section ‹Winding Numbers› text‹By John Harrison et al. Ported from HOL Light by L C Paulson (2017)› theory Winding_Numbers imports Polytope Jordan_Curve Riemann_Mapping begin lemma simply_connected_inside_simple_path: fixes p :: "real ⇒ complex" shows "simple_path p ⟹ simply_connected(inside(path_image p))" using Jordan_inside_outside connected_simple_path_image inside_simple_curve_imp_closed simply_connected_eq_frontier_properties by fastforce lemma simply_connected_Int: fixes S :: "complex set" assumes "open S" "open T" "simply_connected S" "simply_connected T" "connected (S ∩ T)" shows "simply_connected (S ∩ T)" using assms by (force simp: simply_connected_eq_winding_number_zero open_Int) subsection‹Winding number for a triangle› lemma wn_triangle1: assumes "0 ∈ interior(convex hull {a,b,c})" shows "~ (Im(a/b) ≤ 0 ∧ 0 ≤ Im(b/c))" proof - { assume 0: "Im(a/b) ≤ 0" "0 ≤ Im(b/c)" have "0 ∉ interior (convex hull {a,b,c})" proof (cases "a=0 ∨ b=0 ∨ c=0") case True then show ?thesis by (auto simp: not_in_interior_convex_hull_3) next case False then have "b ≠ 0" by blast { fix x y::complex and u::real assume eq_f': "Im x * Re b ≤ Im b * Re x" "Im y * Re b ≤ Im b * Re y" "0 ≤ u" "u ≤ 1" then have "((1 - u) * Im x) * Re b ≤ Im b * ((1 - u) * Re x)" by (simp add: mult_left_mono mult.assoc mult.left_commute [of "Im b"]) then have "((1 - u) * Im x + u * Im y) * Re b ≤ Im b * ((1 - u) * Re x + u * Re y)" using eq_f' ordered_comm_semiring_class.comm_mult_left_mono by (fastforce simp add: algebra_simps) } with False 0 have "convex hull {a,b,c} ≤ {z. Im z * Re b ≤ Im b * Re z}" apply (simp add: Complex.Im_divide divide_simps complex_neq_0 [symmetric]) apply (simp add: algebra_simps) apply (rule hull_minimal) apply (auto simp: algebra_simps convex_alt) done moreover have "0 ∉ interior({z. Im z * Re b ≤ Im b * Re z})" proof assume "0 ∈ interior {z. Im z * Re b ≤ Im b * Re z}" then obtain e where "e>0" and e: "ball 0 e ⊆ {z. Im z * Re b ≤ Im b * Re z}" by (meson mem_interior) define z where "z ≡ - sgn (Im b) * (e/3) + sgn (Re b) * (e/3) * 𝗂" have "z ∈ ball 0 e" using ‹e>0› apply (simp add: z_def dist_norm) apply (rule le_less_trans [OF norm_triangle_ineq4]) apply (simp add: norm_mult abs_sgn_eq) done then have "z ∈ {z. Im z * Re b ≤ Im b * Re z}" using e by blast then show False using ‹e>0› ‹b ≠ 0› apply (simp add: z_def dist_norm sgn_if less_eq_real_def mult_less_0_iff complex.expand split: if_split_asm) apply (auto simp: algebra_simps) apply (meson less_asym less_trans mult_pos_pos neg_less_0_iff_less) by (metis less_asym mult_pos_pos neg_less_0_iff_less) qed ultimately show ?thesis using interior_mono by blast qed } with assms show ?thesis by blast qed lemma wn_triangle2_0: assumes "0 ∈ interior(convex hull {a,b,c})" shows "0 < Im((b - a) * cnj (b)) ∧ 0 < Im((c - b) * cnj (c)) ∧ 0 < Im((a - c) * cnj (a)) ∨ Im((b - a) * cnj (b)) < 0 ∧ 0 < Im((b - c) * cnj (b)) ∧ 0 < Im((a - b) * cnj (a)) ∧ 0 < Im((c - a) * cnj (c))" proof - have [simp]: "{b,c,a} = {a,b,c}" "{c,a,b} = {a,b,c}" by auto show ?thesis using wn_triangle1 [OF assms] wn_triangle1 [of b c a] wn_triangle1 [of c a b] assms by (auto simp: algebra_simps Im_complex_div_gt_0 Im_complex_div_lt_0 not_le not_less) qed lemma wn_triangle2: assumes "z ∈ interior(convex hull {a,b,c})" shows "0 < Im((b - a) * cnj (b - z)) ∧ 0 < Im((c - b) * cnj (c - z)) ∧ 0 < Im((a - c) * cnj (a - z)) ∨ Im((b - a) * cnj (b - z)) < 0 ∧ 0 < Im((b - c) * cnj (b - z)) ∧ 0 < Im((a - b) * cnj (a - z)) ∧ 0 < Im((c - a) * cnj (c - z))" proof - have 0: "0 ∈ interior(convex hull {a-z, b-z, c-z})" using assms convex_hull_translation [of "-z" "{a,b,c}"] interior_translation [of "-z"] by simp show ?thesis using wn_triangle2_0 [OF 0] by simp qed lemma wn_triangle3: assumes z: "z ∈ interior(convex hull {a,b,c})" and "0 < Im((b-a) * cnj (b-z))" "0 < Im((c-b) * cnj (c-z))" "0 < Im((a-c) * cnj (a-z))" shows "winding_number (linepath a b +++ linepath b c +++ linepath c a) z = 1" proof - have znot[simp]: "z ∉ closed_segment a b" "z ∉ closed_segment b c" "z ∉ closed_segment c a" using z interior_of_triangle [of a b c] by (auto simp: closed_segment_def) have gt0: "0 < Re (winding_number (linepath a b +++ linepath b c +++ linepath c a) z)" using assms by (simp add: winding_number_linepath_pos_lt path_image_join winding_number_join_pos_combined) have lt2: "Re (winding_number (linepath a b +++ linepath b c +++ linepath c a) z) < 2" using winding_number_lt_half_linepath [of _ a b] using winding_number_lt_half_linepath [of _ b c] using winding_number_lt_half_linepath [of _ c a] znot apply (fastforce simp add: winding_number_join path_image_join) done show ?thesis by (rule winding_number_eq_1) (simp_all add: path_image_join gt0 lt2) qed proposition winding_number_triangle: assumes z: "z ∈ interior(convex hull {a,b,c})" shows "winding_number(linepath a b +++ linepath b c +++ linepath c a) z = (if 0 < Im((b - a) * cnj (b - z)) then 1 else -1)" proof - have [simp]: "{a,c,b} = {a,b,c}" by auto have znot[simp]: "z ∉ closed_segment a b" "z ∉ closed_segment b c" "z ∉ closed_segment c a" using z interior_of_triangle [of a b c] by (auto simp: closed_segment_def) then have [simp]: "z ∉ closed_segment b a" "z ∉ closed_segment c b" "z ∉ closed_segment a c" using closed_segment_commute by blast+ have *: "winding_number (linepath a b +++ linepath b c +++ linepath c a) z = winding_number (reversepath (linepath a c +++ linepath c b +++ linepath b a)) z" by (simp add: reversepath_joinpaths winding_number_join not_in_path_image_join) show ?thesis using wn_triangle2 [OF z] apply (rule disjE) apply (simp add: wn_triangle3 z) apply (simp add: path_image_join winding_number_reversepath * wn_triangle3 z) done qed subsection‹Winding numbers for simple closed paths› lemma winding_number_from_innerpath: assumes "simple_path c1" and c1: "pathstart c1 = a" "pathfinish c1 = b" and "simple_path c2" and c2: "pathstart c2 = a" "pathfinish c2 = b" and "simple_path c" and c: "pathstart c = a" "pathfinish c = b" and c1c2: "path_image c1 ∩ path_image c2 = {a,b}" and c1c: "path_image c1 ∩ path_image c = {a,b}" and c2c: "path_image c2 ∩ path_image c = {a,b}" and ne_12: "path_image c ∩ inside(path_image c1 ∪ path_image c2) ≠ {}" and z: "z ∈ inside(path_image c1 ∪ path_image c)" and wn_d: "winding_number (c1 +++ reversepath c) z = d" and "a ≠ b" "d ≠ 0" obtains "z ∈ inside(path_image c1 ∪ path_image c2)" "winding_number (c1 +++ reversepath c2) z = d" proof - obtain 0: "inside(path_image c1 ∪ path_image c) ∩ inside(path_image c2 ∪ path_image c) = {}" and 1: "inside(path_image c1 ∪ path_image c) ∪ inside(path_image c2 ∪ path_image c) ∪ (path_image c - {a,b}) = inside(path_image c1 ∪ path_image c2)" by (rule split_inside_simple_closed_curve [OF ‹simple_path c1› c1 ‹simple_path c2› c2 ‹simple_path c› c ‹a ≠ b› c1c2 c1c c2c ne_12]) have znot: "z ∉ path_image c" "z ∉ path_image c1" "z ∉ path_image c2" using union_with_outside z 1 by auto have wn_cc2: "winding_number (c +++ reversepath c2) z = 0" apply (rule winding_number_zero_in_outside) apply (simp_all add: ‹simple_path c2› c c2 ‹simple_path c› simple_path_imp_path path_image_join) by (metis "0" ComplI UnE disjoint_iff_not_equal sup.commute union_with_inside z znot) show ?thesis proof show "z ∈ inside (path_image c1 ∪ path_image c2)" using "1" z by blast have "winding_number c1 z - winding_number c z = d " using assms znot by (metis wn_d winding_number_join simple_path_imp_path winding_number_reversepath add.commute path_image_reversepath path_reversepath pathstart_reversepath uminus_add_conv_diff) then show "winding_number (c1 +++ reversepath c2) z = d" using wn_cc2 by (simp add: winding_number_join simple_path_imp_path assms znot winding_number_reversepath) qed qed lemma simple_closed_path_wn1: fixes a::complex and e::real assumes "0 < e" and sp_pl: "simple_path(p +++ linepath (a - e) (a + e))" and psp: "pathstart p = a + e" and pfp: "pathfinish p = a - e" and disj: "ball a e ∩ path_image p = {}" obtains z where "z ∈ inside (path_image (p +++ linepath (a - e) (a + e)))" "cmod (winding_number (p +++ linepath (a - e) (a + e)) z) = 1" proof - have "arc p" and arc_lp: "arc (linepath (a - e) (a + e))" and pap: "path_image p ∩ path_image (linepath (a - e) (a + e)) ⊆ {pathstart p, a-e}" using simple_path_join_loop_eq [of "linepath (a - e) (a + e)" p] assms by auto have mid_eq_a: "midpoint (a - e) (a + e) = a" by (simp add: midpoint_def) then have "a ∈ path_image(p +++ linepath (a - e) (a + e))" apply (simp add: assms path_image_join) by (metis midpoint_in_closed_segment) have "a ∈ frontier(inside (path_image(p +++ linepath (a - e) (a + e))))" apply (simp add: assms Jordan_inside_outside) apply (simp_all add: assms path_image_join) by (metis mid_eq_a midpoint_in_closed_segment) with ‹0 < e› obtain c where c: "c ∈ inside (path_image(p +++ linepath (a - e) (a + e)))" and dac: "dist a c < e" by (auto simp: frontier_straddle) then have "c ∉ path_image(p +++ linepath (a - e) (a + e))" using inside_no_overlap by blast then have "c ∉ path_image p" "c ∉ closed_segment (a - of_real e) (a + of_real e)" by (simp_all add: assms path_image_join) with ‹0 < e› dac have "c ∉ affine hull {a - of_real e, a + of_real e}" by (simp add: segment_as_ball not_le) with ‹0 < e› have *: "~collinear{a - e, c,a + e}" using collinear_3_affine_hull [of "a-e" "a+e"] by (auto simp: insert_commute) have 13: "1/3 + 1/3 + 1/3 = (1::real)" by simp have "(1/3) *⇩_{R}(a - of_real e) + (1/3) *⇩_{R}c + (1/3) *⇩_{R}(a + of_real e) ∈ interior(convex hull {a - e, c, a + e})" using interior_convex_hull_3_minimal [OF * DIM_complex] by clarsimp (metis 13 zero_less_divide_1_iff zero_less_numeral) then obtain z where z: "z ∈ interior(convex hull {a - e, c, a + e})" by force have [simp]: "z ∉ closed_segment (a - e) c" by (metis DIM_complex Diff_iff IntD2 inf_sup_absorb interior_of_triangle z) have [simp]: "z ∉ closed_segment (a + e) (a - e)" by (metis DIM_complex DiffD2 Un_iff interior_of_triangle z) have [simp]: "z ∉ closed_segment c (a + e)" by (metis (no_types, lifting) DIM_complex DiffD2 Un_insert_right inf_sup_aci(5) insertCI interior_of_triangle mk_disjoint_insert z) show thesis proof have "norm (winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z) = 1" using winding_number_triangle [OF z] by simp have zin: "z ∈ inside (path_image (linepath (a + e) (a - e)) ∪ path_image p)" and zeq: "winding_number (linepath (a + e) (a - e) +++ reversepath p) z = winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z" proof (rule winding_number_from_innerpath [of "linepath (a + e) (a - e)" "a+e" "a-e" p "linepath (a + e) c +++ linepath c (a - e)" z "winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"]) show sp_aec: "simple_path (linepath (a + e) c +++ linepath c (a - e))" proof (rule arc_imp_simple_path [OF arc_join]) show "arc (linepath (a + e) c)" by (metis ‹c ∉ path_image p› arc_linepath pathstart_in_path_image psp) show "arc (linepath c (a - e))" by (metis ‹c ∉ path_image p› arc_linepath pathfinish_in_path_image pfp) show "path_image (linepath (a + e) c) ∩ path_image (linepath c (a - e)) ⊆ {pathstart (linepath c (a - e))}" by clarsimp (metis "*" IntI Int_closed_segment closed_segment_commute singleton_iff) qed auto show "simple_path p" using ‹arc p› arc_simple_path by blast show sp_ae2: "simple_path (linepath (a + e) (a - e))" using ‹arc p› arc_distinct_ends pfp psp by fastforce show pa: "pathfinish (linepath (a + e) (a - e)) = a - e" "pathstart (linepath (a + e) c +++ linepath c (a - e)) = a + e" "pathfinish (linepath (a + e) c +++ linepath c (a - e)) = a - e" "pathstart p = a + e" "pathfinish p = a - e" "pathstart (linepath (a + e) (a - e)) = a + e" by (simp_all add: assms) show 1: "path_image (linepath (a + e) (a - e)) ∩ path_image p = {a + e, a - e}" proof show "path_image (linepath (a + e) (a - e)) ∩ path_image p ⊆ {a + e, a - e}" using pap closed_segment_commute psp segment_convex_hull by fastforce show "{a + e, a - e} ⊆ path_image (linepath (a + e) (a - e)) ∩ path_image p" using pap pathfinish_in_path_image pathstart_in_path_image pfp psp by fastforce qed show 2: "path_image (linepath (a + e) (a - e)) ∩ path_image (linepath (a + e) c +++ linepath c (a - e)) = {a + e, a - e}" (is "?lhs = ?rhs") proof have "¬ collinear {c, a + e, a - e}" using * by (simp add: insert_commute) then have "convex hull {a + e, a - e} ∩ convex hull {a + e, c} = {a + e}" "convex hull {a + e, a - e} ∩ convex hull {c, a - e} = {a - e}" by (metis (full_types) Int_closed_segment insert_commute segment_convex_hull)+ then show "?lhs ⊆ ?rhs" by (metis Int_Un_distrib equalityD1 insert_is_Un path_image_join path_image_linepath path_join_eq path_linepath segment_convex_hull simple_path_def sp_aec) show "?rhs ⊆ ?lhs" using segment_convex_hull by (simp add: path_image_join) qed have "path_image p ∩ path_image (linepath (a + e) c) ⊆ {a + e}" proof (clarsimp simp: path_image_join) fix x assume "x ∈ path_image p" and x_ac: "x ∈ closed_segment (a + e) c" then have "dist x a ≥ e" by (metis IntI all_not_in_conv disj dist_commute mem_ball not_less) with x_ac dac ‹e > 0› show "x = a + e" by (auto simp: norm_minus_commute dist_norm closed_segment_eq_open dest: open_segment_furthest_le [where y=a]) qed moreover have "path_image p ∩ path_image (linepath c (a - e)) ⊆ {a - e}" proof (clarsimp simp: path_image_join) fix x assume "x ∈ path_image p" and x_ac: "x ∈ closed_segment c (a - e)" then have "dist x a ≥ e" by (metis IntI all_not_in_conv disj dist_commute mem_ball not_less) with x_ac dac ‹e > 0› show "x = a - e" by (auto simp: norm_minus_commute dist_norm closed_segment_eq_open dest: open_segment_furthest_le [where y=a]) qed ultimately have "path_image p ∩ path_image (linepath (a + e) c +++ linepath c (a - e)) ⊆ {a + e, a - e}" by (force simp: path_image_join) then show 3: "path_image p ∩ path_image (linepath (a + e) c +++ linepath c (a - e)) = {a + e, a - e}" apply (rule equalityI) apply (clarsimp simp: path_image_join) apply (metis pathstart_in_path_image psp pathfinish_in_path_image pfp) done show 4: "path_image (linepath (a + e) c +++ linepath c (a - e)) ∩ inside (path_image (linepath (a + e) (a - e)) ∪ path_image p) ≠ {}" apply (clarsimp simp: path_image_join segment_convex_hull disjoint_iff_not_equal) by (metis (no_types, hide_lams) UnI1 Un_commute c closed_segment_commute ends_in_segment(1) path_image_join path_image_linepath pathstart_linepath pfp segment_convex_hull) show zin_inside: "z ∈ inside (path_image (linepath (a + e) (a - e)) ∪ path_image (linepath (a + e) c +++ linepath c (a - e)))" apply (simp add: path_image_join) by (metis z inside_of_triangle DIM_complex Un_commute closed_segment_commute) show 5: "winding_number (linepath (a + e) (a - e) +++ reversepath (linepath (a + e) c +++ linepath c (a - e))) z = winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z" by (simp add: reversepath_joinpaths path_image_join winding_number_join) show 6: "winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z ≠ 0" by (simp add: winding_number_triangle z) show "winding_number (linepath (a + e) (a - e) +++ reversepath p) z = winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z" by (metis 1 2 3 4 5 6 pa sp_aec sp_ae2 ‹arc p› ‹simple_path p› arc_distinct_ends winding_number_from_innerpath zin_inside) qed (use assms ‹e > 0› in auto) show "z ∈ inside (path_image (p +++ linepath (a - e) (a + e)))" using zin by (simp add: assms path_image_join Un_commute closed_segment_commute) then have "cmod (winding_number (p +++ linepath (a - e) (a + e)) z) = cmod ((winding_number(reversepath (p +++ linepath (a - e) (a + e))) z))" apply (subst winding_number_reversepath) using simple_path_imp_path sp_pl apply blast apply (metis IntI emptyE inside_no_overlap) by (simp add: inside_def) also have "... = cmod (winding_number(linepath (a + e) (a - e) +++ reversepath p) z)" by (simp add: pfp reversepath_joinpaths) also have "... = cmod (winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z)" by (simp add: zeq) also have "... = 1" using z by (simp add: interior_of_triangle winding_number_triangle) finally show "cmod (winding_number (p +++ linepath (a - e) (a + e)) z) = 1" . qed qed lemma simple_closed_path_wn2: fixes a::complex and d e::real assumes "0 < d" "0 < e" and sp_pl: "simple_path(p +++ linepath (a - d) (a + e))" and psp: "pathstart p = a + e" and pfp: "pathfinish p = a - d" obtains z where "z ∈ inside (path_image (p +++ linepath (a - d) (a + e)))" "cmod (winding_number (p +++ linepath (a - d) (a + e)) z) = 1" proof - have [simp]: "a + of_real x ∈ closed_segment (a - α) (a - β) ⟷ x ∈ closed_segment (-α) (-β)" for x α β::real using closed_segment_translation_eq [of a] by (metis (no_types, hide_lams) add_uminus_conv_diff of_real_minus of_real_closed_segment) have [simp]: "a - of_real x ∈ closed_segment (a + α) (a + β) ⟷ -x ∈ closed_segment α β" for x α β::real by (metis closed_segment_translation_eq diff_conv_add_uminus of_real_closed_segment of_real_minus) have "arc p" and arc_lp: "arc (linepath (a - d) (a + e))" and "path p" and pap: "path_image p ∩ closed_segment (a - d) (a + e) ⊆ {a+e, a-d}" using simple_path_join_loop_eq [of "linepath (a - d) (a + e)" p] assms arc_imp_path by auto have "0 ∈ closed_segment (-d) e" using ‹0 < d› ‹0 < e› closed_segment_eq_real_ivl by auto then have "a ∈ path_image (linepath (a - d) (a + e))" using of_real_closed_segment [THEN iffD2] by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment) then have "a ∉ path_image p" using ‹0 < d› ‹0 < e› pap by auto then obtain k where "0 < k" and k: "ball a k ∩ (path_image p) = {}" using ‹0 < e› ‹path p› not_on_path_ball by blast define kde where "kde ≡ (min k (min d e)) / 2" have "0 < kde" "kde < k" "kde < d" "kde < e" using ‹0 < k› ‹0 < d› ‹0 < e› by (auto simp: kde_def) let ?q = "linepath (a + kde) (a + e) +++ p +++ linepath (a - d) (a - kde)" have "- kde ∈ closed_segment (-d) e" using ‹0 < kde› ‹kde < d› ‹kde < e› closed_segment_eq_real_ivl by auto then have a_diff_kde: "a - kde ∈ closed_segment (a - d) (a + e)" using of_real_closed_segment [THEN iffD2] by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment) then have clsub2: "closed_segment (a - d) (a - kde) ⊆ closed_segment (a - d) (a + e)" by (simp add: subset_closed_segment) then have "path_image p ∩ closed_segment (a - d) (a - kde) ⊆ {a + e, a - d}" using pap by force moreover have "a + e ∉ path_image p ∩ closed_segment (a - d) (a - kde)" using ‹0 < kde› ‹kde < d› ‹0 < e› by (auto simp: closed_segment_eq_real_ivl) ultimately have sub_a_diff_d: "path_image p ∩ closed_segment (a - d) (a - kde) ⊆ {a - d}" by blast have "kde ∈ closed_segment (-d) e" using ‹0 < kde› ‹kde < d› ‹kde < e› closed_segment_eq_real_ivl by auto then have a_diff_kde: "a + kde ∈ closed_segment (a - d) (a + e)" using of_real_closed_segment [THEN iffD2] by (force dest: closed_segment_translation_eq [of "a", THEN iffD2] simp del: of_real_closed_segment) then have clsub1: "closed_segment (a + kde) (a + e) ⊆ closed_segment (a - d) (a + e)" by (simp add: subset_closed_segment) then have "closed_segment (a + kde) (a + e) ∩ path_image p ⊆ {a + e, a - d}" using pap by force moreover have "closed_segment (a + kde) (a + e) ∩ closed_segment (a - d) (a - kde) = {}" proof (clarsimp intro!: equals0I) fix y assume y1: "y ∈ closed_segment (a + kde) (a + e)" and y2: "y ∈ closed_segment (a - d) (a - kde)" obtain u where u: "y = a + of_real u" and "0 < u" using y1 ‹0 < kde› ‹kde < e› ‹0 < e› apply (clarsimp simp: in_segment) apply (rule_tac u = "(1 - u)*kde + u*e" in that) apply (auto simp: scaleR_conv_of_real algebra_simps) by (meson le_less_trans less_add_same_cancel2 less_eq_real_def mult_left_mono) moreover obtain v where v: "y = a + of_real v" and "v ≤ 0" using y2 ‹0 < kde› ‹0 < d› ‹0 < e› apply (clarsimp simp: in_segment) apply (rule_tac v = "- ((1 - u)*d + u*kde)" in that) apply (force simp: scaleR_conv_of_real algebra_simps) by (meson less_eq_real_def neg_le_0_iff_le segment_bound_lemma) ultimately show False by auto qed moreover have "a - d ∉ closed_segment (a + kde) (a + e)" using ‹0 < kde› ‹kde < d› ‹0 < e› by (auto simp: closed_segment_eq_real_ivl) ultimately have sub_a_plus_e: "closed_segment (a + kde) (a + e) ∩ (path_image p ∪ closed_segment (a - d) (a - kde)) ⊆ {a + e}" by auto have "kde ∈ closed_segment (-kde) e" using ‹0 < kde› ‹kde < d› ‹kde < e› closed_segment_eq_real_ivl by auto then have a_add_kde: "a + kde ∈ closed_segment (a - kde) (a + e)" using of_real_closed_segment [THEN iffD2] by (force dest: closed_segment_translation_eq [of "a", THEN iffD2] simp del: of_real_closed_segment) have "closed_segment (a - kde) (a + kde) ∩ closed_segment (a + kde) (a + e) = {a + kde}" by (metis a_add_kde Int_closed_segment) moreover have "path_image p ∩ closed_segment (a - kde) (a + kde) = {}" proof (rule equals0I, clarify) fix y assume "y ∈ path_image p" "y ∈ closed_segment (a - kde) (a + kde)" with equals0D [OF k, of y] ‹0 < kde› ‹kde < k› show False by (auto simp: dist_norm dest: dist_decreases_closed_segment [where c=a]) qed moreover have "- kde ∈ closed_segment (-d) kde" using ‹0 < kde› ‹kde < d› ‹kde < e› closed_segment_eq_real_ivl by auto then have a_diff_kde': "a - kde ∈ closed_segment (a - d) (a + kde)" using of_real_closed_segment [THEN iffD2] by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment) then have "closed_segment (a - d) (a - kde) ∩ closed_segment (a - kde) (a + kde) = {a - kde}" by (metis Int_closed_segment) ultimately have pa_subset_pm_kde: "path_image ?q ∩ closed_segment (a - kde) (a + kde) ⊆ {a - kde, a + kde}" by (auto simp: path_image_join assms) have ge_kde1: "∃y. x = a + y ∧ y ≥ kde" if "x ∈ closed_segment (a + kde) (a + e)" for x using that ‹kde < e› mult_le_cancel_left apply (auto simp: in_segment) apply (rule_tac x="(1-u)*kde + u*e" in exI) apply (fastforce simp: algebra_simps scaleR_conv_of_real) done have ge_kde2: "∃y. x = a + y ∧ y ≤ -kde" if "x ∈ closed_segment (a - d) (a - kde)" for x using that ‹kde < d› affine_ineq apply (auto simp: in_segment) apply (rule_tac x="- ((1-u)*d + u*kde)" in exI) apply (fastforce simp: algebra_simps scaleR_conv_of_real) done have notin_paq: "x ∉ path_image ?q" if "dist a x < kde" for x using that using ‹0 < kde› ‹kde < d› ‹kde < k› apply (auto simp: path_image_join assms dist_norm dest!: ge_kde1 ge_kde2) by (meson k disjoint_iff_not_equal le_less_trans less_eq_real_def mem_ball that) obtain z where zin: "z ∈ inside (path_image (?q +++ linepath (a - kde) (a + kde)))" and z1: "cmod (winding_number (?q +++ linepath (a - kde) (a + kde)) z) = 1" proof (rule simple_closed_path_wn1 [of kde ?q a]) show "simple_path (?q +++ linepath (a - kde) (a + kde))" proof (intro simple_path_join_loop conjI) show "arc ?q" proof (rule arc_join) show "arc (linepath (a + kde) (a + e))" using ‹kde < e› ‹arc p› by (force simp: pfp) show "arc (p +++ linepath (a - d) (a - kde))" using ‹kde < d› ‹kde < e› ‹arc p› sub_a_diff_d by (force simp: pfp intro: arc_join) qed (auto simp: psp pfp path_image_join sub_a_plus_e) show "arc (linepath (a - kde) (a + kde))" using ‹0 < kde› by auto qed (use pa_subset_pm_kde in auto) qed (use ‹0 < kde› notin_paq in auto) have eq: "path_image (?q +++ linepath (a - kde) (a + kde)) = path_image (p +++ linepath (a - d) (a + e))" (is "?lhs = ?rhs") proof show "?lhs ⊆ ?rhs" using clsub1 clsub2 apply (auto simp: path_image_join assms) by (meson subsetCE subset_closed_segment) show "?rhs ⊆ ?lhs" apply (simp add: path_image_join assms Un_ac) by (metis Un_closed_segment Un_assoc a_diff_kde a_diff_kde' le_supI2 subset_refl) qed show thesis proof show zzin: "z ∈ inside (path_image (p +++ linepath (a - d) (a + e)))" by (metis eq zin) then have znotin: "z ∉ path_image p" by (metis ComplD Un_iff inside_Un_outside path_image_join pathfinish_linepath pathstart_reversepath pfp reversepath_linepath) have znotin_de: "z ∉ closed_segment (a - d) (a + kde)" by (metis ComplD Un_iff Un_closed_segment a_diff_kde inside_Un_outside path_image_join path_image_linepath pathstart_linepath pfp zzin) have "winding_number (linepath (a - d) (a + e)) z = winding_number (linepath (a - d) (a + kde)) z + winding_number (linepath (a + kde) (a + e)) z" apply (rule winding_number_split_linepath) apply (simp add: a_diff_kde) by (metis ComplD Un_iff inside_Un_outside path_image_join path_image_linepath pathstart_linepath pfp zzin) also have "... = winding_number (linepath (a + kde) (a + e)) z + (winding_number (linepath (a - d) (a - kde)) z + winding_number (linepath (a - kde) (a + kde)) z)" by (simp add: winding_number_split_linepath [of "a-kde", symmetric] znotin_de a_diff_kde') finally have "winding_number (p +++ linepath (a - d) (a + e)) z = winding_number p z + winding_number (linepath (a + kde) (a + e)) z + (winding_number (linepath (a - d) (a - kde)) z + winding_number (linepath (a - kde) (a + kde)) z)" by (metis (no_types, lifting) ComplD Un_iff ‹arc p› add.assoc arc_imp_path eq path_image_join path_join_path_ends path_linepath simple_path_imp_path sp_pl union_with_outside winding_number_join zin) also have "... = winding_number ?q z + winding_number (linepath (a - kde) (a + kde)) z" using ‹path p› znotin assms zzin clsub1 apply (subst winding_number_join, auto) apply (metis (no_types, hide_lams) ComplD Un_iff contra_subsetD inside_Un_outside path_image_join path_image_linepath pathstart_linepath) apply (metis Un_iff Un_closed_segment a_diff_kde' not_in_path_image_join path_image_linepath znotin_de) by (metis Un_iff Un_closed_segment a_diff_kde' path_image_linepath path_linepath pathstart_linepath winding_number_join znotin_de) also have "... = winding_number (?q +++ linepath (a - kde) (a + kde)) z" using ‹path p› assms zin apply (subst winding_number_join [symmetric], auto) apply (metis ComplD Un_iff path_image_join pathfinish_join pathfinish_linepath pathstart_linepath union_with_outside) by (metis Un_iff Un_closed_segment a_diff_kde' znotin_de) finally have "winding_number (p +++ linepath (a - d) (a + e)) z = winding_number (?q +++ linepath (a - kde) (a + kde)) z" . then show "cmod (winding_number (p +++ linepath (a - d) (a + e)) z) = 1" by (simp add: z1) qed qed proposition simple_closed_path_wn3: fixes p :: "real ⇒ complex" assumes "simple_path p" and loop: "pathfinish p = pathstart p" obtains z where "z ∈ inside (path_image p)" "cmod (winding_number p z) = 1" proof - have ins: "inside(path_image p) ≠ {}" "open(inside(path_image p))" "connected(inside(path_image p))" and out: "outside(path_image p) ≠ {}" "open(outside(path_image p))" "connected(outside(path_image p))" and bo: "bounded(inside(path_image p))" "¬ bounded(outside(path_image p))" and ins_out: "inside(path_image p) ∩ outside(path_image p) = {}" "inside(path_image p) ∪ outside(path_image p) = - path_image p" and fro: "frontier(inside(path_image p)) = path_image p" "frontier(outside(path_image p)) = path_image p" using Jordan_inside_outside [OF assms] by auto obtain a where a: "a ∈ inside(path_image p)" using ‹inside (path_image p) ≠ {}› by blast obtain d::real where "0 < d" and d_fro: "a - d ∈ frontier (inside (path_image p))" and d_int: "⋀ε. ⟦0 ≤ ε; ε < d⟧ ⟹ (a - ε) ∈ inside (path_image p)" apply (rule ray_to_frontier [of "inside (path_image p)" a "-1"]) using ‹bounded (inside (path_image p))› ‹open (inside (path_image p))› a interior_eq apply (auto simp: of_real_def) done obtain e::real where "0 < e" and e_fro: "a + e ∈ frontier (inside (path_image p))" and e_int: "⋀ε. ⟦0 ≤ ε; ε < e⟧ ⟹ (a + ε) ∈ inside (path_image p)" apply (rule ray_to_frontier [of "inside (path_image p)" a 1]) using ‹bounded (inside (path_image p))› ‹open (inside (path_image p))› a interior_eq apply (auto simp: of_real_def) done obtain t0 where "0 ≤ t0" "t0 ≤ 1" and pt: "p t0 = a - d" using a d_fro fro by (auto simp: path_image_def) obtain q where "simple_path q" and q_ends: "pathstart q = a - d" "pathfinish q = a - d" and q_eq_p: "path_image q = path_image p" and wn_q_eq_wn_p: "⋀z. z ∈ inside(path_image p) ⟹ winding_number q z = winding_number p z" proof show "simple_path (shiftpath t0 p)" by (simp add: pathstart_shiftpath pathfinish_shiftpath simple_path_shiftpath path_image_shiftpath ‹0 ≤ t0› ‹t0 ≤ 1› assms) show "pathstart (shiftpath t0 p) = a - d" using pt by (simp add: ‹t0 ≤ 1› pathstart_shiftpath) show "pathfinish (shiftpath t0 p) = a - d" by (simp add: ‹0 ≤ t0› loop pathfinish_shiftpath pt) show "path_image (shiftpath t0 p) = path_image p" by (simp add: ‹0 ≤ t0› ‹t0 ≤ 1› loop path_image_shiftpath) show "winding_number (shiftpath t0 p) z = winding_number p z" if "z ∈ inside (path_image p)" for z by (metis ComplD Un_iff ‹0 ≤ t0› ‹t0 ≤ 1› ‹simple_path p› atLeastAtMost_iff inside_Un_outside loop simple_path_imp_path that winding_number_shiftpath) qed have ad_not_ae: "a - d ≠ a + e" by (metis ‹0 < d› ‹0 < e› add.left_inverse add_left_cancel add_uminus_conv_diff le_add_same_cancel2 less_eq_real_def not_less of_real_add of_real_def of_real_eq_0_iff pt) have ad_ae_q: "{a - d, a + e} ⊆ path_image q" using ‹path_image q = path_image p› d_fro e_fro fro(1) by auto have ada: "open_segment (a - d) a ⊆ inside (path_image p)" proof (clarsimp simp: in_segment) fix u::real assume "0 < u" "u < 1" with d_int have "a - (1 - u) * d ∈ inside (path_image p)" by (metis ‹0 < d› add.commute diff_add_cancel left_diff_distrib' less_add_same_cancel2 less_eq_real_def mult.left_neutral zero_less_mult_iff) then show "(1 - u) *⇩_{R}(a - d) + u *⇩_{R}a ∈ inside (path_image p)" by (simp add: diff_add_eq of_real_def real_vector.scale_right_diff_distrib) qed have aae: "open_segment a (a + e) ⊆ inside (path_image p)" proof (clarsimp simp: in_segment) fix u::real assume "0 < u" "u < 1" with e_int have "a + u * e ∈ inside (path_image p)" by (meson ‹0 < e› less_eq_real_def mult_less_cancel_right2 not_less zero_less_mult_iff) then show "(1 - u) *⇩_{R}a + u *⇩_{R}(a + e) ∈ inside (path_image p)" apply (simp add: algebra_simps) by (simp add: diff_add_eq of_real_def real_vector.scale_right_diff_distrib) qed have "complex_of_real (d * d + (e * e + d * (e + e))) ≠ 0" using ad_not_ae by (metis ‹0 < d› ‹0 < e› add_strict_left_mono less_add_same_cancel1 not_sum_squares_lt_zero of_real_eq_0_iff zero_less_double_add_iff_zero_less_single_add zero_less_mult_iff) then have a_in_de: "a ∈ open_segment (a - d) (a + e)" using ad_not_ae ‹0 < d› ‹0 < e› apply (auto simp: in_segment algebra_simps scaleR_conv_of_real) apply (rule_tac x="d / (d+e)" in exI) apply (auto simp: field_simps) done then have "open_segment (a - d) (a + e) ⊆ inside (path_image p)" using ada a aae Un_open_segment [of a "a-d" "a+e"] by blast then have "path_image q ∩ open_segment (a - d) (a + e) = {}" using inside_no_overlap by (fastforce simp: q_eq_p) with ad_ae_q have paq_Int_cs: "path_image q ∩ closed_segment (a - d) (a + e) = {a - d, a + e}" by (simp add: closed_segment_eq_open) obtain t where "0 ≤ t" "t ≤ 1" and qt: "q t = a + e" using a e_fro fro ad_ae_q by (auto simp: path_defs) then have "t ≠ 0" by (metis ad_not_ae pathstart_def q_ends(1)) then have "t ≠ 1" by (metis ad_not_ae pathfinish_def q_ends(2) qt) have q01: "q 0 = a - d" "q 1 = a - d" using q_ends by (auto simp: pathstart_def pathfinish_def) obtain z where zin: "z ∈ inside (path_image (subpath t 0 q +++ linepath (a - d) (a + e)))" and z1: "cmod (winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z) = 1" proof (rule simple_closed_path_wn2 [of d e "subpath t 0 q" a], simp_all add: q01) show "simple_path (subpath t 0 q +++ linepath (a - d) (a + e))" proof (rule simple_path_join_loop, simp_all add: qt q01) have "inj_on q (closed_segment t 0)" using ‹0 ≤ t› ‹simple_path q› ‹t ≤ 1› ‹t ≠ 0› ‹t ≠ 1› by (fastforce simp: simple_path_def inj_on_def closed_segment_eq_real_ivl) then show "arc (subpath t 0 q)" using ‹0 ≤ t› ‹simple_path q› ‹t ≤ 1› ‹t ≠ 0› by (simp add: arc_subpath_eq simple_path_imp_path) show "arc (linepath (a - d) (a + e))" by (simp add: ad_not_ae) show "path_image (subpath t 0 q) ∩ closed_segment (a - d) (a + e) ⊆ {a + e, a - d}" using qt paq_Int_cs ‹simple_path q› ‹0 ≤ t› ‹t ≤ 1› by (force simp: dest: rev_subsetD [OF _ path_image_subpath_subset] intro: simple_path_imp_path) qed qed (auto simp: ‹0 < d› ‹0 < e› qt) have pa01_Un: "path_image (subpath 0 t q) ∪ path_image (subpath 1 t q) = path_image q" unfolding path_image_subpath using ‹0 ≤ t› ‹t ≤ 1› by (force simp: path_image_def image_iff) with paq_Int_cs have pa_01q: "(path_image (subpath 0 t q) ∪ path_image (subpath 1 t q)) ∩ closed_segment (a - d) (a + e) = {a - d, a + e}" by metis have z_notin_ed: "z ∉ closed_segment (a + e) (a - d)" using zin q01 by (simp add: path_image_join closed_segment_commute inside_def) have z_notin_0t: "z ∉ path_image (subpath 0 t q)" by (metis (no_types, hide_lams) IntI Un_upper1 subsetD empty_iff inside_no_overlap path_image_join path_image_subpath_commute pathfinish_subpath pathstart_def pathstart_linepath q_ends(1) qt subpath_trivial zin) have [simp]: "- winding_number (subpath t 0 q) z = winding_number (subpath 0 t q) z" by (metis ‹0 ≤ t› ‹simple_path q› ‹t ≤ 1› atLeastAtMost_iff zero_le_one path_image_subpath_commute path_subpath real_eq_0_iff_le_ge_0 reversepath_subpath simple_path_imp_path winding_number_reversepath z_notin_0t) obtain z_in_q: "z ∈ inside(path_image q)" and wn_q: "winding_number (subpath 0 t q +++ subpath t 1 q) z = - winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z" proof (rule winding_number_from_innerpath [of "subpath 0 t q" "a-d" "a+e" "subpath 1 t q" "linepath (a - d) (a + e)" z "- winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z"], simp_all add: q01 qt pa01_Un reversepath_subpath) show "simple_path (subpath 0 t q)" "simple_path (subpath 1 t q)" by (simp_all add: ‹0 ≤ t› ‹simple_path q› ‹t ≤ 1› ‹t ≠ 0› ‹t ≠ 1› simple_path_subpath) show "simple_path (linepath (a - d) (a + e))" using ad_not_ae by blast show "path_image (subpath 0 t q) ∩ path_image (subpath 1 t q) = {a - d, a + e}" (is "?lhs = ?rhs") proof show "?lhs ⊆ ?rhs" using ‹0 ≤ t› ‹simple_path q› ‹t ≤ 1› ‹t ≠ 1› q_ends qt q01 by (force simp: pathfinish_def qt simple_path_def path_image_subpath) show "?rhs ⊆ ?lhs" using ‹0 ≤ t› ‹t ≤ 1› q01 qt by (force simp: path_image_subpath) qed show "path_image (subpath 0 t q) ∩ closed_segment (a - d) (a + e) = {a - d, a + e}" (is "?lhs = ?rhs") proof show "?lhs ⊆ ?rhs" using paq_Int_cs pa01_Un by fastforce show "?rhs ⊆ ?lhs" using ‹0 ≤ t› ‹t ≤ 1› q01 qt by (force simp: path_image_subpath) qed show "path_image (subpath 1 t q) ∩ closed_segment (a - d) (a + e) = {a - d, a + e}" (is "?lhs = ?rhs") proof show "?lhs ⊆ ?rhs" by (auto simp: pa_01q [symmetric]) show "?rhs ⊆ ?lhs" using ‹0 ≤ t› ‹t ≤ 1› q01 qt by (force simp: path_image_subpath) qed show "closed_segment (a - d) (a + e) ∩ inside (path_image q) ≠ {}" using a a_in_de open_closed_segment pa01_Un q_eq_p by fastforce show "z ∈ inside (path_image (subpath 0 t q) ∪ closed_segment (a - d) (a + e))" by (metis path_image_join path_image_linepath path_image_subpath_commute pathfinish_subpath pathstart_linepath q01(1) zin) show "winding_number (subpath 0 t q +++ linepath (a + e) (a - d)) z = - winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z" using z_notin_ed z_notin_0t ‹0 ≤ t› ‹simple_path q› ‹t ≤ 1› by (simp add: simple_path_imp_path qt q01 path_image_subpath_commute closed_segment_commute winding_number_join winding_number_reversepath [symmetric]) show "- d ≠ e" using ad_not_ae by auto show "winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z ≠ 0" using z1 by auto qed show ?thesis proof show "z ∈ inside (path_image p)" using q_eq_p z_in_q by auto then have [simp]: "z ∉ path_image q" by (metis disjoint_iff_not_equal inside_no_overlap q_eq_p) have [simp]: "z ∉ path_image (subpath 1 t q)" using inside_def pa01_Un z_in_q by fastforce have "winding_number(subpath 0 t q +++ subpath t 1 q) z = winding_number(subpath 0 1 q) z" using z_notin_0t ‹0 ≤ t› ‹simple_path q› ‹t ≤ 1› by (simp add: simple_path_imp_path qt path_image_subpath_commute winding_number_join winding_number_subpath_combine) with wn_q have "winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z = - winding_number q z" by auto with z1 have "cmod (winding_number q z) = 1" by simp with z1 wn_q_eq_wn_p show "cmod (winding_number p z) = 1" using z1 wn_q_eq_wn_p by (simp add: ‹z ∈ inside (path_image p)›) qed qed theorem simple_closed_path_winding_number_inside: assumes "simple_path γ" obtains "⋀z. z ∈ inside(path_image γ) ⟹ winding_number γ z = 1" | "⋀z. z ∈ inside(path_image γ) ⟹ winding_number γ z = -1" proof (cases "pathfinish γ = pathstart γ") case True have "path γ" by (simp add: assms simple_path_imp_path) then have const: "winding_number γ constant_on inside(path_image γ)" proof (rule winding_number_constant) show "connected (inside(path_image γ))" by (simp add: Jordan_inside_outside True assms) qed (use inside_no_overlap True in auto) obtain z where zin: "z ∈ inside (path_image γ)" and z1: "cmod (winding_number γ z) = 1" using simple_closed_path_wn3 [of γ] True assms by blast have "winding_number γ z ∈ ℤ" using zin integer_winding_number [OF ‹path γ› True] inside_def by blast with z1 consider "winding_number γ z = 1" | "winding_number γ z = -1" apply (auto simp: Ints_def abs_if split: if_split_asm) by (metis of_int_1 of_int_eq_iff of_int_minus) with that const zin show ?thesis unfolding constant_on_def by metis next case False then show ?thesis using inside_simple_curve_imp_closed assms that(2) by blast qed corollary simple_closed_path_abs_winding_number_inside: assumes "simple_path γ" "z ∈ inside(path_image γ)" shows "¦Re (winding_number γ z)¦ = 1" by (metis assms norm_minus_cancel norm_one one_complex.simps(1) real_norm_def simple_closed_path_winding_number_inside uminus_complex.simps(1)) corollary simple_closed_path_norm_winding_number_inside: assumes "simple_path γ" "z ∈ inside(path_image γ)" shows "norm (winding_number γ z) = 1" proof - have "pathfinish γ = pathstart γ" using assms inside_simple_curve_imp_closed by blast with assms integer_winding_number have "winding_number γ z ∈ ℤ" by (simp add: inside_def simple_path_def) then show ?thesis by (metis assms norm_minus_cancel norm_one simple_closed_path_winding_number_inside) qed corollary simple_closed_path_winding_number_cases: "⟦simple_path γ; pathfinish γ = pathstart γ; z ∉ path_image γ⟧ ⟹ winding_number γ z ∈ {-1,0,1}" apply (simp add: inside_Un_outside [of "path_image γ", symmetric, unfolded set_eq_iff Set.Compl_iff] del: inside_Un_outside) apply (rule simple_closed_path_winding_number_inside) using simple_path_def winding_number_zero_in_outside by blast+ corollary simple_closed_path_winding_number_pos: "⟦simple_path γ; pathfinish γ = pathstart γ; z ∉ path_image γ; 0 < Re(winding_number γ z)⟧ ⟹ winding_number γ z = 1" using simple_closed_path_winding_number_cases by fastforce subsection ‹Winding number for rectangular paths› (* TODO: Move *) lemma closed_segmentI: "u ∈ {0..1} ⟹ z = (1 - u) *⇩_{R}a + u *⇩_{R}b ⟹ z ∈ closed_segment a b" by (auto simp: closed_segment_def) lemma in_cbox_complex_iff: "x ∈ cbox a b ⟷ Re x ∈ {Re a..Re b} ∧ Im x ∈ {Im a..Im b}" by (cases x; cases a; cases b) (auto simp: cbox_Complex_eq) lemma box_Complex_eq: "box (Complex a c) (Complex b d) = (λ(x,y). Complex x y) ` (box a b × box c d)" by (auto simp: box_def Basis_complex_def image_iff complex_eq_iff) lemma in_box_complex_iff: "x ∈ box a b ⟷ Re x ∈ {Re a<..<Re b} ∧ Im x ∈ {Im a<..<Im b}" by (cases x; cases a; cases b) (auto simp: box_Complex_eq) (* END TODO *) lemma closed_segment_same_Re: assumes "Re a = Re b" shows "closed_segment a b = {z. Re z = Re a ∧ Im z ∈ closed_segment (Im a) (Im b)}" proof safe fix z assume "z ∈ closed_segment a b" then obtain u where u: "u ∈ {0..1}" "z = a + of_real u * (b - a)" by (auto simp: closed_segment_def scaleR_conv_of_real algebra_simps) from assms show "Re z = Re a" by (auto simp: u) from u(1) show "Im z ∈ closed_segment (Im a) (Im b)" by (intro closed_segmentI[of u]) (auto simp: u algebra_simps) next fix z assume [simp]: "Re z = Re a" and "Im z ∈ closed_segment (Im a) (Im b)" then obtain u where u: "u ∈ {0..1}" "Im z = Im a + of_real u * (Im b - Im a)" by (auto simp: closed_segment_def scaleR_conv_of_real algebra_simps) from u(1) show "z ∈ closed_segment a b" using assms by (intro closed_segmentI[of u]) (auto simp: u algebra_simps scaleR_conv_of_real complex_eq_iff) qed lemma closed_segment_same_Im: assumes "Im a = Im b" shows "closed_segment a b = {z. Im z = Im a ∧ Re z ∈ closed_segment (Re a) (Re b)}" proof safe fix z assume "z ∈ closed_segment a b" then obtain u where u: "u ∈ {0..1}" "z = a + of_real u * (b - a)" by (auto simp: closed_segment_def scaleR_conv_of_real algebra_simps) from assms show "Im z = Im a" by (auto simp: u) from u(1) show "Re z ∈ closed_segment (Re a) (Re b)" by (intro closed_segmentI[of u]) (auto simp: u algebra_simps) next fix z assume [simp]: "Im z = Im a" and "Re z ∈ closed_segment (Re a) (Re b)" then obtain u where u: "u ∈ {0..1}" "Re z = Re a + of_real u * (Re b - Re a)" by (auto simp: closed_segment_def scaleR_conv_of_real algebra_simps) from u(1) show "z ∈ closed_segment a b" using assms by (intro closed_segmentI[of u]) (auto simp: u algebra_simps scaleR_conv_of_real complex_eq_iff) qed definition rectpath where "rectpath a1 a3 = (let a2 = Complex (Re a3) (Im a1); a4 = Complex (Re a1) (Im a3) in linepath a1 a2 +++ linepath a2 a3 +++ linepath a3 a4 +++ linepath a4 a1)" lemma path_rectpath [simp, intro]: "path (rectpath a b)" by (simp add: Let_def rectpath_def) lemma valid_path_rectpath [simp, intro]: "valid_path (rectpath a b)" by (simp add: Let_def rectpath_def) lemma pathstart_rectpath [simp]: "pathstart (rectpath a1 a3) = a1" by (simp add: rectpath_def Let_def) lemma pathfinish_rectpath [simp]: "pathfinish (rectpath a1 a3) = a1" by (simp add: rectpath_def Let_def) lemma simple_path_rectpath [simp, intro]: assumes "Re a1 ≠ Re a3" "Im a1 ≠ Im a3" shows "simple_path (rectpath a1 a3)" unfolding rectpath_def Let_def using assms by (intro simple_path_join_loop arc_join arc_linepath) (auto simp: complex_eq_iff path_image_join closed_segment_same_Re closed_segment_same_Im) lemma path_image_rectpath: assumes "Re a1 ≤ Re a3" "Im a1 ≤ Im a3" shows "path_image (rectpath a1 a3) = {z. Re z ∈ {Re a1, Re a3} ∧ Im z ∈ {Im a1..Im a3}} ∪ {z. Im z ∈ {Im a1, Im a3} ∧ Re z ∈ {Re a1..Re a3}}" (is "?lhs = ?rhs") proof - define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)" have "?lhs = closed_segment a1 a2 ∪ closed_segment a2 a3 ∪ closed_segment a4 a3 ∪ closed_segment a1 a4" by (simp_all add: rectpath_def Let_def path_image_join closed_segment_commute a2_def a4_def Un_assoc) also have "… = ?rhs" using assms by (auto simp: rectpath_def Let_def path_image_join a2_def a4_def closed_segment_same_Re closed_segment_same_Im closed_segment_eq_real_ivl) finally show ?thesis . qed lemma path_image_rectpath_subset_cbox: assumes "Re a ≤ Re b" "Im a ≤ Im b" shows "path_image (rectpath a b) ⊆ cbox a b" using assms by (auto simp: path_image_rectpath in_cbox_complex_iff) lemma path_image_rectpath_inter_box: assumes "Re a ≤ Re b" "Im a ≤ Im b" shows "path_image (rectpath a b) ∩ box a b = {}" using assms by (auto simp: path_image_rectpath in_box_complex_iff) lemma path_image_rectpath_cbox_minus_box: assumes "Re a ≤ Re b" "Im a ≤ Im b" shows "path_image (rectpath a b) = cbox a b - box a b" using assms by (auto simp: path_image_rectpath in_cbox_complex_iff in_box_complex_iff) lemma winding_number_rectpath: assumes "z ∈ box a1 a3" shows "winding_number (rectpath a1 a3) z = 1" proof - from assms have less: "Re a1 < Re a3" "Im a1 < Im a3" by (auto simp: in_box_complex_iff) define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)" let ?l1 = "linepath a1 a2" and ?l2 = "linepath a2 a3" and ?l3 = "linepath a3 a4" and ?l4 = "linepath a4 a1" from assms and less have "z ∉ path_image (rectpath a1 a3)" by (auto simp: path_image_rectpath_cbox_minus_box) also have "path_image (rectpath a1 a3) = path_image ?l1 ∪ path_image ?l2 ∪ path_image ?l3 ∪ path_image ?l4" by (simp add: rectpath_def Let_def path_image_join Un_assoc a2_def a4_def) finally have "z ∉ …" . moreover have "∀l∈{?l1,?l2,?l3,?l4}. Re (winding_number l z) > 0" unfolding ball_simps HOL.simp_thms a2_def a4_def by (intro conjI; (rule winding_number_linepath_pos_lt; (insert assms, auto simp: a2_def a4_def in_box_complex_iff mult_neg_neg))+) ultimately have "Re (winding_number (rectpath a1 a3) z) > 0" by (simp add: winding_number_join path_image_join rectpath_def Let_def a2_def a4_def) thus "winding_number (rectpath a1 a3) z = 1" using assms less by (intro simple_closed_path_winding_number_pos simple_path_rectpath) (auto simp: path_image_rectpath_cbox_minus_box) qed lemma winding_number_rectpath_outside: assumes "Re a1 ≤ Re a3" "Im a1 ≤ Im a3" assumes "z ∉ cbox a1 a3" shows "winding_number (rectpath a1 a3) z = 0" using assms by (intro winding_number_zero_outside[OF _ _ _ assms(3)] path_image_rectpath_subset_cbox) simp_all text‹A per-function version for continuous logs, a kind of monodromy› proposition winding_number_compose_exp: assumes "path p" shows "winding_number (exp ∘ p) 0 = (pathfinish p - pathstart p) / (2 * of_real pi * 𝗂)" proof - obtain e where "0 < e" and e: "⋀t. t ∈ {0..1} ⟹ e ≤ norm(exp(p t))" proof have "closed (path_image (exp ∘ p))" by (simp add: assms closed_path_image holomorphic_on_exp holomorphic_on_imp_continuous_on path_continuous_image) then show "0 < setdist {0} (path_image (exp ∘ p))" by (metis (mono_tags, lifting) compact_sing exp_not_eq_zero imageE path_image_compose path_image_nonempty setdist_eq_0_compact_closed setdist_gt_0_compact_closed setdist_eq_0_closed) next fix t::real assume "t ∈ {0..1}" have "setdist {0} (path_image (exp ∘ p)) ≤ dist 0 (exp (p t))" apply (rule setdist_le_dist) using ‹t ∈ {0..1}› path_image_def by fastforce+ then show "setdist {0} (path_image (exp ∘ p)) ≤ cmod (exp (p t))" by simp qed have "bounded (path_image p)" by (simp add: assms bounded_path_image) then obtain B where "0 < B" and B: "path_image p ⊆ cball 0 B" by (meson bounded_pos mem_cball_0 subsetI) let ?B = "cball (0::complex) (B+1)" have "uniformly_continuous_on ?B exp" using holomorphic_on_exp holomorphic_on_imp_continuous_on by (force intro: compact_uniformly_continuous) then obtain d where "d > 0" and d: "⋀x x'. ⟦x∈?B; x'∈?B; dist x' x < d⟧ ⟹ norm (exp x' - exp x) < e" using ‹e > 0› by (auto simp: uniformly_continuous_on_def dist_norm) then have "min 1 d > 0" by force then obtain g where pfg: "polynomial_function g" and "g 0 = p 0" "g 1 = p 1" and gless: "⋀t. t ∈ {0..1} ⟹ norm(g t - p t) < min 1 d" using path_approx_polynomial_function [OF ‹path p›] ‹d > 0› ‹0 < e› unfolding pathfinish_def pathstart_def by meson have "winding_number (exp ∘ p) 0 = winding_number (exp ∘ g) 0" proof (rule winding_number_nearby_paths_eq [symmetric]) show "path (exp ∘ p)" "path (exp ∘ g)" by (simp_all add: pfg assms holomorphic_on_exp holomorphic_on_imp_continuous_on path_continuous_image path_polynomial_function) next fix t :: "real" assume t: "t ∈ {0..1}" with gless have "norm(g t - p t) < 1" using min_less_iff_conj by blast moreover have ptB: "norm (p t) ≤ B" using B t by (force simp: path_image_def) ultimately have "cmod (g t) ≤ B + 1" by (meson add_mono_thms_linordered_field(4) le_less_trans less_imp_le norm_triangle_sub) with ptB gless t have "cmod ((exp ∘ g) t - (exp ∘ p) t) < e" by (auto simp: dist_norm d) with e t show "cmod ((exp ∘ g) t - (exp ∘ p) t) < cmod ((exp ∘ p) t - 0)" by fastforce qed (use ‹g 0 = p 0› ‹g 1 = p 1› in ‹auto simp: pathfinish_def pathstart_def›) also have "... = 1 / (of_real (2 * pi) * 𝗂) * contour_integral (exp ∘ g) (λw. 1 / (w - 0))" proof (rule winding_number_valid_path) have "continuous_on (path_image g) (deriv exp)" by (metis DERIV_exp DERIV_imp_deriv continuous_on_cong holomorphic_on_exp holomorphic_on_imp_continuous_on) then show "valid_path (exp ∘ g)" by (simp add: field_differentiable_within_exp pfg valid_path_compose valid_path_polynomial_function) show "0 ∉ path_image (exp ∘ g)" by (auto simp: path_image_def) qed also have "... = 1 / (of_real (2 * pi) * 𝗂) * integral {0..1} (λx. vector_derivative g (at x))" proof (simp add: contour_integral_integral, rule integral_cong) fix t :: "real" assume t: "t ∈ {0..1}" show "vector_derivative (exp ∘ g) (at t) / exp (g t) = vector_derivative g (at t)" proof (simp add: divide_simps, rule vector_derivative_unique_at) show "(exp ∘ g has_vector_derivative vector_derivative (exp ∘ g) (at t)) (at t)" by (meson DERIV_exp differentiable_def field_vector_diff_chain_at has_vector_derivative_def has_vector_derivative_polynomial_function pfg vector_derivative_works) show "(exp ∘ g has_vector_derivative vector_derivative g (at t) * exp (g t)) (at t)" apply (rule field_vector_diff_chain_at) apply (metis has_vector_derivative_polynomial_function pfg vector_derivative_at) using DERIV_exp has_field_derivative_def apply blast done qed qed also have "... = (pathfinish p - pathstart p) / (2 * of_real pi * 𝗂)" proof - have "((λx. vector_derivative g (at x)) has_integral g 1 - g 0) {0..1}" apply (rule fundamental_theorem_of_calculus [OF zero_le_one]) by (metis has_vector_derivative_at_within has_vector_derivative_polynomial_function pfg vector_derivative_at) then show ?thesis apply (simp add: pathfinish_def pathstart_def) using ‹g 0 = p 0› ‹g 1 = p 1› by auto qed finally show ?thesis . qed subsection‹The winding number defines a continuous logarithm for the path itself› lemma winding_number_as_continuous_log: assumes "path p" and ζ: "ζ ∉ path_image p" obtains q where "path q" "pathfinish q - pathstart q = 2 * of_real pi * 𝗂 * winding_number p ζ" "⋀t. t ∈ {0..1} ⟹ p t = ζ + exp(q t)" proof - let ?q = "λt. 2 * of_real pi * 𝗂 * winding_number(subpath 0 t p) ζ + Ln(pathstart p - ζ)" show ?thesis proof have *: "continuous (at t within {0..1}) (λx. winding_number (subpath 0 x p) ζ)" if t: "t ∈ {0..1}" for t proof - let ?B = "ball (p t) (norm(p t - ζ))" have "p t ≠ ζ" using path_image_def that ζ by blast then have "simply_connected ?B" by (simp add: convex_imp_simply_connected) then have "⋀f::complex⇒complex. continuous_on ?B f ∧ (∀ζ ∈ ?B. f ζ ≠ 0) ⟶ (∃g. continuous_on ?B g ∧ (∀ζ ∈ ?B. f ζ = exp (g ζ)))" by (simp add: simply_connected_eq_continuous_log) moreover have "continuous_on ?B (λw. w - ζ)" by (intro continuous_intros) moreover have "(∀z ∈ ?B. z - ζ ≠ 0)" by (auto simp: dist_norm) ultimately obtain g where contg: "continuous_on ?B g" and geq: "⋀z. z ∈ ?B ⟹ z - ζ = exp (g z)" by blast obtain d where "0 < d" and d: "⋀x. ⟦x ∈ {0..1}; dist x t < d⟧ ⟹ dist (p x) (p t) < cmod (p t - ζ)" using ‹path p› t unfolding path_def continuous_on_iff by (metis ‹p t ≠ ζ› right_minus_eq zero_less_norm_iff) have "((λx. winding_number (λw. subpath 0 x p w - ζ) 0 - winding_number (λw. subpath 0 t p w - ζ) 0) ⤏ 0) (at t within {0..1})" proof (rule Lim_transform_within [OF _ ‹d > 0›]) have "continuous (at t within {0..1}) (g o p)" proof (rule continuous_within_compose) show "continuous (at t within {0..1}) p" using ‹path p› continuous_on_eq_continuous_within path_def that by blast show "continuous (at (p t) within p ` {0..1}) g" by (metis (no_types, lifting) open_ball UNIV_I ‹p t ≠ ζ› centre_in_ball contg continuous_on_eq_continuous_at continuous_within_topological right_minus_eq zero_less_norm_iff) qed with LIM_zero have "((λu. (g (subpath t u p 1) - g (subpath t u p 0))) ⤏ 0) (at t within {0..1})" by (auto simp: subpath_def continuous_within o_def) then show "((λu. (g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * 𝗂)) ⤏ 0) (at t within {0..1})" by (simp add: tendsto_divide_zero) show "(g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * 𝗂) = winding_number (λw. subpath 0 u p w - ζ) 0 - winding_number (λw. subpath 0 t p w - ζ) 0" if "u ∈ {0..1}" "0 < dist u t" "dist u t < d" for u proof - have "closed_segment t u ⊆ {0..1}" using closed_segment_eq_real_ivl t that by auto then have piB: "path_image(subpath t u p) ⊆ ?B" apply (clarsimp simp add: path_image_subpath_gen) by (metis subsetD le_less_trans ‹dist u t < d› d dist_commute dist_in_closed_segment) have *: "path (g ∘ subpath t u p)" apply (rule path_continuous_image) using ‹path p› t that apply auto[1] using piB contg continuous_on_subset by blast have "(g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * 𝗂) = winding_number (exp ∘ g ∘ subpath t u p) 0" using winding_number_compose_exp [OF *] by (simp add: pathfinish_def pathstart_def o_assoc) also have "... = winding_number (λw. subpath t u p w - ζ) 0" proof (rule winding_number_cong) have "exp(g y) = y - ζ" if "y ∈ (subpath t u p) ` {0..1}" for y by (metis that geq path_image_def piB subset_eq) then show "⋀x. ⟦0 ≤ x; x ≤ 1⟧ ⟹ (exp ∘ g ∘ subpath t u p) x = subpath t u p x - ζ" by auto qed also have "... = winding_number (λw. subpath 0 u p w - ζ) 0 - winding_number (λw. subpath 0 t p w - ζ) 0" apply (simp add: winding_number_offset [symmetric]) using winding_number_subpath_combine [OF ‹path p› ζ, of 0 t u] ‹t ∈ {0..1}› ‹u ∈ {0..1}› by (simp add: add.commute eq_diff_eq) finally show ?thesis . qed qed then show ?thesis by (subst winding_number_offset) (simp add: continuous_within LIM_zero_iff) qed show "path ?q" unfolding path_def by (intro continuous_intros) (simp add: continuous_on_eq_continuous_within *) have "ζ ≠ p 0" by (metis ζ pathstart_def pathstart_in_path_image) then show "pathfinish ?q - pathstart ?q = 2 * of_real pi * 𝗂 * winding_number p ζ" by (simp add: pathfinish_def pathstart_def) show "p t = ζ + exp (?q t)" if "t ∈ {0..1}" for t proof - have "path (subpath 0 t p)" using ‹path p› that by auto moreover have "ζ ∉ path_image (subpath 0 t p)" using ζ [unfolded path_image_def] that by (auto simp: path_image_subpath) ultimately show ?thesis using winding_number_exp_2pi [of "subpath 0 t p" ζ] ‹ζ ≠ p 0› by (auto simp: exp_add algebra_simps pathfinish_def pathstart_def subpath_def) qed qed qed subsection‹Winding number equality is the same as path/loop homotopy in C - {0}› lemma winding_number_homotopic_loops_null_eq: assumes "path p" and ζ: "ζ ∉ path_image p" shows "winding_number p ζ = 0 ⟷ (∃a. homotopic_loops (-{ζ}) p (λt. a))" (is "?lhs = ?rhs") proof assume [simp]: ?lhs obtain q where "path q" and qeq: "pathfinish q - pathstart q = 2 * of_real pi * 𝗂 * winding_number p ζ" and peq: "⋀t. t ∈ {0..1} ⟹ p t = ζ + exp(q t)" using winding_number_as_continuous_log [OF assms] by blast have *: "homotopic_with (λr. pathfinish r = pathstart r) {0..1} (-{ζ}) ((λw. ζ + exp w) ∘ q) ((λw. ζ + exp w) ∘ (λt. 0))" proof (rule homotopic_with_compose_continuous_left) show "homotopic_with (λf. pathfinish ((λw. ζ + exp w) ∘ f) = pathstart ((λw. ζ + exp w) ∘ f)) {0..1} UNIV q (λt. 0)" proof (rule homotopic_with_mono, simp_all add: pathfinish_def pathstart_def) have "homotopic_loops UNIV q (λt. 0)" by (rule homotopic_loops_linear) (use qeq ‹path q› in ‹auto simp: continuous_on_const path_defs›) then show "homotopic_with (λh. exp (h 1) = exp (h 0)) {0..1} UNIV q (λt. 0)" by (simp add: homotopic_loops_def homotopic_with_mono pathfinish_def pathstart_def) qed show "continuous_on UNIV (λw. ζ + exp w)" by (rule continuous_intros)+ show "range (λw. ζ + exp w) ⊆ -{ζ}" by auto qed then have "homotopic_with (λr. pathfinish r = pathstart r) {0..1} (-{ζ}) p (λx. ζ + 1)" by (rule homotopic_with_eq) (auto simp: o_def peq pathfinish_def pathstart_def) then have "homotopic_loops (-{ζ}) p (λt. ζ + 1)" by (simp add: homotopic_loops_def) then show ?rhs .. next assume ?rhs then obtain a where "homotopic_loops (-{ζ}) p (λt. a)" .. then have "winding_number p ζ = winding_number (λt. a) ζ" "a ≠ ζ" using winding_number_homotopic_loops homotopic_loops_imp_subset by (force simp:)+ moreover have "winding_number (λt. a) ζ = 0" by (metis winding_number_zero_const ‹a ≠ ζ›) ultimately show ?lhs by metis qed lemma winding_number_homotopic_paths_null_explicit_eq: assumes "path p" and ζ: "ζ ∉ path_image p" shows "winding_number p ζ = 0 ⟷ homotopic_paths (-{ζ}) p (linepath (pathstart p) (pathstart p))" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs apply (auto simp: winding_number_homotopic_loops_null_eq [OF assms]) apply (rule homotopic_loops_imp_homotopic_paths_null) apply (simp add: linepath_refl) done next assume ?rhs then show ?lhs by (metis ζ pathstart_in_path_image winding_number_homotopic_paths winding_number_trivial) qed lemma winding_number_homotopic_paths_null_eq: assumes "path p" and ζ: "ζ ∉ path_image p" shows "winding_number p ζ = 0 ⟷ (∃a. homotopic_paths (-{ζ}) p (λt. a))" (is "?lhs = ?rhs") proof assume ?lhs then show ?rhs by (auto simp: winding_number_homotopic_paths_null_explicit_eq [OF assms] linepath_refl) next assume ?rhs then show ?lhs by (metis ζ homotopic_paths_imp_pathfinish pathfinish_def pathfinish_in_path_image winding_number_homotopic_paths winding_number_zero_const) qed lemma winding_number_homotopic_paths_eq: assumes "path p" and ζp: "ζ ∉ path_image p" and "path q" and ζq: "ζ ∉ path_image q" and qp: "pathstart q = pathstart p" "pathfinish q = pathfinish p" shows "winding_number p ζ = winding_number q ζ ⟷ homotopic_paths (-{ζ}) p q" (is "?lhs = ?rhs") proof assume ?lhs then have "winding_number (p +++ reversepath q) ζ = 0" using assms by (simp add: winding_number_join winding_number_reversepath) moreover have "path (p +++ reversepath q)" "ζ ∉ path_image (p +++ reversepath q)" using assms by (auto simp: not_in_path_image_join) ultimately obtain a where "homotopic_paths (- {ζ}) (p +++ reversepath q) (linepath a a)" using winding_number_homotopic_paths_null_explicit_eq by blast then show ?rhs using homotopic_paths_imp_pathstart assms by (fastforce simp add: dest: homotopic_paths_imp_homotopic_loops homotopic_paths_loop_parts) next assume ?rhs then show ?lhs by (simp add: winding_number_homotopic_paths) qed lemma winding_number_homotopic_loops_eq: assumes "path p" and ζp: "ζ ∉ path_image p" and "path q" and ζq: "ζ ∉ path_image q" and loops: "pathfinish p = pathstart p" "pathfinish q = pathstart q" shows "winding_number p ζ = winding_number q ζ ⟷ homotopic_loops (-{ζ}) p q" (is "?lhs = ?rhs") proof assume L: ?lhs have "pathstart p ≠ ζ" "pathstart q ≠ ζ" using ζp ζq by blast+ moreover have "path_connected (-{ζ})" by (simp add: path_connected_punctured_universe) ultimately obtain r where "path r" and rim: "path_image r ⊆ -{ζ}" and pas: "pathstart r = pathstart p" and paf: "pathfinish r = pathstart q" by (auto simp: path_connected_def) then have "pathstart r ≠ ζ" by blast have "homotopic_loops (- {ζ}) p (r +++ q +++ reversepath r)" proof (rule homotopic_paths_imp_homotopic_loops) show "homotopic_paths (- {ζ}) p (r +++ q +++ reversepath r)" by (metis (mono_tags, hide_lams) ‹path r› L ζp ζq ‹path p› ‹path q› homotopic_loops_conjugate loops not_in_path_image_join paf pas path_image_reversepath path_imp_reversepath path_join_eq pathfinish_join pathfinish_reversepath pathstart_join pathstart_reversepath rim subset_Compl_singleton winding_number_homotopic_loops winding_number_homotopic_paths_eq) qed (use loops pas in auto) moreover have "homotopic_loops (- {ζ}) (r +++ q +++ reversepath r) q" using rim ζq by (auto simp: homotopic_loops_conjugate paf ‹path q› ‹path r› loops) ultimately show ?rhs using homotopic_loops_trans by metis next assume ?rhs then show ?lhs by (simp add: winding_number_homotopic_loops) qed end