author wenzelm Tue Aug 01 17:33:04 2017 +0200 (23 months ago) changeset 66304 cde6ceffcbc7 parent 66303 210dae34b8bc child 66305 7454317f883c
isabelle update_cartouches -c -t;
```     1.1 --- a/src/HOL/Analysis/Improper_Integral.thy	Tue Aug 01 17:30:02 2017 +0200
1.2 +++ b/src/HOL/Analysis/Improper_Integral.thy	Tue Aug 01 17:33:04 2017 +0200
1.3 @@ -1498,7 +1498,7 @@
1.4        using bounded_integrals_over_subintervals [OF int_gab] unfolding bounded_pos real_norm_def by blast
1.5      show "(\<lambda>x. f x \<bullet> j) absolutely_integrable_on cbox a b"
1.6        using g
1.7 -    proof     --\<open>A lot of duplication in the two proofs\<close>
1.8 +    proof     \<comment>\<open>A lot of duplication in the two proofs\<close>
1.9        assume fg [rule_format]: "\<forall>x\<in>cbox a b. f x \<bullet> j \<le> g x"
1.10        have "(\<lambda>x. (f x \<bullet> j)) = (\<lambda>x. g x - (g x - (f x \<bullet> j)))"
1.11          by simp
```
```     2.1 --- a/src/HOL/Analysis/Topology_Euclidean_Space.thy	Tue Aug 01 17:30:02 2017 +0200
2.2 +++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy	Tue Aug 01 17:33:04 2017 +0200
2.3 @@ -5300,7 +5300,7 @@
2.4    "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"
2.5    using compact_imp_seq_compact seq_compact_imp_heine_borel by blast
2.6
2.7 -lemma compact_def: --\<open>this is the definition of compactness in HOL Light\<close>
2.8 +lemma compact_def: \<comment>\<open>this is the definition of compactness in HOL Light\<close>
2.9    "compact (S :: 'a::metric_space set) \<longleftrightarrow>
2.10     (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
2.11    unfolding compact_eq_seq_compact_metric seq_compact_def by auto
```
```     3.1 --- a/src/HOL/Analysis/Winding_Numbers.thy	Tue Aug 01 17:30:02 2017 +0200
3.2 +++ b/src/HOL/Analysis/Winding_Numbers.thy	Tue Aug 01 17:33:04 2017 +0200
3.3 @@ -41,7 +41,7 @@
3.4            by (meson mem_interior)
3.5          define z where "z \<equiv> - sgn (Im b) * (e/3) + sgn (Re b) * (e/3) * \<i>"
3.6          have "z \<in> ball 0 e"
3.7 -          using `e>0`
3.8 +          using \<open>e>0\<close>
3.9            apply (simp add: z_def dist_norm)
3.10            apply (rule le_less_trans [OF norm_triangle_ineq4])
3.11            apply (simp add: norm_mult abs_sgn_eq)
3.12 @@ -49,7 +49,7 @@
3.13          then have "z \<in> {z. Im z * Re b \<le> Im b * Re z}"
3.14            using e by blast
3.15          then show False
3.16 -          using `e>0` `b \<noteq> 0`
3.17 +          using \<open>e>0\<close> \<open>b \<noteq> 0\<close>
3.18            apply (simp add: z_def dist_norm sgn_if less_eq_real_def mult_less_0_iff complex.expand split: if_split_asm)
3.19            apply (auto simp: algebra_simps)
3.20            apply (meson less_asym less_trans mult_pos_pos neg_less_0_iff_less)
```
```     4.1 --- a/src/HOL/Number_Theory/Residues.thy	Tue Aug 01 17:30:02 2017 +0200
4.2 +++ b/src/HOL/Number_Theory/Residues.thy	Tue Aug 01 17:33:04 2017 +0200
4.3 @@ -378,9 +378,9 @@
4.4      by (metis residues_prime.wilson_theorem1 residues_prime.intro le_eq_less_or_eq)
4.5  qed
4.6
4.7 -text {*
4.8 +text \<open>
4.9    This result can be transferred to the multiplicative group of
4.10 -  \$\mathbb{Z}/p\mathbb{Z}\$ for \$p\$ prime. *}
4.11 +  \$\mathbb{Z}/p\mathbb{Z}\$ for \$p\$ prime.\<close>
4.12
4.13  lemma mod_nat_int_pow_eq:
4.14    fixes n :: nat and p a :: int
4.15 @@ -409,22 +409,22 @@
4.16    have **:"nat ` {1 .. int p - 1} = {1 .. p - 1}" (is "?L = ?R")
4.17    proof
4.18      { fix n assume n: "n \<in> ?L"
4.19 -      then have "n \<in> ?R" using `p\<ge>2` by force
4.20 +      then have "n \<in> ?R" using \<open>p\<ge>2\<close> by force
4.21      } thus "?L \<subseteq> ?R" by blast
4.22      { fix n assume n: "n \<in> ?R"
4.23 -      then have "n \<in> ?L" using `p\<ge>2` Set_Interval.transfer_nat_int_set_functions(2) by fastforce
4.24 +      then have "n \<in> ?L" using \<open>p\<ge>2\<close> Set_Interval.transfer_nat_int_set_functions(2) by fastforce
4.25      } thus "?R \<subseteq> ?L" by blast
4.26    qed
4.27    have "nat ` {a^i mod (int p) | i::nat. i \<in> UNIV} = {nat a^i mod p | i . i \<in> UNIV}" (is "?L = ?R")
4.28    proof
4.29      { fix x assume x: "x \<in> ?L"
4.30        then obtain i where i:"x = nat (a^i mod (int p))" by blast
4.31 -      hence "x = nat a ^ i mod p" using mod_nat_int_pow_eq[of a "int p" i] a `p\<ge>2` by auto
4.32 +      hence "x = nat a ^ i mod p" using mod_nat_int_pow_eq[of a "int p" i] a \<open>p\<ge>2\<close> by auto
4.33        hence "x \<in> ?R" using i by blast
4.34      } thus "?L \<subseteq> ?R" by blast
4.35      { fix x assume x: "x \<in> ?R"
4.36        then obtain i where i:"x = nat a^i mod p" by blast
4.37 -      hence "x \<in> ?L" using mod_nat_int_pow_eq[of a "int p" i] a `p\<ge>2` by auto
4.38 +      hence "x \<in> ?L" using mod_nat_int_pow_eq[of a "int p" i] a \<open>p\<ge>2\<close> by auto
4.39      } thus "?R \<subseteq> ?L" by blast
4.40    qed
4.41    hence "{1 .. p - 1} = {nat a^i mod p | i. i \<in> UNIV}"
```