--- a/src/HOL/Complex_Analysis/Contour_Integration.thy Sat Aug 12 10:09:29 2023 +0100
+++ b/src/HOL/Complex_Analysis/Contour_Integration.thy Mon Aug 21 18:38:25 2023 +0100
@@ -44,14 +44,14 @@
unfolding contour_integrable_on_def contour_integral_def by blast
lemma contour_integral_unique: "(f has_contour_integral i) g \<Longrightarrow> contour_integral g f = i"
- apply (simp add: contour_integral_def has_contour_integral_def contour_integrable_on_def)
+ unfolding contour_integral_def has_contour_integral_def contour_integrable_on_def
using has_integral_unique by blast
lemma has_contour_integral_eqpath:
- "\<lbrakk>(f has_contour_integral y) p; f contour_integrable_on \<gamma>;
+ "\<lbrakk>(f has_contour_integral y) p; f contour_integrable_on \<gamma>;
contour_integral p f = contour_integral \<gamma> f\<rbrakk>
\<Longrightarrow> (f has_contour_integral y) \<gamma>"
-using contour_integrable_on_def contour_integral_unique by auto
+ using contour_integrable_on_def contour_integral_unique by auto
lemma has_contour_integral_integral:
"f contour_integrable_on i \<Longrightarrow> (f has_contour_integral (contour_integral i f)) i"
@@ -329,12 +329,12 @@
qed
lemma contour_integrable_join [simp]:
- "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
+ "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
\<Longrightarrow> f contour_integrable_on (g1 +++ g2) \<longleftrightarrow> f contour_integrable_on g1 \<and> f contour_integrable_on g2"
-using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast
+ using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast
lemma contour_integral_join [simp]:
- "\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
+ "\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
\<Longrightarrow> contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f"
by (simp add: has_contour_integral_integral has_contour_integral_join contour_integral_unique)
@@ -353,11 +353,7 @@
using assms by (auto simp: has_contour_integral)
then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) +
integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
- apply (rule has_integral_unique)
- apply (subst add.commute)
- apply (subst Henstock_Kurzweil_Integration.integral_combine)
- using assms * integral_unique by auto
-
+ by (smt (verit, ccfv_threshold) Henstock_Kurzweil_Integration.integral_combine a add.commute atLeastAtMost_iff has_integral_iff)
have vd1: "vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
if "0 \<le> x" "x + a < 1" "x \<notin> (\<lambda>x. x - a) ` S" for x
unfolding shiftpath_def
@@ -371,8 +367,7 @@
then show "(g has_vector_derivative vector_derivative g (at (a + x))) (at (x + a))"
by (metis add.commute vector_derivative_works)
qed
- then
- show "((\<lambda>x. g (a + x)) has_vector_derivative vector_derivative g (at (x + a))) (at x)"
+ then show "((\<lambda>x. g (a + x)) has_vector_derivative vector_derivative g (at (x + a))) (at x)"
by (auto simp: field_simps)
show "0 < dist (1 - a) x"
using that by auto
@@ -474,8 +469,8 @@
lemma contour_integrable_on_shiftpath_eq:
assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
- shows "f contour_integrable_on (shiftpath a g) \<longleftrightarrow> f contour_integrable_on g"
-using assms contour_integrable_on_def has_contour_integral_shiftpath_eq by auto
+ shows "f contour_integrable_on (shiftpath a g) \<longleftrightarrow> f contour_integrable_on g"
+ using assms contour_integrable_on_def has_contour_integral_shiftpath_eq by auto
lemma contour_integral_shiftpath:
assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
@@ -556,26 +551,17 @@
lemma contour_integrable_subpath:
assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
shows "f contour_integrable_on (subpath u v g)"
-proof (cases u v rule: linorder_class.le_cases)
- case le
- then show ?thesis
- by (metis contour_integrable_on_def has_contour_integral_subpath [OF assms])
-next
- case ge
- with assms show ?thesis
- by (metis (no_types, lifting) contour_integrable_on_def contour_integrable_reversepath_eq has_contour_integral_subpath reversepath_subpath valid_path_subpath)
-qed
+ by (smt (verit, ccfv_threshold) assms contour_integrable_on_def contour_integrable_reversepath_eq
+ has_contour_integral_subpath reversepath_subpath valid_path_subpath)
lemma has_integral_contour_integral_subpath:
assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
- shows "(((\<lambda>x. f(g x) * vector_derivative g (at x)))
+ shows "((\<lambda>x. f(g x) * vector_derivative g (at x))
has_integral contour_integral (subpath u v g) f) {u..v}"
- using assms
+ (is "(?fg has_integral _)_")
proof -
- have "(\<lambda>r. f (g r) * vector_derivative g (at r)) integrable_on {u..v}"
- by (metis (full_types) assms(1) assms(3) assms(4) atLeastAtMost_iff atLeastatMost_subset_iff contour_integrable_on integrable_on_subinterval)
- then have "((\<lambda>r. f (g r) * vector_derivative g (at r)) has_integral integral {u..v} (\<lambda>r. f (g r) * vector_derivative g (at r))) {u..v}"
- by blast
+ have "(?fg has_integral integral {u..v} ?fg) {u..v}"
+ using assms contour_integrable_on integrable_on_subinterval by fastforce
then show ?thesis
by (metis (full_types) assms contour_integral_unique has_contour_integral_subpath)
qed
@@ -653,9 +639,9 @@
next
fix x assume "x \<in> {0..1} - ({0, 1} \<union> g -` A \<inter> {0<..<1})"
hence "g x \<in> path_image g - A" by (auto simp: path_image_def)
- from assms(4)[OF this] and assms(3)
- show "f' (g' x) * vector_derivative g' (at x) = f (g x) * vector_derivative g (at x)" by simp
- qed
+ with assms show "f' (g' x) * vector_derivative g' (at x) = f (g x) * vector_derivative g (at x)"
+ by simp
+qed
text \<open>Contour integral along a segment on the real axis\<close>
@@ -664,7 +650,7 @@
fixes a b :: complex and f :: "complex \<Rightarrow> complex"
assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
shows "(f has_contour_integral I) (linepath a b) \<longleftrightarrow>
- ((\<lambda>x. f (of_real x)) has_integral I) {Re a..Re b}"
+ ((\<lambda>x. f (of_real x)) has_integral I) {Re a..Re b}"
proof -
from assms have [simp]: "of_real (Re a) = a" "of_real (Re b) = b"
by (simp_all add: complex_eq_iff)
@@ -738,9 +724,8 @@
have "((\<lambda>x. f (g x)) has_vector_derivative f' (g x) * vector_derivative g (at x within {a..b})) (at x within {a..b})"
using diff_chain_within [OF gdiff fdiff]
by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
- } note * = this
- show ?thesis
- using assms cfg *
+ } then show ?thesis
+ using assms cfg
by (force simp: at_within_Icc_at intro: fundamental_theorem_of_calculus_interior_strong [OF \<open>finite K\<close>])
qed
@@ -759,7 +744,7 @@
shows "(f' has_contour_integral 0) g"
using assms by (metis diff_self contour_integral_primitive)
-text\<open>Existence of path integral for continuous function\<close>
+
lemma contour_integrable_continuous_linepath:
assumes "continuous_on (closed_segment a b) f"
shows "f contour_integrable_on (linepath a b)"
@@ -951,10 +936,9 @@
by fastforce
lemma contour_integrable_sum:
- "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
+ "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
\<Longrightarrow> (\<lambda>x. sum (\<lambda>a. f a x) s) contour_integrable_on p"
- unfolding contour_integrable_on_def
- by (metis has_contour_integral_sum)
+ unfolding contour_integrable_on_def by (metis has_contour_integral_sum)
lemma contour_integrable_neg_iff:
"(\<lambda>x. -f x) contour_integrable_on g \<longleftrightarrow> f contour_integrable_on g"
@@ -1027,10 +1011,10 @@
apply (simp add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
apply (auto dest: has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
done
- } note fj = this
- show ?thesis
+ }
+ then show ?thesis
using f k unfolding has_contour_integral_linepath
- by (simp add: linepath_def has_integral_combine [OF _ _ fi fj])
+ by (simp add: linepath_def has_integral_combine [OF _ _ fi])
qed
lemma continuous_on_closed_segment_transform:
@@ -1107,10 +1091,10 @@
then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>z. vector_derivative g (at (fst z)))"
and hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
by auto
- have "continuous_on (cbox (0, 0) (1, 1)) ((\<lambda>(y1, y2). f y1 y2) \<circ> (\<lambda>w. ((g \<circ> fst) w, (h \<circ> snd) w)))"
- apply (intro gcon hcon continuous_intros | simp)+
- apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
- done
+ have "continuous_on ((\<lambda>x. (g (fst x), h (snd x))) ` cbox (0,0) (1,1)) (\<lambda>(y1, y2). f y1 y2)"
+ by (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
+ then have "continuous_on (cbox (0, 0) (1, 1)) ((\<lambda>(y1, y2). f y1 y2) \<circ> (\<lambda>w. ((g \<circ> fst) w, (h \<circ> snd) w)))"
+ by (intro gcon hcon continuous_intros | simp)+
then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
by auto
have "integral {0..1} (\<lambda>x. contour_integral h (f (g x)) * vector_derivative g (at x)) =
@@ -1186,7 +1170,7 @@
lemma valid_path_polynomial_function:
fixes p :: "real \<Rightarrow> 'a::euclidean_space"
shows "polynomial_function p \<Longrightarrow> valid_path p"
-by (force simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_polymonial_function C1_differentiable_polynomial_function)
+ by (force simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_polymonial_function C1_differentiable_polynomial_function)
lemma valid_path_subpath_trivial [simp]:
fixes g :: "real \<Rightarrow> 'a::euclidean_space"
@@ -1199,15 +1183,15 @@
where "part_circlepath z r s t \<equiv> \<lambda>x. z + of_real r * exp (\<i> * of_real (linepath s t x))"
lemma pathstart_part_circlepath [simp]:
- "pathstart(part_circlepath z r s t) = z + r*exp(\<i> * s)"
-by (metis part_circlepath_def pathstart_def pathstart_linepath)
+ "pathstart(part_circlepath z r s t) = z + r*exp(\<i> * s)"
+ by (metis part_circlepath_def pathstart_def pathstart_linepath)
lemma pathfinish_part_circlepath [simp]:
- "pathfinish(part_circlepath z r s t) = z + r*exp(\<i>*t)"
-by (metis part_circlepath_def pathfinish_def pathfinish_linepath)
+ "pathfinish(part_circlepath z r s t) = z + r*exp(\<i>*t)"
+ by (metis part_circlepath_def pathfinish_def pathfinish_linepath)
lemma reversepath_part_circlepath[simp]:
- "reversepath (part_circlepath z r s t) = part_circlepath z r t s"
+ "reversepath (part_circlepath z r s t) = part_circlepath z r t s"
unfolding part_circlepath_def reversepath_def linepath_def
by (auto simp:algebra_simps)
@@ -1284,24 +1268,12 @@
lemma in_path_image_part_circlepath:
assumes "w \<in> path_image(part_circlepath z r s t)" "s \<le> t" "0 \<le> r"
shows "norm(w - z) = r"
-proof -
- have "w \<in> {c. dist z c = r}"
- by (metis (no_types) path_image_part_circlepath_subset sphere_def subset_eq assms)
- thus ?thesis
- by (simp add: dist_norm norm_minus_commute)
-qed
+ by (smt (verit) assms dist_norm mem_Collect_eq norm_minus_commute path_image_part_circlepath_subset sphere_def subsetD)
lemma path_image_part_circlepath_subset':
assumes "r \<ge> 0"
shows "path_image (part_circlepath z r s t) \<subseteq> sphere z r"
-proof (cases "s \<le> t")
- case True
- thus ?thesis using path_image_part_circlepath_subset[of s t r z] assms by simp
-next
- case False
- thus ?thesis using path_image_part_circlepath_subset[of t s r z] assms
- by (subst reversepath_part_circlepath [symmetric], subst path_image_reversepath) simp_all
-qed
+ by (smt (verit) assms path_image_part_circlepath_subset reversepath_part_circlepath reversepath_simps(2))
lemma part_circlepath_cnj: "cnj (part_circlepath c r a b x) = part_circlepath (cnj c) r (-a) (-b) x"
by (simp add: part_circlepath_def exp_cnj linepath_def algebra_simps)
@@ -1458,7 +1430,7 @@
have *: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1; part_circlepath z r s t x \<notin> k\<rbrakk> \<Longrightarrow> cmod (f (part_circlepath z r s t x)) \<le> B"
by (auto intro!: B [unfolded path_image_def image_def])
show ?thesis
- apply (rule has_integral_bound [where 'a=real, simplified, OF _ **, simplified])
+ using has_integral_bound [where 'a=real, simplified, OF _ **]
using assms le * "2" \<open>r > 0\<close> by (auto simp add: norm_mult vector_derivative_part_circlepath)
qed
qed
@@ -1520,13 +1492,16 @@
qed
have abs_away: "\<And>P. (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. P \<bar>x - y\<bar>) \<longleftrightarrow> (\<forall>x::real. 0 \<le> x \<and> x \<le> 1 \<longrightarrow> P x)"
by force
+ have "\<And>x n. \<lbrakk>s \<noteq> t; \<bar>s - t\<bar> \<le> 2 * pi; 0 \<le> x; x < 1;
+ x * (t - s) = 2 * (real_of_int n * pi)\<rbrakk>
+ \<Longrightarrow> x = 0"
+ by (rule ccontr) (auto simp: 2 field_split_simps abs_mult dest: of_int_leD)
+ then
show ?thesis using False
apply (simp add: simple_path_def loop_free_def)
apply (simp add: part_circlepath_def linepath_def exp_eq * ** abs01 del: Set.insert_iff)
apply (subst abs_away)
apply (auto simp: 1)
- apply (rule ccontr)
- apply (auto simp: 2 field_split_simps abs_mult dest: of_int_leD)
done
qed
@@ -1719,16 +1694,18 @@
and inta: "(\<lambda>t. f a (\<gamma> t) * vector_derivative \<gamma> (at t)) integrable_on {0..1}"
using eventually_happens [OF eventually_conj]
by (fastforce simp: contour_integrable_on path_image_def)
- have Ble: "B * e / (\<bar>B\<bar> + 1) \<le> e"
- using \<open>0 \<le> B\<close> \<open>0 < e\<close> by (simp add: field_split_simps)
have "\<exists>h. (\<forall>x\<in>{0..1}. cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - h x) \<le> e) \<and> h integrable_on {0..1}"
proof (intro exI conjI ballI)
show "cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - f a (\<gamma> x) * vector_derivative \<gamma> (at x)) \<le> e"
if "x \<in> {0..1}" for x
- apply (rule order_trans [OF _ Ble])
- using noleB [OF that] fga [OF that] \<open>0 \<le> B\<close> \<open>0 < e\<close>
- apply (fastforce simp: mult_ac dest: mult_mono [OF less_imp_le] simp add: norm_mult left_diff_distrib [symmetric] norm_minus_commute divide_simps)
- done
+ proof -
+ have "cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - f a (\<gamma> x) * vector_derivative \<gamma> (at x)) \<le> B * e / (\<bar>B\<bar> + 1)"
+ using noleB [OF that] fga [OF that] \<open>0 \<le> B\<close> \<open>0 < e\<close>
+ by (fastforce simp: mult_ac dest: mult_mono [OF less_imp_le] simp add: norm_mult left_diff_distrib [symmetric] norm_minus_commute divide_simps)
+ also have "\<dots> \<le> e"
+ using \<open>0 \<le> B\<close> \<open>0 < e\<close> by (simp add: field_split_simps)
+ finally show ?thesis .
+ qed
qed (rule inta)
}
then show lintg: "l contour_integrable_on \<gamma>"