--- a/src/HOL/Analysis/Complex_Transcendental.thy Thu Jan 05 15:03:37 2017 +0000
+++ b/src/HOL/Analysis/Complex_Transcendental.thy Thu Jan 05 16:03:23 2017 +0000
@@ -3559,4 +3559,70 @@
shows "S homotopy_eqv T \<Longrightarrow> simply_connected S \<longleftrightarrow> simply_connected T"
by (simp add: simply_connected_eq_homotopic_circlemaps homotopy_eqv_homotopic_triviality)
+
+subsection\<open>Homeomorphism of simple closed curves to circles\<close>
+
+proposition homeomorphic_simple_path_image_circle:
+ fixes a :: complex and \<gamma> :: "real \<Rightarrow> 'a::t2_space"
+ assumes "simple_path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and "0 < r"
+ shows "(path_image \<gamma>) homeomorphic sphere a r"
+proof -
+ have "homotopic_loops (path_image \<gamma>) \<gamma> \<gamma>"
+ by (simp add: assms homotopic_loops_refl simple_path_imp_path)
+ then have hom: "homotopic_with (\<lambda>h. True) (sphere 0 1) (path_image \<gamma>)
+ (\<gamma> \<circ> (\<lambda>z. Arg z / (2*pi))) (\<gamma> \<circ> (\<lambda>z. Arg z / (2*pi)))"
+ by (rule homotopic_loops_imp_homotopic_circlemaps)
+ have "\<exists>g. homeomorphism (sphere 0 1) (path_image \<gamma>) (\<gamma> \<circ> (\<lambda>z. Arg z / (2*pi))) g"
+ proof (rule homeomorphism_compact)
+ show "continuous_on (sphere 0 1) (\<gamma> \<circ> (\<lambda>z. Arg z / (2*pi)))"
+ using hom homotopic_with_imp_continuous by blast
+ show "inj_on (\<gamma> \<circ> (\<lambda>z. Arg z / (2*pi))) (sphere 0 1)"
+ proof
+ fix x y
+ assume xy: "x \<in> sphere 0 1" "y \<in> sphere 0 1"
+ and eq: "(\<gamma> \<circ> (\<lambda>z. Arg z / (2*pi))) x = (\<gamma> \<circ> (\<lambda>z. Arg z / (2*pi))) y"
+ then have "(Arg x / (2*pi)) = (Arg y / (2*pi))"
+ proof -
+ have "(Arg x / (2*pi)) \<in> {0..1}" "(Arg y / (2*pi)) \<in> {0..1}"
+ using Arg_ge_0 Arg_lt_2pi dual_order.strict_iff_order by fastforce+
+ with eq show ?thesis
+ using \<open>simple_path \<gamma>\<close> Arg_lt_2pi unfolding simple_path_def o_def
+ by (metis eq_divide_eq_1 not_less_iff_gr_or_eq)
+ qed
+ with xy show "x = y"
+ by (metis Arg Arg_0 dist_0_norm divide_cancel_right dual_order.strict_iff_order mem_sphere)
+ qed
+ have "\<And>z. cmod z = 1 \<Longrightarrow> \<exists>x\<in>{0..1}. \<gamma> (Arg z / (2*pi)) = \<gamma> x"
+ by (metis Arg_ge_0 Arg_lt_2pi atLeastAtMost_iff divide_less_eq_1 less_eq_real_def zero_less_mult_iff pi_gt_zero zero_le_divide_iff zero_less_numeral)
+ moreover have "\<exists>z\<in>sphere 0 1. \<gamma> x = \<gamma> (Arg z / (2*pi))" if "0 \<le> x" "x \<le> 1" for x
+ proof (cases "x=1")
+ case True
+ then show ?thesis
+ apply (rule_tac x=1 in bexI)
+ apply (metis loop Arg_of_real divide_eq_0_iff of_real_1 pathfinish_def pathstart_def \<open>0 \<le> x\<close>, auto)
+ done
+ next
+ case False
+ then have *: "(Arg (exp (\<i>*(2* of_real pi* of_real x))) / (2*pi)) = x"
+ using that by (auto simp: Arg_exp divide_simps)
+ show ?thesis
+ by (rule_tac x="exp(ii* of_real(2*pi*x))" in bexI) (auto simp: *)
+ qed
+ ultimately show "(\<gamma> \<circ> (\<lambda>z. Arg z / (2*pi))) ` sphere 0 1 = path_image \<gamma>"
+ by (auto simp: path_image_def image_iff)
+ qed auto
+ then have "path_image \<gamma> homeomorphic sphere (0::complex) 1"
+ using homeomorphic_def homeomorphic_sym by blast
+ also have "... homeomorphic sphere a r"
+ by (simp add: assms homeomorphic_spheres)
+ finally show ?thesis .
+qed
+
+lemma homeomorphic_simple_path_images:
+ fixes \<gamma>1 :: "real \<Rightarrow> 'a::t2_space" and \<gamma>2 :: "real \<Rightarrow> 'b::t2_space"
+ assumes "simple_path \<gamma>1" and loop: "pathfinish \<gamma>1 = pathstart \<gamma>1"
+ assumes "simple_path \<gamma>2" and loop: "pathfinish \<gamma>2 = pathstart \<gamma>2"
+ shows "(path_image \<gamma>1) homeomorphic (path_image \<gamma>2)"
+ by (meson assms homeomorphic_simple_path_image_circle homeomorphic_sym homeomorphic_trans loop pi_gt_zero)
+
end