section \<open>Complex path integrals and Cauchy's integral theorem\<close>
theory Cauchy_Integral_Thm
imports Complex_Transcendental Weierstrass
begin
(*FIXME migrate into libraries*)
lemma inj_mult_left [simp]: "inj (op* x) \<longleftrightarrow> x \<noteq> (0::'a::idom)"
by (metis injI mult_cancel_left the_inv_f_f zero_neq_one)
lemma continuous_on_strong_cong:
"s = t \<Longrightarrow> (\<And>x. x \<in> t =simp=> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"
by (simp add: simp_implies_def)
thm image_affinity_atLeastAtMost_div_diff
lemma image_affinity_atLeastAtMost_div:
fixes c :: "'a::linordered_field"
shows "((\<lambda>x. x/m + c) ` {a..b}) = (if {a..b}={} then {}
else if 0 \<le> m then {a/m + c .. b/m + c}
else {b/m + c .. a/m + c})"
using image_affinity_atLeastAtMost [of "inverse m" c a b]
by (simp add: field_class.field_divide_inverse algebra_simps)
thm continuous_on_closed_Un
lemma continuous_on_open_Un:
"open s \<Longrightarrow> open t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
using continuous_on_open_Union [of "{s,t}"] by auto
thm continuous_on_eq (*REPLACE*)
lemma continuous_on_eq:
"\<lbrakk>continuous_on s f; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> continuous_on s g"
unfolding continuous_on_def tendsto_def eventually_at_topological
by simp
thm vector_derivative_add_at
lemma vector_derivative_mult_at [simp]:
fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
shows "\<lbrakk>f differentiable at a; g differentiable at a\<rbrakk>
\<Longrightarrow> vector_derivative (\<lambda>x. f x * g x) (at a) = f a * vector_derivative g (at a) + vector_derivative f (at a) * g a"
by (simp add: vector_derivative_at has_vector_derivative_mult vector_derivative_works [symmetric])
lemma vector_derivative_scaleR_at [simp]:
"\<lbrakk>f differentiable at a; g differentiable at a\<rbrakk>
\<Longrightarrow> vector_derivative (\<lambda>x. f x *\<^sub>R g x) (at a) = f a *\<^sub>R vector_derivative g (at a) + vector_derivative f (at a) *\<^sub>R g a"
apply (rule vector_derivative_at)
apply (rule has_vector_derivative_scaleR)
apply (auto simp: vector_derivative_works has_vector_derivative_def has_field_derivative_def mult_commute_abs)
done
thm Derivative.vector_diff_chain_at
lemma vector_derivative_chain_at:
assumes "f differentiable at x" "(g differentiable at (f x))"
shows "vector_derivative (g \<circ> f) (at x) =
vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"
by (metis Derivative.vector_diff_chain_at vector_derivative_at vector_derivative_works assms)
subsection \<open>Piecewise differentiable functions\<close>
definition piecewise_differentiable_on
(infixr "piecewise'_differentiable'_on" 50)
where "f piecewise_differentiable_on i \<equiv>
continuous_on i f \<and>
(\<exists>s. finite s \<and> (\<forall>x \<in> i - s. f differentiable (at x within i)))"
lemma piecewise_differentiable_on_imp_continuous_on:
"f piecewise_differentiable_on s \<Longrightarrow> continuous_on s f"
by (simp add: piecewise_differentiable_on_def)
lemma piecewise_differentiable_on_subset:
"f piecewise_differentiable_on s \<Longrightarrow> t \<le> s \<Longrightarrow> f piecewise_differentiable_on t"
using continuous_on_subset
unfolding piecewise_differentiable_on_def
apply safe
apply (blast intro: elim: continuous_on_subset)
by (meson Diff_iff differentiable_within_subset subsetCE)
lemma differentiable_on_imp_piecewise_differentiable:
fixes a:: "'a::{linorder_topology,real_normed_vector}"
shows "f differentiable_on {a..b} \<Longrightarrow> f piecewise_differentiable_on {a..b}"
apply (simp add: piecewise_differentiable_on_def differentiable_imp_continuous_on)
apply (rule_tac x="{a,b}" in exI, simp add: differentiable_on_def)
done
lemma differentiable_imp_piecewise_differentiable:
"(\<And>x. x \<in> s \<Longrightarrow> f differentiable (at x within s))
\<Longrightarrow> f piecewise_differentiable_on s"
by (auto simp: piecewise_differentiable_on_def differentiable_imp_continuous_on differentiable_on_def
intro: differentiable_within_subset)
lemma piecewise_differentiable_compose:
"\<lbrakk>f piecewise_differentiable_on s; g piecewise_differentiable_on (f ` s);
\<And>x. finite (s \<inter> f-`{x})\<rbrakk>
\<Longrightarrow> (g o f) piecewise_differentiable_on s"
apply (simp add: piecewise_differentiable_on_def, safe)
apply (blast intro: continuous_on_compose2)
apply (rename_tac A B)
apply (rule_tac x="A \<union> (\<Union>x\<in>B. s \<inter> f-`{x})" in exI)
apply (blast intro: differentiable_chain_within)
done
lemma piecewise_differentiable_affine:
fixes m::real
assumes "f piecewise_differentiable_on ((\<lambda>x. m *\<^sub>R x + c) ` s)"
shows "(f o (\<lambda>x. m *\<^sub>R x + c)) piecewise_differentiable_on s"
proof (cases "m = 0")
case True
then show ?thesis
unfolding o_def
by (force intro: differentiable_imp_piecewise_differentiable differentiable_const)
next
case False
show ?thesis
apply (rule piecewise_differentiable_compose [OF differentiable_imp_piecewise_differentiable])
apply (rule assms derivative_intros | simp add: False vimage_def real_vector_affinity_eq)+
done
qed
lemma piecewise_differentiable_cases:
fixes c::real
assumes "f piecewise_differentiable_on {a..c}"
"g piecewise_differentiable_on {c..b}"
"a \<le> c" "c \<le> b" "f c = g c"
shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_differentiable_on {a..b}"
proof -
obtain s t where st: "finite s" "finite t"
"\<forall>x\<in>{a..c} - s. f differentiable at x within {a..c}"
"\<forall>x\<in>{c..b} - t. g differentiable at x within {c..b}"
using assms
by (auto simp: piecewise_differentiable_on_def)
have finabc: "finite ({a,b,c} \<union> (s \<union> t))"
by (metis `finite s` `finite t` finite_Un finite_insert finite.emptyI)
have "continuous_on {a..c} f" "continuous_on {c..b} g"
using assms piecewise_differentiable_on_def by auto
then have "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
OF closed_real_atLeastAtMost [of c b],
of f g "\<lambda>x. x\<le>c"] assms
by (force simp: ivl_disj_un_two_touch)
moreover
{ fix x
assume x: "x \<in> {a..b} - ({a,b,c} \<union> (s \<union> t))"
have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b}" (is "?diff_fg")
proof (cases x c rule: le_cases)
case le show ?diff_fg
apply (rule differentiable_transform_within [where d = "dist x c" and f = f])
using x le st
apply (simp_all add: dist_real_def dist_nz [symmetric])
apply (rule differentiable_at_withinI)
apply (rule differentiable_within_open [where s = "{a<..<c} - s", THEN iffD1], simp_all)
apply (blast intro: open_greaterThanLessThan finite_imp_closed)
apply (force elim!: differentiable_subset)
done
next
case ge show ?diff_fg
apply (rule differentiable_transform_within [where d = "dist x c" and f = g])
using x ge st
apply (simp_all add: dist_real_def dist_nz [symmetric])
apply (rule differentiable_at_withinI)
apply (rule differentiable_within_open [where s = "{c<..<b} - t", THEN iffD1], simp_all)
apply (blast intro: open_greaterThanLessThan finite_imp_closed)
apply (force elim!: differentiable_subset)
done
qed
}
then have "\<exists>s. finite s \<and>
(\<forall>x\<in>{a..b} - s. (\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b})"
by (meson finabc)
ultimately show ?thesis
by (simp add: piecewise_differentiable_on_def)
qed
lemma piecewise_differentiable_neg:
"f piecewise_differentiable_on s \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_differentiable_on s"
by (auto simp: piecewise_differentiable_on_def continuous_on_minus)
lemma piecewise_differentiable_add:
assumes "f piecewise_differentiable_on i"
"g piecewise_differentiable_on i"
shows "(\<lambda>x. f x + g x) piecewise_differentiable_on i"
proof -
obtain s t where st: "finite s" "finite t"
"\<forall>x\<in>i - s. f differentiable at x within i"
"\<forall>x\<in>i - t. g differentiable at x within i"
using assms by (auto simp: piecewise_differentiable_on_def)
then have "finite (s \<union> t) \<and> (\<forall>x\<in>i - (s \<union> t). (\<lambda>x. f x + g x) differentiable at x within i)"
by auto
moreover have "continuous_on i f" "continuous_on i g"
using assms piecewise_differentiable_on_def by auto
ultimately show ?thesis
by (auto simp: piecewise_differentiable_on_def continuous_on_add)
qed
lemma piecewise_differentiable_diff:
"\<lbrakk>f piecewise_differentiable_on s; g piecewise_differentiable_on s\<rbrakk>
\<Longrightarrow> (\<lambda>x. f x - g x) piecewise_differentiable_on s"
unfolding diff_conv_add_uminus
by (metis piecewise_differentiable_add piecewise_differentiable_neg)
lemma continuous_on_joinpaths_D1:
"continuous_on {0..1} (g1 +++ g2) \<Longrightarrow> continuous_on {0..1} g1"
apply (rule continuous_on_eq [of _ "(g1 +++ g2) o (op*(inverse 2))"])
apply (rule continuous_intros | simp)+
apply (auto elim!: continuous_on_subset simp: joinpaths_def)
done
lemma continuous_on_joinpaths_D2:
"\<lbrakk>continuous_on {0..1} (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> continuous_on {0..1} g2"
apply (rule continuous_on_eq [of _ "(g1 +++ g2) o (\<lambda>x. inverse 2*x + 1/2)"])
apply (rule continuous_intros | simp)+
apply (auto elim!: continuous_on_subset simp add: joinpaths_def pathfinish_def pathstart_def Ball_def)
done
lemma piecewise_differentiable_D1:
"(g1 +++ g2) piecewise_differentiable_on {0..1} \<Longrightarrow> g1 piecewise_differentiable_on {0..1}"
apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D1)
apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
apply simp
apply (intro ballI)
apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))"
in differentiable_transform_within)
apply (auto simp: dist_real_def joinpaths_def)
apply (rule differentiable_chain_within derivative_intros | simp)+
apply (rule differentiable_subset)
apply (force simp:)+
done
lemma piecewise_differentiable_D2:
"\<lbrakk>(g1 +++ g2) piecewise_differentiable_on {0..1}; pathfinish g1 = pathstart g2\<rbrakk>
\<Longrightarrow> g2 piecewise_differentiable_on {0..1}"
apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D2)
apply (rule_tac x="insert 0 ((\<lambda>x. 2*x-1)`s)" in exI)
apply simp
apply (intro ballI)
apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\<lambda>x. (x+1)/2)"
in differentiable_transform_within)
apply (auto simp: dist_real_def joinpaths_def abs_if field_simps split: split_if_asm)
apply (rule differentiable_chain_within derivative_intros | simp)+
apply (rule differentiable_subset)
apply (force simp: divide_simps)+
done
subsubsection\<open>The concept of continuously differentiable\<close>
definition C1_differentiable_on :: "(real \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> real set \<Rightarrow> bool"
(infix "C1'_differentiable'_on" 50)
where
"f C1_differentiable_on s \<longleftrightarrow>
(\<exists>D. (\<forall>x \<in> s. (f has_vector_derivative (D x)) (at x)) \<and> continuous_on s D)"
lemma C1_differentiable_on_eq:
"f C1_differentiable_on s \<longleftrightarrow>
(\<forall>x \<in> s. f differentiable at x) \<and> continuous_on s (\<lambda>x. vector_derivative f (at x))"
unfolding C1_differentiable_on_def
apply safe
using differentiable_def has_vector_derivative_def apply blast
apply (erule continuous_on_eq)
using vector_derivative_at apply fastforce
using vector_derivative_works apply fastforce
done
lemma C1_differentiable_on_subset:
"f C1_differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f C1_differentiable_on s"
unfolding C1_differentiable_on_def continuous_on_eq_continuous_within
by (blast intro: continuous_within_subset)
lemma C1_differentiable_compose:
"\<lbrakk>f C1_differentiable_on s; g C1_differentiable_on (f ` s);
\<And>x. finite (s \<inter> f-`{x})\<rbrakk>
\<Longrightarrow> (g o f) C1_differentiable_on s"
apply (simp add: C1_differentiable_on_eq, safe)
using differentiable_chain_at apply blast
apply (rule continuous_on_eq [of _ "\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"])
apply (rule Limits.continuous_on_scaleR, assumption)
apply (metis (mono_tags, lifting) continuous_on_eq continuous_at_imp_continuous_on continuous_on_compose differentiable_imp_continuous_within o_def)
by (simp add: vector_derivative_chain_at)
lemma C1_diff_imp_diff: "f C1_differentiable_on s \<Longrightarrow> f differentiable_on s"
by (simp add: C1_differentiable_on_eq differentiable_at_imp_differentiable_on)
lemma C1_differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) C1_differentiable_on s"
by (auto simp: C1_differentiable_on_eq continuous_on_const)
lemma C1_differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. a) C1_differentiable_on s"
by (auto simp: C1_differentiable_on_eq continuous_on_const)
lemma C1_differentiable_on_add [simp, derivative_intros]:
"f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x + g x) C1_differentiable_on s"
unfolding C1_differentiable_on_eq by (auto intro: continuous_intros)
lemma C1_differentiable_on_minus [simp, derivative_intros]:
"f C1_differentiable_on s \<Longrightarrow> (\<lambda>x. - f x) C1_differentiable_on s"
unfolding C1_differentiable_on_eq by (auto intro: continuous_intros)
lemma C1_differentiable_on_diff [simp, derivative_intros]:
"f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x - g x) C1_differentiable_on s"
unfolding C1_differentiable_on_eq by (auto intro: continuous_intros)
lemma C1_differentiable_on_mult [simp, derivative_intros]:
fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
shows "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x * g x) C1_differentiable_on s"
unfolding C1_differentiable_on_eq
by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within)
lemma C1_differentiable_on_scaleR [simp, derivative_intros]:
"f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) C1_differentiable_on s"
unfolding C1_differentiable_on_eq
by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+
definition piecewise_C1_differentiable_on
(infixr "piecewise'_C1'_differentiable'_on" 50)
where "f piecewise_C1_differentiable_on i \<equiv>
continuous_on i f \<and>
(\<exists>s. finite s \<and> (f C1_differentiable_on (i - s)))"
lemma C1_differentiable_imp_piecewise:
"f C1_differentiable_on s \<Longrightarrow> f piecewise_C1_differentiable_on s"
by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within)
lemma piecewise_C1_imp_differentiable:
"f piecewise_C1_differentiable_on i \<Longrightarrow> f piecewise_differentiable_on i"
by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def
C1_differentiable_on_def differentiable_def has_vector_derivative_def
intro: has_derivative_at_within)
lemma piecewise_C1_differentiable_compose:
"\<lbrakk>f piecewise_C1_differentiable_on s; g piecewise_C1_differentiable_on (f ` s);
\<And>x. finite (s \<inter> f-`{x})\<rbrakk>
\<Longrightarrow> (g o f) piecewise_C1_differentiable_on s"
apply (simp add: piecewise_C1_differentiable_on_def, safe)
apply (blast intro: continuous_on_compose2)
apply (rename_tac A B)
apply (rule_tac x="A \<union> (\<Union>x\<in>B. s \<inter> f-`{x})" in exI)
apply (rule conjI, blast)
apply (rule C1_differentiable_compose)
apply (blast intro: C1_differentiable_on_subset)
apply (blast intro: C1_differentiable_on_subset)
by (simp add: Diff_Int_distrib2)
lemma piecewise_C1_differentiable_on_subset:
"f piecewise_C1_differentiable_on s \<Longrightarrow> t \<le> s \<Longrightarrow> f piecewise_C1_differentiable_on t"
by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset)
lemma C1_differentiable_imp_continuous_on:
"f C1_differentiable_on s \<Longrightarrow> continuous_on s f"
unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within
using differentiable_at_withinI differentiable_imp_continuous_within by blast
lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}"
unfolding C1_differentiable_on_def
by auto
lemma piecewise_C1_differentiable_affine:
fixes m::real
assumes "f piecewise_C1_differentiable_on ((\<lambda>x. m * x + c) ` s)"
shows "(f o (\<lambda>x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on s"
proof (cases "m = 0")
case True
then show ?thesis
unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def continuous_on_const)
next
case False
show ?thesis
apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise])
apply (rule assms derivative_intros | simp add: False vimage_def)+
using real_vector_affinity_eq [OF False, where c=c, unfolded scaleR_conv_of_real]
apply simp
done
qed
lemma piecewise_C1_differentiable_cases:
fixes c::real
assumes "f piecewise_C1_differentiable_on {a..c}"
"g piecewise_C1_differentiable_on {c..b}"
"a \<le> c" "c \<le> b" "f c = g c"
shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}"
proof -
obtain s t where st: "f C1_differentiable_on ({a..c} - s)"
"g C1_differentiable_on ({c..b} - t)"
"finite s" "finite t"
using assms
by (force simp: piecewise_C1_differentiable_on_def)
then have f_diff: "f differentiable_on {a..<c} - s"
and g_diff: "g differentiable_on {c<..b} - t"
by (simp_all add: C1_differentiable_on_eq differentiable_at_withinI differentiable_on_def)
have "continuous_on {a..c} f" "continuous_on {c..b} g"
using assms piecewise_C1_differentiable_on_def by auto
then have cab: "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
OF closed_real_atLeastAtMost [of c b],
of f g "\<lambda>x. x\<le>c"] assms
by (force simp: ivl_disj_un_two_touch)
{ fix x
assume x: "x \<in> {a..b} - insert c (s \<union> t)"
have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x" (is "?diff_fg")
proof (cases x c rule: le_cases)
case le show ?diff_fg
apply (rule differentiable_transform_at [of "dist x c" _ f])
using x dist_real_def le st by (auto simp: C1_differentiable_on_eq)
next
case ge show ?diff_fg
apply (rule differentiable_transform_at [of "dist x c" _ g])
using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq)
qed
}
then have "(\<forall>x \<in> {a..b} - insert c (s \<union> t). (\<lambda>x. if x \<le> c then f x else g x) differentiable at x)"
by auto
moreover
{ assume fcon: "continuous_on ({a<..<c} - s) (\<lambda>x. vector_derivative f (at x))"
and gcon: "continuous_on ({c<..<b} - t) (\<lambda>x. vector_derivative g (at x))"
have "open ({a<..<c} - s)" "open ({c<..<b} - t)"
using st by (simp_all add: open_Diff finite_imp_closed)
moreover have "continuous_on ({a<..<c} - s) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
apply (rule continuous_on_eq [OF fcon])
apply (simp add:)
apply (rule vector_derivative_at [symmetric])
apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_at)
apply (simp_all add: dist_norm vector_derivative_works [symmetric])
using f_diff
by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_imp_le st(1))
moreover have "continuous_on ({c<..<b} - t) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
apply (rule continuous_on_eq [OF gcon])
apply (simp add:)
apply (rule vector_derivative_at [symmetric])
apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_at)
apply (simp_all add: dist_norm vector_derivative_works [symmetric])
using g_diff
by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_imp_le st(2))
ultimately have "continuous_on ({a<..<b} - insert c (s \<union> t))
(\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
apply (rule continuous_on_subset [OF continuous_on_open_Un], auto)
done
} note * = this
have "continuous_on ({a<..<b} - insert c (s \<union> t)) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
using st
by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *)
ultimately have "\<exists>s. finite s \<and> ((\<lambda>x. if x \<le> c then f x else g x) C1_differentiable_on {a..b} - s)"
apply (rule_tac x="{a,b,c} \<union> s \<union> t" in exI)
using st by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset)
with cab show ?thesis
by (simp add: piecewise_C1_differentiable_on_def)
qed
lemma piecewise_C1_differentiable_neg:
"f piecewise_C1_differentiable_on s \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_C1_differentiable_on s"
unfolding piecewise_C1_differentiable_on_def
by (auto intro!: continuous_on_minus C1_differentiable_on_minus)
lemma piecewise_C1_differentiable_add:
assumes "f piecewise_C1_differentiable_on i"
"g piecewise_C1_differentiable_on i"
shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i"
proof -
obtain s t where st: "finite s" "finite t"
"f C1_differentiable_on (i-s)"
"g C1_differentiable_on (i-t)"
using assms by (auto simp: piecewise_C1_differentiable_on_def)
then have "finite (s \<union> t) \<and> (\<lambda>x. f x + g x) C1_differentiable_on i - (s \<union> t)"
by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset)
moreover have "continuous_on i f" "continuous_on i g"
using assms piecewise_C1_differentiable_on_def by auto
ultimately show ?thesis
by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add)
qed
lemma piecewise_C1_differentiable_C1_diff:
"\<lbrakk>f piecewise_C1_differentiable_on s; g piecewise_C1_differentiable_on s\<rbrakk>
\<Longrightarrow> (\<lambda>x. f x - g x) piecewise_C1_differentiable_on s"
unfolding diff_conv_add_uminus
by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg)
lemma piecewise_C1_differentiable_D1:
fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}"
shows "g1 piecewise_C1_differentiable_on {0..1}"
proof -
obtain s where "finite s"
and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
using assms by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
then have g1D: "g1 differentiable at x" if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))" in differentiable_transform_at)
using that
apply (simp_all add: dist_real_def joinpaths_def)
apply (rule differentiable_chain_at derivative_intros | force)+
done
have [simp]: "vector_derivative (g1 \<circ> op * 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)"
if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
apply (subst vector_derivative_chain_at)
using that
apply (rule derivative_eq_intros g1D | simp)+
done
have "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
using co12 by (rule continuous_on_subset) force
then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 o op*2) (at x))"
apply (rule continuous_on_eq [OF _ vector_derivative_at])
apply (rule_tac f="g1 o op*2" and d="dist x (1/2)" in has_vector_derivative_transform_at)
apply (simp_all add: dist_norm joinpaths_def vector_derivative_works [symmetric])
apply (force intro: g1D differentiable_chain_at)
done
have "continuous_on ({0..1} - insert 1 (op * 2 ` s))
((\<lambda>x. 1/2 * vector_derivative (g1 o op*2) (at x)) o op*(1/2))"
apply (rule continuous_intros)+
using coDhalf
apply (simp add: scaleR_conv_of_real image_set_diff image_image)
done
then have con_g1: "continuous_on ({0..1} - insert 1 (op * 2 ` s)) (\<lambda>x. vector_derivative g1 (at x))"
by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
have "continuous_on {0..1} g1"
using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast
with `finite s` show ?thesis
apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
apply (simp add: g1D con_g1)
done
qed
lemma piecewise_C1_differentiable_D2:
fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" "pathfinish g1 = pathstart g2"
shows "g2 piecewise_C1_differentiable_on {0..1}"
proof -
obtain s where "finite s"
and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
using assms by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
then have g2D: "g2 differentiable at x" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)" for x
apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\<lambda>x. (x+1)/2)" in differentiable_transform_at)
using that
apply (simp_all add: dist_real_def joinpaths_def)
apply (auto simp: dist_real_def joinpaths_def field_simps)
apply (rule differentiable_chain_at derivative_intros | force)+
apply (drule_tac x= "(x + 1) / 2" in bspec, force simp: divide_simps)
apply assumption
done
have [simp]: "vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at ((x+1)/2)) = 2 *\<^sub>R vector_derivative g2 (at x)"
if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)" for x
using that by (auto simp: vector_derivative_chain_at divide_simps g2D)
have "continuous_on ({1/2..1} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
using co12 by (rule continuous_on_subset) force
then have coDhalf: "continuous_on ({1/2..1} - insert (1/2) s) (\<lambda>x. vector_derivative (g2 o (\<lambda>x. 2*x-1)) (at x))"
apply (rule continuous_on_eq [OF _ vector_derivative_at])
apply (rule_tac f="g2 o (\<lambda>x. 2*x-1)" and d="dist (3/4) ((x+1)/2)" in has_vector_derivative_transform_at)
apply (auto simp: dist_real_def field_simps joinpaths_def vector_derivative_works [symmetric]
intro!: g2D differentiable_chain_at)
done
have [simp]: "((\<lambda>x. (x + 1) / 2) ` ({0..1} - insert 0 ((\<lambda>x. 2 * x - 1) ` s))) = ({1/2..1} - insert (1/2) s)"
apply (simp add: image_set_diff inj_on_def image_image)
apply (auto simp: image_affinity_atLeastAtMost_div add_divide_distrib)
done
have "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s))
((\<lambda>x. 1/2 * vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x)) o (\<lambda>x. (x+1)/2))"
by (rule continuous_intros | simp add: coDhalf)+
then have con_g2: "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)) (\<lambda>x. vector_derivative g2 (at x))"
by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
have "continuous_on {0..1} g2"
using continuous_on_joinpaths_D2 assms piecewise_C1_differentiable_on_def by blast
with `finite s` show ?thesis
apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
apply (rule_tac x="insert 0 ((\<lambda>x. 2 * x - 1) ` s)" in exI)
apply (simp add: g2D con_g2)
done
qed
subsection \<open>Valid paths, and their start and finish\<close>
lemma Diff_Un_eq: "A - (B \<union> C) = A - B - C"
by blast
definition valid_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
where "valid_path f \<equiv> f piecewise_C1_differentiable_on {0..1::real}"
definition closed_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
where "closed_path g \<equiv> g 0 = g 1"
subsubsection\<open>In particular, all results for paths apply\<close>
lemma valid_path_imp_path: "valid_path g \<Longrightarrow> path g"
by (simp add: path_def piecewise_C1_differentiable_on_def valid_path_def)
lemma connected_valid_path_image: "valid_path g \<Longrightarrow> connected(path_image g)"
by (metis connected_path_image valid_path_imp_path)
lemma compact_valid_path_image: "valid_path g \<Longrightarrow> compact(path_image g)"
by (metis compact_path_image valid_path_imp_path)
lemma bounded_valid_path_image: "valid_path g \<Longrightarrow> bounded(path_image g)"
by (metis bounded_path_image valid_path_imp_path)
lemma closed_valid_path_image: "valid_path g \<Longrightarrow> closed(path_image g)"
by (metis closed_path_image valid_path_imp_path)
subsection\<open>Contour Integrals along a path\<close>
text\<open>This definition is for complex numbers only, and does not generalise to line integrals in a vector field\<close>
text\<open>piecewise differentiable function on [0,1]\<close>
definition has_path_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool"
(infixr "has'_path'_integral" 50)
where "(f has_path_integral i) g \<equiv>
((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1}))
has_integral i) {0..1}"
definition path_integrable_on
(infixr "path'_integrable'_on" 50)
where "f path_integrable_on g \<equiv> \<exists>i. (f has_path_integral i) g"
definition path_integral
where "path_integral g f \<equiv> @i. (f has_path_integral i) g"
lemma path_integral_unique: "(f has_path_integral i) g \<Longrightarrow> path_integral g f = i"
by (auto simp: path_integral_def has_path_integral_def integral_def [symmetric])
lemma has_path_integral_integral:
"f path_integrable_on i \<Longrightarrow> (f has_path_integral (path_integral i f)) i"
by (metis path_integral_unique path_integrable_on_def)
lemma has_path_integral_unique:
"(f has_path_integral i) g \<Longrightarrow> (f has_path_integral j) g \<Longrightarrow> i = j"
using has_integral_unique
by (auto simp: has_path_integral_def)
lemma has_path_integral_integrable: "(f has_path_integral i) g \<Longrightarrow> f path_integrable_on g"
using path_integrable_on_def by blast
(* Show that we can forget about the localized derivative.*)
lemma vector_derivative_within_interior:
"\<lbrakk>x \<in> interior s; NO_MATCH UNIV s\<rbrakk>
\<Longrightarrow> vector_derivative f (at x within s) = vector_derivative f (at x)"
apply (simp add: vector_derivative_def has_vector_derivative_def has_derivative_def netlimit_within_interior)
apply (subst lim_within_interior, auto)
done
lemma has_integral_localized_vector_derivative:
"((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
proof -
have "{a..b} - {a,b} = interior {a..b}"
by (simp add: atLeastAtMost_diff_ends)
show ?thesis
apply (rule has_integral_spike_eq [of "{a,b}"])
apply (auto simp: vector_derivative_within_interior)
done
qed
lemma integrable_on_localized_vector_derivative:
"(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow>
(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
by (simp add: integrable_on_def has_integral_localized_vector_derivative)
lemma has_path_integral:
"(f has_path_integral i) g \<longleftrightarrow>
((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
by (simp add: has_integral_localized_vector_derivative has_path_integral_def)
lemma path_integrable_on:
"f path_integrable_on g \<longleftrightarrow>
(\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
by (simp add: has_path_integral integrable_on_def path_integrable_on_def)
subsection\<open>Reversing a path\<close>
lemma valid_path_imp_reverse:
assumes "valid_path g"
shows "valid_path(reversepath g)"
proof -
obtain s where "finite s" "g C1_differentiable_on ({0..1} - s)"
using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
then have "finite (op - 1 ` s)" "(reversepath g C1_differentiable_on ({0..1} - op - 1 ` s))"
apply (auto simp: reversepath_def)
apply (rule C1_differentiable_compose [of "\<lambda>x::real. 1-x" _ g, unfolded o_def])
apply (auto simp: C1_differentiable_on_eq)
apply (rule continuous_intros, force)
apply (force elim!: continuous_on_subset)
apply (simp add: finite_vimageI inj_on_def)
done
then show ?thesis using assms
by (auto simp: valid_path_def piecewise_C1_differentiable_on_def path_def [symmetric])
qed
lemma valid_path_reversepath: "valid_path(reversepath g) \<longleftrightarrow> valid_path g"
using valid_path_imp_reverse by force
lemma has_path_integral_reversepath:
assumes "valid_path g" "(f has_path_integral i) g"
shows "(f has_path_integral (-i)) (reversepath g)"
proof -
{ fix s x
assume xs: "g C1_differentiable_on ({0..1} - s)" "x \<notin> op - 1 ` s" "0 \<le> x" "x \<le> 1"
have "vector_derivative (\<lambda>x. 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 o (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
apply (rule vector_diff_chain_within)
apply (intro vector_diff_chain_within derivative_eq_intros | simp)+
apply (rule has_vector_derivative_at_within [OF f'])
done
then have mf': "((\<lambda>x. 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
have 01: "{0..1::real} = cbox 0 1"
by simp
show ?thesis using assms
apply (auto simp: has_path_integral_def)
apply (drule has_integral_affinity01 [where m= "-1" and c=1])
apply (auto simp: reversepath_def valid_path_def piecewise_C1_differentiable_on_def)
apply (drule has_integral_neg)
apply (rule_tac s = "(\<lambda>x. 1 - x) ` s" in has_integral_spike_finite)
apply (auto simp: *)
done
qed
lemma path_integrable_reversepath:
"valid_path g \<Longrightarrow> f path_integrable_on g \<Longrightarrow> f path_integrable_on (reversepath g)"
using has_path_integral_reversepath path_integrable_on_def by blast
lemma path_integrable_reversepath_eq:
"valid_path g \<Longrightarrow> (f path_integrable_on (reversepath g) \<longleftrightarrow> f path_integrable_on g)"
using path_integrable_reversepath valid_path_reversepath by fastforce
lemma path_integral_reversepath:
"\<lbrakk>valid_path g; f path_integrable_on g\<rbrakk> \<Longrightarrow> path_integral (reversepath g) f = -(path_integral g f)"
using has_path_integral_reversepath path_integrable_on_def path_integral_unique by blast
subsection\<open>Joining two paths together\<close>
lemma valid_path_join:
assumes "valid_path g1" "valid_path g2" "pathfinish g1 = pathstart g2"
shows "valid_path(g1 +++ g2)"
proof -
have "g1 1 = g2 0"
using assms by (auto simp: pathfinish_def pathstart_def)
moreover have "(g1 o (\<lambda>x. 2*x)) piecewise_C1_differentiable_on {0..1/2}"
apply (rule piecewise_C1_differentiable_compose)
using assms
apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths)
apply (rule continuous_intros | simp)+
apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
done
moreover have "(g2 o (\<lambda>x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}"
apply (rule piecewise_C1_differentiable_compose)
using assms unfolding valid_path_def piecewise_C1_differentiable_on_def
by (auto intro!: continuous_intros finite_vimageI [where h = "(\<lambda>x. 2*x - 1)"] inj_onI
simp: image_affinity_atLeastAtMost_diff continuous_on_joinpaths)
ultimately show ?thesis
apply (simp only: valid_path_def continuous_on_joinpaths joinpaths_def)
apply (rule piecewise_C1_differentiable_cases)
apply (auto simp: o_def)
done
qed
lemma valid_path_join_D1:
fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
shows "valid_path (g1 +++ g2) \<Longrightarrow> valid_path g1"
unfolding valid_path_def
by (rule piecewise_C1_differentiable_D1)
lemma valid_path_join_D2:
fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
shows "\<lbrakk>valid_path (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> valid_path g2"
unfolding valid_path_def
by (rule piecewise_C1_differentiable_D2)
lemma valid_path_join_eq [simp]:
fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
shows "pathfinish g1 = pathstart g2 \<Longrightarrow> (valid_path(g1 +++ g2) \<longleftrightarrow> valid_path g1 \<and> valid_path g2)"
using valid_path_join_D1 valid_path_join_D2 valid_path_join by blast
lemma has_path_integral_join:
assumes "(f has_path_integral i1) g1" "(f has_path_integral i2) g2"
"valid_path g1" "valid_path g2"
shows "(f has_path_integral (i1 + i2)) (g1 +++ g2)"
proof -
obtain s1 s2
where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
and s2: "finite s2" "\<forall>x\<in>{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: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
using assms
by (auto simp: has_path_integral)
have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
and i2: "((\<lambda>x. (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: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
2 *\<^sub>R vector_derivative g1 (at (z*2))" for z
apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\<bar>z - 1/2\<bar>" _ "(\<lambda>x. g1(2*x))"]])
apply (simp_all add: dist_real_def abs_if split: split_if_asm)
apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
using s1
apply (auto simp: algebra_simps vector_derivative_works)
done
have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z
apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\<bar>z - 1/2\<bar>" _ "(\<lambda>x. g2 (2*x - 1))"]])
apply (simp_all add: dist_real_def abs_if split: split_if_asm)
apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
using s2
apply (auto simp: algebra_simps vector_derivative_works)
done
have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) (op*2 -` s1)"])
using s1
apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
done
moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
using s2
apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
done
ultimately
show ?thesis
apply (simp add: has_path_integral)
apply (rule has_integral_combine [where c = "1/2"], auto)
done
qed
lemma path_integrable_joinI:
assumes "f path_integrable_on g1" "f path_integrable_on g2"
"valid_path g1" "valid_path g2"
shows "f path_integrable_on (g1 +++ g2)"
using assms
by (meson has_path_integral_join path_integrable_on_def)
lemma path_integrable_joinD1:
assumes "f path_integrable_on (g1 +++ g2)" "valid_path g1"
shows "f path_integrable_on g1"
proof -
obtain s1
where s1: "finite s1" "\<forall>x\<in>{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 "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
using assms
apply (auto simp: path_integrable_on)
apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
done
then have *: "(\<lambda>x. (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: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow>
vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
2 *\<^sub>R vector_derivative g1 (at z)" for z
apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\<bar>(z-1)/2\<bar>" _ "(\<lambda>x. g1(2*x))"]])
apply (simp_all add: field_simps dist_real_def abs_if split: split_if_asm)
apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def])
using s1
apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
done
show ?thesis
using s1
apply (auto simp: path_integrable_on)
apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
done
qed
lemma path_integrable_joinD2:
assumes "f path_integrable_on (g1 +++ g2)" "valid_path g2"
shows "f path_integrable_on g2"
proof -
obtain s2
where s2: "finite s2" "\<forall>x\<in>{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 "(\<lambda>x. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}"
using assms
apply (auto simp: path_integrable_on)
apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
apply (simp add: image_affinity_atLeastAtMost_diff)
done
then have *: "(\<lambda>x. (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: "\<lbrakk>0 < z; z < 1; z \<notin> s2\<rbrakk> \<Longrightarrow>
vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) =
2 *\<^sub>R vector_derivative g2 (at z)" for z
apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\<bar>z/2\<bar>" _ "(\<lambda>x. g2(2*x-1))"]])
apply (simp_all add: field_simps dist_real_def abs_if split: split_if_asm)
apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def])
using s2
apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left
vector_derivative_works add_divide_distrib)
done
show ?thesis
using s2
apply (auto simp: path_integrable_on)
apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
done
qed
lemma path_integrable_join [simp]:
shows
"\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
\<Longrightarrow> f path_integrable_on (g1 +++ g2) \<longleftrightarrow> f path_integrable_on g1 \<and> f path_integrable_on g2"
using path_integrable_joinD1 path_integrable_joinD2 path_integrable_joinI by blast
lemma path_integral_join [simp]:
shows
"\<lbrakk>f path_integrable_on g1; f path_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
\<Longrightarrow> path_integral (g1 +++ g2) f = path_integral g1 f + path_integral g2 f"
by (simp add: has_path_integral_integral has_path_integral_join path_integral_unique)
subsection\<open>Shifting the starting point of a (closed) path\<close>
lemma shiftpath_alt_def: "shiftpath a f = (\<lambda>x. if x \<le> 1-a then f (a + x) else f (a + x - 1))"
by (auto simp: shiftpath_def)
lemma valid_path_shiftpath [intro]:
assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
shows "valid_path(shiftpath a g)"
using assms
apply (auto simp: valid_path_def shiftpath_alt_def)
apply (rule piecewise_C1_differentiable_cases)
apply (auto simp: algebra_simps)
apply (rule piecewise_C1_differentiable_affine [of g 1 a, simplified o_def scaleR_one])
apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
apply (rule piecewise_C1_differentiable_affine [of g 1 "a-1", simplified o_def scaleR_one algebra_simps])
apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
done
lemma has_path_integral_shiftpath:
assumes f: "(f has_path_integral i) g" "valid_path g"
and a: "a \<in> {0..1}"
shows "(f has_path_integral i) (shiftpath a g)"
proof -
obtain s
where s: "finite s" and g: "\<forall>x\<in>{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 *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
using assms by (auto simp: has_path_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 Integration.integral_combine)
using assms * integral_unique by auto
{ fix x
have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow>
vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
unfolding shiftpath_def
apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "dist(1-a) x" _ "(\<lambda>x. g(a+x))"]])
apply (auto simp: field_simps dist_real_def abs_if split: split_if_asm)
apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one])
apply (intro derivative_eq_intros | simp)+
using g
apply (drule_tac x="x+a" in bspec)
using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
done
} note vd1 = this
{ fix x
have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow>
vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
unfolding shiftpath_def
apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "dist (1-a) x" _ "(\<lambda>x. g(a+x-1))"]])
apply (auto simp: field_simps dist_real_def abs_if split: split_if_asm)
apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one])
apply (intro derivative_eq_intros | simp)+
using g
apply (drule_tac x="x+a-1" in bspec)
using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
done
} note vd2 = this
have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
using * a by (fastforce intro: integrable_subinterval_real)
have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
apply (rule integrable_subinterval_real)
using * a by auto
have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
has_integral integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x))) {0..1 - a}"
apply (rule has_integral_spike_finite
[where s = "{1-a} \<union> (\<lambda>x. x-a) ` s" and f = "\<lambda>x. f(g(a+x)) * vector_derivative g (at(a+x))"])
using s apply blast
using a apply (auto simp: algebra_simps vd1)
apply (force simp: shiftpath_def add.commute)
using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]]
apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute)
done
moreover
have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
has_integral integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))) {1 - a..1}"
apply (rule has_integral_spike_finite
[where s = "{1-a} \<union> (\<lambda>x. x-a+1) ` s" and f = "\<lambda>x. f(g(a+x-1)) * vector_derivative g (at(a+x-1))"])
using s apply blast
using a apply (auto simp: algebra_simps vd2)
apply (force simp: shiftpath_def add.commute)
using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified])
apply (simp add: algebra_simps)
done
ultimately show ?thesis
using a
by (auto simp: i has_path_integral intro: has_integral_combine [where c = "1-a"])
qed
lemma has_path_integral_shiftpath_D:
assumes "(f has_path_integral i) (shiftpath a g)"
"valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
shows "(f has_path_integral i) g"
proof -
obtain s
where s: "finite s" and g: "\<forall>x\<in>{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 \<notin> s"
then have gx: "g differentiable at x"
using g by auto
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
[of "{0<..<1}-s" _ "(shiftpath (1 - a) (shiftpath a g))"]])
using s g assms x
apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
vector_derivative_within_interior vector_derivative_works [symmetric])
apply (rule Derivative.differentiable_transform_at [of "min x (1-x)", OF _ _ gx])
apply (auto simp: dist_real_def shiftpath_shiftpath abs_if)
done
} note vd = this
have fi: "(f has_path_integral i) (shiftpath (1 - a) (shiftpath a g))"
using assms by (auto intro!: has_path_integral_shiftpath)
show ?thesis
apply (simp add: has_path_integral_def)
apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _ fi [unfolded has_path_integral_def]])
using s assms vd
apply (auto simp: Path_Connected.shiftpath_shiftpath)
done
qed
lemma has_path_integral_shiftpath_eq:
assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
shows "(f has_path_integral i) (shiftpath a g) \<longleftrightarrow> (f has_path_integral i) g"
using assms has_path_integral_shiftpath has_path_integral_shiftpath_D by blast
lemma path_integral_shiftpath:
assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
shows "path_integral (shiftpath a g) f = path_integral g f"
using assms by (simp add: path_integral_def has_path_integral_shiftpath_eq)
subsection\<open>More about straight-line paths\<close>
lemma has_vector_derivative_linepath_within:
"(linepath a b has_vector_derivative (b - a)) (at x within s)"
apply (simp add: linepath_def has_vector_derivative_def algebra_simps)
apply (rule derivative_eq_intros | simp)+
done
lemma vector_derivative_linepath_within:
"x \<in> {0..1} \<Longrightarrow> vector_derivative (linepath a b) (at x within {0..1}) = b - a"
apply (rule vector_derivative_within_closed_interval [of 0 "1::real", simplified])
apply (auto simp: has_vector_derivative_linepath_within)
done
lemma vector_derivative_linepath_at [simp]: "vector_derivative (linepath a b) (at x) = b - a"
by (simp add: has_vector_derivative_linepath_within vector_derivative_at)
lemma valid_path_linepath [iff]: "valid_path (linepath a b)"
apply (simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_on_linepath)
apply (rule_tac x="{}" in exI)
apply (simp add: differentiable_on_def differentiable_def)
using has_vector_derivative_def has_vector_derivative_linepath_within
apply (fastforce simp add: continuous_on_eq_continuous_within)
done
lemma has_path_integral_linepath:
shows "(f has_path_integral i) (linepath a b) \<longleftrightarrow>
((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
by (simp add: has_path_integral vector_derivative_linepath_at)
lemma linepath_in_path:
shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b"
by (auto simp: segment linepath_def)
lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
by (auto simp: segment linepath_def)
lemma linepath_in_convex_hull:
fixes x::real
assumes a: "a \<in> convex hull s"
and b: "b \<in> convex hull s"
and x: "0\<le>x" "x\<le>1"
shows "linepath a b x \<in> convex hull s"
apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD])
using x
apply (auto simp: linepath_image_01 [symmetric])
done
lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
by (simp add: linepath_def)
lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
by (simp add: linepath_def)
lemma linepath_of_real: "(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)"
by (simp add: scaleR_conv_of_real linepath_def)
lemma of_real_linepath: "of_real (linepath a b x) = linepath (of_real a) (of_real b) x"
by (metis linepath_of_real mult.right_neutral of_real_def real_scaleR_def)
lemma has_path_integral_trivial [iff]: "(f has_path_integral 0) (linepath a a)"
by (simp add: has_path_integral_linepath)
lemma path_integral_trivial [simp]: "path_integral (linepath a a) f = 0"
using has_path_integral_trivial path_integral_unique by blast
subsection\<open>Relation to subpath construction\<close>
lemma valid_path_subpath:
fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
assumes "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
shows "valid_path(subpath u v g)"
proof (cases "v=u")
case True
then show ?thesis
unfolding valid_path_def subpath_def
by (force intro: C1_differentiable_on_const C1_differentiable_imp_piecewise)
next
case False
have "(g o (\<lambda>x. ((v-u) * x + u))) piecewise_C1_differentiable_on {0..1}"
apply (rule piecewise_C1_differentiable_compose)
apply (simp add: C1_differentiable_imp_piecewise)
apply (simp add: image_affinity_atLeastAtMost)
using assms False
apply (auto simp: algebra_simps valid_path_def piecewise_C1_differentiable_on_subset)
apply (subst Int_commute)
apply (auto simp: inj_on_def algebra_simps crossproduct_eq finite_vimage_IntI)
done
then show ?thesis
by (auto simp: o_def valid_path_def subpath_def)
qed
lemma has_path_integral_subpath_refl [iff]: "(f has_path_integral 0) (subpath u u g)"
by (simp add: has_path_integral subpath_def)
lemma path_integrable_subpath_refl [iff]: "f path_integrable_on (subpath u u g)"
using has_path_integral_subpath_refl path_integrable_on_def by blast
lemma path_integral_subpath_refl [simp]: "path_integral (subpath u u g) f = 0"
by (simp add: has_path_integral_subpath_refl path_integral_unique)
lemma has_path_integral_subpath:
assumes f: "f path_integrable_on g" and g: "valid_path g"
and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
shows "(f has_path_integral integral {u..v} (\<lambda>x. 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: path_integrable_on_def subpath_def has_path_integral)
next
case False
obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> 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 *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t)))
{0..1}"
using f uv
apply (simp add: path_integrable_on subpath_def has_path_integral)
apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
apply (simp_all add: has_integral_integral)
apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
apply (simp add: divide_simps False)
done
{ fix x
have "x \<in> {0..1} \<Longrightarrow>
x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u))"
apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
apply (intro derivative_eq_intros | simp)+
apply (cut_tac s [of "(v - u) * x + u"])
using uv mult_left_le [of x "v-u"]
apply (auto simp: vector_derivative_works)
done
} note vd = this
show ?thesis
apply (cut_tac has_integral_cmul [OF *, where c = "v-u"])
using fs assms
apply (simp add: False subpath_def has_path_integral)
apply (rule_tac s = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real)
done
qed
lemma path_integrable_subpath:
assumes "f path_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
shows "f path_integrable_on (subpath u v g)"
apply (cases u v rule: linorder_class.le_cases)
apply (metis path_integrable_on_def has_path_integral_subpath [OF assms])
apply (subst reversepath_subpath [symmetric])
apply (rule path_integrable_reversepath)
using assms apply (blast intro: valid_path_subpath)
apply (simp add: path_integrable_on_def)
using assms apply (blast intro: has_path_integral_subpath)
done
lemma has_integral_integrable_integral: "(f has_integral i) s \<longleftrightarrow> f integrable_on s \<and> integral s f = i"
by blast
lemma has_integral_path_integral_subpath:
assumes "f path_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)))
has_integral path_integral (subpath u v g) f) {u..v}"
using assms
apply (auto simp: has_integral_integrable_integral)
apply (rule integrable_on_subcbox [where a=u and b=v and s = "{0..1}", simplified])
apply (auto simp: path_integral_unique [OF has_path_integral_subpath] path_integrable_on)
done
lemma path_integral_subpath_integral:
assumes "f path_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
shows "path_integral (subpath u v g) f =
integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))"
using assms has_path_integral_subpath path_integral_unique by blast
lemma path_integral_subpath_combine_less:
assumes "f path_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
"u<v" "v<w"
shows "path_integral (subpath u v g) f + path_integral (subpath v w g) f =
path_integral (subpath u w g) f"
using assms apply (auto simp: path_integral_subpath_integral)
apply (rule integral_combine, auto)
apply (rule integrable_on_subcbox [where a=u and b=w and s = "{0..1}", simplified])
apply (auto simp: path_integrable_on)
done
lemma path_integral_subpath_combine:
assumes "f path_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
shows "path_integral (subpath u v g) f + path_integral (subpath v w g) f =
path_integral (subpath u w g) f"
proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w")
case True
have *: "subpath v u g = reversepath(subpath u v g) \<and>
subpath w u g = reversepath(subpath u w g) \<and>
subpath w v g = reversepath(subpath v w g)"
by (auto simp: reversepath_subpath)
have "u < v \<and> v < w \<or>
u < w \<and> w < v \<or>
v < u \<and> u < w \<or>
v < w \<and> w < u \<or>
w < u \<and> u < v \<or>
w < v \<and> v < u"
using True assms by linarith
with assms show ?thesis
using path_integral_subpath_combine_less [of f g u v w]
path_integral_subpath_combine_less [of f g u w v]
path_integral_subpath_combine_less [of f g v u w]
path_integral_subpath_combine_less [of f g v w u]
path_integral_subpath_combine_less [of f g w u v]
path_integral_subpath_combine_less [of f g w v u]
apply simp
apply (elim disjE)
apply (auto simp: * path_integral_reversepath path_integrable_subpath
valid_path_reversepath valid_path_subpath algebra_simps)
done
next
case False
then show ?thesis
apply (auto simp: path_integral_subpath_refl)
using assms
by (metis eq_neg_iff_add_eq_0 path_integrable_subpath path_integral_reversepath reversepath_subpath valid_path_subpath)
qed
lemma path_integral_integral:
shows "path_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
by (simp add: path_integral_def integral_def has_path_integral)
subsection\<open>Segments via convex hulls\<close>
lemma segments_subset_convex_hull:
"closed_segment a b \<subseteq> (convex hull {a,b,c})"
"closed_segment a c \<subseteq> (convex hull {a,b,c})"
"closed_segment b c \<subseteq> (convex hull {a,b,c})"
"closed_segment b a \<subseteq> (convex hull {a,b,c})"
"closed_segment c a \<subseteq> (convex hull {a,b,c})"
"closed_segment c b \<subseteq> (convex hull {a,b,c})"
by (auto simp: segment_convex_hull linepath_of_real elim!: rev_subsetD [OF _ hull_mono])
lemma midpoints_in_convex_hull:
assumes "x \<in> convex hull s" "y \<in> convex hull s"
shows "midpoint x y \<in> convex hull s"
proof -
have "(1 - inverse(2)) *\<^sub>R x + inverse(2) *\<^sub>R y \<in> convex hull s"
apply (rule mem_convex)
using assms
apply (auto simp: convex_convex_hull)
done
then show ?thesis
by (simp add: midpoint_def algebra_simps)
qed
lemma convex_hull_subset:
"s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
by (simp add: convex_convex_hull subset_hull)
lemma not_in_interior_convex_hull_3:
fixes a :: "complex"
shows "a \<notin> interior(convex hull {a,b,c})"
"b \<notin> interior(convex hull {a,b,c})"
"c \<notin> interior(convex hull {a,b,c})"
by (auto simp: card_insert_le_m1 not_in_interior_convex_hull)
text\<open>Cauchy's theorem where there's a primitive\<close>
lemma path_integral_primitive_lemma:
fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
assumes "a \<le> b"
and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
and "g piecewise_differentiable_on {a..b}" "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
shows "((\<lambda>x. 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 k: "finite k" "\<forall>x\<in>{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 cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
apply (rule continuous_on_compose [OF cg, unfolded o_def])
using assms
apply (metis complex_differentiable_def complex_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
done
{ fix x::real
assume a: "a < x" and b: "x < b" and xk: "x \<notin> 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 (\<lambda>u. 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 op * (f' (g x))) (at (g x) within g ` {a..b})"
by (simp add: has_field_derivative_def)
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
apply (rule fundamental_theorem_of_calculus_interior_strong)
using k assms cfg *
apply (auto simp: at_within_closed_interval)
done
qed
lemma path_integral_primitive:
assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
and "valid_path g" "path_image g \<subseteq> s"
shows "(f' has_path_integral (f(pathfinish g) - f(pathstart g))) g"
using assms
apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_path_integral_def)
apply (auto intro!: piecewise_C1_imp_differentiable path_integral_primitive_lemma [of 0 1 s])
done
corollary Cauchy_theorem_primitive:
assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
and "valid_path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g"
shows "(f' has_path_integral 0) g"
using assms
by (metis diff_self path_integral_primitive)
text\<open>Existence of path integral for continuous function\<close>
lemma path_integrable_continuous_linepath:
assumes "continuous_on (closed_segment a b) f"
shows "f path_integrable_on (linepath a b)"
proof -
have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) o linepath a b)"
apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01)
apply (rule continuous_intros | simp add: assms)+
done
then show ?thesis
apply (simp add: path_integrable_on_def has_path_integral_def integrable_on_def [symmetric])
apply (rule integrable_continuous [of 0 "1::real", simplified])
apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"])
apply (auto simp: vector_derivative_linepath_within)
done
qed
lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)"
by (rule has_derivative_imp_has_field_derivative)
(rule derivative_intros | simp)+
lemma path_integral_id [simp]: "path_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2"
apply (rule path_integral_unique)
using path_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"]
apply (auto simp: field_simps has_field_der_id)
done
lemma path_integrable_on_const [iff]: "(\<lambda>x. c) path_integrable_on (linepath a b)"
by (simp add: continuous_on_const path_integrable_continuous_linepath)
lemma path_integrable_on_id [iff]: "(\<lambda>x. x) path_integrable_on (linepath a b)"
by (simp add: continuous_on_id path_integrable_continuous_linepath)
subsection\<open>Arithmetical combining theorems\<close>
lemma has_path_integral_neg:
"(f has_path_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_path_integral (-i)) g"
by (simp add: has_integral_neg has_path_integral_def)
lemma has_path_integral_add:
"\<lbrakk>(f1 has_path_integral i1) g; (f2 has_path_integral i2) g\<rbrakk>
\<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_path_integral (i1 + i2)) g"
by (simp add: has_integral_add has_path_integral_def algebra_simps)
lemma has_path_integral_diff:
"\<lbrakk>(f1 has_path_integral i1) g; (f2 has_path_integral i2) g\<rbrakk>
\<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_path_integral (i1 - i2)) g"
by (simp add: has_integral_sub has_path_integral_def algebra_simps)
lemma has_path_integral_lmul:
"(f has_path_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_path_integral (c*i)) g"
apply (simp add: has_path_integral_def)
apply (drule has_integral_mult_right)
apply (simp add: algebra_simps)
done
lemma has_path_integral_rmul:
"(f has_path_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_path_integral (i*c)) g"
apply (drule has_path_integral_lmul)
apply (simp add: mult.commute)
done
lemma has_path_integral_div:
"(f has_path_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_path_integral (i/c)) g"
by (simp add: field_class.field_divide_inverse) (metis has_path_integral_rmul)
lemma has_path_integral_eq:
"\<lbrakk>(f has_path_integral y) p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> (g has_path_integral y) p"
apply (simp add: path_image_def has_path_integral_def)
by (metis (no_types, lifting) image_eqI has_integral_eq)
lemma has_path_integral_bound_linepath:
assumes "(f has_path_integral i) (linepath a b)"
"0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
shows "norm i \<le> B * norm(b - a)"
proof -
{ fix x::real
assume x: "0 \<le> x" "x \<le> 1"
have "norm (f (linepath a b x)) *
norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)"
by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x)
} note * = this
have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
apply (rule has_integral_bound
[of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
using assms * unfolding has_path_integral_def
apply (auto simp: norm_mult)
done
then show ?thesis
by (auto simp: content_real)
qed
(*UNUSED
lemma has_path_integral_bound_linepath_strong:
fixes a :: real and f :: "complex \<Rightarrow> real"
assumes "(f has_path_integral i) (linepath a b)"
"finite k"
"0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B"
shows "norm i \<le> B*norm(b - a)"
*)
lemma has_path_integral_const_linepath: "((\<lambda>x. c) has_path_integral c*(b - a))(linepath a b)"
unfolding has_path_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_path_integral_0: "((\<lambda>x. 0) has_path_integral 0) g"
by (simp add: has_path_integral_def)
lemma has_path_integral_is_0:
"(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_path_integral 0) g"
by (rule has_path_integral_eq [OF has_path_integral_0]) auto
lemma has_path_integral_setsum:
"\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_path_integral i a) p\<rbrakk>
\<Longrightarrow> ((\<lambda>x. setsum (\<lambda>a. f a x) s) has_path_integral setsum i s) p"
by (induction s rule: finite_induct) (auto simp: has_path_integral_0 has_path_integral_add)
subsection \<open>Operations on path integrals\<close>
lemma path_integral_const_linepath [simp]: "path_integral (linepath a b) (\<lambda>x. c) = c*(b - a)"
by (rule path_integral_unique [OF has_path_integral_const_linepath])
lemma path_integral_neg:
"f path_integrable_on g \<Longrightarrow> path_integral g (\<lambda>x. -(f x)) = -(path_integral g f)"
by (simp add: path_integral_unique has_path_integral_integral has_path_integral_neg)
lemma path_integral_add:
"f1 path_integrable_on g \<Longrightarrow> f2 path_integrable_on g \<Longrightarrow> path_integral g (\<lambda>x. f1 x + f2 x) =
path_integral g f1 + path_integral g f2"
by (simp add: path_integral_unique has_path_integral_integral has_path_integral_add)
lemma path_integral_diff:
"f1 path_integrable_on g \<Longrightarrow> f2 path_integrable_on g \<Longrightarrow> path_integral g (\<lambda>x. f1 x - f2 x) =
path_integral g f1 - path_integral g f2"
by (simp add: path_integral_unique has_path_integral_integral has_path_integral_diff)
lemma path_integral_lmul:
shows "f path_integrable_on g
\<Longrightarrow> path_integral g (\<lambda>x. c * f x) = c*path_integral g f"
by (simp add: path_integral_unique has_path_integral_integral has_path_integral_lmul)
lemma path_integral_rmul:
shows "f path_integrable_on g
\<Longrightarrow> path_integral g (\<lambda>x. f x * c) = path_integral g f * c"
by (simp add: path_integral_unique has_path_integral_integral has_path_integral_rmul)
lemma path_integral_div:
shows "f path_integrable_on g
\<Longrightarrow> path_integral g (\<lambda>x. f x / c) = path_integral g f / c"
by (simp add: path_integral_unique has_path_integral_integral has_path_integral_div)
lemma path_integral_eq:
"(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> path_integral p f = path_integral p g"
by (simp add: path_integral_def) (metis has_path_integral_eq)
lemma path_integral_eq_0:
"(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> path_integral g f = 0"
by (simp add: has_path_integral_is_0 path_integral_unique)
lemma path_integral_bound_linepath:
shows
"\<lbrakk>f path_integrable_on (linepath a b);
0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
\<Longrightarrow> norm(path_integral (linepath a b) f) \<le> B*norm(b - a)"
apply (rule has_path_integral_bound_linepath [of f])
apply (auto simp: has_path_integral_integral)
done
lemma path_integral_0: "path_integral g (\<lambda>x. 0) = 0"
by (simp add: path_integral_unique has_path_integral_0)
lemma path_integral_setsum:
"\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) path_integrable_on p\<rbrakk>
\<Longrightarrow> path_integral p (\<lambda>x. setsum (\<lambda>a. f a x) s) = setsum (\<lambda>a. path_integral p (f a)) s"
by (auto simp: path_integral_unique has_path_integral_setsum has_path_integral_integral)
lemma path_integrable_eq:
"\<lbrakk>f path_integrable_on p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g path_integrable_on p"
unfolding path_integrable_on_def
by (metis has_path_integral_eq)
subsection \<open>Arithmetic theorems for path integrability\<close>
lemma path_integrable_neg:
"f path_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) path_integrable_on g"
using has_path_integral_neg path_integrable_on_def by blast
lemma path_integrable_add:
"\<lbrakk>f1 path_integrable_on g; f2 path_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x + f2 x) path_integrable_on g"
using has_path_integral_add path_integrable_on_def
by fastforce
lemma path_integrable_diff:
"\<lbrakk>f1 path_integrable_on g; f2 path_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x - f2 x) path_integrable_on g"
using has_path_integral_diff path_integrable_on_def
by fastforce
lemma path_integrable_lmul:
"f path_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) path_integrable_on g"
using has_path_integral_lmul path_integrable_on_def
by fastforce
lemma path_integrable_rmul:
"f path_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) path_integrable_on g"
using has_path_integral_rmul path_integrable_on_def
by fastforce
lemma path_integrable_div:
"f path_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) path_integrable_on g"
using has_path_integral_div path_integrable_on_def
by fastforce
lemma path_integrable_setsum:
"\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) path_integrable_on p\<rbrakk>
\<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) s) path_integrable_on p"
unfolding path_integrable_on_def
by (metis has_path_integral_setsum)
subsection\<open>Reversing a path integral\<close>
lemma has_path_integral_reverse_linepath:
"(f has_path_integral i) (linepath a b)
\<Longrightarrow> (f has_path_integral (-i)) (linepath b a)"
using has_path_integral_reversepath valid_path_linepath by fastforce
lemma path_integral_reverse_linepath:
"continuous_on (closed_segment a b) f
\<Longrightarrow> path_integral (linepath a b) f = - (path_integral(linepath b a) f)"
apply (rule path_integral_unique)
apply (rule has_path_integral_reverse_linepath)
by (simp add: closed_segment_commute path_integrable_continuous_linepath has_path_integral_integral)
(* Splitting a path integral in a flat way.*)
lemma has_path_integral_split:
assumes f: "(f has_path_integral i) (linepath a c)" "(f has_path_integral j) (linepath c b)"
and k: "0 \<le> k" "k \<le> 1"
and c: "c - a = k *\<^sub>R (b - a)"
shows "(f has_path_integral (i + j)) (linepath a b)"
proof (cases "k = 0 \<or> k = 1")
case True
then show ?thesis
using assms
apply auto
apply (metis add.left_neutral has_path_integral_trivial has_path_integral_unique)
apply (metis add.right_neutral has_path_integral_trivial has_path_integral_unique)
done
next
case False
then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1"
using assms apply auto
using of_real_eq_iff by fastforce
have c': "c = k *\<^sub>R (b - a) + a"
by (metis diff_add_cancel c)
have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)"
by (simp add: algebra_simps c')
{ assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b"
using False
apply (simp add: c' algebra_simps)
apply (simp add: real_vector.scale_left_distrib [symmetric] divide_simps)
done
have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
using * k
apply -
apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse k" "0", simplified])
apply (simp_all add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c)
apply (drule Integration.has_integral_cmul [where c = "inverse k"])
apply (simp add: Integration.has_integral_cmul)
done
} note fi = this
{ assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}"
have **: "\<And>x. (((1 - x) / (1 - k)) *\<^sub>R c + ((x - k) / (1 - k)) *\<^sub>R b) = ((1 - x) *\<^sub>R a + x *\<^sub>R b)"
using k
apply (simp add: c' field_simps)
apply (simp add: scaleR_conv_of_real divide_simps)
apply (simp add: field_simps)
done
have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
using * k
apply -
apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse(1 - k)" "-(k/(1 - k))", simplified])
apply (simp_all add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
apply (drule Integration.has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
apply (simp add: Integration.has_integral_cmul)
done
} note fj = this
show ?thesis
using f k
apply (simp add: has_path_integral_linepath)
apply (simp add: linepath_def)
apply (rule has_integral_combine [OF _ _ fi fj], simp_all)
done
qed
lemma continuous_on_closed_segment_transform:
assumes f: "continuous_on (closed_segment a b) f"
and k: "0 \<le> k" "k \<le> 1"
and c: "c - a = k *\<^sub>R (b - a)"
shows "continuous_on (closed_segment a c) f"
proof -
have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
using c by (simp add: algebra_simps)
show "continuous_on (closed_segment a c) f"
apply (rule continuous_on_subset [OF f])
apply (simp add: segment_convex_hull)
apply (rule convex_hull_subset)
using assms
apply (auto simp: hull_inc c' Convex.mem_convex)
done
qed
lemma path_integral_split:
assumes f: "continuous_on (closed_segment a b) f"
and k: "0 \<le> k" "k \<le> 1"
and c: "c - a = k *\<^sub>R (b - a)"
shows "path_integral(linepath a b) f = path_integral(linepath a c) f + path_integral(linepath c b) f"
proof -
have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
using c by (simp add: algebra_simps)
have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
apply (rule_tac [!] continuous_on_subset [OF f])
apply (simp_all add: segment_convex_hull)
apply (rule_tac [!] convex_hull_subset)
using assms
apply (auto simp: hull_inc c' Convex.mem_convex)
done
show ?thesis
apply (rule path_integral_unique)
apply (rule has_path_integral_split [OF has_path_integral_integral has_path_integral_integral k c])
apply (rule path_integrable_continuous_linepath *)+
done
qed
lemma path_integral_split_linepath:
assumes f: "continuous_on (closed_segment a b) f"
and c: "c \<in> closed_segment a b"
shows "path_integral(linepath a b) f = path_integral(linepath a c) f + path_integral(linepath c b) f"
using c
by (auto simp: closed_segment_def algebra_simps intro!: path_integral_split [OF f])
(* The special case of midpoints used in the main quadrisection.*)
lemma has_path_integral_midpoint:
assumes "(f has_path_integral i) (linepath a (midpoint a b))"
"(f has_path_integral j) (linepath (midpoint a b) b)"
shows "(f has_path_integral (i + j)) (linepath a b)"
apply (rule has_path_integral_split [where c = "midpoint a b" and k = "1/2"])
using assms
apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
done
lemma path_integral_midpoint:
"continuous_on (closed_segment a b) f
\<Longrightarrow> path_integral (linepath a b) f =
path_integral (linepath a (midpoint a b)) f + path_integral (linepath (midpoint a b) b) f"
apply (rule path_integral_split [where c = "midpoint a b" and k = "1/2"])
using assms
apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
done
text\<open>A couple of special case lemmas that are useful below\<close>
lemma triangle_linear_has_chain_integral:
"((\<lambda>x. m*x + d) has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
apply (rule Cauchy_theorem_primitive [of UNIV "\<lambda>x. m/2 * x^2 + d*x"])
apply (auto intro!: derivative_eq_intros)
done
lemma has_chain_integral_chain_integral3:
"(f has_path_integral i) (linepath a b +++ linepath b c +++ linepath c d)
\<Longrightarrow> path_integral (linepath a b) f + path_integral (linepath b c) f + path_integral (linepath c d) f = i"
apply (subst path_integral_unique [symmetric], assumption)
apply (drule has_path_integral_integrable)
apply (simp add: valid_path_join)
done
subsection\<open>Reversing the order in a double path integral\<close>
text\<open>The condition is stronger than needed but it's often true in typical situations\<close>
lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b"
by (auto simp: cbox_Pair_eq)
lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d"
by (auto simp: cbox_Pair_eq)
lemma path_integral_swap:
assumes fcon: "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)"
and vp: "valid_path g" "valid_path h"
and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
shows "path_integral g (\<lambda>w. path_integral h (f w)) =
path_integral h (\<lambda>z. path_integral g (\<lambda>w. 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: "\<And>x. (\<lambda>t. f (g x) (h t)) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>t. (g x, h t))"
by (rule ext) simp
have fgh2: "\<And>x. (\<lambda>t. f (g t) (h x)) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>t. (g t, h x))"
by (rule ext) simp
have fcon_im1: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g x, h t)) ` {0..1}) (\<lambda>(x, y). f x y)"
by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
have fcon_im2: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g t, h x)) ` {0..1}) (\<lambda>(x, y). f x y)"
by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
have vdg: "\<And>y. y \<in> {0..1} \<Longrightarrow> (\<lambda>x. f (g x) (h y) * vector_derivative g (at x)) integrable_on {0..1}"
apply (rule integrable_continuous_real)
apply (rule continuous_on_mult [OF _ gvcon])
apply (subst fgh2)
apply (rule fcon_im2 gcon continuous_intros | simp)+
done
have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) o fst"
by auto
then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))"
apply (rule ssubst)
apply (rule continuous_intros | simp add: gvcon)+
done
have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) o snd"
by auto
then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
apply (rule ssubst)
apply (rule continuous_intros | simp add: hvcon)+
done
have "(\<lambda>x. f (g (fst x)) (h (snd x))) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>w. ((g o fst) w, (h o snd) w))"
by auto
then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
apply (rule ssubst)
apply (rule gcon hcon continuous_intros | simp)+
apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
done
have "integral {0..1} (\<lambda>x. path_integral h (f (g x)) * vector_derivative g (at x)) =
integral {0..1} (\<lambda>x. path_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))"
apply (rule integral_cong [OF path_integral_rmul [symmetric]])
apply (clarsimp simp: path_integrable_on)
apply (rule integrable_continuous_real)
apply (rule continuous_on_mult [OF _ hvcon])
apply (subst fgh1)
apply (rule fcon_im1 hcon continuous_intros | simp)+
done
also have "... = integral {0..1}
(\<lambda>y. path_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))"
apply (simp add: path_integral_integral)
apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
apply (simp add: algebra_simps)
done
also have "... = path_integral h (\<lambda>z. path_integral g (\<lambda>w. f w z))"
apply (simp add: path_integral_integral)
apply (rule integral_cong)
apply (subst integral_mult_left [symmetric])
apply (blast intro: vdg)
apply (simp add: algebra_simps)
done
finally show ?thesis
by (simp add: path_integral_integral)
qed
subsection\<open>The key quadrisection step\<close>
lemma norm_sum_half:
assumes "norm(a + b) >= e"
shows "norm a >= e/2 \<or> norm b >= e/2"
proof -
have "e \<le> norm (- a - b)"
by (simp add: add.commute assms norm_minus_commute)
thus ?thesis
using norm_triangle_ineq4 order_trans by fastforce
qed
lemma norm_sum_lemma:
assumes "e \<le> norm (a + b + c + d)"
shows "e / 4 \<le> norm a \<or> e / 4 \<le> norm b \<or> e / 4 \<le> norm c \<or> e / 4 \<le> norm d"
proof -
have "e \<le> norm ((a + b) + (c + d))" using assms
by (simp add: algebra_simps)
then show ?thesis
by (auto dest!: norm_sum_half)
qed
lemma Cauchy_theorem_quadrisection:
assumes f: "continuous_on (convex hull {a,b,c}) f"
and dist: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
and e: "e * K^2 \<le>
norm (path_integral(linepath a b) f + path_integral(linepath b c) f + path_integral(linepath c a) f)"
shows "\<exists>a' b' c'.
a' \<in> convex hull {a,b,c} \<and> b' \<in> convex hull {a,b,c} \<and> c' \<in> convex hull {a,b,c} \<and>
dist a' b' \<le> K/2 \<and> dist b' c' \<le> K/2 \<and> dist c' a' \<le> K/2 \<and>
e * (K/2)^2 \<le> norm(path_integral(linepath a' b') f + path_integral(linepath b' c') f + path_integral(linepath c' a') f)"
proof -
note divide_le_eq_numeral1 [simp del]
def a' \<equiv> "midpoint b c"
def b' \<equiv> "midpoint c a"
def c' \<equiv> "midpoint a b"
have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
using f continuous_on_subset segments_subset_convex_hull by metis+
have fcont': "continuous_on (closed_segment c' b') f"
"continuous_on (closed_segment a' c') f"
"continuous_on (closed_segment b' a') f"
unfolding a'_def b'_def c'_def
apply (rule continuous_on_subset [OF f],
metis midpoints_in_convex_hull convex_hull_subset hull_subset insert_subset segment_convex_hull)+
done
let ?pathint = "\<lambda>x y. path_integral(linepath x y) f"
have *: "?pathint a b + ?pathint b c + ?pathint c a =
(?pathint a c' + ?pathint c' b' + ?pathint b' a) +
(?pathint a' c' + ?pathint c' b + ?pathint b a') +
(?pathint a' c + ?pathint c b' + ?pathint b' a') +
(?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
apply (simp add: fcont' path_integral_reverse_linepath)
apply (simp add: a'_def b'_def c'_def path_integral_midpoint fabc)
done
have [simp]: "\<And>x y. cmod (x * 2 - y * 2) = cmod (x - y) * 2"
by (metis left_diff_distrib mult.commute norm_mult_numeral1)
have [simp]: "\<And>x y. cmod (x - y) = cmod (y - x)"
by (simp add: norm_minus_commute)
consider "e * K\<^sup>2 / 4 \<le> cmod (?pathint a c' + ?pathint c' b' + ?pathint b' a)" |
"e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c' + ?pathint c' b + ?pathint b a')" |
"e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c + ?pathint c b' + ?pathint b' a')" |
"e * K\<^sup>2 / 4 \<le> cmod (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
using assms
apply (simp only: *)
apply (blast intro: that dest!: norm_sum_lemma)
done
then show ?thesis
proof cases
case 1 then show ?thesis
apply (rule_tac x=a in exI)
apply (rule exI [where x=c'])
apply (rule exI [where x=b'])
using assms
apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
done
next
case 2 then show ?thesis
apply (rule_tac x=a' in exI)
apply (rule exI [where x=c'])
apply (rule exI [where x=b])
using assms
apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
done
next
case 3 then show ?thesis
apply (rule_tac x=a' in exI)
apply (rule exI [where x=c])
apply (rule exI [where x=b'])
using assms
apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
done
next
case 4 then show ?thesis
apply (rule_tac x=a' in exI)
apply (rule exI [where x=b'])
apply (rule exI [where x=c'])
using assms
apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
done
qed
qed
subsection\<open>Cauchy's theorem for triangles\<close>
lemma triangle_points_closer:
fixes a::complex
shows "\<lbrakk>x \<in> convex hull {a,b,c}; y \<in> convex hull {a,b,c}\<rbrakk>
\<Longrightarrow> norm(x - y) \<le> norm(a - b) \<or>
norm(x - y) \<le> norm(b - c) \<or>
norm(x - y) \<le> norm(c - a)"
using simplex_extremal_le [of "{a,b,c}"]
by (auto simp: norm_minus_commute)
lemma holomorphic_point_small_triangle:
assumes x: "x \<in> s"
and f: "continuous_on s f"
and cd: "f complex_differentiable (at x within s)"
and e: "0 < e"
shows "\<exists>k>0. \<forall>a b c. dist a b \<le> k \<and> dist b c \<le> k \<and> dist c a \<le> k \<and>
x \<in> convex hull {a,b,c} \<and> convex hull {a,b,c} \<subseteq> s
\<longrightarrow> norm(path_integral(linepath a b) f + path_integral(linepath b c) f +
path_integral(linepath c a) f)
\<le> e*(dist a b + dist b c + dist c a)^2"
(is "\<exists>k>0. \<forall>a b c. _ \<longrightarrow> ?normle a b c")
proof -
have le_of_3: "\<And>a x y z. \<lbrakk>0 \<le> x*y; 0 \<le> x*z; 0 \<le> y*z; a \<le> (e*(x + y + z))*x + (e*(x + y + z))*y + (e*(x + y + z))*z\<rbrakk>
\<Longrightarrow> a \<le> e*(x + y + z)^2"
by (simp add: algebra_simps power2_eq_square)
have disj_le: "\<lbrakk>x \<le> a \<or> x \<le> b \<or> x \<le> c; 0 \<le> a; 0 \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> x \<le> a + b + c"
for x::real and a b c
by linarith
have fabc: "f path_integrable_on linepath a b" "f path_integrable_on linepath b c" "f path_integrable_on linepath c a"
if "convex hull {a, b, c} \<subseteq> s" for a b c
using segments_subset_convex_hull that
by (metis continuous_on_subset f path_integrable_continuous_linepath)+
note path_bound = has_path_integral_bound_linepath [simplified norm_minus_commute, OF has_path_integral_integral]
{ fix f' a b c d
assume d: "0 < d"
and f': "\<And>y. \<lbrakk>cmod (y - x) \<le> d; y \<in> s\<rbrakk> \<Longrightarrow> cmod (f y - f x - f' * (y - x)) \<le> e * cmod (y - x)"
and le: "cmod (a - b) \<le> d" "cmod (b - c) \<le> d" "cmod (c - a) \<le> d"
and xc: "x \<in> convex hull {a, b, c}"
and s: "convex hull {a, b, c} \<subseteq> s"
have pa: "path_integral (linepath a b) f + path_integral (linepath b c) f + path_integral (linepath c a) f =
path_integral (linepath a b) (\<lambda>y. f y - f x - f'*(y - x)) +
path_integral (linepath b c) (\<lambda>y. f y - f x - f'*(y - x)) +
path_integral (linepath c a) (\<lambda>y. f y - f x - f'*(y - x))"
apply (simp add: path_integral_diff path_integral_lmul path_integrable_lmul path_integrable_diff fabc [OF s])
apply (simp add: field_simps)
done
{ fix y
assume yc: "y \<in> convex hull {a,b,c}"
have "cmod (f y - f x - f' * (y - x)) \<le> e*norm(y - x)"
apply (rule f')
apply (metis triangle_points_closer [OF xc yc] le norm_minus_commute order_trans)
using s yc by blast
also have "... \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))"
by (simp add: yc e xc disj_le [OF triangle_points_closer])
finally have "cmod (f y - f x - f' * (y - x)) \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" .
} note cm_le = this
have "?normle a b c"
apply (simp add: dist_norm pa)
apply (rule le_of_3)
using f' xc s e
apply simp_all
apply (intro norm_triangle_le add_mono path_bound)
apply (simp_all add: path_integral_diff path_integral_lmul path_integrable_lmul path_integrable_diff fabc)
apply (blast intro: cm_le elim: dest: segments_subset_convex_hull [THEN subsetD])+
done
} note * = this
show ?thesis
using cd e
apply (simp add: complex_differentiable_def has_field_derivative_def has_derivative_within_alt approachable_lt_le2 Ball_def)
apply (clarify dest!: spec mp)
using *
apply (simp add: dist_norm, blast)
done
qed
(* Hence the most basic theorem for a triangle.*)
locale Chain =
fixes x0 At Follows
assumes At0: "At x0 0"
and AtSuc: "\<And>x n. At x n \<Longrightarrow> \<exists>x'. At x' (Suc n) \<and> Follows x' x"
begin
primrec f where
"f 0 = x0"
| "f (Suc n) = (SOME x. At x (Suc n) \<and> Follows x (f n))"
lemma At: "At (f n) n"
proof (induct n)
case 0 show ?case
by (simp add: At0)
next
case (Suc n) show ?case
by (metis (no_types, lifting) AtSuc [OF Suc] f.simps(2) someI_ex)
qed
lemma Follows: "Follows (f(Suc n)) (f n)"
by (metis (no_types, lifting) AtSuc [OF At [of n]] f.simps(2) someI_ex)
declare f.simps(2) [simp del]
end
lemma Chain3:
assumes At0: "At x0 y0 z0 0"
and AtSuc: "\<And>x y z n. At x y z n \<Longrightarrow> \<exists>x' y' z'. At x' y' z' (Suc n) \<and> Follows x' y' z' x y z"
obtains f g h where
"f 0 = x0" "g 0 = y0" "h 0 = z0"
"\<And>n. At (f n) (g n) (h n) n"
"\<And>n. Follows (f(Suc n)) (g(Suc n)) (h(Suc n)) (f n) (g n) (h n)"
proof -
interpret three: Chain "(x0,y0,z0)" "\<lambda>(x,y,z). At x y z" "\<lambda>(x',y',z'). \<lambda>(x,y,z). Follows x' y' z' x y z"
apply unfold_locales
using At0 AtSuc by auto
show ?thesis
apply (rule that [of "\<lambda>n. fst (three.f n)" "\<lambda>n. fst (snd (three.f n))" "\<lambda>n. snd (snd (three.f n))"])
apply simp_all
using three.At three.Follows
apply (simp_all add: split_beta')
done
qed
lemma Cauchy_theorem_triangle:
assumes "f holomorphic_on (convex hull {a,b,c})"
shows "(f has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
proof -
have contf: "continuous_on (convex hull {a,b,c}) f"
by (metis assms holomorphic_on_imp_continuous_on)
let ?pathint = "\<lambda>x y. path_integral(linepath x y) f"
{ fix y::complex
assume fy: "(f has_path_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
and ynz: "y \<noteq> 0"
def K \<equiv> "1 + max (dist a b) (max (dist b c) (dist c a))"
def e \<equiv> "norm y / K^2"
have K1: "K \<ge> 1" by (simp add: K_def max.coboundedI1)
then have K: "K > 0" by linarith
have [iff]: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
by (simp_all add: K_def)
have e: "e > 0"
unfolding e_def using ynz K1 by simp
def At \<equiv> "\<lambda>x y z n. convex hull {x,y,z} \<subseteq> convex hull {a,b,c} \<and>
dist x y \<le> K/2^n \<and> dist y z \<le> K/2^n \<and> dist z x \<le> K/2^n \<and>
norm(?pathint x y + ?pathint y z + ?pathint z x) \<ge> e*(K/2^n)^2"
have At0: "At a b c 0"
using fy
by (simp add: At_def e_def has_chain_integral_chain_integral3)
{ fix x y z n
assume At: "At x y z n"
then have contf': "continuous_on (convex hull {x,y,z}) f"
using contf At_def continuous_on_subset by blast
have "\<exists>x' y' z'. At x' y' z' (Suc n) \<and> convex hull {x',y',z'} \<subseteq> convex hull {x,y,z}"
using At
apply (simp add: At_def)
using Cauchy_theorem_quadrisection [OF contf', of "K/2^n" e]
apply clarsimp
apply (rule_tac x="a'" in exI)
apply (rule_tac x="b'" in exI)
apply (rule_tac x="c'" in exI)
apply (simp add: algebra_simps)
apply (meson convex_hull_subset empty_subsetI insert_subset subsetCE)
done
} note AtSuc = this
obtain fa fb fc
where f0 [simp]: "fa 0 = a" "fb 0 = b" "fc 0 = c"
and cosb: "\<And>n. convex hull {fa n, fb n, fc n} \<subseteq> convex hull {a,b,c}"
and dist: "\<And>n. dist (fa n) (fb n) \<le> K/2^n"
"\<And>n. dist (fb n) (fc n) \<le> K/2^n"
"\<And>n. dist (fc n) (fa n) \<le> K/2^n"
and no: "\<And>n. norm(?pathint (fa n) (fb n) +
?pathint (fb n) (fc n) +
?pathint (fc n) (fa n)) \<ge> e * (K/2^n)^2"
and conv_le: "\<And>n. convex hull {fa(Suc n), fb(Suc n), fc(Suc n)} \<subseteq> convex hull {fa n, fb n, fc n}"
apply (rule Chain3 [of At, OF At0 AtSuc])
apply (auto simp: At_def)
done
have "\<exists>x. \<forall>n. x \<in> convex hull {fa n, fb n, fc n}"
apply (rule bounded_closed_nest)
apply (simp_all add: compact_imp_closed finite_imp_compact_convex_hull finite_imp_bounded_convex_hull)
apply (rule allI)
apply (rule transitive_stepwise_le)
apply (auto simp: conv_le)
done
then obtain x where x: "\<And>n. x \<in> convex hull {fa n, fb n, fc n}" by auto
then have xin: "x \<in> convex hull {a,b,c}"
using assms f0 by blast
then have fx: "f complex_differentiable at x within (convex hull {a,b,c})"
using assms holomorphic_on_def by blast
{ fix k n
assume k: "0 < k"
and le:
"\<And>x' y' z'.
\<lbrakk>dist x' y' \<le> k; dist y' z' \<le> k; dist z' x' \<le> k;
x \<in> convex hull {x',y',z'};
convex hull {x',y',z'} \<subseteq> convex hull {a,b,c}\<rbrakk>
\<Longrightarrow>
cmod (?pathint x' y' + ?pathint y' z' + ?pathint z' x') * 10
\<le> e * (dist x' y' + dist y' z' + dist z' x')\<^sup>2"
and Kk: "K / k < 2 ^ n"
have "K / 2 ^ n < k" using Kk k
by (auto simp: field_simps)
then have DD: "dist (fa n) (fb n) \<le> k" "dist (fb n) (fc n) \<le> k" "dist (fc n) (fa n) \<le> k"
using dist [of n] k
by linarith+
have dle: "(dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2
\<le> (3 * K / 2 ^ n)\<^sup>2"
using dist [of n] e K
by (simp add: abs_le_square_iff [symmetric])
have less10: "\<And>x y::real. 0 < x \<Longrightarrow> y \<le> 9*x \<Longrightarrow> y < x*10"
by linarith
have "e * (dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2 \<le> e * (3 * K / 2 ^ n)\<^sup>2"
using ynz dle e mult_le_cancel_left_pos by blast
also have "... <
cmod (?pathint (fa n) (fb n) + ?pathint (fb n) (fc n) + ?pathint (fc n) (fa n)) * 10"
using no [of n] e K
apply (simp add: e_def field_simps)
apply (simp only: zero_less_norm_iff [symmetric])
done
finally have False
using le [OF DD x cosb] by auto
} then
have ?thesis
using holomorphic_point_small_triangle [OF xin contf fx, of "e/10"] e
apply clarsimp
apply (rule_tac x1="K/k" in exE [OF real_arch_pow2], blast)
done
}
moreover have "f path_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
by simp (meson contf continuous_on_subset path_integrable_continuous_linepath segments_subset_convex_hull(1)
segments_subset_convex_hull(3) segments_subset_convex_hull(5))
ultimately show ?thesis
using has_path_integral_integral by fastforce
qed
subsection\<open>Version needing function holomorphic in interior only\<close>
lemma Cauchy_theorem_flat_lemma:
assumes f: "continuous_on (convex hull {a,b,c}) f"
and c: "c - a = k *\<^sub>R (b - a)"
and k: "0 \<le> k"
shows "path_integral (linepath a b) f + path_integral (linepath b c) f +
path_integral (linepath c a) f = 0"
proof -
have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
using f continuous_on_subset segments_subset_convex_hull by metis+
show ?thesis
proof (cases "k \<le> 1")
case True show ?thesis
by (simp add: path_integral_split [OF fabc(1) k True c] path_integral_reverse_linepath fabc)
next
case False then show ?thesis
using fabc c
apply (subst path_integral_split [of a c f "1/k" b, symmetric])
apply (metis closed_segment_commute fabc(3))
apply (auto simp: k path_integral_reverse_linepath)
done
qed
qed
lemma Cauchy_theorem_flat:
assumes f: "continuous_on (convex hull {a,b,c}) f"
and c: "c - a = k *\<^sub>R (b - a)"
shows "path_integral (linepath a b) f +
path_integral (linepath b c) f +
path_integral (linepath c a) f = 0"
proof (cases "0 \<le> k")
case True with assms show ?thesis
by (blast intro: Cauchy_theorem_flat_lemma)
next
case False
have "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
using f continuous_on_subset segments_subset_convex_hull by metis+
moreover have "path_integral (linepath b a) f + path_integral (linepath a c) f +
path_integral (linepath c b) f = 0"
apply (rule Cauchy_theorem_flat_lemma [of b a c f "1-k"])
using False c
apply (auto simp: f insert_commute scaleR_conv_of_real algebra_simps)
done
ultimately show ?thesis
apply (auto simp: path_integral_reverse_linepath)
using add_eq_0_iff by force
qed
lemma Cauchy_theorem_triangle_interior:
assumes contf: "continuous_on (convex hull {a,b,c}) f"
and holf: "f holomorphic_on interior (convex hull {a,b,c})"
shows "(f has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
proof -
have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
using contf continuous_on_subset segments_subset_convex_hull by metis+
have "bounded (f ` (convex hull {a,b,c}))"
by (simp add: compact_continuous_image compact_convex_hull compact_imp_bounded contf)
then obtain B where "0 < B" and Bnf: "\<And>x. x \<in> convex hull {a,b,c} \<Longrightarrow> norm (f x) \<le> B"
by (auto simp: dest!: bounded_pos [THEN iffD1])
have "bounded (convex hull {a,b,c})"
by (simp add: bounded_convex_hull)
then obtain C where C: "0 < C" and Cno: "\<And>y. y \<in> convex hull {a,b,c} \<Longrightarrow> norm y < C"
using bounded_pos_less by blast
then have diff_2C: "norm(x - y) \<le> 2*C"
if x: "x \<in> convex hull {a, b, c}" and y: "y \<in> convex hull {a, b, c}" for x y
proof -
have "cmod x \<le> C"
using x by (meson Cno not_le not_less_iff_gr_or_eq)
hence "cmod (x - y) \<le> C + C"
using y by (meson Cno add_mono_thms_linordered_field(4) less_eq_real_def norm_triangle_ineq4 order_trans)
thus "cmod (x - y) \<le> 2 * C"
by (metis mult_2)
qed
have contf': "continuous_on (convex hull {b,a,c}) f"
using contf by (simp add: insert_commute)
{ fix y::complex
assume fy: "(f has_path_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
and ynz: "y \<noteq> 0"
have pi_eq_y: "path_integral (linepath a b) f + path_integral (linepath b c) f + path_integral (linepath c a) f = y"
by (rule has_chain_integral_chain_integral3 [OF fy])
have ?thesis
proof (cases "c=a \<or> a=b \<or> b=c")
case True then show ?thesis
using Cauchy_theorem_flat [OF contf, of 0]
using has_chain_integral_chain_integral3 [OF fy] ynz
by (force simp: fabc path_integral_reverse_linepath)
next
case False
then have car3: "card {a, b, c} = Suc (DIM(complex))"
by auto
{ assume "interior(convex hull {a,b,c}) = {}"
then have "collinear{a,b,c}"
using interior_convex_hull_eq_empty [OF car3]
by (simp add: collinear_3_eq_affine_dependent)
then have "False"
using False
apply (clarsimp simp add: collinear_3 collinear_lemma)
apply (drule Cauchy_theorem_flat [OF contf'])
using pi_eq_y ynz
apply (simp add: fabc add_eq_0_iff path_integral_reverse_linepath)
done
}
then obtain d where d: "d \<in> interior (convex hull {a, b, c})"
by blast
{ fix d1
assume d1_pos: "0 < d1"
and d1: "\<And>x x'. \<lbrakk>x\<in>convex hull {a, b, c}; x'\<in>convex hull {a, b, c}; cmod (x' - x) < d1\<rbrakk>
\<Longrightarrow> cmod (f x' - f x) < cmod y / (24 * C)"
def e \<equiv> "min 1 (min (d1/(4*C)) ((norm y / 24 / C) / B))"
def shrink \<equiv> "\<lambda>x. x - e *\<^sub>R (x - d)"
let ?pathint = "\<lambda>x y. path_integral(linepath x y) f"
have e: "0 < e" "e \<le> 1" "e \<le> d1 / (4 * C)" "e \<le> cmod y / 24 / C / B"
using d1_pos `C>0` `B>0` ynz by (simp_all add: e_def)
then have eCB: "24 * e * C * B \<le> cmod y"
using `C>0` `B>0` by (simp add: field_simps)
have e_le_d1: "e * (4 * C) \<le> d1"
using e `C>0` by (simp add: field_simps)
have "shrink a \<in> interior(convex hull {a,b,c})"
"shrink b \<in> interior(convex hull {a,b,c})"
"shrink c \<in> interior(convex hull {a,b,c})"
using d e by (auto simp: hull_inc mem_interior_convex_shrink shrink_def)
then have fhp0: "(f has_path_integral 0)
(linepath (shrink a) (shrink b) +++ linepath (shrink b) (shrink c) +++ linepath (shrink c) (shrink a))"
by (simp add: Cauchy_theorem_triangle holomorphic_on_subset [OF holf] hull_minimal convex_interior)
then have f_0_shrink: "?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a) = 0"
by (simp add: has_chain_integral_chain_integral3)
have fpi_abc: "f path_integrable_on linepath (shrink a) (shrink b)"
"f path_integrable_on linepath (shrink b) (shrink c)"
"f path_integrable_on linepath (shrink c) (shrink a)"
using fhp0 by (auto simp: valid_path_join dest: has_path_integral_integrable)
have cmod_shr: "\<And>x y. cmod (shrink y - shrink x - (y - x)) = e * cmod (x - y)"
using e by (simp add: shrink_def real_vector.scale_right_diff_distrib [symmetric])
have sh_eq: "\<And>a b d::complex. (b - e *\<^sub>R (b - d)) - (a - e *\<^sub>R (a - d)) - (b - a) = e *\<^sub>R (a - b)"
by (simp add: algebra_simps)
have "cmod y / (24 * C) \<le> cmod y / cmod (b - a) / 12"
using False `C>0` diff_2C [of b a] ynz
by (auto simp: divide_simps hull_inc)
have less_C: "\<lbrakk>u \<in> convex hull {a, b, c}; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> x * cmod u < C" for x u
apply (cases "x=0", simp add: `0<C`)
using Cno [of u] mult_left_le_one_le [of "cmod u" x] le_less_trans norm_ge_zero by blast
{ fix u v
assume uv: "u \<in> convex hull {a, b, c}" "v \<in> convex hull {a, b, c}" "u\<noteq>v"
and fpi_uv: "f path_integrable_on linepath (shrink u) (shrink v)"
have shr_uv: "shrink u \<in> interior(convex hull {a,b,c})"
"shrink v \<in> interior(convex hull {a,b,c})"
using d e uv
by (auto simp: hull_inc mem_interior_convex_shrink shrink_def)
have cmod_fuv: "\<And>x. 0\<le>x \<Longrightarrow> x\<le>1 \<Longrightarrow> cmod (f (linepath (shrink u) (shrink v) x)) \<le> B"
using shr_uv by (blast intro: Bnf linepath_in_convex_hull interior_subset [THEN subsetD])
have By_uv: "B * (12 * (e * cmod (u - v))) \<le> cmod y"
apply (rule order_trans [OF _ eCB])
using e `B>0` diff_2C [of u v] uv
by (auto simp: field_simps)
{ fix x::real assume x: "0\<le>x" "x\<le>1"
have cmod_less_4C: "cmod ((1 - x) *\<^sub>R u - (1 - x) *\<^sub>R d) + cmod (x *\<^sub>R v - x *\<^sub>R d) < (C+C) + (C+C)"
apply (rule add_strict_mono; rule norm_triangle_half_l [of _ 0])
using uv x d interior_subset
apply (auto simp: hull_inc intro!: less_C)
done
have ll: "linepath (shrink u) (shrink v) x - linepath u v x = -e * ((1 - x) *\<^sub>R (u - d) + x *\<^sub>R (v - d))"
by (simp add: linepath_def shrink_def algebra_simps scaleR_conv_of_real)
have cmod_less_dt: "cmod (linepath (shrink u) (shrink v) x - linepath u v x) < d1"
using `e>0`
apply (simp add: ll norm_mult scaleR_diff_right)
apply (rule less_le_trans [OF _ e_le_d1])
using cmod_less_4C
apply (force intro: norm_triangle_lt)
done
have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) < cmod y / (24 * C)"
using x uv shr_uv cmod_less_dt
by (auto simp: hull_inc intro: d1 interior_subset [THEN subsetD] linepath_in_convex_hull)
also have "... \<le> cmod y / cmod (v - u) / 12"
using False uv `C>0` diff_2C [of v u] ynz
by (auto simp: divide_simps hull_inc)
finally have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) \<le> cmod y / cmod (v - u) / 12"
by simp
then have cmod_12_le: "cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) * 12 \<le> cmod y"
using uv False by (auto simp: field_simps)
have "cmod (f (linepath (shrink u) (shrink v) x)) * cmod (shrink v - shrink u - (v - u)) +
cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x))
\<le> cmod y / 6"
apply (rule order_trans [of _ "B*((norm y / 24 / C / B)*2*C) + (2*C)*(norm y /24 / C)"])
apply (rule add_mono [OF mult_mono])
using By_uv e `0 < B` `0 < C` x ynz
apply (simp_all add: cmod_fuv cmod_shr cmod_12_le hull_inc)
apply (simp add: field_simps)
done
} note cmod_diff_le = this
have f_uv: "continuous_on (closed_segment u v) f"
by (blast intro: uv continuous_on_subset [OF contf closed_segment_subset_convex_hull])
have **: "\<And>f' x' f x::complex. f'*x' - f*x = f'*(x' - x) + x*(f' - f)"
by (simp add: algebra_simps)
have "norm (?pathint (shrink u) (shrink v) - ?pathint u v) \<le> norm y / 6"
apply (rule order_trans)
apply (rule has_integral_bound
[of "B*(norm y /24/C/B)*2*C + (2*C)*(norm y/24/C)"
"\<lambda>x. f(linepath (shrink u) (shrink v) x) * (shrink v - shrink u) - f(linepath u v x)*(v - u)"
_ 0 1 ])
using ynz `0 < B` `0 < C`
apply (simp_all del: le_divide_eq_numeral1)
apply (simp add: has_integral_sub has_path_integral_linepath [symmetric] has_path_integral_integral
fpi_uv f_uv path_integrable_continuous_linepath, clarify)
apply (simp only: **)
apply (simp add: norm_triangle_le norm_mult cmod_diff_le del: le_divide_eq_numeral1)
done
} note * = this
have "norm (?pathint (shrink a) (shrink b) - ?pathint a b) \<le> norm y / 6"
using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
moreover
have "norm (?pathint (shrink b) (shrink c) - ?pathint b c) \<le> norm y / 6"
using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
moreover
have "norm (?pathint (shrink c) (shrink a) - ?pathint c a) \<le> norm y / 6"
using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
ultimately
have "norm((?pathint (shrink a) (shrink b) - ?pathint a b) +
(?pathint (shrink b) (shrink c) - ?pathint b c) + (?pathint (shrink c) (shrink a) - ?pathint c a))
\<le> norm y / 6 + norm y / 6 + norm y / 6"
by (metis norm_triangle_le add_mono)
also have "... = norm y / 2"
by simp
finally have "norm((?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a)) -
(?pathint a b + ?pathint b c + ?pathint c a))
\<le> norm y / 2"
by (simp add: algebra_simps)
then
have "norm(?pathint a b + ?pathint b c + ?pathint c a) \<le> norm y / 2"
by (simp add: f_0_shrink) (metis (mono_tags) add.commute minus_add_distrib norm_minus_cancel uminus_add_conv_diff)
then have "False"
using pi_eq_y ynz by auto
}
moreover have "uniformly_continuous_on (convex hull {a,b,c}) f"
by (simp add: contf compact_convex_hull compact_uniformly_continuous)
ultimately have "False"
unfolding uniformly_continuous_on_def
by (force simp: ynz `0 < C` dist_norm)
then show ?thesis ..
qed
}
moreover have "f path_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
using fabc path_integrable_continuous_linepath by auto
ultimately show ?thesis
using has_path_integral_integral by fastforce
qed
subsection\<open>Version allowing finite number of exceptional points\<close>
lemma Cauchy_theorem_triangle_cofinite:
assumes "continuous_on (convex hull {a,b,c}) f"
and "finite s"
and "(\<And>x. x \<in> interior(convex hull {a,b,c}) - s \<Longrightarrow> f complex_differentiable (at x))"
shows "(f has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
using assms
proof (induction "card s" arbitrary: a b c s rule: less_induct)
case (less s a b c)
show ?case
proof (cases "s={}")
case True with less show ?thesis
by (fastforce simp: holomorphic_on_def complex_differentiable_at_within
Cauchy_theorem_triangle_interior)
next
case False
then obtain d s' where d: "s = insert d s'" "d \<notin> s'"
by (meson Set.set_insert all_not_in_conv)
then show ?thesis
proof (cases "d \<in> convex hull {a,b,c}")
case False
show "(f has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
apply (rule less.hyps [of "s'"])
using False d `finite s` interior_subset
apply (auto intro!: less.prems)
done
next
case True
have *: "convex hull {a, b, d} \<subseteq> convex hull {a, b, c}"
by (meson True hull_subset insert_subset convex_hull_subset)
have abd: "(f has_path_integral 0) (linepath a b +++ linepath b d +++ linepath d a)"
apply (rule less.hyps [of "s'"])
using True d `finite s` not_in_interior_convex_hull_3
apply (auto intro!: less.prems continuous_on_subset [OF _ *])
apply (metis * insert_absorb insert_subset interior_mono)
done
have *: "convex hull {b, c, d} \<subseteq> convex hull {a, b, c}"
by (meson True hull_subset insert_subset convex_hull_subset)
have bcd: "(f has_path_integral 0) (linepath b c +++ linepath c d +++ linepath d b)"
apply (rule less.hyps [of "s'"])
using True d `finite s` not_in_interior_convex_hull_3
apply (auto intro!: less.prems continuous_on_subset [OF _ *])
apply (metis * insert_absorb insert_subset interior_mono)
done
have *: "convex hull {c, a, d} \<subseteq> convex hull {a, b, c}"
by (meson True hull_subset insert_subset convex_hull_subset)
have cad: "(f has_path_integral 0) (linepath c a +++ linepath a d +++ linepath d c)"
apply (rule less.hyps [of "s'"])
using True d `finite s` not_in_interior_convex_hull_3
apply (auto intro!: less.prems continuous_on_subset [OF _ *])
apply (metis * insert_absorb insert_subset interior_mono)
done
have "f path_integrable_on linepath a b"
using less.prems
by (metis continuous_on_subset insert_commute path_integrable_continuous_linepath segments_subset_convex_hull(3))
moreover have "f path_integrable_on linepath b c"
using less.prems
by (metis continuous_on_subset path_integrable_continuous_linepath segments_subset_convex_hull(3))
moreover have "f path_integrable_on linepath c a"
using less.prems
by (metis continuous_on_subset insert_commute path_integrable_continuous_linepath segments_subset_convex_hull(3))
ultimately have fpi: "f path_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
by auto
{ fix y::complex
assume fy: "(f has_path_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
and ynz: "y \<noteq> 0"
have cont_ad: "continuous_on (closed_segment a d) f"
by (meson "*" continuous_on_subset less.prems(1) segments_subset_convex_hull(3))
have cont_bd: "continuous_on (closed_segment b d) f"
by (meson True closed_segment_subset_convex_hull continuous_on_subset hull_subset insert_subset less.prems(1))
have cont_cd: "continuous_on (closed_segment c d) f"
by (meson "*" continuous_on_subset less.prems(1) segments_subset_convex_hull(2))
have "path_integral (linepath a b) f = - (path_integral (linepath b d) f + (path_integral (linepath d a) f))"
"path_integral (linepath b c) f = - (path_integral (linepath c d) f + (path_integral (linepath d b) f))"
"path_integral (linepath c a) f = - (path_integral (linepath a d) f + path_integral (linepath d c) f)"
using has_chain_integral_chain_integral3 [OF abd]
has_chain_integral_chain_integral3 [OF bcd]
has_chain_integral_chain_integral3 [OF cad]
by (simp_all add: algebra_simps add_eq_0_iff)
then have ?thesis
using cont_ad cont_bd cont_cd fy has_chain_integral_chain_integral3 path_integral_reverse_linepath by fastforce
}
then show ?thesis
using fpi path_integrable_on_def by blast
qed
qed
qed
subsection\<open>Cauchy's theorem for an open starlike set\<close>
lemma starlike_convex_subset:
assumes s: "a \<in> s" "closed_segment b c \<subseteq> s" and subs: "\<And>x. x \<in> s \<Longrightarrow> closed_segment a x \<subseteq> s"
shows "convex hull {a,b,c} \<subseteq> s"
using s
apply (clarsimp simp add: convex_hull_insert [of "{b,c}" a] segment_convex_hull)
apply (meson subs convexD convex_segment ends_in_segment(1) ends_in_segment(2) subsetCE)
done
lemma triangle_path_integrals_starlike_primitive:
assumes contf: "continuous_on s f"
and s: "a \<in> s" "open s"
and x: "x \<in> s"
and subs: "\<And>y. y \<in> s \<Longrightarrow> closed_segment a y \<subseteq> s"
and zer: "\<And>b c. closed_segment b c \<subseteq> s
\<Longrightarrow> path_integral (linepath a b) f + path_integral (linepath b c) f +
path_integral (linepath c a) f = 0"
shows "((\<lambda>x. path_integral(linepath a x) f) has_field_derivative f x) (at x)"
proof -
let ?pathint = "\<lambda>x y. path_integral(linepath x y) f"
{ fix e y
assume e: "0 < e" and bxe: "ball x e \<subseteq> s" and close: "cmod (y - x) < e"
have y: "y \<in> s"
using bxe close by (force simp: dist_norm norm_minus_commute)
have cont_ayf: "continuous_on (closed_segment a y) f"
using contf continuous_on_subset subs y by blast
have xys: "closed_segment x y \<subseteq> s"
apply (rule order_trans [OF _ bxe])
using close
by (auto simp: dist_norm ball_def norm_minus_commute dest: segment_bound)
have "?pathint a y - ?pathint a x = ?pathint x y"
using zer [OF xys] path_integral_reverse_linepath [OF cont_ayf] add_eq_0_iff by force
} note [simp] = this
{ fix e::real
assume e: "0 < e"
have cont_atx: "continuous (at x) f"
using x s contf continuous_on_eq_continuous_at by blast
then obtain d1 where d1: "d1>0" and d1_less: "\<And>y. cmod (y - x) < d1 \<Longrightarrow> cmod (f y - f x) < e/2"
unfolding continuous_at Lim_at dist_norm using e
by (drule_tac x="e/2" in spec) force
obtain d2 where d2: "d2>0" "ball x d2 \<subseteq> s" using `open s` x
by (auto simp: open_contains_ball)
have dpos: "min d1 d2 > 0" using d1 d2 by simp
{ fix y
assume yx: "y \<noteq> x" and close: "cmod (y - x) < min d1 d2"
have y: "y \<in> s"
using d2 close by (force simp: dist_norm norm_minus_commute)
have fxy: "f path_integrable_on linepath x y"
apply (rule path_integrable_continuous_linepath)
apply (rule continuous_on_subset [OF contf])
using close d2
apply (auto simp: dist_norm norm_minus_commute dest!: segment_bound(1))
done
then obtain i where i: "(f has_path_integral i) (linepath x y)"
by (auto simp: path_integrable_on_def)
then have "((\<lambda>w. f w - f x) has_path_integral (i - f x * (y - x))) (linepath x y)"
by (rule has_path_integral_diff [OF _ has_path_integral_const_linepath])
then have "cmod (i - f x * (y - x)) \<le> e / 2 * cmod (y - x)"
apply (rule has_path_integral_bound_linepath [where B = "e/2"])
using e apply simp
apply (rule d1_less [THEN less_imp_le])
using close segment_bound
apply force
done
also have "... < e * cmod (y - x)"
by (simp add: e yx)
finally have "cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
using i yx by (simp add: path_integral_unique divide_less_eq)
}
then have "\<exists>d>0. \<forall>y. y \<noteq> x \<and> cmod (y-x) < d \<longrightarrow> cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
using dpos by blast
}
then have *: "(\<lambda>y. (?pathint x y - f x * (y - x)) /\<^sub>R cmod (y - x)) -- x --> 0"
by (simp add: Lim_at dist_norm inverse_eq_divide)
show ?thesis
apply (simp add: has_field_derivative_def has_derivative_at bounded_linear_mult_right)
apply (rule Lim_transform [OF * Lim_eventually])
apply (simp add: inverse_eq_divide [symmetric] eventually_at)
using `open s` x
apply (force simp: dist_norm open_contains_ball)
done
qed
(** Existence of a primitive.*)
lemma holomorphic_starlike_primitive:
assumes contf: "continuous_on s f"
and s: "starlike s" and os: "open s"
and k: "finite k"
and fcd: "\<And>x. x \<in> s - k \<Longrightarrow> f complex_differentiable at x"
shows "\<exists>g. \<forall>x \<in> s. (g has_field_derivative f x) (at x)"
proof -
obtain a where a: "a\<in>s" and a_cs: "\<And>x. x\<in>s \<Longrightarrow> closed_segment a x \<subseteq> s"
using s by (auto simp: starlike_def)
{ fix x b c
assume "x \<in> s" "closed_segment b c \<subseteq> s"
then have abcs: "convex hull {a, b, c} \<subseteq> s"
by (simp add: a a_cs starlike_convex_subset)
then have *: "continuous_on (convex hull {a, b, c}) f"
by (simp add: continuous_on_subset [OF contf])
have "(f has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
apply (rule Cauchy_theorem_triangle_cofinite [OF _ k])
using abcs apply (simp add: continuous_on_subset [OF contf])
using * abcs interior_subset apply (auto intro: fcd)
done
} note 0 = this
show ?thesis
apply (intro exI ballI)
apply (rule triangle_path_integrals_starlike_primitive [OF contf a os], assumption)
apply (metis a_cs)
apply (metis has_chain_integral_chain_integral3 0)
done
qed
lemma Cauchy_theorem_starlike:
"\<lbrakk>open s; starlike s; finite k; continuous_on s f;
\<And>x. x \<in> s - k \<Longrightarrow> f complex_differentiable at x;
valid_path g; path_image g \<subseteq> s; pathfinish g = pathstart g\<rbrakk>
\<Longrightarrow> (f has_path_integral 0) g"
by (metis holomorphic_starlike_primitive Cauchy_theorem_primitive at_within_open)
lemma Cauchy_theorem_starlike_simple:
"\<lbrakk>open s; starlike s; f holomorphic_on s; valid_path g; path_image g \<subseteq> s; pathfinish g = pathstart g\<rbrakk>
\<Longrightarrow> (f has_path_integral 0) g"
apply (rule Cauchy_theorem_starlike [OF _ _ finite.emptyI])
apply (simp_all add: holomorphic_on_imp_continuous_on)
apply (metis at_within_open holomorphic_on_def)
done
subsection\<open>Cauchy's theorem for a convex set\<close>
text\<open>For a convex set we can avoid assuming openness and boundary analyticity\<close>
lemma triangle_path_integrals_convex_primitive:
assumes contf: "continuous_on s f"
and s: "a \<in> s" "convex s"
and x: "x \<in> s"
and zer: "\<And>b c. \<lbrakk>b \<in> s; c \<in> s\<rbrakk>
\<Longrightarrow> path_integral (linepath a b) f + path_integral (linepath b c) f +
path_integral (linepath c a) f = 0"
shows "((\<lambda>x. path_integral(linepath a x) f) has_field_derivative f x) (at x within s)"
proof -
let ?pathint = "\<lambda>x y. path_integral(linepath x y) f"
{ fix y
assume y: "y \<in> s"
have cont_ayf: "continuous_on (closed_segment a y) f"
using s y by (meson contf continuous_on_subset convex_contains_segment)
have xys: "closed_segment x y \<subseteq> s" (*?*)
using convex_contains_segment s x y by auto
have "?pathint a y - ?pathint a x = ?pathint x y"
using zer [OF x y] path_integral_reverse_linepath [OF cont_ayf] add_eq_0_iff by force
} note [simp] = this
{ fix e::real
assume e: "0 < e"
have cont_atx: "continuous (at x within s) f"
using x s contf by (simp add: continuous_on_eq_continuous_within)
then obtain d1 where d1: "d1>0" and d1_less: "\<And>y. \<lbrakk>y \<in> s; cmod (y - x) < d1\<rbrakk> \<Longrightarrow> cmod (f y - f x) < e/2"
unfolding continuous_within Lim_within dist_norm using e
by (drule_tac x="e/2" in spec) force
{ fix y
assume yx: "y \<noteq> x" and close: "cmod (y - x) < d1" and y: "y \<in> s"
have fxy: "f path_integrable_on linepath x y"
using convex_contains_segment s x y
by (blast intro!: path_integrable_continuous_linepath continuous_on_subset [OF contf])
then obtain i where i: "(f has_path_integral i) (linepath x y)"
by (auto simp: path_integrable_on_def)
then have "((\<lambda>w. f w - f x) has_path_integral (i - f x * (y - x))) (linepath x y)"
by (rule has_path_integral_diff [OF _ has_path_integral_const_linepath])
then have "cmod (i - f x * (y - x)) \<le> e / 2 * cmod (y - x)"
apply (rule has_path_integral_bound_linepath [where B = "e/2"])
using e apply simp
apply (rule d1_less [THEN less_imp_le])
using convex_contains_segment s(2) x y apply blast
using close segment_bound(1) apply fastforce
done
also have "... < e * cmod (y - x)"
by (simp add: e yx)
finally have "cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
using i yx by (simp add: path_integral_unique divide_less_eq)
}
then have "\<exists>d>0. \<forall>y\<in>s. y \<noteq> x \<and> cmod (y-x) < d \<longrightarrow> cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
using d1 by blast
}
then have *: "((\<lambda>y. (path_integral (linepath x y) f - f x * (y - x)) /\<^sub>R cmod (y - x)) ---> 0) (at x within s)"
by (simp add: Lim_within dist_norm inverse_eq_divide)
show ?thesis
apply (simp add: has_field_derivative_def has_derivative_within bounded_linear_mult_right)
apply (rule Lim_transform [OF * Lim_eventually])
using linordered_field_no_ub
apply (force simp: inverse_eq_divide [symmetric] eventually_at)
done
qed
lemma pathintegral_convex_primitive:
"\<lbrakk>convex s; continuous_on s f;
\<And>a b c. \<lbrakk>a \<in> s; b \<in> s; c \<in> s\<rbrakk> \<Longrightarrow> (f has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)\<rbrakk>
\<Longrightarrow> \<exists>g. \<forall>x \<in> s. (g has_field_derivative f x) (at x within s)"
apply (cases "s={}")
apply (simp_all add: ex_in_conv [symmetric])
apply (blast intro: triangle_path_integrals_convex_primitive has_chain_integral_chain_integral3)
done
lemma holomorphic_convex_primitive:
"\<lbrakk>convex s; finite k; continuous_on s f;
\<And>x. x \<in> interior s - k \<Longrightarrow> f complex_differentiable at x\<rbrakk>
\<Longrightarrow> \<exists>g. \<forall>x \<in> s. (g has_field_derivative f x) (at x within s)"
apply (rule pathintegral_convex_primitive [OF _ _ Cauchy_theorem_triangle_cofinite])
prefer 3
apply (erule continuous_on_subset)
apply (simp add: subset_hull continuous_on_subset, assumption+)
by (metis Diff_iff convex_contains_segment insert_absorb insert_subset interior_mono segment_convex_hull subset_hull)
lemma Cauchy_theorem_convex:
"\<lbrakk>continuous_on s f;convex s; finite k;
\<And>x. x \<in> interior s - k \<Longrightarrow> f complex_differentiable at x;
valid_path g; path_image g \<subseteq> s;
pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_path_integral 0) g"
by (metis holomorphic_convex_primitive Cauchy_theorem_primitive)
lemma Cauchy_theorem_convex_simple:
"\<lbrakk>f holomorphic_on s; convex s;
valid_path g; path_image g \<subseteq> s;
pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_path_integral 0) g"
apply (rule Cauchy_theorem_convex)
apply (simp_all add: holomorphic_on_imp_continuous_on)
apply (rule finite.emptyI)
using at_within_interior holomorphic_on_def interior_subset by fastforce
text\<open>In particular for a disc\<close>
lemma Cauchy_theorem_disc:
"\<lbrakk>finite k; continuous_on (cball a e) f;
\<And>x. x \<in> ball a e - k \<Longrightarrow> f complex_differentiable at x;
valid_path g; path_image g \<subseteq> cball a e;
pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_path_integral 0) g"
apply (rule Cauchy_theorem_convex)
apply (auto simp: convex_cball interior_cball)
done
lemma Cauchy_theorem_disc_simple:
"\<lbrakk>f holomorphic_on (ball a e); valid_path g; path_image g \<subseteq> ball a e;
pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_path_integral 0) g"
by (simp add: Cauchy_theorem_convex_simple)
subsection\<open>Generalize integrability to local primitives\<close>
lemma path_integral_local_primitive_lemma:
fixes f :: "complex\<Rightarrow>complex"
shows
"\<lbrakk>g piecewise_differentiable_on {a..b};
\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s);
\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s\<rbrakk>
\<Longrightarrow> (\<lambda>x. f' (g x) * vector_derivative g (at x within {a..b}))
integrable_on {a..b}"
apply (cases "cbox a b = {}", force)
apply (simp add: integrable_on_def)
apply (rule exI)
apply (rule path_integral_primitive_lemma, assumption+)
using atLeastAtMost_iff by blast
lemma path_integral_local_primitive_any:
fixes f :: "complex \<Rightarrow> complex"
assumes gpd: "g piecewise_differentiable_on {a..b}"
and dh: "\<And>x. x \<in> s
\<Longrightarrow> \<exists>d h. 0 < d \<and>
(\<forall>y. norm(y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
and gs: "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
shows "(\<lambda>x. f(g x) * vector_derivative g (at x)) integrable_on {a..b}"
proof -
{ fix x
assume x: "a \<le> x" "x \<le> b"
obtain d h where d: "0 < d"
and h: "(\<And>y. norm(y - g x) < d \<Longrightarrow> (h has_field_derivative f y) (at y within s))"
using x gs dh by (metis atLeastAtMost_iff)
have "continuous_on {a..b} g" using gpd piecewise_differentiable_on_def by blast
then obtain e where e: "e>0" and lessd: "\<And>x'. x' \<in> {a..b} \<Longrightarrow> \<bar>x' - x\<bar> < e \<Longrightarrow> cmod (g x' - g x) < d"
using x d
apply (auto simp: dist_norm continuous_on_iff)
apply (drule_tac x=x in bspec)
using x apply simp
apply (drule_tac x=d in spec, auto)
done
have "\<exists>d>0. \<forall>u v. u \<le> x \<and> x \<le> v \<and> {u..v} \<subseteq> ball x d \<and> (u \<le> v \<longrightarrow> a \<le> u \<and> v \<le> b) \<longrightarrow>
(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {u..v}"
apply (rule_tac x=e in exI)
using e
apply (simp add: integrable_on_localized_vector_derivative [symmetric], clarify)
apply (rule_tac f = h and s = "g ` {u..v}" in path_integral_local_primitive_lemma)
apply (meson atLeastatMost_subset_iff gpd piecewise_differentiable_on_subset)
apply (force simp: ball_def dist_norm intro: lessd gs DERIV_subset [OF h], force)
done
} then
show ?thesis
by (force simp: intro!: integrable_on_little_subintervals [of a b, simplified])
qed
lemma path_integral_local_primitive:
fixes f :: "complex \<Rightarrow> complex"
assumes g: "valid_path g" "path_image g \<subseteq> s"
and dh: "\<And>x. x \<in> s
\<Longrightarrow> \<exists>d h. 0 < d \<and>
(\<forall>y. norm(y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
shows "f path_integrable_on g"
using g
apply (simp add: valid_path_def path_image_def path_integrable_on_def has_path_integral_def
has_integral_localized_vector_derivative integrable_on_def [symmetric])
using path_integral_local_primitive_any [OF _ dh]
by (meson image_subset_iff piecewise_C1_imp_differentiable)
text\<open>In particular if a function is holomorphic\<close>
lemma path_integrable_holomorphic:
assumes contf: "continuous_on s f"
and os: "open s"
and k: "finite k"
and g: "valid_path g" "path_image g \<subseteq> s"
and fcd: "\<And>x. x \<in> s - k \<Longrightarrow> f complex_differentiable at x"
shows "f path_integrable_on g"
proof -
{ fix z
assume z: "z \<in> s"
obtain d where d: "d>0" "ball z d \<subseteq> s" using `open s` z
by (auto simp: open_contains_ball)
then have contfb: "continuous_on (ball z d) f"
using contf continuous_on_subset by blast
obtain h where "\<forall>y\<in>ball z d. (h has_field_derivative f y) (at y within ball z d)"
using holomorphic_convex_primitive [OF convex_ball k contfb fcd] d
interior_subset by force
then have "\<forall>y\<in>ball z d. (h has_field_derivative f y) (at y within s)"
by (metis Topology_Euclidean_Space.open_ball at_within_open d(2) os subsetCE)
then have "\<exists>h. (\<forall>y. cmod (y - z) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
by (force simp: dist_norm norm_minus_commute)
then have "\<exists>d h. 0 < d \<and> (\<forall>y. cmod (y - z) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
using d by blast
}
then show ?thesis
by (rule path_integral_local_primitive [OF g])
qed
lemma path_integrable_holomorphic_simple:
assumes contf: "continuous_on s f"
and os: "open s"
and g: "valid_path g" "path_image g \<subseteq> s"
and fh: "f holomorphic_on s"
shows "f path_integrable_on g"
apply (rule path_integrable_holomorphic [OF contf os Finite_Set.finite.emptyI g])
using fh by (simp add: complex_differentiable_def holomorphic_on_open os)
lemma continuous_on_inversediff:
fixes z:: "'a::real_normed_field" shows "z \<notin> s \<Longrightarrow> continuous_on s (\<lambda>w. 1 / (w - z))"
by (rule continuous_intros | force)+
corollary path_integrable_inversediff:
"\<lbrakk>valid_path g; z \<notin> path_image g\<rbrakk> \<Longrightarrow> (\<lambda>w. 1 / (w-z)) path_integrable_on g"
apply (rule path_integrable_holomorphic_simple [of "UNIV-{z}", OF continuous_on_inversediff])
apply (auto simp: holomorphic_on_open open_delete intro!: derivative_eq_intros)
done
text{*Key fact that path integral is the same for a "nearby" path. This is the
main lemma for the homotopy form of Cauchy's theorem and is also useful
if we want "without loss of generality" to assume some nice properties of a
path (e.g. smoothness). It can also be used to define the integrals of
analytic functions over arbitrary continuous paths. This is just done for
winding numbers now.
*}
text{*This formulation covers two cases: @{term g} and @{term h} share their
start and end points; @{term g} and @{term h} both loop upon themselves. *}
lemma path_integral_nearby:
assumes os: "open s"
and p: "path p" "path_image p \<subseteq> s"
shows
"\<exists>d. 0 < d \<and>
(\<forall>g h. valid_path g \<and> valid_path h \<and>
(\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
(if Ends then pathstart h = pathstart g \<and> pathfinish h = pathfinish g
else pathfinish g = pathstart g \<and> pathfinish h = pathstart h)
\<longrightarrow> path_image g \<subseteq> s \<and> path_image h \<subseteq> s \<and>
(\<forall>f. f holomorphic_on s \<longrightarrow> path_integral h f = path_integral g f))"
proof -
have "\<forall>z. \<exists>e. z \<in> path_image p \<longrightarrow> 0 < e \<and> ball z e \<subseteq> s"
using open_contains_ball os p(2) by blast
then obtain ee where ee: "\<And>z. z \<in> path_image p \<Longrightarrow> 0 < ee z \<and> ball z (ee z) \<subseteq> s"
by metis
def cover \<equiv> "(\<lambda>z. ball z (ee z/3)) ` (path_image p)"
have "compact (path_image p)"
by (metis p(1) compact_path_image)
moreover have "path_image p \<subseteq> (\<Union>c\<in>path_image p. ball c (ee c / 3))"
using ee by auto
ultimately have "\<exists>D \<subseteq> cover. finite D \<and> path_image p \<subseteq> \<Union>D"
by (simp add: compact_eq_heine_borel cover_def)
then obtain D where D: "D \<subseteq> cover" "finite D" "path_image p \<subseteq> \<Union>D"
by blast
then obtain k where k: "k \<subseteq> {0..1}" "finite k" and D_eq: "D = ((\<lambda>z. ball z (ee z / 3)) \<circ> p) ` k"
apply (simp add: cover_def path_image_def image_comp)
apply (blast dest!: finite_subset_image [OF `finite D`])
done
then have kne: "k \<noteq> {}"
using D by auto
have pi: "\<And>i. i \<in> k \<Longrightarrow> p i \<in> path_image p"
using k by (auto simp: path_image_def)
then have eepi: "\<And>i. i \<in> k \<Longrightarrow> 0 < ee((p i))"
by (metis ee)
def e \<equiv> "Min((ee o p) ` k)"
have fin_eep: "finite ((ee o p) ` k)"
using k by blast
have enz: "0 < e"
using ee k by (simp add: kne e_def Min_gr_iff [OF fin_eep] eepi)
have "uniformly_continuous_on {0..1} p"
using p by (simp add: path_def compact_uniformly_continuous)
then obtain d::real where d: "d>0"
and de: "\<And>x x'. \<bar>x' - x\<bar> < d \<Longrightarrow> x\<in>{0..1} \<Longrightarrow> x'\<in>{0..1} \<Longrightarrow> cmod (p x' - p x) < e/3"
unfolding uniformly_continuous_on_def dist_norm real_norm_def
by (metis divide_pos_pos enz zero_less_numeral)
then obtain N::nat where N: "N>0" "inverse N < d"
using real_arch_inv [of d] by auto
{ fix g h
assume g: "valid_path g" and gp: "\<forall>t\<in>{0..1}. cmod (g t - p t) < e / 3"
and h: "valid_path h" and hp: "\<forall>t\<in>{0..1}. cmod (h t - p t) < e / 3"
and joins: "if Ends then pathstart h = pathstart g \<and> pathfinish h = pathfinish g
else pathfinish g = pathstart g \<and> pathfinish h = pathstart h"
{ fix t::real
assume t: "0 \<le> t" "t \<le> 1"
then obtain u where u: "u \<in> k" and ptu: "p t \<in> ball(p u) (ee(p u) / 3)"
using `path_image p \<subseteq> \<Union>D` D_eq by (force simp: path_image_def)
then have ele: "e \<le> ee (p u)" using fin_eep
by (simp add: e_def)
have "cmod (g t - p t) < e / 3" "cmod (h t - p t) < e / 3"
using gp hp t by auto
with ele have "cmod (g t - p t) < ee (p u) / 3"
"cmod (h t - p t) < ee (p u) / 3"
by linarith+
then have "g t \<in> ball(p u) (ee(p u))" "h t \<in> ball(p u) (ee(p u))"
using norm_diff_triangle_ineq [of "g t" "p t" "p t" "p u"]
norm_diff_triangle_ineq [of "h t" "p t" "p t" "p u"] ptu eepi u
by (force simp: dist_norm ball_def norm_minus_commute)+
then have "g t \<in> s" "h t \<in> s" using ee u k
by (auto simp: path_image_def ball_def)
}
then have ghs: "path_image g \<subseteq> s" "path_image h \<subseteq> s"
by (auto simp: path_image_def)
moreover
{ fix f
assume fhols: "f holomorphic_on s"
then have fpa: "f path_integrable_on g" "f path_integrable_on h"
using g ghs h holomorphic_on_imp_continuous_on os path_integrable_holomorphic_simple
by blast+
have contf: "continuous_on s f"
by (simp add: fhols holomorphic_on_imp_continuous_on)
{ fix z
assume z: "z \<in> path_image p"
have "f holomorphic_on ball z (ee z)"
using fhols ee z holomorphic_on_subset by blast
then have "\<exists>ff. (\<forall>w \<in> ball z (ee z). (ff has_field_derivative f w) (at w))"
using holomorphic_convex_primitive [of "ball z (ee z)" "{}" f, simplified]
by (metis open_ball at_within_open holomorphic_on_def holomorphic_on_imp_continuous_on mem_ball)
}
then obtain ff where ff:
"\<And>z w. \<lbrakk>z \<in> path_image p; w \<in> ball z (ee z)\<rbrakk> \<Longrightarrow> (ff z has_field_derivative f w) (at w)"
by metis
{ fix n
assume n: "n \<le> N"
then have "path_integral(subpath 0 (n/N) h) f - path_integral(subpath 0 (n/N) g) f =
path_integral(linepath (g(n/N)) (h(n/N))) f - path_integral(linepath (g 0) (h 0)) f"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n)
obtain t where t: "t \<in> k" and "p (n/N) \<in> ball(p t) (ee(p t) / 3)"
using `path_image p \<subseteq> \<Union>D` [THEN subsetD, where c="p (n/N)"] D_eq N Suc.prems
by (force simp: path_image_def)
then have ptu: "cmod (p t - p (n/N)) < ee (p t) / 3"
by (simp add: dist_norm)
have e3le: "e/3 \<le> ee (p t) / 3" using fin_eep t
by (simp add: e_def)
{ fix x
assume x: "n/N \<le> x" "x \<le> (1 + n)/N"
then have nN01: "0 \<le> n/N" "(1 + n)/N \<le> 1"
using Suc.prems by auto
then have x01: "0 \<le> x" "x \<le> 1"
using x by linarith+
have "cmod (p t - p x) < ee (p t) / 3 + e/3"
apply (rule norm_diff_triangle_less [OF ptu de])
using x N x01 Suc.prems
apply (auto simp: field_simps)
done
then have ptx: "cmod (p t - p x) < 2*ee (p t)/3"
using e3le eepi [OF t] by simp
have "cmod (p t - g x) < 2*ee (p t)/3 + e/3 "
apply (rule norm_diff_triangle_less [OF ptx])
using gp x01 by (simp add: norm_minus_commute)
also have "... \<le> ee (p t)"
using e3le eepi [OF t] by simp
finally have gg: "cmod (p t - g x) < ee (p t)" .
have "cmod (p t - h x) < 2*ee (p t)/3 + e/3 "
apply (rule norm_diff_triangle_less [OF ptx])
using hp x01 by (simp add: norm_minus_commute)
also have "... \<le> ee (p t)"
using e3le eepi [OF t] by simp
finally have "cmod (p t - g x) < ee (p t)"
"cmod (p t - h x) < ee (p t)"
using gg by auto
} note ptgh_ee = this
have pi_hgn: "path_image (linepath (h (n/N)) (g (n/N))) \<subseteq> ball (p t) (ee (p t))"
using ptgh_ee [of "n/N"] Suc.prems
by (auto simp: field_simps real_of_nat_def dist_norm dest: segment_furthest_le [where y="p t"])
then have gh_ns: "closed_segment (g (n/N)) (h (n/N)) \<subseteq> s"
using `N>0` Suc.prems
apply (simp add: real_of_nat_def path_image_join field_simps closed_segment_commute)
apply (erule order_trans)
apply (simp add: ee pi t)
done
have pi_ghn': "path_image (linepath (g ((1 + n) / N)) (h ((1 + n) / N)))
\<subseteq> ball (p t) (ee (p t))"
using ptgh_ee [of "(1+n)/N"] Suc.prems
by (auto simp: field_simps real_of_nat_def dist_norm dest: segment_furthest_le [where y="p t"])
then have gh_n's: "closed_segment (g ((1 + n) / N)) (h ((1 + n) / N)) \<subseteq> s"
using `N>0` Suc.prems ee pi t
by (auto simp: Path_Connected.path_image_join field_simps)
have pi_subset_ball:
"path_image (subpath (n/N) ((1+n) / N) g +++ linepath (g ((1+n) / N)) (h ((1+n) / N)) +++
subpath ((1+n) / N) (n/N) h +++ linepath (h (n/N)) (g (n/N)))
\<subseteq> ball (p t) (ee (p t))"
apply (intro subset_path_image_join pi_hgn pi_ghn')
using `N>0` Suc.prems
apply (auto simp: dist_norm field_simps closed_segment_eq_real_ivl ptgh_ee)
done
have pi0: "(f has_path_integral 0)
(subpath (n/ N) ((Suc n)/N) g +++ linepath(g ((Suc n) / N)) (h((Suc n) / N)) +++
subpath ((Suc n) / N) (n/N) h +++ linepath(h (n/N)) (g (n/N)))"
apply (rule Cauchy_theorem_primitive [of "ball(p t) (ee(p t))" "ff (p t)" "f"])
apply (metis ff open_ball at_within_open pi t)
apply (intro valid_path_join)
using Suc.prems pi_subset_ball apply (simp_all add: valid_path_subpath g h)
done
have fpa1: "f path_integrable_on subpath (real n / real N) (real (Suc n) / real N) g"
using Suc.prems by (simp add: path_integrable_subpath g fpa)
have fpa2: "f path_integrable_on linepath (g (real (Suc n) / real N)) (h (real (Suc n) / real N))"
using gh_n's
by (auto intro!: path_integrable_continuous_linepath continuous_on_subset [OF contf])
have fpa3: "f path_integrable_on linepath (h (real n / real N)) (g (real n / real N))"
using gh_ns
by (auto simp: closed_segment_commute intro!: path_integrable_continuous_linepath continuous_on_subset [OF contf])
have eq0: "path_integral (subpath (n/N) ((Suc n) / real N) g) f +
path_integral (linepath (g ((Suc n) / N)) (h ((Suc n) / N))) f +
path_integral (subpath ((Suc n) / N) (n/N) h) f +
path_integral (linepath (h (n/N)) (g (n/N))) f = 0"
using path_integral_unique [OF pi0] Suc.prems
by (simp add: g h fpa valid_path_subpath path_integrable_subpath
fpa1 fpa2 fpa3 algebra_simps)
have *: "\<And>hn he hn' gn gd gn' hgn ghn gh0 ghn'.
\<lbrakk>hn - gn = ghn - gh0;
gd + ghn' + he + hgn = (0::complex);
hn - he = hn'; gn + gd = gn'; hgn = -ghn\<rbrakk> \<Longrightarrow> hn' - gn' = ghn' - gh0"
by (auto simp: algebra_simps)
have "path_integral (subpath 0 (n/N) h) f - path_integral (subpath ((Suc n) / N) (n/N) h) f =
path_integral (subpath 0 (n/N) h) f + path_integral (subpath (n/N) ((Suc n) / N) h) f"
unfolding reversepath_subpath [symmetric, of "((Suc n) / N)"]
using Suc.prems by (simp add: h fpa path_integral_reversepath valid_path_subpath path_integrable_subpath)
also have "... = path_integral (subpath 0 ((Suc n) / N) h) f"
using Suc.prems by (simp add: path_integral_subpath_combine h fpa)
finally have pi0_eq:
"path_integral (subpath 0 (n/N) h) f - path_integral (subpath ((Suc n) / N) (n/N) h) f =
path_integral (subpath 0 ((Suc n) / N) h) f" .
show ?case
apply (rule * [OF Suc.hyps eq0 pi0_eq])
using Suc.prems
apply (simp_all add: g h fpa path_integral_subpath_combine
path_integral_reversepath [symmetric] path_integrable_continuous_linepath
continuous_on_subset [OF contf gh_ns])
done
qed
} note ind = this
have "path_integral h f = path_integral g f"
using ind [OF order_refl] N joins
by (simp add: pathstart_def pathfinish_def split: split_if_asm)
}
ultimately
have "path_image g \<subseteq> s \<and> path_image h \<subseteq> s \<and> (\<forall>f. f holomorphic_on s \<longrightarrow> path_integral h f = path_integral g f)"
by metis
} note * = this
show ?thesis
apply (rule_tac x="e/3" in exI)
apply (rule conjI)
using enz apply simp
apply (clarsimp simp only: ball_conj_distrib)
apply (rule *; assumption)
done
qed
lemma
assumes "open s" "path p" "path_image p \<subseteq> s"
shows path_integral_nearby_ends:
"\<exists>d. 0 < d \<and>
(\<forall>g h. valid_path g \<and> valid_path h \<and>
(\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
pathstart h = pathstart g \<and> pathfinish h = pathfinish g
\<longrightarrow> path_image g \<subseteq> s \<and>
path_image h \<subseteq> s \<and>
(\<forall>f. f holomorphic_on s
\<longrightarrow> path_integral h f = path_integral g f))"
and path_integral_nearby_loop:
"\<exists>d. 0 < d \<and>
(\<forall>g h. valid_path g \<and> valid_path h \<and>
(\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
pathfinish g = pathstart g \<and> pathfinish h = pathstart h
\<longrightarrow> path_image g \<subseteq> s \<and>
path_image h \<subseteq> s \<and>
(\<forall>f. f holomorphic_on s
\<longrightarrow> path_integral h f = path_integral g f))"
using path_integral_nearby [OF assms, where Ends=True]
using path_integral_nearby [OF assms, where Ends=False]
by simp_all
thm has_vector_derivative_polynomial_function
corollary differentiable_polynomial_function:
fixes p :: "real \<Rightarrow> 'a::euclidean_space"
shows "polynomial_function p \<Longrightarrow> p differentiable_on s"
by (meson has_vector_derivative_polynomial_function differentiable_at_imp_differentiable_on differentiable_def has_vector_derivative_def)
lemma C1_differentiable_polynomial_function:
fixes p :: "real \<Rightarrow> 'a::euclidean_space"
shows "polynomial_function p \<Longrightarrow> 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 \<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)
lemma path_integral_bound_exists:
assumes s: "open s"
and g: "valid_path g"
and pag: "path_image g \<subseteq> s"
shows "\<exists>L. 0 < L \<and>
(\<forall>f B. f holomorphic_on s \<and> (\<forall>z \<in> s. norm(f z) \<le> B)
\<longrightarrow> norm(path_integral g f) \<le> L*B)"
proof -
have "path g" using g
by (simp add: valid_path_imp_path)
then obtain d::real and p
where d: "0 < d"
and p: "polynomial_function p" "path_image p \<subseteq> s"
and pi: "\<And>f. f holomorphic_on s \<Longrightarrow> path_integral g f = path_integral p f"
using path_integral_nearby_ends [OF s `path g` pag]
apply clarify
apply (drule_tac x=g in spec)
apply (simp only: assms)
apply (force simp: valid_path_polynomial_function dest: path_approx_polynomial_function)
done
then obtain p' where p': "polynomial_function p'"
"\<And>x. (p has_vector_derivative (p' x)) (at x)"
using has_vector_derivative_polynomial_function by force
then have "bounded(p' ` {0..1})"
using continuous_on_polymonial_function
by (force simp: intro!: compact_imp_bounded compact_continuous_image)
then obtain L where L: "L>0" and nop': "\<And>x. x \<in> {0..1} \<Longrightarrow> norm (p' x) \<le> L"
by (force simp: bounded_pos)
{ fix f B
assume f: "f holomorphic_on s"
and B: "\<And>z. z\<in>s \<Longrightarrow> cmod (f z) \<le> B"
then have "f path_integrable_on p \<and> valid_path p"
using p s
by (blast intro: valid_path_polynomial_function path_integrable_holomorphic_simple holomorphic_on_imp_continuous_on)
moreover have "\<And>x. x \<in> {0..1} \<Longrightarrow> cmod (vector_derivative p (at x)) * cmod (f (p x)) \<le> L * B"
apply (rule mult_mono)
apply (subst Derivative.vector_derivative_at; force intro: p' nop')
using L B p
apply (auto simp: path_image_def image_subset_iff)
done
ultimately have "cmod (path_integral g f) \<le> L * B"
apply (simp add: pi [OF f])
apply (simp add: path_integral_integral)
apply (rule order_trans [OF integral_norm_bound_integral])
apply (auto simp: mult.commute integral_norm_bound_integral path_integrable_on [symmetric] norm_mult)
done
} then
show ?thesis
by (force simp: L path_integral_integral)
qed
end