src/HOL/Analysis/Cauchy_Integral_Theorem.thy
author paulson <lp15@cam.ac.uk>
Mon Jun 11 22:43:33 2018 +0100 (12 months ago)
changeset 68420 529d6b132c27
parent 68403 223172b97d0b
child 68493 143b4cc8fc74
permissions -rw-r--r--
tidier Cauchy proofs
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section \<open>Complex path integrals and Cauchy's integral theorem\<close>
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text\<open>By John Harrison et al.  Ported from HOL Light by L C Paulson (2015)\<close>
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theory Cauchy_Integral_Theorem
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imports Complex_Transcendental Weierstrass_Theorems Ordered_Euclidean_Space
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begin
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subsection\<open>Homeomorphisms of arc images\<close>
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lemma homeomorphism_arc:
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  fixes g :: "real \<Rightarrow> 'a::t2_space"
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  assumes "arc g"
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  obtains h where "homeomorphism {0..1} (path_image g) g h"
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using assms by (force simp: arc_def homeomorphism_compact path_def path_image_def)
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lemma homeomorphic_arc_image_interval:
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  fixes g :: "real \<Rightarrow> 'a::t2_space" and a::real
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  assumes "arc g" "a < b"
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  shows "(path_image g) homeomorphic {a..b}"
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proof -
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  have "(path_image g) homeomorphic {0..1::real}"
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    by (meson assms(1) homeomorphic_def homeomorphic_sym homeomorphism_arc)
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  also have "\<dots> homeomorphic {a..b}"
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    using assms by (force intro: homeomorphic_closed_intervals_real)
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  finally show ?thesis .
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qed
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lemma homeomorphic_arc_images:
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  fixes g :: "real \<Rightarrow> 'a::t2_space" and h :: "real \<Rightarrow> 'b::t2_space"
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  assumes "arc g" "arc h"
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  shows "(path_image g) homeomorphic (path_image h)"
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proof -
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  have "(path_image g) homeomorphic {0..1::real}"
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    by (meson assms homeomorphic_def homeomorphic_sym homeomorphism_arc)
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  also have "\<dots> homeomorphic (path_image h)"
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    by (meson assms homeomorphic_def homeomorphism_arc)
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  finally show ?thesis .
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qed
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lemma path_connected_arc_complement:
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  fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
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  assumes "arc \<gamma>" "2 \<le> DIM('a)"
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  shows "path_connected(- path_image \<gamma>)"
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proof -
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  have "path_image \<gamma> homeomorphic {0..1::real}"
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    by (simp add: assms homeomorphic_arc_image_interval)
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  then
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  show ?thesis
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    apply (rule path_connected_complement_homeomorphic_convex_compact)
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      apply (auto simp: assms)
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    done
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qed
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lemma connected_arc_complement:
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  fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
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  assumes "arc \<gamma>" "2 \<le> DIM('a)"
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  shows "connected(- path_image \<gamma>)"
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  by (simp add: assms path_connected_arc_complement path_connected_imp_connected)
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lemma inside_arc_empty:
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  fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
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  assumes "arc \<gamma>"
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    shows "inside(path_image \<gamma>) = {}"
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proof (cases "DIM('a) = 1")
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  case True
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  then show ?thesis
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    using assms connected_arc_image connected_convex_1_gen inside_convex by blast
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next
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  case False
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  show ?thesis
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  proof (rule inside_bounded_complement_connected_empty)
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    show "connected (- path_image \<gamma>)"
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      apply (rule connected_arc_complement [OF assms])
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      using False
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      by (metis DIM_ge_Suc0 One_nat_def Suc_1 not_less_eq_eq order_class.order.antisym)
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    show "bounded (path_image \<gamma>)"
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      by (simp add: assms bounded_arc_image)
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  qed
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qed
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lemma inside_simple_curve_imp_closed:
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  fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
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    shows "\<lbrakk>simple_path \<gamma>; x \<in> inside(path_image \<gamma>)\<rbrakk> \<Longrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
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  using arc_simple_path  inside_arc_empty by blast
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subsection \<open>Piecewise differentiable functions\<close>
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definition piecewise_differentiable_on
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           (infixr "piecewise'_differentiable'_on" 50)
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  where "f piecewise_differentiable_on i  \<equiv>
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           continuous_on i f \<and>
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           (\<exists>S. finite S \<and> (\<forall>x \<in> i - S. f differentiable (at x within i)))"
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lemma piecewise_differentiable_on_imp_continuous_on:
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    "f piecewise_differentiable_on S \<Longrightarrow> continuous_on S f"
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by (simp add: piecewise_differentiable_on_def)
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lemma piecewise_differentiable_on_subset:
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    "f piecewise_differentiable_on S \<Longrightarrow> T \<le> S \<Longrightarrow> f piecewise_differentiable_on T"
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  using continuous_on_subset
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  unfolding piecewise_differentiable_on_def
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  apply safe
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  apply (blast elim: continuous_on_subset)
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  by (meson Diff_iff differentiable_within_subset subsetCE)
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lemma differentiable_on_imp_piecewise_differentiable:
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  fixes a:: "'a::{linorder_topology,real_normed_vector}"
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  shows "f differentiable_on {a..b} \<Longrightarrow> f piecewise_differentiable_on {a..b}"
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  apply (simp add: piecewise_differentiable_on_def differentiable_imp_continuous_on)
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  apply (rule_tac x="{a,b}" in exI, simp add: differentiable_on_def)
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  done
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lemma differentiable_imp_piecewise_differentiable:
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    "(\<And>x. x \<in> S \<Longrightarrow> f differentiable (at x within S))
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         \<Longrightarrow> f piecewise_differentiable_on S"
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by (auto simp: piecewise_differentiable_on_def differentiable_imp_continuous_on differentiable_on_def
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         intro: differentiable_within_subset)
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lemma piecewise_differentiable_const [iff]: "(\<lambda>x. z) piecewise_differentiable_on S"
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  by (simp add: differentiable_imp_piecewise_differentiable)
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lemma piecewise_differentiable_compose:
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    "\<lbrakk>f piecewise_differentiable_on S; g piecewise_differentiable_on (f ` S);
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      \<And>x. finite (S \<inter> f-`{x})\<rbrakk>
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      \<Longrightarrow> (g \<circ> f) piecewise_differentiable_on S"
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  apply (simp add: piecewise_differentiable_on_def, safe)
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  apply (blast intro: continuous_on_compose2)
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  apply (rename_tac A B)
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  apply (rule_tac x="A \<union> (\<Union>x\<in>B. S \<inter> f-`{x})" in exI)
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  apply (blast intro!: differentiable_chain_within)
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  done
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lemma piecewise_differentiable_affine:
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  fixes m::real
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  assumes "f piecewise_differentiable_on ((\<lambda>x. m *\<^sub>R x + c) ` S)"
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  shows "(f \<circ> (\<lambda>x. m *\<^sub>R x + c)) piecewise_differentiable_on S"
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proof (cases "m = 0")
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  case True
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  then show ?thesis
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    unfolding o_def
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    by (force intro: differentiable_imp_piecewise_differentiable differentiable_const)
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next
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  case False
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  show ?thesis
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    apply (rule piecewise_differentiable_compose [OF differentiable_imp_piecewise_differentiable])
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    apply (rule assms derivative_intros | simp add: False vimage_def real_vector_affinity_eq)+
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    done
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qed
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lemma piecewise_differentiable_cases:
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  fixes c::real
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  assumes "f piecewise_differentiable_on {a..c}"
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          "g piecewise_differentiable_on {c..b}"
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           "a \<le> c" "c \<le> b" "f c = g c"
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  shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_differentiable_on {a..b}"
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proof -
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  obtain S T where st: "finite S" "finite T"
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               and fd: "\<And>x. x \<in> {a..c} - S \<Longrightarrow> f differentiable at x within {a..c}"
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               and gd: "\<And>x. x \<in> {c..b} - T \<Longrightarrow> g differentiable at x within {c..b}"
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    using assms
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    by (auto simp: piecewise_differentiable_on_def)
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  have finabc: "finite ({a,b,c} \<union> (S \<union> T))"
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    by (metis \<open>finite S\<close> \<open>finite T\<close> finite_Un finite_insert finite.emptyI)
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  have "continuous_on {a..c} f" "continuous_on {c..b} g"
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    using assms piecewise_differentiable_on_def by auto
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  then have "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
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    using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
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                               OF closed_real_atLeastAtMost [of c b],
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                               of f g "\<lambda>x. x\<le>c"]  assms
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    by (force simp: ivl_disj_un_two_touch)
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  moreover
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  { fix x
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    assume x: "x \<in> {a..b} - ({a,b,c} \<union> (S \<union> T))"
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    have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b}" (is "?diff_fg")
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    proof (cases x c rule: le_cases)
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      case le show ?diff_fg
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      proof (rule differentiable_transform_within [where d = "dist x c"])
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        have "f differentiable at x"
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          using x le fd [of x] at_within_interior [of x "{a..c}"] by simp
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        then show "f differentiable at x within {a..b}"
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          by (simp add: differentiable_at_withinI)
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      qed (use x le st dist_real_def in auto)
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    next
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      case ge show ?diff_fg
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      proof (rule differentiable_transform_within [where d = "dist x c"])
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        have "g differentiable at x"
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          using x ge gd [of x] at_within_interior [of x "{c..b}"] by simp
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        then show "g differentiable at x within {a..b}"
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          by (simp add: differentiable_at_withinI)
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      qed (use x ge st dist_real_def in auto)
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    qed
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  }
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  then have "\<exists>S. finite S \<and>
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                 (\<forall>x\<in>{a..b} - S. (\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b})"
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    by (meson finabc)
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  ultimately show ?thesis
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    by (simp add: piecewise_differentiable_on_def)
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qed
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lemma piecewise_differentiable_neg:
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    "f piecewise_differentiable_on S \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_differentiable_on S"
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  by (auto simp: piecewise_differentiable_on_def continuous_on_minus)
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lemma piecewise_differentiable_add:
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  assumes "f piecewise_differentiable_on i"
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          "g piecewise_differentiable_on i"
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    shows "(\<lambda>x. f x + g x) piecewise_differentiable_on i"
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proof -
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  obtain S T where st: "finite S" "finite T"
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                       "\<forall>x\<in>i - S. f differentiable at x within i"
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                       "\<forall>x\<in>i - T. g differentiable at x within i"
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    using assms by (auto simp: piecewise_differentiable_on_def)
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  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)"
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    by auto
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  moreover have "continuous_on i f" "continuous_on i g"
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    using assms piecewise_differentiable_on_def by auto
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  ultimately show ?thesis
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    by (auto simp: piecewise_differentiable_on_def continuous_on_add)
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qed
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lemma piecewise_differentiable_diff:
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    "\<lbrakk>f piecewise_differentiable_on S;  g piecewise_differentiable_on S\<rbrakk>
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     \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_differentiable_on S"
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  unfolding diff_conv_add_uminus
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  by (metis piecewise_differentiable_add piecewise_differentiable_neg)
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lemma continuous_on_joinpaths_D1:
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    "continuous_on {0..1} (g1 +++ g2) \<Longrightarrow> continuous_on {0..1} g1"
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  apply (rule continuous_on_eq [of _ "(g1 +++ g2) \<circ> (( *)(inverse 2))"])
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  apply (rule continuous_intros | simp)+
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  apply (auto elim!: continuous_on_subset simp: joinpaths_def)
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  done
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lemma continuous_on_joinpaths_D2:
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    "\<lbrakk>continuous_on {0..1} (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> continuous_on {0..1} g2"
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  apply (rule continuous_on_eq [of _ "(g1 +++ g2) \<circ> (\<lambda>x. inverse 2*x + 1/2)"])
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  apply (rule continuous_intros | simp)+
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  apply (auto elim!: continuous_on_subset simp add: joinpaths_def pathfinish_def pathstart_def Ball_def)
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  done
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lemma piecewise_differentiable_D1:
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  assumes "(g1 +++ g2) piecewise_differentiable_on {0..1}"
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  shows "g1 piecewise_differentiable_on {0..1}"
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proof -
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  obtain S where cont: "continuous_on {0..1} g1" and "finite S"
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    and S: "\<And>x. x \<in> {0..1} - S \<Longrightarrow> g1 +++ g2 differentiable at x within {0..1}"
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    using assms unfolding piecewise_differentiable_on_def
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    by (blast dest!: continuous_on_joinpaths_D1)
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  show ?thesis
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    unfolding piecewise_differentiable_on_def
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  proof (intro exI conjI ballI cont)
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    show "finite (insert 1 ((( *)2) ` S))"
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      by (simp add: \<open>finite S\<close>)
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    show "g1 differentiable at x within {0..1}" if "x \<in> {0..1} - insert 1 (( *) 2 ` S)" for x
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    proof (rule_tac d="dist (x/2) (1/2)" in differentiable_transform_within)
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      have "g1 +++ g2 differentiable at (x / 2) within {0..1/2}"
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        by (rule differentiable_subset [OF S [of "x/2"]] | use that in force)+
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      then show "g1 +++ g2 \<circ> ( *) (inverse 2) differentiable at x within {0..1}"
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        by (auto intro: differentiable_chain_within)
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    qed (use that in \<open>auto simp: joinpaths_def\<close>)
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  qed
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qed
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lemma piecewise_differentiable_D2:
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  assumes "(g1 +++ g2) piecewise_differentiable_on {0..1}" and eq: "pathfinish g1 = pathstart g2"
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  shows "g2 piecewise_differentiable_on {0..1}"
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proof -
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  have [simp]: "g1 1 = g2 0"
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    using eq by (simp add: pathfinish_def pathstart_def)
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  obtain S where cont: "continuous_on {0..1} g2" and "finite S"
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    and S: "\<And>x. x \<in> {0..1} - S \<Longrightarrow> g1 +++ g2 differentiable at x within {0..1}"
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    using assms unfolding piecewise_differentiable_on_def
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    by (blast dest!: continuous_on_joinpaths_D2)
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  show ?thesis
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    unfolding piecewise_differentiable_on_def
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  proof (intro exI conjI ballI cont)
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    show "finite (insert 0 ((\<lambda>x. 2*x-1)`S))"
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      by (simp add: \<open>finite S\<close>)
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    show "g2 differentiable at x within {0..1}" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1)`S)" for x
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    proof (rule_tac d="dist ((x+1)/2) (1/2)" in differentiable_transform_within)
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      have x2: "(x + 1) / 2 \<notin> S"
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        using that
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        apply (clarsimp simp: image_iff)
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        by (metis add.commute add_diff_cancel_left' mult_2 real_sum_of_halves)
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      have "g1 +++ g2 \<circ> (\<lambda>x. (x+1) / 2) differentiable at x within {0..1}"
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        by (rule differentiable_chain_within differentiable_subset [OF S [of "(x+1)/2"]] | use x2 that in force)+
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      then show "g1 +++ g2 \<circ> (\<lambda>x. (x+1) / 2) differentiable at x within {0..1}"
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        by (auto intro: differentiable_chain_within)
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      show "(g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2)) x' = g2 x'" if "x' \<in> {0..1}" "dist x' x < dist ((x + 1) / 2) (1/2)" for x'
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      proof -
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   293
        have [simp]: "(2*x'+2)/2 = x'+1"
lp15@68284
   294
          by (simp add: divide_simps)
lp15@68284
   295
        show ?thesis
lp15@68284
   296
          using that by (auto simp: joinpaths_def)
lp15@68284
   297
      qed
lp15@68284
   298
    qed (use that in \<open>auto simp: joinpaths_def\<close>)
lp15@68284
   299
  qed
lp15@68284
   300
qed
lp15@61190
   301
lp15@61190
   302
lp15@61190
   303
subsubsection\<open>The concept of continuously differentiable\<close>
lp15@61190
   304
lp15@62408
   305
text \<open>
lp15@62408
   306
John Harrison writes as follows:
lp15@62408
   307
wenzelm@62456
   308
``The usual assumption in complex analysis texts is that a path \<open>\<gamma>\<close> should be piecewise
lp15@62408
   309
continuously differentiable, which ensures that the path integral exists at least for any continuous
lp15@62408
   310
f, since all piecewise continuous functions are integrable. However, our notion of validity is
lp15@68341
   311
weaker, just piecewise differentiability\ldots{} [namely] continuity plus differentiability except on a
lp15@68341
   312
finite set\ldots{} [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to
lp15@62408
   313
the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this
lp15@62408
   314
can integrate all derivatives.''
lp15@62408
   315
lp15@62534
   316
"Formalizing basic complex analysis." From Insight to Proof: Festschrift in Honour of Andrzej Trybulec.
lp15@62408
   317
Studies in Logic, Grammar and Rhetoric 10.23 (2007): 151-165.
lp15@62408
   318
lp15@62408
   319
And indeed he does not assume that his derivatives are continuous, but the penalty is unreasonably
lp15@62408
   320
difficult proofs concerning winding numbers. We need a self-contained and straightforward theorem
lp15@62408
   321
asserting that all derivatives can be integrated before we can adopt Harrison's choice.\<close>
lp15@62408
   322
lp15@61190
   323
definition C1_differentiable_on :: "(real \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> real set \<Rightarrow> bool"
lp15@61190
   324
           (infix "C1'_differentiable'_on" 50)
lp15@61190
   325
  where
lp15@68296
   326
  "f C1_differentiable_on S \<longleftrightarrow>
lp15@68296
   327
   (\<exists>D. (\<forall>x \<in> S. (f has_vector_derivative (D x)) (at x)) \<and> continuous_on S D)"
lp15@61190
   328
lp15@61190
   329
lemma C1_differentiable_on_eq:
lp15@68296
   330
    "f C1_differentiable_on S \<longleftrightarrow>
lp15@68296
   331
     (\<forall>x \<in> S. f differentiable at x) \<and> continuous_on S (\<lambda>x. vector_derivative f (at x))"
lp15@68296
   332
     (is "?lhs = ?rhs")
lp15@68296
   333
proof
lp15@68296
   334
  assume ?lhs
lp15@68296
   335
  then show ?rhs
lp15@68296
   336
    unfolding C1_differentiable_on_def
lp15@68296
   337
    by (metis (no_types, lifting) continuous_on_eq  differentiableI_vector vector_derivative_at)
lp15@68296
   338
next
lp15@68296
   339
  assume ?rhs
lp15@68296
   340
  then show ?lhs
lp15@68296
   341
    using C1_differentiable_on_def vector_derivative_works by fastforce
lp15@68296
   342
qed
lp15@61190
   343
lp15@61190
   344
lemma C1_differentiable_on_subset:
lp15@68296
   345
  "f C1_differentiable_on T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> f C1_differentiable_on S"
lp15@61190
   346
  unfolding C1_differentiable_on_def  continuous_on_eq_continuous_within
lp15@61190
   347
  by (blast intro:  continuous_within_subset)
lp15@61190
   348
lp15@61190
   349
lemma C1_differentiable_compose:
lp15@68296
   350
  assumes fg: "f C1_differentiable_on S" "g C1_differentiable_on (f ` S)" and fin: "\<And>x. finite (S \<inter> f-`{x})"
lp15@68339
   351
  shows "(g \<circ> f) C1_differentiable_on S"
lp15@68296
   352
proof -
lp15@68296
   353
  have "\<And>x. x \<in> S \<Longrightarrow> g \<circ> f differentiable at x"
lp15@68296
   354
    by (meson C1_differentiable_on_eq assms differentiable_chain_at imageI)
lp15@68296
   355
  moreover have "continuous_on S (\<lambda>x. vector_derivative (g \<circ> f) (at x))"
lp15@68296
   356
  proof (rule continuous_on_eq [of _ "\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"])
lp15@68296
   357
    show "continuous_on S (\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x)))"
lp15@68296
   358
      using fg
lp15@68296
   359
      apply (clarsimp simp add: C1_differentiable_on_eq)
lp15@68296
   360
      apply (rule Limits.continuous_on_scaleR, assumption)
lp15@68296
   361
      by (metis (mono_tags, lifting) continuous_at_imp_continuous_on continuous_on_compose continuous_on_cong differentiable_imp_continuous_within o_def)
lp15@68296
   362
    show "\<And>x. x \<in> S \<Longrightarrow> vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x)) = vector_derivative (g \<circ> f) (at x)"
lp15@68296
   363
      by (metis (mono_tags, hide_lams) C1_differentiable_on_eq fg imageI vector_derivative_chain_at)
lp15@68296
   364
  qed
lp15@68296
   365
  ultimately show ?thesis
lp15@68296
   366
    by (simp add: C1_differentiable_on_eq)
lp15@68296
   367
qed
lp15@68296
   368
lp15@68296
   369
lemma C1_diff_imp_diff: "f C1_differentiable_on S \<Longrightarrow> f differentiable_on S"
lp15@61190
   370
  by (simp add: C1_differentiable_on_eq differentiable_at_imp_differentiable_on)
lp15@61190
   371
lp15@68296
   372
lemma C1_differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) C1_differentiable_on S"
lp15@61190
   373
  by (auto simp: C1_differentiable_on_eq continuous_on_const)
lp15@61190
   374
lp15@68296
   375
lemma C1_differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. a) C1_differentiable_on S"
lp15@61190
   376
  by (auto simp: C1_differentiable_on_eq continuous_on_const)
lp15@61190
   377
lp15@61190
   378
lemma C1_differentiable_on_add [simp, derivative_intros]:
lp15@68296
   379
  "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x + g x) C1_differentiable_on S"
lp15@61190
   380
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
lp15@61190
   381
lp15@61190
   382
lemma C1_differentiable_on_minus [simp, derivative_intros]:
lp15@68296
   383
  "f C1_differentiable_on S \<Longrightarrow> (\<lambda>x. - f x) C1_differentiable_on S"
lp15@61190
   384
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
lp15@61190
   385
lp15@61190
   386
lemma C1_differentiable_on_diff [simp, derivative_intros]:
lp15@68296
   387
  "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x - g x) C1_differentiable_on S"
lp15@61190
   388
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
lp15@61190
   389
lp15@61190
   390
lemma C1_differentiable_on_mult [simp, derivative_intros]:
lp15@61190
   391
  fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
lp15@68296
   392
  shows "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x * g x) C1_differentiable_on S"
lp15@61190
   393
  unfolding C1_differentiable_on_eq
lp15@61190
   394
  by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within)
lp15@61190
   395
lp15@61190
   396
lemma C1_differentiable_on_scaleR [simp, derivative_intros]:
lp15@68296
   397
  "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) C1_differentiable_on S"
lp15@61190
   398
  unfolding C1_differentiable_on_eq
lp15@61190
   399
  by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+
lp15@61190
   400
lp15@61190
   401
lp15@61190
   402
definition piecewise_C1_differentiable_on
lp15@61190
   403
           (infixr "piecewise'_C1'_differentiable'_on" 50)
lp15@61190
   404
  where "f piecewise_C1_differentiable_on i  \<equiv>
lp15@61190
   405
           continuous_on i f \<and>
lp15@68296
   406
           (\<exists>S. finite S \<and> (f C1_differentiable_on (i - S)))"
lp15@61190
   407
lp15@61190
   408
lemma C1_differentiable_imp_piecewise:
lp15@68296
   409
    "f C1_differentiable_on S \<Longrightarrow> f piecewise_C1_differentiable_on S"
lp15@61190
   410
  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within)
lp15@61190
   411
lp15@61190
   412
lemma piecewise_C1_imp_differentiable:
lp15@61190
   413
    "f piecewise_C1_differentiable_on i \<Longrightarrow> f piecewise_differentiable_on i"
lp15@61190
   414
  by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def
lp15@61190
   415
           C1_differentiable_on_def differentiable_def has_vector_derivative_def
lp15@67979
   416
           intro: has_derivative_at_withinI)
lp15@61190
   417
lp15@61190
   418
lemma piecewise_C1_differentiable_compose:
lp15@68296
   419
  assumes fg: "f piecewise_C1_differentiable_on S" "g piecewise_C1_differentiable_on (f ` S)" and fin: "\<And>x. finite (S \<inter> f-`{x})"
lp15@68339
   420
  shows "(g \<circ> f) piecewise_C1_differentiable_on S"
lp15@68296
   421
proof -
lp15@68296
   422
  have "continuous_on S (\<lambda>x. g (f x))"
lp15@68296
   423
    by (metis continuous_on_compose2 fg order_refl piecewise_C1_differentiable_on_def)
lp15@68296
   424
  moreover have "\<exists>T. finite T \<and> g \<circ> f C1_differentiable_on S - T"
lp15@68296
   425
  proof -
lp15@68296
   426
    obtain F where "finite F" and F: "f C1_differentiable_on S - F" and f: "f piecewise_C1_differentiable_on S"
lp15@68296
   427
      using fg by (auto simp: piecewise_C1_differentiable_on_def)
lp15@68296
   428
    obtain G where "finite G" and G: "g C1_differentiable_on f ` S - G" and g: "g piecewise_C1_differentiable_on f ` S"
lp15@68296
   429
      using fg by (auto simp: piecewise_C1_differentiable_on_def)
lp15@68296
   430
    show ?thesis
lp15@68296
   431
    proof (intro exI conjI)
lp15@68296
   432
      show "finite (F \<union> (\<Union>x\<in>G. S \<inter> f-`{x}))"
lp15@68296
   433
        using fin by (auto simp only: Int_Union \<open>finite F\<close> \<open>finite G\<close> finite_UN finite_imageI)
lp15@68296
   434
      show "g \<circ> f C1_differentiable_on S - (F \<union> (\<Union>x\<in>G. S \<inter> f -` {x}))"
lp15@68296
   435
        apply (rule C1_differentiable_compose)
lp15@68296
   436
          apply (blast intro: C1_differentiable_on_subset [OF F])
lp15@68296
   437
          apply (blast intro: C1_differentiable_on_subset [OF G])
lp15@68296
   438
        by (simp add:  C1_differentiable_on_subset G Diff_Int_distrib2 fin)
lp15@68296
   439
    qed
lp15@68296
   440
  qed
lp15@68296
   441
  ultimately show ?thesis
lp15@68296
   442
    by (simp add: piecewise_C1_differentiable_on_def)
lp15@68296
   443
qed
lp15@61190
   444
lp15@61190
   445
lemma piecewise_C1_differentiable_on_subset:
lp15@68296
   446
    "f piecewise_C1_differentiable_on S \<Longrightarrow> T \<le> S \<Longrightarrow> f piecewise_C1_differentiable_on T"
lp15@61190
   447
  by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset)
lp15@61190
   448
lp15@61190
   449
lemma C1_differentiable_imp_continuous_on:
lp15@68296
   450
  "f C1_differentiable_on S \<Longrightarrow> continuous_on S f"
lp15@61190
   451
  unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within
lp15@61190
   452
  using differentiable_at_withinI differentiable_imp_continuous_within by blast
lp15@61190
   453
lp15@61190
   454
lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}"
lp15@61190
   455
  unfolding C1_differentiable_on_def
lp15@61190
   456
  by auto
lp15@61190
   457
lp15@61190
   458
lemma piecewise_C1_differentiable_affine:
lp15@61190
   459
  fixes m::real
lp15@68296
   460
  assumes "f piecewise_C1_differentiable_on ((\<lambda>x. m * x + c) ` S)"
lp15@68339
   461
  shows "(f \<circ> (\<lambda>x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on S"
lp15@61190
   462
proof (cases "m = 0")
lp15@61190
   463
  case True
lp15@61190
   464
  then show ?thesis
lp15@61190
   465
    unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def continuous_on_const)
lp15@61190
   466
next
lp15@61190
   467
  case False
lp15@68296
   468
  have *: "\<And>x. finite (S \<inter> {y. m * y + c = x})"
lp15@68296
   469
    using False not_finite_existsD by fastforce 
lp15@61190
   470
  show ?thesis
lp15@61190
   471
    apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise])
lp15@68296
   472
    apply (rule * assms derivative_intros | simp add: False vimage_def)+
lp15@61190
   473
    done
lp15@61190
   474
qed
lp15@61190
   475
lp15@61190
   476
lemma piecewise_C1_differentiable_cases:
lp15@61190
   477
  fixes c::real
lp15@61190
   478
  assumes "f piecewise_C1_differentiable_on {a..c}"
lp15@61190
   479
          "g piecewise_C1_differentiable_on {c..b}"
lp15@61190
   480
           "a \<le> c" "c \<le> b" "f c = g c"
lp15@61190
   481
  shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}"
lp15@61190
   482
proof -
lp15@68296
   483
  obtain S T where st: "f C1_differentiable_on ({a..c} - S)"
lp15@68296
   484
                       "g C1_differentiable_on ({c..b} - T)"
lp15@68296
   485
                       "finite S" "finite T"
lp15@61190
   486
    using assms
lp15@61190
   487
    by (force simp: piecewise_C1_differentiable_on_def)
lp15@68296
   488
  then have f_diff: "f differentiable_on {a..<c} - S"
lp15@68296
   489
        and g_diff: "g differentiable_on {c<..b} - T"
lp15@61190
   490
    by (simp_all add: C1_differentiable_on_eq differentiable_at_withinI differentiable_on_def)
lp15@61190
   491
  have "continuous_on {a..c} f" "continuous_on {c..b} g"
lp15@61190
   492
    using assms piecewise_C1_differentiable_on_def by auto
lp15@61190
   493
  then have cab: "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
lp15@61190
   494
    using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
lp15@61190
   495
                               OF closed_real_atLeastAtMost [of c b],
lp15@61190
   496
                               of f g "\<lambda>x. x\<le>c"]  assms
lp15@61190
   497
    by (force simp: ivl_disj_un_two_touch)
lp15@61190
   498
  { fix x
lp15@68296
   499
    assume x: "x \<in> {a..b} - insert c (S \<union> T)"
lp15@61190
   500
    have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x" (is "?diff_fg")
lp15@61190
   501
    proof (cases x c rule: le_cases)
lp15@61190
   502
      case le show ?diff_fg
paulson@62087
   503
        apply (rule differentiable_transform_within [where f=f and d = "dist x c"])
lp15@61190
   504
        using x dist_real_def le st by (auto simp: C1_differentiable_on_eq)
lp15@61190
   505
    next
lp15@61190
   506
      case ge show ?diff_fg
paulson@62087
   507
        apply (rule differentiable_transform_within [where f=g and d = "dist x c"])
lp15@61190
   508
        using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq)
lp15@61190
   509
    qed
lp15@61190
   510
  }
lp15@68296
   511
  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)"
lp15@61190
   512
    by auto
lp15@61190
   513
  moreover
lp15@68296
   514
  { assume fcon: "continuous_on ({a<..<c} - S) (\<lambda>x. vector_derivative f (at x))"
lp15@68296
   515
       and gcon: "continuous_on ({c<..<b} - T) (\<lambda>x. vector_derivative g (at x))"
lp15@68296
   516
    have "open ({a<..<c} - S)"  "open ({c<..<b} - T)"
lp15@61190
   517
      using st by (simp_all add: open_Diff finite_imp_closed)
lp15@68296
   518
    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))"
lp15@68296
   519
    proof -
lp15@68296
   520
      have "((\<lambda>x. if x \<le> c then f x else g x) has_vector_derivative vector_derivative f (at x))            (at x)"
lp15@68296
   521
        if "a < x" "x < c" "x \<notin> S" for x
lp15@68296
   522
      proof -
lp15@68296
   523
        have f: "f differentiable at x"
lp15@68296
   524
          by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_eq_real_def st(1) that)
lp15@68296
   525
        show ?thesis
lp15@68296
   526
          using that
lp15@68296
   527
          apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_within)
lp15@68339
   528
             apply (auto simp: dist_norm vector_derivative_works [symmetric] f)
lp15@68296
   529
          done
lp15@68296
   530
      qed
lp15@68296
   531
      then show ?thesis
lp15@68296
   532
        by (metis (no_types, lifting) continuous_on_eq [OF fcon] DiffE greaterThanLessThan_iff vector_derivative_at)
lp15@68296
   533
    qed
lp15@68296
   534
    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))"
lp15@68296
   535
    proof -
lp15@68296
   536
      have "((\<lambda>x. if x \<le> c then f x else g x) has_vector_derivative vector_derivative g (at x))            (at x)"
lp15@68296
   537
        if "c < x" "x < b" "x \<notin> T" for x
lp15@68296
   538
      proof -
lp15@68296
   539
        have g: "g differentiable at x"
lp15@68296
   540
          by (metis C1_differentiable_on_eq DiffD1 DiffI atLeastAtMost_diff_ends greaterThanLessThan_iff st(2) that)
lp15@68296
   541
        show ?thesis
lp15@68296
   542
          using that
lp15@68296
   543
          apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_within)
lp15@68339
   544
             apply (auto simp: dist_norm vector_derivative_works [symmetric] g)
lp15@68296
   545
          done
lp15@68296
   546
      qed
lp15@68296
   547
      then show ?thesis
lp15@68296
   548
        by (metis (no_types, lifting) continuous_on_eq [OF gcon] DiffE greaterThanLessThan_iff vector_derivative_at)
lp15@68296
   549
    qed
lp15@68296
   550
    ultimately have "continuous_on ({a<..<b} - insert c (S \<union> T))
lp15@61190
   551
        (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
lp15@68296
   552
      by (rule continuous_on_subset [OF continuous_on_open_Un], auto)
lp15@61190
   553
  } note * = this
lp15@68296
   554
  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))"
lp15@61190
   555
    using st
lp15@61190
   556
    by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *)
lp15@68296
   557
  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)"
lp15@68296
   558
    apply (rule_tac x="{a,b,c} \<union> S \<union> T" in exI)
lp15@61190
   559
    using st  by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset)
lp15@61190
   560
  with cab show ?thesis
lp15@61190
   561
    by (simp add: piecewise_C1_differentiable_on_def)
lp15@61190
   562
qed
lp15@61190
   563
lp15@61190
   564
lemma piecewise_C1_differentiable_neg:
lp15@68296
   565
    "f piecewise_C1_differentiable_on S \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_C1_differentiable_on S"
lp15@61190
   566
  unfolding piecewise_C1_differentiable_on_def
lp15@61190
   567
  by (auto intro!: continuous_on_minus C1_differentiable_on_minus)
lp15@61190
   568
lp15@61190
   569
lemma piecewise_C1_differentiable_add:
lp15@61190
   570
  assumes "f piecewise_C1_differentiable_on i"
lp15@61190
   571
          "g piecewise_C1_differentiable_on i"
lp15@61190
   572
    shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i"
lp15@61190
   573
proof -
lp15@68296
   574
  obtain S t where st: "finite S" "finite t"
lp15@68296
   575
                       "f C1_differentiable_on (i-S)"
lp15@61190
   576
                       "g C1_differentiable_on (i-t)"
lp15@61190
   577
    using assms by (auto simp: piecewise_C1_differentiable_on_def)
lp15@68296
   578
  then have "finite (S \<union> t) \<and> (\<lambda>x. f x + g x) C1_differentiable_on i - (S \<union> t)"
lp15@61190
   579
    by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset)
lp15@61190
   580
  moreover have "continuous_on i f" "continuous_on i g"
lp15@61190
   581
    using assms piecewise_C1_differentiable_on_def by auto
lp15@61190
   582
  ultimately show ?thesis
lp15@61190
   583
    by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add)
lp15@61190
   584
qed
lp15@61190
   585
paulson@61204
   586
lemma piecewise_C1_differentiable_diff:
lp15@68296
   587
    "\<lbrakk>f piecewise_C1_differentiable_on S;  g piecewise_C1_differentiable_on S\<rbrakk>
lp15@68296
   588
     \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_C1_differentiable_on S"
lp15@61190
   589
  unfolding diff_conv_add_uminus
lp15@61190
   590
  by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg)
lp15@61190
   591
lp15@61190
   592
lemma piecewise_C1_differentiable_D1:
lp15@61190
   593
  fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   594
  assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}"
lp15@61190
   595
    shows "g1 piecewise_C1_differentiable_on {0..1}"
lp15@61190
   596
proof -
lp15@68296
   597
  obtain S where "finite S"
lp15@68296
   598
             and co12: "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@68296
   599
             and g12D: "\<forall>x\<in>{0..1} - S. g1 +++ g2 differentiable at x"
lp15@61190
   600
    using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@68339
   601
  have g1D: "g1 differentiable at x" if "x \<in> {0..1} - insert 1 (( *) 2 ` S)" for x
lp15@68296
   602
  proof (rule differentiable_transform_within)
lp15@68339
   603
    show "g1 +++ g2 \<circ> ( *) (inverse 2) differentiable at x"
lp15@68296
   604
      using that g12D 
lp15@68296
   605
      apply (simp only: joinpaths_def)
lp15@68296
   606
      by (rule differentiable_chain_at derivative_intros | force)+
lp15@68296
   607
    show "\<And>x'. \<lbrakk>dist x' x < dist (x/2) (1/2)\<rbrakk>
lp15@68339
   608
          \<Longrightarrow> (g1 +++ g2 \<circ> ( *) (inverse 2)) x' = g1 x'"
lp15@68339
   609
      using that by (auto simp: dist_real_def joinpaths_def)
lp15@68296
   610
  qed (use that in \<open>auto simp: dist_real_def\<close>)
lp15@68339
   611
  have [simp]: "vector_derivative (g1 \<circ> ( *) 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)"
lp15@68339
   612
               if "x \<in> {0..1} - insert 1 (( *) 2 ` S)" for x
lp15@61190
   613
    apply (subst vector_derivative_chain_at)
lp15@61190
   614
    using that
lp15@61190
   615
    apply (rule derivative_eq_intros g1D | simp)+
lp15@61190
   616
    done
lp15@68296
   617
  have "continuous_on ({0..1/2} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   618
    using co12 by (rule continuous_on_subset) force
lp15@68339
   619
  then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 \<circ> ( *)2) (at x))"
lp15@68296
   620
  proof (rule continuous_on_eq [OF _ vector_derivative_at])
lp15@68339
   621
    show "(g1 +++ g2 has_vector_derivative vector_derivative (g1 \<circ> ( *) 2) (at x)) (at x)"
lp15@68296
   622
      if "x \<in> {0..1/2} - insert (1/2) S" for x
lp15@68296
   623
    proof (rule has_vector_derivative_transform_within)
lp15@68339
   624
      show "(g1 \<circ> ( *) 2 has_vector_derivative vector_derivative (g1 \<circ> ( *) 2) (at x)) (at x)"
lp15@68296
   625
        using that
lp15@68296
   626
        by (force intro: g1D differentiable_chain_at simp: vector_derivative_works [symmetric])
lp15@68339
   627
      show "\<And>x'. \<lbrakk>dist x' x < dist x (1/2)\<rbrakk> \<Longrightarrow> (g1 \<circ> ( *) 2) x' = (g1 +++ g2) x'"
lp15@68296
   628
        using that by (auto simp: dist_norm joinpaths_def)
lp15@68296
   629
    qed (use that in \<open>auto simp: dist_norm\<close>)
lp15@68296
   630
  qed
lp15@68339
   631
  have "continuous_on ({0..1} - insert 1 (( *) 2 ` S))
lp15@68339
   632
                      ((\<lambda>x. 1/2 * vector_derivative (g1 \<circ> ( *)2) (at x)) \<circ> ( *)(1/2))"
lp15@61190
   633
    apply (rule continuous_intros)+
lp15@61190
   634
    using coDhalf
lp15@61190
   635
    apply (simp add: scaleR_conv_of_real image_set_diff image_image)
lp15@61190
   636
    done
lp15@68339
   637
  then have con_g1: "continuous_on ({0..1} - insert 1 (( *) 2 ` S)) (\<lambda>x. vector_derivative g1 (at x))"
lp15@61190
   638
    by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
lp15@61190
   639
  have "continuous_on {0..1} g1"
lp15@61190
   640
    using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast
lp15@68296
   641
  with \<open>finite S\<close> show ?thesis
lp15@61190
   642
    apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@68339
   643
    apply (rule_tac x="insert 1 ((( *)2)`S)" in exI)
lp15@61190
   644
    apply (simp add: g1D con_g1)
lp15@61190
   645
  done
lp15@61190
   646
qed
lp15@61190
   647
lp15@61190
   648
lemma piecewise_C1_differentiable_D2:
lp15@61190
   649
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   650
  assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" "pathfinish g1 = pathstart g2"
lp15@61190
   651
    shows "g2 piecewise_C1_differentiable_on {0..1}"
lp15@61190
   652
proof -
lp15@68296
   653
  obtain S where "finite S"
lp15@68296
   654
             and co12: "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@68296
   655
             and g12D: "\<forall>x\<in>{0..1} - S. g1 +++ g2 differentiable at x"
lp15@61190
   656
    using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@68296
   657
  have g2D: "g2 differentiable at x" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)" for x
lp15@68296
   658
  proof (rule differentiable_transform_within)
lp15@68296
   659
    show "g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2) differentiable at x"
lp15@68296
   660
      using g12D that
lp15@68296
   661
      apply (simp only: joinpaths_def)
lp15@68296
   662
      apply (drule_tac x= "(x+1) / 2" in bspec, force simp: divide_simps)
lp15@68296
   663
      apply (rule differentiable_chain_at derivative_intros | force)+
lp15@68296
   664
      done
lp15@68296
   665
    show "\<And>x'. dist x' x < dist ((x + 1) / 2) (1/2) \<Longrightarrow> (g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2)) x' = g2 x'"
lp15@68296
   666
      using that by (auto simp: dist_real_def joinpaths_def field_simps)
lp15@68296
   667
    qed (use that in \<open>auto simp: dist_norm\<close>)
lp15@61190
   668
  have [simp]: "vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at ((x+1)/2)) = 2 *\<^sub>R vector_derivative g2 (at x)"
lp15@68296
   669
               if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)" for x
lp15@61190
   670
    using that  by (auto simp: vector_derivative_chain_at divide_simps g2D)
lp15@68296
   671
  have "continuous_on ({1/2..1} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   672
    using co12 by (rule continuous_on_subset) force
lp15@68339
   673
  then have coDhalf: "continuous_on ({1/2..1} - insert (1/2) S) (\<lambda>x. vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x))"
lp15@68296
   674
  proof (rule continuous_on_eq [OF _ vector_derivative_at])
lp15@68296
   675
    show "(g1 +++ g2 has_vector_derivative vector_derivative (g2 \<circ> (\<lambda>x. 2 * x - 1)) (at x))
lp15@68296
   676
          (at x)"
lp15@68296
   677
      if "x \<in> {1 / 2..1} - insert (1 / 2) S" for x
lp15@68339
   678
    proof (rule_tac f="g2 \<circ> (\<lambda>x. 2*x-1)" and d="dist (3/4) ((x+1)/2)" in has_vector_derivative_transform_within)
lp15@68296
   679
      show "(g2 \<circ> (\<lambda>x. 2 * x - 1) has_vector_derivative vector_derivative (g2 \<circ> (\<lambda>x. 2 * x - 1)) (at x))
lp15@68296
   680
            (at x)"
lp15@68296
   681
        using that by (force intro: g2D differentiable_chain_at simp: vector_derivative_works [symmetric])
lp15@68296
   682
      show "\<And>x'. \<lbrakk>dist x' x < dist (3 / 4) ((x + 1) / 2)\<rbrakk> \<Longrightarrow> (g2 \<circ> (\<lambda>x. 2 * x - 1)) x' = (g1 +++ g2) x'"
lp15@68296
   683
        using that by (auto simp: dist_norm joinpaths_def add_divide_distrib)
lp15@68296
   684
    qed (use that in \<open>auto simp: dist_norm\<close>)
lp15@68296
   685
  qed
lp15@68296
   686
  have [simp]: "((\<lambda>x. (x+1) / 2) ` ({0..1} - insert 0 ((\<lambda>x. 2 * x - 1) ` S))) = ({1/2..1} - insert (1/2) S)"
lp15@61190
   687
    apply (simp add: image_set_diff inj_on_def image_image)
lp15@61190
   688
    apply (auto simp: image_affinity_atLeastAtMost_div add_divide_distrib)
lp15@61190
   689
    done
lp15@68296
   690
  have "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S))
lp15@68339
   691
                      ((\<lambda>x. 1/2 * vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x)) \<circ> (\<lambda>x. (x+1)/2))"
lp15@61190
   692
    by (rule continuous_intros | simp add:  coDhalf)+
lp15@68296
   693
  then have con_g2: "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)) (\<lambda>x. vector_derivative g2 (at x))"
lp15@61190
   694
    by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
lp15@61190
   695
  have "continuous_on {0..1} g2"
lp15@61190
   696
    using continuous_on_joinpaths_D2 assms piecewise_C1_differentiable_on_def by blast
lp15@68296
   697
  with \<open>finite S\<close> show ?thesis
lp15@61190
   698
    apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@68296
   699
    apply (rule_tac x="insert 0 ((\<lambda>x. 2 * x - 1) ` S)" in exI)
lp15@61190
   700
    apply (simp add: g2D con_g2)
lp15@61190
   701
  done
lp15@61190
   702
qed
lp15@60809
   703
lp15@60809
   704
subsection \<open>Valid paths, and their start and finish\<close>
lp15@60809
   705
lp15@60809
   706
definition valid_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
lp15@61190
   707
  where "valid_path f \<equiv> f piecewise_C1_differentiable_on {0..1::real}"
lp15@60809
   708
lp15@60809
   709
definition closed_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
lp15@60809
   710
  where "closed_path g \<equiv> g 0 = g 1"
lp15@60809
   711
lp15@60809
   712
subsubsection\<open>In particular, all results for paths apply\<close>
lp15@60809
   713
lp15@60809
   714
lemma valid_path_imp_path: "valid_path g \<Longrightarrow> path g"
lp15@61190
   715
by (simp add: path_def piecewise_C1_differentiable_on_def valid_path_def)
lp15@60809
   716
lp15@60809
   717
lemma connected_valid_path_image: "valid_path g \<Longrightarrow> connected(path_image g)"
lp15@60809
   718
  by (metis connected_path_image valid_path_imp_path)
lp15@60809
   719
lp15@60809
   720
lemma compact_valid_path_image: "valid_path g \<Longrightarrow> compact(path_image g)"
lp15@60809
   721
  by (metis compact_path_image valid_path_imp_path)
lp15@60809
   722
lp15@60809
   723
lemma bounded_valid_path_image: "valid_path g \<Longrightarrow> bounded(path_image g)"
lp15@60809
   724
  by (metis bounded_path_image valid_path_imp_path)
lp15@60809
   725
lp15@60809
   726
lemma closed_valid_path_image: "valid_path g \<Longrightarrow> closed(path_image g)"
lp15@60809
   727
  by (metis closed_path_image valid_path_imp_path)
lp15@60809
   728
lp15@62540
   729
proposition valid_path_compose:
lp15@62623
   730
  assumes "valid_path g"
lp15@64394
   731
      and der: "\<And>x. x \<in> path_image g \<Longrightarrow> f field_differentiable (at x)"
lp15@62540
   732
      and con: "continuous_on (path_image g) (deriv f)"
lp15@68339
   733
    shows "valid_path (f \<circ> g)"
lp15@62408
   734
proof -
lp15@68296
   735
  obtain S where "finite S" and g_diff: "g C1_differentiable_on {0..1} - S"
wenzelm@62837
   736
    using \<open>valid_path g\<close> unfolding valid_path_def piecewise_C1_differentiable_on_def by auto
lp15@68296
   737
  have "f \<circ> g differentiable at t" when "t\<in>{0..1} - S" for t
lp15@62408
   738
    proof (rule differentiable_chain_at)
wenzelm@62837
   739
      show "g differentiable at t" using \<open>valid_path g\<close>
lp15@68296
   740
        by (meson C1_differentiable_on_eq \<open>g C1_differentiable_on {0..1} - S\<close> that)
lp15@62408
   741
    next
lp15@62408
   742
      have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis
lp15@64394
   743
      then show "f differentiable at (g t)" 
lp15@64394
   744
        using der[THEN field_differentiable_imp_differentiable] by auto
lp15@62408
   745
    qed
lp15@68296
   746
  moreover have "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (f \<circ> g) (at x))"
lp15@62540
   747
    proof (rule continuous_on_eq [where f = "\<lambda>x. vector_derivative g (at x) * deriv f (g x)"],
lp15@62540
   748
        rule continuous_intros)
lp15@68296
   749
      show "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative g (at x))"
lp15@62540
   750
        using g_diff C1_differentiable_on_eq by auto
lp15@62540
   751
    next
lp15@62623
   752
      have "continuous_on {0..1} (\<lambda>x. deriv f (g x))"
lp15@62623
   753
        using continuous_on_compose[OF _ con[unfolded path_image_def],unfolded comp_def]
wenzelm@62837
   754
          \<open>valid_path g\<close> piecewise_C1_differentiable_on_def valid_path_def
lp15@62540
   755
        by blast
lp15@68296
   756
      then show "continuous_on ({0..1} - S) (\<lambda>x. deriv f (g x))"
lp15@62540
   757
        using continuous_on_subset by blast
lp15@62408
   758
    next
lp15@62540
   759
      show "vector_derivative g (at t) * deriv f (g t) = vector_derivative (f \<circ> g) (at t)"
lp15@68296
   760
          when "t \<in> {0..1} - S" for t
lp15@62540
   761
        proof (rule vector_derivative_chain_at_general[symmetric])
lp15@62540
   762
          show "g differentiable at t" by (meson C1_differentiable_on_eq g_diff that)
lp15@62540
   763
        next
lp15@62540
   764
          have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis
lp15@64394
   765
          then show "f field_differentiable at (g t)" using der by auto
lp15@62540
   766
        qed
lp15@62408
   767
    qed
lp15@68339
   768
  ultimately have "f \<circ> g C1_differentiable_on {0..1} - S"
lp15@62408
   769
    using C1_differentiable_on_eq by blast
lp15@64394
   770
  moreover have "path (f \<circ> g)" 
lp15@64394
   771
    apply (rule path_continuous_image[OF valid_path_imp_path[OF \<open>valid_path g\<close>]])
lp15@64394
   772
    using der
lp15@64394
   773
    by (simp add: continuous_at_imp_continuous_on field_differentiable_imp_continuous_at)
lp15@62408
   774
  ultimately show ?thesis unfolding valid_path_def piecewise_C1_differentiable_on_def path_def
lp15@68296
   775
    using \<open>finite S\<close> by auto
lp15@62408
   776
qed
lp15@62408
   777
lp15@60809
   778
lp15@60809
   779
subsection\<open>Contour Integrals along a path\<close>
lp15@60809
   780
lp15@60809
   781
text\<open>This definition is for complex numbers only, and does not generalise to line integrals in a vector field\<close>
lp15@60809
   782
lp15@61190
   783
text\<open>piecewise differentiable function on [0,1]\<close>
lp15@60809
   784
lp15@61738
   785
definition has_contour_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool"
lp15@61738
   786
           (infixr "has'_contour'_integral" 50)
lp15@61738
   787
  where "(f has_contour_integral i) g \<equiv>
lp15@60809
   788
           ((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1}))
lp15@60809
   789
            has_integral i) {0..1}"
lp15@60809
   790
lp15@61738
   791
definition contour_integrable_on
lp15@61738
   792
           (infixr "contour'_integrable'_on" 50)
lp15@61738
   793
  where "f contour_integrable_on g \<equiv> \<exists>i. (f has_contour_integral i) g"
lp15@61738
   794
lp15@61738
   795
definition contour_integral
wenzelm@67613
   796
  where "contour_integral g f \<equiv> SOME i. (f has_contour_integral i) g \<or> \<not> f contour_integrable_on g \<and> i=0"
wenzelm@67613
   797
wenzelm@67613
   798
lemma not_integrable_contour_integral: "\<not> f contour_integrable_on g \<Longrightarrow> contour_integral g f = 0"
lp15@62534
   799
  unfolding contour_integrable_on_def contour_integral_def by blast
lp15@62463
   800
lp15@62463
   801
lemma contour_integral_unique: "(f has_contour_integral i) g \<Longrightarrow> contour_integral g f = i"
lp15@62463
   802
  apply (simp add: contour_integral_def has_contour_integral_def contour_integrable_on_def)
lp15@62463
   803
  using has_integral_unique by blast
lp15@61738
   804
paulson@62131
   805
corollary has_contour_integral_eqpath:
lp15@62397
   806
     "\<lbrakk>(f has_contour_integral y) p; f contour_integrable_on \<gamma>;
paulson@62131
   807
       contour_integral p f = contour_integral \<gamma> f\<rbrakk>
paulson@62131
   808
      \<Longrightarrow> (f has_contour_integral y) \<gamma>"
paulson@62131
   809
using contour_integrable_on_def contour_integral_unique by auto
paulson@62131
   810
lp15@61738
   811
lemma has_contour_integral_integral:
lp15@61738
   812
    "f contour_integrable_on i \<Longrightarrow> (f has_contour_integral (contour_integral i f)) i"
lp15@61738
   813
  by (metis contour_integral_unique contour_integrable_on_def)
lp15@61738
   814
lp15@61738
   815
lemma has_contour_integral_unique:
lp15@61738
   816
    "(f has_contour_integral i) g \<Longrightarrow> (f has_contour_integral j) g \<Longrightarrow> i = j"
lp15@60809
   817
  using has_integral_unique
lp15@61738
   818
  by (auto simp: has_contour_integral_def)
lp15@61738
   819
lp15@61738
   820
lemma has_contour_integral_integrable: "(f has_contour_integral i) g \<Longrightarrow> f contour_integrable_on g"
lp15@61738
   821
  using contour_integrable_on_def by blast
lp15@60809
   822
lp15@68296
   823
subsubsection\<open>Show that we can forget about the localized derivative.\<close>
lp15@60809
   824
lp15@60809
   825
lemma has_integral_localized_vector_derivative:
lp15@60809
   826
    "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
lp15@60809
   827
     ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
lp15@60809
   828
proof -
lp15@68296
   829
  have *: "{a..b} - {a,b} = interior {a..b}"
lp15@60809
   830
    by (simp add: atLeastAtMost_diff_ends)
lp15@60809
   831
  show ?thesis
lp15@60809
   832
    apply (rule has_integral_spike_eq [of "{a,b}"])
lp15@68296
   833
    apply (auto simp: at_within_interior [of _ "{a..b}"])
lp15@60809
   834
    done
lp15@60809
   835
qed
lp15@60809
   836
lp15@60809
   837
lemma integrable_on_localized_vector_derivative:
lp15@60809
   838
    "(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow>
lp15@60809
   839
     (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
lp15@60809
   840
  by (simp add: integrable_on_def has_integral_localized_vector_derivative)
lp15@60809
   841
lp15@61738
   842
lemma has_contour_integral:
lp15@61738
   843
     "(f has_contour_integral i) g \<longleftrightarrow>
lp15@60809
   844
      ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
lp15@61738
   845
  by (simp add: has_integral_localized_vector_derivative has_contour_integral_def)
lp15@61738
   846
lp15@61738
   847
lemma contour_integrable_on:
lp15@61738
   848
     "f contour_integrable_on g \<longleftrightarrow>
lp15@60809
   849
      (\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
lp15@61738
   850
  by (simp add: has_contour_integral integrable_on_def contour_integrable_on_def)
lp15@60809
   851
lp15@60809
   852
subsection\<open>Reversing a path\<close>
lp15@60809
   853
lp15@60809
   854
lemma valid_path_imp_reverse:
lp15@60809
   855
  assumes "valid_path g"
lp15@60809
   856
    shows "valid_path(reversepath g)"
lp15@60809
   857
proof -
lp15@68296
   858
  obtain S where "finite S" and S: "g C1_differentiable_on ({0..1} - S)"
lp15@61190
   859
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
lp15@68296
   860
  then have "finite ((-) 1 ` S)"
lp15@68296
   861
    by auto
lp15@68296
   862
  moreover have "(reversepath g C1_differentiable_on ({0..1} - (-) 1 ` S))"
lp15@68296
   863
    unfolding reversepath_def
lp15@61190
   864
    apply (rule C1_differentiable_compose [of "\<lambda>x::real. 1-x" _ g, unfolded o_def])
lp15@68296
   865
    using S
lp15@68296
   866
    by (force simp: finite_vimageI inj_on_def C1_differentiable_on_eq continuous_on_const elim!: continuous_on_subset)+
lp15@68296
   867
  ultimately show ?thesis using assms
lp15@61190
   868
    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def path_def [symmetric])
lp15@60809
   869
qed
lp15@60809
   870
lp15@62540
   871
lemma valid_path_reversepath [simp]: "valid_path(reversepath g) \<longleftrightarrow> valid_path g"
lp15@60809
   872
  using valid_path_imp_reverse by force
lp15@60809
   873
lp15@61738
   874
lemma has_contour_integral_reversepath:
lp15@68296
   875
  assumes "valid_path g" and f: "(f has_contour_integral i) g"
lp15@61738
   876
    shows "(f has_contour_integral (-i)) (reversepath g)"
lp15@60809
   877
proof -
lp15@68296
   878
  { fix S x
lp15@68296
   879
    assume xs: "g C1_differentiable_on ({0..1} - S)" "x \<notin> (-) 1 ` S" "0 \<le> x" "x \<le> 1"
lp15@68296
   880
    have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
lp15@60809
   881
            - vector_derivative g (at (1 - x) within {0..1})"
lp15@68296
   882
    proof -
lp15@68296
   883
      obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
lp15@68296
   884
        using xs
lp15@68296
   885
        by (force simp: has_vector_derivative_def C1_differentiable_on_def)
lp15@68339
   886
      have "(g \<circ> (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
lp15@68296
   887
        by (intro vector_diff_chain_within has_vector_derivative_at_within [OF f'] derivative_eq_intros | simp)+
lp15@68296
   888
      then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)"
lp15@68296
   889
        by (simp add: o_def)
lp15@68296
   890
      show ?thesis
lp15@68296
   891
        using xs
lp15@68296
   892
        by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
lp15@68296
   893
    qed
lp15@60809
   894
  } note * = this
lp15@68296
   895
  obtain S where S: "continuous_on {0..1} g" "finite S" "g C1_differentiable_on {0..1} - S"
lp15@68296
   896
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
lp15@68296
   897
  have "((\<lambda>x. - (f (g (1 - x)) * vector_derivative g (at (1 - x) within {0..1}))) has_integral -i)
lp15@68296
   898
       {0..1}"
lp15@68296
   899
    using has_integral_affinity01 [where m= "-1" and c=1, OF f [unfolded has_contour_integral_def]]
lp15@68296
   900
    by (simp add: has_integral_neg)
lp15@68296
   901
  then show ?thesis 
lp15@68296
   902
    using S
lp15@68296
   903
    apply (clarsimp simp: reversepath_def has_contour_integral_def)
lp15@68296
   904
    apply (rule_tac S = "(\<lambda>x. 1 - x) ` S" in has_integral_spike_finite)
lp15@68296
   905
      apply (auto simp: *)
lp15@60809
   906
    done
lp15@60809
   907
qed
lp15@60809
   908
lp15@61738
   909
lemma contour_integrable_reversepath:
lp15@61738
   910
    "valid_path g \<Longrightarrow> f contour_integrable_on g \<Longrightarrow> f contour_integrable_on (reversepath g)"
lp15@61738
   911
  using has_contour_integral_reversepath contour_integrable_on_def by blast
lp15@61738
   912
lp15@61738
   913
lemma contour_integrable_reversepath_eq:
lp15@61738
   914
    "valid_path g \<Longrightarrow> (f contour_integrable_on (reversepath g) \<longleftrightarrow> f contour_integrable_on g)"
lp15@61738
   915
  using contour_integrable_reversepath valid_path_reversepath by fastforce
lp15@61738
   916
lp15@61738
   917
lemma contour_integral_reversepath:
lp15@62463
   918
  assumes "valid_path g"
lp15@62463
   919
    shows "contour_integral (reversepath g) f = - (contour_integral g f)"
lp15@62463
   920
proof (cases "f contour_integrable_on g")
lp15@62463
   921
  case True then show ?thesis
lp15@62463
   922
    by (simp add: assms contour_integral_unique has_contour_integral_integral has_contour_integral_reversepath)
lp15@62463
   923
next
lp15@62463
   924
  case False then have "~ f contour_integrable_on (reversepath g)"
lp15@62463
   925
    by (simp add: assms contour_integrable_reversepath_eq)
lp15@62463
   926
  with False show ?thesis by (simp add: not_integrable_contour_integral)
lp15@62463
   927
qed
lp15@60809
   928
lp15@60809
   929
lp15@60809
   930
subsection\<open>Joining two paths together\<close>
lp15@60809
   931
lp15@60809
   932
lemma valid_path_join:
lp15@60809
   933
  assumes "valid_path g1" "valid_path g2" "pathfinish g1 = pathstart g2"
lp15@60809
   934
    shows "valid_path(g1 +++ g2)"
lp15@60809
   935
proof -
lp15@60809
   936
  have "g1 1 = g2 0"
lp15@60809
   937
    using assms by (auto simp: pathfinish_def pathstart_def)
lp15@68339
   938
  moreover have "(g1 \<circ> (\<lambda>x. 2*x)) piecewise_C1_differentiable_on {0..1/2}"
lp15@61190
   939
    apply (rule piecewise_C1_differentiable_compose)
lp15@60809
   940
    using assms
lp15@61190
   941
    apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths)
lp15@68339
   942
    apply (force intro: finite_vimageI [where h = "( *)2"] inj_onI)
lp15@60809
   943
    done
lp15@68339
   944
  moreover have "(g2 \<circ> (\<lambda>x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}"
lp15@61190
   945
    apply (rule piecewise_C1_differentiable_compose)
lp15@61190
   946
    using assms unfolding valid_path_def piecewise_C1_differentiable_on_def
lp15@61190
   947
    by (auto intro!: continuous_intros finite_vimageI [where h = "(\<lambda>x. 2*x - 1)"] inj_onI
lp15@61190
   948
             simp: image_affinity_atLeastAtMost_diff continuous_on_joinpaths)
lp15@60809
   949
  ultimately show ?thesis
lp15@60809
   950
    apply (simp only: valid_path_def continuous_on_joinpaths joinpaths_def)
lp15@61190
   951
    apply (rule piecewise_C1_differentiable_cases)
lp15@60809
   952
    apply (auto simp: o_def)
lp15@60809
   953
    done
lp15@60809
   954
qed
lp15@60809
   955
lp15@61190
   956
lemma valid_path_join_D1:
lp15@61190
   957
  fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   958
  shows "valid_path (g1 +++ g2) \<Longrightarrow> valid_path g1"
lp15@61190
   959
  unfolding valid_path_def
lp15@61190
   960
  by (rule piecewise_C1_differentiable_D1)
lp15@60809
   961
lp15@61190
   962
lemma valid_path_join_D2:
lp15@61190
   963
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   964
  shows "\<lbrakk>valid_path (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> valid_path g2"
lp15@61190
   965
  unfolding valid_path_def
lp15@61190
   966
  by (rule piecewise_C1_differentiable_D2)
lp15@60809
   967
lp15@60809
   968
lemma valid_path_join_eq [simp]:
lp15@61190
   969
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   970
  shows "pathfinish g1 = pathstart g2 \<Longrightarrow> (valid_path(g1 +++ g2) \<longleftrightarrow> valid_path g1 \<and> valid_path g2)"
lp15@60809
   971
  using valid_path_join_D1 valid_path_join_D2 valid_path_join by blast
lp15@60809
   972
lp15@61738
   973
lemma has_contour_integral_join:
lp15@61738
   974
  assumes "(f has_contour_integral i1) g1" "(f has_contour_integral i2) g2"
lp15@60809
   975
          "valid_path g1" "valid_path g2"
lp15@61738
   976
    shows "(f has_contour_integral (i1 + i2)) (g1 +++ g2)"
lp15@60809
   977
proof -
lp15@60809
   978
  obtain s1 s2
lp15@60809
   979
    where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
lp15@60809
   980
      and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
lp15@60809
   981
    using assms
lp15@61190
   982
    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   983
  have 1: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
lp15@60809
   984
   and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
lp15@60809
   985
    using assms
lp15@61738
   986
    by (auto simp: has_contour_integral)
lp15@60809
   987
  have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
lp15@60809
   988
   and i2: "((\<lambda>x. (2*f (g2 (2*x - 1))) * vector_derivative g2 (at (2*x - 1))) has_integral i2) {1/2..1}"
lp15@60809
   989
    using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
lp15@60809
   990
          has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
lp15@60809
   991
    by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
lp15@60809
   992
  have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
lp15@60809
   993
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
lp15@60809
   994
            2 *\<^sub>R vector_derivative g1 (at (z*2))" for z
paulson@62087
   995
    apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>z - 1/2\<bar>"]])
nipkow@62390
   996
    apply (simp_all add: dist_real_def abs_if split: if_split_asm)
lp15@60809
   997
    apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
lp15@60809
   998
    apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
lp15@60809
   999
    using s1
lp15@60809
  1000
    apply (auto simp: algebra_simps vector_derivative_works)
lp15@60809
  1001
    done
lp15@60809
  1002
  have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
lp15@60809
  1003
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
lp15@60809
  1004
            2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z
paulson@62087
  1005
    apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2 (2*x - 1))" and d = "\<bar>z - 1/2\<bar>"]])
nipkow@62390
  1006
    apply (simp_all add: dist_real_def abs_if split: if_split_asm)
lp15@60809
  1007
    apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
lp15@60809
  1008
    apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
lp15@60809
  1009
    using s2
lp15@60809
  1010
    apply (auto simp: algebra_simps vector_derivative_works)
lp15@60809
  1011
    done
lp15@60809
  1012
  have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
lp15@68339
  1013
    apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) (( *)2 -` s1)"])
lp15@60809
  1014
    using s1
lp15@68339
  1015
    apply (force intro: finite_vimageI [where h = "( *)2"] inj_onI)
lp15@60809
  1016
    apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
lp15@60809
  1017
    done
lp15@60809
  1018
  moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
lp15@60809
  1019
    apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
lp15@60809
  1020
    using s2
lp15@60809
  1021
    apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
lp15@60809
  1022
    apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
lp15@60809
  1023
    done
lp15@60809
  1024
  ultimately
lp15@60809
  1025
  show ?thesis
lp15@61738
  1026
    apply (simp add: has_contour_integral)
lp15@60809
  1027
    apply (rule has_integral_combine [where c = "1/2"], auto)
lp15@60809
  1028
    done
lp15@60809
  1029
qed
lp15@60809
  1030
lp15@61738
  1031
lemma contour_integrable_joinI:
lp15@61738
  1032
  assumes "f contour_integrable_on g1" "f contour_integrable_on g2"
lp15@60809
  1033
          "valid_path g1" "valid_path g2"
lp15@61738
  1034
    shows "f contour_integrable_on (g1 +++ g2)"
lp15@60809
  1035
  using assms
lp15@61738
  1036
  by (meson has_contour_integral_join contour_integrable_on_def)
lp15@61738
  1037
lp15@61738
  1038
lemma contour_integrable_joinD1:
lp15@61738
  1039
  assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g1"
lp15@61738
  1040
    shows "f contour_integrable_on g1"
lp15@60809
  1041
proof -
lp15@60809
  1042
  obtain s1
lp15@60809
  1043
    where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
lp15@61190
  1044
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
  1045
  have "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
lp15@60809
  1046
    using assms
lp15@61738
  1047
    apply (auto simp: contour_integrable_on)
lp15@60809
  1048
    apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
lp15@60809
  1049
    apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
lp15@60809
  1050
    done
lp15@60809
  1051
  then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
lp15@61190
  1052
    by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
lp15@60809
  1053
  have g1: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow>
lp15@60809
  1054
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
lp15@60809
  1055
            2 *\<^sub>R vector_derivative g1 (at z)"  for z
paulson@62087
  1056
    apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>(z-1)/2\<bar>"]])
nipkow@62390
  1057
    apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
lp15@60809
  1058
    apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def])
lp15@60809
  1059
    using s1
lp15@60809
  1060
    apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
lp15@60809
  1061
    done
lp15@60809
  1062
  show ?thesis
lp15@60809
  1063
    using s1
lp15@61738
  1064
    apply (auto simp: contour_integrable_on)
lp15@60809
  1065
    apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
lp15@60809
  1066
    apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
lp15@60809
  1067
    done
lp15@60809
  1068
qed
lp15@60809
  1069
lp15@61738
  1070
lemma contour_integrable_joinD2:
lp15@61738
  1071
  assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g2"
lp15@61738
  1072
    shows "f contour_integrable_on g2"
lp15@60809
  1073
proof -
lp15@60809
  1074
  obtain s2
lp15@60809
  1075
    where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
lp15@61190
  1076
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
  1077
  have "(\<lambda>x. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}"
lp15@60809
  1078
    using assms
lp15@61738
  1079
    apply (auto simp: contour_integrable_on)
lp15@60809
  1080
    apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
lp15@60809
  1081
    apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
lp15@60809
  1082
    apply (simp add: image_affinity_atLeastAtMost_diff)
lp15@60809
  1083
    done
lp15@60809
  1084
  then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2)))
lp15@60809
  1085
                integrable_on {0..1}"
lp15@60809
  1086
    by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
lp15@60809
  1087
  have g2: "\<lbrakk>0 < z; z < 1; z \<notin> s2\<rbrakk> \<Longrightarrow>
lp15@60809
  1088
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) =
lp15@60809
  1089
            2 *\<^sub>R vector_derivative g2 (at z)" for z
paulson@62087
  1090
    apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2(2*x-1))" and d = "\<bar>z/2\<bar>"]])
nipkow@62390
  1091
    apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
lp15@60809
  1092
    apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def])
lp15@60809
  1093
    using s2
lp15@60809
  1094
    apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left
lp15@60809
  1095
                      vector_derivative_works add_divide_distrib)
lp15@60809
  1096
    done
lp15@60809
  1097
  show ?thesis
lp15@60809
  1098
    using s2
lp15@61738
  1099
    apply (auto simp: contour_integrable_on)
lp15@60809
  1100
    apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
lp15@60809
  1101
    apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
lp15@60809
  1102
    done
lp15@60809
  1103
qed
lp15@60809
  1104
lp15@61738
  1105
lemma contour_integrable_join [simp]:
lp15@60809
  1106
  shows
lp15@60809
  1107
    "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
lp15@61738
  1108
     \<Longrightarrow> f contour_integrable_on (g1 +++ g2) \<longleftrightarrow> f contour_integrable_on g1 \<and> f contour_integrable_on g2"
lp15@61738
  1109
using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast
lp15@61738
  1110
lp15@61738
  1111
lemma contour_integral_join [simp]:
lp15@60809
  1112
  shows
lp15@61738
  1113
    "\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
lp15@61738
  1114
        \<Longrightarrow> contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f"
lp15@61738
  1115
  by (simp add: has_contour_integral_integral has_contour_integral_join contour_integral_unique)
lp15@60809
  1116
lp15@60809
  1117
lp15@60809
  1118
subsection\<open>Shifting the starting point of a (closed) path\<close>
lp15@60809
  1119
lp15@60809
  1120
lemma shiftpath_alt_def: "shiftpath a f = (\<lambda>x. if x \<le> 1-a then f (a + x) else f (a + x - 1))"
lp15@60809
  1121
  by (auto simp: shiftpath_def)
lp15@60809
  1122
lp15@60809
  1123
lemma valid_path_shiftpath [intro]:
lp15@60809
  1124
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@60809
  1125
    shows "valid_path(shiftpath a g)"
lp15@60809
  1126
  using assms
lp15@60809
  1127
  apply (auto simp: valid_path_def shiftpath_alt_def)
lp15@61190
  1128
  apply (rule piecewise_C1_differentiable_cases)
lp15@60809
  1129
  apply (auto simp: algebra_simps)
lp15@61190
  1130
  apply (rule piecewise_C1_differentiable_affine [of g 1 a, simplified o_def scaleR_one])
lp15@61190
  1131
  apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
lp15@61190
  1132
  apply (rule piecewise_C1_differentiable_affine [of g 1 "a-1", simplified o_def scaleR_one algebra_simps])
lp15@61190
  1133
  apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
lp15@60809
  1134
  done
lp15@60809
  1135
lp15@61738
  1136
lemma has_contour_integral_shiftpath:
lp15@61738
  1137
  assumes f: "(f has_contour_integral i) g" "valid_path g"
lp15@60809
  1138
      and a: "a \<in> {0..1}"
lp15@61738
  1139
    shows "(f has_contour_integral i) (shiftpath a g)"
lp15@60809
  1140
proof -
lp15@60809
  1141
  obtain s
lp15@60809
  1142
    where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
lp15@61190
  1143
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
  1144
  have *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
lp15@61738
  1145
    using assms by (auto simp: has_contour_integral)
lp15@60809
  1146
  then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) +
lp15@60809
  1147
                    integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
lp15@60809
  1148
    apply (rule has_integral_unique)
lp15@60809
  1149
    apply (subst add.commute)
hoelzl@63594
  1150
    apply (subst integral_combine)
lp15@60809
  1151
    using assms * integral_unique by auto
lp15@60809
  1152
  { fix x
lp15@60809
  1153
    have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow>
lp15@60809
  1154
         vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
lp15@60809
  1155
      unfolding shiftpath_def
paulson@62087
  1156
      apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x))" and d = "dist(1-a) x"]])
nipkow@62390
  1157
        apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
lp15@60809
  1158
      apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one])
lp15@60809
  1159
       apply (intro derivative_eq_intros | simp)+
lp15@60809
  1160
      using g
lp15@60809
  1161
       apply (drule_tac x="x+a" in bspec)
lp15@60809
  1162
      using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
lp15@60809
  1163
      done
lp15@60809
  1164
  } note vd1 = this
lp15@60809
  1165
  { fix x
lp15@60809
  1166
    have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow>
lp15@60809
  1167
          vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
lp15@60809
  1168
      unfolding shiftpath_def
paulson@62087
  1169
      apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x-1))" and d = "dist (1-a) x"]])
nipkow@62390
  1170
        apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
lp15@60809
  1171
      apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one])
lp15@60809
  1172
       apply (intro derivative_eq_intros | simp)+
lp15@60809
  1173
      using g
lp15@60809
  1174
      apply (drule_tac x="x+a-1" in bspec)
lp15@60809
  1175
      using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
lp15@60809
  1176
      done
lp15@60809
  1177
  } note vd2 = this
lp15@60809
  1178
  have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
lp15@60809
  1179
    using * a   by (fastforce intro: integrable_subinterval_real)
lp15@60809
  1180
  have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
lp15@60809
  1181
    apply (rule integrable_subinterval_real)
lp15@60809
  1182
    using * a by auto
lp15@60809
  1183
  have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
lp15@60809
  1184
        has_integral  integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {0..1 - a}"
lp15@60809
  1185
    apply (rule has_integral_spike_finite
lp15@65587
  1186
             [where S = "{1-a} \<union> (\<lambda>x. x-a) ` s" and f = "\<lambda>x. f(g(a+x)) * vector_derivative g (at(a+x))"])
lp15@60809
  1187
      using s apply blast
lp15@60809
  1188
     using a apply (auto simp: algebra_simps vd1)
lp15@60809
  1189
     apply (force simp: shiftpath_def add.commute)
lp15@60809
  1190
    using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]]
lp15@60809
  1191
    apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute)
lp15@60809
  1192
    done
lp15@60809
  1193
  moreover
lp15@60809
  1194
  have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
lp15@60809
  1195
        has_integral  integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {1 - a..1}"
lp15@60809
  1196
    apply (rule has_integral_spike_finite
lp15@65587
  1197
             [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))"])
lp15@60809
  1198
      using s apply blast
lp15@60809
  1199
     using a apply (auto simp: algebra_simps vd2)
lp15@60809
  1200
     apply (force simp: shiftpath_def add.commute)
lp15@60809
  1201
    using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
lp15@60809
  1202
    apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified])
lp15@60809
  1203
    apply (simp add: algebra_simps)
lp15@60809
  1204
    done
lp15@60809
  1205
  ultimately show ?thesis
lp15@60809
  1206
    using a
lp15@61738
  1207
    by (auto simp: i has_contour_integral intro: has_integral_combine [where c = "1-a"])
lp15@60809
  1208
qed
lp15@60809
  1209
lp15@61738
  1210
lemma has_contour_integral_shiftpath_D:
lp15@61738
  1211
  assumes "(f has_contour_integral i) (shiftpath a g)"
lp15@60809
  1212
          "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@61738
  1213
    shows "(f has_contour_integral i) g"
lp15@60809
  1214
proof -
lp15@60809
  1215
  obtain s
lp15@60809
  1216
    where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
lp15@61190
  1217
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
  1218
  { fix x
lp15@60809
  1219
    assume x: "0 < x" "x < 1" "x \<notin> s"
lp15@60809
  1220
    then have gx: "g differentiable at x"
lp15@60809
  1221
      using g by auto
lp15@60809
  1222
    have "vector_derivative g (at x within {0..1}) =
lp15@60809
  1223
          vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
lp15@60809
  1224
      apply (rule vector_derivative_at_within_ivl
lp15@60809
  1225
                  [OF has_vector_derivative_transform_within_open
lp15@68239
  1226
                      [where f = "(shiftpath (1 - a) (shiftpath a g))" and S = "{0<..<1}-s"]])
lp15@60809
  1227
      using s g assms x
lp15@60809
  1228
      apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
lp15@68296
  1229
                        at_within_interior [of _ "{0..1}"] vector_derivative_works [symmetric])
paulson@62087
  1230
      apply (rule differentiable_transform_within [OF gx, of "min x (1-x)"])
nipkow@62390
  1231
      apply (auto simp: dist_real_def shiftpath_shiftpath abs_if split: if_split_asm)
lp15@60809
  1232
      done
lp15@60809
  1233
  } note vd = this
lp15@61738
  1234
  have fi: "(f has_contour_integral i) (shiftpath (1 - a) (shiftpath a g))"
lp15@61738
  1235
    using assms  by (auto intro!: has_contour_integral_shiftpath)
lp15@60809
  1236
  show ?thesis
lp15@61738
  1237
    apply (simp add: has_contour_integral_def)
lp15@61738
  1238
    apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _  fi [unfolded has_contour_integral_def]])
lp15@60809
  1239
    using s assms vd
lp15@60809
  1240
    apply (auto simp: Path_Connected.shiftpath_shiftpath)
lp15@60809
  1241
    done
lp15@60809
  1242
qed
lp15@60809
  1243
lp15@61738
  1244
lemma has_contour_integral_shiftpath_eq:
lp15@60809
  1245
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@61738
  1246
    shows "(f has_contour_integral i) (shiftpath a g) \<longleftrightarrow> (f has_contour_integral i) g"
lp15@61738
  1247
  using assms has_contour_integral_shiftpath has_contour_integral_shiftpath_D by blast
lp15@61738
  1248
lp15@62463
  1249
lemma contour_integrable_on_shiftpath_eq:
lp15@62463
  1250
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@62463
  1251
    shows "f contour_integrable_on (shiftpath a g) \<longleftrightarrow> f contour_integrable_on g"
lp15@62463
  1252
using assms contour_integrable_on_def has_contour_integral_shiftpath_eq by auto
lp15@62463
  1253
lp15@61738
  1254
lemma contour_integral_shiftpath:
lp15@60809
  1255
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@61738
  1256
    shows "contour_integral (shiftpath a g) f = contour_integral g f"
lp15@62534
  1257
   using assms
lp15@62463
  1258
   by (simp add: contour_integral_def contour_integrable_on_def has_contour_integral_shiftpath_eq)
lp15@60809
  1259
lp15@60809
  1260
lp15@60809
  1261
subsection\<open>More about straight-line paths\<close>
lp15@60809
  1262
lp15@60809
  1263
lemma has_vector_derivative_linepath_within:
lp15@60809
  1264
    "(linepath a b has_vector_derivative (b - a)) (at x within s)"
lp15@60809
  1265
apply (simp add: linepath_def has_vector_derivative_def algebra_simps)
lp15@60809
  1266
apply (rule derivative_eq_intros | simp)+
lp15@60809
  1267
done
lp15@60809
  1268
lp15@60809
  1269
lemma vector_derivative_linepath_within:
lp15@60809
  1270
    "x \<in> {0..1} \<Longrightarrow> vector_derivative (linepath a b) (at x within {0..1}) = b - a"
immler@67685
  1271
  apply (rule vector_derivative_within_cbox [of 0 "1::real", simplified])
lp15@60809
  1272
  apply (auto simp: has_vector_derivative_linepath_within)
lp15@60809
  1273
  done
lp15@60809
  1274
lp15@61190
  1275
lemma vector_derivative_linepath_at [simp]: "vector_derivative (linepath a b) (at x) = b - a"
lp15@60809
  1276
  by (simp add: has_vector_derivative_linepath_within vector_derivative_at)
lp15@60809
  1277
lp15@61190
  1278
lemma valid_path_linepath [iff]: "valid_path (linepath a b)"
lp15@61190
  1279
  apply (simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_on_linepath)
lp15@61190
  1280
  apply (rule_tac x="{}" in exI)
lp15@61190
  1281
  apply (simp add: differentiable_on_def differentiable_def)
lp15@61190
  1282
  using has_vector_derivative_def has_vector_derivative_linepath_within
lp15@61190
  1283
  apply (fastforce simp add: continuous_on_eq_continuous_within)
lp15@61190
  1284
  done
lp15@61190
  1285
lp15@61738
  1286
lemma has_contour_integral_linepath:
lp15@61738
  1287
  shows "(f has_contour_integral i) (linepath a b) \<longleftrightarrow>
lp15@60809
  1288
         ((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
lp15@61738
  1289
  by (simp add: has_contour_integral vector_derivative_linepath_at)
lp15@60809
  1290
lp15@60809
  1291
lemma linepath_in_path:
lp15@60809
  1292
  shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b"
lp15@60809
  1293
  by (auto simp: segment linepath_def)
lp15@60809
  1294
lp15@60809
  1295
lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
lp15@60809
  1296
  by (auto simp: segment linepath_def)
lp15@60809
  1297
lp15@60809
  1298
lemma linepath_in_convex_hull:
lp15@60809
  1299
    fixes x::real
lp15@60809
  1300
    assumes a: "a \<in> convex hull s"
lp15@60809
  1301
        and b: "b \<in> convex hull s"
lp15@60809
  1302
        and x: "0\<le>x" "x\<le>1"
lp15@60809
  1303
       shows "linepath a b x \<in> convex hull s"
lp15@60809
  1304
  apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD])
lp15@60809
  1305
  using x
lp15@60809
  1306
  apply (auto simp: linepath_image_01 [symmetric])
lp15@60809
  1307
  done
lp15@60809
  1308
lp15@60809
  1309
lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
lp15@60809
  1310
  by (simp add: linepath_def)
lp15@60809
  1311
lp15@60809
  1312
lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
lp15@60809
  1313
  by (simp add: linepath_def)
lp15@60809
  1314
lp15@61738
  1315
lemma has_contour_integral_trivial [iff]: "(f has_contour_integral 0) (linepath a a)"
lp15@61738
  1316
  by (simp add: has_contour_integral_linepath)
lp15@61738
  1317
lp15@68296
  1318
lemma has_contour_integral_trivial_iff [simp]: "(f has_contour_integral i) (linepath a a) \<longleftrightarrow> i=0"
lp15@68296
  1319
  using has_contour_integral_unique by blast
lp15@68296
  1320
lp15@61738
  1321
lemma contour_integral_trivial [simp]: "contour_integral (linepath a a) f = 0"
lp15@61738
  1322
  using has_contour_integral_trivial contour_integral_unique by blast
lp15@60809
  1323
lp15@60809
  1324
lp15@60809
  1325
subsection\<open>Relation to subpath construction\<close>
lp15@60809
  1326
lp15@60809
  1327
lemma valid_path_subpath:
lp15@60809
  1328
  fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
lp15@60809
  1329
  assumes "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
lp15@60809
  1330
    shows "valid_path(subpath u v g)"
lp15@60809
  1331
proof (cases "v=u")
lp15@60809
  1332
  case True
lp15@60809
  1333
  then show ?thesis
lp15@61190
  1334
    unfolding valid_path_def subpath_def
lp15@61190
  1335
    by (force intro: C1_differentiable_on_const C1_differentiable_imp_piecewise)
lp15@60809
  1336
next
lp15@60809
  1337
  case False
lp15@68339
  1338
  have "(g \<circ> (\<lambda>x. ((v-u) * x + u))) piecewise_C1_differentiable_on {0..1}"
lp15@61190
  1339
    apply (rule piecewise_C1_differentiable_compose)
lp15@61190
  1340
    apply (simp add: C1_differentiable_imp_piecewise)
lp15@60809
  1341
     apply (simp add: image_affinity_atLeastAtMost)
lp15@60809
  1342
    using assms False
lp15@61190
  1343
    apply (auto simp: algebra_simps valid_path_def piecewise_C1_differentiable_on_subset)
lp15@60809
  1344
    apply (subst Int_commute)
lp15@60809
  1345
    apply (auto simp: inj_on_def algebra_simps crossproduct_eq finite_vimage_IntI)
lp15@60809
  1346
    done
lp15@60809
  1347
  then show ?thesis
lp15@60809
  1348
    by (auto simp: o_def valid_path_def subpath_def)
lp15@60809
  1349
qed
lp15@60809
  1350
lp15@61738
  1351
lemma has_contour_integral_subpath_refl [iff]: "(f has_contour_integral 0) (subpath u u g)"
lp15@61738
  1352
  by (simp add: has_contour_integral subpath_def)
lp15@61738
  1353
lp15@61738
  1354
lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)"
lp15@61738
  1355
  using has_contour_integral_subpath_refl contour_integrable_on_def by blast
lp15@61738
  1356
lp15@61738
  1357
lemma contour_integral_subpath_refl [simp]: "contour_integral (subpath u u g) f = 0"
lp15@61738
  1358
  by (simp add: has_contour_integral_subpath_refl contour_integral_unique)
lp15@61738
  1359
lp15@61738
  1360
lemma has_contour_integral_subpath:
lp15@61738
  1361
  assumes f: "f contour_integrable_on g" and g: "valid_path g"
lp15@60809
  1362
      and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
lp15@61738
  1363
    shows "(f has_contour_integral  integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x)))
lp15@60809
  1364
           (subpath u v g)"
lp15@60809
  1365
proof (cases "v=u")
lp15@60809
  1366
  case True
lp15@60809
  1367
  then show ?thesis
lp15@61738
  1368
    using f   by (simp add: contour_integrable_on_def subpath_def has_contour_integral)
lp15@60809
  1369
next
lp15@60809
  1370
  case False
lp15@60809
  1371
  obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s"
lp15@61190
  1372
    using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
lp15@60809
  1373
  have *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
lp15@60809
  1374
            has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t)))
lp15@60809
  1375
           {0..1}"
lp15@60809
  1376
    using f uv
lp15@61738
  1377
    apply (simp add: contour_integrable_on subpath_def has_contour_integral)
lp15@60809
  1378
    apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
lp15@60809
  1379
    apply (simp_all add: has_integral_integral)
lp15@60809
  1380
    apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
lp15@60809
  1381
    apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
lp15@60809
  1382
    apply (simp add: divide_simps False)
lp15@60809
  1383
    done
lp15@60809
  1384
  { fix x
lp15@60809
  1385
    have "x \<in> {0..1} \<Longrightarrow>
lp15@60809
  1386
           x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
lp15@60809
  1387
           vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u))"
lp15@60809
  1388
      apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
lp15@60809
  1389
      apply (intro derivative_eq_intros | simp)+
lp15@60809
  1390
      apply (cut_tac s [of "(v - u) * x + u"])
lp15@60809
  1391
      using uv mult_left_le [of x "v-u"]
lp15@60809
  1392
      apply (auto simp:  vector_derivative_works)
lp15@60809
  1393
      done
lp15@60809
  1394
  } note vd = this
lp15@60809
  1395
  show ?thesis
lp15@60809
  1396
    apply (cut_tac has_integral_cmul [OF *, where c = "v-u"])
lp15@60809
  1397
    using fs assms
lp15@61738
  1398
    apply (simp add: False subpath_def has_contour_integral)
lp15@65587
  1399
    apply (rule_tac S = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
lp15@60809
  1400
    apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real)
lp15@60809
  1401
    done
lp15@60809
  1402
qed
lp15@60809
  1403
lp15@61738
  1404
lemma contour_integrable_subpath:
lp15@61738
  1405
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
lp15@61738
  1406
    shows "f contour_integrable_on (subpath u v g)"
lp15@60809
  1407
  apply (cases u v rule: linorder_class.le_cases)
lp15@61738
  1408
   apply (metis contour_integrable_on_def has_contour_integral_subpath [OF assms])
lp15@60809
  1409
  apply (subst reversepath_subpath [symmetric])
lp15@61738
  1410
  apply (rule contour_integrable_reversepath)
lp15@60809
  1411
   using assms apply (blast intro: valid_path_subpath)
lp15@61738
  1412
  apply (simp add: contour_integrable_on_def)
lp15@61738
  1413
  using assms apply (blast intro: has_contour_integral_subpath)
lp15@60809
  1414
  done
lp15@60809
  1415
lp15@61738
  1416
lemma has_integral_contour_integral_subpath:
lp15@61738
  1417
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
lp15@60809
  1418
    shows "(((\<lambda>x. f(g x) * vector_derivative g (at x)))
lp15@61738
  1419
            has_integral  contour_integral (subpath u v g) f) {u..v}"
lp15@60809
  1420
  using assms
lp15@60809
  1421
  apply (auto simp: has_integral_integrable_integral)
lp15@66507
  1422
  apply (rule integrable_on_subcbox [where a=u and b=v and S = "{0..1}", simplified])
lp15@61738
  1423
  apply (auto simp: contour_integral_unique [OF has_contour_integral_subpath] contour_integrable_on)
lp15@60809
  1424
  done
lp15@60809
  1425
lp15@61738
  1426
lemma contour_integral_subcontour_integral:
lp15@61738
  1427
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
lp15@61738
  1428
    shows "contour_integral (subpath u v g) f =
lp15@60809
  1429
           integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))"
lp15@61738
  1430
  using assms has_contour_integral_subpath contour_integral_unique by blast
lp15@61738
  1431
lp15@61738
  1432
lemma contour_integral_subpath_combine_less:
lp15@61738
  1433
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
lp15@60809
  1434
          "u<v" "v<w"
lp15@61738
  1435
    shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
lp15@61738
  1436
           contour_integral (subpath u w g) f"
lp15@61738
  1437
  using assms apply (auto simp: contour_integral_subcontour_integral)
lp15@60809
  1438
  apply (rule integral_combine, auto)
lp15@66507
  1439
  apply (rule integrable_on_subcbox [where a=u and b=w and S = "{0..1}", simplified])
lp15@61738
  1440
  apply (auto simp: contour_integrable_on)
lp15@60809
  1441
  done
lp15@60809
  1442
lp15@61738
  1443
lemma contour_integral_subpath_combine:
lp15@61738
  1444
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
lp15@61738
  1445
    shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
lp15@61738
  1446
           contour_integral (subpath u w g) f"
lp15@60809
  1447
proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w")
lp15@60809
  1448
  case True
lp15@60809
  1449
    have *: "subpath v u g = reversepath(subpath u v g) \<and>
lp15@60809
  1450
             subpath w u g = reversepath(subpath u w g) \<and>
lp15@60809
  1451
             subpath w v g = reversepath(subpath v w g)"
lp15@60809
  1452
      by (auto simp: reversepath_subpath)
lp15@60809
  1453
    have "u < v \<and> v < w \<or>
lp15@60809
  1454
          u < w \<and> w < v \<or>
lp15@60809
  1455
          v < u \<and> u < w \<or>
lp15@60809
  1456
          v < w \<and> w < u \<or>
lp15@60809
  1457
          w < u \<and> u < v \<or>
lp15@60809
  1458
          w < v \<and> v < u"
lp15@60809
  1459
      using True assms by linarith
lp15@60809
  1460
    with assms show ?thesis
lp15@61738
  1461
      using contour_integral_subpath_combine_less [of f g u v w]
lp15@61738
  1462
            contour_integral_subpath_combine_less [of f g u w v]
lp15@61738
  1463
            contour_integral_subpath_combine_less [of f g v u w]
lp15@61738
  1464
            contour_integral_subpath_combine_less [of f g v w u]
lp15@61738
  1465
            contour_integral_subpath_combine_less [of f g w u v]
lp15@61738
  1466
            contour_integral_subpath_combine_less [of f g w v u]
lp15@60809
  1467
      apply simp
lp15@60809
  1468
      apply (elim disjE)
lp15@61738
  1469
      apply (auto simp: * contour_integral_reversepath contour_integrable_subpath
lp15@60809
  1470
                   valid_path_reversepath valid_path_subpath algebra_simps)
lp15@60809
  1471
      done
lp15@60809
  1472
next
lp15@60809
  1473
  case False
lp15@60809
  1474
  then show ?thesis
lp15@61738
  1475
    apply (auto simp: contour_integral_subpath_refl)
lp15@60809
  1476
    using assms
lp15@61738
  1477
    by (metis eq_neg_iff_add_eq_0 contour_integrable_subpath contour_integral_reversepath reversepath_subpath valid_path_subpath)
lp15@60809
  1478
qed
lp15@60809
  1479
lp15@61738
  1480
lemma contour_integral_integral:
lp15@62463
  1481
     "contour_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
lp15@62463
  1482
  by (simp add: contour_integral_def integral_def has_contour_integral contour_integrable_on)
lp15@60809
  1483
lp15@60809
  1484
lp15@60809
  1485
text\<open>Cauchy's theorem where there's a primitive\<close>
lp15@60809
  1486
lp15@61738
  1487
lemma contour_integral_primitive_lemma:
lp15@60809
  1488
  fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
lp15@60809
  1489
  assumes "a \<le> b"
lp15@60809
  1490
      and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
lp15@60809
  1491
      and "g piecewise_differentiable_on {a..b}"  "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
lp15@60809
  1492
    shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b}))
lp15@60809
  1493
             has_integral (f(g b) - f(g a))) {a..b}"
lp15@60809
  1494
proof -
lp15@61190
  1495
  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"
lp15@60809
  1496
    using assms by (auto simp: piecewise_differentiable_on_def)
lp15@60809
  1497
  have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
lp15@60809
  1498
    apply (rule continuous_on_compose [OF cg, unfolded o_def])
lp15@60809
  1499
    using assms
lp15@62534
  1500
    apply (metis field_differentiable_def field_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
lp15@60809
  1501
    done
lp15@60809
  1502
  { fix x::real
lp15@60809
  1503
    assume a: "a < x" and b: "x < b" and xk: "x \<notin> k"
lp15@60809
  1504
    then have "g differentiable at x within {a..b}"
lp15@60809
  1505
      using k by (simp add: differentiable_at_withinI)
lp15@60809
  1506
    then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
lp15@60809
  1507
      by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
lp15@60809
  1508
    then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
lp15@60809
  1509
      by (simp add: has_vector_derivative_def scaleR_conv_of_real)
lp15@60809
  1510
    have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
lp15@60809
  1511
      using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
lp15@68339
  1512
    then have fdiff: "(f has_derivative ( *) (f' (g x))) (at (g x) within g ` {a..b})"
lp15@60809
  1513
      by (simp add: has_field_derivative_def)
lp15@60809
  1514
    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})"
lp15@60809
  1515
      using diff_chain_within [OF gdiff fdiff]
lp15@60809
  1516
      by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
lp15@60809
  1517
  } note * = this
lp15@60809
  1518
  show ?thesis
lp15@60809
  1519
    apply (rule fundamental_theorem_of_calculus_interior_strong)
lp15@60809
  1520
    using k assms cfg *
lp15@66793
  1521
    apply (auto simp: at_within_Icc_at)
lp15@60809
  1522
    done
lp15@60809
  1523
qed
lp15@60809
  1524
lp15@61738
  1525
lemma contour_integral_primitive:
lp15@60809
  1526
  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
lp15@60809
  1527
      and "valid_path g" "path_image g \<subseteq> s"
lp15@61738
  1528
    shows "(f' has_contour_integral (f(pathfinish g) - f(pathstart g))) g"
lp15@60809
  1529
  using assms
lp15@61738
  1530
  apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_contour_integral_def)
lp15@61738
  1531
  apply (auto intro!: piecewise_C1_imp_differentiable contour_integral_primitive_lemma [of 0 1 s])
lp15@60809
  1532
  done
lp15@60809
  1533
lp15@60809
  1534
corollary Cauchy_theorem_primitive:
lp15@60809
  1535
  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
lp15@60809
  1536
      and "valid_path g"  "path_image g \<subseteq> s" "pathfinish g = pathstart g"
lp15@61738
  1537
    shows "(f' has_contour_integral 0) g"
lp15@60809
  1538
  using assms
lp15@61738
  1539
  by (metis diff_self contour_integral_primitive)
lp15@60809
  1540
lp15@60809
  1541
lp15@60809
  1542
text\<open>Existence of path integral for continuous function\<close>
lp15@61738
  1543
lemma contour_integrable_continuous_linepath:
lp15@60809
  1544
  assumes "continuous_on (closed_segment a b) f"
lp15@61738
  1545
  shows "f contour_integrable_on (linepath a b)"
lp15@60809
  1546
proof -
lp15@68339
  1547
  have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) \<circ> linepath a b)"
lp15@60809
  1548
    apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01)
lp15@60809
  1549
    apply (rule continuous_intros | simp add: assms)+
lp15@60809
  1550
    done
lp15@60809
  1551
  then show ?thesis
lp15@61738
  1552
    apply (simp add: contour_integrable_on_def has_contour_integral_def integrable_on_def [symmetric])
lp15@60809
  1553
    apply (rule integrable_continuous [of 0 "1::real", simplified])
lp15@60809
  1554
    apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"])
lp15@60809
  1555
    apply (auto simp: vector_derivative_linepath_within)
lp15@60809
  1556
    done
lp15@60809
  1557
qed
lp15@60809
  1558
lp15@60809
  1559
lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)"
lp15@60809
  1560
  by (rule has_derivative_imp_has_field_derivative)
lp15@60809
  1561
     (rule derivative_intros | simp)+
lp15@60809
  1562
lp15@61738
  1563
lemma contour_integral_id [simp]: "contour_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2"
lp15@61738
  1564
  apply (rule contour_integral_unique)
lp15@61738
  1565
  using contour_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"]
lp15@60809
  1566
  apply (auto simp: field_simps has_field_der_id)
lp15@60809
  1567
  done
lp15@60809
  1568
lp15@61738
  1569
lemma contour_integrable_on_const [iff]: "(\<lambda>x. c) contour_integrable_on (linepath a b)"
lp15@61738
  1570
  by (simp add: continuous_on_const contour_integrable_continuous_linepath)
lp15@61738
  1571
lp15@61738
  1572
lemma contour_integrable_on_id [iff]: "(\<lambda>x. x) contour_integrable_on (linepath a b)"
lp15@61738
  1573
  by (simp add: continuous_on_id contour_integrable_continuous_linepath)
lp15@60809
  1574
lp15@60809
  1575
lp15@60809
  1576
subsection\<open>Arithmetical combining theorems\<close>
lp15@60809
  1577
lp15@61738
  1578
lemma has_contour_integral_neg:
lp15@61738
  1579
    "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_contour_integral (-i)) g"
lp15@61738
  1580
  by (simp add: has_integral_neg has_contour_integral_def)
lp15@61738
  1581
lp15@61738
  1582
lemma has_contour_integral_add:
lp15@61738
  1583
    "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
lp15@61738
  1584
     \<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_contour_integral (i1 + i2)) g"
lp15@61738
  1585
  by (simp add: has_integral_add has_contour_integral_def algebra_simps)
lp15@61738
  1586
lp15@61738
  1587
lemma has_contour_integral_diff:
lp15@61738
  1588
  "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
lp15@61738
  1589
         \<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_contour_integral (i1 - i2)) g"
lp15@66112
  1590
  by (simp add: has_integral_diff has_contour_integral_def algebra_simps)
lp15@61738
  1591
lp15@61738
  1592
lemma has_contour_integral_lmul:
lp15@61738
  1593
  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_contour_integral (c*i)) g"
lp15@61738
  1594
apply (simp add: has_contour_integral_def)
lp15@60809
  1595
apply (drule has_integral_mult_right)
lp15@60809
  1596
apply (simp add: algebra_simps)
lp15@60809
  1597
done
lp15@60809
  1598
lp15@61738
  1599
lemma has_contour_integral_rmul:
lp15@61738
  1600
  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_contour_integral (i*c)) g"
lp15@61738
  1601
apply (drule has_contour_integral_lmul)
lp15@60809
  1602
apply (simp add: mult.commute)
lp15@60809
  1603
done
lp15@60809
  1604
lp15@61738
  1605
lemma has_contour_integral_div:
lp15@61738
  1606
  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_contour_integral (i/c)) g"
lp15@61738
  1607
  by (simp add: field_class.field_divide_inverse) (metis has_contour_integral_rmul)
lp15@61738
  1608
lp15@61738
  1609
lemma has_contour_integral_eq:
lp15@61738
  1610
    "\<lbrakk>(f has_contour_integral y) p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> (g has_contour_integral y) p"
lp15@61738
  1611
apply (simp add: path_image_def has_contour_integral_def)
lp15@60809
  1612
by (metis (no_types, lifting) image_eqI has_integral_eq)
lp15@60809
  1613
lp15@61738
  1614
lemma has_contour_integral_bound_linepath:
lp15@61738
  1615
  assumes "(f has_contour_integral i) (linepath a b)"
lp15@60809
  1616
          "0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
lp15@60809
  1617
    shows "norm i \<le> B * norm(b - a)"
lp15@60809
  1618
proof -
lp15@60809
  1619
  { fix x::real
lp15@60809
  1620
    assume x: "0 \<le> x" "x \<le> 1"
lp15@60809
  1621
  have "norm (f (linepath a b x)) *
lp15@60809
  1622
        norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)"
lp15@60809
  1623
    by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x)
lp15@60809
  1624
  } note * = this
lp15@60809
  1625
  have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
lp15@60809
  1626
    apply (rule has_integral_bound
lp15@60809
  1627
       [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
lp15@61738
  1628
    using assms * unfolding has_contour_integral_def
lp15@60809
  1629
    apply (auto simp: norm_mult)
lp15@60809
  1630
    done
lp15@60809
  1631
  then show ?thesis
lp15@60809
  1632
    by (auto simp: content_real)
lp15@60809
  1633
qed
lp15@60809
  1634
lp15@60809
  1635
(*UNUSED
lp15@61738
  1636
lemma has_contour_integral_bound_linepath_strong:
lp15@60809
  1637
  fixes a :: real and f :: "complex \<Rightarrow> real"
lp15@61738
  1638
  assumes "(f has_contour_integral i) (linepath a b)"
lp15@60809
  1639
          "finite k"
lp15@60809
  1640
          "0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B"
lp15@60809
  1641
    shows "norm i \<le> B*norm(b - a)"
lp15@60809
  1642
*)
lp15@60809
  1643
lp15@61738
  1644
lemma has_contour_integral_const_linepath: "((\<lambda>x. c) has_contour_integral c*(b - a))(linepath a b)"
lp15@61738
  1645
  unfolding has_contour_integral_linepath
lp15@60809
  1646
  by (metis content_real diff_0_right has_integral_const_real lambda_one of_real_1 scaleR_conv_of_real zero_le_one)
lp15@60809
  1647
lp15@61738
  1648
lemma has_contour_integral_0: "((\<lambda>x. 0) has_contour_integral 0) g"
lp15@61738
  1649
  by (simp add: has_contour_integral_def)
lp15@61738
  1650
lp15@61738
  1651
lemma has_contour_integral_is_0:
lp15@61738
  1652
    "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_contour_integral 0) g"
lp15@61738
  1653
  by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto
lp15@61738
  1654
nipkow@64267
  1655
lemma has_contour_integral_sum:
lp15@61738
  1656
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p\<rbrakk>
nipkow@64267
  1657
     \<Longrightarrow> ((\<lambda>x. sum (\<lambda>a. f a x) s) has_contour_integral sum i s) p"
lp15@61738
  1658
  by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add)
lp15@60809
  1659
lp15@60809
  1660
lp15@60809
  1661
subsection \<open>Operations on path integrals\<close>
lp15@60809
  1662
lp15@61738
  1663
lemma contour_integral_const_linepath [simp]: "contour_integral (linepath a b) (\<lambda>x. c) = c*(b - a)"
lp15@61738
  1664
  by (rule contour_integral_unique [OF has_contour_integral_const_linepath])
lp15@61738
  1665
lp15@61738
  1666
lemma contour_integral_neg:
lp15@61738
  1667
    "f contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. -(f x)) = -(contour_integral g f)"
lp15@61738
  1668
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_neg)
lp15@61738
  1669
lp15@61738
  1670
lemma contour_integral_add:
lp15@61738
  1671
    "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x + f2 x) =
lp15@61738
  1672
                contour_integral g f1 + contour_integral g f2"
lp15@61738
  1673
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_add)
lp15@61738
  1674
lp15@61738
  1675
lemma contour_integral_diff:
lp15@61738
  1676
    "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x - f2 x) =
lp15@61738
  1677
                contour_integral g f1 - contour_integral g f2"
lp15@61738
  1678
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_diff)
lp15@61738
  1679
lp15@61738
  1680
lemma contour_integral_lmul:
lp15@61738
  1681
  shows "f contour_integrable_on g
lp15@61738
  1682
           \<Longrightarrow> contour_integral g (\<lambda>x. c * f x) = c*contour_integral g f"
lp15@61738
  1683
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_lmul)
lp15@61738
  1684
lp15@61738
  1685
lemma contour_integral_rmul:
lp15@61738
  1686
  shows "f contour_integrable_on g
lp15@61738
  1687
        \<Longrightarrow> contour_integral g (\<lambda>x. f x * c) = contour_integral g f * c"
lp15@61738
  1688
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_rmul)
lp15@61738
  1689
lp15@61738
  1690
lemma contour_integral_div:
lp15@61738
  1691
  shows "f contour_integrable_on g
lp15@61738
  1692
        \<Longrightarrow> contour_integral g (\<lambda>x. f x / c) = contour_integral g f / c"
lp15@61738
  1693
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_div)
lp15@61738
  1694
lp15@61738
  1695
lemma contour_integral_eq:
lp15@61738
  1696
    "(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> contour_integral p f = contour_integral p g"
lp15@62463
  1697
  apply (simp add: contour_integral_def)
lp15@62463
  1698
  using has_contour_integral_eq
lp15@62463
  1699
  by (metis contour_integral_unique has_contour_integral_integrable has_contour_integral_integral)
lp15@61738
  1700
lp15@61738
  1701
lemma contour_integral_eq_0:
lp15@61738
  1702
    "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> contour_integral g f = 0"
lp15@61738
  1703
  by (simp add: has_contour_integral_is_0 contour_integral_unique)
lp15@61738
  1704
lp15@61738
  1705
lemma contour_integral_bound_linepath:
lp15@60809
  1706
  shows
lp15@61738
  1707
    "\<lbrakk>f contour_integrable_on (linepath a b);
lp15@60809
  1708
      0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
lp15@61738
  1709
     \<Longrightarrow> norm(contour_integral (linepath a b) f) \<le> B*norm(b - a)"
lp15@61738
  1710
  apply (rule has_contour_integral_bound_linepath [of f])
lp15@61738
  1711
  apply (auto simp: has_contour_integral_integral)
lp15@60809
  1712
  done
lp15@60809
  1713
lp15@61806
  1714
lemma contour_integral_0 [simp]: "contour_integral g (\<lambda>x. 0) = 0"
lp15@61738
  1715
  by (simp add: contour_integral_unique has_contour_integral_0)
lp15@61738
  1716
nipkow@64267
  1717
lemma contour_integral_sum:
lp15@61738
  1718
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
nipkow@64267
  1719
     \<Longrightarrow> contour_integral p (\<lambda>x. sum (\<lambda>a. f a x) s) = sum (\<lambda>a. contour_integral p (f a)) s"
nipkow@64267
  1720
  by (auto simp: contour_integral_unique has_contour_integral_sum has_contour_integral_integral)
lp15@61738
  1721
lp15@61738
  1722
lemma contour_integrable_eq:
lp15@61738
  1723
    "\<lbrakk>f contour_integrable_on p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g contour_integrable_on p"
lp15@61738
  1724
  unfolding contour_integrable_on_def
lp15@61738
  1725
  by (metis has_contour_integral_eq)
lp15@60809
  1726
lp15@60809
  1727
lp15@60809
  1728
subsection \<open>Arithmetic theorems for path integrability\<close>
lp15@60809
  1729
lp15@61738
  1730
lemma contour_integrable_neg:
lp15@61738
  1731
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) contour_integrable_on g"
lp15@61738
  1732
  using has_contour_integral_neg contour_integrable_on_def by blast
lp15@61738
  1733
lp15@61738
  1734
lemma contour_integrable_add:
lp15@61738
  1735
    "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x + f2 x) contour_integrable_on g"
lp15@61738
  1736
  using has_contour_integral_add contour_integrable_on_def
lp15@60809
  1737
  by fastforce
lp15@60809
  1738
lp15@61738
  1739
lemma contour_integrable_diff:
lp15@61738
  1740
    "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x - f2 x) contour_integrable_on g"
lp15@61738
  1741
  using has_contour_integral_diff contour_integrable_on_def
lp15@60809
  1742
  by fastforce
lp15@60809
  1743
lp15@61738
  1744
lemma contour_integrable_lmul:
lp15@61738
  1745
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) contour_integrable_on g"
lp15@61738
  1746
  using has_contour_integral_lmul contour_integrable_on_def
lp15@60809
  1747
  by fastforce
lp15@60809
  1748
lp15@61738
  1749
lemma contour_integrable_rmul:
lp15@61738
  1750
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) contour_integrable_on g"
lp15@61738
  1751
  using has_contour_integral_rmul contour_integrable_on_def
lp15@60809
  1752
  by fastforce
lp15@60809
  1753
lp15@61738
  1754
lemma contour_integrable_div:
lp15@61738
  1755
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) contour_integrable_on g"
lp15@61738
  1756
  using has_contour_integral_div contour_integrable_on_def
lp15@60809
  1757
  by fastforce
lp15@60809
  1758
nipkow@64267
  1759
lemma contour_integrable_sum:
lp15@61738
  1760
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
nipkow@64267
  1761
     \<Longrightarrow> (\<lambda>x. sum (\<lambda>a. f a x) s) contour_integrable_on p"
lp15@61738
  1762
   unfolding contour_integrable_on_def
nipkow@64267
  1763
   by (metis has_contour_integral_sum)
lp15@60809
  1764
lp15@60809
  1765
lp15@60809
  1766
subsection\<open>Reversing a path integral\<close>
lp15@60809
  1767
lp15@61738
  1768
lemma has_contour_integral_reverse_linepath:
lp15@61738
  1769
    "(f has_contour_integral i) (linepath a b)
lp15@61738
  1770
     \<Longrightarrow> (f has_contour_integral (-i)) (linepath b a)"
lp15@61738
  1771
  using has_contour_integral_reversepath valid_path_linepath by fastforce
lp15@61738
  1772
lp15@61738
  1773
lemma contour_integral_reverse_linepath:
lp15@60809
  1774
    "continuous_on (closed_segment a b) f
lp15@61738
  1775
     \<Longrightarrow> contour_integral (linepath a b) f = - (contour_integral(linepath b a) f)"
lp15@61738
  1776
apply (rule contour_integral_unique)
lp15@61738
  1777
apply (rule has_contour_integral_reverse_linepath)
lp15@61738
  1778
by (simp add: closed_segment_commute contour_integrable_continuous_linepath has_contour_integral_integral)
lp15@60809
  1779
lp15@60809
  1780
lp15@60809
  1781
(* Splitting a path integral in a flat way.*)
lp15@60809
  1782
lp15@61738
  1783
lemma has_contour_integral_split:
lp15@61738
  1784
  assumes f: "(f has_contour_integral i) (linepath a c)" "(f has_contour_integral j) (linepath c b)"
lp15@60809
  1785
      and k: "0 \<le> k" "k \<le> 1"
lp15@60809
  1786
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@61738
  1787
    shows "(f has_contour_integral (i + j)) (linepath a b)"
lp15@60809
  1788
proof (cases "k = 0 \<or> k = 1")
lp15@60809
  1789
  case True
lp15@60809
  1790
  then show ?thesis
lp15@68296
  1791
    using assms by auto
lp15@60809
  1792
next
lp15@60809
  1793
  case False
lp15@60809
  1794
  then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1"
lp15@65578
  1795
    using assms by auto
lp15@60809
  1796
  have c': "c = k *\<^sub>R (b - a) + a"
lp15@60809
  1797
    by (metis diff_add_cancel c)
lp15@60809
  1798
  have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)"
lp15@60809
  1799
    by (simp add: algebra_simps c')
lp15@60809
  1800
  { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
lp15@60809
  1801
    have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b"
lp15@68302
  1802
      using False apply (simp add: c' algebra_simps)
lp15@60809
  1803
      apply (simp add: real_vector.scale_left_distrib [symmetric] divide_simps)
lp15@60809
  1804
      done
lp15@60809
  1805
    have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
lp15@68296
  1806
      using k has_integral_affinity01 [OF *, of "inverse k" "0"]
lp15@68296
  1807
      apply (simp add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c)
lp15@68296
  1808
      apply (auto dest: has_integral_cmul [where c = "inverse k"])
lp15@60809
  1809
      done
lp15@60809
  1810
  } note fi = this
lp15@60809
  1811
  { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}"
lp15@60809
  1812
    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)"
lp15@60809
  1813
      using k
lp15@60809
  1814
      apply (simp add: c' field_simps)
lp15@60809
  1815
      apply (simp add: scaleR_conv_of_real divide_simps)
lp15@60809
  1816
      apply (simp add: field_simps)
lp15@60809
  1817
      done
lp15@60809
  1818
    have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
lp15@68296
  1819
      using k has_integral_affinity01 [OF *, of "inverse(1 - k)" "-(k/(1 - k))"]
lp15@68296
  1820
      apply (simp add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
lp15@68296
  1821
      apply (auto dest: has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
lp15@60809
  1822
      done
lp15@60809
  1823
  } note fj = this
lp15@60809
  1824
  show ?thesis
lp15@60809
  1825
    using f k
lp15@61738
  1826
    apply (simp add: has_contour_integral_linepath)
lp15@60809
  1827
    apply (simp add: linepath_def)
lp15@60809
  1828
    apply (rule has_integral_combine [OF _ _ fi fj], simp_all)
lp15@60809
  1829
    done
lp15@60809
  1830
qed
lp15@60809
  1831
lp15@60809
  1832
lemma continuous_on_closed_segment_transform:
lp15@60809
  1833
  assumes f: "continuous_on (closed_segment a b) f"
lp15@60809
  1834
      and k: "0 \<le> k" "k \<le> 1"
lp15@60809
  1835
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@60809
  1836
    shows "continuous_on (closed_segment a c) f"
lp15@60809
  1837
proof -
lp15@60809
  1838
  have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
lp15@60809
  1839
    using c by (simp add: algebra_simps)
lp15@68302
  1840
  have "closed_segment a c \<subseteq> closed_segment a b"
lp15@68302
  1841
    by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
lp15@68302
  1842
  then show "continuous_on (closed_segment a c) f"
lp15@68302
  1843
    by (rule continuous_on_subset [OF f])
lp15@60809
  1844
qed
lp15@60809
  1845
lp15@61738
  1846
lemma contour_integral_split:
lp15@60809
  1847
  assumes f: "continuous_on (closed_segment a b) f"
lp15@60809
  1848
      and k: "0 \<le> k" "k \<le> 1"
lp15@60809
  1849
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@61738
  1850
    shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
lp15@60809
  1851
proof -
lp15@60809
  1852
  have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
lp15@60809
  1853
    using c by (simp add: algebra_simps)
lp15@68302
  1854
  have "closed_segment a c \<subseteq> closed_segment a b"
lp15@68302
  1855
    by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
lp15@68302
  1856
  moreover have "closed_segment c b \<subseteq> closed_segment a b"
lp15@68302
  1857
    by (metis c' ends_in_segment(2) in_segment(1) k subset_closed_segment)
lp15@68302
  1858
  ultimately
lp15@60809
  1859
  have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
lp15@68302
  1860
    by (auto intro: continuous_on_subset [OF f])
lp15@60809
  1861
  show ?thesis
lp15@68302
  1862
    by (rule contour_integral_unique) (meson "*" c contour_integrable_continuous_linepath has_contour_integral_integral has_contour_integral_split k)
lp15@60809
  1863
qed
lp15@60809
  1864
lp15@61738
  1865
lemma contour_integral_split_linepath:
lp15@60809
  1866
  assumes f: "continuous_on (closed_segment a b) f"
lp15@60809
  1867
      and c: "c \<in> closed_segment a b"
lp15@61738
  1868
    shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
lp15@68302
  1869
  using c by (auto simp: closed_segment_def algebra_simps intro!: contour_integral_split [OF f])
lp15@60809
  1870
lp15@68296
  1871
text\<open>The special case of midpoints used in the main quadrisection\<close>
lp15@60809
  1872
lp15@61738
  1873
lemma has_contour_integral_midpoint:
lp15@61738
  1874
  assumes "(f has_contour_integral i) (linepath a (midpoint a b))"
lp15@61738
  1875
          "(f has_contour_integral j) (linepath (midpoint a b) b)"
lp15@61738
  1876
    shows "(f has_contour_integral (i + j)) (linepath a b)"
lp15@61738
  1877
  apply (rule has_contour_integral_split [where c = "midpoint a b" and k = "1/2"])
lp15@60809
  1878
  using assms
lp15@60809
  1879
  apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
lp15@60809
  1880
  done
lp15@60809
  1881
lp15@61738
  1882
lemma contour_integral_midpoint:
lp15@60809
  1883
   "continuous_on (closed_segment a b) f
lp15@61738
  1884
    \<Longrightarrow> contour_integral (linepath a b) f =
lp15@61738
  1885
        contour_integral (linepath a (midpoint a b)) f + contour_integral (linepath (midpoint a b) b) f"
lp15@61738
  1886
  apply (rule contour_integral_split [where c = "midpoint a b" and k = "1/2"])
lp15@60809
  1887
  apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
lp15@60809
  1888
  done
lp15@60809
  1889
lp15@60809
  1890
lp15@60809
  1891
text\<open>A couple of special case lemmas that are useful below\<close>
lp15@60809
  1892
lp15@60809
  1893
lemma triangle_linear_has_chain_integral:
lp15@61738
  1894
    "((\<lambda>x. m*x + d) has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  1895
  apply (rule Cauchy_theorem_primitive [of UNIV "\<lambda>x. m/2 * x^2 + d*x"])
lp15@60809
  1896
  apply (auto intro!: derivative_eq_intros)
lp15@60809
  1897
  done
lp15@60809
  1898
lp15@60809
  1899
lemma has_chain_integral_chain_integral3:
lp15@61738
  1900
     "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d)
lp15@61738
  1901
      \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f = i"
lp15@61738
  1902
  apply (subst contour_integral_unique [symmetric], assumption)
lp15@61738
  1903
  apply (drule has_contour_integral_integrable)
lp15@60809
  1904
  apply (simp add: valid_path_join)
lp15@60809
  1905
  done
lp15@60809
  1906
lp15@62397
  1907
lemma has_chain_integral_chain_integral4:
lp15@62397
  1908
     "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d +++ linepath d e)
lp15@62397
  1909
      \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f + contour_integral (linepath d e) f = i"
lp15@62397
  1910
  apply (subst contour_integral_unique [symmetric], assumption)
lp15@62397
  1911
  apply (drule has_contour_integral_integrable)
lp15@62397
  1912
  apply (simp add: valid_path_join)
lp15@62397
  1913
  done
lp15@62397
  1914
lp15@60809
  1915
subsection\<open>Reversing the order in a double path integral\<close>
lp15@60809
  1916
lp15@60809
  1917
text\<open>The condition is stronger than needed but it's often true in typical situations\<close>
lp15@60809
  1918
lp15@60809
  1919
lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b"
lp15@60809
  1920
  by (auto simp: cbox_Pair_eq)
lp15@60809
  1921
lp15@60809
  1922
lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d"
lp15@60809
  1923
  by (auto simp: cbox_Pair_eq)
lp15@60809
  1924
lp15@61738
  1925
lemma contour_integral_swap:
lp15@60809
  1926
  assumes fcon:  "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)"
lp15@60809
  1927
      and vp:    "valid_path g" "valid_path h"
lp15@60809
  1928
      and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
lp15@60809
  1929
      and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
lp15@61738
  1930
  shows "contour_integral g (\<lambda>w. contour_integral h (f w)) =
lp15@61738
  1931
         contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
lp15@60809
  1932
proof -
lp15@60809
  1933
  have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h"
lp15@61190
  1934
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
lp15@68339
  1935
  have fgh1: "\<And>x. (\<lambda>t. f (g x) (h t)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g x, h t))"
lp15@60809
  1936
    by (rule ext) simp
lp15@68339
  1937
  have fgh2: "\<And>x. (\<lambda>t. f (g t) (h x)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g t, h x))"
lp15@60809
  1938
    by (rule ext) simp
lp15@60809
  1939
  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)"
lp15@60809
  1940
    by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
lp15@60809
  1941
  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)"
lp15@60809
  1942
    by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
lp15@68302
  1943
  have "\<And>y. y \<in> {0..1} \<Longrightarrow> continuous_on {0..1} (\<lambda>x. f (g x) (h y))"
lp15@68302
  1944
    by (subst fgh2) (rule fcon_im2 gcon continuous_intros | simp)+
lp15@68302
  1945
  then 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}"
lp15@68302
  1946
    using continuous_on_mult gvcon integrable_continuous_real by blast
lp15@68339
  1947
  have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) \<circ> fst"
lp15@60809
  1948
    by auto
lp15@60809
  1949
  then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))"
lp15@60809
  1950
    apply (rule ssubst)
lp15@60809
  1951
    apply (rule continuous_intros | simp add: gvcon)+
lp15@60809
  1952
    done
lp15@68339
  1953
  have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) \<circ> snd"
lp15@60809
  1954
    by auto
lp15@60809
  1955
  then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
lp15@60809
  1956
    apply (rule ssubst)
lp15@60809
  1957
    apply (rule continuous_intros | simp add: hvcon)+
lp15@60809
  1958
    done
lp15@68339
  1959
  have "(\<lambda>x. f (g (fst x)) (h (snd x))) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>w. ((g \<circ> fst) w, (h \<circ> snd) w))"
lp15@60809
  1960
    by auto
lp15@60809
  1961
  then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
lp15@60809
  1962
    apply (rule ssubst)
lp15@60809
  1963
    apply (rule gcon hcon continuous_intros | simp)+
lp15@60809
  1964
    apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
lp15@60809
  1965
    done
lp15@61738
  1966
  have "integral {0..1} (\<lambda>x. contour_integral h (f (g x)) * vector_derivative g (at x)) =
lp15@61738
  1967
        integral {0..1} (\<lambda>x. contour_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))"
lp15@68302
  1968
  proof (rule integral_cong [OF contour_integral_rmul [symmetric]])
lp15@68302
  1969
    show "\<And>x. x \<in> {0..1} \<Longrightarrow> f (g x) contour_integrable_on h"
lp15@68302
  1970
      unfolding contour_integrable_on
lp15@60809
  1971
    apply (rule integrable_continuous_real)
lp15@60809
  1972
    apply (rule continuous_on_mult [OF _ hvcon])
lp15@60809
  1973
    apply (subst fgh1)
lp15@60809
  1974
    apply (rule fcon_im1 hcon continuous_intros | simp)+
lp15@68302
  1975
      done
lp15@68302
  1976
  qed
lp15@68339
  1977
  also have "\<dots> = integral {0..1}
lp15@61738
  1978
                     (\<lambda>y. contour_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))"
lp15@68302
  1979
    unfolding contour_integral_integral
lp15@60809
  1980
    apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
lp15@62463
  1981
     apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
lp15@62463
  1982
    unfolding integral_mult_left [symmetric]
lp15@62463
  1983
    apply (simp only: mult_ac)
lp15@60809
  1984
    done
lp15@68339
  1985
  also have "\<dots> = contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
lp15@68302
  1986
    unfolding contour_integral_integral
lp15@60809
  1987
    apply (rule integral_cong)
lp15@62463
  1988
    unfolding integral_mult_left [symmetric]
lp15@60809
  1989
    apply (simp add: algebra_simps)
lp15@60809
  1990
    done
lp15@60809
  1991
  finally show ?thesis
lp15@61738
  1992
    by (simp add: contour_integral_integral)
lp15@60809
  1993
qed
lp15@60809
  1994
lp15@60809
  1995
lp15@60809
  1996
subsection\<open>The key quadrisection step\<close>
lp15@60809
  1997
lp15@60809
  1998
lemma norm_sum_half:
lp15@68302
  1999
  assumes "norm(a + b) \<ge> e"
lp15@68302
  2000
    shows "norm a \<ge> e/2 \<or> norm b \<ge> e/2"
lp15@60809
  2001
proof -
lp15@60809
  2002
  have "e \<le> norm (- a - b)"
lp15@60809
  2003
    by (simp add: add.commute assms norm_minus_commute)
lp15@60809
  2004
  thus ?thesis
lp15@60809
  2005
    using norm_triangle_ineq4 order_trans by fastforce
lp15@60809
  2006
qed
lp15@60809
  2007
lp15@60809
  2008
lemma norm_sum_lemma:
lp15@60809
  2009
  assumes "e \<le> norm (a + b + c + d)"
lp15@60809
  2010
    shows "e / 4 \<le> norm a \<or> e / 4 \<le> norm b \<or> e / 4 \<le> norm c \<or> e / 4 \<le> norm d"
lp15@60809
  2011
proof -
lp15@60809
  2012
  have "e \<le> norm ((a + b) + (c + d))" using assms
lp15@60809
  2013
    by (simp add: algebra_simps)
lp15@60809
  2014
  then show ?thesis
lp15@60809
  2015
    by (auto dest!: norm_sum_half)
lp15@60809
  2016
qed
lp15@60809
  2017
lp15@60809
  2018
lemma Cauchy_theorem_quadrisection:
lp15@60809
  2019
  assumes f: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  2020
      and dist: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
lp15@60809
  2021
      and e: "e * K^2 \<le>
lp15@61738
  2022
              norm (contour_integral(linepath a b) f + contour_integral(linepath b c) f + contour_integral(linepath c a) f)"
lp15@60809
  2023
  shows "\<exists>a' b' c'.
lp15@60809
  2024
           a' \<in> convex hull {a,b,c} \<and> b' \<in> convex hull {a,b,c} \<and> c' \<in> convex hull {a,b,c} \<and>
lp15@60809
  2025
           dist a' b' \<le> K/2  \<and>  dist b' c' \<le> K/2  \<and>  dist c' a' \<le> K/2  \<and>
lp15@61738
  2026
           e * (K/2)^2 \<le> norm(contour_integral(linepath a' b') f + contour_integral(linepath b' c') f + contour_integral(linepath c' a') f)"
lp15@68302
  2027
         (is "\<exists>x y z. ?\<Phi> x y z")
lp15@60809
  2028
proof -
lp15@60809
  2029
  note divide_le_eq_numeral1 [simp del]
wenzelm@63040
  2030
  define a' where "a' = midpoint b c"
wenzelm@63040
  2031
  define b' where "b' = midpoint c a"
wenzelm@63040
  2032
  define c' where "c' = midpoint a b"
lp15@60809
  2033
  have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
lp15@60809
  2034
    using f continuous_on_subset segments_subset_convex_hull by metis+
lp15@60809
  2035
  have fcont': "continuous_on (closed_segment c' b') f"
lp15@60809
  2036
               "continuous_on (closed_segment a' c') f"
lp15@60809
  2037
               "continuous_on (closed_segment b' a') f"
lp15@60809
  2038
    unfolding a'_def b'_def c'_def
lp15@68302
  2039
    by (rule continuous_on_subset [OF f],
lp15@60809
  2040
           metis midpoints_in_convex_hull convex_hull_subset hull_subset insert_subset segment_convex_hull)+
lp15@61738
  2041
  let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
lp15@60809
  2042
  have *: "?pathint a b + ?pathint b c + ?pathint c a =
lp15@60809
  2043
          (?pathint a c' + ?pathint c' b' + ?pathint b' a) +
lp15@60809
  2044
          (?pathint a' c' + ?pathint c' b + ?pathint b a') +
lp15@60809
  2045
          (?pathint a' c + ?pathint c b' + ?pathint b' a') +
lp15@60809
  2046
          (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
lp15@68302
  2047
    by (simp add: fcont' contour_integral_reverse_linepath) (simp add: a'_def b'_def c'_def contour_integral_midpoint fabc)
lp15@60809
  2048
  have [simp]: "\<And>x y. cmod (x * 2 - y * 2) = cmod (x - y) * 2"
lp15@60809
  2049
    by (metis left_diff_distrib mult.commute norm_mult_numeral1)
lp15@60809
  2050
  have [simp]: "\<And>x y. cmod (x - y) = cmod (y - x)"
lp15@60809
  2051
    by (simp add: norm_minus_commute)
lp15@60809
  2052
  consider "e * K\<^sup>2 / 4 \<le> cmod (?pathint a c' + ?pathint c' b' + ?pathint b' a)" |
lp15@60809
  2053
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c' + ?pathint c' b + ?pathint b a')" |
lp15@60809
  2054
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c + ?pathint c b' + ?pathint b' a')" |
lp15@60809
  2055
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
lp15@68302
  2056
    using assms unfolding * by (blast intro: that dest!: norm_sum_lemma)
lp15@60809
  2057
  then show ?thesis
lp15@60809
  2058
  proof cases
lp15@68302
  2059
    case 1 then have "?\<Phi> a c' b'"
lp15@60809
  2060
      using assms
lp15@68302
  2061
      apply (clarsimp simp: c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@68302
  2062
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@68302
  2063
      done
lp15@68302
  2064
    then show ?thesis by blast
lp15@68302
  2065
  next
lp15@68302
  2066
    case 2 then  have "?\<Phi> a' c' b"
lp15@68302
  2067
      using assms
lp15@68302
  2068
      apply (clarsimp simp: a'_def c'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  2069
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  2070
      done
lp15@68302
  2071
    then show ?thesis by blast
lp15@60809
  2072
  next
lp15@68302
  2073
    case 3 then have "?\<Phi> a' c b'"
lp15@60809
  2074
      using assms
lp15@68302
  2075
      apply (clarsimp simp: a'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  2076
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  2077
      done
lp15@68302
  2078
    then show ?thesis by blast
lp15@60809
  2079
  next
lp15@68302
  2080
    case 4 then have "?\<Phi> a' b' c'"
lp15@60809
  2081
      using assms
lp15@68302
  2082
      apply (clarsimp simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  2083
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  2084
      done
lp15@68302
  2085
    then show ?thesis by blast
lp15@60809
  2086
  qed
lp15@60809
  2087
qed
lp15@60809
  2088
lp15@60809
  2089
subsection\<open>Cauchy's theorem for triangles\<close>
lp15@60809
  2090
lp15@60809
  2091
lemma triangle_points_closer:
lp15@60809
  2092
  fixes a::complex
lp15@60809
  2093
  shows "\<lbrakk>x \<in> convex hull {a,b,c};  y \<in> convex hull {a,b,c}\<rbrakk>
lp15@60809
  2094
         \<Longrightarrow> norm(x - y) \<le> norm(a - b) \<or>
lp15@60809
  2095
             norm(x - y) \<le> norm(b - c) \<or>
lp15@60809
  2096
             norm(x - y) \<le> norm(c - a)"
lp15@60809
  2097
  using simplex_extremal_le [of "{a,b,c}"]
lp15@60809
  2098
  by (auto simp: norm_minus_commute)
lp15@60809
  2099
lp15@60809
  2100
lemma holomorphic_point_small_triangle:
lp15@68302
  2101
  assumes x: "x \<in> S"
lp15@68302
  2102
      and f: "continuous_on S f"
lp15@68302
  2103
      and cd: "f field_differentiable (at x within S)"
lp15@60809
  2104
      and e: "0 < e"
lp15@60809
  2105
    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>
lp15@68302
  2106
              x \<in> convex hull {a,b,c} \<and> convex hull {a,b,c} \<subseteq> S
lp15@61738
  2107
              \<longrightarrow> norm(contour_integral(linepath a b) f + contour_integral(linepath b c) f +
lp15@61738
  2108
                       contour_integral(linepath c a) f)
lp15@60809
  2109
                  \<le> e*(dist a b + dist b c + dist c a)^2"
lp15@60809
  2110
           (is "\<exists>k>0. \<forall>a b c. _ \<longrightarrow> ?normle a b c")
lp15@60809
  2111
proof -
lp15@60809
  2112
  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>
lp15@60809
  2113
                     \<Longrightarrow> a \<le> e*(x + y + z)^2"
lp15@60809
  2114
    by (simp add: algebra_simps power2_eq_square)
lp15@60809
  2115
  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"
lp15@60809
  2116
             for x::real and a b c
lp15@60809
  2117
    by linarith
lp15@61738
  2118
  have fabc: "f contour_integrable_on linepath a b" "f contour_integrable_on linepath b c" "f contour_integrable_on linepath c a"
lp15@68302
  2119
              if "convex hull {a, b, c} \<subseteq> S" for a b c
lp15@60809
  2120
    using segments_subset_convex_hull that
lp15@61738
  2121
    by (metis continuous_on_subset f contour_integrable_continuous_linepath)+
lp15@61738
  2122
  note path_bound = has_contour_integral_bound_linepath [simplified norm_minus_commute, OF has_contour_integral_integral]
lp15@60809
  2123
  { fix f' a b c d
lp15@60809
  2124
    assume d: "0 < d"
lp15@68302
  2125
       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)"
lp15@60809
  2126
       and le: "cmod (a - b) \<le> d" "cmod (b - c) \<le> d" "cmod (c - a) \<le> d"
lp15@60809
  2127
       and xc: "x \<in> convex hull {a, b, c}"
lp15@68302
  2128
       and S: "convex hull {a, b, c} \<subseteq> S"
lp15@61738
  2129
    have pa: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f =
lp15@61738
  2130
              contour_integral (linepath a b) (\<lambda>y. f y - f x - f'*(y - x)) +
lp15@61738
  2131
              contour_integral (linepath b c) (\<lambda>y. f y - f x - f'*(y - x)) +
lp15@61738
  2132
              contour_integral (linepath c a) (\<lambda>y. f y - f x - f'*(y - x))"
lp15@68302
  2133
      apply (simp add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc [OF S])
lp15@60809
  2134
      apply (simp add: field_simps)
lp15@60809
  2135
      done
lp15@60809
  2136
    { fix y
lp15@60809
  2137
      assume yc: "y \<in> convex hull {a,b,c}"
lp15@60809
  2138
      have "cmod (f y - f x - f' * (y - x)) \<le> e*norm(y - x)"
lp15@68302
  2139
      proof (rule f')
lp15@68302
  2140
        show "cmod (y - x) \<le> d"
lp15@68302
  2141
          by (metis triangle_points_closer [OF xc yc] le norm_minus_commute order_trans)
lp15@68302
  2142
      qed (use S yc in blast)
lp15@68339
  2143
      also have "\<dots> \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))"
lp15@60809
  2144
        by (simp add: yc e xc disj_le [OF triangle_points_closer])
lp15@60809
  2145
      finally have "cmod (f y - f x - f' * (y - x)) \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" .
lp15@60809
  2146
    } note cm_le = this
lp15@60809
  2147
    have "?normle a b c"
lp15@68302
  2148
      unfolding dist_norm pa
lp15@60809
  2149
      apply (rule le_of_3)
lp15@68302
  2150
      using f' xc S e
lp15@60809
  2151
      apply simp_all
lp15@60809
  2152
      apply (intro norm_triangle_le add_mono path_bound)
lp15@61738
  2153
      apply (simp_all add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc)
lp15@60809
  2154
      apply (blast intro: cm_le elim: dest: segments_subset_convex_hull [THEN subsetD])+
lp15@60809
  2155
      done
lp15@60809
  2156
  } note * = this
lp15@60809
  2157
  show ?thesis
lp15@68302
  2158
    using cd e 
lp15@62534
  2159
    apply (simp add: field_differentiable_def has_field_derivative_def has_derivative_within_alt approachable_lt_le2 Ball_def)
lp15@60809
  2160
    apply (clarify dest!: spec mp)
lp15@68302
  2161
    using * unfolding dist_norm
lp15@68339
  2162
    apply blast
lp15@60809
  2163
    done
lp15@60809
  2164
qed
lp15@60809
  2165
lp15@60809
  2166
lp15@68310
  2167
text\<open>Hence the most basic theorem for a triangle.\<close>
lp15@68310
  2168
lp15@60809
  2169
locale Chain =
lp15@60809
  2170
  fixes x0 At Follows
lp15@60809
  2171
  assumes At0: "At x0 0"
lp15@60809
  2172
      and AtSuc: "\<And>x n. At x n \<Longrightarrow> \<exists>x'. At x' (Suc n) \<and> Follows x' x"
lp15@60809
  2173
begin
lp15@60809
  2174
  primrec f where
lp15@60809
  2175
    "f 0 = x0"
lp15@60809
  2176
  | "f (Suc n) = (SOME x. At x (Suc n) \<and> Follows x (f n))"
lp15@60809
  2177
lp15@60809
  2178
  lemma At: "At (f n) n"
lp15@60809
  2179
  proof (induct n)
lp15@60809
  2180
    case 0 show ?case
lp15@60809
  2181
      by (simp add: At0)
lp15@60809
  2182
  next
lp15@60809
  2183
    case (Suc n) show ?case
lp15@60809
  2184
      by (metis (no_types, lifting) AtSuc [OF Suc] f.simps(2) someI_ex)
lp15@60809
  2185
  qed
lp15@60809
  2186
lp15@60809
  2187
  lemma Follows: "Follows (f(Suc n)) (f n)"
lp15@60809
  2188
    by (metis (no_types, lifting) AtSuc [OF At [of n]] f.simps(2) someI_ex)
lp15@60809
  2189
lp15@60809
  2190
  declare f.simps(2) [simp del]
lp15@60809
  2191
end
lp15@60809
  2192
lp15@60809
  2193
lemma Chain3:
lp15@60809
  2194
  assumes At0: "At x0 y0 z0 0"
lp15@60809
  2195
      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"
lp15@60809
  2196
  obtains f g h where
lp15@60809
  2197
    "f 0 = x0" "g 0 = y0" "h 0 = z0"
lp15@60809
  2198
                      "\<And>n. At (f n) (g n) (h n) n"
lp15@60809
  2199
                       "\<And>n. Follows (f(Suc n)) (g(Suc n)) (h(Suc n)) (f n) (g n) (h n)"
lp15@60809
  2200
proof -
lp15@60809
  2201
  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"
lp15@60809
  2202
    apply unfold_locales
lp15@60809
  2203
    using At0 AtSuc by auto
lp15@60809
  2204
  show ?thesis
lp15@60809
  2205
  apply (rule that [of "\<lambda>n. fst (three.f n)"  "\<lambda>n. fst (snd (three.f n))" "\<lambda>n. snd (snd (three.f n))"])
lp15@68302
  2206
  using three.At three.Follows
lp15@60809
  2207
  apply simp_all
lp15@60809
  2208
  apply (simp_all add: split_beta')
lp15@60809
  2209
  done
lp15@60809
  2210
qed
lp15@60809
  2211
lp15@68310
  2212
proposition Cauchy_theorem_triangle:
lp15@60809
  2213
  assumes "f holomorphic_on (convex hull {a,b,c})"
lp15@61738
  2214
    shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2215
proof -
lp15@60809
  2216
  have contf: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  2217
    by (metis assms holomorphic_on_imp_continuous_on)
lp15@61738
  2218
  let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
lp15@60809
  2219
  { fix y::complex
lp15@61738
  2220
    assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2221
       and ynz: "y \<noteq> 0"
wenzelm@63040
  2222
    define K where "K = 1 + max (dist a b) (max (dist b c) (dist c a))"
wenzelm@63040
  2223
    define e where "e = norm y / K^2"
lp15@60809
  2224
    have K1: "K \<ge> 1"  by (simp add: K_def max.coboundedI1)
lp15@60809
  2225
    then have K: "K > 0" by linarith
lp15@60809
  2226
    have [iff]: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
lp15@60809
  2227
      by (simp_all add: K_def)
lp15@60809
  2228
    have e: "e > 0"
lp15@60809
  2229
      unfolding e_def using ynz K1 by simp
wenzelm@63040
  2230
    define At where "At x y z n \<longleftrightarrow>
wenzelm@63040
  2231
        convex hull {x,y,z} \<subseteq> convex hull {a,b,c} \<and>
wenzelm@63040
  2232
        dist x y \<le> K/2^n \<and> dist y z \<le> K/2^n \<and> dist z x \<le> K/2^n \<and>
wenzelm@63040
  2233
        norm(?pathint x y + ?pathint y z + ?pathint z x) \<ge> e*(K/2^n)^2"
wenzelm@63040
  2234
      for x y z n
lp15@60809
  2235
    have At0: "At a b c 0"
lp15@60809
  2236
      using fy
lp15@60809
  2237
      by (simp add: At_def e_def has_chain_integral_chain_integral3)
lp15@60809
  2238
    { fix x y z n
lp15@60809
  2239
      assume At: "At x y z n"
lp15@60809
  2240
      then have contf': "continuous_on (convex hull {x,y,z}) f"
lp15@63938
  2241
        using contf At_def continuous_on_subset by metis
lp15@60809
  2242
      have "\<exists>x' y' z'. At x' y' z' (Suc n) \<and> convex hull {x',y',z'} \<subseteq> convex hull {x,y,z}"
lp15@68302
  2243
        using At Cauchy_theorem_quadrisection [OF contf', of "K/2^n" e]
lp15@68302
  2244
        apply (simp add: At_def algebra_simps)
lp15@60809
  2245
        apply (meson convex_hull_subset empty_subsetI insert_subset subsetCE)
lp15@60809
  2246
        done
lp15@60809
  2247
    } note AtSuc = this
lp15@60809
  2248
    obtain fa fb fc
lp15@60809
  2249
      where f0 [simp]: "fa 0 = a" "fb 0 = b" "fc 0 = c"
lp15@60809
  2250
        and cosb: "\<And>n. convex hull {fa n, fb n, fc n} \<subseteq> convex hull {a,b,c}"
lp15@60809
  2251
        and dist: "\<And>n. dist (fa n) (fb n) \<le> K/2^n"
lp15@60809
  2252
                  "\<And>n. dist (fb n) (fc n) \<le> K/2^n"
lp15@60809
  2253
                  "\<And>n. dist (fc n) (fa n) \<le> K/2^n"
lp15@60809
  2254
        and no: "\<And>n. norm(?pathint (fa n) (fb n) +
lp15@60809
  2255
                           ?pathint (fb n) (fc n) +
lp15@60809
  2256
                           ?pathint (fc n) (fa n)) \<ge> e * (K/2^n)^2"
lp15@60809
  2257
        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}"