
and scaleR_one: "scaleR 1 x = x"

-interpretation real_vector:
-  vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector"
-apply unfold_locales
-apply (rule scaleR_scaleR)
-apply (rule scaleR_one)
-done
+interpretation real_vector: vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector"
+  apply unfold_locales
+   apply (rule scaleR_scaleR)
+  apply (rule scaleR_one)
+  done

text \<open>Recover original theorem names\<close>

@@ -183,14 +177,13 @@
lemmas scaleR_left_diff_distrib = scaleR_diff_left
lemmas scaleR_right_diff_distrib = scaleR_diff_right

-lemma scaleR_minus1_left [simp]:
-  fixes x :: "'a::real_vector"
-  shows "scaleR (-1) x = - x"
+lemma scaleR_minus1_left [simp]: "scaleR (-1) x = - x"
+  for x :: "'a::real_vector"
using scaleR_minus_left [of 1 x] by simp

class real_algebra = real_vector + ring +
assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
-  and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
+    and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"

class real_algebra_1 = real_algebra + ring_1

@@ -201,125 +194,122 @@
instantiation real :: real_field
begin

-definition
-  real_scaleR_def [simp]: "scaleR a x = a * x"
+definition real_scaleR_def [simp]: "scaleR a x = a * x"

-instance proof
+instance
+  by standard (simp_all add: algebra_simps)

end

-interpretation scaleR_left: additive "(\<lambda>a. scaleR a x::'a::real_vector)"
-proof qed (rule scaleR_left_distrib)
+interpretation scaleR_left: additive "(\<lambda>a. scaleR a x :: 'a::real_vector)"
+  by standard (rule scaleR_left_distrib)

-interpretation scaleR_right: additive "(\<lambda>x. scaleR a x::'a::real_vector)"
-proof qed (rule scaleR_right_distrib)
+interpretation scaleR_right: additive "(\<lambda>x. scaleR a x :: 'a::real_vector)"
+  by standard (rule scaleR_right_distrib)

lemma nonzero_inverse_scaleR_distrib:
-  fixes x :: "'a::real_div_algebra" shows
-  "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
-by (rule inverse_unique, simp)
+  "a \<noteq> 0 \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
+  for x :: "'a::real_div_algebra"
+  by (rule inverse_unique) simp

-lemma inverse_scaleR_distrib:
-  fixes x :: "'a::{real_div_algebra, division_ring}"
-  shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
-apply (case_tac "a = 0", simp)
-apply (case_tac "x = 0", simp)
-apply (erule (1) nonzero_inverse_scaleR_distrib)
-done
-
-lemma setsum_constant_scaleR:
-  fixes y :: "'a::real_vector"
-  shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
-  apply (cases "finite A")
-  apply (induct set: finite)
+lemma inverse_scaleR_distrib: "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
+  for x :: "'a::{real_div_algebra,division_ring}"
+  apply (cases "a = 0")
+   apply simp
+  apply (cases "x = 0")
+   apply simp
+  apply (erule (1) nonzero_inverse_scaleR_distrib)
done

-  fixes v :: "'a :: real_vector"
-  shows "v + (b / z) *\<^sub>R w = (if z = 0 then v else (z *\<^sub>R v + b *\<^sub>R w) /\<^sub>R z)"
-        "a *\<^sub>R v + (b / z) *\<^sub>R w = (if z = 0 then a *\<^sub>R v else ((a * z) *\<^sub>R v + b *\<^sub>R w) /\<^sub>R z)"
-        "(a / z) *\<^sub>R v + w = (if z = 0 then w else (a *\<^sub>R v + z *\<^sub>R w) /\<^sub>R z)"
-        "(a / z) *\<^sub>R v + b *\<^sub>R w = (if z = 0 then b *\<^sub>R w else (a *\<^sub>R v + (b * z) *\<^sub>R w) /\<^sub>R z)"
-        "v - (b / z) *\<^sub>R w = (if z = 0 then v else (z *\<^sub>R v - b *\<^sub>R w) /\<^sub>R z)"
-        "a *\<^sub>R v - (b / z) *\<^sub>R w = (if z = 0 then a *\<^sub>R v else ((a * z) *\<^sub>R v - b *\<^sub>R w) /\<^sub>R z)"
-        "(a / z) *\<^sub>R v - w = (if z = 0 then -w else (a *\<^sub>R v - z *\<^sub>R w) /\<^sub>R z)"
-        "(a / z) *\<^sub>R v - b *\<^sub>R w = (if z = 0 then -b *\<^sub>R w else (a *\<^sub>R v - (b * z) *\<^sub>R w) /\<^sub>R z)"
+lemma setsum_constant_scaleR: "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
+  for y :: "'a::real_vector"
+  apply (cases "finite A")
+   apply (induct set: finite)
+  done
+
+  "v + (b / z) *\<^sub>R w = (if z = 0 then v else (z *\<^sub>R v + b *\<^sub>R w) /\<^sub>R z)"
+  "a *\<^sub>R v + (b / z) *\<^sub>R w = (if z = 0 then a *\<^sub>R v else ((a * z) *\<^sub>R v + b *\<^sub>R w) /\<^sub>R z)"
+  "(a / z) *\<^sub>R v + w = (if z = 0 then w else (a *\<^sub>R v + z *\<^sub>R w) /\<^sub>R z)"
+  "(a / z) *\<^sub>R v + b *\<^sub>R w = (if z = 0 then b *\<^sub>R w else (a *\<^sub>R v + (b * z) *\<^sub>R w) /\<^sub>R z)"
+  "v - (b / z) *\<^sub>R w = (if z = 0 then v else (z *\<^sub>R v - b *\<^sub>R w) /\<^sub>R z)"
+  "a *\<^sub>R v - (b / z) *\<^sub>R w = (if z = 0 then a *\<^sub>R v else ((a * z) *\<^sub>R v - b *\<^sub>R w) /\<^sub>R z)"
+  "(a / z) *\<^sub>R v - w = (if z = 0 then -w else (a *\<^sub>R v - z *\<^sub>R w) /\<^sub>R z)"
+  "(a / z) *\<^sub>R v - b *\<^sub>R w = (if z = 0 then -b *\<^sub>R w else (a *\<^sub>R v - (b * z) *\<^sub>R w) /\<^sub>R z)"
+  for v :: "'a :: real_vector"

lemma real_vector_affinity_eq:
fixes x :: "'a :: real_vector"
assumes m0: "m \<noteq> 0"
shows "m *\<^sub>R x + c = y \<longleftrightarrow> x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)"
+    (is "?lhs \<longleftrightarrow> ?rhs")
proof
-  assume h: "m *\<^sub>R x + c = y"
-  hence "m *\<^sub>R x = y - c" by (simp add: field_simps)
-  hence "inverse m *\<^sub>R (m *\<^sub>R x) = inverse m *\<^sub>R (y - c)" by simp
+  assume ?lhs
+  then have "m *\<^sub>R x = y - c" by (simp add: field_simps)
+  then have "inverse m *\<^sub>R (m *\<^sub>R x) = inverse m *\<^sub>R (y - c)" by simp
then show "x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)"
using m0
next
-  assume h: "x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)"
-  show "m *\<^sub>R x + c = y" unfolding h
-    using m0  by (simp add: real_vector.scale_right_diff_distrib)
+  assume ?rhs
+  with m0 show "m *\<^sub>R x + c = y"
qed

-lemma real_vector_eq_affinity:
-  fixes x :: "'a :: real_vector"
-  shows "m \<noteq> 0 ==> (y = m *\<^sub>R x + c \<longleftrightarrow> inverse m *\<^sub>R y - (inverse m *\<^sub>R c) = x)"
+lemma real_vector_eq_affinity: "m \<noteq> 0 \<Longrightarrow> y = m *\<^sub>R x + c \<longleftrightarrow> inverse m *\<^sub>R y - (inverse m *\<^sub>R c) = x"
+  for x :: "'a::real_vector"
using real_vector_affinity_eq[where m=m and x=x and y=y and c=c]
by metis

-lemma scaleR_eq_iff [simp]:
-  fixes a :: "'a :: real_vector"
-  shows "b + u *\<^sub>R a = a + u *\<^sub>R b \<longleftrightarrow> a=b \<or> u=1"
-proof (cases "u=1")
-  case True then show ?thesis by auto
+lemma scaleR_eq_iff [simp]: "b + u *\<^sub>R a = a + u *\<^sub>R b \<longleftrightarrow> a = b \<or> u = 1"
+  for a :: "'a::real_vector"
+proof (cases "u = 1")
+  case True
+  then show ?thesis by auto
next
case False
-  { assume "b + u *\<^sub>R a = a + u *\<^sub>R b"
-    then have "(u - 1) *\<^sub>R a = (u - 1) *\<^sub>R b"
+  have "a = b" if "b + u *\<^sub>R a = a + u *\<^sub>R b"
+  proof -
+    from that have "(u - 1) *\<^sub>R a = (u - 1) *\<^sub>R b"
-    with False have "a=b"
+    with False show ?thesis
by auto
-  }
+  qed
then show ?thesis by auto
qed

-lemma scaleR_collapse [simp]:
-  fixes a :: "'a :: real_vector"
-  shows "(1 - u) *\<^sub>R a + u *\<^sub>R a = a"
+lemma scaleR_collapse [simp]: "(1 - u) *\<^sub>R a + u *\<^sub>R a = a"
+  for a :: "'a::real_vector"

-subsection \<open>Embedding of the Reals into any \<open>real_algebra_1\<close>:
-@{term of_real}\<close>
+subsection \<open>Embedding of the Reals into any \<open>real_algebra_1\<close>: \<open>of_real\<close>\<close>

-definition
-  of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
-  "of_real r = scaleR r 1"
+definition of_real :: "real \<Rightarrow> 'a::real_algebra_1"
+  where "of_real r = scaleR r 1"

lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"

lemma of_real_0 [simp]: "of_real 0 = 0"

lemma of_real_1 [simp]: "of_real 1 = 1"

lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
+  by (simp add: of_real_def scaleR_left_distrib)

lemma of_real_minus [simp]: "of_real (- x) = - of_real x"

lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
+  by (simp add: of_real_def scaleR_left_diff_distrib)

lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
+  by (simp add: of_real_def mult.commute)

lemma of_real_setsum[simp]: "of_real (setsum f s) = (\<Sum>x\<in>s. of_real (f x))"
by (induct s rule: infinite_finite_induct) auto
@@ -328,63 +318,56 @@
by (induct s rule: infinite_finite_induct) auto

lemma nonzero_of_real_inverse:
-  "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
-   inverse (of_real x :: 'a::real_div_algebra)"
+  "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) = inverse (of_real x :: 'a::real_div_algebra)"
+  by (simp add: of_real_def nonzero_inverse_scaleR_distrib)

lemma of_real_inverse [simp]:
-  "of_real (inverse x) =
-   inverse (of_real x :: 'a::{real_div_algebra, division_ring})"
+  "of_real (inverse x) = inverse (of_real x :: 'a::{real_div_algebra,division_ring})"
+  by (simp add: of_real_def inverse_scaleR_distrib)

lemma nonzero_of_real_divide:
-  "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
-   (of_real x / of_real y :: 'a::real_field)"
+  "y \<noteq> 0 \<Longrightarrow> of_real (x / y) = (of_real x / of_real y :: 'a::real_field)"
+  by (simp add: divide_inverse nonzero_of_real_inverse)

lemma of_real_divide [simp]:
"of_real (x / y) = (of_real x / of_real y :: 'a::real_div_algebra)"

lemma of_real_power [simp]:
"of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"
-by (induct n) simp_all
+  by (induct n) simp_all

-lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
+lemma of_real_eq_iff [simp]: "of_real x = of_real y \<longleftrightarrow> x = y"

-lemma inj_of_real:
-  "inj of_real"
+lemma inj_of_real: "inj of_real"
by (auto intro: injI)

lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]

lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
-proof
-  fix r
-  show "of_real r = id r"
-qed
+  by (rule ext) (simp add: of_real_def)

-text\<open>Collapse nested embeddings\<close>
+text \<open>Collapse nested embeddings.\<close>
lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
-by (induct n) auto
+  by (induct n) auto

lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
-by (cases z rule: int_diff_cases, simp)
+  by (cases z rule: int_diff_cases) simp

lemma of_real_numeral [simp]: "of_real (numeral w) = numeral w"
-using of_real_of_int_eq [of "numeral w"] by simp
+  using of_real_of_int_eq [of "numeral w"] by simp

lemma of_real_neg_numeral [simp]: "of_real (- numeral w) = - numeral w"
-using of_real_of_int_eq [of "- numeral w"] by simp
+  using of_real_of_int_eq [of "- numeral w"] by simp

-text\<open>Every real algebra has characteristic zero\<close>
-
+text \<open>Every real algebra has characteristic zero.\<close>
instance real_algebra_1 < ring_char_0
proof
-  from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" by (rule inj_comp)
-  then show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: comp_def)
+  from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)"
+    by (rule inj_comp)
+  then show "inj (of_nat :: nat \<Rightarrow> 'a)"
qed

instance real_field < field_char_0 ..
@@ -396,97 +379,91 @@
where "\<real> = range of_real"

lemma Reals_of_real [simp]: "of_real r \<in> \<real>"

lemma Reals_of_int [simp]: "of_int z \<in> \<real>"
-by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
+  by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)

lemma Reals_of_nat [simp]: "of_nat n \<in> \<real>"
-by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
+  by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)

lemma Reals_numeral [simp]: "numeral w \<in> \<real>"
-by (subst of_real_numeral [symmetric], rule Reals_of_real)
+  by (subst of_real_numeral [symmetric], rule Reals_of_real)

lemma Reals_0 [simp]: "0 \<in> \<real>"
-apply (unfold Reals_def)
-apply (rule range_eqI)
-apply (rule of_real_0 [symmetric])
-done
+  apply (unfold Reals_def)
+  apply (rule range_eqI)
+  apply (rule of_real_0 [symmetric])
+  done

lemma Reals_1 [simp]: "1 \<in> \<real>"
-apply (unfold Reals_def)
-apply (rule range_eqI)
-apply (rule of_real_1 [symmetric])
-done
+  apply (unfold Reals_def)
+  apply (rule range_eqI)
+  apply (rule of_real_1 [symmetric])
+  done

-lemma Reals_add [simp]: "\<lbrakk>a \<in> \<real>; b \<in> \<real>\<rbrakk> \<Longrightarrow> a + b \<in> \<real>"
-apply (rule range_eqI)
-done
+lemma Reals_add [simp]: "a \<in> \<real> \<Longrightarrow> b \<in> \<real> \<Longrightarrow> a + b \<in> \<real>"
+  apply (auto simp add: Reals_def)
+  apply (rule range_eqI)
+  done

lemma Reals_minus [simp]: "a \<in> \<real> \<Longrightarrow> - a \<in> \<real>"
-apply (rule range_eqI)
-apply (rule of_real_minus [symmetric])
-done
+  apply (auto simp add: Reals_def)
+  apply (rule range_eqI)
+  apply (rule of_real_minus [symmetric])
+  done

-lemma Reals_diff [simp]: "\<lbrakk>a \<in> \<real>; b \<in> \<real>\<rbrakk> \<Longrightarrow> a - b \<in> \<real>"
-apply (rule range_eqI)
-apply (rule of_real_diff [symmetric])
-done
+lemma Reals_diff [simp]: "a \<in> \<real> \<Longrightarrow> b \<in> \<real> \<Longrightarrow> a - b \<in> \<real>"
+  apply (auto simp add: Reals_def)
+  apply (rule range_eqI)
+  apply (rule of_real_diff [symmetric])
+  done

-lemma Reals_mult [simp]: "\<lbrakk>a \<in> \<real>; b \<in> \<real>\<rbrakk> \<Longrightarrow> a * b \<in> \<real>"
-apply (rule range_eqI)
-apply (rule of_real_mult [symmetric])
-done
+lemma Reals_mult [simp]: "a \<in> \<real> \<Longrightarrow> b \<in> \<real> \<Longrightarrow> a * b \<in> \<real>"
+  apply (auto simp add: Reals_def)
+  apply (rule range_eqI)
+  apply (rule of_real_mult [symmetric])
+  done

-lemma nonzero_Reals_inverse:
-  fixes a :: "'a::real_div_algebra"
-  shows "\<lbrakk>a \<in> \<real>; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> \<real>"
-apply (rule range_eqI)
-apply (erule nonzero_of_real_inverse [symmetric])
-done
+lemma nonzero_Reals_inverse: "a \<in> \<real> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> inverse a \<in> \<real>"
+  for a :: "'a::real_div_algebra"
+  apply (auto simp add: Reals_def)
+  apply (rule range_eqI)
+  apply (erule nonzero_of_real_inverse [symmetric])
+  done

-lemma Reals_inverse:
-  fixes a :: "'a::{real_div_algebra, division_ring}"
-  shows "a \<in> \<real> \<Longrightarrow> inverse a \<in> \<real>"
-apply (rule range_eqI)
-apply (rule of_real_inverse [symmetric])
-done
+lemma Reals_inverse: "a \<in> \<real> \<Longrightarrow> inverse a \<in> \<real>"
+  for a :: "'a::{real_div_algebra,division_ring}"
+  apply (auto simp add: Reals_def)
+  apply (rule range_eqI)
+  apply (rule of_real_inverse [symmetric])
+  done

-lemma Reals_inverse_iff [simp]:
-  fixes x:: "'a :: {real_div_algebra, division_ring}"
-  shows "inverse x \<in> \<real> \<longleftrightarrow> x \<in> \<real>"
-by (metis Reals_inverse inverse_inverse_eq)
+lemma Reals_inverse_iff [simp]: "inverse x \<in> \<real> \<longleftrightarrow> x \<in> \<real>"
+  for x :: "'a::{real_div_algebra,division_ring}"
+  by (metis Reals_inverse inverse_inverse_eq)

-lemma nonzero_Reals_divide:
-  fixes a b :: "'a::real_field"
-  shows "\<lbrakk>a \<in> \<real>; b \<in> \<real>; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> \<real>"
-apply (rule range_eqI)
-apply (erule nonzero_of_real_divide [symmetric])
-done
+lemma nonzero_Reals_divide: "a \<in> \<real> \<Longrightarrow> b \<in> \<real> \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a / b \<in> \<real>"
+  for a b :: "'a::real_field"
+  apply (auto simp add: Reals_def)
+  apply (rule range_eqI)
+  apply (erule nonzero_of_real_divide [symmetric])
+  done

-lemma Reals_divide [simp]:
-  fixes a b :: "'a::{real_field, field}"
-  shows "\<lbrakk>a \<in> \<real>; b \<in> \<real>\<rbrakk> \<Longrightarrow> a / b \<in> \<real>"
-apply (rule range_eqI)
-apply (rule of_real_divide [symmetric])
-done
+lemma Reals_divide [simp]: "a \<in> \<real> \<Longrightarrow> b \<in> \<real> \<Longrightarrow> a / b \<in> \<real>"
+  for a b :: "'a::{real_field,field}"
+  apply (auto simp add: Reals_def)
+  apply (rule range_eqI)
+  apply (rule of_real_divide [symmetric])
+  done

-lemma Reals_power [simp]:
-  fixes a :: "'a::{real_algebra_1}"
-  shows "a \<in> \<real> \<Longrightarrow> a ^ n \<in> \<real>"
-apply (rule range_eqI)
-apply (rule of_real_power [symmetric])
-done
+lemma Reals_power [simp]: "a \<in> \<real> \<Longrightarrow> a ^ n \<in> \<real>"
+  for a :: "'a::real_algebra_1"
+  apply (auto simp add: Reals_def)
+  apply (rule range_eqI)
+  apply (rule of_real_power [symmetric])
+  done

lemma Reals_cases [cases set: Reals]:
assumes "q \<in> \<real>"
@@ -499,49 +476,56 @@
qed

lemma setsum_in_Reals [intro,simp]:
-  assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>" shows "setsum f s \<in> \<real>"
+  assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>"
+  shows "setsum f s \<in> \<real>"
proof (cases "finite s")
-  case True then show ?thesis using assms
-    by (induct s rule: finite_induct) auto
+  case True
+  then show ?thesis
+    using assms by (induct s rule: finite_induct) auto
next
-  case False then show ?thesis using assms
-    by (metis Reals_0 setsum.infinite)
+  case False
+  then show ?thesis
+    using assms by (metis Reals_0 setsum.infinite)
qed

lemma setprod_in_Reals [intro,simp]:
-  assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>" shows "setprod f s \<in> \<real>"
+  assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>"
+  shows "setprod f s \<in> \<real>"
proof (cases "finite s")
-  case True then show ?thesis using assms
-    by (induct s rule: finite_induct) auto
+  case True
+  then show ?thesis
+    using assms by (induct s rule: finite_induct) auto
next
-  case False then show ?thesis using assms
-    by (metis Reals_1 setprod.infinite)
+  case False
+  then show ?thesis
+    using assms by (metis Reals_1 setprod.infinite)
qed

lemma Reals_induct [case_names of_real, induct set: Reals]:
"q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
by (rule Reals_cases) auto

+
subsection \<open>Ordered real vector spaces\<close>

class ordered_real_vector = real_vector + ordered_ab_group_add +
assumes scaleR_left_mono: "x \<le> y \<Longrightarrow> 0 \<le> a \<Longrightarrow> a *\<^sub>R x \<le> a *\<^sub>R y"
-  assumes scaleR_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R x"
+    and scaleR_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R x"
begin

-lemma scaleR_mono:
-  "a \<le> b \<Longrightarrow> x \<le> y \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R y"
-apply (erule scaleR_right_mono [THEN order_trans], assumption)
-apply (erule scaleR_left_mono, assumption)
-done
+lemma scaleR_mono: "a \<le> b \<Longrightarrow> x \<le> y \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R y"
+  apply (erule scaleR_right_mono [THEN order_trans])
+   apply assumption
+  apply (erule scaleR_left_mono)
+  apply assumption
+  done

-lemma scaleR_mono':
-  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R d"
+lemma scaleR_mono': "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R d"
by (rule scaleR_mono) (auto intro: order.trans)

lemma pos_le_divideRI:
assumes "0 < c"
-  assumes "c *\<^sub>R a \<le> b"
+    and "c *\<^sub>R a \<le> b"
shows "a \<le> b /\<^sub>R c"
proof -
from scaleR_left_mono[OF assms(2)] assms(1)
@@ -554,20 +538,22 @@
lemma pos_le_divideR_eq:
assumes "0 < c"
shows "a \<le> b /\<^sub>R c \<longleftrightarrow> c *\<^sub>R a \<le> b"
-proof rule
-  assume "a \<le> b /\<^sub>R c"
-  from scaleR_left_mono[OF this] assms
-  have "c *\<^sub>R a \<le> c *\<^sub>R (b /\<^sub>R c)"
+    (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+  assume ?lhs
+  from scaleR_left_mono[OF this] assms have "c *\<^sub>R a \<le> c *\<^sub>R (b /\<^sub>R c)"
by simp
-  with assms show "c *\<^sub>R a \<le> b"
+  with assms show ?rhs
by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)
-qed (rule pos_le_divideRI[OF assms])
+next
+  assume ?rhs
+  with assms show ?lhs by (rule pos_le_divideRI)
+qed

-lemma scaleR_image_atLeastAtMost:
-  "c > 0 \<Longrightarrow> scaleR c ` {x..y} = {c *\<^sub>R x..c *\<^sub>R y}"
+lemma scaleR_image_atLeastAtMost: "c > 0 \<Longrightarrow> scaleR c ` {x..y} = {c *\<^sub>R x..c *\<^sub>R y}"
apply (auto intro!: scaleR_left_mono)
apply (rule_tac x = "inverse c *\<^sub>R xa" in image_eqI)
-  apply (simp_all add: pos_le_divideR_eq[symmetric] scaleR_scaleR scaleR_one)
+   apply (simp_all add: pos_le_divideR_eq[symmetric] scaleR_scaleR scaleR_one)
done

end
@@ -576,103 +562,105 @@
fixes a :: "'a :: ordered_real_vector"
assumes "c < 0"
shows "a \<le> b /\<^sub>R c \<longleftrightarrow> b \<le> c *\<^sub>R a"
-  using pos_le_divideR_eq [of "-c" a "-b"] assms
-  by simp
+  using pos_le_divideR_eq [of "-c" a "-b"] assms by simp

-lemma scaleR_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> (x::'a::ordered_real_vector) \<Longrightarrow> 0 \<le> a *\<^sub>R x"
-  using scaleR_left_mono [of 0 x a]
-  by simp
+lemma scaleR_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> a *\<^sub>R x"
+  for x :: "'a::ordered_real_vector"
+  using scaleR_left_mono [of 0 x a] by simp

-lemma scaleR_nonneg_nonpos: "0 \<le> a \<Longrightarrow> (x::'a::ordered_real_vector) \<le> 0 \<Longrightarrow> a *\<^sub>R x \<le> 0"
+lemma scaleR_nonneg_nonpos: "0 \<le> a \<Longrightarrow> x \<le> 0 \<Longrightarrow> a *\<^sub>R x \<le> 0"
+  for x :: "'a::ordered_real_vector"
using scaleR_left_mono [of x 0 a] by simp

-lemma scaleR_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> (x::'a::ordered_real_vector) \<Longrightarrow> a *\<^sub>R x \<le> 0"
+lemma scaleR_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> 0"
+  for x :: "'a::ordered_real_vector"
using scaleR_right_mono [of a 0 x] by simp

-lemma split_scaleR_neg_le: "(0 \<le> a & x \<le> 0) | (a \<le> 0 & 0 \<le> x) \<Longrightarrow>
-  a *\<^sub>R (x::'a::ordered_real_vector) \<le> 0"
+lemma split_scaleR_neg_le: "(0 \<le> a \<and> x \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> x) \<Longrightarrow> a *\<^sub>R x \<le> 0"
+  for x :: "'a::ordered_real_vector"
by (auto simp add: scaleR_nonneg_nonpos scaleR_nonpos_nonneg)

-  fixes c d e::"'a::ordered_real_vector"
-  shows "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> (a - b) *\<^sub>R e + c \<le> d"
+lemma le_add_iff1: "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> (a - b) *\<^sub>R e + c \<le> d"
+  for c d e :: "'a::ordered_real_vector"

-  fixes c d e::"'a::ordered_real_vector"
-  shows "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> c \<le> (b - a) *\<^sub>R e + d"
+lemma le_add_iff2: "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> c \<le> (b - a) *\<^sub>R e + d"
+  for c d e :: "'a::ordered_real_vector"

-lemma scaleR_left_mono_neg:
-  fixes a b::"'a::ordered_real_vector"
-  shows "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b"
+lemma scaleR_left_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b"
+  for a b :: "'a::ordered_real_vector"
apply (drule scaleR_left_mono [of _ _ "- c"])
-  apply simp_all
+   apply simp_all
done

-lemma scaleR_right_mono_neg:
-  fixes c::"'a::ordered_real_vector"
-  shows "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R c"
+lemma scaleR_right_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R c"
+  for c :: "'a::ordered_real_vector"
apply (drule scaleR_right_mono [of _ _ "- c"])
-  apply simp_all
+   apply simp_all
done

-lemma scaleR_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> (b::'a::ordered_real_vector) \<le> 0 \<Longrightarrow> 0 \<le> a *\<^sub>R b"
-using scaleR_right_mono_neg [of a 0 b] by simp
+lemma scaleR_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a *\<^sub>R b"
+  for b :: "'a::ordered_real_vector"
+  using scaleR_right_mono_neg [of a 0 b] by simp

-lemma split_scaleR_pos_le:
-  fixes b::"'a::ordered_real_vector"
-  shows "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a *\<^sub>R b"
+lemma split_scaleR_pos_le: "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a *\<^sub>R b"
+  for b :: "'a::ordered_real_vector"
by (auto simp add: scaleR_nonneg_nonneg scaleR_nonpos_nonpos)

lemma zero_le_scaleR_iff:
-  fixes b::"'a::ordered_real_vector"
-  shows "0 \<le> a *\<^sub>R b \<longleftrightarrow> 0 < a \<and> 0 \<le> b \<or> a < 0 \<and> b \<le> 0 \<or> a = 0" (is "?lhs = ?rhs")
-proof cases
-  assume "a \<noteq> 0"
+  fixes b :: "'a::ordered_real_vector"
+  shows "0 \<le> a *\<^sub>R b \<longleftrightarrow> 0 < a \<and> 0 \<le> b \<or> a < 0 \<and> b \<le> 0 \<or> a = 0"
+    (is "?lhs = ?rhs")
+proof (cases "a = 0")
+  case True
+  then show ?thesis by simp
+next
+  case False
show ?thesis
proof
-    assume lhs: ?lhs
-    {
-      assume "0 < a"
-      with lhs have "inverse a *\<^sub>R 0 \<le> inverse a *\<^sub>R (a *\<^sub>R b)"
+    assume ?lhs
+    from \<open>a \<noteq> 0\<close> consider "a > 0" | "a < 0" by arith
+    then show ?rhs
+    proof cases
+      case 1
+      with \<open>?lhs\<close> have "inverse a *\<^sub>R 0 \<le> inverse a *\<^sub>R (a *\<^sub>R b)"
by (intro scaleR_mono) auto
-      hence ?rhs using \<open>0 < a\<close>
+      with 1 show ?thesis
by simp
-    } moreover {
-      assume "0 > a"
-      with lhs have "- inverse a *\<^sub>R 0 \<le> - inverse a *\<^sub>R (a *\<^sub>R b)"
+    next
+      case 2
+      with \<open>?lhs\<close> have "- inverse a *\<^sub>R 0 \<le> - inverse a *\<^sub>R (a *\<^sub>R b)"
by (intro scaleR_mono) auto
-      hence ?rhs using \<open>0 > a\<close>
+      with 2 show ?thesis
by simp
-    } ultimately show ?rhs using \<open>a \<noteq> 0\<close> by arith
-  qed (auto simp: not_le \<open>a \<noteq> 0\<close> intro!: split_scaleR_pos_le)
-qed simp
+    qed
+  next
+    assume ?rhs
+    then show ?lhs
+      by (auto simp: not_le \<open>a \<noteq> 0\<close> intro!: split_scaleR_pos_le)
+  qed
+qed

-lemma scaleR_le_0_iff:
-  fixes b::"'a::ordered_real_vector"
-  shows "a *\<^sub>R b \<le> 0 \<longleftrightarrow> 0 < a \<and> b \<le> 0 \<or> a < 0 \<and> 0 \<le> b \<or> a = 0"
+lemma scaleR_le_0_iff: "a *\<^sub>R b \<le> 0 \<longleftrightarrow> 0 < a \<and> b \<le> 0 \<or> a < 0 \<and> 0 \<le> b \<or> a = 0"
+  for b::"'a::ordered_real_vector"
by (insert zero_le_scaleR_iff [of "-a" b]) force

-lemma scaleR_le_cancel_left:
-  fixes b::"'a::ordered_real_vector"
-  shows "c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
+lemma scaleR_le_cancel_left: "c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
+  for b :: "'a::ordered_real_vector"
by (auto simp add: neq_iff scaleR_left_mono scaleR_left_mono_neg
-    dest: scaleR_left_mono[where a="inverse c"] scaleR_left_mono_neg[where c="inverse c"])
+      dest: scaleR_left_mono[where a="inverse c"] scaleR_left_mono_neg[where c="inverse c"])

-lemma scaleR_le_cancel_left_pos:
-  fixes b::"'a::ordered_real_vector"
-  shows "0 < c \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> a \<le> b"
+lemma scaleR_le_cancel_left_pos: "0 < c \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> a \<le> b"
+  for b :: "'a::ordered_real_vector"
by (auto simp: scaleR_le_cancel_left)

-lemma scaleR_le_cancel_left_neg:
-  fixes b::"'a::ordered_real_vector"
-  shows "c < 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> b \<le> a"
+lemma scaleR_le_cancel_left_neg: "c < 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> b \<le> a"
+  for b :: "'a::ordered_real_vector"
by (auto simp: scaleR_le_cancel_left)

-lemma scaleR_left_le_one_le:
-  fixes x::"'a::ordered_real_vector" and a::real
-  shows "0 \<le> x \<Longrightarrow> a \<le> 1 \<Longrightarrow> a *\<^sub>R x \<le> x"
+lemma scaleR_left_le_one_le: "0 \<le> x \<Longrightarrow> a \<le> 1 \<Longrightarrow> a *\<^sub>R x \<le> x"
+  for x :: "'a::ordered_real_vector" and a :: real
using scaleR_right_mono[of a 1 x] by simp

@@ -704,8 +692,8 @@

class real_normed_vector = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +
assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
-  and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
-  and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
+    and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
+    and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
begin

lemma norm_ge_zero [simp]: "0 \<le> norm x"
@@ -724,9 +712,8 @@
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
assumes norm_one [simp]: "norm 1 = 1"

-lemma (in real_normed_algebra_1) scaleR_power [simp]:
-  "(scaleR x y) ^ n = scaleR (x^n) (y^n)"
-  by (induction n) (simp_all add: scaleR_one scaleR_scaleR mult_ac)
+lemma (in real_normed_algebra_1) scaleR_power [simp]: "(scaleR x y) ^ n = scaleR (x^n) (y^n)"
+  by (induct n) (simp_all add: scaleR_one scaleR_scaleR mult_ac)

class real_normed_div_algebra = real_div_algebra + real_normed_vector +
assumes norm_mult: "norm (x * y) = norm x * norm y"
@@ -735,36 +722,31 @@

instance real_normed_div_algebra < real_normed_algebra_1
proof
-  fix x y :: 'a
-  show "norm (x * y) \<le> norm x * norm y"
+  show "norm (x * y) \<le> norm x * norm y" for x y :: 'a
next
have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
by (rule norm_mult)
-  thus "norm (1::'a) = 1" by simp
+  then show "norm (1::'a) = 1" by simp
qed

lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
-by simp
+  by simp

-lemma zero_less_norm_iff [simp]:
-  fixes x :: "'a::real_normed_vector"
-  shows "(0 < norm x) = (x \<noteq> 0)"
+lemma zero_less_norm_iff [simp]: "norm x > 0 \<longleftrightarrow> x \<noteq> 0"
+  for x :: "'a::real_normed_vector"

-lemma norm_not_less_zero [simp]:
-  fixes x :: "'a::real_normed_vector"
-  shows "\<not> norm x < 0"
+lemma norm_not_less_zero [simp]: "\<not> norm x < 0"
+  for x :: "'a::real_normed_vector"

-lemma norm_le_zero_iff [simp]:
-  fixes x :: "'a::real_normed_vector"
-  shows "(norm x \<le> 0) = (x = 0)"
+lemma norm_le_zero_iff [simp]: "norm x \<le> 0 \<longleftrightarrow> x = 0"
+  for x :: "'a::real_normed_vector"

-lemma norm_minus_cancel [simp]:
-  fixes x :: "'a::real_normed_vector"
-  shows "norm (- x) = norm x"
+lemma norm_minus_cancel [simp]: "norm (- x) = norm x"
+  for x :: "'a::real_normed_vector"
proof -
have "norm (- x) = norm (scaleR (- 1) x)"
by (simp only: scaleR_minus_left scaleR_one)
@@ -773,78 +755,68 @@
finally show ?thesis by simp
qed

-lemma norm_minus_commute:
-  fixes a b :: "'a::real_normed_vector"
-  shows "norm (a - b) = norm (b - a)"
+lemma norm_minus_commute: "norm (a - b) = norm (b - a)"
+  for a b :: "'a::real_normed_vector"
proof -
have "norm (- (b - a)) = norm (b - a)"
by (rule norm_minus_cancel)
-  thus ?thesis by simp
+  then show ?thesis by simp
qed
-
-  fixes a :: "'a::real_normed_vector"
-  shows "dist (a + b) (a + c) = dist b c"
+
+lemma dist_add_cancel [simp]: "dist (a + b) (a + c) = dist b c"
+  for a :: "'a::real_normed_vector"

-  fixes a :: "'a::real_normed_vector"
-  shows "dist (b + a) (c + a) = dist b c"
+lemma dist_add_cancel2 [simp]: "dist (b + a) (c + a) = dist b c"
+  for a :: "'a::real_normed_vector"

-lemma dist_scaleR [simp]:
-  fixes a :: "'a::real_normed_vector"
-  shows "dist (x *\<^sub>R a) (y *\<^sub>R a) = abs (x-y) * norm a"
-by (metis dist_norm norm_scaleR scaleR_left.diff)
+lemma dist_scaleR [simp]: "dist (x *\<^sub>R a) (y *\<^sub>R a) = \<bar>x - y\<bar> * norm a"
+  for a :: "'a::real_normed_vector"
+  by (metis dist_norm norm_scaleR scaleR_left.diff)

-lemma norm_uminus_minus: "norm (-x - y :: 'a :: real_normed_vector) = norm (x + y)"
+lemma norm_uminus_minus: "norm (- x - y :: 'a :: real_normed_vector) = norm (x + y)"
by (subst (2) norm_minus_cancel[symmetric], subst minus_add_distrib) simp

-lemma norm_triangle_ineq2:
-  fixes a b :: "'a::real_normed_vector"
-  shows "norm a - norm b \<le> norm (a - b)"
+lemma norm_triangle_ineq2: "norm a - norm b \<le> norm (a - b)"
+  for a b :: "'a::real_normed_vector"
proof -
have "norm (a - b + b) \<le> norm (a - b) + norm b"
by (rule norm_triangle_ineq)
-  thus ?thesis by simp
+  then show ?thesis by simp
qed

-lemma norm_triangle_ineq3:
-  fixes a b :: "'a::real_normed_vector"
-  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
-apply (subst abs_le_iff)
-apply auto
-apply (rule norm_triangle_ineq2)
-apply (subst norm_minus_commute)
-apply (rule norm_triangle_ineq2)
-done
+lemma norm_triangle_ineq3: "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
+  for a b :: "'a::real_normed_vector"
+  apply (subst abs_le_iff)
+  apply auto
+   apply (rule norm_triangle_ineq2)
+  apply (subst norm_minus_commute)
+  apply (rule norm_triangle_ineq2)
+  done

-lemma norm_triangle_ineq4:
-  fixes a b :: "'a::real_normed_vector"
-  shows "norm (a - b) \<le> norm a + norm b"
+lemma norm_triangle_ineq4: "norm (a - b) \<le> norm a + norm b"
+  for a b :: "'a::real_normed_vector"
proof -
have "norm (a + - b) \<le> norm a + norm (- b)"
by (rule norm_triangle_ineq)
then show ?thesis by simp
qed

-lemma norm_diff_ineq:
-  fixes a b :: "'a::real_normed_vector"
-  shows "norm a - norm b \<le> norm (a + b)"
+lemma norm_diff_ineq: "norm a - norm b \<le> norm (a + b)"
+  for a b :: "'a::real_normed_vector"
proof -
have "norm a - norm (- b) \<le> norm (a - - b)"
by (rule norm_triangle_ineq2)
-  thus ?thesis by simp
+  then show ?thesis by simp
qed

-  fixes a b :: "'a::real_normed_vector"
-  shows "norm (a + b) \<le> c \<Longrightarrow> norm b \<le> norm a + c"
-    by (metis add.commute diff_le_eq norm_diff_ineq order.trans)
+lemma norm_add_leD: "norm (a + b) \<le> c \<Longrightarrow> norm b \<le> norm a + c"
+  for a b :: "'a::real_normed_vector"
+  by (metis add.commute diff_le_eq norm_diff_ineq order.trans)

-lemma norm_diff_triangle_ineq:
-  fixes a b c d :: "'a::real_normed_vector"
-  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
+lemma norm_diff_triangle_ineq: "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
+  for a b c d :: "'a::real_normed_vector"
proof -
have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
@@ -856,19 +828,19 @@
lemma norm_diff_triangle_le:
fixes x y z :: "'a::real_normed_vector"
assumes "norm (x - y) \<le> e1"  "norm (y - z) \<le> e2"
-    shows "norm (x - z) \<le> e1 + e2"
+  shows "norm (x - z) \<le> e1 + e2"
using norm_diff_triangle_ineq [of x y y z] assms by simp

lemma norm_diff_triangle_less:
fixes x y z :: "'a::real_normed_vector"
assumes "norm (x - y) < e1"  "norm (y - z) < e2"
-    shows "norm (x - z) < e1 + e2"
+  shows "norm (x - z) < e1 + e2"
using norm_diff_triangle_ineq [of x y y z] assms by simp

lemma norm_triangle_mono:
fixes a b :: "'a::real_normed_vector"
-  shows "\<lbrakk>norm a \<le> r; norm b \<le> s\<rbrakk> \<Longrightarrow> norm (a + b) \<le> r + s"
+  shows "norm a \<le> r \<Longrightarrow> norm b \<le> s \<Longrightarrow> norm (a + b) \<le> r + s"
+  by (metis add_mono_thms_linordered_semiring(1) norm_triangle_ineq order.trans)

lemma norm_setsum:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
@@ -881,82 +853,68 @@
shows "norm (setsum f S) \<le> setsum g S"
by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)

-lemma abs_norm_cancel [simp]:
-  fixes a :: "'a::real_normed_vector"
-  shows "\<bar>norm a\<bar> = norm a"
-by (rule abs_of_nonneg [OF norm_ge_zero])
+lemma abs_norm_cancel [simp]: "\<bar>norm a\<bar> = norm a"
+  for a :: "'a::real_normed_vector"
+  by (rule abs_of_nonneg [OF norm_ge_zero])

-  fixes x y :: "'a::real_normed_vector"
-  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
-by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
+lemma norm_add_less: "norm x < r \<Longrightarrow> norm y < s \<Longrightarrow> norm (x + y) < r + s"
+  for x y :: "'a::real_normed_vector"
+  by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])

-lemma norm_mult_less:
-  fixes x y :: "'a::real_normed_algebra"
-  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
-apply (rule order_le_less_trans [OF norm_mult_ineq])
-done
+lemma norm_mult_less: "norm x < r \<Longrightarrow> norm y < s \<Longrightarrow> norm (x * y) < r * s"
+  for x y :: "'a::real_normed_algebra"
+  by (rule order_le_less_trans [OF norm_mult_ineq]) (simp add: mult_strict_mono')

-lemma norm_of_real [simp]:
-  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
-unfolding of_real_def by simp
+lemma norm_of_real [simp]: "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"

-lemma norm_numeral [simp]:
-  "norm (numeral w::'a::real_normed_algebra_1) = numeral w"
-by (subst of_real_numeral [symmetric], subst norm_of_real, simp)
+lemma norm_numeral [simp]: "norm (numeral w::'a::real_normed_algebra_1) = numeral w"
+  by (subst of_real_numeral [symmetric], subst norm_of_real, simp)

-lemma norm_neg_numeral [simp]:
-  "norm (- numeral w::'a::real_normed_algebra_1) = numeral w"
-by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)
+lemma norm_neg_numeral [simp]: "norm (- numeral w::'a::real_normed_algebra_1) = numeral w"
+  by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)

-     "norm (of_real x + 1 :: 'a :: real_normed_div_algebra) = abs (x + 1)"
+lemma norm_of_real_add1 [simp]: "norm (of_real x + 1 :: 'a :: real_normed_div_algebra) = \<bar>x + 1\<bar>"

-     "norm (of_real x + numeral b :: 'a :: real_normed_div_algebra) = abs (x + numeral b)"
+  "norm (of_real x + numeral b :: 'a :: real_normed_div_algebra) = \<bar>x + numeral b\<bar>"

-lemma norm_of_int [simp]:
-  "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
-by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
+lemma norm_of_int [simp]: "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
+  by (subst of_real_of_int_eq [symmetric], rule norm_of_real)

-lemma norm_of_nat [simp]:
-  "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
-apply (subst of_real_of_nat_eq [symmetric])
-apply (subst norm_of_real, simp)
-done
+lemma norm_of_nat [simp]: "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
+  apply (subst of_real_of_nat_eq [symmetric])
+  apply (subst norm_of_real, simp)
+  done

-lemma nonzero_norm_inverse:
-  fixes a :: "'a::real_normed_div_algebra"
-  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
-apply (rule inverse_unique [symmetric])
-done
+lemma nonzero_norm_inverse: "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
+  for a :: "'a::real_normed_div_algebra"
+  apply (rule inverse_unique [symmetric])
+  apply (simp add: norm_mult [symmetric])
+  done

-lemma norm_inverse:
-  fixes a :: "'a::{real_normed_div_algebra, division_ring}"
-  shows "norm (inverse a) = inverse (norm a)"
-apply (case_tac "a = 0", simp)
-apply (erule nonzero_norm_inverse)
-done
+lemma norm_inverse: "norm (inverse a) = inverse (norm a)"
+  for a :: "'a::{real_normed_div_algebra,division_ring}"
+  apply (cases "a = 0")
+   apply simp
+  apply (erule nonzero_norm_inverse)
+  done

-lemma nonzero_norm_divide:
-  fixes a b :: "'a::real_normed_field"
-  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
-by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
+lemma nonzero_norm_divide: "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
+  for a b :: "'a::real_normed_field"
+  by (simp add: divide_inverse norm_mult nonzero_norm_inverse)

-lemma norm_divide:
-  fixes a b :: "'a::{real_normed_field, field}"
-  shows "norm (a / b) = norm a / norm b"
-by (simp add: divide_inverse norm_mult norm_inverse)
+lemma norm_divide: "norm (a / b) = norm a / norm b"
+  for a b :: "'a::{real_normed_field,field}"
+  by (simp add: divide_inverse norm_mult norm_inverse)

-lemma norm_power_ineq:
-  fixes x :: "'a::{real_normed_algebra_1}"
-  shows "norm (x ^ n) \<le> norm x ^ n"
+lemma norm_power_ineq: "norm (x ^ n) \<le> norm x ^ n"
+  for x :: "'a::real_normed_algebra_1"
proof (induct n)
-  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
+  case 0
+  show "norm (x ^ 0) \<le> norm x ^ 0" by simp
next
case (Suc n)
have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
@@ -967,10 +925,9 @@
by simp
qed

-lemma norm_power:
-  fixes x :: "'a::real_normed_div_algebra"
-  shows "norm (x ^ n) = norm x ^ n"
-by (induct n) (simp_all add: norm_mult)
+lemma norm_power: "norm (x ^ n) = norm x ^ n"
+  for x :: "'a::real_normed_div_algebra"
+  by (induct n) (simp_all add: norm_mult)

lemma power_eq_imp_eq_norm:
fixes w :: "'a::real_normed_div_algebra"
@@ -983,34 +940,31 @@
using assms by (force intro: power_eq_imp_eq_base)
qed

-lemma norm_mult_numeral1 [simp]:
-  fixes a b :: "'a::{real_normed_field, field}"
-  shows "norm (numeral w * a) = numeral w * norm a"
+lemma norm_mult_numeral1 [simp]: "norm (numeral w * a) = numeral w * norm a"
+  for a b :: "'a::{real_normed_field,field}"

-lemma norm_mult_numeral2 [simp]:
-  fixes a b :: "'a::{real_normed_field, field}"
-  shows "norm (a * numeral w) = norm a * numeral w"
+lemma norm_mult_numeral2 [simp]: "norm (a * numeral w) = norm a * numeral w"
+  for a b :: "'a::{real_normed_field,field}"

-lemma norm_divide_numeral [simp]:
-  fixes a b :: "'a::{real_normed_field, field}"
-  shows "norm (a / numeral w) = norm a / numeral w"
+lemma norm_divide_numeral [simp]: "norm (a / numeral w) = norm a / numeral w"
+  for a b :: "'a::{real_normed_field,field}"

lemma norm_of_real_diff [simp]:
-    "norm (of_real b - of_real a :: 'a::real_normed_algebra_1) \<le> \<bar>b - a\<bar>"
+  "norm (of_real b - of_real a :: 'a::real_normed_algebra_1) \<le> \<bar>b - a\<bar>"
by (metis norm_of_real of_real_diff order_refl)

-text\<open>Despite a superficial resemblance, \<open>norm_eq_1\<close> is not relevant.\<close>
+text \<open>Despite a superficial resemblance, \<open>norm_eq_1\<close> is not relevant.\<close>
lemma square_norm_one:
fixes x :: "'a::real_normed_div_algebra"
-  assumes "x^2 = 1" shows "norm x = 1"
+  assumes "x\<^sup>2 = 1"
+  shows "norm x = 1"
by (metis assms norm_minus_cancel norm_one power2_eq_1_iff)

-lemma norm_less_p1:
-  fixes x :: "'a::real_normed_algebra_1"
-  shows "norm x < norm (of_real (norm x) + 1 :: 'a)"
+lemma norm_less_p1: "norm x < norm (of_real (norm x) + 1 :: 'a)"
+  for x :: "'a::real_normed_algebra_1"
proof -
have "norm x < norm (of_real (norm x + 1) :: 'a)"
@@ -1018,14 +972,16 @@
by simp
qed

-lemma setprod_norm:
-  fixes f :: "'a \<Rightarrow> 'b::{comm_semiring_1,real_normed_div_algebra}"
-  shows "setprod (\<lambda>x. norm(f x)) A = norm (setprod f A)"
+lemma setprod_norm: "setprod (\<lambda>x. norm (f x)) A = norm (setprod f A)"
+  for f :: "'a \<Rightarrow> 'b::{comm_semiring_1,real_normed_div_algebra}"
by (induct A rule: infinite_finite_induct) (auto simp: norm_mult)

lemma norm_setprod_le:
-  "norm (setprod f A) \<le> (\<Prod>a\<in>A. norm (f a :: 'a :: {real_normed_algebra_1, comm_monoid_mult}))"
-proof (induction A rule: infinite_finite_induct)
+  "norm (setprod f A) \<le> (\<Prod>a\<in>A. norm (f a :: 'a :: {real_normed_algebra_1,comm_monoid_mult}))"
+proof (induct A rule: infinite_finite_induct)
+  case empty
+  then show ?case by simp
+next
case (insert a A)
then have "norm (setprod f (insert a A)) \<le> norm (f a) * norm (setprod f A)"
@@ -1033,13 +989,19 @@
by (rule insert)
finally show ?case
-qed simp_all
+next
+  case infinite
+  then show ?case by simp
+qed

lemma norm_setprod_diff:
fixes z w :: "'i \<Rightarrow> 'a::{real_normed_algebra_1, comm_monoid_mult}"
shows "(\<And>i. i \<in> I \<Longrightarrow> norm (z i) \<le> 1) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> norm (w i) \<le> 1) \<Longrightarrow>
norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) \<le> (\<Sum>i\<in>I. norm (z i - w i))"
proof (induction I rule: infinite_finite_induct)
+  case empty
+  then show ?case by simp
+next
case (insert i I)
note insert.hyps[simp]

@@ -1047,7 +1009,7 @@
norm ((\<Prod>i\<in>I. z i) * (z i - w i) + ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) * w i)"
(is "_ = norm (?t1 + ?t2)")
-  also have "... \<le> norm ?t1 + norm ?t2"
+  also have "\<dots> \<le> norm ?t1 + norm ?t2"
by (rule norm_triangle_ineq)
also have "norm ?t1 \<le> norm (\<Prod>i\<in>I. z i) * norm (z i - w i)"
by (rule norm_mult_ineq)
@@ -1063,7 +1025,10 @@
using insert by auto
finally show ?case
by (auto simp add: ac_simps mult_right_mono mult_left_mono)
-qed simp_all
+next
+  case infinite
+  then show ?case by simp
+qed

lemma norm_power_diff:
fixes z w :: "'a::{real_normed_algebra_1, comm_monoid_mult}"
@@ -1079,27 +1044,28 @@
finally show ?thesis .
qed

+
subsection \<open>Metric spaces\<close>

class metric_space = uniformity_dist + open_uniformity +
assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
-  assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
+    and dist_triangle2: "dist x y \<le> dist x z + dist y z"
begin

lemma dist_self [simp]: "dist x x = 0"
-by simp
+  by simp

lemma zero_le_dist [simp]: "0 \<le> dist x y"
-using dist_triangle2 [of x x y] by simp
+  using dist_triangle2 [of x x y] by simp

lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"

lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"

lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"

lemma dist_commute: "dist x y = dist y x"
proof (rule order_antisym)
@@ -1118,56 +1084,52 @@
lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
using dist_triangle2 [of x y a] by (simp add: dist_commute)

-lemma dist_pos_lt:
-  shows "x \<noteq> y ==> 0 < dist x y"
+lemma dist_pos_lt: "x \<noteq> y \<Longrightarrow> 0 < dist x y"

-lemma dist_nz:
-  shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
+lemma dist_nz: "x \<noteq> y \<longleftrightarrow> 0 < dist x y"

declare dist_nz [symmetric, simp]

-lemma dist_triangle_le:
-  shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
-by (rule order_trans [OF dist_triangle2])
+lemma dist_triangle_le: "dist x z + dist y z \<le> e \<Longrightarrow> dist x y \<le> e"
+  by (rule order_trans [OF dist_triangle2])

-lemma dist_triangle_lt:
-  shows "dist x z + dist y z < e ==> dist x y < e"
-by (rule le_less_trans [OF dist_triangle2])
+lemma dist_triangle_lt: "dist x z + dist y z < e \<Longrightarrow> dist x y < e"
+  by (rule le_less_trans [OF dist_triangle2])

-   "\<lbrakk>dist x1 y < e1; dist x2 y < e2\<rbrakk> \<Longrightarrow> dist x1 x2 < e1 + e2"
-by (rule dist_triangle_lt [where z=y], simp)
+lemma dist_triangle_less_add: "dist x1 y < e1 \<Longrightarrow> dist x2 y < e2 \<Longrightarrow> dist x1 x2 < e1 + e2"
+  by (rule dist_triangle_lt [where z=y]) simp

-lemma dist_triangle_half_l:
-  shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
-by (rule dist_triangle_lt [where z=y], simp)
+lemma dist_triangle_half_l: "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
+  by (rule dist_triangle_lt [where z=y]) simp

-lemma dist_triangle_half_r:
-  shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
-by (rule dist_triangle_half_l, simp_all add: dist_commute)
+lemma dist_triangle_half_r: "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
+  by (rule dist_triangle_half_l) (simp_all add: dist_commute)

subclass uniform_space
proof
-  fix E x assume "eventually E uniformity"
+  fix E x
+  assume "eventually E uniformity"
then obtain e where E: "0 < e" "\<And>x y. dist x y < e \<Longrightarrow> E (x, y)"
-    unfolding eventually_uniformity_metric by auto
+    by (auto simp: eventually_uniformity_metric)
then show "E (x, x)" "\<forall>\<^sub>F (x, y) in uniformity. E (y, x)"
-    unfolding eventually_uniformity_metric by (auto simp: dist_commute)
-
+    by (auto simp: eventually_uniformity_metric dist_commute)
show "\<exists>D. eventually D uniformity \<and> (\<forall>x y z. D (x, y) \<longrightarrow> D (y, z) \<longrightarrow> E (x, z))"
-    using E dist_triangle_half_l[where e=e] unfolding eventually_uniformity_metric
+    using E dist_triangle_half_l[where e=e]
+    unfolding eventually_uniformity_metric
by (intro exI[of _ "\<lambda>(x, y). dist x y < e / 2"] exI[of _ "e/2"] conjI)
-       (auto simp: dist_commute)
+      (auto simp: dist_commute)
qed

lemma open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
-  unfolding open_uniformity eventually_uniformity_metric by (simp add: dist_commute)
+  by (simp add: dist_commute open_uniformity eventually_uniformity_metric)

lemma open_ball: "open {y. dist x y < d}"
-proof (unfold open_dist, intro ballI)
-  fix y assume *: "y \<in> {y. dist x y < d}"
+  unfolding open_dist
+proof (intro ballI)
+  fix y
+  assume *: "y \<in> {y. dist x y < d}"
then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
qed
@@ -1177,7 +1139,8 @@
fix x
show "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])
-    fix S assume "open S" "x \<in> S"
+    fix S
+    assume "open S" "x \<in> S"
then obtain e where e: "0 < e" and "{y. dist x y < e} \<subseteq> S"
by (auto simp: open_dist subset_eq dist_commute)
moreover
@@ -1198,34 +1161,33 @@
assume xy: "x \<noteq> y"
let ?U = "{y'. dist x y' < dist x y / 2}"
let ?V = "{x'. dist y x' < dist x y / 2}"
-  have th0: "\<And>d x y z. (d x z :: real) \<le> d x y + d y z \<Longrightarrow> d y z = d z y
-               \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
+  have *: "d x z \<le> d x y + d y z \<Longrightarrow> d y z = d z y \<Longrightarrow> \<not> (d x y * 2 < d x z \<and> d z y * 2 < d x z)"
+    for d :: "'a \<Rightarrow> 'a \<Rightarrow> real" and x y z :: 'a
+    by arith
have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
-    using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
+    using dist_pos_lt[OF xy] *[of dist, OF dist_triangle dist_commute]
using open_ball[of _ "dist x y / 2"] by auto
then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
by blast
qed

text \<open>Every normed vector space is a metric space.\<close>
-
instance real_normed_vector < metric_space
proof
-  fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
-    unfolding dist_norm by simp
-next
-  fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
-    unfolding dist_norm
-    using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
+  fix x y z :: 'a
+  show "dist x y = 0 \<longleftrightarrow> x = y"
+  show "dist x y \<le> dist x z + dist y z"
+    using norm_triangle_ineq4 [of "x - z" "y - z"] by (simp add: dist_norm)
qed

+
subsection \<open>Class instances for real numbers\<close>

instantiation real :: real_normed_field
begin

-definition dist_real_def:
-  "dist x y = \<bar>x - y\<bar>"
+definition dist_real_def: "dist x y = \<bar>x - y\<bar>"

definition uniformity_real_def [code del]:
"(uniformity :: (real \<times> real) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"
@@ -1233,29 +1195,28 @@
definition open_real_def [code del]:
"open (U :: real set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"

-definition real_norm_def [simp]:
-  "norm r = \<bar>r\<bar>"
+definition real_norm_def [simp]: "norm r = \<bar>r\<bar>"

instance
-apply (intro_classes, unfold real_norm_def real_scaleR_def)
-apply (rule dist_real_def)
-apply (rule uniformity_real_def)
-apply (rule open_real_def)
-apply (rule abs_eq_0)
-apply (rule abs_triangle_ineq)
-apply (rule abs_mult)
-apply (rule abs_mult)
-done
+  apply intro_classes
+         apply (unfold real_norm_def real_scaleR_def)
+         apply (rule dist_real_def)
+       apply (rule uniformity_real_def)
+      apply (rule open_real_def)
+     apply (rule abs_eq_0)
+    apply (rule abs_triangle_ineq)
+   apply (rule abs_mult)
+  apply (rule abs_mult)
+  done

end

declare uniformity_Abort[where 'a=real, code]

-lemma dist_of_real [simp]:
-  fixes a :: "'a::real_normed_div_algebra"
-  shows "dist (of_real x :: 'a) (of_real y) = dist x y"
-by (metis dist_norm norm_of_real of_real_diff real_norm_def)
+lemma dist_of_real [simp]: "dist (of_real x :: 'a) (of_real y) = dist x y"
+  for a :: "'a::real_normed_div_algebra"
+  by (metis dist_norm norm_of_real of_real_diff real_norm_def)

declare [[code abort: "open :: real set \<Rightarrow> bool"]]

@@ -1263,7 +1224,8 @@
proof
show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"
proof (rule ext, safe)
-    fix S :: "real set" assume "open S"
+    fix S :: "real set"
+    assume "open S"
then obtain f where "\<forall>x\<in>S. 0 < f x \<and> (\<forall>y. dist y x < f x \<longrightarrow> y \<in> S)"
unfolding open_dist bchoice_iff ..
then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"
@@ -1271,23 +1233,26 @@
show "generate_topology (range lessThan \<union> range greaterThan) S"
apply (subst *)
apply (intro generate_topology_Union generate_topology.Int)
-      apply (auto intro: generate_topology.Basis)
+       apply (auto intro: generate_topology.Basis)
done
next
-    fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"
+    fix S :: "real set"
+    assume "generate_topology (range lessThan \<union> range greaterThan) S"
moreover have "\<And>a::real. open {..<a}"
unfolding open_dist dist_real_def
proof clarify
-      fix x a :: real assume "x < a"
-      hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
-      thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
+      fix x a :: real
+      assume "x < a"
+      then have "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
+      then show "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
qed
moreover have "\<And>a::real. open {a <..}"
unfolding open_dist dist_real_def
proof clarify
-      fix x a :: real assume "a < x"
-      hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
-      thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
+      fix x a :: real
+      assume "a < x"
+      then have "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
+      then show "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
qed
ultimately show "open S"
by induct auto
@@ -1303,66 +1268,67 @@
lemmas closed_real_atLeast = closed_atLeast[where 'a=real]
lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]

+
subsection \<open>Extra type constraints\<close>

text \<open>Only allow @{term "open"} in class \<open>topological_space\<close>.\<close>
-
(@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"})\<close>

text \<open>Only allow @{term "uniformity"} in class \<open>uniform_space\<close>.\<close>
-
(@{const_name "uniformity"}, SOME @{typ "('a::uniformity \<times> 'a) filter"})\<close>

text \<open>Only allow @{term dist} in class \<open>metric_space\<close>.\<close>
-
(@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"})\<close>

text \<open>Only allow @{term norm} in class \<open>real_normed_vector\<close>.\<close>
-
(@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"})\<close>

+
subsection \<open>Sign function\<close>

-lemma norm_sgn:
-  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
+lemma norm_sgn: "norm (sgn x) = (if x = 0 then 0 else 1)"
+  for x :: "'a::real_normed_vector"

-lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
+lemma sgn_zero [simp]: "sgn (0::'a::real_normed_vector) = 0"

-lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
+lemma sgn_zero_iff: "sgn x = 0 \<longleftrightarrow> x = 0"
+  for x :: "'a::real_normed_vector"

-lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
+lemma sgn_minus: "sgn (- x) = - sgn x"
+  for x :: "'a::real_normed_vector"

-lemma sgn_scaleR:
-  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
+lemma sgn_scaleR: "sgn (scaleR r x) = scaleR (sgn r) (sgn x)"
+  for x :: "'a::real_normed_vector"
+  by (simp add: sgn_div_norm ac_simps)

lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"

-lemma sgn_of_real:
-  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
-unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
+lemma sgn_of_real: "sgn (of_real r :: 'a::real_normed_algebra_1) = of_real (sgn r)"
+  unfolding of_real_def by (simp only: sgn_scaleR sgn_one)

-lemma sgn_mult:
-  fixes x y :: "'a::real_normed_div_algebra"
-  shows "sgn (x * y) = sgn x * sgn y"
-by (simp add: sgn_div_norm norm_mult mult.commute)
+lemma sgn_mult: "sgn (x * y) = sgn x * sgn y"
+  for x y :: "'a::real_normed_div_algebra"
+  by (simp add: sgn_div_norm norm_mult mult.commute)

-lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
+lemma real_sgn_eq: "sgn x = x / \<bar>x\<bar>"
+  for x :: real

-lemma zero_le_sgn_iff [simp]: "0 \<le> sgn x \<longleftrightarrow> 0 \<le> (x::real)"
+lemma zero_le_sgn_iff [simp]: "0 \<le> sgn x \<longleftrightarrow> 0 \<le> x"
+  for x :: real
by (cases "0::real" x rule: linorder_cases) simp_all

-lemma sgn_le_0_iff [simp]: "sgn x \<le> 0 \<longleftrightarrow> (x::real) \<le> 0"
+lemma sgn_le_0_iff [simp]: "sgn x \<le> 0 \<longleftrightarrow> x \<le> 0"
+  for x :: real
by (cases "0::real" x rule: linorder_cases) simp_all

lemma norm_conv_dist: "norm x = dist x 0"
@@ -1370,10 +1336,9 @@

declare norm_conv_dist [symmetric, simp]

-lemma dist_0_norm [simp]:
-  fixes x :: "'a::real_normed_vector"
-  shows "dist 0 x = norm x"
-unfolding dist_norm by simp
+lemma dist_0_norm [simp]: "dist 0 x = norm x"
+  for x :: "'a::real_normed_vector"

lemma dist_diff [simp]: "dist a (a - b) = norm b"  "dist (a - b) a = norm b"
@@ -1390,23 +1355,25 @@
"dist (of_nat m) (of_nat n :: 'a :: real_normed_algebra_1) = of_int \<bar>int m - int n\<bar>"
by (subst (1 2) of_int_of_nat_eq [symmetric]) (rule dist_of_int)

+
subsection \<open>Bounded Linear and Bilinear Operators\<close>

locale linear = additive f for f :: "'a::real_vector \<Rightarrow> 'b::real_vector" +
assumes scaleR: "f (scaleR r x) = scaleR r (f x)"

lemma linear_imp_scaleR:
-  assumes "linear D" obtains d where "D = (\<lambda>x. x *\<^sub>R d)"
+  assumes "linear D"
+  obtains d where "D = (\<lambda>x. x *\<^sub>R d)"
by (metis assms linear.scaleR mult.commute mult.left_neutral real_scaleR_def)

corollary real_linearD:
fixes f :: "real \<Rightarrow> real"
assumes "linear f" obtains c where "f = op* c"
-by (rule linear_imp_scaleR [OF assms]) (force simp: scaleR_conv_of_real)
+  by (rule linear_imp_scaleR [OF assms]) (force simp: scaleR_conv_of_real)

lemma linearI:
assumes "\<And>x y. f (x + y) = f x + f y"
-  assumes "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
+    and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
shows "linear f"
by standard (rule assms)+

@@ -1414,8 +1381,7 @@
assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
begin

-lemma pos_bounded:
-  "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
+lemma pos_bounded: "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
proof -
obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
using bounded by blast
@@ -1432,118 +1398,107 @@
qed
qed

-lemma nonneg_bounded:
-  "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
-proof -
-  from pos_bounded
-  show ?thesis by (auto intro: order_less_imp_le)
-qed
+lemma nonneg_bounded: "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
+  using pos_bounded by (auto intro: order_less_imp_le)

-lemma linear: "linear f"
+lemma linear: "linear f"
by (fact local.linear_axioms)

end

lemma bounded_linear_intro:
assumes "\<And>x y. f (x + y) = f x + f y"
-  assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
-  assumes "\<And>x. norm (f x) \<le> norm x * K"
+    and "\<And>r x. f (scaleR r x) = scaleR r (f x)"
+    and "\<And>x. norm (f x) \<le> norm x * K"
shows "bounded_linear f"
by standard (blast intro: assms)+

locale bounded_bilinear =
-  fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
-                 \<Rightarrow> 'c::real_normed_vector"
+  fixes prod :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector \<Rightarrow> 'c::real_normed_vector"
(infixl "**" 70)
assumes add_left: "prod (a + a') b = prod a b + prod a' b"
-  assumes add_right: "prod a (b + b') = prod a b + prod a b'"
-  assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
-  assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
-  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
+    and add_right: "prod a (b + b') = prod a b + prod a b'"
+    and scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
+    and scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
+    and bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
begin

-lemma pos_bounded:
-  "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
-apply (cut_tac bounded, erule exE)
-apply (rule_tac x="max 1 K" in exI, safe)
-apply (rule order_less_le_trans [OF zero_less_one max.cobounded1])
-apply (drule spec, drule spec, erule order_trans)
-apply (rule mult_left_mono [OF max.cobounded2])
-apply (intro mult_nonneg_nonneg norm_ge_zero)
-done
+lemma pos_bounded: "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
+  apply (insert bounded)
+  apply (erule exE)
+  apply (rule_tac x="max 1 K" in exI)
+  apply safe
+   apply (rule order_less_le_trans [OF zero_less_one max.cobounded1])
+  apply (drule spec)
+  apply (drule spec)
+  apply (erule order_trans)
+  apply (rule mult_left_mono [OF max.cobounded2])
+  apply (intro mult_nonneg_nonneg norm_ge_zero)
+  done

-lemma nonneg_bounded:
-  "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
-proof -
-  from pos_bounded
-  show ?thesis by (auto intro: order_less_imp_le)
-qed
+lemma nonneg_bounded: "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
+  using pos_bounded by (auto intro: order_less_imp_le)

lemma zero_left: "prod 0 b = 0"

lemma zero_right: "prod a 0 = 0"

lemma minus_left: "prod (- a) b = - prod a b"

lemma minus_right: "prod a (- b) = - prod a b"

-lemma diff_left:
-  "prod (a - a') b = prod a b - prod a' b"
+lemma diff_left: "prod (a - a') b = prod a b - prod a' b"

-lemma diff_right:
-  "prod a (b - b') = prod a b - prod a b'"
+lemma diff_right: "prod a (b - b') = prod a b - prod a b'"

-lemma setsum_left:
-  "prod (setsum g S) x = setsum ((\<lambda>i. prod (g i) x)) S"
+lemma setsum_left: "prod (setsum g S) x = setsum ((\<lambda>i. prod (g i) x)) S"

-lemma setsum_right:
-  "prod x (setsum g S) = setsum ((\<lambda>i. (prod x (g i)))) S"
+lemma setsum_right: "prod x (setsum g S) = setsum ((\<lambda>i. (prod x (g i)))) S"

-lemma bounded_linear_left:
-  "bounded_linear (\<lambda>a. a ** b)"
-apply (cut_tac bounded, safe)
-apply (rule_tac K="norm b * K" in bounded_linear_intro)
-apply (rule scaleR_left)
-done
+lemma bounded_linear_left: "bounded_linear (\<lambda>a. a ** b)"
+  apply (insert bounded)
+  apply safe
+  apply (rule_tac K="norm b * K" in bounded_linear_intro)
+   apply (rule scaleR_left)
+  done

-lemma bounded_linear_right:
-  "bounded_linear (\<lambda>b. a ** b)"
-apply (cut_tac bounded, safe)
-apply (rule_tac K="norm a * K" in bounded_linear_intro)
-apply (rule scaleR_right)
-done
+lemma bounded_linear_right: "bounded_linear (\<lambda>b. a ** b)"
+  apply (insert bounded)
+  apply safe
+  apply (rule_tac K="norm a * K" in bounded_linear_intro)
+   apply (rule scaleR_right)
+  done

-lemma prod_diff_prod:
-  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
+lemma prod_diff_prod: "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
+  by (simp add: diff_left diff_right)

lemma flip: "bounded_bilinear (\<lambda>x y. y ** x)"
apply standard
-  apply (rule scaleR_right)
-  apply (rule scaleR_left)
+    apply (rule scaleR_right)
+   apply (rule scaleR_left)
apply (subst mult.commute)
-  using bounded
+  apply (insert bounded)
apply blast
done

@@ -1557,11 +1512,10 @@
"\<And>r a b. g (r *\<^sub>R a) ** b = r *\<^sub>R (g a ** b)"
"\<And>a r b. g a ** (r *\<^sub>R b) = r *\<^sub>R (g a ** b)"
-  from g.nonneg_bounded nonneg_bounded
-  obtain K L
-  where nn: "0 \<le> K" "0 \<le> L"
-    and K: "\<And>x. norm (g x) \<le> norm x * K"
-    and L: "\<And>a b. norm (a ** b) \<le> norm a * norm b * L"
+  from g.nonneg_bounded nonneg_bounded obtain K L
+    where nn: "0 \<le> K" "0 \<le> L"
+      and K: "\<And>x. norm (g x) \<le> norm x * K"
+      and L: "\<And>a b. norm (a ** b) \<le> norm a * norm b * L"
by auto
have "norm (g a ** b) \<le> norm a * K * norm b * L" for a b
by (auto intro!:  order_trans[OF K] order_trans[OF L] mult_mono simp: nn)
@@ -1569,8 +1523,7 @@
by (auto intro!: exI[where x="K * L"] simp: ac_simps)
qed

-lemma comp:
-  "bounded_linear f \<Longrightarrow> bounded_linear g \<Longrightarrow> bounded_bilinear (\<lambda>x y. f x ** g y)"
+lemma comp: "bounded_linear f \<Longrightarrow> bounded_linear g \<Longrightarrow> bounded_bilinear (\<lambda>x y. f x ** g y)"
by (rule bounded_bilinear.flip[OF bounded_bilinear.comp1[OF bounded_bilinear.flip[OF comp1]]])

end
@@ -1583,15 +1536,17 @@

assumes "bounded_linear f"
-  assumes "bounded_linear g"
+    and "bounded_linear g"
shows "bounded_linear (\<lambda>x. f x + g x)"
proof -
interpret f: bounded_linear f by fact
interpret g: bounded_linear g by fact
show ?thesis
proof
-    from f.bounded obtain Kf where Kf: "\<And>x. norm (f x) \<le> norm x * Kf" by blast
-    from g.bounded obtain Kg where Kg: "\<And>x. norm (g x) \<le> norm x * Kg" by blast
+    from f.bounded obtain Kf where Kf: "norm (f x) \<le> norm x * Kf" for x
+      by blast
+    from g.bounded obtain Kg where Kg: "norm (g x) \<le> norm x * Kg" for x
+      by blast
show "\<exists>K. \<forall>x. norm (f x + g x) \<le> norm x * K"
by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans)
@@ -1603,9 +1558,10 @@
shows "bounded_linear (\<lambda>x. - f x)"
proof -
interpret f: bounded_linear f by fact
-  show ?thesis apply (unfold_locales)
+  show ?thesis
+    apply unfold_locales
done
qed
@@ -1618,31 +1574,32 @@
fixes f :: "'i \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes "\<And>i. i \<in> I \<Longrightarrow> bounded_linear (f i)"
shows "bounded_linear (\<lambda>x. \<Sum>i\<in>I. f i x)"
-proof cases
-  assume "finite I"
-  from this show ?thesis
-    using assms
-    by (induct I) (auto intro!: bounded_linear_add)
-qed simp
+proof (cases "finite I")
+  case True
+  then show ?thesis
+    using assms by induct (auto intro!: bounded_linear_add)
+next
+  case False
+  then show ?thesis by simp
+qed

lemma bounded_linear_compose:
assumes "bounded_linear f"
-  assumes "bounded_linear g"
+    and "bounded_linear g"
shows "bounded_linear (\<lambda>x. f (g x))"
proof -
interpret f: bounded_linear f by fact
interpret g: bounded_linear g by fact
-  show ?thesis proof (unfold_locales)
-    fix x y show "f (g (x + y)) = f (g x) + f (g y)"
+  show ?thesis
+  proof unfold_locales
+    show "f (g (x + y)) = f (g x) + f (g y)" for x y
-  next
-    fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))"
+    show "f (g (scaleR r x)) = scaleR r (f (g x))" for r x
by (simp only: f.scaleR g.scaleR)
-  next
-    from f.pos_bounded
-    obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf" by blast
-    from g.pos_bounded
-    obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg" by blast
+    from f.pos_bounded obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf"
+      by blast
+    from g.pos_bounded obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg"
+      by blast
show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
proof (intro exI allI)
fix x
@@ -1657,24 +1614,21 @@
qed
qed

-lemma bounded_bilinear_mult:
-  "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
-apply (rule bounded_bilinear.intro)
-apply (rule distrib_right)
-apply (rule distrib_left)
-apply (rule mult_scaleR_left)
-apply (rule mult_scaleR_right)
-apply (rule_tac x="1" in exI)
-done
+lemma bounded_bilinear_mult: "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
+  apply (rule bounded_bilinear.intro)
+      apply (rule distrib_right)
+     apply (rule distrib_left)
+    apply (rule mult_scaleR_left)
+   apply (rule mult_scaleR_right)
+  apply (rule_tac x="1" in exI)
+  done

-lemma bounded_linear_mult_left:
-  "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
+lemma bounded_linear_mult_left: "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
using bounded_bilinear_mult
by (rule bounded_bilinear.bounded_linear_left)

-lemma bounded_linear_mult_right:
-  "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
+lemma bounded_linear_mult_right: "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
using bounded_bilinear_mult
by (rule bounded_bilinear.bounded_linear_right)

@@ -1684,18 +1638,19 @@
lemmas bounded_linear_const_mult =
bounded_linear_mult_right [THEN bounded_linear_compose]

-lemma bounded_linear_divide:
-  "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"
+lemma bounded_linear_divide: "bounded_linear (\<lambda>x. x / y)"
+  for y :: "'a::real_normed_field"
unfolding divide_inverse by (rule bounded_linear_mult_left)

lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
-apply (rule bounded_bilinear.intro)
-apply (rule scaleR_left_distrib)
-apply (rule scaleR_right_distrib)
-apply simp
-apply (rule scaleR_left_commute)
-apply (rule_tac x="1" in exI, simp)
-done
+  apply (rule bounded_bilinear.intro)
+      apply (rule scaleR_left_distrib)
+     apply (rule scaleR_right_distrib)
+    apply simp
+   apply (rule scaleR_left_commute)
+  apply (rule_tac x="1" in exI)
+  apply simp
+  done

lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"
using bounded_bilinear_scaleR
@@ -1714,48 +1669,53 @@
lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
unfolding of_real_def by (rule bounded_linear_scaleR_left)

-lemma real_bounded_linear:
-  fixes f :: "real \<Rightarrow> real"
-  shows "bounded_linear f \<longleftrightarrow> (\<exists>c::real. f = (\<lambda>x. x * c))"
+lemma real_bounded_linear: "bounded_linear f \<longleftrightarrow> (\<exists>c::real. f = (\<lambda>x. x * c))"
+  for f :: "real \<Rightarrow> real"
proof -
-  { fix x assume "bounded_linear f"
+  {
+    fix x
+    assume "bounded_linear f"
then interpret bounded_linear f .
from scaleR[of x 1] have "f x = x * f 1"
-      by simp }
+      by simp
+  }
then show ?thesis
by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left)
qed

-lemma bij_linear_imp_inv_linear:
-  assumes "linear f" "bij f" shows "linear (inv f)"
-  using assms unfolding linear_def linear_axioms_def additive_def
-  by (auto simp: bij_is_surj bij_is_inj surj_f_inv_f intro!:  Hilbert_Choice.inv_f_eq)
+lemma bij_linear_imp_inv_linear: "linear f \<Longrightarrow> bij f \<Longrightarrow> linear (inv f)"
+  by (auto simp: linear_def linear_axioms_def additive_def bij_is_surj bij_is_inj surj_f_inv_f
+      intro!:  Hilbert_Choice.inv_f_eq)

instance real_normed_algebra_1 \<subseteq> perfect_space
proof
-  fix x::'a
-  show "\<not> open {x}"
-    unfolding open_dist dist_norm
-    by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
+  show "\<not> open {x}" for x :: 'a
+    apply (simp only: open_dist dist_norm)
+    apply clarsimp
+    apply (rule_tac x = "x + of_real (e/2)" in exI)
+    apply simp
+    done
qed

+
subsection \<open>Filters and Limits on Metric Space\<close>

lemma (in metric_space) nhds_metric: "nhds x = (INF e:{0 <..}. principal {y. dist y x < e})"
unfolding nhds_def
proof (safe intro!: INF_eq)
-  fix S assume "open S" "x \<in> S"
+  fix S
+  assume "open S" "x \<in> S"
then obtain e where "{y. dist y x < e} \<subseteq> S" "0 < e"
by (auto simp: open_dist subset_eq)
then show "\<exists>e\<in>{0<..}. principal {y. dist y x < e} \<le> principal S"
by auto
qed (auto intro!: exI[of _ "{y. dist x y < e}" for e] open_ball simp: dist_commute)

-lemma (in metric_space) tendsto_iff:
-  "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
+lemma (in metric_space) tendsto_iff: "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
unfolding nhds_metric filterlim_INF filterlim_principal by auto

-lemma (in metric_space) tendstoI [intro?]: "(\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F) \<Longrightarrow> (f \<longlongrightarrow> l) F"
+lemma (in metric_space) tendstoI [intro?]:
+  "(\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F) \<Longrightarrow> (f \<longlongrightarrow> l) F"
by (auto simp: tendsto_iff)

lemma (in metric_space) tendstoD: "(f \<longlongrightarrow> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
@@ -1767,15 +1727,13 @@
by (subst eventually_INF_base)
(auto simp: eventually_principal Bex_def subset_eq intro: exI[of _ "min a b" for a b])

-lemma eventually_at:
-  fixes a :: "'a :: metric_space"
-  shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
-  unfolding eventually_at_filter eventually_nhds_metric by auto
+lemma eventually_at: "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
+  for a :: "'a :: metric_space"
+  by (auto simp: eventually_at_filter eventually_nhds_metric)

-lemma eventually_at_le:
-  fixes a :: "'a::metric_space"
-  shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a \<le> d \<longrightarrow> P x)"
-  unfolding eventually_at_filter eventually_nhds_metric
+lemma eventually_at_le: "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a \<le> d \<longrightarrow> P x)"
+  for a :: "'a::metric_space"
+  apply (simp only: eventually_at_filter eventually_nhds_metric)
apply auto
apply (rule_tac x="d / 2" in exI)
apply auto
@@ -1788,19 +1746,21 @@
by (subst eventually_at, rule exI[of _ "b - a"]) (force simp: dist_real_def)

lemma metric_tendsto_imp_tendsto:
-  fixes a :: "'a :: metric_space" and b :: "'b :: metric_space"
+  fixes a :: "'a :: metric_space"
+    and b :: "'b :: metric_space"
assumes f: "(f \<longlongrightarrow> a) F"
-  assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
+    and le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
shows "(g \<longlongrightarrow> b) F"
proof (rule tendstoI)
-  fix e :: real assume "0 < e"
+  fix e :: real
+  assume "0 < e"
with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
with le show "eventually (\<lambda>x. dist (g x) b < e) F"
using le_less_trans by (rule eventually_elim2)
qed

lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
-  unfolding filterlim_at_top
+  apply (simp only: filterlim_at_top)
apply (intro allI)
apply (rule_tac c="nat \<lceil>Z + 1\<rceil>" in eventually_sequentiallyI)
apply linarith
@@ -1809,73 +1769,79 @@

subsubsection \<open>Limits of Sequences\<close>

-lemma lim_sequentially: "X \<longlonglongrightarrow> (L::'a::metric_space) \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
+lemma lim_sequentially: "X \<longlonglongrightarrow> L \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
+  for L :: "'a::metric_space"
unfolding tendsto_iff eventually_sequentially ..

lemmas LIMSEQ_def = lim_sequentially  (*legacy binding*)

-lemma LIMSEQ_iff_nz: "X \<longlonglongrightarrow> (L::'a::metric_space) = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
+lemma LIMSEQ_iff_nz: "X \<longlonglongrightarrow> L \<longleftrightarrow> (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
+  for L :: "'a::metric_space"
unfolding lim_sequentially by (metis Suc_leD zero_less_Suc)

-lemma metric_LIMSEQ_I:
-  "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X \<longlonglongrightarrow> (L::'a::metric_space)"
+lemma metric_LIMSEQ_I: "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X \<longlonglongrightarrow> L"
+  for L :: "'a::metric_space"

-lemma metric_LIMSEQ_D:
-  "\<lbrakk>X \<longlonglongrightarrow> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
+lemma metric_LIMSEQ_D: "X \<longlonglongrightarrow> L \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
+  for L :: "'a::metric_space"

subsubsection \<open>Limits of Functions\<close>

-lemma LIM_def: "f \<midarrow>(a::'a::metric_space)\<rightarrow> (L::'b::metric_space) =
-     (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
-        --> dist (f x) L < r)"
+lemma LIM_def: "f \<midarrow>a\<rightarrow> L \<longleftrightarrow> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)"
+  for a :: "'a::metric_space" and L :: "'b::metric_space"
unfolding tendsto_iff eventually_at by simp

lemma metric_LIM_I:
-  "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
-    \<Longrightarrow> f \<midarrow>(a::'a::metric_space)\<rightarrow> (L::'b::metric_space)"
+  "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r) \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
+  for a :: "'a::metric_space" and L :: "'b::metric_space"

-lemma metric_LIM_D:
-  "\<lbrakk>f \<midarrow>(a::'a::metric_space)\<rightarrow> (L::'b::metric_space); 0 < r\<rbrakk>
-    \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
+lemma metric_LIM_D: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
+  for a :: "'a::metric_space" and L :: "'b::metric_space"

lemma metric_LIM_imp_LIM:
-  assumes f: "f \<midarrow>a\<rightarrow> (l::'a::metric_space)"
-  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
-  shows "g \<midarrow>a\<rightarrow> (m::'b::metric_space)"
+  fixes l :: "'a::metric_space"
+    and m :: "'b::metric_space"
+  assumes f: "f \<midarrow>a\<rightarrow> l"
+    and le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
+  shows "g \<midarrow>a\<rightarrow> m"
by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le)

lemma metric_LIM_equal2:
-  assumes 1: "0 < R"
-  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
-  shows "g \<midarrow>a\<rightarrow> l \<Longrightarrow> f \<midarrow>(a::'a::metric_space)\<rightarrow> l"
-apply (rule topological_tendstoI)
-apply (drule (2) topological_tendstoD)
-apply (rule_tac x="min d R" in exI, safe)
-done
+  fixes a :: "'a::metric_space"
+  assumes "0 < R"
+    and "\<And>x. x \<noteq> a \<Longrightarrow> dist x a < R \<Longrightarrow> f x = g x"
+  shows "g \<midarrow>a\<rightarrow> l \<Longrightarrow> f \<midarrow>a\<rightarrow> l"
+  apply (rule topological_tendstoI)
+  apply (drule (2) topological_tendstoD)
+  apply safe
+  apply (rule_tac x="min d R" in exI)
+  apply safe
+  done

lemma metric_LIM_compose2:
-  assumes f: "f \<midarrow>(a::'a::metric_space)\<rightarrow> b"
-  assumes g: "g \<midarrow>b\<rightarrow> c"
-  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
+  fixes a :: "'a::metric_space"
+  assumes f: "f \<midarrow>a\<rightarrow> b"
+    and g: "g \<midarrow>b\<rightarrow> c"
+    and inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
-  using inj
-  by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at)
+  using inj by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at)

lemma metric_isCont_LIM_compose2:
fixes f :: "'a :: metric_space \<Rightarrow> _"
assumes f [unfolded isCont_def]: "isCont f a"
-  assumes g: "g \<midarrow>f a\<rightarrow> l"
-  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
+    and g: "g \<midarrow>f a\<rightarrow> l"
+    and inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> l"
-by (rule metric_LIM_compose2 [OF f g inj])
+  by (rule metric_LIM_compose2 [OF f g inj])
+

subsection \<open>Complete metric spaces\<close>

@@ -1883,12 +1849,14 @@

lemma (in metric_space) Cauchy_def: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)"
proof -
-  have *: "eventually P (INF M. principal {(X m, X n) | n m. m \<ge> M \<and> n \<ge> M}) =
+  have *: "eventually P (INF M. principal {(X m, X n) | n m. m \<ge> M \<and> n \<ge> M}) \<longleftrightarrow>
(\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. P (X m, X n))" for P
-  proof (subst eventually_INF_base, goal_cases)
-    case (2 a b) then show ?case
+    apply (subst eventually_INF_base)
+    subgoal by simp
+    subgoal for a b
by (intro bexI[of _ "max a b"]) (auto simp: eventually_principal subset_eq)
-  qed (auto simp: eventually_principal, blast)
+    subgoal by (auto simp: eventually_principal, blast)
+    done
have "Cauchy X \<longleftrightarrow> (INF M. principal {(X m, X n) | n m. m \<ge> M \<and> n \<ge> M}) \<le> uniformity"
unfolding Cauchy_uniform_iff le_filter_def * ..
also have "\<dots> = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)"
@@ -1896,26 +1864,31 @@
finally show ?thesis .
qed

-lemma (in metric_space) Cauchy_altdef:
-  "Cauchy f = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e)"
+lemma (in metric_space) Cauchy_altdef: "Cauchy f \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e)"
+  (is "?lhs \<longleftrightarrow> ?rhs")
proof
-  assume A: "\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e"
-  show "Cauchy f" unfolding Cauchy_def
+  assume ?rhs
+  show ?lhs
+    unfolding Cauchy_def
proof (intro allI impI)
fix e :: real assume e: "e > 0"
-    with A obtain M where M: "\<And>m n. m \<ge> M \<Longrightarrow> n > m \<Longrightarrow> dist (f m) (f n) < e" by blast
+    with \<open>?rhs\<close> obtain M where M: "m \<ge> M \<Longrightarrow> n > m \<Longrightarrow> dist (f m) (f n) < e" for m n
+      by blast
have "dist (f m) (f n) < e" if "m \<ge> M" "n \<ge> M" for m n
using M[of m n] M[of n m] e that by (cases m n rule: linorder_cases) (auto simp: dist_commute)
-    thus "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m) (f n) < e" by blast
+    then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m) (f n) < e"
+      by blast
qed
next
-  assume "Cauchy f"
-  show "\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e"
+  assume ?lhs
+  show ?rhs
proof (intro allI impI)
-    fix e :: real assume e: "e > 0"
+    fix e :: real
+    assume e: "e > 0"
with \<open>Cauchy f\<close> obtain M where "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> dist (f m) (f n) < e"
unfolding Cauchy_def by blast
-    thus "\<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e" by (intro exI[of _ M]) force
+    then show "\<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e"
+      by (intro exI[of _ M]) force
qed
qed

@@ -1923,7 +1896,8 @@
"(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"

-lemma (in metric_space) CauchyI': "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
+lemma (in metric_space) CauchyI':
+  "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
unfolding Cauchy_altdef by blast

lemma (in metric_space) metric_CauchyD:
@@ -1932,53 +1906,63 @@

lemma (in metric_space) metric_Cauchy_iff2:
"Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < inverse(real (Suc j))))"
-apply (drule reals_Archimedean, safe)
-apply (drule_tac x = n in spec, auto)
-apply (rule_tac x = M in exI, auto)
-apply (drule_tac x = m in spec, simp)
-apply (drule_tac x = na in spec, auto)
-done
+  apply auto
+  apply (drule reals_Archimedean)
+  apply safe
+  apply (drule_tac x = n in spec)
+  apply auto
+  apply (rule_tac x = M in exI)
+  apply auto
+  apply (drule_tac x = m in spec)
+  apply simp
+  apply (drule_tac x = na in spec)
+  apply auto
+  done

-lemma Cauchy_iff2:
-  "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
-  unfolding metric_Cauchy_iff2 dist_real_def ..
+lemma Cauchy_iff2: "Cauchy X \<longleftrightarrow> (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse (real (Suc j))))"
+  by (simp only: metric_Cauchy_iff2 dist_real_def)

lemma lim_1_over_n: "((\<lambda>n. 1 / of_nat n) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially"
proof (subst lim_sequentially, intro allI impI exI)
-  fix e :: real assume e: "e > 0"
-  fix n :: nat assume n: "n \<ge> nat \<lceil>inverse e + 1\<rceil>"
+  fix e :: real
+  assume e: "e > 0"
+  fix n :: nat
+  assume n: "n \<ge> nat \<lceil>inverse e + 1\<rceil>"
have "inverse e < of_nat (nat \<lceil>inverse e + 1\<rceil>)" by linarith
also note n
-  finally show "dist (1 / of_nat n :: 'a) 0 < e" using e
-    by (simp add: divide_simps mult.commute norm_divide)
+  finally show "dist (1 / of_nat n :: 'a) 0 < e"
+    using e by (simp add: divide_simps mult.commute norm_divide)
qed

lemma (in metric_space) complete_def:
shows "complete S = (\<forall>f. (\<forall>n. f n \<in> S) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>S. f \<longlonglongrightarrow> l))"
unfolding complete_uniform
proof safe
-  fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> S" "Cauchy f"
+  fix f :: "nat \<Rightarrow> 'a"
+  assume f: "\<forall>n. f n \<in> S" "Cauchy f"
and *: "\<forall>F\<le>principal S. F \<noteq> bot \<longrightarrow> cauchy_filter F \<longrightarrow> (\<exists>x\<in>S. F \<le> nhds x)"
then show "\<exists>l\<in>S. f \<longlonglongrightarrow> l"
unfolding filterlim_def using f
by (intro *[rule_format])
(auto simp: filtermap_sequentually_ne_bot le_principal eventually_filtermap Cauchy_uniform)
next
-  fix F :: "'a filter" assume "F \<le> principal S" "F \<noteq> bot" "cauchy_filter F"
+  fix F :: "'a filter"
+  assume "F \<le> principal S" "F \<noteq> bot" "cauchy_filter F"
assume seq: "\<forall>f. (\<forall>n. f n \<in> S) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>S. f \<longlonglongrightarrow> l)"

-  from \<open>F \<le> principal S\<close> \<open>cauchy_filter F\<close> have FF_le: "F \<times>\<^sub>F F \<le> uniformity_on S"
+  from \<open>F \<le> principal S\<close> \<open>cauchy_filter F\<close>
+  have FF_le: "F \<times>\<^sub>F F \<le> uniformity_on S"
by (simp add: cauchy_filter_def principal_prod_principal[symmetric] prod_filter_mono)

let ?P = "\<lambda>P e. eventually P F \<and> (\<forall>x. P x \<longrightarrow> x \<in> S) \<and> (\<forall>x y. P x \<longrightarrow> P y \<longrightarrow> dist x y < e)"
-
-  { fix \<epsilon> :: real assume "0 < \<epsilon>"
-    then have "eventually (\<lambda>(x, y). x \<in> S \<and> y \<in> S \<and> dist x y < \<epsilon>) (uniformity_on S)"
-      unfolding eventually_inf_principal eventually_uniformity_metric by auto
-    from filter_leD[OF FF_le this] have "\<exists>P. ?P P \<epsilon>"
-      unfolding eventually_prod_same by auto }
-  note P = this
+  have P: "\<exists>P. ?P P \<epsilon>" if "0 < \<epsilon>" for \<epsilon> :: real
+  proof -
+    from that have "eventually (\<lambda>(x, y). x \<in> S \<and> y \<in> S \<and> dist x y < \<epsilon>) (uniformity_on S)"
+      by (auto simp: eventually_inf_principal eventually_uniformity_metric)
+    from filter_leD[OF FF_le this] show ?thesis
+      by (auto simp: eventually_prod_same)
+  qed

have "\<exists>P. \<forall>n. ?P (P n) (1 / Suc n) \<and> P (Suc n) \<le> P n"
proof (rule dependent_nat_choice)
@@ -1991,18 +1975,20 @@
ultimately show "\<exists>Q. ?P Q (1 / Suc (Suc n)) \<and> Q \<le> P"
by (intro exI[of _ "\<lambda>x. P x \<and> Q x"]) (auto simp: eventually_conj_iff)
qed
-  then obtain P where P: "\<And>n. eventually (P n) F" "\<And>n x. P n x \<Longrightarrow> x \<in> S"
-    "\<And>n x y. P n x \<Longrightarrow> P n y \<Longrightarrow> dist x y < 1 / Suc n" "\<And>n. P (Suc n) \<le> P n"
+  then obtain P where P: "eventually (P n) F" "P n x \<Longrightarrow> x \<in> S"
+    "P n x \<Longrightarrow> P n y \<Longrightarrow> dist x y < 1 / Suc n" "P (Suc n) \<le> P n"
+    for n x y
by metis
have "antimono P"
using P(4) unfolding decseq_Suc_iff le_fun_def by blast

-  obtain X where X: "\<And>n. P n (X n)"
+  obtain X where X: "P n (X n)" for n
using P(1)[THEN eventually_happens'[OF \<open>F \<noteq> bot\<close>]] by metis
have "Cauchy X"
unfolding metric_Cauchy_iff2 inverse_eq_divide
proof (intro exI allI impI)
-    fix j m n :: nat assume "j \<le> m" "j \<le> n"
+    fix j m n :: nat
+    assume "j \<le> m" "j \<le> n"
with \<open>antimono P\<close> X have "P j (X m)" "P j (X n)"
by (auto simp: antimono_def)
then show "dist (X m) (X n) < 1 / Suc j"
@@ -2015,23 +2001,27 @@

show "\<exists>x\<in>S. F \<le> nhds x"
proof (rule bexI)
-    { fix e :: real assume "0 < e"
-      then have "(\<lambda>n. 1 / Suc n :: real) \<longlonglongrightarrow> 0 \<and> 0 < e / 2"
+    have "eventually (\<lambda>y. dist y x < e) F" if "0 < e" for e :: real
+    proof -
+      from that have "(\<lambda>n. 1 / Suc n :: real) \<longlonglongrightarrow> 0 \<and> 0 < e / 2"
by (subst LIMSEQ_Suc_iff) (auto intro!: lim_1_over_n)
then have "\<forall>\<^sub>F n in sequentially. dist (X n) x < e / 2 \<and> 1 / Suc n < e / 2"
-        using \<open>X \<longlonglongrightarrow> x\<close> unfolding tendsto_iff order_tendsto_iff[where 'a=real] eventually_conj_iff by blast
+        using \<open>X \<longlonglongrightarrow> x\<close>
+        unfolding tendsto_iff order_tendsto_iff[where 'a=real] eventually_conj_iff
+        by blast
then obtain n where "dist x (X n) < e / 2" "1 / Suc n < e / 2"
by (auto simp: eventually_sequentially dist_commute)
-      have "eventually (\<lambda>y. dist y x < e) F"
+      show ?thesis
using \<open>eventually (P n) F\<close>
proof eventually_elim
-        fix y assume "P n y"
+        case (elim y)
then have "dist y (X n) < 1 / Suc n"
by (intro X P)
also have "\<dots> < e / 2" by fact
finally show "dist y x < e"
by (rule dist_triangle_half_l) fact
-      qed }
+      qed
+    qed
then show "F \<le> nhds x"
unfolding nhds_metric le_INF_iff le_principal by auto
qed fact
@@ -2039,7 +2029,7 @@

lemma (in metric_space) totally_bounded_metric:
"totally_bounded S \<longleftrightarrow> (\<forall>e>0. \<exists>k. finite k \<and> S \<subseteq> (\<Union>x\<in>k. {y. dist x y < e}))"
-  unfolding totally_bounded_def eventually_uniformity_metric imp_ex
+  apply (simp only: totally_bounded_def eventually_uniformity_metric imp_ex)
apply (subst all_comm)
apply (intro arg_cong[where f=All] ext)
apply safe
@@ -2053,45 +2043,47 @@
done
done

+
subsubsection \<open>Cauchy Sequences are Convergent\<close>

(* TODO: update to uniform_space *)
class complete_space = metric_space +
assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"

-lemma Cauchy_convergent_iff:
-  fixes X :: "nat \<Rightarrow> 'a::complete_space"
-  shows "Cauchy X = convergent X"
-by (blast intro: Cauchy_convergent convergent_Cauchy)
+lemma Cauchy_convergent_iff: "Cauchy X \<longleftrightarrow> convergent X"
+  for X :: "nat \<Rightarrow> 'a::complete_space"
+  by (blast intro: Cauchy_convergent convergent_Cauchy)
+

subsection \<open>The set of real numbers is a complete metric space\<close>

text \<open>
-Proof that Cauchy sequences converge based on the one from
-@{url "http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html"}
+  Proof that Cauchy sequences converge based on the one from
+  @{url "http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html"}
\<close>

text \<open>
If sequence @{term "X"} is Cauchy, then its limit is the lub of
@{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
\<close>
-
lemma increasing_LIMSEQ:
fixes f :: "nat \<Rightarrow> real"
assumes inc: "\<And>n. f n \<le> f (Suc n)"
-      and bdd: "\<And>n. f n \<le> l"
-      and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
+    and bdd: "\<And>n. f n \<le> l"
+    and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
shows "f \<longlonglongrightarrow> l"
proof (rule increasing_tendsto)
-  fix x assume "x < l"
+  fix x
+  assume "x < l"
with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"
by auto
from en[OF \<open>0 < e\<close>] obtain n where "l - e \<le> f n"
by (auto simp: field_simps)
-  with \<open>e < l - x\<close> \<open>0 < e\<close> have "x < f n" by simp
+  with \<open>e < l - x\<close> \<open>0 < e\<close> have "x < f n"
+    by simp
with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially"
by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)
-qed (insert bdd, auto)
+qed (use bdd in auto)

lemma real_Cauchy_convergent:
fixes X :: "nat \<Rightarrow> real"
@@ -2099,63 +2091,66 @@
shows "convergent X"
proof -
define S :: "real set" where "S = {x. \<exists>N. \<forall>n\<ge>N. x < X n}"
-  then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
+  then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S"
+    by auto

-  { fix N x assume N: "\<forall>n\<ge>N. X n < x"
-  fix y::real assume "y \<in> S"
-  hence "\<exists>M. \<forall>n\<ge>M. y < X n"
-  then obtain M where "\<forall>n\<ge>M. y < X n" ..
-  hence "y < X (max M N)" by simp
-  also have "\<dots> < x" using N by simp
-  finally have "y \<le> x"
-    by (rule order_less_imp_le) }
-  note bound_isUb = this
+  have bound_isUb: "y \<le> x" if N: "\<forall>n\<ge>N. X n < x" and "y \<in> S" for N and x y :: real
+  proof -
+    from that have "\<exists>M. \<forall>n\<ge>M. y < X n"
+    then obtain M where "\<forall>n\<ge>M. y < X n" ..
+    then have "y < X (max M N)" by simp
+    also have "\<dots> < x" using N by simp
+    finally show ?thesis by (rule order_less_imp_le)
+  qed

obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < 1"
using X[THEN metric_CauchyD, OF zero_less_one] by auto
-  hence N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp
+  then have N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp
have [simp]: "S \<noteq> {}"
proof (intro exI ex_in_conv[THEN iffD1])
from N have "\<forall>n\<ge>N. X N - 1 < X n"
-    thus "X N - 1 \<in> S" by (rule mem_S)
+    then show "X N - 1 \<in> S" by (rule mem_S)
qed
have [simp]: "bdd_above S"
proof
from N have "\<forall>n\<ge>N. X n < X N + 1"
-    thus "\<And>s. s \<in> S \<Longrightarrow>  s \<le> X N + 1"
+    then show "\<And>s. s \<in> S \<Longrightarrow>  s \<le> X N + 1"
by (rule bound_isUb)
qed
have "X \<longlonglongrightarrow> Sup S"
proof (rule metric_LIMSEQ_I)
-  fix r::real assume "0 < r"
-  hence r: "0 < r/2" by simp
-  obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2"
-    using metric_CauchyD [OF X r] by auto
-  hence "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp
-  hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
-    by (simp only: dist_real_def abs_diff_less_iff)
+    fix r :: real
+    assume "0 < r"
+    then have r: "0 < r/2" by simp
+    obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2"
+      using metric_CauchyD [OF X r] by auto
+    then have "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp
+    then have N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
+      by (simp only: dist_real_def abs_diff_less_iff)

-  from N have "\<forall>n\<ge>N. X N - r/2 < X n" by blast
-  hence "X N - r/2 \<in> S" by (rule mem_S)
-  hence 1: "X N - r/2 \<le> Sup S" by (simp add: cSup_upper)
+    from N have "\<forall>n\<ge>N. X N - r/2 < X n" by blast
+    then have "X N - r/2 \<in> S" by (rule mem_S)
+    then have 1: "X N - r/2 \<le> Sup S" by (simp add: cSup_upper)

-  from N have "\<forall>n\<ge>N. X n < X N + r/2" by blast
-  from bound_isUb[OF this]
-  have 2: "Sup S \<le> X N + r/2"
-    by (intro cSup_least) simp_all
+    from N have "\<forall>n\<ge>N. X n < X N + r/2" by blast
+    from bound_isUb[OF this]
+    have 2: "Sup S \<le> X N + r/2"
+      by (intro cSup_least) simp_all

-  show "\<exists>N. \<forall>n\<ge>N. dist (X n) (Sup S) < r"
-  proof (intro exI allI impI)
-    fix n assume n: "N \<le> n"
-    from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
-    thus "dist (X n) (Sup S) < r" using 1 2
-      by (simp add: abs_diff_less_iff dist_real_def)
+    show "\<exists>N. \<forall>n\<ge>N. dist (X n) (Sup S) < r"
+    proof (intro exI allI impI)
+      fix n
+      assume n: "N \<le> n"
+      from N n have "X n < X N + r/2" and "X N - r/2 < X n"
+        by simp_all
+      then show "dist (X n) (Sup S) < r" using 1 2
+        by (simp add: abs_diff_less_iff dist_real_def)
+    qed
qed
-  qed
-  then show ?thesis unfolding convergent_def by auto
+  then show ?thesis by (auto simp: convergent_def)
qed

instance real :: complete_space
@@ -2170,7 +2165,8 @@
assumes *: "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y"
shows "(f \<longlongrightarrow> y) at_top"
proof -
-  from nhds_countable[of y] guess A . note A = this
+  obtain A where A: "decseq A" "open (A n)" "y \<in> A n" "nhds y = (INF n. principal (A n))" for n
+    by (rule nhds_countable[of y]) (rule that)

have "\<forall>m. \<exists>k. \<forall>x\<ge>k. f x \<in> A m"
proof (rule ccontr)
@@ -2181,45 +2177,46 @@
by (intro dependent_nat_choice) (auto simp del: max.bounded_iff)
then obtain X where X: "\<And>n. f (X n) \<notin> A m" "\<And>n. max n (X n) + 1 \<le> X (Suc n)"
by auto
-    { fix n have "1 \<le> n \<longrightarrow> real n \<le> X n"
-        using X[of "n - 1"] by auto }
+    have "1 \<le> n \<Longrightarrow> real n \<le> X n" for n
+      using X[of "n - 1"] by auto
then have "filterlim X at_top sequentially"
by (force intro!: filterlim_at_top_mono[OF filterlim_real_sequentially]
-                simp: eventually_sequentially)
+          simp: eventually_sequentially)
from topological_tendstoD[OF *[OF this] A(2, 3), of m] X(1) show False
by auto
qed
-  then obtain k where "\<And>m x. k m \<le> x \<Longrightarrow> f x \<in> A m"
+  then obtain k where "k m \<le> x \<Longrightarrow> f x \<in> A m" for m x
by metis
then show ?thesis
-    unfolding at_top_def A
-    by (intro filterlim_base[where i=k]) auto
+    unfolding at_top_def A by (intro filterlim_base[where i=k]) auto
qed

lemma tendsto_at_topI_sequentially_real:
fixes f :: "real \<Rightarrow> real"
assumes mono: "mono f"
-  assumes limseq: "(\<lambda>n. f (real n)) \<longlonglongrightarrow> y"
+    and limseq: "(\<lambda>n. f (real n)) \<longlonglongrightarrow> y"
shows "(f \<longlongrightarrow> y) at_top"
proof (rule tendstoI)
-  fix e :: real assume "0 < e"
-  with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
+  fix e :: real
+  assume "0 < e"
+  with limseq obtain N :: nat where N: "N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e" for n
by (auto simp: lim_sequentially dist_real_def)
-  { fix x :: real
+  have le: "f x \<le> y" for x :: real
+  proof -
obtain n where "x \<le> real_of_nat n"
using real_arch_simple[of x] ..
note monoD[OF mono this]
also have "f (real_of_nat n) \<le> y"
by (rule LIMSEQ_le_const[OF limseq]) (auto intro!: exI[of _ n] monoD[OF mono])
-    finally have "f x \<le> y" . }
-  note le = this
+    finally show ?thesis .
+  qed
have "eventually (\<lambda>x. real N \<le> x) at_top"
by (rule eventually_ge_at_top)
then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
proof eventually_elim
-    fix x assume N': "real N \<le> x"
+    case (elim x)
with N[of N] le have "y - f (real N) < e" by auto
-    moreover note monoD[OF mono N']
+    moreover note monoD[OF mono elim]
ultimately show "dist (f x) y < e"
using le[of x] by (auto simp: dist_real_def field_simps)
qed```