Theory Contour_Integration
section ‹Contour integration›
theory Contour_Integration
imports "HOL-Analysis.Analysis"
begin
lemma lhopital_complex_simple:
assumes "(f has_field_derivative f') (at z)"
assumes "(g has_field_derivative g') (at z)"
assumes "f z = 0" "g z = 0" "g' ≠ 0" "f' / g' = c"
shows "((λw. f w / g w) ⤏ c) (at z)"
proof -
have "eventually (λw. w ≠ z) (at z)"
by (auto simp: eventually_at_filter)
hence "eventually (λw. ((f w - f z) / (w - z)) / ((g w - g z) / (w - z)) = f w / g w) (at z)"
by eventually_elim (simp add: assms field_split_simps)
moreover have "((λw. ((f w - f z) / (w - z)) / ((g w - g z) / (w - z))) ⤏ f' / g') (at z)"
by (intro tendsto_divide has_field_derivativeD assms)
ultimately have "((λw. f w / g w) ⤏ f' / g') (at z)"
by (blast intro: Lim_transform_eventually)
with assms show ?thesis by simp
qed
subsection‹Definition›
text‹
This definition is for complex numbers only, and does not generalise to
line integrals in a vector field
›
definition has_contour_integral :: "(complex ⇒ complex) ⇒ complex ⇒ (real ⇒ complex) ⇒ bool"
(infixr ‹has'_contour'_integral› 50)
where "(f has_contour_integral i) g ≡
((λx. f(g x) * vector_derivative g (at x within {0..1}))
has_integral i) {0..1}"
definition contour_integrable_on
(infixr ‹contour'_integrable'_on› 50)
where "f contour_integrable_on g ≡ ∃i. (f has_contour_integral i) g"
definition contour_integral
where "contour_integral g f ≡ SOME i. (f has_contour_integral i) g ∨ ¬ f contour_integrable_on g ∧ i=0"
lemma not_integrable_contour_integral: "¬ f contour_integrable_on g ⟹ contour_integral g f = 0"
unfolding contour_integrable_on_def contour_integral_def by blast
lemma contour_integral_unique: "(f has_contour_integral i) g ⟹ contour_integral g f = i"
unfolding contour_integral_def has_contour_integral_def contour_integrable_on_def
using has_integral_unique by blast
lemma has_contour_integral_eqpath:
"⟦(f has_contour_integral y) p; f contour_integrable_on γ;
contour_integral p f = contour_integral γ f⟧
⟹ (f has_contour_integral y) γ"
using contour_integrable_on_def contour_integral_unique by auto
lemma has_contour_integral_integral:
"f contour_integrable_on i ⟹ (f has_contour_integral (contour_integral i f)) i"
by (metis contour_integral_unique contour_integrable_on_def)
lemma has_contour_integral_unique:
"(f has_contour_integral i) g ⟹ (f has_contour_integral j) g ⟹ i = j"
using has_integral_unique
by (auto simp: has_contour_integral_def)
lemma has_contour_integral_integrable: "(f has_contour_integral i) g ⟹ f contour_integrable_on g"
using contour_integrable_on_def by blast
text‹Show that we can forget about the localized derivative.›
lemma has_integral_localized_vector_derivative:
"((λx. f (g x) * vector_derivative p (at x within {a..b})) has_integral i) {a..b} ⟷
((λx. f (g x) * vector_derivative p (at x)) has_integral i) {a..b}"
proof -
have *: "{a..b} - {a,b} = interior {a..b}"
by (simp add: atLeastAtMost_diff_ends)
show ?thesis
by (rule has_integral_spike_eq [of "{a,b}"]) (auto simp: at_within_interior [of _ "{a..b}"])
qed
lemma integrable_on_localized_vector_derivative:
"(λx. f (g x) * vector_derivative p (at x within {a..b})) integrable_on {a..b} ⟷
(λx. f (g x) * vector_derivative p (at x)) integrable_on {a..b}"
by (simp add: integrable_on_def has_integral_localized_vector_derivative)
lemma has_contour_integral:
"(f has_contour_integral i) g ⟷
((λx. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
by (simp add: has_integral_localized_vector_derivative has_contour_integral_def)
lemma contour_integrable_on:
"f contour_integrable_on g ⟷
(λt. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
by (simp add: has_contour_integral integrable_on_def contour_integrable_on_def)
lemma has_contour_integral_mirror_iff:
assumes "valid_path g"
shows "(f has_contour_integral I) (-g) ⟷ ((λx. -f (- x)) has_contour_integral I) g"
proof -
from assms have "g piecewise_differentiable_on {0..1}"
by (auto simp: valid_path_def piecewise_C1_imp_differentiable)
then obtain S where "finite S" and S: "⋀x. x ∈ {0..1} - S ⟹ g differentiable at x within {0..1}"
unfolding piecewise_differentiable_on_def by blast
have S': "g differentiable at x" if "x ∈ {0..1} - ({0, 1} ∪ S)" for x
proof -
from that have "x ∈ interior {0..1}" by auto
with S[of x] that show ?thesis by (auto simp: at_within_interior[of _ "{0..1}"])
qed
have "(f has_contour_integral I) (-g) ⟷
((λx. f (- g x) * vector_derivative (-g) (at x)) has_integral I) {0..1}"
by (simp add: has_contour_integral)
also have "… ⟷ ((λx. -f (- g x) * vector_derivative g (at x)) has_integral I) {0..1}"
by (intro has_integral_spike_finite_eq[of "S ∪ {0, 1}"])
(insert ‹finite S› S', auto simp: o_def fun_Compl_def)
also have "… ⟷ ((λx. -f (-x)) has_contour_integral I) g"
by (simp add: has_contour_integral)
finally show ?thesis .
qed
lemma contour_integral_on_mirror_iff:
assumes "valid_path g"
shows "f contour_integrable_on (-g) ⟷ (λx. -f (- x)) contour_integrable_on g"
by (auto simp: contour_integrable_on_def has_contour_integral_mirror_iff assms)
lemma contour_integral_mirror:
assumes "valid_path g"
shows "contour_integral (-g) f = contour_integral g (λx. -f (- x))"
proof (cases "f contour_integrable_on (-g)")
case True with contour_integral_unique assms show ?thesis
by (auto simp: contour_integrable_on_def has_contour_integral_mirror_iff)
next
case False then show ?thesis
by (simp add: assms contour_integral_on_mirror_iff not_integrable_contour_integral)
qed
subsection ‹Reversing a path›
lemma has_contour_integral_reversepath:
assumes "valid_path g" and f: "(f has_contour_integral i) g"
shows "(f has_contour_integral (-i)) (reversepath g)"
proof -
{ fix S x
assume xs: "g C1_differentiable_on ({0..1} - S)" "x ∉ (-) 1 ` S" "0 ≤ x" "x ≤ 1"
have "vector_derivative (λx. g (1 - x)) (at x within {0..1}) =
- vector_derivative g (at (1 - x) within {0..1})"
proof -
obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
using xs
by (force simp: has_vector_derivative_def C1_differentiable_on_def)
have "(g ∘ (λx. 1 - x) has_vector_derivative -1 *⇩R f') (at x)"
by (intro vector_diff_chain_within has_vector_derivative_at_within [OF f'] derivative_eq_intros | simp)+
then have mf': "((λx. g (1 - x)) has_vector_derivative -f') (at x)"
by (simp add: o_def)
show ?thesis
using xs
by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
qed
} note * = this
obtain S where S: "continuous_on {0..1} g" "finite S" "g C1_differentiable_on {0..1} - S"
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
have "((λx. - (f (g (1 - x)) * vector_derivative g (at (1 - x) within {0..1}))) has_integral -i)
{0..1}"
using has_integral_affinity01 [where m= "-1" and c=1, OF f [unfolded has_contour_integral_def]]
by (simp add: has_integral_neg)
then show ?thesis
using S
unfolding reversepath_def has_contour_integral_def
by (rule_tac S = "(λx. 1 - x) ` S" in has_integral_spike_finite) (auto simp: *)
qed
lemma contour_integrable_reversepath:
"valid_path g ⟹ f contour_integrable_on g ⟹ f contour_integrable_on (reversepath g)"
using has_contour_integral_reversepath contour_integrable_on_def by blast
lemma contour_integrable_reversepath_eq:
"valid_path g ⟹ (f contour_integrable_on (reversepath g) ⟷ f contour_integrable_on g)"
using contour_integrable_reversepath valid_path_reversepath by fastforce
lemma contour_integral_reversepath:
assumes "valid_path g"
shows "contour_integral (reversepath g) f = - (contour_integral g f)"
proof (cases "f contour_integrable_on g")
case True then show ?thesis
by (simp add: assms contour_integral_unique has_contour_integral_integral has_contour_integral_reversepath)
next
case False then have "¬ f contour_integrable_on (reversepath g)"
by (simp add: assms contour_integrable_reversepath_eq)
with False show ?thesis by (simp add: not_integrable_contour_integral)
qed
subsection ‹Joining two paths together›
lemma has_contour_integral_join:
assumes "(f has_contour_integral i1) g1" "(f has_contour_integral i2) g2"
"valid_path g1" "valid_path g2"
shows "(f has_contour_integral (i1 + i2)) (g1 +++ g2)"
proof -
obtain s1 s2
where s1: "finite s1" "∀x∈{0..1} - s1. g1 differentiable at x"
and s2: "finite s2" "∀x∈{0..1} - s2. g2 differentiable at x"
using assms
by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
have 1: "((λx. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
and 2: "((λx. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
using assms
by (auto simp: has_contour_integral)
have i1: "((λx. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
and i2: "((λx. (2*f (g2 (2*x - 1))) * vector_derivative g2 (at (2*x - 1))) has_integral i2) {1/2..1}"
using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
have g1: "vector_derivative (λx. if x*2 ≤ 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
2 *⇩R vector_derivative g1 (at (z*2))"
if "0 ≤ z" "z*2 < 1" "z*2 ∉ s1" for z
proof (rule vector_derivative_at [OF has_vector_derivative_transform_within])
show "0 < ¦z - 1/2¦"
using that by auto
have "((*) 2 has_vector_derivative 2) (at z)"
by (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
moreover have "(g1 has_vector_derivative vector_derivative g1 (at (z * 2))) (at (2 * z))"
using s1 that by (auto simp: algebra_simps vector_derivative_works)
ultimately
show "((λx. g1 (2 * x)) has_vector_derivative 2 *⇩R vector_derivative g1 (at (z * 2))) (at z)"
by (intro vector_diff_chain_at [simplified o_def])
qed (use that in ‹simp_all add: dist_real_def abs_if split: if_split_asm›)
have g2: "vector_derivative (λx. if x*2 ≤ 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
2 *⇩R vector_derivative g2 (at (z*2 - 1))"
if "1 < z*2" "z ≤ 1" "z*2 - 1 ∉ s2" for z
proof (rule vector_derivative_at [OF has_vector_derivative_transform_within])
show "0 < ¦z - 1/2¦"
using that by auto
have "((λx. 2 * x - 1) has_vector_derivative 2) (at z)"
by (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
moreover have "(g2 has_vector_derivative vector_derivative g2 (at (z * 2 - 1))) (at (2 * z - 1))"
using s2 that by (auto simp: algebra_simps vector_derivative_works)
ultimately
show "((λx. g2 (2 * x - 1)) has_vector_derivative 2 *⇩R vector_derivative g2 (at (z * 2 - 1))) (at z)"
by (intro vector_diff_chain_at [simplified o_def])
qed (use that in ‹simp_all add: dist_real_def abs_if split: if_split_asm›)
have "((λx. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
proof (rule has_integral_spike_finite [OF _ _ i1])
show "finite (insert (1/2) ((*) 2 -` s1))"
using s1 by (force intro: finite_vimageI [where h = "(*)2"] inj_onI)
qed (auto simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
moreover have "((λx. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
proof (rule has_integral_spike_finite [OF _ _ i2])
show "finite (insert (1/2) ((λx. 2 * x - 1) -` s2))"
using s2 by (force intro: finite_vimageI [where h = "λx. 2*x-1"] inj_onI)
qed (auto simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
ultimately
show ?thesis
by (simp add: has_contour_integral has_integral_combine [where c = "1/2"])
qed
lemma contour_integrable_joinI:
assumes "f contour_integrable_on g1" "f contour_integrable_on g2"
"valid_path g1" "valid_path g2"
shows "f contour_integrable_on (g1 +++ g2)"
using assms
by (meson has_contour_integral_join contour_integrable_on_def)
lemma contour_integrable_joinD1:
assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g1"
shows "f contour_integrable_on g1"
proof -
obtain s1
where s1: "finite s1" "∀x∈{0..1} - s1. g1 differentiable at x"
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
have "(λx. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
using assms integrable_affinity [of _ 0 "1/2" "1/2" 0] integrable_on_subcbox [where a=0 and b="1/2"]
by (fastforce simp: contour_integrable_on)
then have *: "(λx. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
have g1: "vector_derivative (λx. if x*2 ≤ 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
2 *⇩R vector_derivative g1 (at z)"
if "0 < z" "z < 1" "z ∉ s1" for z
proof (rule vector_derivative_at [OF has_vector_derivative_transform_within])
show "0 < ¦(z - 1)/2¦"
using that by auto
have §: "((λx. x * 2) has_vector_derivative 2) (at (z/2))"
using s1 by (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
have "(g1 has_vector_derivative vector_derivative g1 (at z)) (at z)"
using s1 that by (auto simp: vector_derivative_works)
then show "((λx. g1 (2 * x)) has_vector_derivative 2 *⇩R vector_derivative g1 (at z)) (at (z/2))"
using vector_diff_chain_at [OF §] by (auto simp: field_simps o_def)
qed (use that in ‹simp_all add: field_simps dist_real_def abs_if split: if_split_asm›)
have fin01: "finite ({0, 1} ∪ s1)"
by (simp add: s1)
show ?thesis
unfolding contour_integrable_on
by (intro integrable_spike_finite [OF fin01 _ *]) (auto simp: joinpaths_def scaleR_conv_of_real g1)
qed
lemma contour_integrable_joinD2:
assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g2"
shows "f contour_integrable_on g2"
proof -
obtain s2
where s2: "finite s2" "∀x∈{0..1} - s2. g2 differentiable at x"
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
have "(λx. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}"
using assms integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2"]
integrable_on_subcbox [where a="1/2" and b=1]
by (fastforce simp: contour_integrable_on image_affinity_atLeastAtMost_diff)
then have *: "(λx. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2)))
integrable_on {0..1}"
by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
have g2: "vector_derivative (λx. if x*2 ≤ 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) =
2 *⇩R vector_derivative g2 (at z)"
if "0 < z" "z < 1" "z ∉ s2" for z
proof (rule vector_derivative_at [OF has_vector_derivative_transform_within])
show "0 < ¦z/2¦"
using that by auto
have §: "((λx. x * 2 - 1) has_vector_derivative 2) (at ((1 + z)/2))"
using s2 by (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
have "(g2 has_vector_derivative vector_derivative g2 (at z)) (at z)"
using s2 that by (auto simp: vector_derivative_works)
then show "((λx. g2 (2*x - 1)) has_vector_derivative 2 *⇩R vector_derivative g2 (at z)) (at (z/2 + 1/2))"
using vector_diff_chain_at [OF §] by (auto simp: field_simps o_def)
qed (use that in ‹simp_all add: field_simps dist_real_def abs_if split: if_split_asm›)
have fin01: "finite ({0, 1} ∪ s2)"
by (simp add: s2)
show ?thesis
unfolding contour_integrable_on
by (intro integrable_spike_finite [OF fin01 _ *]) (auto simp: joinpaths_def scaleR_conv_of_real g2)
qed
lemma contour_integrable_join [simp]:
"⟦valid_path g1; valid_path g2⟧
⟹ f contour_integrable_on (g1 +++ g2) ⟷ f contour_integrable_on g1 ∧ f contour_integrable_on g2"
using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast
lemma contour_integral_join [simp]:
"⟦f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2⟧
⟹ 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)
subsection ‹Shifting the starting point of a (closed) path›
lemma has_contour_integral_shiftpath:
assumes f: "(f has_contour_integral i) g" "valid_path g"
and a: "a ∈ {0..1}"
shows "(f has_contour_integral i) (shiftpath a g)"
proof -
obtain S
where S: "finite S" and g: "∀x∈{0..1} - S. g differentiable at x"
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
have *: "((λx. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
using assms by (auto simp: has_contour_integral)
then have i: "i = integral {a..1} (λx. f (g x) * vector_derivative g (at x)) +
integral {0..a} (λx. f (g x) * vector_derivative g (at x))"
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 ≤ x" "x + a < 1" "x ∉ (λx. x - a) ` S" for x
unfolding shiftpath_def
proof (rule vector_derivative_at [OF has_vector_derivative_transform_within])
have "((λx. g (x + a)) has_vector_derivative vector_derivative g (at (a + x))) (at x)"
proof (rule vector_diff_chain_at [of _ 1, simplified o_def scaleR_one])
show "((λx. x + a) has_vector_derivative 1) (at x)"
by (rule derivative_eq_intros | simp)+
have "g differentiable at (x + a)"
using g a that by force
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 "((λx. 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
qed (use that in ‹auto simp: dist_real_def›)
have vd2: "vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
if "x ≤ 1" "1 < x + a" "x ∉ (λx. x - a + 1) ` S" for x
unfolding shiftpath_def
proof (rule vector_derivative_at [OF has_vector_derivative_transform_within])
have "((λx. g (x + a - 1)) has_vector_derivative vector_derivative g (at (a+x-1))) (at x)"
proof (rule vector_diff_chain_at [of _ 1, simplified o_def scaleR_one])
show "((λx. x + a - 1) has_vector_derivative 1) (at x)"
by (rule derivative_eq_intros | simp)+
have "g differentiable at (x+a-1)"
using g a that by force
then show "(g has_vector_derivative vector_derivative g (at (a+x-1))) (at (x + a - 1))"
by (metis add.commute vector_derivative_works)
qed
then show "((λx. g (a + x - 1)) has_vector_derivative vector_derivative g (at (x + a - 1))) (at x)"
by (auto simp: field_simps)
show "0 < dist (1 - a) x"
using that by auto
qed (use that in ‹auto simp: dist_real_def›)
have va1: "(λx. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
using * a by (fastforce intro: integrable_subinterval_real)
have v0a: "(λx. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
using * a by (force intro: integrable_subinterval_real)
have "finite ({1 - a} ∪ (λx. x - a) ` S)"
using S by blast
then have "((λx. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
has_integral integral {a..1} (λx. f (g x) * vector_derivative g (at x))) {0..1 - a}"
apply (rule has_integral_spike_finite
[where f = "λx. f(g(a+x)) * vector_derivative g (at(a+x))"])
subgoal
using a by (simp add: vd1) (force simp: shiftpath_def add.commute)
subgoal
using has_integral_affinity [where m=1 and c=a] integrable_integral [OF va1]
by (force simp add: add.commute)
done
moreover
have "finite ({1 - a} ∪ (λx. x - a + 1) ` S)"
using S by blast
then have "((λx. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
has_integral integral {0..a} (λx. f (g x) * vector_derivative g (at x))) {1 - a..1}"
apply (rule has_integral_spike_finite
[where f = "λx. f(g(a+x-1)) * vector_derivative g (at(a+x-1))"])
subgoal
using a by (simp add: vd2) (force simp: shiftpath_def add.commute)
subgoal
using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
by (force simp add: algebra_simps)
done
ultimately show ?thesis
using a
by (auto simp: i has_contour_integral intro: has_integral_combine [where c = "1-a"])
qed
lemma has_contour_integral_shiftpath_D:
assumes "(f has_contour_integral i) (shiftpath a g)"
"valid_path g" "pathfinish g = pathstart g" "a ∈ {0..1}"
shows "(f has_contour_integral i) g"
proof -
obtain S
where S: "finite S" and g: "∀x∈{0..1} - S. g differentiable at x"
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
{ fix x
assume x: "0 < x" "x < 1" "x ∉ S"
then have gx: "g differentiable at x"
using g by auto
have §: "shiftpath (1 - a) (shiftpath a g) differentiable at x"
using assms x
by (intro differentiable_transform_within [OF gx, of "min x (1-x)"])
(auto simp: dist_real_def shiftpath_shiftpath abs_if split: if_split_asm)
have "vector_derivative g (at x within {0..1}) =
vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
apply (rule vector_derivative_at_within_ivl
[OF has_vector_derivative_transform_within_open
[where f = "(shiftpath (1 - a) (shiftpath a g))" and S = "{0<..<1}-S"]])
using S assms x §
apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
at_within_interior [of _ "{0..1}"] vector_derivative_works [symmetric])
done
} note vd = this
have fi: "(f has_contour_integral i) (shiftpath (1 - a) (shiftpath a g))"
using assms by (auto intro!: has_contour_integral_shiftpath)
show ?thesis
unfolding has_contour_integral_def
proof (rule has_integral_spike_finite [of "{0,1} ∪ S", OF _ _ fi [unfolded has_contour_integral_def]])
show "finite ({0, 1} ∪ S)"
by (simp add: S)
qed (use S assms vd in ‹auto simp: shiftpath_shiftpath›)
qed
lemma has_contour_integral_shiftpath_eq:
assumes "valid_path g" "pathfinish g = pathstart g" "a ∈ {0..1}"
shows "(f has_contour_integral i) (shiftpath a g) ⟷ (f has_contour_integral i) g"
using assms has_contour_integral_shiftpath has_contour_integral_shiftpath_D by blast
lemma contour_integrable_on_shiftpath_eq:
assumes "valid_path g" "pathfinish g = pathstart g" "a ∈ {0..1}"
shows "f contour_integrable_on (shiftpath a g) ⟷ 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 ∈ {0..1}"
shows "contour_integral (shiftpath a g) f = contour_integral g f"
using assms
by (simp add: contour_integral_def contour_integrable_on_def has_contour_integral_shiftpath_eq)
subsection ‹More about straight-line paths›
lemma has_contour_integral_linepath:
shows "(f has_contour_integral i) (linepath a b) ⟷
((λx. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
by (simp add: has_contour_integral)
lemma has_contour_integral_trivial [iff]: "(f has_contour_integral 0) (linepath a a)"
by (simp add: has_contour_integral_linepath)
lemma has_contour_integral_trivial_iff [simp]: "(f has_contour_integral i) (linepath a a) ⟷ i=0"
using has_contour_integral_unique by blast
lemma contour_integral_trivial [simp]: "contour_integral (linepath a a) f = 0"
using has_contour_integral_trivial contour_integral_unique by blast
subsection‹Relation to subpath construction›
lemma has_contour_integral_subpath_refl [iff]: "(f has_contour_integral 0) (subpath u u g)"
by (simp add: has_contour_integral subpath_def)
lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)"
using has_contour_integral_subpath_refl contour_integrable_on_def by blast
lemma contour_integral_subpath_refl [simp]: "contour_integral (subpath u u g) f = 0"
by (simp add: contour_integral_unique)
lemma has_contour_integral_subpath:
assumes f: "f contour_integrable_on g" and g: "valid_path g"
and uv: "u ∈ {0..1}" "v ∈ {0..1}" "u ≤ v"
shows "(f has_contour_integral integral {u..v} (λx. f(g x) * vector_derivative g (at x)))
(subpath u v g)"
proof (cases "v=u")
case True
then show ?thesis
using f by (simp add: contour_integrable_on_def subpath_def has_contour_integral)
next
case False
obtain S where S: "⋀x. x ∈ {0..1} - S ⟹ g differentiable at x" and fs: "finite S"
using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
have §: "(λt. f (g t) * vector_derivative g (at t)) integrable_on {u..v}"
using contour_integrable_on f integrable_on_subinterval uv by fastforce
then have *: "((λx. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
has_integral (1 / (v - u)) * integral {u..v} (λt. f (g t) * vector_derivative g (at t)))
{0..1}"
using uv False unfolding has_integral_integral
apply simp
apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
apply (simp_all add: image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
apply (simp add: divide_simps)
done
have vd: "vector_derivative (λx. g ((v-u) * x + u)) (at x) = (v-u) *⇩R vector_derivative g (at ((v-u) * x + u))"
if "x ∈ {0..1}" "x ∉ (λt. (v-u) *⇩R t + u) -` S" for x
proof (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
show "((λx. (v - u) * x + u) has_vector_derivative v - u) (at x)"
by (intro derivative_eq_intros | simp)+
qed (use S uv mult_left_le [of x "v-u"] that in ‹auto simp: vector_derivative_works›)
have fin: "finite ((λt. (v - u) *⇩R t + u) -` S)"
using fs by (auto simp: inj_on_def False finite_vimageI)
show ?thesis
unfolding subpath_def has_contour_integral
apply (rule has_integral_spike_finite [OF fin])
using has_integral_cmul [OF *, where c = "v-u"] fs assms
by (auto simp: False vd scaleR_conv_of_real)
qed
lemma contour_integrable_subpath:
assumes "f contour_integrable_on g" "valid_path g" "u ∈ {0..1}" "v ∈ {0..1}"
shows "f contour_integrable_on (subpath u v g)"
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 ∈ {0..1}" "v ∈ {0..1}" "u ≤ v"
shows "((λx. f(g x) * vector_derivative g (at x))
has_integral contour_integral (subpath u v g) f) {u..v}"
(is "(?fg has_integral _)_")
proof -
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
lemma contour_integral_subcontour_integral:
assumes "f contour_integrable_on g" "valid_path g" "u ∈ {0..1}" "v ∈ {0..1}" "u ≤ v"
shows "contour_integral (subpath u v g) f =
integral {u..v} (λx. f(g x) * vector_derivative g (at x))"
using assms has_contour_integral_subpath contour_integral_unique by blast
lemma contour_integral_subpath_combine_less:
assumes "f contour_integrable_on g" "valid_path g" "u ∈ {0..1}" "v ∈ {0..1}" "w ∈ {0..1}"
"u<v" "v<w"
shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
contour_integral (subpath u w g) f"
proof -
have "(λx. f (g x) * vector_derivative g (at x)) integrable_on {u..w}"
using integrable_on_subcbox [where a=u and b=w and S = "{0..1}"] assms
by (auto simp: contour_integrable_on)
with assms show ?thesis
by (auto simp: contour_integral_subcontour_integral Henstock_Kurzweil_Integration.integral_combine)
qed
lemma contour_integral_subpath_combine:
assumes "f contour_integrable_on g" "valid_path g" "u ∈ {0..1}" "v ∈ {0..1}" "w ∈ {0..1}"
shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
contour_integral (subpath u w g) f"
proof (cases "u≠v ∧ v≠w ∧ u≠w")
case True
have *: "subpath v u g = reversepath(subpath u v g) ∧
subpath w u g = reversepath(subpath u w g) ∧
subpath w v g = reversepath(subpath v w g)"
by (auto simp: reversepath_subpath)
have "u < v ∧ v < w ∨
u < w ∧ w < v ∨
v < u ∧ u < w ∨
v < w ∧ w < u ∨
w < u ∧ u < v ∨
w < v ∧ v < u"
using True assms by linarith
with assms show ?thesis
using contour_integral_subpath_combine_less [of f g u v w]
contour_integral_subpath_combine_less [of f g u w v]
contour_integral_subpath_combine_less [of f g v u w]
contour_integral_subpath_combine_less [of f g v w u]
contour_integral_subpath_combine_less [of f g w u v]
contour_integral_subpath_combine_less [of f g w v u]
by (elim disjE) (auto simp: * contour_integral_reversepath contour_integrable_subpath
valid_path_subpath algebra_simps)
next
case False
with assms show ?thesis
by (metis add.right_neutral contour_integral_reversepath contour_integral_subpath_refl diff_0 eq_diff_eq add_0 reversepath_subpath valid_path_subpath)
qed
lemma contour_integral_integral:
"contour_integral g f = integral {0..1} (λx. f (g x) * vector_derivative g (at x))"
by (simp add: contour_integral_def integral_def has_contour_integral contour_integrable_on)
lemma contour_integral_cong:
assumes "g = g'" "⋀x. x ∈ path_image g ⟹ f x = f' x"
shows "contour_integral g f = contour_integral g' f'"
unfolding contour_integral_integral using assms
by (intro integral_cong) (auto simp: path_image_def)
lemma contour_integral_spike_finite_simple_path:
assumes "finite A" "simple_path g" "g = g'" "⋀x. x ∈ path_image g - A ⟹ f x = f' x"
shows "contour_integral g f = contour_integral g' f'"
unfolding contour_integral_integral
proof (rule integral_spike)
have "finite (g -` A ∩ {0<..<1})" using ‹simple_path g› ‹finite A›
by (intro finite_vimage_IntI simple_path_inj_on) auto
hence "finite ({0, 1} ∪ g -` A ∩ {0<..<1})" by auto
thus "negligible ({0, 1} ∪ g -` A ∩ {0<..<1})" by (rule negligible_finite)
next
fix x assume "x ∈ {0..1} - ({0, 1} ∪ g -` A ∩ {0<..<1})"
hence "g x ∈ path_image g - A" by (auto simp: path_image_def)
with assms show "f' (g' x) * vector_derivative g' (at x) = f (g x) * vector_derivative g (at x)"
by simp
qed
text ‹Contour integral along a segment on the real axis›
lemma has_contour_integral_linepath_Reals_iff:
fixes a b :: complex and f :: "complex ⇒ complex"
assumes "a ∈ Reals" "b ∈ Reals" "Re a < Re b"
shows "(f has_contour_integral I) (linepath a b) ⟷
((λx. 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)
from assms have "a ≠ b" by auto
have "((λx. f (of_real x)) has_integral I) (cbox (Re a) (Re b)) ⟷
((λx. f (a + b * of_real x - a * of_real x)) has_integral I /⇩R (Re b - Re a)) {0..1}"
by (subst has_integral_affinity_iff [of "Re b - Re a" _ "Re a", symmetric])
(insert assms, simp_all add: field_simps scaleR_conv_of_real)
also have "(λx. f (a + b * of_real x - a * of_real x)) =
(λx. (f (a + b * of_real x - a * of_real x) * (b - a)) /⇩R (Re b - Re a))"
using ‹a ≠ b› by (auto simp: field_simps fun_eq_iff scaleR_conv_of_real)
also have "(… has_integral I /⇩R (Re b - Re a)) {0..1} ⟷
((λx. f (linepath a b x) * (b - a)) has_integral I) {0..1}" using assms
by (subst has_integral_cmul_iff) (auto simp: linepath_def scaleR_conv_of_real algebra_simps)
also have "… ⟷ (f has_contour_integral I) (linepath a b)" unfolding has_contour_integral_def
by (intro has_integral_cong) (simp add: vector_derivative_linepath_within)
finally show ?thesis by simp
qed
lemma contour_integrable_linepath_Reals_iff:
fixes a b :: complex and f :: "complex ⇒ complex"
assumes "a ∈ Reals" "b ∈ Reals" "Re a < Re b"
shows "(f contour_integrable_on linepath a b) ⟷
(λx. f (of_real x)) integrable_on {Re a..Re b}"
using has_contour_integral_linepath_Reals_iff[OF assms, of f]
by (auto simp: contour_integrable_on_def integrable_on_def)
lemma contour_integral_linepath_Reals_eq:
fixes a b :: complex and f :: "complex ⇒ complex"
assumes "a ∈ Reals" "b ∈ Reals" "Re a < Re b"
shows "contour_integral (linepath a b) f = integral {Re a..Re b} (λx. f (of_real x))"
proof (cases "f contour_integrable_on linepath a b")
case True
thus ?thesis using has_contour_integral_linepath_Reals_iff[OF assms, of f]
using has_contour_integral_integral has_contour_integral_unique by blast
next
case False
thus ?thesis using contour_integrable_linepath_Reals_iff[OF assms, of f]
by (simp add: not_integrable_contour_integral not_integrable_integral)
qed
subsection ‹Cauchy's theorem where there's a primitive›
lemma contour_integral_primitive_lemma:
fixes f :: "complex ⇒ complex" and g :: "real ⇒ complex"
assumes "a ≤ b"
and "⋀x. x ∈ S ⟹ (f has_field_derivative f' x) (at x within S)"
and "g piecewise_differentiable_on {a..b}" "⋀x. x ∈ {a..b} ⟹ g x ∈ S"
shows "((λx. f'(g x) * vector_derivative g (at x within {a..b}))
has_integral (f(g b) - f(g a))) {a..b}"
proof -
obtain K where "finite K" and K: "∀x∈{a..b} - K. g differentiable (at x within {a..b})" and cg: "continuous_on {a..b} g"
using assms by (auto simp: piecewise_differentiable_on_def)
have "continuous_on (g ` {a..b}) f"
using assms
by (metis field_differentiable_def field_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
then have cfg: "continuous_on {a..b} (λx. f (g x))"
by (rule continuous_on_compose [OF cg, unfolded o_def])
{ fix x::real
assume a: "a < x" and b: "x < b" and xk: "x ∉ K"
then have "g differentiable at x within {a..b}"
using K by (simp add: differentiable_at_withinI)
then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
then have gdiff: "(g has_derivative (λu. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
by (simp add: has_vector_derivative_def scaleR_conv_of_real)
have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
then have fdiff: "(f has_derivative (*) (f' (g x))) (at (g x) within g ` {a..b})"
by (simp add: has_field_derivative_def)
have "((λx. 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)
} then show ?thesis
using assms cfg
by (force simp: at_within_Icc_at intro: fundamental_theorem_of_calculus_interior_strong [OF ‹finite K›])
qed
lemma contour_integral_primitive:
assumes "⋀x. x ∈ S ⟹ (f has_field_derivative f' x) (at x within S)"
and "valid_path g" "path_image g ⊆ S"
shows "(f' has_contour_integral (f(pathfinish g) - f(pathstart g))) g"
using assms
apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_contour_integral_def)
apply (auto intro!: piecewise_C1_imp_differentiable contour_integral_primitive_lemma [of 0 1 S])
done
corollary Cauchy_theorem_primitive:
assumes "⋀x. x ∈ S ⟹ (f has_field_derivative f' x) (at x within S)"
and "valid_path g" "path_image g ⊆ S" "pathfinish g = pathstart g"
shows "(f' has_contour_integral 0) g"
using assms by (metis diff_self contour_integral_primitive)
lemma contour_integrable_continuous_linepath:
assumes "continuous_on (closed_segment a b) f"
shows "f contour_integrable_on (linepath a b)"
proof -
have "continuous_on (closed_segment a b) (λx. f x * (b - a))"
by (rule continuous_intros | simp add: assms)+
then have "continuous_on {0..1} (λx. f (linepath a b x) * (b - a))"
by (metis (no_types, lifting) continuous_on_compose continuous_on_cong continuous_on_linepath linepath_image_01 o_apply)
then have "(λx. f (linepath a b x) *
vector_derivative (linepath a b)
(at x within {0..1})) integrable_on
{0..1}"
by (metis (no_types, lifting) continuous_on_cong integrable_continuous_real vector_derivative_linepath_within)
then show ?thesis
by (simp add: contour_integrable_on_def has_contour_integral_def integrable_on_def [symmetric])
qed
lemma has_field_der_id: "((λx. x⇧2/2) has_field_derivative x) (at x)"
by (rule has_derivative_imp_has_field_derivative)
(rule derivative_intros | simp)+
lemma contour_integral_id [simp]: "contour_integral (linepath a b) (λy. y) = (b^2 - a^2)/2"
using contour_integral_primitive [of UNIV "λx. x^2/2" "λx. x" "linepath a b"] contour_integral_unique
by (simp add: has_field_der_id)
lemma contour_integrable_on_const [iff]: "(λx. c) contour_integrable_on (linepath a b)"
by (simp add: contour_integrable_continuous_linepath)
lemma contour_integrable_on_id [iff]: "(λx. x) contour_integrable_on (linepath a b)"
by (simp add: contour_integrable_continuous_linepath)
subsection ‹Arithmetical combining theorems›
lemma has_contour_integral_neg:
"(f has_contour_integral i) g ⟹ ((λx. -(f x)) has_contour_integral (-i)) g"
by (simp add: has_integral_neg has_contour_integral_def)
lemma has_contour_integral_add:
"⟦(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g⟧
⟹ ((λx. f1 x + f2 x) has_contour_integral (i1 + i2)) g"
by (simp add: has_integral_add has_contour_integral_def algebra_simps)
lemma has_contour_integral_diff:
"⟦(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g⟧
⟹ ((λx. f1 x - f2 x) has_contour_integral (i1 - i2)) g"
by (simp add: has_integral_diff has_contour_integral_def algebra_simps)
lemma has_contour_integral_lmul:
"(f has_contour_integral i) g ⟹ ((λx. c * (f x)) has_contour_integral (c*i)) g"
by (simp add: has_contour_integral_def algebra_simps has_integral_mult_right)
lemma has_contour_integral_rmul:
"(f has_contour_integral i) g ⟹ ((λx. (f x) * c) has_contour_integral (i*c)) g"
by (simp add: mult.commute has_contour_integral_lmul)
lemma has_contour_integral_div:
"(f has_contour_integral i) g ⟹ ((λx. f x/c) has_contour_integral (i/c)) g"
by (simp add: field_class.field_divide_inverse) (metis has_contour_integral_rmul)
lemma has_contour_integral_eq:
"⟦(f has_contour_integral y) p; ⋀x. x ∈ path_image p ⟹ f x = g x⟧ ⟹ (g has_contour_integral y) p"
by (metis (mono_tags, lifting) has_contour_integral_def has_integral_eq image_eqI path_image_def)
lemma has_contour_integral_bound_linepath:
assumes "(f has_contour_integral i) (linepath a b)"
"0 ≤ B" and B: "⋀x. x ∈ closed_segment a b ⟹ norm(f x) ≤ B"
shows "norm i ≤ B * norm(b - a)"
proof -
have "norm i ≤ (B * norm (b - a)) * content (cbox 0 (1::real))"
proof (rule has_integral_bound
[of _ "λx. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
show "cmod (f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1}))
≤ B * cmod (b - a)"
if "x ∈ cbox 0 1" for x::real
using that box_real(2) norm_mult
by (metis B linepath_in_path mult_right_mono norm_ge_zero vector_derivative_linepath_within)
qed (use assms has_contour_integral_def in auto)
then show ?thesis
by (auto simp: content_real)
qed
lemma has_contour_integral_const_linepath: "((λx. c) has_contour_integral c*(b - a))(linepath a b)"
unfolding has_contour_integral_linepath
by (metis content_real diff_0_right has_integral_const_real lambda_one of_real_1 scaleR_conv_of_real zero_le_one)
lemma has_contour_integral_0: "((λx. 0) has_contour_integral 0) g"
by (simp add: has_contour_integral_def)
lemma has_contour_integral_is_0:
"(⋀z. z ∈ path_image g ⟹ f z = 0) ⟹ (f has_contour_integral 0) g"
by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto
lemma has_contour_integral_sum:
"⟦finite s; ⋀a. a ∈ s ⟹ (f a has_contour_integral i a) p⟧
⟹ ((λx. sum (λa. f a x) s) has_contour_integral sum i s) p"
by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add)
subsection ‹Operations on path integrals›
lemma contour_integral_const_linepath [simp]: "contour_integral (linepath a b) (λx. c) = c*(b - a)"
by (rule contour_integral_unique [OF has_contour_integral_const_linepath])
lemma contour_integral_neg: "contour_integral g (λz. -f z) = -contour_integral g f"
by (simp add: contour_integral_integral)
lemma contour_integral_add:
"f1 contour_integrable_on g ⟹ f2 contour_integrable_on g ⟹ contour_integral g (λx. f1 x + f2 x) =
contour_integral g f1 + contour_integral g f2"
by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_add)
lemma contour_integral_diff:
"f1 contour_integrable_on g ⟹ f2 contour_integrable_on g ⟹ contour_integral g (λx. f1 x - f2 x) =
contour_integral g f1 - contour_integral g f2"
by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_diff)
lemma contour_integral_lmul:
shows "f contour_integrable_on g
⟹ contour_integral g (λx. c * f x) = c*contour_integral g f"
by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_lmul)
lemma contour_integral_rmul:
shows "f contour_integrable_on g
⟹ contour_integral g (λx. f x * c) = contour_integral g f * c"
by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_rmul)
lemma contour_integral_div:
shows "f contour_integrable_on g
⟹ contour_integral g (λx. f x / c) = contour_integral g f / c"
by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_div)
lemma contour_integral_eq:
"(⋀x. x ∈ path_image p ⟹ f x = g x) ⟹ contour_integral p f = contour_integral p g"
using contour_integral_cong contour_integral_def by fastforce
lemma contour_integral_eq_0:
"(⋀z. z ∈ path_image g ⟹ f z = 0) ⟹ contour_integral g f = 0"
by (simp add: has_contour_integral_is_0 contour_integral_unique)
lemma contour_integral_bound_linepath:
shows
"⟦f contour_integrable_on (linepath a b);
0 ≤ B; ⋀x. x ∈ closed_segment a b ⟹ norm(f x) ≤ B⟧
⟹ norm(contour_integral (linepath a b) f) ≤ B*norm(b - a)"
by (meson has_contour_integral_bound_linepath has_contour_integral_integral)
lemma contour_integral_0 [simp]: "contour_integral g (λx. 0) = 0"
by (simp add: contour_integral_unique has_contour_integral_0)
lemma contour_integral_sum:
"⟦finite s; ⋀a. a ∈ s ⟹ (f a) contour_integrable_on p⟧
⟹ contour_integral p (λx. sum (λa. f a x) s) = sum (λa. contour_integral p (f a)) s"
by (auto simp: contour_integral_unique has_contour_integral_sum has_contour_integral_integral)
lemma contour_integrable_eq:
"⟦f contour_integrable_on p; ⋀x. x ∈ path_image p ⟹ f x = g x⟧ ⟹ g contour_integrable_on p"
unfolding contour_integrable_on_def
by (metis has_contour_integral_eq)
subsection ‹Arithmetic theorems for path integrability›
lemma contour_integrable_neg:
"f contour_integrable_on g ⟹ (λx. -(f x)) contour_integrable_on g"
using has_contour_integral_neg contour_integrable_on_def by blast
lemma contour_integrable_add:
"⟦f1 contour_integrable_on g; f2 contour_integrable_on g⟧ ⟹ (λx. f1 x + f2 x) contour_integrable_on g"
using has_contour_integral_add contour_integrable_on_def
by fastforce
lemma contour_integrable_diff:
"⟦f1 contour_integrable_on g; f2 contour_integrable_on g⟧ ⟹ (λx. f1 x - f2 x) contour_integrable_on g"
using has_contour_integral_diff contour_integrable_on_def
by fastforce
lemma contour_integrable_lmul:
"f contour_integrable_on g ⟹ (λx. c * f x) contour_integrable_on g"
using has_contour_integral_lmul contour_integrable_on_def
by fastforce
lemma contour_integrable_rmul:
"f contour_integrable_on g ⟹ (λx. f x * c) contour_integrable_on g"
using has_contour_integral_rmul contour_integrable_on_def
by fastforce
lemma contour_integrable_div:
"f contour_integrable_on g ⟹ (λx. f x / c) contour_integrable_on g"
using has_contour_integral_div contour_integrable_on_def
by fastforce
lemma contour_integrable_sum:
"⟦finite s; ⋀a. a ∈ s ⟹ (f a) contour_integrable_on p⟧
⟹ (λx. sum (λa. f a x) s) contour_integrable_on p"
unfolding contour_integrable_on_def by (metis has_contour_integral_sum)
lemma contour_integrable_neg_iff:
"(λx. -f x) contour_integrable_on g ⟷ f contour_integrable_on g"
using contour_integrable_neg[of f g] contour_integrable_neg[of "λx. -f x" g] by auto
lemma contour_integrable_lmul_iff:
"c ≠ 0 ⟹ (λx. c * f x) contour_integrable_on g ⟷ f contour_integrable_on g"
using contour_integrable_lmul[of f g c] contour_integrable_lmul[of "λx. c * f x" g "inverse c"]
by (auto simp: field_simps)
lemma contour_integrable_rmul_iff:
"c ≠ 0 ⟹ (λx. f x * c) contour_integrable_on g ⟷ f contour_integrable_on g"
using contour_integrable_rmul[of f g c] contour_integrable_rmul[of "λx. c * f x" g "inverse c"]
by (auto simp: field_simps)
lemma contour_integrable_div_iff:
"c ≠ 0 ⟹ (λx. f x / c) contour_integrable_on g ⟷ f contour_integrable_on g"
using contour_integrable_rmul_iff[of "inverse c"] by (simp add: field_simps)
subsection ‹Reversing a path integral›
lemma has_contour_integral_reverse_linepath:
"(f has_contour_integral i) (linepath a b)
⟹ (f has_contour_integral (-i)) (linepath b a)"
using has_contour_integral_reversepath valid_path_linepath by fastforce
lemma contour_integral_reverse_linepath:
"continuous_on (closed_segment a b) f ⟹ contour_integral (linepath a b) f = - (contour_integral(linepath b a) f)"
using contour_integral_reversepath by fastforce
text ‹Splitting a path integral in a flat way.*)›
lemma has_contour_integral_split:
assumes f: "(f has_contour_integral i) (linepath a c)" "(f has_contour_integral j) (linepath c b)"
and k: "0 ≤ k" "k ≤ 1"
and c: "c - a = k *⇩R (b - a)"
shows "(f has_contour_integral (i + j)) (linepath a b)"
proof (cases "k = 0 ∨ k = 1")
case True
then show ?thesis
using assms by auto
next
case False
then have k: "0 < k" "k < 1" "complex_of_real k ≠ 1"
using assms by auto
have c': "c = k *⇩R (b - a) + a"
by (metis diff_add_cancel c)
have bc: "(b - c) = (1 - k) *⇩R (b - a)"
by (simp add: algebra_simps c')
{ assume *: "((λx. f ((1 - x) *⇩R a + x *⇩R c) * (c - a)) has_integral i) {0..1}"
have "⋀x. (x / k) *⇩R a + ((k - x) / k) *⇩R a = a"
using False by (simp add: field_split_simps flip: real_vector.scale_left_distrib)
then have "⋀x. ((k - x) / k) *⇩R a + (x / k) *⇩R c = (1 - x) *⇩R a + x *⇩R b"
using False by (simp add: c' algebra_simps)
then have "((λx. f ((1 - x) *⇩R a + x *⇩R b) * (b - a)) has_integral i) {0..k}"
using k has_integral_affinity01 [OF *, of "inverse k" "0"]
by (force dest: has_integral_cmul [where c = "inverse k"]
simp add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost c)
} note fi = this
{ assume *: "((λx. f ((1 - x) *⇩R c + x *⇩R b) * (b - c)) has_integral j) {0..1}"
have **: "⋀x. (((1 - x) / (1 - k)) *⇩R c + ((x - k) / (1 - k)) *⇩R b) = ((1 - x) *⇩R a + x *⇩R b)"
using k unfolding c' scaleR_conv_of_real
apply (simp add: divide_simps)
apply (simp add: distrib_right distrib_left right_diff_distrib left_diff_distrib)
done
have "((λx. f ((1 - x) *⇩R a + x *⇩R b) * (b - a)) has_integral j) {k..1}"
using k has_integral_affinity01 [OF *, of "inverse(1 - k)" "-(k/(1 - k))"]
apply (simp add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
apply (auto dest: has_integral_cmul [where k = "(1 - k) *⇩R j" and c = "inverse (1 - k)"])
done
}
then show ?thesis
using f k unfolding has_contour_integral_linepath
by (simp add: linepath_def has_integral_combine [OF _ _ fi])
qed
lemma continuous_on_closed_segment_transform:
assumes f: "continuous_on (closed_segment a b) f"
and k: "0 ≤ k" "k ≤ 1"
and c: "c - a = k *⇩R (b - a)"
shows "continuous_on (closed_segment a c) f"
proof -
have c': "c = (1 - k) *⇩R a + k *⇩R b"
using c by (simp add: algebra_simps)
have "closed_segment a c ⊆ closed_segment a b"
by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
then show "continuous_on (closed_segment a c) f"
by (rule continuous_on_subset [OF f])
qed
lemma contour_integral_split:
assumes f: "continuous_on (closed_segment a b) f"
and k: "0 ≤ k" "k ≤ 1"
and c: "c - a = k *⇩R (b - a)"
shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
proof -
have c': "c = (1 - k) *⇩R a + k *⇩R b"
using c by (simp add: algebra_simps)
have "closed_segment a c ⊆ closed_segment a b"
by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
moreover have "closed_segment c b ⊆ closed_segment a b"
by (metis c' ends_in_segment(2) in_segment(1) k subset_closed_segment)
ultimately
have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
by (auto intro: continuous_on_subset [OF f])
show ?thesis
by (rule contour_integral_unique) (meson "*" c contour_integrable_continuous_linepath has_contour_integral_integral has_contour_integral_split k)
qed
lemma contour_integral_split_linepath:
assumes f: "continuous_on (closed_segment a b) f"
and c: "c ∈ closed_segment a b"
shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
using c by (auto simp: closed_segment_def algebra_simps intro!: contour_integral_split [OF f])
subsection‹Reversing the order in a double path integral›
text‹The condition is stronger than needed but it's often true in typical situations›
lemma fst_im_cbox [simp]: "cbox c d ≠ {} ⟹ (fst ` cbox (a,c) (b,d)) = cbox a b"
by (auto simp: cbox_Pair_eq)
lemma snd_im_cbox [simp]: "cbox a b ≠ {} ⟹ (snd ` cbox (a,c) (b,d)) = cbox c d"
by (auto simp: cbox_Pair_eq)
proposition contour_integral_swap:
assumes fcon: "continuous_on (path_image g × path_image h) (λ(y1,y2). f y1 y2)"
and vp: "valid_path g" "valid_path h"
and gvcon: "continuous_on {0..1} (λt. vector_derivative g (at t))"
and hvcon: "continuous_on {0..1} (λt. vector_derivative h (at t))"
shows "contour_integral g (λw. contour_integral h (f w)) =
contour_integral h (λz. contour_integral g (λw. f w z))"
proof -
have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h"
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
have fgh1: "⋀x. (λt. f (g x) (h t)) = (λ(y1,y2). f y1 y2) ∘ (λt. (g x, h t))"
by (rule ext) simp
have fgh2: "⋀x. (λt. f (g t) (h x)) = (λ(y1,y2). f y1 y2) ∘ (λt. (g t, h x))"
by (rule ext) simp
have fcon_im1: "⋀x. 0 ≤ x ⟹ x ≤ 1 ⟹ continuous_on ((λt. (g x, h t)) ` {0..1}) (λ(x, y). f x y)"
by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
have fcon_im2: "⋀x. 0 ≤ x ⟹ x ≤ 1 ⟹ continuous_on ((λt. (g t, h x)) ` {0..1}) (λ(x, y). f x y)"
by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
have "continuous_on (cbox (0, 0) (1, 1::real)) ((λx. vector_derivative g (at x)) ∘ fst)"
"continuous_on (cbox (0, 0) (1::real, 1)) ((λx. vector_derivative h (at x)) ∘ snd)"
by (rule continuous_intros | simp add: gvcon hvcon)+
then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (λz. vector_derivative g (at (fst z)))"
and hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (λx. vector_derivative h (at (snd x)))"
by auto
have "continuous_on ((λx. (g (fst x), h (snd x))) ` cbox (0,0) (1,1)) (λ(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)) ((λ(y1, y2). f y1 y2) ∘ (λw. ((g ∘ fst) w, (h ∘ snd) w)))"
by (intro gcon hcon continuous_intros | simp)+
then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (λx. f (g (fst x)) (h (snd x)))"
by auto
have "integral {0..1} (λx. contour_integral h (f (g x)) * vector_derivative g (at x)) =
integral {0..1} (λx. contour_integral h (λy. f (g x) y * vector_derivative g (at x)))"
proof (rule integral_cong [OF contour_integral_rmul [symmetric]])
have "⋀x. x ∈ {0..1} ⟹
continuous_on {0..1} (λxa. f (g x) (h xa))"
by (subst fgh1) (rule fcon_im1 hcon continuous_intros | simp)+
then show "⋀x. x ∈ {0..1} ⟹ f (g x) contour_integrable_on h"
unfolding contour_integrable_on
using continuous_on_mult hvcon integrable_continuous_real by blast
qed
also have "… = integral {0..1}
(λy. contour_integral g (λx. f x (h y) * vector_derivative h (at y)))"
unfolding contour_integral_integral
apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
subgoal
by (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
subgoal
unfolding integral_mult_left [symmetric]
by (simp only: mult_ac)
done
also have "… = contour_integral h (λz. contour_integral g (λw. f w z))"
unfolding contour_integral_integral integral_mult_left [symmetric]
by (simp add: algebra_simps)
finally show ?thesis
by (simp add: contour_integral_integral)
qed
lemma valid_path_negatepath: "valid_path γ ⟹ valid_path (uminus ∘ γ)"
unfolding o_def using piecewise_C1_differentiable_neg valid_path_def by blast
lemma has_contour_integral_negatepath:
assumes γ: "valid_path γ" and cint: "((λz. f (- z)) has_contour_integral - i) γ"
shows "(f has_contour_integral i) (uminus ∘ γ)"
proof -
obtain S where cont: "continuous_on {0..1} γ" and "finite S" and diff: "γ C1_differentiable_on {0..1} - S"
using γ by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
have "((λx. - (f (- γ x) * vector_derivative γ (at x within {0..1}))) has_integral i) {0..1}"
using cint by (auto simp: has_contour_integral_def dest: has_integral_neg)
then
have "((λx. f (- γ x) * vector_derivative (uminus ∘ γ) (at x within {0..1})) has_integral i) {0..1}"
proof (rule rev_iffD1 [OF _ has_integral_spike_eq])
show "negligible S"
by (simp add: ‹finite S› negligible_finite)
show "f (- γ x) * vector_derivative (uminus ∘ γ) (at x within {0..1}) =
- (f (- γ x) * vector_derivative γ (at x within {0..1}))"
if "x ∈ {0..1} - S" for x
proof -
have "vector_derivative (uminus ∘ γ) (at x within cbox 0 1) = - vector_derivative γ (at x within cbox 0 1)"
proof (rule vector_derivative_within_cbox)
show "(uminus ∘ γ has_vector_derivative - vector_derivative γ (at x within cbox 0 1)) (at x within cbox 0 1)"
using that unfolding o_def
by (metis C1_differentiable_on_eq UNIV_I diff differentiable_subset has_vector_derivative_minus subsetI that vector_derivative_works)
qed (use that in auto)
then show ?thesis
by simp
qed
qed
then show ?thesis by (simp add: has_contour_integral_def)
qed
lemma contour_integrable_negatepath:
assumes γ: "valid_path γ" and pi: "(λz. f (- z)) contour_integrable_on γ"
shows "f contour_integrable_on (uminus ∘ γ)"
by (metis γ add.inverse_inverse contour_integrable_on_def has_contour_integral_negatepath pi)
lemma C1_differentiable_polynomial_function:
fixes p :: "real ⇒ 'a::euclidean_space"
shows "polynomial_function p ⟹ p C1_differentiable_on S"
by (metis continuous_on_polymonial_function C1_differentiable_on_def has_vector_derivative_polynomial_function)
lemma valid_path_polynomial_function:
fixes p :: "real ⇒ 'a::euclidean_space"
shows "polynomial_function p ⟹ valid_path p"
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 ⇒ 'a::euclidean_space"
shows "z ≠ g x ⟹ valid_path (subpath x x g)"
by (simp add: subpath_def valid_path_polynomial_function)
subsection‹Partial circle path›
definition part_circlepath :: "[complex, real, real, real, real] ⇒ complex"
where "part_circlepath z r s t ≡ λx. 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)
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)
lemma reversepath_part_circlepath[simp]:
"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)
lemma has_vector_derivative_part_circlepath [derivative_intros]:
"((part_circlepath z r s t) has_vector_derivative
(𝗂 * r * (of_real t - of_real s) * exp(𝗂 * linepath s t x)))
(at x within X)"
unfolding part_circlepath_def linepath_def scaleR_conv_of_real
by (rule has_vector_derivative_real_field derivative_eq_intros | simp)+
lemma differentiable_part_circlepath:
"part_circlepath c r a b differentiable at x within A"
using has_vector_derivative_part_circlepath[of c r a b x A] differentiableI_vector by blast
lemma vector_derivative_part_circlepath:
"vector_derivative (part_circlepath z r s t) (at x) =
𝗂 * r * (of_real t - of_real s) * exp(𝗂 * linepath s t x)"
using has_vector_derivative_part_circlepath vector_derivative_at by blast
lemma vector_derivative_part_circlepath01:
"⟦0 ≤ x; x ≤ 1⟧
⟹ vector_derivative (part_circlepath z r s t) (at x within {0..1}) =
𝗂 * r * (of_real t - of_real s) * exp(𝗂 * linepath s t x)"
using has_vector_derivative_part_circlepath
by (auto simp: vector_derivative_at_within_ivl)
lemma valid_path_part_circlepath [simp]: "valid_path (part_circlepath z r s t)"
unfolding valid_path_def
by (auto simp: C1_differentiable_on_eq vector_derivative_works vector_derivative_part_circlepath has_vector_derivative_part_circlepath
intro!: C1_differentiable_imp_piecewise continuous_intros)
lemma path_part_circlepath [simp]: "path (part_circlepath z r s t)"
by (simp add: valid_path_imp_path)
proposition path_image_part_circlepath:
assumes "s ≤ t"
shows "path_image (part_circlepath z r s t) = {z + r * exp(𝗂 * of_real x) | x. s ≤ x ∧ x ≤ t}"
proof -
{ fix z::real
assume "0 ≤ z" "z ≤ 1"
with ‹s ≤ t› have "∃x. (exp (𝗂 * linepath s t z) = exp (𝗂 * of_real x)) ∧ s ≤ x ∧ x ≤ t"
apply (rule_tac x="(1 - z) * s + z * t" in exI)
apply (simp add: linepath_def scaleR_conv_of_real algebra_simps)
by (metis (no_types) affine_ineq mult.commute mult_left_mono)
}
moreover
{ fix z
assume "s ≤ z" "z ≤ t"
then have "z + of_real r * exp (𝗂 * of_real z) ∈ (λx. z + of_real r * exp (𝗂 * linepath s t x)) ` {0..1}"
apply (rule_tac x="(z - s)/(t - s)" in image_eqI)
apply (simp add: linepath_def scaleR_conv_of_real divide_simps exp_eq)
apply (auto simp: field_split_simps)
done
}
ultimately show ?thesis
by (fastforce simp add: path_image_def part_circlepath_def)
qed
lemma path_image_part_circlepath':
"path_image (part_circlepath z r s t) = (λx. z + r * cis x) ` closed_segment s t"
proof -
have "path_image (part_circlepath z r s t) =
(λx. z + r * exp(𝗂 * of_real x)) ` linepath s t ` {0..1}"
by (simp add: image_image path_image_def part_circlepath_def)
also have "linepath s t ` {0..1} = closed_segment s t"
by (rule linepath_image_01)
finally show ?thesis by (simp add: cis_conv_exp)
qed
lemma path_image_part_circlepath_subset:
"⟦s ≤ t; 0 ≤ r⟧ ⟹ path_image(part_circlepath z r s t) ⊆ sphere z r"
by (auto simp: path_image_part_circlepath sphere_def dist_norm algebra_simps norm_mult)
lemma in_path_image_part_circlepath:
assumes "w ∈ path_image(part_circlepath z r s t)" "s ≤ t" "0 ≤ r"
shows "norm(w - z) = r"
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 ≥ 0"
shows "path_image (part_circlepath z r s t) ⊆ sphere z r"
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)
lemma contour_integrable_on_compose_cnj_iff:
assumes "valid_path γ"
shows "f contour_integrable_on (cnj ∘ γ) ⟷ (cnj ∘ f ∘ cnj) contour_integrable_on γ"
proof -
from assms obtain S where S: "finite S" "γ C1_differentiable_on {0..1} - S"
unfolding valid_path_def piecewise_C1_differentiable_on_def by blast
have "f contour_integrable_on (cnj ∘ γ) ⟷
((λt. cnj (cnj (f (cnj (γ t))) * vector_derivative γ (at t))) integrable_on {0..1})"
unfolding contour_integrable_on o_def
proof (intro integrable_spike_finite_eq [OF S(1)])
fix t :: real assume "t ∈ {0..1} - S"
hence "γ differentiable at t"
using S(2) by (meson C1_differentiable_on_eq)
hence "vector_derivative (λx. cnj (γ x)) (at t) = cnj (vector_derivative γ (at t))"
by (rule vector_derivative_cnj)
thus "f (cnj (γ t)) * vector_derivative (λx. cnj (γ x)) (at t) =
cnj (cnj (f (cnj (γ t))) * vector_derivative γ (at t))"
by simp
qed
also have "… ⟷ ((λt. cnj (f (cnj (γ t))) * vector_derivative γ (at t)) integrable_on {0..1})"
by (rule integrable_on_cnj_iff)
also have "… ⟷ (cnj ∘ f ∘ cnj) contour_integrable_on γ"
by (simp add: contour_integrable_on o_def)
finally show ?thesis .
qed
lemma contour_integral_cnj:
assumes "valid_path γ"
shows "contour_integral (cnj ∘ γ) f = cnj (contour_integral γ (cnj ∘ f ∘ cnj))"
proof -
from assms obtain S where S: "finite S" "γ C1_differentiable_on {0..1} - S"
unfolding valid_path_def piecewise_C1_differentiable_on_def by blast
have "contour_integral (cnj ∘ γ) f =
integral {0..1} (λt. cnj (cnj (f (cnj (γ t))) * vector_derivative γ (at t)))"
unfolding contour_integral_integral
proof (intro integral_spike)
fix t assume "t ∈ {0..1} - S"
hence "γ differentiable at t"
using S(2) by (meson C1_differentiable_on_eq)
hence "vector_derivative (λx. cnj (γ x)) (at t) = cnj (vector_derivative γ (at t))"
by (rule vector_derivative_cnj)
thus "cnj (cnj (f (cnj (γ t))) * vector_derivative γ (at t)) =
f ((cnj ∘ γ) t) * vector_derivative (cnj ∘ γ) (at t)"
by (simp add: o_def)
qed (use S(1) in auto)
also have "… = cnj (integral {0..1} (λt. cnj (f (cnj (γ t))) * vector_derivative γ (at t)))"
by (subst integral_cnj [symmetric]) auto
also have "… = cnj (contour_integral γ (cnj ∘ f ∘ cnj))"
by (simp add: contour_integral_integral)
finally show ?thesis .
qed
lemma contour_integral_negatepath:
assumes "valid_path γ"
shows "contour_integral (uminus ∘ γ) f = -(contour_integral γ (λx. f (-x)))" (is "?lhs = ?rhs")
proof (cases "f contour_integrable_on (uminus ∘ γ)")
case True
hence *: "(f has_contour_integral ?lhs) (uminus ∘ γ)"
using has_contour_integral_integral by blast
have "((λz. f (-z)) has_contour_integral - contour_integral (uminus ∘ γ) f)
(uminus ∘ (uminus ∘ γ))"
by (rule has_contour_integral_negatepath) (use * assms in auto)
hence "((λx. f (-x)) has_contour_integral -?lhs) γ"
by (simp add: o_def)
thus ?thesis
by (simp add: contour_integral_unique)
next
case False
hence "¬(λz. f (- z)) contour_integrable_on γ"
using contour_integrable_negatepath[of γ f] assms by auto
with False show ?thesis
by (simp add: not_integrable_contour_integral)
qed
lemma contour_integral_bound_part_circlepath:
assumes "f contour_integrable_on part_circlepath c r a b"
assumes "B ≥ 0" "r ≥ 0" "⋀x. x ∈ path_image (part_circlepath c r a b) ⟹ norm (f x) ≤ B"
shows "norm (contour_integral (part_circlepath c r a b) f) ≤ B * r * ¦b - a¦"
proof -
let ?I = "integral {0..1} (λx. f (part_circlepath c r a b x) * 𝗂 * of_real (r * (b - a)) *
exp (𝗂 * linepath a b x))"
have "norm ?I ≤ integral {0..1} (λx::real. B * 1 * (r * ¦b - a¦) * 1)"
proof (rule integral_norm_bound_integral, goal_cases)
case 1
with assms(1) show ?case
by (simp add: contour_integrable_on vector_derivative_part_circlepath mult_ac)
next
case (3 x)
with assms(2-) show ?case unfolding norm_mult norm_of_real abs_mult
by (intro mult_mono) (auto simp: path_image_def)
qed auto
also have "?I = contour_integral (part_circlepath c r a b) f"
by (simp add: contour_integral_integral vector_derivative_part_circlepath mult_ac)
finally show ?thesis by simp
qed
lemma has_contour_integral_part_circlepath_iff:
assumes "a < b"
shows "(f has_contour_integral I) (part_circlepath c r a b) ⟷
((λt. f (c + r * cis t) * r * 𝗂 * cis t) has_integral I) {a..b}"
proof -
have "(f has_contour_integral I) (part_circlepath c r a b) ⟷
((λx. f (part_circlepath c r a b x) * vector_derivative (part_circlepath c r a b)
(at x within {0..1})) has_integral I) {0..1}"
unfolding has_contour_integral_def ..
also have "… ⟷ ((λx. f (part_circlepath c r a b x) * r * (b - a) * 𝗂 *
cis (linepath a b x)) has_integral I) {0..1}"
by (intro has_integral_cong, subst vector_derivative_part_circlepath01)
(simp_all add: cis_conv_exp)
also have "… ⟷ ((λx. f (c + r * exp (𝗂 * linepath (of_real a) (of_real b) x)) *
r * 𝗂 * exp (𝗂 * linepath (of_real a) (of_real b) x) *
vector_derivative (linepath (of_real a) (of_real b))
(at x within {0..1})) has_integral I) {0..1}"
by (intro has_integral_cong, subst vector_derivative_linepath_within)
(auto simp: part_circlepath_def cis_conv_exp of_real_linepath [symmetric])
also have "… ⟷ ((λz. f (c + r * exp (𝗂 * z)) * r * 𝗂 * exp (𝗂 * z)) has_contour_integral I)
(linepath (of_real a) (of_real b))"
by (simp add: has_contour_integral_def)
also have "… ⟷ ((λt. f (c + r * cis t) * r * 𝗂 * cis t) has_integral I) {a..b}" using assms
by (subst has_contour_integral_linepath_Reals_iff) (simp_all add: cis_conv_exp)
finally show ?thesis .
qed
lemma contour_integrable_part_circlepath_iff:
assumes "a < b"
shows "f contour_integrable_on (part_circlepath c r a b) ⟷
(λt. f (c + r * cis t) * r * 𝗂 * cis t) integrable_on {a..b}"
using assms by (auto simp: contour_integrable_on_def integrable_on_def
has_contour_integral_part_circlepath_iff)
lemma contour_integral_part_circlepath_eq:
assumes "a < b"
shows "contour_integral (part_circlepath c r a b) f =
integral {a..b} (λt. f (c + r * cis t) * r * 𝗂 * cis t)"
proof (cases "f contour_integrable_on part_circlepath c r a b")
case True
hence "(λt. f (c + r * cis t) * r * 𝗂 * cis t) integrable_on {a..b}"
using assms by (simp add: contour_integrable_part_circlepath_iff)
with True show ?thesis
using has_contour_integral_part_circlepath_iff[OF assms]
contour_integral_unique has_integral_integrable_integral by blast
next
case False
hence "¬(λt. f (c + r * cis t) * r * 𝗂 * cis t) integrable_on {a..b}"
using assms by (simp add: contour_integrable_part_circlepath_iff)
with False show ?thesis
by (simp add: not_integrable_contour_integral not_integrable_integral)
qed
lemma contour_integral_part_circlepath_reverse:
"contour_integral (part_circlepath c r a b) f = -contour_integral (part_circlepath c r b a) f"
by (metis contour_integral_reversepath reversepath_part_circlepath valid_path_part_circlepath)
lemma contour_integral_part_circlepath_reverse':
"b < a ⟹ contour_integral (part_circlepath c r a b) f =
-contour_integral (part_circlepath c r b a) f"
by (rule contour_integral_part_circlepath_reverse)
lemma finite_bounded_log: "finite {z::complex. norm z ≤ b ∧ exp z = w}"
proof (cases "w = 0")
case True then show ?thesis by auto
next
case False
have *: "finite {x. cmod ((2 * real_of_int x * pi) * 𝗂) ≤ b + cmod (Ln w)}"
proof (simp add: norm_mult finite_int_iff_bounded_le)
show "∃k. abs ` {x. 2 * ¦of_int x¦ * pi ≤ b + cmod (Ln w)} ⊆ {..k}"
apply (rule_tac x="⌊(b + cmod (Ln w)) / (2*pi)⌋" in exI)
apply (auto simp: field_split_simps le_floor_iff)
done
qed
have [simp]: "⋀P f. {z. P z ∧ (∃n. z = f n)} = f ` {n. P (f n)}"
by blast
have "finite {z. cmod z ≤ b ∧ exp z = exp (Ln w)}"
using norm_add_leD by (fastforce intro: finite_subset [OF _ *] simp: exp_eq)
then show ?thesis
using False by auto
qed
lemma finite_bounded_log2:
fixes a::complex
assumes "a ≠ 0"
shows "finite {z. norm z ≤ b ∧ exp(a*z) = w}"
proof -
have *: "finite ((λz. z / a) ` {z. cmod z ≤ b * cmod a ∧ exp z = w})"
by (rule finite_imageI [OF finite_bounded_log])
show ?thesis
by (rule finite_subset [OF _ *]) (force simp: assms norm_mult)
qed
lemma has_contour_integral_bound_part_circlepath_strong:
assumes fi: "(f has_contour_integral i) (part_circlepath z r s t)"
and "finite k" and le: "0 ≤ B" "0 < r" "s ≤ t"
and B: "⋀x. x ∈ path_image(part_circlepath z r s t) - k ⟹ norm(f x) ≤ B"
shows "cmod i ≤ B * r * (t - s)"
proof -
consider "s = t" | "s < t" using ‹s ≤ t› by linarith
then show ?thesis
proof cases
case 1 with fi [unfolded has_contour_integral]
have "i = 0" by (simp add: vector_derivative_part_circlepath)
with assms show ?thesis by simp
next
case 2
have [simp]: "¦r¦ = r" using ‹r > 0› by linarith
have [simp]: "cmod (complex_of_real t - complex_of_real s) = t-s"
by (metis "2" abs_of_pos diff_gt_0_iff_gt norm_of_real of_real_diff)
have "finite (part_circlepath z r s t -` {y} ∩ {0..1})" if "y ∈ k" for y
proof -
let ?w = "(y - z)/of_real r / exp(𝗂 * of_real s)"
have fin: "finite (of_real -` {z. cmod z ≤ 1 ∧ exp (𝗂 * complex_of_real (t - s) * z) = ?w})"
using ‹s < t›
by (intro finite_vimageI [OF finite_bounded_log2]) (auto simp: inj_of_real)
show ?thesis
unfolding part_circlepath_def linepath_def vimage_def
using le
by (intro finite_subset [OF _ fin]) (auto simp: algebra_simps scaleR_conv_of_real exp_add exp_diff)
qed
then have fin01: "finite ((part_circlepath z r s t) -` k ∩ {0..1})"
by (rule finite_finite_vimage_IntI [OF ‹finite k›])
have **: "((λx. if (part_circlepath z r s t x) ∈ k then 0
else f(part_circlepath z r s t x) *
vector_derivative (part_circlepath z r s t) (at x)) has_integral i) {0..1}"
by (rule has_integral_spike [OF negligible_finite [OF fin01]]) (use fi has_contour_integral in auto)
have *: "⋀x. ⟦0 ≤ x; x ≤ 1; part_circlepath z r s t x ∉ k⟧ ⟹ cmod (f (part_circlepath z r s t x)) ≤ B"
by (auto intro!: B [unfolded path_image_def image_def])
show ?thesis
using has_integral_bound [where 'a=real, simplified, OF _ **]
using assms le * "2" ‹r > 0› by (auto simp add: norm_mult vector_derivative_part_circlepath)
qed
qed
corollary contour_integral_bound_part_circlepath_strong:
assumes "f contour_integrable_on part_circlepath z r s t"
and "finite k" and "0 ≤ B" "0 < r" "s ≤ t"
and "⋀x. x ∈ path_image(part_circlepath z r s t) - k ⟹ norm(f x) ≤ B"
shows "cmod (contour_integral (part_circlepath z r s t) f) ≤ B * r * (t - s)"
using assms has_contour_integral_bound_part_circlepath_strong has_contour_integral_integral by blast
lemma has_contour_integral_bound_part_circlepath:
"⟦(f has_contour_integral i) (part_circlepath z r s t);
0 ≤ B; 0 < r; s ≤ t;
⋀x. x ∈ path_image(part_circlepath z r s t) ⟹ norm(f x) ≤ B⟧
⟹ norm i ≤ B*r*(t - s)"
by (auto intro: has_contour_integral_bound_part_circlepath_strong)
lemma contour_integrable_continuous_part_circlepath:
"continuous_on (path_image (part_circlepath z r s t)) f
⟹ f contour_integrable_on (part_circlepath z r s t)"
unfolding contour_integrable_on has_contour_integral_def vector_derivative_part_circlepath path_image_def
by (best intro: integrable_continuous_real path_part_circlepath [unfolded path_def] continuous_intros
continuous_on_compose2 [where g=f, OF _ _ order_refl])
lemma simple_path_part_circlepath:
"simple_path(part_circlepath z r s t) ⟷ (r ≠ 0 ∧ s ≠ t ∧ ¦s - t¦ ≤ 2*pi)"
proof (cases "r = 0 ∨ s = t")
case True
then show ?thesis
unfolding part_circlepath_def simple_path_def loop_free_def
by (rule disjE) (force intro: bexI [where x = "1/4"] bexI [where x = "1/3"])+
next
case False then have "r ≠ 0" "s ≠ t" by auto
have *: "⋀x y z s t. 𝗂*((1 - x) * s + x * t) = 𝗂*(((1 - y) * s + y * t)) + z ⟷ 𝗂*(x - y) * (t - s) = z"
by (simp add: algebra_simps)
have abs01: "⋀x y::real. 0 ≤ x ∧ x ≤ 1 ∧ 0 ≤ y ∧ y ≤ 1
⟹ (x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0 ⟷ ¦x - y¦ ∈ {0,1})"
by auto
have **: "⋀x y. (∃n. (complex_of_real x - of_real y) * (of_real t - of_real s) = 2 * (of_int n * of_real pi)) ⟷
(∃n. ¦x - y¦ * (t - s) = 2 * (of_int n * pi))"
by (force simp: algebra_simps abs_if dest: arg_cong [where f=Re] arg_cong [where f=complex_of_real]
intro: exI [where x = "-n" for n])
have 1: "¦s - t¦ ≤ 2 * pi"
if "⋀x. 0 ≤ x ∧ x ≤ 1 ⟹ (∃n. x * (t - s) = 2 * (real_of_int n * pi)) ⟶ x = 0 ∨ x = 1"
proof (rule ccontr)
assume "¬ ¦s - t¦ ≤ 2 * pi"
then have *: "⋀n. t - s ≠ of_int n * ¦s - t¦"
using False that [of "2*pi / ¦t - s¦"]
by (simp add: abs_minus_commute divide_simps)
show False
using * [of 1] * [of "-1"] by auto
qed
have 2: "¦s - t¦ = ¦2 * (real_of_int n * pi) / x¦" if "x ≠ 0" "x * (t - s) = 2 * (real_of_int n * pi)" for x n
proof -
have "t-s = 2 * (real_of_int n * pi)/x"
using that by (simp add: field_simps)
then show ?thesis by (metis abs_minus_commute)
qed
have abs_away: "⋀P. (∀x∈{0..1}. ∀y∈{0..1}. P ¦x - y¦) ⟷ (∀x::real. 0 ≤ x ∧ x ≤ 1 ⟶ P x)"
by force
have "⋀x n. ⟦s ≠ t; ¦s - t¦ ≤ 2 * pi; 0 ≤ x; x < 1;
x * (t - s) = 2 * (real_of_int n * pi)⟧
⟹ 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)
done
qed
lemma arc_part_circlepath:
assumes "r ≠ 0" "s ≠ t" "¦s - t¦ < 2*pi"
shows "arc (part_circlepath z r s t)"
proof -
have *: "x = y" if eq: "𝗂 * (linepath s t x) = 𝗂 * (linepath s t y) + 2 * of_int n * complex_of_real pi * 𝗂"
and x: "x ∈ {0..1}" and y: "y ∈ {0..1}" for x y n
proof (rule ccontr)
assume "x ≠ y"
have "(linepath s t x) = (linepath s t y) + 2 * of_int n * complex_of_real pi"
by (metis add_divide_eq_iff complex_i_not_zero mult.commute nonzero_mult_div_cancel_left eq)
then have "s*y + t*x = s*x + (t*y + of_int n * (pi * 2))"
by (force simp: algebra_simps linepath_def dest: arg_cong [where f=Re])
with ‹x ≠ y› have st: "s-t = (of_int n * (pi * 2) / (y-x))"
by (force simp: field_simps)
have "¦real_of_int n¦ < ¦y - x¦"
using assms ‹x ≠ y› by (simp add: st abs_mult field_simps)
then show False
using assms x y st by (auto dest: of_int_lessD)
qed
then have "inj_on (part_circlepath z r s t) {0..1}"
using assms by (force simp add: part_circlepath_def inj_on_def exp_eq)
then show ?thesis
by (simp add: arc_def)
qed
subsection‹Special case of one complete circle›
definition circlepath :: "[complex, real, real] ⇒ complex"
where "circlepath z r ≡ part_circlepath z r 0 (2*pi)"
lemma circlepath: "circlepath z r = (λx. z + r * exp(2 * of_real pi * 𝗂 * of_real x))"
by (simp add: circlepath_def part_circlepath_def linepath_def algebra_simps)
lemma pathstart_circlepath [simp]: "pathstart (circlepath z r) = z + r"
by (simp add: circlepath_def)
lemma pathfinish_circlepath [simp]: "pathfinish (circlepath z r) = z + r"
by (simp add: circlepath_def) (metis exp_two_pi_i mult.commute)
lemma circlepath_minus: "circlepath z (-r) x = circlepath z r (x + 1/2)"
proof -
have "z + of_real r * exp (2 * pi * 𝗂 * (x + 1/2)) =
z + of_real r * exp (2 * pi * 𝗂 * x + pi * 𝗂)"
by (simp add: divide_simps) (simp add: algebra_simps)
also have "… = z - r * exp (2 * pi * 𝗂 * x)"
by (simp add: exp_add)
finally show ?thesis
by (simp add: circlepath path_image_def sphere_def dist_norm)
qed
lemma circlepath_add1: "circlepath z r (x+1) = circlepath z r x"
using circlepath_minus [of z r "x+1/2"] circlepath_minus [of z "-r" x]
by (simp add: add.commute)
lemma circlepath_add_half: "circlepath z r (x + 1/2) = circlepath z r (x - 1/2)"
using circlepath_add1 [of z r "x-1/2"]
by (simp add: add.commute)
lemma path_image_circlepath_minus_subset:
"path_image (circlepath z (-r)) ⊆ path_image (circlepath z r)"
proof -
have "∃x∈{0..1}. circlepath z r (y + 1/2) = circlepath z r x"
if "0 ≤ y" "y ≤ 1" for y
proof (cases "y ≤ 1/2")
case False
with that show ?thesis
by (force simp: circlepath_add_half)
qed (use that in force)
then show ?thesis
by (auto simp add: path_image_def image_def circlepath_minus)
qed
lemma path_image_circlepath_minus: "path_image (circlepath z (-r)) = path_image (circlepath z r)"
using path_image_circlepath_minus_subset by fastforce
lemma has_vector_derivative_circlepath [derivative_intros]:
"((circlepath z r) has_vector_derivative (2 * pi * 𝗂 * r * exp (2 * of_real pi * 𝗂 * x)))
(at x within X)"
unfolding circlepath_def scaleR_conv_of_real
by (rule derivative_eq_intros) (simp add: algebra_simps)
lemma vector_derivative_circlepath:
"vector_derivative (circlepath z r) (at x) =
2 * pi * 𝗂 * r * exp(2 * of_real pi * 𝗂 * x)"
using has_vector_derivative_circlepath vector_derivative_at by blast
lemma vector_derivative_circlepath01:
"⟦0 ≤ x; x ≤ 1⟧
⟹ vector_derivative (circlepath z r) (at x within {0..1}) =
2 * pi * 𝗂 * r * exp(2 * of_real pi * 𝗂 * x)"
using has_vector_derivative_circlepath
by (auto simp: vector_derivative_at_within_ivl)
lemma valid_path_circlepath [simp]: "valid_path (circlepath z r)"
by (simp add: circlepath_def)
lemma path_circlepath [simp]: "path (circlepath z r)"
by (simp add: valid_path_imp_path)
lemma path_image_circlepath_nonneg:
assumes "0 ≤ r" shows "path_image (circlepath z r) = sphere z r"
proof -
have *: "x ∈ (λu. z + (cmod (x - z)) * exp (𝗂 * (of_real u * (of_real pi * 2)))) ` {0..1}" for x
proof (cases "x = z")
case True then show ?thesis by force
next
case False
define w where "w = x - z"
then have "w ≠ 0" by (simp add: False)
have **: "⋀t. ⟦Re w = cos t * cmod w; Im w = sin t * cmod w⟧ ⟹ w = of_real (cmod w) * exp (𝗂 * t)"
using cis_conv_exp complex_eq_iff by auto
obtain t where "0 ≤ t" "t < 2*pi" "Re(w/norm w) = cos t" "Im(w/norm w) = sin t"
apply (rule sincos_total_2pi [of "Re(w/(norm w))" "Im(w/(norm w))"])
by (auto simp add: divide_simps ‹w ≠ 0› cmod_power2 [symmetric])
then
show ?thesis
using False ** w_def ‹w ≠ 0›
by (rule_tac x="t / (2*pi)" in image_eqI) (auto simp add: field_simps)
qed
show ?thesis
unfolding circlepath path_image_def sphere_def dist_norm
by (force simp: assms algebra_simps norm_mult norm_minus_commute intro: *)
qed
lemma path_image_circlepath [simp]:
"path_image (circlepath z r) = sphere z ¦r¦"
using path_image_circlepath_minus
by (force simp: path_image_circlepath_nonneg abs_if)
lemma has_contour_integral_bound_circlepath_strong:
"⟦(f has_contour_integral i) (circlepath z r);
finite k; 0 ≤ B; 0 < r;
⋀x. ⟦norm(x - z) = r; x ∉ k⟧ ⟹ norm(f x) ≤ B⟧
⟹ norm i ≤ B*(2*pi*r)"
unfolding circlepath_def
by (auto simp: algebra_simps in_path_image_part_circlepath dest!: has_contour_integral_bound_part_circlepath_strong)
lemma has_contour_integral_bound_circlepath:
"⟦(f has_contour_integral i) (circlepath z r);
0 ≤ B; 0 < r; ⋀x. norm(x - z) = r ⟹ norm(f x) ≤ B⟧
⟹ norm i ≤ B*(2*pi*r)"
by (auto intro: has_contour_integral_bound_circlepath_strong)
lemma contour_integrable_continuous_circlepath:
"continuous_on (path_image (circlepath z r)) f
⟹ f contour_integrable_on (circlepath z r)"
by (simp add: circlepath_def contour_integrable_continuous_part_circlepath)
lemma simple_path_circlepath: "simple_path(circlepath z r) ⟷ (r ≠ 0)"
by (simp add: circlepath_def simple_path_part_circlepath)
lemma notin_path_image_circlepath [simp]: "cmod (w - z) < r ⟹ w ∉ path_image (circlepath z r)"
by (simp add: sphere_def dist_norm norm_minus_commute)
lemma contour_integral_circlepath:
assumes "r > 0"
shows "contour_integral (circlepath z r) (λw. 1 / (w - z)) = 2 * complex_of_real pi * 𝗂"
proof (rule contour_integral_unique)
show "((λw. 1 / (w - z)) has_contour_integral 2 * complex_of_real pi * 𝗂) (circlepath z r)"
unfolding has_contour_integral_def using assms has_integral_const_real [of _ 0 1]
apply (subst has_integral_cong)
apply (simp add: vector_derivative_circlepath01)
apply (force simp: circlepath)
done
qed
subsection‹ Uniform convergence of path integral›
text‹Uniform convergence when the derivative of the path is bounded, and in particular for the special case of a circle.›
proposition contour_integral_uniform_limit:
assumes ev_fint: "eventually (λn::'a. (f n) contour_integrable_on γ) F"
and ul_f: "uniform_limit (path_image γ) f l F"
and noleB: "⋀t. t ∈ {0..1} ⟹ norm (vector_derivative γ (at t)) ≤ B"
and γ: "valid_path γ"
and [simp]: "¬ trivial_limit F"
shows "l contour_integrable_on γ" "((λn. contour_integral γ (f n)) ⤏ contour_integral γ l) F"
proof -
have "0 ≤ B" by (meson noleB [of 0] atLeastAtMost_iff norm_ge_zero order_refl order_trans zero_le_one)
{ fix e::real
assume "0 < e"
then have "0 < e / (¦B¦ + 1)" by simp
then have §: "∀⇩F n in F. ∀x∈path_image γ. cmod (f n x - l x) < e / (¦B¦ + 1)"
using ul_f [unfolded uniform_limit_iff dist_norm] by auto
obtain a where fga: "⋀x. x ∈ {0..1} ⟹ cmod (f a (γ x) - l (γ x)) < e / (¦B¦ + 1)"
and inta: "(λt. f a (γ t) * vector_derivative γ (at t)) integrable_on {0..1}"
using eventually_happens [OF eventually_conj [OF ev_fint §]]
by (fastforce simp: contour_integrable_on path_image_def)
have "∃h. (∀x∈{0..1}. cmod (l (γ x) * vector_derivative γ (at x) - h x) ≤ e) ∧ h integrable_on {0..1}"
proof (intro exI conjI ballI)
show "cmod (l (γ x) * vector_derivative γ (at x) - f a (γ x) * vector_derivative γ (at x)) ≤ e"
if "x ∈ {0..1}" for x
proof -
have "cmod (l (γ x) * vector_derivative γ (at x) - f a (γ x) * vector_derivative γ (at x)) ≤ B * e / (¦B¦ + 1)"
using noleB [OF that] fga [OF that] ‹0 ≤ B› ‹0 < e›
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 "… ≤ e"
using ‹0 ≤ B› ‹0 < e› by (simp add: field_split_simps)
finally show ?thesis .
qed
qed (rule inta)
}
then show lintg: "l contour_integrable_on γ"
unfolding contour_integrable_on by (metis (mono_tags, lifting)integrable_uniform_limit_real)
{ fix e::real
define B' where "B' = B + 1"
have B': "B' > 0" "B' > B" using ‹0 ≤ B› by (auto simp: B'_def)
assume "0 < e"
then have ev_no': "∀⇩F n in F. ∀x∈path_image γ. 2 * cmod (f n x - l x) < e / B'"
using ul_f [unfolded uniform_limit_iff dist_norm, rule_format, of "e / B'/2"] B'
by (simp add: field_simps)
have ie: "integral {0..1::real} (λx. e/2) < e" using ‹0 < e› by simp
have *: "cmod (f x (γ t) * vector_derivative γ (at t) - l (γ t) * vector_derivative γ (at t)) ≤ e/2"
if t: "t∈{0..1}" and leB': "2 * cmod (f x (γ t) - l (γ t)) < e / B'" for x t
proof -
have "2 * cmod (f x (γ t) - l (γ t)) * cmod (vector_derivative γ (at t)) ≤ e * (B/ B')"
using mult_mono [OF less_imp_le [OF leB'] noleB] B' ‹0 < e› t by auto
also have "… < e"
by (simp add: B' ‹0 < e› mult_imp_div_pos_less)
finally have "2 * cmod (f x (γ t) - l (γ t)) * cmod (vector_derivative γ (at t)) < e" .
then show ?thesis
by (simp add: left_diff_distrib [symmetric] norm_mult)
qed
have le_e: "⋀x. ⟦∀u∈{0..1}. 2 * cmod (f x (γ u) - l (γ u)) < e / B'; f x contour_integrable_on γ⟧
⟹ cmod (integral {0..1}
(λu. f x (γ u) * vector_derivative γ (at u) - l (γ u) * vector_derivative γ (at u))) < e"
apply (rule le_less_trans [OF integral_norm_bound_integral ie])
apply (simp add: lintg integrable_diff contour_integrable_on [symmetric])
apply (blast intro: *)+
done
have "∀⇩F x in F. dist (contour_integral γ (f x)) (contour_integral γ l) < e"
apply (rule eventually_mono [OF eventually_conj [OF ev_no' ev_fint]])
apply (simp add: dist_norm contour_integrable_on path_image_def contour_integral_integral)
apply (simp add: lintg integral_diff [symmetric] contour_integrable_on [symmetric] le_e)
done
}
then show "((λn. contour_integral γ (f n)) ⤏ contour_integral γ l) F"
by (rule tendstoI)
qed
corollary contour_integral_uniform_limit_circlepath:
assumes "∀⇩F n::'a in F. (f n) contour_integrable_on (circlepath z r)"
and "uniform_limit (sphere z r) f l F"
and "¬ trivial_limit F" "0 < r"
shows "l contour_integrable_on (circlepath z r)"
"((λn. contour_integral (circlepath z r) (f n)) ⤏ contour_integral (circlepath z r) l) F"
using assms by (auto simp: vector_derivative_circlepath norm_mult intro!: contour_integral_uniform_limit)
end