src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy
author paulson <lp15@cam.ac.uk>
Mon Mar 14 15:58:02 2016 +0000 (2016-03-14)
changeset 62620 d21dab28b3f9
parent 62618 f7f2467ab854
child 62623 dbc62f86a1a9
permissions -rw-r--r--
New results about paths, segments, etc. The notion of simply_connected.
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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_Thm
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imports Complex_Transcendental Weierstrass 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 add: 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 "... 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 "... 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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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 intro: 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 o 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 o (\<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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                       "\<forall>x\<in>{a..c} - s. f differentiable at x within {a..c}"
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                       "\<forall>x\<in>{c..b} - t. 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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        apply (rule differentiable_transform_within [where d = "dist x c" and f = f])
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        using x le st
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        apply (simp_all add: dist_real_def)
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        apply (rule differentiable_at_withinI)
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        apply (rule differentiable_within_open [where s = "{a<..<c} - s", THEN iffD1], simp_all)
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        apply (blast intro: open_greaterThanLessThan finite_imp_closed)
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        apply (force elim!: differentiable_subset)+
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        done
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    next
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      case ge show ?diff_fg
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        apply (rule differentiable_transform_within [where d = "dist x c" and f = g])
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        using x ge st
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        apply (simp_all add: dist_real_def)
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        apply (rule differentiable_at_withinI)
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        apply (rule differentiable_within_open [where s = "{c<..<b} - t", THEN iffD1], simp_all)
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        apply (blast intro: open_greaterThanLessThan finite_imp_closed)
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        apply (force elim!: differentiable_subset)+
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        done
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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) o (op*(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) o (\<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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    "(g1 +++ g2) piecewise_differentiable_on {0..1} \<Longrightarrow> g1 piecewise_differentiable_on {0..1}"
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  apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D1)
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  apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
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  apply simp
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  apply (intro ballI)
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  apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))"
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       in differentiable_transform_within)
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  apply (auto simp: dist_real_def joinpaths_def)
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  apply (rule differentiable_chain_within derivative_intros | simp)+
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  apply (rule differentiable_subset)
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  apply (force simp:)+
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  done
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lemma piecewise_differentiable_D2:
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    "\<lbrakk>(g1 +++ g2) piecewise_differentiable_on {0..1}; pathfinish g1 = pathstart g2\<rbrakk>
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    \<Longrightarrow> g2 piecewise_differentiable_on {0..1}"
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  apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D2)
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  apply (rule_tac x="insert 0 ((\<lambda>x. 2*x-1)`s)" in exI)
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  apply simp
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  apply (intro ballI)
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  apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\<lambda>x. (x+1)/2)"
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          in differentiable_transform_within)
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  apply (auto simp: dist_real_def joinpaths_def abs_if field_simps split: if_split_asm)
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  apply (rule differentiable_chain_within derivative_intros | simp)+
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  apply (rule differentiable_subset)
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  apply (force simp: divide_simps)+
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  done
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subsubsection\<open>The concept of continuously differentiable\<close>
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text \<open>
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John Harrison writes as follows:
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``The usual assumption in complex analysis texts is that a path \<open>\<gamma>\<close> should be piecewise
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continuously differentiable, which ensures that the path integral exists at least for any continuous
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f, since all piecewise continuous functions are integrable. However, our notion of validity is
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weaker, just piecewise differentiability... [namely] continuity plus differentiability except on a
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finite set ... [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to
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the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this
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can integrate all derivatives.''
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"Formalizing basic complex analysis." From Insight to Proof: Festschrift in Honour of Andrzej Trybulec.
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Studies in Logic, Grammar and Rhetoric 10.23 (2007): 151-165.
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And indeed he does not assume that his derivatives are continuous, but the penalty is unreasonably
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difficult proofs concerning winding numbers. We need a self-contained and straightforward theorem
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asserting that all derivatives can be integrated before we can adopt Harrison's choice.\<close>
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definition C1_differentiable_on :: "(real \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> real set \<Rightarrow> bool"
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           (infix "C1'_differentiable'_on" 50)
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  where
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  "f C1_differentiable_on s \<longleftrightarrow>
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   (\<exists>D. (\<forall>x \<in> s. (f has_vector_derivative (D x)) (at x)) \<and> continuous_on s D)"
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lemma C1_differentiable_on_eq:
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    "f C1_differentiable_on s \<longleftrightarrow>
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     (\<forall>x \<in> s. f differentiable at x) \<and> continuous_on s (\<lambda>x. vector_derivative f (at x))"
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  unfolding C1_differentiable_on_def
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  apply safe
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  using differentiable_def has_vector_derivative_def apply blast
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  apply (erule continuous_on_eq)
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  using vector_derivative_at apply fastforce
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  using vector_derivative_works apply fastforce
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  done
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lemma C1_differentiable_on_subset:
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  "f C1_differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f C1_differentiable_on s"
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  unfolding C1_differentiable_on_def  continuous_on_eq_continuous_within
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  by (blast intro:  continuous_within_subset)
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lemma C1_differentiable_compose:
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    "\<lbrakk>f C1_differentiable_on s; g C1_differentiable_on (f ` s);
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      \<And>x. finite (s \<inter> f-`{x})\<rbrakk>
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      \<Longrightarrow> (g o f) C1_differentiable_on s"
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  apply (simp add: C1_differentiable_on_eq, safe)
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   using differentiable_chain_at apply blast
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  apply (rule continuous_on_eq [of _ "\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"])
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   apply (rule Limits.continuous_on_scaleR, assumption)
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   apply (metis (mono_tags, lifting) continuous_on_eq continuous_at_imp_continuous_on continuous_on_compose differentiable_imp_continuous_within o_def)
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  by (simp add: vector_derivative_chain_at)
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lemma C1_diff_imp_diff: "f C1_differentiable_on s \<Longrightarrow> f differentiable_on s"
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  by (simp add: C1_differentiable_on_eq differentiable_at_imp_differentiable_on)
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lemma C1_differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) C1_differentiable_on s"
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  by (auto simp: C1_differentiable_on_eq continuous_on_const)
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lemma C1_differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. a) C1_differentiable_on s"
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  by (auto simp: C1_differentiable_on_eq continuous_on_const)
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lemma C1_differentiable_on_add [simp, derivative_intros]:
lp15@61190
   293
  "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x + g x) C1_differentiable_on s"
lp15@61190
   294
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
lp15@61190
   295
lp15@61190
   296
lemma C1_differentiable_on_minus [simp, derivative_intros]:
lp15@61190
   297
  "f C1_differentiable_on s \<Longrightarrow> (\<lambda>x. - f x) C1_differentiable_on s"
lp15@61190
   298
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
lp15@61190
   299
lp15@61190
   300
lemma C1_differentiable_on_diff [simp, derivative_intros]:
lp15@61190
   301
  "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x - g x) C1_differentiable_on s"
lp15@61190
   302
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
lp15@61190
   303
lp15@61190
   304
lemma C1_differentiable_on_mult [simp, derivative_intros]:
lp15@61190
   305
  fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
lp15@61190
   306
  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
   307
  unfolding C1_differentiable_on_eq
lp15@61190
   308
  by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within)
lp15@61190
   309
lp15@61190
   310
lemma C1_differentiable_on_scaleR [simp, derivative_intros]:
lp15@61190
   311
  "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
   312
  unfolding C1_differentiable_on_eq
lp15@61190
   313
  by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+
lp15@61190
   314
lp15@61190
   315
lp15@61190
   316
definition piecewise_C1_differentiable_on
lp15@61190
   317
           (infixr "piecewise'_C1'_differentiable'_on" 50)
lp15@61190
   318
  where "f piecewise_C1_differentiable_on i  \<equiv>
lp15@61190
   319
           continuous_on i f \<and>
lp15@61190
   320
           (\<exists>s. finite s \<and> (f C1_differentiable_on (i - s)))"
lp15@61190
   321
lp15@61190
   322
lemma C1_differentiable_imp_piecewise:
lp15@61190
   323
    "f C1_differentiable_on s \<Longrightarrow> f piecewise_C1_differentiable_on s"
lp15@61190
   324
  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within)
lp15@61190
   325
lp15@61190
   326
lemma piecewise_C1_imp_differentiable:
lp15@61190
   327
    "f piecewise_C1_differentiable_on i \<Longrightarrow> f piecewise_differentiable_on i"
lp15@61190
   328
  by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def
lp15@61190
   329
           C1_differentiable_on_def differentiable_def has_vector_derivative_def
lp15@61190
   330
           intro: has_derivative_at_within)
lp15@61190
   331
lp15@61190
   332
lemma piecewise_C1_differentiable_compose:
lp15@61190
   333
    "\<lbrakk>f piecewise_C1_differentiable_on s; g piecewise_C1_differentiable_on (f ` s);
lp15@61190
   334
      \<And>x. finite (s \<inter> f-`{x})\<rbrakk>
lp15@61190
   335
      \<Longrightarrow> (g o f) piecewise_C1_differentiable_on s"
lp15@61190
   336
  apply (simp add: piecewise_C1_differentiable_on_def, safe)
lp15@61190
   337
  apply (blast intro: continuous_on_compose2)
lp15@61190
   338
  apply (rename_tac A B)
lp15@61190
   339
  apply (rule_tac x="A \<union> (\<Union>x\<in>B. s \<inter> f-`{x})" in exI)
lp15@61190
   340
  apply (rule conjI, blast)
lp15@61190
   341
  apply (rule C1_differentiable_compose)
lp15@61190
   342
  apply (blast intro: C1_differentiable_on_subset)
lp15@61190
   343
  apply (blast intro: C1_differentiable_on_subset)
lp15@61190
   344
  by (simp add: Diff_Int_distrib2)
lp15@61190
   345
lp15@61190
   346
lemma piecewise_C1_differentiable_on_subset:
lp15@61190
   347
    "f piecewise_C1_differentiable_on s \<Longrightarrow> t \<le> s \<Longrightarrow> f piecewise_C1_differentiable_on t"
lp15@61190
   348
  by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset)
lp15@61190
   349
lp15@61190
   350
lemma C1_differentiable_imp_continuous_on:
lp15@61190
   351
  "f C1_differentiable_on s \<Longrightarrow> continuous_on s f"
lp15@61190
   352
  unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within
lp15@61190
   353
  using differentiable_at_withinI differentiable_imp_continuous_within by blast
lp15@61190
   354
lp15@61190
   355
lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}"
lp15@61190
   356
  unfolding C1_differentiable_on_def
lp15@61190
   357
  by auto
lp15@61190
   358
lp15@61190
   359
lemma piecewise_C1_differentiable_affine:
lp15@61190
   360
  fixes m::real
lp15@61190
   361
  assumes "f piecewise_C1_differentiable_on ((\<lambda>x. m * x + c) ` s)"
lp15@61190
   362
  shows "(f o (\<lambda>x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on s"
lp15@61190
   363
proof (cases "m = 0")
lp15@61190
   364
  case True
lp15@61190
   365
  then show ?thesis
lp15@61190
   366
    unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def continuous_on_const)
lp15@61190
   367
next
lp15@61190
   368
  case False
lp15@61190
   369
  show ?thesis
lp15@61190
   370
    apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise])
lp15@61190
   371
    apply (rule assms derivative_intros | simp add: False vimage_def)+
lp15@61190
   372
    using real_vector_affinity_eq [OF False, where c=c, unfolded scaleR_conv_of_real]
lp15@61190
   373
    apply simp
lp15@61190
   374
    done
lp15@61190
   375
qed
lp15@61190
   376
lp15@61190
   377
lemma piecewise_C1_differentiable_cases:
lp15@61190
   378
  fixes c::real
lp15@61190
   379
  assumes "f piecewise_C1_differentiable_on {a..c}"
lp15@61190
   380
          "g piecewise_C1_differentiable_on {c..b}"
lp15@61190
   381
           "a \<le> c" "c \<le> b" "f c = g c"
lp15@61190
   382
  shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}"
lp15@61190
   383
proof -
lp15@61190
   384
  obtain s t where st: "f C1_differentiable_on ({a..c} - s)"
lp15@61190
   385
                       "g C1_differentiable_on ({c..b} - t)"
lp15@61190
   386
                       "finite s" "finite t"
lp15@61190
   387
    using assms
lp15@61190
   388
    by (force simp: piecewise_C1_differentiable_on_def)
lp15@61190
   389
  then have f_diff: "f differentiable_on {a..<c} - s"
lp15@61190
   390
        and g_diff: "g differentiable_on {c<..b} - t"
lp15@61190
   391
    by (simp_all add: C1_differentiable_on_eq differentiable_at_withinI differentiable_on_def)
lp15@61190
   392
  have "continuous_on {a..c} f" "continuous_on {c..b} g"
lp15@61190
   393
    using assms piecewise_C1_differentiable_on_def by auto
lp15@61190
   394
  then have cab: "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
lp15@61190
   395
    using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
lp15@61190
   396
                               OF closed_real_atLeastAtMost [of c b],
lp15@61190
   397
                               of f g "\<lambda>x. x\<le>c"]  assms
lp15@61190
   398
    by (force simp: ivl_disj_un_two_touch)
lp15@61190
   399
  { fix x
lp15@61190
   400
    assume x: "x \<in> {a..b} - insert c (s \<union> t)"
lp15@61190
   401
    have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x" (is "?diff_fg")
lp15@61190
   402
    proof (cases x c rule: le_cases)
lp15@61190
   403
      case le show ?diff_fg
paulson@62087
   404
        apply (rule differentiable_transform_within [where f=f and d = "dist x c"])
lp15@61190
   405
        using x dist_real_def le st by (auto simp: C1_differentiable_on_eq)
lp15@61190
   406
    next
lp15@61190
   407
      case ge show ?diff_fg
paulson@62087
   408
        apply (rule differentiable_transform_within [where f=g and d = "dist x c"])
lp15@61190
   409
        using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq)
lp15@61190
   410
    qed
lp15@61190
   411
  }
lp15@61190
   412
  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
   413
    by auto
lp15@61190
   414
  moreover
lp15@61190
   415
  { assume fcon: "continuous_on ({a<..<c} - s) (\<lambda>x. vector_derivative f (at x))"
lp15@61190
   416
       and gcon: "continuous_on ({c<..<b} - t) (\<lambda>x. vector_derivative g (at x))"
lp15@61190
   417
    have "open ({a<..<c} - s)"  "open ({c<..<b} - t)"
lp15@61190
   418
      using st by (simp_all add: open_Diff finite_imp_closed)
lp15@61190
   419
    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@61190
   420
      apply (rule continuous_on_eq [OF fcon])
lp15@61190
   421
      apply (simp add:)
lp15@61190
   422
      apply (rule vector_derivative_at [symmetric])
paulson@62087
   423
      apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_within)
lp15@61190
   424
      apply (simp_all add: dist_norm vector_derivative_works [symmetric])
paulson@62087
   425
      apply (metis (full_types) C1_differentiable_on_eq Diff_iff Groups.add_ac(2) add_mono_thms_linordered_field(5) atLeastAtMost_iff linorder_not_le order_less_irrefl st(1))
paulson@62087
   426
      apply auto
paulson@62087
   427
      done
lp15@61190
   428
    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@61190
   429
      apply (rule continuous_on_eq [OF gcon])
lp15@61190
   430
      apply (simp add:)
lp15@61190
   431
      apply (rule vector_derivative_at [symmetric])
paulson@62087
   432
      apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_within)
lp15@61190
   433
      apply (simp_all add: dist_norm vector_derivative_works [symmetric])
paulson@62087
   434
      apply (metis (full_types) C1_differentiable_on_eq Diff_iff Groups.add_ac(2) add_mono_thms_linordered_field(5) atLeastAtMost_iff less_irrefl not_le st(2))
paulson@62087
   435
      apply auto
paulson@62087
   436
      done
lp15@61190
   437
    ultimately have "continuous_on ({a<..<b} - insert c (s \<union> t))
lp15@61190
   438
        (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
lp15@61190
   439
      apply (rule continuous_on_subset [OF continuous_on_open_Un], auto)
lp15@61190
   440
      done
lp15@61190
   441
  } note * = this
lp15@61190
   442
  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
   443
    using st
lp15@61190
   444
    by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *)
lp15@61190
   445
  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@61190
   446
    apply (rule_tac x="{a,b,c} \<union> s \<union> t" in exI)
lp15@61190
   447
    using st  by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset)
lp15@61190
   448
  with cab show ?thesis
lp15@61190
   449
    by (simp add: piecewise_C1_differentiable_on_def)
lp15@61190
   450
qed
lp15@61190
   451
lp15@61190
   452
lemma piecewise_C1_differentiable_neg:
lp15@61190
   453
    "f piecewise_C1_differentiable_on s \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_C1_differentiable_on s"
lp15@61190
   454
  unfolding piecewise_C1_differentiable_on_def
lp15@61190
   455
  by (auto intro!: continuous_on_minus C1_differentiable_on_minus)
lp15@61190
   456
lp15@61190
   457
lemma piecewise_C1_differentiable_add:
lp15@61190
   458
  assumes "f piecewise_C1_differentiable_on i"
lp15@61190
   459
          "g piecewise_C1_differentiable_on i"
lp15@61190
   460
    shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i"
lp15@61190
   461
proof -
lp15@61190
   462
  obtain s t where st: "finite s" "finite t"
lp15@61190
   463
                       "f C1_differentiable_on (i-s)"
lp15@61190
   464
                       "g C1_differentiable_on (i-t)"
lp15@61190
   465
    using assms by (auto simp: piecewise_C1_differentiable_on_def)
lp15@61190
   466
  then have "finite (s \<union> t) \<and> (\<lambda>x. f x + g x) C1_differentiable_on i - (s \<union> t)"
lp15@61190
   467
    by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset)
lp15@61190
   468
  moreover have "continuous_on i f" "continuous_on i g"
lp15@61190
   469
    using assms piecewise_C1_differentiable_on_def by auto
lp15@61190
   470
  ultimately show ?thesis
lp15@61190
   471
    by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add)
lp15@61190
   472
qed
lp15@61190
   473
paulson@61204
   474
lemma piecewise_C1_differentiable_diff:
lp15@61190
   475
    "\<lbrakk>f piecewise_C1_differentiable_on s;  g piecewise_C1_differentiable_on s\<rbrakk>
lp15@61190
   476
     \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_C1_differentiable_on s"
lp15@61190
   477
  unfolding diff_conv_add_uminus
lp15@61190
   478
  by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg)
lp15@61190
   479
lp15@61190
   480
lemma piecewise_C1_differentiable_D1:
lp15@61190
   481
  fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   482
  assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}"
lp15@61190
   483
    shows "g1 piecewise_C1_differentiable_on {0..1}"
lp15@61190
   484
proof -
lp15@61190
   485
  obtain s where "finite s"
lp15@61190
   486
             and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   487
             and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
lp15@61190
   488
    using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@61190
   489
  then have g1D: "g1 differentiable at x" if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
paulson@62087
   490
    apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))" in differentiable_transform_within)
lp15@61190
   491
    using that
lp15@61190
   492
    apply (simp_all add: dist_real_def joinpaths_def)
lp15@61190
   493
    apply (rule differentiable_chain_at derivative_intros | force)+
lp15@61190
   494
    done
lp15@61190
   495
  have [simp]: "vector_derivative (g1 \<circ> op * 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)"
lp15@61190
   496
               if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
lp15@61190
   497
    apply (subst vector_derivative_chain_at)
lp15@61190
   498
    using that
lp15@61190
   499
    apply (rule derivative_eq_intros g1D | simp)+
lp15@61190
   500
    done
lp15@61190
   501
  have "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   502
    using co12 by (rule continuous_on_subset) force
lp15@61190
   503
  then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 o op*2) (at x))"
lp15@61190
   504
    apply (rule continuous_on_eq [OF _ vector_derivative_at])
paulson@62087
   505
    apply (rule_tac f="g1 o op*2" and d="dist x (1/2)" in has_vector_derivative_transform_within)
lp15@61190
   506
    apply (simp_all add: dist_norm joinpaths_def vector_derivative_works [symmetric])
lp15@61190
   507
    apply (force intro: g1D differentiable_chain_at)
paulson@62087
   508
    apply auto
lp15@61190
   509
    done
lp15@61190
   510
  have "continuous_on ({0..1} - insert 1 (op * 2 ` s))
lp15@61190
   511
                      ((\<lambda>x. 1/2 * vector_derivative (g1 o op*2) (at x)) o op*(1/2))"
lp15@61190
   512
    apply (rule continuous_intros)+
lp15@61190
   513
    using coDhalf
lp15@61190
   514
    apply (simp add: scaleR_conv_of_real image_set_diff image_image)
lp15@61190
   515
    done
lp15@61190
   516
  then have con_g1: "continuous_on ({0..1} - insert 1 (op * 2 ` s)) (\<lambda>x. vector_derivative g1 (at x))"
lp15@61190
   517
    by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
lp15@61190
   518
  have "continuous_on {0..1} g1"
lp15@61190
   519
    using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast
wenzelm@61222
   520
  with \<open>finite s\<close> show ?thesis
lp15@61190
   521
    apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@61190
   522
    apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
lp15@61190
   523
    apply (simp add: g1D con_g1)
lp15@61190
   524
  done
lp15@61190
   525
qed
lp15@61190
   526
lp15@61190
   527
lemma piecewise_C1_differentiable_D2:
lp15@61190
   528
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   529
  assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" "pathfinish g1 = pathstart g2"
lp15@61190
   530
    shows "g2 piecewise_C1_differentiable_on {0..1}"
lp15@61190
   531
proof -
lp15@61190
   532
  obtain s where "finite s"
lp15@61190
   533
             and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   534
             and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
lp15@61190
   535
    using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@61190
   536
  then have g2D: "g2 differentiable at x" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)" for x
paulson@62087
   537
    apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\<lambda>x. (x+1)/2)" in differentiable_transform_within)
lp15@61190
   538
    using that
lp15@61190
   539
    apply (simp_all add: dist_real_def joinpaths_def)
lp15@61190
   540
    apply (auto simp: dist_real_def joinpaths_def field_simps)
lp15@61190
   541
    apply (rule differentiable_chain_at derivative_intros | force)+
lp15@61190
   542
    apply (drule_tac x= "(x + 1) / 2" in bspec, force simp: divide_simps)
lp15@61190
   543
    apply assumption
lp15@61190
   544
    done
lp15@61190
   545
  have [simp]: "vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at ((x+1)/2)) = 2 *\<^sub>R vector_derivative g2 (at x)"
lp15@61190
   546
               if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)" for x
lp15@61190
   547
    using that  by (auto simp: vector_derivative_chain_at divide_simps g2D)
lp15@61190
   548
  have "continuous_on ({1/2..1} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   549
    using co12 by (rule continuous_on_subset) force
lp15@61190
   550
  then have coDhalf: "continuous_on ({1/2..1} - insert (1/2) s) (\<lambda>x. vector_derivative (g2 o (\<lambda>x. 2*x-1)) (at x))"
lp15@61190
   551
    apply (rule continuous_on_eq [OF _ vector_derivative_at])
paulson@62087
   552
    apply (rule_tac f="g2 o (\<lambda>x. 2*x-1)" and d="dist (3/4) ((x+1)/2)" in has_vector_derivative_transform_within)
lp15@61190
   553
    apply (auto simp: dist_real_def field_simps joinpaths_def vector_derivative_works [symmetric]
lp15@61190
   554
                intro!: g2D differentiable_chain_at)
lp15@61190
   555
    done
lp15@61190
   556
  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
   557
    apply (simp add: image_set_diff inj_on_def image_image)
lp15@61190
   558
    apply (auto simp: image_affinity_atLeastAtMost_div add_divide_distrib)
lp15@61190
   559
    done
lp15@61190
   560
  have "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s))
lp15@61190
   561
                      ((\<lambda>x. 1/2 * vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x)) o (\<lambda>x. (x+1)/2))"
lp15@61190
   562
    by (rule continuous_intros | simp add:  coDhalf)+
lp15@61190
   563
  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
   564
    by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
lp15@61190
   565
  have "continuous_on {0..1} g2"
lp15@61190
   566
    using continuous_on_joinpaths_D2 assms piecewise_C1_differentiable_on_def by blast
wenzelm@61222
   567
  with \<open>finite s\<close> show ?thesis
lp15@61190
   568
    apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@61190
   569
    apply (rule_tac x="insert 0 ((\<lambda>x. 2 * x - 1) ` s)" in exI)
lp15@61190
   570
    apply (simp add: g2D con_g2)
lp15@61190
   571
  done
lp15@61190
   572
qed
lp15@60809
   573
lp15@60809
   574
subsection \<open>Valid paths, and their start and finish\<close>
lp15@60809
   575
lp15@60809
   576
lemma Diff_Un_eq: "A - (B \<union> C) = A - B - C"
lp15@60809
   577
  by blast
lp15@60809
   578
lp15@60809
   579
definition valid_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
lp15@61190
   580
  where "valid_path f \<equiv> f piecewise_C1_differentiable_on {0..1::real}"
lp15@60809
   581
lp15@60809
   582
definition closed_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
lp15@60809
   583
  where "closed_path g \<equiv> g 0 = g 1"
lp15@60809
   584
lp15@60809
   585
subsubsection\<open>In particular, all results for paths apply\<close>
lp15@60809
   586
lp15@60809
   587
lemma valid_path_imp_path: "valid_path g \<Longrightarrow> path g"
lp15@61190
   588
by (simp add: path_def piecewise_C1_differentiable_on_def valid_path_def)
lp15@60809
   589
lp15@60809
   590
lemma connected_valid_path_image: "valid_path g \<Longrightarrow> connected(path_image g)"
lp15@60809
   591
  by (metis connected_path_image valid_path_imp_path)
lp15@60809
   592
lp15@60809
   593
lemma compact_valid_path_image: "valid_path g \<Longrightarrow> compact(path_image g)"
lp15@60809
   594
  by (metis compact_path_image valid_path_imp_path)
lp15@60809
   595
lp15@60809
   596
lemma bounded_valid_path_image: "valid_path g \<Longrightarrow> bounded(path_image g)"
lp15@60809
   597
  by (metis bounded_path_image valid_path_imp_path)
lp15@60809
   598
lp15@60809
   599
lemma closed_valid_path_image: "valid_path g \<Longrightarrow> closed(path_image g)"
lp15@60809
   600
  by (metis closed_path_image valid_path_imp_path)
lp15@60809
   601
lp15@62540
   602
proposition valid_path_compose:
lp15@62540
   603
  assumes "valid_path g" 
lp15@62540
   604
      and der: "\<And>x. x \<in> path_image g \<Longrightarrow> \<exists>f'. (f has_field_derivative f') (at x)"
lp15@62540
   605
      and con: "continuous_on (path_image g) (deriv f)"
lp15@62408
   606
    shows "valid_path (f o g)"
lp15@62408
   607
proof -
lp15@62408
   608
  obtain s where "finite s" and g_diff: "g C1_differentiable_on {0..1} - s"
lp15@62408
   609
    using `valid_path g` unfolding valid_path_def piecewise_C1_differentiable_on_def by auto
lp15@62540
   610
  have "f \<circ> g differentiable at t" when "t\<in>{0..1} - s" for t
lp15@62408
   611
    proof (rule differentiable_chain_at)
lp15@62540
   612
      show "g differentiable at t" using `valid_path g` 
lp15@62408
   613
        by (meson C1_differentiable_on_eq \<open>g C1_differentiable_on {0..1} - s\<close> that)
lp15@62408
   614
    next
lp15@62408
   615
      have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis
lp15@62408
   616
      then obtain f' where "(f has_field_derivative f') (at (g t))"
lp15@62408
   617
        using der by auto
lp15@62408
   618
      then have " (f has_derivative op * f') (at (g t))"
lp15@62408
   619
        using has_field_derivative_imp_has_derivative[of f f' "at (g t)"] by auto
lp15@62408
   620
      then show "f differentiable at (g t)" using differentiableI by auto
lp15@62408
   621
    qed
lp15@62408
   622
  moreover have "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (f \<circ> g) (at x))"
lp15@62540
   623
    proof (rule continuous_on_eq [where f = "\<lambda>x. vector_derivative g (at x) * deriv f (g x)"],
lp15@62540
   624
        rule continuous_intros)
lp15@62540
   625
      show "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative g (at x))"
lp15@62540
   626
        using g_diff C1_differentiable_on_eq by auto
lp15@62540
   627
    next
lp15@62540
   628
      have "continuous_on {0..1} (\<lambda>x. deriv f (g x))" 
lp15@62540
   629
        using continuous_on_compose[OF _ con[unfolded path_image_def],unfolded comp_def] 
lp15@62540
   630
          `valid_path g` piecewise_C1_differentiable_on_def valid_path_def 
lp15@62540
   631
        by blast
lp15@62540
   632
      then show "continuous_on ({0..1} - s) (\<lambda>x. deriv f (g x))" 
lp15@62540
   633
        using continuous_on_subset by blast
lp15@62408
   634
    next
lp15@62540
   635
      show "vector_derivative g (at t) * deriv f (g t) = vector_derivative (f \<circ> g) (at t)"
lp15@62540
   636
          when "t \<in> {0..1} - s" for t
lp15@62540
   637
        proof (rule vector_derivative_chain_at_general[symmetric])
lp15@62540
   638
          show "g differentiable at t" by (meson C1_differentiable_on_eq g_diff that)
lp15@62540
   639
        next
lp15@62540
   640
          have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis
lp15@62540
   641
          then obtain f' where "(f has_field_derivative f') (at (g t))"
lp15@62540
   642
            using der by auto
lp15@62540
   643
          then show "\<exists>g'. (f has_field_derivative g') (at (g t))" by auto
lp15@62540
   644
        qed
lp15@62408
   645
    qed
lp15@62408
   646
  ultimately have "f o g C1_differentiable_on {0..1} - s"
lp15@62408
   647
    using C1_differentiable_on_eq by blast
lp15@62540
   648
  moreover have "path (f o g)" 
lp15@62540
   649
    proof -
lp15@62540
   650
      have "isCont f x" when "x\<in>path_image g" for x 
lp15@62540
   651
        proof -
lp15@62540
   652
          obtain f' where "(f has_field_derivative f') (at x)" 
lp15@62540
   653
            using der[rule_format] `x\<in>path_image g` by auto
lp15@62540
   654
          thus ?thesis using DERIV_isCont by auto
lp15@62540
   655
        qed
lp15@62540
   656
      then have "continuous_on (path_image g) f" using continuous_at_imp_continuous_on by auto
lp15@62540
   657
      then show ?thesis using path_continuous_image `valid_path g` valid_path_imp_path by auto
lp15@62540
   658
    qed
lp15@62408
   659
  ultimately show ?thesis unfolding valid_path_def piecewise_C1_differentiable_on_def path_def
lp15@62408
   660
    using `finite s` by auto
lp15@62408
   661
qed
lp15@62408
   662
lp15@60809
   663
lp15@60809
   664
subsection\<open>Contour Integrals along a path\<close>
lp15@60809
   665
lp15@60809
   666
text\<open>This definition is for complex numbers only, and does not generalise to line integrals in a vector field\<close>
lp15@60809
   667
lp15@61190
   668
text\<open>piecewise differentiable function on [0,1]\<close>
lp15@60809
   669
lp15@61738
   670
definition has_contour_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool"
lp15@61738
   671
           (infixr "has'_contour'_integral" 50)
lp15@61738
   672
  where "(f has_contour_integral i) g \<equiv>
lp15@60809
   673
           ((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1}))
lp15@60809
   674
            has_integral i) {0..1}"
lp15@60809
   675
lp15@61738
   676
definition contour_integrable_on
lp15@61738
   677
           (infixr "contour'_integrable'_on" 50)
lp15@61738
   678
  where "f contour_integrable_on g \<equiv> \<exists>i. (f has_contour_integral i) g"
lp15@61738
   679
lp15@61738
   680
definition contour_integral
lp15@62463
   681
  where "contour_integral g f \<equiv> @i. (f has_contour_integral i) g \<or> ~ f contour_integrable_on g \<and> i=0"
lp15@62463
   682
lp15@62463
   683
lemma not_integrable_contour_integral: "~ f contour_integrable_on g \<Longrightarrow> contour_integral g f = 0"
lp15@62534
   684
  unfolding contour_integrable_on_def contour_integral_def by blast
lp15@62463
   685
lp15@62463
   686
lemma contour_integral_unique: "(f has_contour_integral i) g \<Longrightarrow> contour_integral g f = i"
lp15@62463
   687
  apply (simp add: contour_integral_def has_contour_integral_def contour_integrable_on_def)
lp15@62463
   688
  using has_integral_unique by blast
lp15@61738
   689
paulson@62131
   690
corollary has_contour_integral_eqpath:
lp15@62397
   691
     "\<lbrakk>(f has_contour_integral y) p; f contour_integrable_on \<gamma>;
paulson@62131
   692
       contour_integral p f = contour_integral \<gamma> f\<rbrakk>
paulson@62131
   693
      \<Longrightarrow> (f has_contour_integral y) \<gamma>"
paulson@62131
   694
using contour_integrable_on_def contour_integral_unique by auto
paulson@62131
   695
lp15@61738
   696
lemma has_contour_integral_integral:
lp15@61738
   697
    "f contour_integrable_on i \<Longrightarrow> (f has_contour_integral (contour_integral i f)) i"
lp15@61738
   698
  by (metis contour_integral_unique contour_integrable_on_def)
lp15@61738
   699
lp15@61738
   700
lemma has_contour_integral_unique:
lp15@61738
   701
    "(f has_contour_integral i) g \<Longrightarrow> (f has_contour_integral j) g \<Longrightarrow> i = j"
lp15@60809
   702
  using has_integral_unique
lp15@61738
   703
  by (auto simp: has_contour_integral_def)
lp15@61738
   704
lp15@61738
   705
lemma has_contour_integral_integrable: "(f has_contour_integral i) g \<Longrightarrow> f contour_integrable_on g"
lp15@61738
   706
  using contour_integrable_on_def by blast
lp15@60809
   707
lp15@60809
   708
(* Show that we can forget about the localized derivative.*)
lp15@60809
   709
lp15@60809
   710
lemma vector_derivative_within_interior:
lp15@60809
   711
     "\<lbrakk>x \<in> interior s; NO_MATCH UNIV s\<rbrakk>
lp15@60809
   712
      \<Longrightarrow> vector_derivative f (at x within s) = vector_derivative f (at x)"
lp15@60809
   713
  apply (simp add: vector_derivative_def has_vector_derivative_def has_derivative_def netlimit_within_interior)
lp15@60809
   714
  apply (subst lim_within_interior, auto)
lp15@60809
   715
  done
lp15@60809
   716
lp15@60809
   717
lemma has_integral_localized_vector_derivative:
lp15@60809
   718
    "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
lp15@60809
   719
     ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
lp15@60809
   720
proof -
lp15@60809
   721
  have "{a..b} - {a,b} = interior {a..b}"
lp15@60809
   722
    by (simp add: atLeastAtMost_diff_ends)
lp15@60809
   723
  show ?thesis
lp15@60809
   724
    apply (rule has_integral_spike_eq [of "{a,b}"])
lp15@60809
   725
    apply (auto simp: vector_derivative_within_interior)
lp15@60809
   726
    done
lp15@60809
   727
qed
lp15@60809
   728
lp15@60809
   729
lemma integrable_on_localized_vector_derivative:
lp15@60809
   730
    "(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow>
lp15@60809
   731
     (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
lp15@60809
   732
  by (simp add: integrable_on_def has_integral_localized_vector_derivative)
lp15@60809
   733
lp15@61738
   734
lemma has_contour_integral:
lp15@61738
   735
     "(f has_contour_integral i) g \<longleftrightarrow>
lp15@60809
   736
      ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
lp15@61738
   737
  by (simp add: has_integral_localized_vector_derivative has_contour_integral_def)
lp15@61738
   738
lp15@61738
   739
lemma contour_integrable_on:
lp15@61738
   740
     "f contour_integrable_on g \<longleftrightarrow>
lp15@60809
   741
      (\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
lp15@61738
   742
  by (simp add: has_contour_integral integrable_on_def contour_integrable_on_def)
lp15@60809
   743
lp15@60809
   744
subsection\<open>Reversing a path\<close>
lp15@60809
   745
lp15@60809
   746
lemma valid_path_imp_reverse:
lp15@60809
   747
  assumes "valid_path g"
lp15@60809
   748
    shows "valid_path(reversepath g)"
lp15@60809
   749
proof -
lp15@61190
   750
  obtain s where "finite s" "g C1_differentiable_on ({0..1} - s)"
lp15@61190
   751
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
lp15@61190
   752
  then have "finite (op - 1 ` s)" "(reversepath g C1_differentiable_on ({0..1} - op - 1 ` s))"
lp15@60809
   753
    apply (auto simp: reversepath_def)
lp15@61190
   754
    apply (rule C1_differentiable_compose [of "\<lambda>x::real. 1-x" _ g, unfolded o_def])
lp15@61190
   755
    apply (auto simp: C1_differentiable_on_eq)
lp15@61190
   756
    apply (rule continuous_intros, force)
lp15@61190
   757
    apply (force elim!: continuous_on_subset)
lp15@61190
   758
    apply (simp add: finite_vimageI inj_on_def)
lp15@60809
   759
    done
lp15@60809
   760
  then show ?thesis using assms
lp15@61190
   761
    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def path_def [symmetric])
lp15@60809
   762
qed
lp15@60809
   763
lp15@62540
   764
lemma valid_path_reversepath [simp]: "valid_path(reversepath g) \<longleftrightarrow> valid_path g"
lp15@60809
   765
  using valid_path_imp_reverse by force
lp15@60809
   766
lp15@61738
   767
lemma has_contour_integral_reversepath:
lp15@61738
   768
  assumes "valid_path g" "(f has_contour_integral i) g"
lp15@61738
   769
    shows "(f has_contour_integral (-i)) (reversepath g)"
lp15@60809
   770
proof -
lp15@60809
   771
  { fix s x
lp15@61190
   772
    assume xs: "g C1_differentiable_on ({0..1} - s)" "x \<notin> op - 1 ` s" "0 \<le> x" "x \<le> 1"
lp15@60809
   773
      have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
lp15@60809
   774
            - vector_derivative g (at (1 - x) within {0..1})"
lp15@60809
   775
      proof -
lp15@60809
   776
        obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
lp15@60809
   777
          using xs
lp15@61190
   778
          by (force simp: has_vector_derivative_def C1_differentiable_on_def)
lp15@60809
   779
        have "(g o (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
lp15@60809
   780
          apply (rule vector_diff_chain_within)
lp15@60809
   781
          apply (intro vector_diff_chain_within derivative_eq_intros | simp)+
lp15@60809
   782
          apply (rule has_vector_derivative_at_within [OF f'])
lp15@60809
   783
          done
lp15@60809
   784
        then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)"
lp15@60809
   785
          by (simp add: o_def)
lp15@60809
   786
        show ?thesis
lp15@60809
   787
          using xs
lp15@60809
   788
          by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
lp15@60809
   789
      qed
lp15@60809
   790
  } note * = this
lp15@60809
   791
  have 01: "{0..1::real} = cbox 0 1"
lp15@60809
   792
    by simp
lp15@60809
   793
  show ?thesis using assms
lp15@61738
   794
    apply (auto simp: has_contour_integral_def)
lp15@60809
   795
    apply (drule has_integral_affinity01 [where m= "-1" and c=1])
lp15@61190
   796
    apply (auto simp: reversepath_def valid_path_def piecewise_C1_differentiable_on_def)
lp15@60809
   797
    apply (drule has_integral_neg)
lp15@60809
   798
    apply (rule_tac s = "(\<lambda>x. 1 - x) ` s" in has_integral_spike_finite)
lp15@60809
   799
    apply (auto simp: *)
lp15@60809
   800
    done
lp15@60809
   801
qed
lp15@60809
   802
lp15@61738
   803
lemma contour_integrable_reversepath:
lp15@61738
   804
    "valid_path g \<Longrightarrow> f contour_integrable_on g \<Longrightarrow> f contour_integrable_on (reversepath g)"
lp15@61738
   805
  using has_contour_integral_reversepath contour_integrable_on_def by blast
lp15@61738
   806
lp15@61738
   807
lemma contour_integrable_reversepath_eq:
lp15@61738
   808
    "valid_path g \<Longrightarrow> (f contour_integrable_on (reversepath g) \<longleftrightarrow> f contour_integrable_on g)"
lp15@61738
   809
  using contour_integrable_reversepath valid_path_reversepath by fastforce
lp15@61738
   810
lp15@61738
   811
lemma contour_integral_reversepath:
lp15@62463
   812
  assumes "valid_path g"
lp15@62463
   813
    shows "contour_integral (reversepath g) f = - (contour_integral g f)"
lp15@62463
   814
proof (cases "f contour_integrable_on g")
lp15@62463
   815
  case True then show ?thesis
lp15@62463
   816
    by (simp add: assms contour_integral_unique has_contour_integral_integral has_contour_integral_reversepath)
lp15@62463
   817
next
lp15@62463
   818
  case False then have "~ f contour_integrable_on (reversepath g)"
lp15@62463
   819
    by (simp add: assms contour_integrable_reversepath_eq)
lp15@62463
   820
  with False show ?thesis by (simp add: not_integrable_contour_integral)
lp15@62463
   821
qed
lp15@60809
   822
lp15@60809
   823
lp15@60809
   824
subsection\<open>Joining two paths together\<close>
lp15@60809
   825
lp15@60809
   826
lemma valid_path_join:
lp15@60809
   827
  assumes "valid_path g1" "valid_path g2" "pathfinish g1 = pathstart g2"
lp15@60809
   828
    shows "valid_path(g1 +++ g2)"
lp15@60809
   829
proof -
lp15@60809
   830
  have "g1 1 = g2 0"
lp15@60809
   831
    using assms by (auto simp: pathfinish_def pathstart_def)
lp15@61190
   832
  moreover have "(g1 o (\<lambda>x. 2*x)) piecewise_C1_differentiable_on {0..1/2}"
lp15@61190
   833
    apply (rule piecewise_C1_differentiable_compose)
lp15@60809
   834
    using assms
lp15@61190
   835
    apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths)
lp15@60809
   836
    apply (rule continuous_intros | simp)+
lp15@60809
   837
    apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
lp15@60809
   838
    done
lp15@61190
   839
  moreover have "(g2 o (\<lambda>x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}"
lp15@61190
   840
    apply (rule piecewise_C1_differentiable_compose)
lp15@61190
   841
    using assms unfolding valid_path_def piecewise_C1_differentiable_on_def
lp15@61190
   842
    by (auto intro!: continuous_intros finite_vimageI [where h = "(\<lambda>x. 2*x - 1)"] inj_onI
lp15@61190
   843
             simp: image_affinity_atLeastAtMost_diff continuous_on_joinpaths)
lp15@60809
   844
  ultimately show ?thesis
lp15@60809
   845
    apply (simp only: valid_path_def continuous_on_joinpaths joinpaths_def)
lp15@61190
   846
    apply (rule piecewise_C1_differentiable_cases)
lp15@60809
   847
    apply (auto simp: o_def)
lp15@60809
   848
    done
lp15@60809
   849
qed
lp15@60809
   850
lp15@61190
   851
lemma valid_path_join_D1:
lp15@61190
   852
  fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   853
  shows "valid_path (g1 +++ g2) \<Longrightarrow> valid_path g1"
lp15@61190
   854
  unfolding valid_path_def
lp15@61190
   855
  by (rule piecewise_C1_differentiable_D1)
lp15@60809
   856
lp15@61190
   857
lemma valid_path_join_D2:
lp15@61190
   858
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   859
  shows "\<lbrakk>valid_path (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> valid_path g2"
lp15@61190
   860
  unfolding valid_path_def
lp15@61190
   861
  by (rule piecewise_C1_differentiable_D2)
lp15@60809
   862
lp15@60809
   863
lemma valid_path_join_eq [simp]:
lp15@61190
   864
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   865
  shows "pathfinish g1 = pathstart g2 \<Longrightarrow> (valid_path(g1 +++ g2) \<longleftrightarrow> valid_path g1 \<and> valid_path g2)"
lp15@60809
   866
  using valid_path_join_D1 valid_path_join_D2 valid_path_join by blast
lp15@60809
   867
lp15@61738
   868
lemma has_contour_integral_join:
lp15@61738
   869
  assumes "(f has_contour_integral i1) g1" "(f has_contour_integral i2) g2"
lp15@60809
   870
          "valid_path g1" "valid_path g2"
lp15@61738
   871
    shows "(f has_contour_integral (i1 + i2)) (g1 +++ g2)"
lp15@60809
   872
proof -
lp15@60809
   873
  obtain s1 s2
lp15@60809
   874
    where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
lp15@60809
   875
      and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
lp15@60809
   876
    using assms
lp15@61190
   877
    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   878
  have 1: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
lp15@60809
   879
   and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
lp15@60809
   880
    using assms
lp15@61738
   881
    by (auto simp: has_contour_integral)
lp15@60809
   882
  have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
lp15@60809
   883
   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
   884
    using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
lp15@60809
   885
          has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
lp15@60809
   886
    by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
lp15@60809
   887
  have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
lp15@60809
   888
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
lp15@60809
   889
            2 *\<^sub>R vector_derivative g1 (at (z*2))" for z
paulson@62087
   890
    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
   891
    apply (simp_all add: dist_real_def abs_if split: if_split_asm)
lp15@60809
   892
    apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
lp15@60809
   893
    apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
lp15@60809
   894
    using s1
lp15@60809
   895
    apply (auto simp: algebra_simps vector_derivative_works)
lp15@60809
   896
    done
lp15@60809
   897
  have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
lp15@60809
   898
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
lp15@60809
   899
            2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z
paulson@62087
   900
    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
   901
    apply (simp_all add: dist_real_def abs_if split: if_split_asm)
lp15@60809
   902
    apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
lp15@60809
   903
    apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
lp15@60809
   904
    using s2
lp15@60809
   905
    apply (auto simp: algebra_simps vector_derivative_works)
lp15@60809
   906
    done
lp15@60809
   907
  have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
lp15@60809
   908
    apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) (op*2 -` s1)"])
lp15@60809
   909
    using s1
lp15@60809
   910
    apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
lp15@60809
   911
    apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
lp15@60809
   912
    done
lp15@60809
   913
  moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
lp15@60809
   914
    apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
lp15@60809
   915
    using s2
lp15@60809
   916
    apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
lp15@60809
   917
    apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
lp15@60809
   918
    done
lp15@60809
   919
  ultimately
lp15@60809
   920
  show ?thesis
lp15@61738
   921
    apply (simp add: has_contour_integral)
lp15@60809
   922
    apply (rule has_integral_combine [where c = "1/2"], auto)
lp15@60809
   923
    done
lp15@60809
   924
qed
lp15@60809
   925
lp15@61738
   926
lemma contour_integrable_joinI:
lp15@61738
   927
  assumes "f contour_integrable_on g1" "f contour_integrable_on g2"
lp15@60809
   928
          "valid_path g1" "valid_path g2"
lp15@61738
   929
    shows "f contour_integrable_on (g1 +++ g2)"
lp15@60809
   930
  using assms
lp15@61738
   931
  by (meson has_contour_integral_join contour_integrable_on_def)
lp15@61738
   932
lp15@61738
   933
lemma contour_integrable_joinD1:
lp15@61738
   934
  assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g1"
lp15@61738
   935
    shows "f contour_integrable_on g1"
lp15@60809
   936
proof -
lp15@60809
   937
  obtain s1
lp15@60809
   938
    where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
lp15@61190
   939
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   940
  have "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
lp15@60809
   941
    using assms
lp15@61738
   942
    apply (auto simp: contour_integrable_on)
lp15@60809
   943
    apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
lp15@60809
   944
    apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
lp15@60809
   945
    done
lp15@60809
   946
  then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
lp15@61190
   947
    by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
lp15@60809
   948
  have g1: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow>
lp15@60809
   949
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
lp15@60809
   950
            2 *\<^sub>R vector_derivative g1 (at z)"  for z
paulson@62087
   951
    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
   952
    apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
lp15@60809
   953
    apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def])
lp15@60809
   954
    using s1
lp15@60809
   955
    apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
lp15@60809
   956
    done
lp15@60809
   957
  show ?thesis
lp15@60809
   958
    using s1
lp15@61738
   959
    apply (auto simp: contour_integrable_on)
lp15@60809
   960
    apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
lp15@60809
   961
    apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
lp15@60809
   962
    done
lp15@60809
   963
qed
lp15@60809
   964
lp15@61738
   965
lemma contour_integrable_joinD2:
lp15@61738
   966
  assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g2"
lp15@61738
   967
    shows "f contour_integrable_on g2"
lp15@60809
   968
proof -
lp15@60809
   969
  obtain s2
lp15@60809
   970
    where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
lp15@61190
   971
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   972
  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
   973
    using assms
lp15@61738
   974
    apply (auto simp: contour_integrable_on)
lp15@60809
   975
    apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
lp15@60809
   976
    apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
lp15@60809
   977
    apply (simp add: image_affinity_atLeastAtMost_diff)
lp15@60809
   978
    done
lp15@60809
   979
  then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2)))
lp15@60809
   980
                integrable_on {0..1}"
lp15@60809
   981
    by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
lp15@60809
   982
  have g2: "\<lbrakk>0 < z; z < 1; z \<notin> s2\<rbrakk> \<Longrightarrow>
lp15@60809
   983
            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
   984
            2 *\<^sub>R vector_derivative g2 (at z)" for z
paulson@62087
   985
    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
   986
    apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
lp15@60809
   987
    apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def])
lp15@60809
   988
    using s2
lp15@60809
   989
    apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left
lp15@60809
   990
                      vector_derivative_works add_divide_distrib)
lp15@60809
   991
    done
lp15@60809
   992
  show ?thesis
lp15@60809
   993
    using s2
lp15@61738
   994
    apply (auto simp: contour_integrable_on)
lp15@60809
   995
    apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
lp15@60809
   996
    apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
lp15@60809
   997
    done
lp15@60809
   998
qed
lp15@60809
   999
lp15@61738
  1000
lemma contour_integrable_join [simp]:
lp15@60809
  1001
  shows
lp15@60809
  1002
    "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
lp15@61738
  1003
     \<Longrightarrow> f contour_integrable_on (g1 +++ g2) \<longleftrightarrow> f contour_integrable_on g1 \<and> f contour_integrable_on g2"
lp15@61738
  1004
using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast
lp15@61738
  1005
lp15@61738
  1006
lemma contour_integral_join [simp]:
lp15@60809
  1007
  shows
lp15@61738
  1008
    "\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
lp15@61738
  1009
        \<Longrightarrow> contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f"
lp15@61738
  1010
  by (simp add: has_contour_integral_integral has_contour_integral_join contour_integral_unique)
lp15@60809
  1011
lp15@60809
  1012
lp15@60809
  1013
subsection\<open>Shifting the starting point of a (closed) path\<close>
lp15@60809
  1014
lp15@60809
  1015
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
  1016
  by (auto simp: shiftpath_def)
lp15@60809
  1017
lp15@60809
  1018
lemma valid_path_shiftpath [intro]:
lp15@60809
  1019
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@60809
  1020
    shows "valid_path(shiftpath a g)"
lp15@60809
  1021
  using assms
lp15@60809
  1022
  apply (auto simp: valid_path_def shiftpath_alt_def)
lp15@61190
  1023
  apply (rule piecewise_C1_differentiable_cases)
lp15@60809
  1024
  apply (auto simp: algebra_simps)
lp15@61190
  1025
  apply (rule piecewise_C1_differentiable_affine [of g 1 a, simplified o_def scaleR_one])
lp15@61190
  1026
  apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
lp15@61190
  1027
  apply (rule piecewise_C1_differentiable_affine [of g 1 "a-1", simplified o_def scaleR_one algebra_simps])
lp15@61190
  1028
  apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
lp15@60809
  1029
  done
lp15@60809
  1030
lp15@61738
  1031
lemma has_contour_integral_shiftpath:
lp15@61738
  1032
  assumes f: "(f has_contour_integral i) g" "valid_path g"
lp15@60809
  1033
      and a: "a \<in> {0..1}"
lp15@61738
  1034
    shows "(f has_contour_integral i) (shiftpath a g)"
lp15@60809
  1035
proof -
lp15@60809
  1036
  obtain s
lp15@60809
  1037
    where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
lp15@61190
  1038
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
  1039
  have *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
lp15@61738
  1040
    using assms by (auto simp: has_contour_integral)
lp15@60809
  1041
  then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) +
lp15@60809
  1042
                    integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
lp15@60809
  1043
    apply (rule has_integral_unique)
lp15@60809
  1044
    apply (subst add.commute)
lp15@60809
  1045
    apply (subst Integration.integral_combine)
lp15@60809
  1046
    using assms * integral_unique by auto
lp15@60809
  1047
  { fix x
lp15@60809
  1048
    have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow>
lp15@60809
  1049
         vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
lp15@60809
  1050
      unfolding shiftpath_def
paulson@62087
  1051
      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
  1052
        apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
lp15@60809
  1053
      apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one])
lp15@60809
  1054
       apply (intro derivative_eq_intros | simp)+
lp15@60809
  1055
      using g
lp15@60809
  1056
       apply (drule_tac x="x+a" in bspec)
lp15@60809
  1057
      using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
lp15@60809
  1058
      done
lp15@60809
  1059
  } note vd1 = this
lp15@60809
  1060
  { fix x
lp15@60809
  1061
    have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow>
lp15@60809
  1062
          vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
lp15@60809
  1063
      unfolding shiftpath_def
paulson@62087
  1064
      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
  1065
        apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
lp15@60809
  1066
      apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one])
lp15@60809
  1067
       apply (intro derivative_eq_intros | simp)+
lp15@60809
  1068
      using g
lp15@60809
  1069
      apply (drule_tac x="x+a-1" in bspec)
lp15@60809
  1070
      using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
lp15@60809
  1071
      done
lp15@60809
  1072
  } note vd2 = this
lp15@60809
  1073
  have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
lp15@60809
  1074
    using * a   by (fastforce intro: integrable_subinterval_real)
lp15@60809
  1075
  have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
lp15@60809
  1076
    apply (rule integrable_subinterval_real)
lp15@60809
  1077
    using * a by auto
lp15@60809
  1078
  have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
lp15@60809
  1079
        has_integral  integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {0..1 - a}"
lp15@60809
  1080
    apply (rule has_integral_spike_finite
lp15@60809
  1081
             [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
  1082
      using s apply blast
lp15@60809
  1083
     using a apply (auto simp: algebra_simps vd1)
lp15@60809
  1084
     apply (force simp: shiftpath_def add.commute)
lp15@60809
  1085
    using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]]
lp15@60809
  1086
    apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute)
lp15@60809
  1087
    done
lp15@60809
  1088
  moreover
lp15@60809
  1089
  have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
lp15@60809
  1090
        has_integral  integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {1 - a..1}"
lp15@60809
  1091
    apply (rule has_integral_spike_finite
lp15@60809
  1092
             [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
  1093
      using s apply blast
lp15@60809
  1094
     using a apply (auto simp: algebra_simps vd2)
lp15@60809
  1095
     apply (force simp: shiftpath_def add.commute)
lp15@60809
  1096
    using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
lp15@60809
  1097
    apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified])
lp15@60809
  1098
    apply (simp add: algebra_simps)
lp15@60809
  1099
    done
lp15@60809
  1100
  ultimately show ?thesis
lp15@60809
  1101
    using a
lp15@61738
  1102
    by (auto simp: i has_contour_integral intro: has_integral_combine [where c = "1-a"])
lp15@60809
  1103
qed
lp15@60809
  1104
lp15@61738
  1105
lemma has_contour_integral_shiftpath_D:
lp15@61738
  1106
  assumes "(f has_contour_integral i) (shiftpath a g)"
lp15@60809
  1107
          "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@61738
  1108
    shows "(f has_contour_integral i) g"
lp15@60809
  1109
proof -
lp15@60809
  1110
  obtain s
lp15@60809
  1111
    where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
lp15@61190
  1112
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
  1113
  { fix x
lp15@60809
  1114
    assume x: "0 < x" "x < 1" "x \<notin> s"
lp15@60809
  1115
    then have gx: "g differentiable at x"
lp15@60809
  1116
      using g by auto
lp15@60809
  1117
    have "vector_derivative g (at x within {0..1}) =
lp15@60809
  1118
          vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
lp15@60809
  1119
      apply (rule vector_derivative_at_within_ivl
lp15@60809
  1120
                  [OF has_vector_derivative_transform_within_open
paulson@62087
  1121
                      [where f = "(shiftpath (1 - a) (shiftpath a g))" and s = "{0<..<1}-s"]])
lp15@60809
  1122
      using s g assms x
lp15@60809
  1123
      apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
lp15@60809
  1124
                        vector_derivative_within_interior vector_derivative_works [symmetric])
paulson@62087
  1125
      apply (rule differentiable_transform_within [OF gx, of "min x (1-x)"])
nipkow@62390
  1126
      apply (auto simp: dist_real_def shiftpath_shiftpath abs_if split: if_split_asm)
lp15@60809
  1127
      done
lp15@60809
  1128
  } note vd = this
lp15@61738
  1129
  have fi: "(f has_contour_integral i) (shiftpath (1 - a) (shiftpath a g))"
lp15@61738
  1130
    using assms  by (auto intro!: has_contour_integral_shiftpath)
lp15@60809
  1131
  show ?thesis
lp15@61738
  1132
    apply (simp add: has_contour_integral_def)
lp15@61738
  1133
    apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _  fi [unfolded has_contour_integral_def]])
lp15@60809
  1134
    using s assms vd
lp15@60809
  1135
    apply (auto simp: Path_Connected.shiftpath_shiftpath)
lp15@60809
  1136
    done
lp15@60809
  1137
qed
lp15@60809
  1138
lp15@61738
  1139
lemma has_contour_integral_shiftpath_eq:
lp15@60809
  1140
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@61738
  1141
    shows "(f has_contour_integral i) (shiftpath a g) \<longleftrightarrow> (f has_contour_integral i) g"
lp15@61738
  1142
  using assms has_contour_integral_shiftpath has_contour_integral_shiftpath_D by blast
lp15@61738
  1143
lp15@62463
  1144
lemma contour_integrable_on_shiftpath_eq:
lp15@62463
  1145
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@62463
  1146
    shows "f contour_integrable_on (shiftpath a g) \<longleftrightarrow> f contour_integrable_on g"
lp15@62463
  1147
using assms contour_integrable_on_def has_contour_integral_shiftpath_eq by auto
lp15@62463
  1148
lp15@61738
  1149
lemma contour_integral_shiftpath:
lp15@60809
  1150
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@61738
  1151
    shows "contour_integral (shiftpath a g) f = contour_integral g f"
lp15@62534
  1152
   using assms
lp15@62463
  1153
   by (simp add: contour_integral_def contour_integrable_on_def has_contour_integral_shiftpath_eq)
lp15@60809
  1154
lp15@60809
  1155
lp15@60809
  1156
subsection\<open>More about straight-line paths\<close>
lp15@60809
  1157
lp15@60809
  1158
lemma has_vector_derivative_linepath_within:
lp15@60809
  1159
    "(linepath a b has_vector_derivative (b - a)) (at x within s)"
lp15@60809
  1160
apply (simp add: linepath_def has_vector_derivative_def algebra_simps)
lp15@60809
  1161
apply (rule derivative_eq_intros | simp)+
lp15@60809
  1162
done
lp15@60809
  1163
lp15@60809
  1164
lemma vector_derivative_linepath_within:
lp15@60809
  1165
    "x \<in> {0..1} \<Longrightarrow> vector_derivative (linepath a b) (at x within {0..1}) = b - a"
lp15@60809
  1166
  apply (rule vector_derivative_within_closed_interval [of 0 "1::real", simplified])
lp15@60809
  1167
  apply (auto simp: has_vector_derivative_linepath_within)
lp15@60809
  1168
  done
lp15@60809
  1169
lp15@61190
  1170
lemma vector_derivative_linepath_at [simp]: "vector_derivative (linepath a b) (at x) = b - a"
lp15@60809
  1171
  by (simp add: has_vector_derivative_linepath_within vector_derivative_at)
lp15@60809
  1172
lp15@61190
  1173
lemma valid_path_linepath [iff]: "valid_path (linepath a b)"
lp15@61190
  1174
  apply (simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_on_linepath)
lp15@61190
  1175
  apply (rule_tac x="{}" in exI)
lp15@61190
  1176
  apply (simp add: differentiable_on_def differentiable_def)
lp15@61190
  1177
  using has_vector_derivative_def has_vector_derivative_linepath_within
lp15@61190
  1178
  apply (fastforce simp add: continuous_on_eq_continuous_within)
lp15@61190
  1179
  done
lp15@61190
  1180
lp15@61738
  1181
lemma has_contour_integral_linepath:
lp15@61738
  1182
  shows "(f has_contour_integral i) (linepath a b) \<longleftrightarrow>
lp15@60809
  1183
         ((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
lp15@61738
  1184
  by (simp add: has_contour_integral vector_derivative_linepath_at)
lp15@60809
  1185
lp15@60809
  1186
lemma linepath_in_path:
lp15@60809
  1187
  shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b"
lp15@60809
  1188
  by (auto simp: segment linepath_def)
lp15@60809
  1189
lp15@60809
  1190
lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
lp15@60809
  1191
  by (auto simp: segment linepath_def)
lp15@60809
  1192
lp15@60809
  1193
lemma linepath_in_convex_hull:
lp15@60809
  1194
    fixes x::real
lp15@60809
  1195
    assumes a: "a \<in> convex hull s"
lp15@60809
  1196
        and b: "b \<in> convex hull s"
lp15@60809
  1197
        and x: "0\<le>x" "x\<le>1"
lp15@60809
  1198
       shows "linepath a b x \<in> convex hull s"
lp15@60809
  1199
  apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD])
lp15@60809
  1200
  using x
lp15@60809
  1201
  apply (auto simp: linepath_image_01 [symmetric])
lp15@60809
  1202
  done
lp15@60809
  1203
lp15@60809
  1204
lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
lp15@60809
  1205
  by (simp add: linepath_def)
lp15@60809
  1206
lp15@60809
  1207
lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
lp15@60809
  1208
  by (simp add: linepath_def)
lp15@60809
  1209
lp15@61738
  1210
lemma has_contour_integral_trivial [iff]: "(f has_contour_integral 0) (linepath a a)"
lp15@61738
  1211
  by (simp add: has_contour_integral_linepath)
lp15@61738
  1212
lp15@61738
  1213
lemma contour_integral_trivial [simp]: "contour_integral (linepath a a) f = 0"
lp15@61738
  1214
  using has_contour_integral_trivial contour_integral_unique by blast
lp15@60809
  1215
lp15@60809
  1216
lp15@60809
  1217
subsection\<open>Relation to subpath construction\<close>
lp15@60809
  1218
lp15@60809
  1219
lemma valid_path_subpath:
lp15@60809
  1220
  fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
lp15@60809
  1221
  assumes "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
lp15@60809
  1222
    shows "valid_path(subpath u v g)"
lp15@60809
  1223
proof (cases "v=u")
lp15@60809
  1224
  case True
lp15@60809
  1225
  then show ?thesis
lp15@61190
  1226
    unfolding valid_path_def subpath_def
lp15@61190
  1227
    by (force intro: C1_differentiable_on_const C1_differentiable_imp_piecewise)
lp15@60809
  1228
next
lp15@60809
  1229
  case False
lp15@61190
  1230
  have "(g o (\<lambda>x. ((v-u) * x + u))) piecewise_C1_differentiable_on {0..1}"
lp15@61190
  1231
    apply (rule piecewise_C1_differentiable_compose)
lp15@61190
  1232
    apply (simp add: C1_differentiable_imp_piecewise)
lp15@60809
  1233
     apply (simp add: image_affinity_atLeastAtMost)
lp15@60809
  1234
    using assms False
lp15@61190
  1235
    apply (auto simp: algebra_simps valid_path_def piecewise_C1_differentiable_on_subset)
lp15@60809
  1236
    apply (subst Int_commute)
lp15@60809
  1237
    apply (auto simp: inj_on_def algebra_simps crossproduct_eq finite_vimage_IntI)
lp15@60809
  1238
    done
lp15@60809
  1239
  then show ?thesis
lp15@60809
  1240
    by (auto simp: o_def valid_path_def subpath_def)
lp15@60809
  1241
qed
lp15@60809
  1242
lp15@61738
  1243
lemma has_contour_integral_subpath_refl [iff]: "(f has_contour_integral 0) (subpath u u g)"
lp15@61738
  1244
  by (simp add: has_contour_integral subpath_def)
lp15@61738
  1245
lp15@61738
  1246
lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)"
lp15@61738
  1247
  using has_contour_integral_subpath_refl contour_integrable_on_def by blast
lp15@61738
  1248
lp15@61738
  1249
lemma contour_integral_subpath_refl [simp]: "contour_integral (subpath u u g) f = 0"
lp15@61738
  1250
  by (simp add: has_contour_integral_subpath_refl contour_integral_unique)
lp15@61738
  1251
lp15@61738
  1252
lemma has_contour_integral_subpath:
lp15@61738
  1253
  assumes f: "f contour_integrable_on g" and g: "valid_path g"
lp15@60809
  1254
      and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
lp15@61738
  1255
    shows "(f has_contour_integral  integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x)))
lp15@60809
  1256
           (subpath u v g)"
lp15@60809
  1257
proof (cases "v=u")
lp15@60809
  1258
  case True
lp15@60809
  1259
  then show ?thesis
lp15@61738
  1260
    using f   by (simp add: contour_integrable_on_def subpath_def has_contour_integral)
lp15@60809
  1261
next
lp15@60809
  1262
  case False
lp15@60809
  1263
  obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s"
lp15@61190
  1264
    using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
lp15@60809
  1265
  have *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
lp15@60809
  1266
            has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t)))
lp15@60809
  1267
           {0..1}"
lp15@60809
  1268
    using f uv
lp15@61738
  1269
    apply (simp add: contour_integrable_on subpath_def has_contour_integral)
lp15@60809
  1270
    apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
lp15@60809
  1271
    apply (simp_all add: has_integral_integral)
lp15@60809
  1272
    apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
lp15@60809
  1273
    apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
lp15@60809
  1274
    apply (simp add: divide_simps False)
lp15@60809
  1275
    done
lp15@60809
  1276
  { fix x
lp15@60809
  1277
    have "x \<in> {0..1} \<Longrightarrow>
lp15@60809
  1278
           x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
lp15@60809
  1279
           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
  1280
      apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
lp15@60809
  1281
      apply (intro derivative_eq_intros | simp)+
lp15@60809
  1282
      apply (cut_tac s [of "(v - u) * x + u"])
lp15@60809
  1283
      using uv mult_left_le [of x "v-u"]
lp15@60809
  1284
      apply (auto simp:  vector_derivative_works)
lp15@60809
  1285
      done
lp15@60809
  1286
  } note vd = this
lp15@60809
  1287
  show ?thesis
lp15@60809
  1288
    apply (cut_tac has_integral_cmul [OF *, where c = "v-u"])
lp15@60809
  1289
    using fs assms
lp15@61738
  1290
    apply (simp add: False subpath_def has_contour_integral)
lp15@60809
  1291
    apply (rule_tac s = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
lp15@60809
  1292
    apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real)
lp15@60809
  1293
    done
lp15@60809
  1294
qed
lp15@60809
  1295
lp15@61738
  1296
lemma contour_integrable_subpath:
lp15@61738
  1297
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
lp15@61738
  1298
    shows "f contour_integrable_on (subpath u v g)"
lp15@60809
  1299
  apply (cases u v rule: linorder_class.le_cases)
lp15@61738
  1300
   apply (metis contour_integrable_on_def has_contour_integral_subpath [OF assms])
lp15@60809
  1301
  apply (subst reversepath_subpath [symmetric])
lp15@61738
  1302
  apply (rule contour_integrable_reversepath)
lp15@60809
  1303
   using assms apply (blast intro: valid_path_subpath)
lp15@61738
  1304
  apply (simp add: contour_integrable_on_def)
lp15@61738
  1305
  using assms apply (blast intro: has_contour_integral_subpath)
lp15@60809
  1306
  done
lp15@60809
  1307
lp15@60809
  1308
lemma has_integral_integrable_integral: "(f has_integral i) s \<longleftrightarrow> f integrable_on s \<and> integral s f = i"
lp15@60809
  1309
  by blast
lp15@60809
  1310
lp15@61738
  1311
lemma has_integral_contour_integral_subpath:
lp15@61738
  1312
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
lp15@60809
  1313
    shows "(((\<lambda>x. f(g x) * vector_derivative g (at x)))
lp15@61738
  1314
            has_integral  contour_integral (subpath u v g) f) {u..v}"
lp15@60809
  1315
  using assms
lp15@60809
  1316
  apply (auto simp: has_integral_integrable_integral)
lp15@60809
  1317
  apply (rule integrable_on_subcbox [where a=u and b=v and s = "{0..1}", simplified])
lp15@61738
  1318
  apply (auto simp: contour_integral_unique [OF has_contour_integral_subpath] contour_integrable_on)
lp15@60809
  1319
  done
lp15@60809
  1320
lp15@61738
  1321
lemma contour_integral_subcontour_integral:
lp15@61738
  1322
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
lp15@61738
  1323
    shows "contour_integral (subpath u v g) f =
lp15@60809
  1324
           integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))"
lp15@61738
  1325
  using assms has_contour_integral_subpath contour_integral_unique by blast
lp15@61738
  1326
lp15@61738
  1327
lemma contour_integral_subpath_combine_less:
lp15@61738
  1328
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
lp15@60809
  1329
          "u<v" "v<w"
lp15@61738
  1330
    shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
lp15@61738
  1331
           contour_integral (subpath u w g) f"
lp15@61738
  1332
  using assms apply (auto simp: contour_integral_subcontour_integral)
lp15@60809
  1333
  apply (rule integral_combine, auto)
lp15@60809
  1334
  apply (rule integrable_on_subcbox [where a=u and b=w and s = "{0..1}", simplified])
lp15@61738
  1335
  apply (auto simp: contour_integrable_on)
lp15@60809
  1336
  done
lp15@60809
  1337
lp15@61738
  1338
lemma contour_integral_subpath_combine:
lp15@61738
  1339
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
lp15@61738
  1340
    shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
lp15@61738
  1341
           contour_integral (subpath u w g) f"
lp15@60809
  1342
proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w")
lp15@60809
  1343
  case True
lp15@60809
  1344
    have *: "subpath v u g = reversepath(subpath u v g) \<and>
lp15@60809
  1345
             subpath w u g = reversepath(subpath u w g) \<and>
lp15@60809
  1346
             subpath w v g = reversepath(subpath v w g)"
lp15@60809
  1347
      by (auto simp: reversepath_subpath)
lp15@60809
  1348
    have "u < v \<and> v < w \<or>
lp15@60809
  1349
          u < w \<and> w < v \<or>
lp15@60809
  1350
          v < u \<and> u < w \<or>
lp15@60809
  1351
          v < w \<and> w < u \<or>
lp15@60809
  1352
          w < u \<and> u < v \<or>
lp15@60809
  1353
          w < v \<and> v < u"
lp15@60809
  1354
      using True assms by linarith
lp15@60809
  1355
    with assms show ?thesis
lp15@61738
  1356
      using contour_integral_subpath_combine_less [of f g u v w]
lp15@61738
  1357
            contour_integral_subpath_combine_less [of f g u w v]
lp15@61738
  1358
            contour_integral_subpath_combine_less [of f g v u w]
lp15@61738
  1359
            contour_integral_subpath_combine_less [of f g v w u]
lp15@61738
  1360
            contour_integral_subpath_combine_less [of f g w u v]
lp15@61738
  1361
            contour_integral_subpath_combine_less [of f g w v u]
lp15@60809
  1362
      apply simp
lp15@60809
  1363
      apply (elim disjE)
lp15@61738
  1364
      apply (auto simp: * contour_integral_reversepath contour_integrable_subpath
lp15@60809
  1365
                   valid_path_reversepath valid_path_subpath algebra_simps)
lp15@60809
  1366
      done
lp15@60809
  1367
next
lp15@60809
  1368
  case False
lp15@60809
  1369
  then show ?thesis
lp15@61738
  1370
    apply (auto simp: contour_integral_subpath_refl)
lp15@60809
  1371
    using assms
lp15@61738
  1372
    by (metis eq_neg_iff_add_eq_0 contour_integrable_subpath contour_integral_reversepath reversepath_subpath valid_path_subpath)
lp15@60809
  1373
qed
lp15@60809
  1374
lp15@61738
  1375
lemma contour_integral_integral:
lp15@62463
  1376
     "contour_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
lp15@62463
  1377
  by (simp add: contour_integral_def integral_def has_contour_integral contour_integrable_on)
lp15@60809
  1378
lp15@60809
  1379
lp15@60809
  1380
text\<open>Cauchy's theorem where there's a primitive\<close>
lp15@60809
  1381
lp15@61738
  1382
lemma contour_integral_primitive_lemma:
lp15@60809
  1383
  fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
lp15@60809
  1384
  assumes "a \<le> b"
lp15@60809
  1385
      and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
lp15@60809
  1386
      and "g piecewise_differentiable_on {a..b}"  "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
lp15@60809
  1387
    shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b}))
lp15@60809
  1388
             has_integral (f(g b) - f(g a))) {a..b}"
lp15@60809
  1389
proof -
lp15@61190
  1390
  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
  1391
    using assms by (auto simp: piecewise_differentiable_on_def)
lp15@60809
  1392
  have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
lp15@60809
  1393
    apply (rule continuous_on_compose [OF cg, unfolded o_def])
lp15@60809
  1394
    using assms
lp15@62534
  1395
    apply (metis field_differentiable_def field_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
lp15@60809
  1396
    done
lp15@60809
  1397
  { fix x::real
lp15@60809
  1398
    assume a: "a < x" and b: "x < b" and xk: "x \<notin> k"
lp15@60809
  1399
    then have "g differentiable at x within {a..b}"
lp15@60809
  1400
      using k by (simp add: differentiable_at_withinI)
lp15@60809
  1401
    then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
lp15@60809
  1402
      by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
lp15@60809
  1403
    then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
lp15@60809
  1404
      by (simp add: has_vector_derivative_def scaleR_conv_of_real)
lp15@60809
  1405
    have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
lp15@60809
  1406
      using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
lp15@60809
  1407
    then have fdiff: "(f has_derivative op * (f' (g x))) (at (g x) within g ` {a..b})"
lp15@60809
  1408
      by (simp add: has_field_derivative_def)
lp15@60809
  1409
    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
  1410
      using diff_chain_within [OF gdiff fdiff]
lp15@60809
  1411
      by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
lp15@60809
  1412
  } note * = this
lp15@60809
  1413
  show ?thesis
lp15@60809
  1414
    apply (rule fundamental_theorem_of_calculus_interior_strong)
lp15@60809
  1415
    using k assms cfg *
lp15@60809
  1416
    apply (auto simp: at_within_closed_interval)
lp15@60809
  1417
    done
lp15@60809
  1418
qed
lp15@60809
  1419
lp15@61738
  1420
lemma contour_integral_primitive:
lp15@60809
  1421
  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
lp15@60809
  1422
      and "valid_path g" "path_image g \<subseteq> s"
lp15@61738
  1423
    shows "(f' has_contour_integral (f(pathfinish g) - f(pathstart g))) g"
lp15@60809
  1424
  using assms
lp15@61738
  1425
  apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_contour_integral_def)
lp15@61738
  1426
  apply (auto intro!: piecewise_C1_imp_differentiable contour_integral_primitive_lemma [of 0 1 s])
lp15@60809
  1427
  done
lp15@60809
  1428
lp15@60809
  1429
corollary Cauchy_theorem_primitive:
lp15@60809
  1430
  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
lp15@60809
  1431
      and "valid_path g"  "path_image g \<subseteq> s" "pathfinish g = pathstart g"
lp15@61738
  1432
    shows "(f' has_contour_integral 0) g"
lp15@60809
  1433
  using assms
lp15@61738
  1434
  by (metis diff_self contour_integral_primitive)
lp15@60809
  1435
lp15@60809
  1436
lp15@60809
  1437
text\<open>Existence of path integral for continuous function\<close>
lp15@61738
  1438
lemma contour_integrable_continuous_linepath:
lp15@60809
  1439
  assumes "continuous_on (closed_segment a b) f"
lp15@61738
  1440
  shows "f contour_integrable_on (linepath a b)"
lp15@60809
  1441
proof -
lp15@60809
  1442
  have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) o linepath a b)"
lp15@60809
  1443
    apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01)
lp15@60809
  1444
    apply (rule continuous_intros | simp add: assms)+
lp15@60809
  1445
    done
lp15@60809
  1446
  then show ?thesis
lp15@61738
  1447
    apply (simp add: contour_integrable_on_def has_contour_integral_def integrable_on_def [symmetric])
lp15@60809
  1448
    apply (rule integrable_continuous [of 0 "1::real", simplified])
lp15@60809
  1449
    apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"])
lp15@60809
  1450
    apply (auto simp: vector_derivative_linepath_within)
lp15@60809
  1451
    done
lp15@60809
  1452
qed
lp15@60809
  1453
lp15@60809
  1454
lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)"
lp15@60809
  1455
  by (rule has_derivative_imp_has_field_derivative)
lp15@60809
  1456
     (rule derivative_intros | simp)+
lp15@60809
  1457
lp15@61738
  1458
lemma contour_integral_id [simp]: "contour_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2"
lp15@61738
  1459
  apply (rule contour_integral_unique)
lp15@61738
  1460
  using contour_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"]
lp15@60809
  1461
  apply (auto simp: field_simps has_field_der_id)
lp15@60809
  1462
  done
lp15@60809
  1463
lp15@61738
  1464
lemma contour_integrable_on_const [iff]: "(\<lambda>x. c) contour_integrable_on (linepath a b)"
lp15@61738
  1465
  by (simp add: continuous_on_const contour_integrable_continuous_linepath)
lp15@61738
  1466
lp15@61738
  1467
lemma contour_integrable_on_id [iff]: "(\<lambda>x. x) contour_integrable_on (linepath a b)"
lp15@61738
  1468
  by (simp add: continuous_on_id contour_integrable_continuous_linepath)
lp15@60809
  1469
lp15@60809
  1470
lp15@60809
  1471
subsection\<open>Arithmetical combining theorems\<close>
lp15@60809
  1472
lp15@61738
  1473
lemma has_contour_integral_neg:
lp15@61738
  1474
    "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_contour_integral (-i)) g"
lp15@61738
  1475
  by (simp add: has_integral_neg has_contour_integral_def)
lp15@61738
  1476
lp15@61738
  1477
lemma has_contour_integral_add:
lp15@61738
  1478
    "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
lp15@61738
  1479
     \<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_contour_integral (i1 + i2)) g"
lp15@61738
  1480
  by (simp add: has_integral_add has_contour_integral_def algebra_simps)
lp15@61738
  1481
lp15@61738
  1482
lemma has_contour_integral_diff:
lp15@61738
  1483
  "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
lp15@61738
  1484
         \<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_contour_integral (i1 - i2)) g"
lp15@61738
  1485
  by (simp add: has_integral_sub has_contour_integral_def algebra_simps)
lp15@61738
  1486
lp15@61738
  1487
lemma has_contour_integral_lmul:
lp15@61738
  1488
  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_contour_integral (c*i)) g"
lp15@61738
  1489
apply (simp add: has_contour_integral_def)
lp15@60809
  1490
apply (drule has_integral_mult_right)
lp15@60809
  1491
apply (simp add: algebra_simps)
lp15@60809
  1492
done
lp15@60809
  1493
lp15@61738
  1494
lemma has_contour_integral_rmul:
lp15@61738
  1495
  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_contour_integral (i*c)) g"
lp15@61738
  1496
apply (drule has_contour_integral_lmul)
lp15@60809
  1497
apply (simp add: mult.commute)
lp15@60809
  1498
done
lp15@60809
  1499
lp15@61738
  1500
lemma has_contour_integral_div:
lp15@61738
  1501
  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_contour_integral (i/c)) g"
lp15@61738
  1502
  by (simp add: field_class.field_divide_inverse) (metis has_contour_integral_rmul)
lp15@61738
  1503
lp15@61738
  1504
lemma has_contour_integral_eq:
lp15@61738
  1505
    "\<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
  1506
apply (simp add: path_image_def has_contour_integral_def)
lp15@60809
  1507
by (metis (no_types, lifting) image_eqI has_integral_eq)
lp15@60809
  1508
lp15@61738
  1509
lemma has_contour_integral_bound_linepath:
lp15@61738
  1510
  assumes "(f has_contour_integral i) (linepath a b)"
lp15@60809
  1511
          "0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
lp15@60809
  1512
    shows "norm i \<le> B * norm(b - a)"
lp15@60809
  1513
proof -
lp15@60809
  1514
  { fix x::real
lp15@60809
  1515
    assume x: "0 \<le> x" "x \<le> 1"
lp15@60809
  1516
  have "norm (f (linepath a b x)) *
lp15@60809
  1517
        norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)"
lp15@60809
  1518
    by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x)
lp15@60809
  1519
  } note * = this
lp15@60809
  1520
  have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
lp15@60809
  1521
    apply (rule has_integral_bound
lp15@60809
  1522
       [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
lp15@61738
  1523
    using assms * unfolding has_contour_integral_def
lp15@60809
  1524
    apply (auto simp: norm_mult)
lp15@60809
  1525
    done
lp15@60809
  1526
  then show ?thesis
lp15@60809
  1527
    by (auto simp: content_real)
lp15@60809
  1528
qed
lp15@60809
  1529
lp15@60809
  1530
(*UNUSED
lp15@61738
  1531
lemma has_contour_integral_bound_linepath_strong:
lp15@60809
  1532
  fixes a :: real and f :: "complex \<Rightarrow> real"
lp15@61738
  1533
  assumes "(f has_contour_integral i) (linepath a b)"
lp15@60809
  1534
          "finite k"
lp15@60809
  1535
          "0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B"
lp15@60809
  1536
    shows "norm i \<le> B*norm(b - a)"
lp15@60809
  1537
*)
lp15@60809
  1538
lp15@61738
  1539
lemma has_contour_integral_const_linepath: "((\<lambda>x. c) has_contour_integral c*(b - a))(linepath a b)"
lp15@61738
  1540
  unfolding has_contour_integral_linepath
lp15@60809
  1541
  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
  1542
lp15@61738
  1543
lemma has_contour_integral_0: "((\<lambda>x. 0) has_contour_integral 0) g"
lp15@61738
  1544
  by (simp add: has_contour_integral_def)
lp15@61738
  1545
lp15@61738
  1546
lemma has_contour_integral_is_0:
lp15@61738
  1547
    "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_contour_integral 0) g"
lp15@61738
  1548
  by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto
lp15@61738
  1549
lp15@61738
  1550
lemma has_contour_integral_setsum:
lp15@61738
  1551
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p\<rbrakk>
lp15@61738
  1552
     \<Longrightarrow> ((\<lambda>x. setsum (\<lambda>a. f a x) s) has_contour_integral setsum i s) p"
lp15@61738
  1553
  by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add)
lp15@60809
  1554
lp15@60809
  1555
lp15@60809
  1556
subsection \<open>Operations on path integrals\<close>
lp15@60809
  1557
lp15@61738
  1558
lemma contour_integral_const_linepath [simp]: "contour_integral (linepath a b) (\<lambda>x. c) = c*(b - a)"
lp15@61738
  1559
  by (rule contour_integral_unique [OF has_contour_integral_const_linepath])
lp15@61738
  1560
lp15@61738
  1561
lemma contour_integral_neg:
lp15@61738
  1562
    "f contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. -(f x)) = -(contour_integral g f)"
lp15@61738
  1563
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_neg)
lp15@61738
  1564
lp15@61738
  1565
lemma contour_integral_add:
lp15@61738
  1566
    "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x + f2 x) =
lp15@61738
  1567
                contour_integral g f1 + contour_integral g f2"
lp15@61738
  1568
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_add)
lp15@61738
  1569
lp15@61738
  1570
lemma contour_integral_diff:
lp15@61738
  1571
    "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x - f2 x) =
lp15@61738
  1572
                contour_integral g f1 - contour_integral g f2"
lp15@61738
  1573
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_diff)
lp15@61738
  1574
lp15@61738
  1575
lemma contour_integral_lmul:
lp15@61738
  1576
  shows "f contour_integrable_on g
lp15@61738
  1577
           \<Longrightarrow> contour_integral g (\<lambda>x. c * f x) = c*contour_integral g f"
lp15@61738
  1578
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_lmul)
lp15@61738
  1579
lp15@61738
  1580
lemma contour_integral_rmul:
lp15@61738
  1581
  shows "f contour_integrable_on g
lp15@61738
  1582
        \<Longrightarrow> contour_integral g (\<lambda>x. f x * c) = contour_integral g f * c"
lp15@61738
  1583
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_rmul)
lp15@61738
  1584
lp15@61738
  1585
lemma contour_integral_div:
lp15@61738
  1586
  shows "f contour_integrable_on g
lp15@61738
  1587
        \<Longrightarrow> contour_integral g (\<lambda>x. f x / c) = contour_integral g f / c"
lp15@61738
  1588
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_div)
lp15@61738
  1589
lp15@61738
  1590
lemma contour_integral_eq:
lp15@61738
  1591
    "(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> contour_integral p f = contour_integral p g"
lp15@62463
  1592
  apply (simp add: contour_integral_def)
lp15@62463
  1593
  using has_contour_integral_eq
lp15@62463
  1594
  by (metis contour_integral_unique has_contour_integral_integrable has_contour_integral_integral)
lp15@61738
  1595
lp15@61738
  1596
lemma contour_integral_eq_0:
lp15@61738
  1597
    "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> contour_integral g f = 0"
lp15@61738
  1598
  by (simp add: has_contour_integral_is_0 contour_integral_unique)
lp15@61738
  1599
lp15@61738
  1600
lemma contour_integral_bound_linepath:
lp15@60809
  1601
  shows
lp15@61738
  1602
    "\<lbrakk>f contour_integrable_on (linepath a b);
lp15@60809
  1603
      0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
lp15@61738
  1604
     \<Longrightarrow> norm(contour_integral (linepath a b) f) \<le> B*norm(b - a)"
lp15@61738
  1605
  apply (rule has_contour_integral_bound_linepath [of f])
lp15@61738
  1606
  apply (auto simp: has_contour_integral_integral)
lp15@60809
  1607
  done
lp15@60809
  1608
lp15@61806
  1609
lemma contour_integral_0 [simp]: "contour_integral g (\<lambda>x. 0) = 0"
lp15@61738
  1610
  by (simp add: contour_integral_unique has_contour_integral_0)
lp15@61738
  1611
lp15@61738
  1612
lemma contour_integral_setsum:
lp15@61738
  1613
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
lp15@61738
  1614
     \<Longrightarrow> contour_integral p (\<lambda>x. setsum (\<lambda>a. f a x) s) = setsum (\<lambda>a. contour_integral p (f a)) s"
lp15@61738
  1615
  by (auto simp: contour_integral_unique has_contour_integral_setsum has_contour_integral_integral)
lp15@61738
  1616
lp15@61738
  1617
lemma contour_integrable_eq:
lp15@61738
  1618
    "\<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
  1619
  unfolding contour_integrable_on_def
lp15@61738
  1620
  by (metis has_contour_integral_eq)
lp15@60809
  1621
lp15@60809
  1622
lp15@60809
  1623
subsection \<open>Arithmetic theorems for path integrability\<close>
lp15@60809
  1624
lp15@61738
  1625
lemma contour_integrable_neg:
lp15@61738
  1626
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) contour_integrable_on g"
lp15@61738
  1627
  using has_contour_integral_neg contour_integrable_on_def by blast
lp15@61738
  1628
lp15@61738
  1629
lemma contour_integrable_add:
lp15@61738
  1630
    "\<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
  1631
  using has_contour_integral_add contour_integrable_on_def
lp15@60809
  1632
  by fastforce
lp15@60809
  1633
lp15@61738
  1634
lemma contour_integrable_diff:
lp15@61738
  1635
    "\<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
  1636
  using has_contour_integral_diff contour_integrable_on_def
lp15@60809
  1637
  by fastforce
lp15@60809
  1638
lp15@61738
  1639
lemma contour_integrable_lmul:
lp15@61738
  1640
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) contour_integrable_on g"
lp15@61738
  1641
  using has_contour_integral_lmul contour_integrable_on_def
lp15@60809
  1642
  by fastforce
lp15@60809
  1643
lp15@61738
  1644
lemma contour_integrable_rmul:
lp15@61738
  1645
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) contour_integrable_on g"
lp15@61738
  1646
  using has_contour_integral_rmul contour_integrable_on_def
lp15@60809
  1647
  by fastforce
lp15@60809
  1648
lp15@61738
  1649
lemma contour_integrable_div:
lp15@61738
  1650
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) contour_integrable_on g"
lp15@61738
  1651
  using has_contour_integral_div contour_integrable_on_def
lp15@60809
  1652
  by fastforce
lp15@60809
  1653
lp15@61738
  1654
lemma contour_integrable_setsum:
lp15@61738
  1655
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
lp15@61738
  1656
     \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) s) contour_integrable_on p"
lp15@61738
  1657
   unfolding contour_integrable_on_def
lp15@61738
  1658
   by (metis has_contour_integral_setsum)
lp15@60809
  1659
lp15@60809
  1660
lp15@60809
  1661
subsection\<open>Reversing a path integral\<close>
lp15@60809
  1662
lp15@61738
  1663
lemma has_contour_integral_reverse_linepath:
lp15@61738
  1664
    "(f has_contour_integral i) (linepath a b)
lp15@61738
  1665
     \<Longrightarrow> (f has_contour_integral (-i)) (linepath b a)"
lp15@61738
  1666
  using has_contour_integral_reversepath valid_path_linepath by fastforce
lp15@61738
  1667
lp15@61738
  1668
lemma contour_integral_reverse_linepath:
lp15@60809
  1669
    "continuous_on (closed_segment a b) f
lp15@61738
  1670
     \<Longrightarrow> contour_integral (linepath a b) f = - (contour_integral(linepath b a) f)"
lp15@61738
  1671
apply (rule contour_integral_unique)
lp15@61738
  1672
apply (rule has_contour_integral_reverse_linepath)
lp15@61738
  1673
by (simp add: closed_segment_commute contour_integrable_continuous_linepath has_contour_integral_integral)
lp15@60809
  1674
lp15@60809
  1675
lp15@60809
  1676
(* Splitting a path integral in a flat way.*)
lp15@60809
  1677
lp15@61738
  1678
lemma has_contour_integral_split:
lp15@61738
  1679
  assumes f: "(f has_contour_integral i) (linepath a c)" "(f has_contour_integral j) (linepath c b)"
lp15@60809
  1680
      and k: "0 \<le> k" "k \<le> 1"
lp15@60809
  1681
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@61738
  1682
    shows "(f has_contour_integral (i + j)) (linepath a b)"
lp15@60809
  1683
proof (cases "k = 0 \<or> k = 1")
lp15@60809
  1684
  case True
lp15@60809
  1685
  then show ?thesis
lp15@60809
  1686
    using assms
lp15@60809
  1687
    apply auto
lp15@61738
  1688
    apply (metis add.left_neutral has_contour_integral_trivial has_contour_integral_unique)
lp15@61738
  1689
    apply (metis add.right_neutral has_contour_integral_trivial has_contour_integral_unique)
lp15@60809
  1690
    done
lp15@60809
  1691
next
lp15@60809
  1692
  case False
lp15@60809
  1693
  then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1"
lp15@60809
  1694
    using assms apply auto
lp15@60809
  1695
    using of_real_eq_iff by fastforce
lp15@60809
  1696
  have c': "c = k *\<^sub>R (b - a) + a"
lp15@60809
  1697
    by (metis diff_add_cancel c)
lp15@60809
  1698
  have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)"
lp15@60809
  1699
    by (simp add: algebra_simps c')
lp15@60809
  1700
  { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
lp15@60809
  1701
    have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b"
lp15@60809
  1702
      using False
lp15@60809
  1703
      apply (simp add: c' algebra_simps)
lp15@60809
  1704
      apply (simp add: real_vector.scale_left_distrib [symmetric] divide_simps)
lp15@60809
  1705
      done
lp15@60809
  1706
    have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
lp15@60809
  1707
      using * k
lp15@60809
  1708
      apply -
lp15@60809
  1709
      apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse k" "0", simplified])
lp15@60809
  1710
      apply (simp_all add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c)
lp15@60809
  1711
      apply (drule Integration.has_integral_cmul [where c = "inverse k"])
lp15@60809
  1712
      apply (simp add: Integration.has_integral_cmul)
lp15@60809
  1713
      done
lp15@60809
  1714
  } note fi = this
lp15@60809
  1715
  { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}"
lp15@60809
  1716
    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
  1717
      using k
lp15@60809
  1718
      apply (simp add: c' field_simps)
lp15@60809
  1719
      apply (simp add: scaleR_conv_of_real divide_simps)
lp15@60809
  1720
      apply (simp add: field_simps)
lp15@60809
  1721
      done
lp15@60809
  1722
    have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
lp15@60809
  1723
      using * k
lp15@60809
  1724
      apply -
lp15@60809
  1725
      apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse(1 - k)" "-(k/(1 - k))", simplified])
lp15@60809
  1726
      apply (simp_all add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
lp15@60809
  1727
      apply (drule Integration.has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
lp15@60809
  1728
      apply (simp add: Integration.has_integral_cmul)
lp15@60809
  1729
      done
lp15@60809
  1730
  } note fj = this
lp15@60809
  1731
  show ?thesis
lp15@60809
  1732
    using f k
lp15@61738
  1733
    apply (simp add: has_contour_integral_linepath)
lp15@60809
  1734
    apply (simp add: linepath_def)
lp15@60809
  1735
    apply (rule has_integral_combine [OF _ _ fi fj], simp_all)
lp15@60809
  1736
    done
lp15@60809
  1737
qed
lp15@60809
  1738
lp15@60809
  1739
lemma continuous_on_closed_segment_transform:
lp15@60809
  1740
  assumes f: "continuous_on (closed_segment a b) f"
lp15@60809
  1741
      and k: "0 \<le> k" "k \<le> 1"
lp15@60809
  1742
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@60809
  1743
    shows "continuous_on (closed_segment a c) f"
lp15@60809
  1744
proof -
lp15@60809
  1745
  have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
lp15@60809
  1746
    using c by (simp add: algebra_simps)
lp15@60809
  1747
  show "continuous_on (closed_segment a c) f"
lp15@60809
  1748
    apply (rule continuous_on_subset [OF f])
lp15@60809
  1749
    apply (simp add: segment_convex_hull)
lp15@60809
  1750
    apply (rule convex_hull_subset)
lp15@60809
  1751
    using assms
lp15@61426
  1752
    apply (auto simp: hull_inc c' Convex.convexD_alt)
lp15@60809
  1753
    done
lp15@60809
  1754
qed
lp15@60809
  1755
lp15@61738
  1756
lemma contour_integral_split:
lp15@60809
  1757
  assumes f: "continuous_on (closed_segment a b) f"
lp15@60809
  1758
      and k: "0 \<le> k" "k \<le> 1"
lp15@60809
  1759
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@61738
  1760
    shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
lp15@60809
  1761
proof -
lp15@60809
  1762
  have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
lp15@60809
  1763
    using c by (simp add: algebra_simps)
lp15@60809
  1764
  have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
lp15@60809
  1765
    apply (rule_tac [!] continuous_on_subset [OF f])
lp15@60809
  1766
    apply (simp_all add: segment_convex_hull)
lp15@60809
  1767
    apply (rule_tac [!] convex_hull_subset)
lp15@60809
  1768
    using assms
lp15@61426
  1769
    apply (auto simp: hull_inc c' Convex.convexD_alt)
lp15@60809
  1770
    done
lp15@60809
  1771
  show ?thesis
lp15@61738
  1772
    apply (rule contour_integral_unique)
lp15@61738
  1773
    apply (rule has_contour_integral_split [OF has_contour_integral_integral has_contour_integral_integral k c])
lp15@61738
  1774
    apply (rule contour_integrable_continuous_linepath *)+
lp15@60809
  1775
    done
lp15@60809
  1776
qed
lp15@60809
  1777
lp15@61738
  1778
lemma contour_integral_split_linepath:
lp15@60809
  1779
  assumes f: "continuous_on (closed_segment a b) f"
lp15@60809
  1780
      and c: "c \<in> closed_segment a b"
lp15@61738
  1781
    shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
lp15@60809
  1782
  using c
lp15@61738
  1783
  by (auto simp: closed_segment_def algebra_simps intro!: contour_integral_split [OF f])
lp15@60809
  1784
lp15@60809
  1785
(* The special case of midpoints used in the main quadrisection.*)
lp15@60809
  1786
lp15@61738
  1787
lemma has_contour_integral_midpoint:
lp15@61738
  1788
  assumes "(f has_contour_integral i) (linepath a (midpoint a b))"
lp15@61738
  1789
          "(f has_contour_integral j) (linepath (midpoint a b) b)"
lp15@61738
  1790
    shows "(f has_contour_integral (i + j)) (linepath a b)"
lp15@61738
  1791
  apply (rule has_contour_integral_split [where c = "midpoint a b" and k = "1/2"])
lp15@60809
  1792
  using assms
lp15@60809
  1793
  apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
lp15@60809
  1794
  done
lp15@60809
  1795
lp15@61738
  1796
lemma contour_integral_midpoint:
lp15@60809
  1797
   "continuous_on (closed_segment a b) f
lp15@61738
  1798
    \<Longrightarrow> contour_integral (linepath a b) f =
lp15@61738
  1799
        contour_integral (linepath a (midpoint a b)) f + contour_integral (linepath (midpoint a b) b) f"
lp15@61738
  1800
  apply (rule contour_integral_split [where c = "midpoint a b" and k = "1/2"])
lp15@60809
  1801
  using assms
lp15@60809
  1802
  apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
lp15@60809
  1803
  done
lp15@60809
  1804
lp15@60809
  1805
lp15@60809
  1806
text\<open>A couple of special case lemmas that are useful below\<close>
lp15@60809
  1807
lp15@60809
  1808
lemma triangle_linear_has_chain_integral:
lp15@61738
  1809
    "((\<lambda>x. m*x + d) has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  1810
  apply (rule Cauchy_theorem_primitive [of UNIV "\<lambda>x. m/2 * x^2 + d*x"])
lp15@60809
  1811
  apply (auto intro!: derivative_eq_intros)
lp15@60809
  1812
  done
lp15@60809
  1813
lp15@60809
  1814
lemma has_chain_integral_chain_integral3:
lp15@61738
  1815
     "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d)
lp15@61738
  1816
      \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f = i"
lp15@61738
  1817
  apply (subst contour_integral_unique [symmetric], assumption)
lp15@61738
  1818
  apply (drule has_contour_integral_integrable)
lp15@60809
  1819
  apply (simp add: valid_path_join)
lp15@60809
  1820
  done
lp15@60809
  1821
lp15@62397
  1822
lemma has_chain_integral_chain_integral4:
lp15@62397
  1823
     "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d +++ linepath d e)
lp15@62397
  1824
      \<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
  1825
  apply (subst contour_integral_unique [symmetric], assumption)
lp15@62397
  1826
  apply (drule has_contour_integral_integrable)
lp15@62397
  1827
  apply (simp add: valid_path_join)
lp15@62397
  1828
  done
lp15@62397
  1829
lp15@60809
  1830
subsection\<open>Reversing the order in a double path integral\<close>
lp15@60809
  1831
lp15@60809
  1832
text\<open>The condition is stronger than needed but it's often true in typical situations\<close>
lp15@60809
  1833
lp15@60809
  1834
lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b"
lp15@60809
  1835
  by (auto simp: cbox_Pair_eq)
lp15@60809
  1836
lp15@60809
  1837
lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d"
lp15@60809
  1838
  by (auto simp: cbox_Pair_eq)
lp15@60809
  1839
lp15@61738
  1840
lemma contour_integral_swap:
lp15@60809
  1841
  assumes fcon:  "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)"
lp15@60809
  1842
      and vp:    "valid_path g" "valid_path h"
lp15@60809
  1843
      and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
lp15@60809
  1844
      and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
lp15@61738
  1845
  shows "contour_integral g (\<lambda>w. contour_integral h (f w)) =
lp15@61738
  1846
         contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
lp15@60809
  1847
proof -
lp15@60809
  1848
  have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h"
lp15@61190
  1849
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
lp15@60809
  1850
  have fgh1: "\<And>x. (\<lambda>t. f (g x) (h t)) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>t. (g x, h t))"
lp15@60809
  1851
    by (rule ext) simp
lp15@60809
  1852
  have fgh2: "\<And>x. (\<lambda>t. f (g t) (h x)) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>t. (g t, h x))"
lp15@60809
  1853
    by (rule ext) simp
lp15@60809
  1854
  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
  1855
    by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
lp15@60809
  1856
  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
  1857
    by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
lp15@60809
  1858
  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@60809
  1859
    apply (rule integrable_continuous_real)
lp15@60809
  1860
    apply (rule continuous_on_mult [OF _ gvcon])
lp15@60809
  1861
    apply (subst fgh2)
lp15@60809
  1862
    apply (rule fcon_im2 gcon continuous_intros | simp)+
lp15@60809
  1863
    done
lp15@60809
  1864
  have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) o fst"
lp15@60809
  1865
    by auto
lp15@60809
  1866
  then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))"
lp15@60809
  1867
    apply (rule ssubst)
lp15@60809
  1868
    apply (rule continuous_intros | simp add: gvcon)+
lp15@60809
  1869
    done
lp15@60809
  1870
  have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) o snd"
lp15@60809
  1871
    by auto
lp15@60809
  1872
  then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
lp15@60809
  1873
    apply (rule ssubst)
lp15@60809
  1874
    apply (rule continuous_intros | simp add: hvcon)+
lp15@60809
  1875
    done
lp15@60809
  1876
  have "(\<lambda>x. f (g (fst x)) (h (snd x))) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>w. ((g o fst) w, (h o snd) w))"
lp15@60809
  1877
    by auto
lp15@60809
  1878
  then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
lp15@60809
  1879
    apply (rule ssubst)
lp15@60809
  1880
    apply (rule gcon hcon continuous_intros | simp)+
lp15@60809
  1881
    apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
lp15@60809
  1882
    done
lp15@61738
  1883
  have "integral {0..1} (\<lambda>x. contour_integral h (f (g x)) * vector_derivative g (at x)) =
lp15@61738
  1884
        integral {0..1} (\<lambda>x. contour_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))"
lp15@61738
  1885
    apply (rule integral_cong [OF contour_integral_rmul [symmetric]])
lp15@61738
  1886
    apply (clarsimp simp: contour_integrable_on)
lp15@60809
  1887
    apply (rule integrable_continuous_real)
lp15@60809
  1888
    apply (rule continuous_on_mult [OF _ hvcon])
lp15@60809
  1889
    apply (subst fgh1)
lp15@60809
  1890
    apply (rule fcon_im1 hcon continuous_intros | simp)+
lp15@60809
  1891
    done
lp15@60809
  1892
  also have "... = integral {0..1}
lp15@61738
  1893
                     (\<lambda>y. contour_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))"
lp15@62463
  1894
    apply (simp only: contour_integral_integral)
lp15@60809
  1895
    apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
lp15@62463
  1896
     apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
lp15@62463
  1897
    unfolding integral_mult_left [symmetric]
lp15@62463
  1898
    apply (simp only: mult_ac)
lp15@60809
  1899
    done
lp15@61738
  1900
  also have "... = contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
lp15@61738
  1901
    apply (simp add: contour_integral_integral)
lp15@60809
  1902
    apply (rule integral_cong)
lp15@62463
  1903
    unfolding integral_mult_left [symmetric]
lp15@60809
  1904
    apply (simp add: algebra_simps)
lp15@60809
  1905
    done
lp15@60809
  1906
  finally show ?thesis
lp15@61738
  1907
    by (simp add: contour_integral_integral)
lp15@60809
  1908
qed
lp15@60809
  1909
lp15@60809
  1910
lp15@60809
  1911
subsection\<open>The key quadrisection step\<close>
lp15@60809
  1912
lp15@60809
  1913
lemma norm_sum_half:
lp15@60809
  1914
  assumes "norm(a + b) >= e"
lp15@60809
  1915
    shows "norm a >= e/2 \<or> norm b >= e/2"
lp15@60809
  1916
proof -
lp15@60809
  1917
  have "e \<le> norm (- a - b)"
lp15@60809
  1918
    by (simp add: add.commute assms norm_minus_commute)
lp15@60809
  1919
  thus ?thesis
lp15@60809
  1920
    using norm_triangle_ineq4 order_trans by fastforce
lp15@60809
  1921
qed
lp15@60809
  1922
lp15@60809
  1923
lemma norm_sum_lemma:
lp15@60809
  1924
  assumes "e \<le> norm (a + b + c + d)"
lp15@60809
  1925
    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
  1926
proof -
lp15@60809
  1927
  have "e \<le> norm ((a + b) + (c + d))" using assms
lp15@60809
  1928
    by (simp add: algebra_simps)
lp15@60809
  1929
  then show ?thesis
lp15@60809
  1930
    by (auto dest!: norm_sum_half)
lp15@60809
  1931
qed
lp15@60809
  1932
lp15@60809
  1933
lemma Cauchy_theorem_quadrisection:
lp15@60809
  1934
  assumes f: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  1935
      and dist: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
lp15@60809
  1936
      and e: "e * K^2 \<le>
lp15@61738
  1937
              norm (contour_integral(linepath a b) f + contour_integral(linepath b c) f + contour_integral(linepath c a) f)"
lp15@60809
  1938
  shows "\<exists>a' b' c'.
lp15@60809
  1939
           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
  1940
           dist a' b' \<le> K/2  \<and>  dist b' c' \<le> K/2  \<and>  dist c' a' \<le> K/2  \<and>
lp15@61738
  1941
           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@60809
  1942
proof -
lp15@60809
  1943
  note divide_le_eq_numeral1 [simp del]
lp15@60809
  1944
  def a' \<equiv> "midpoint b c"
lp15@60809
  1945
  def b' \<equiv> "midpoint c a"
lp15@60809
  1946
  def c' \<equiv> "midpoint a b"
lp15@60809
  1947
  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
  1948
    using f continuous_on_subset segments_subset_convex_hull by metis+
lp15@60809
  1949
  have fcont': "continuous_on (closed_segment c' b') f"
lp15@60809
  1950
               "continuous_on (closed_segment a' c') f"
lp15@60809
  1951
               "continuous_on (closed_segment b' a') f"
lp15@60809
  1952
    unfolding a'_def b'_def c'_def
lp15@60809
  1953
    apply (rule continuous_on_subset [OF f],
lp15@60809
  1954
           metis midpoints_in_convex_hull convex_hull_subset hull_subset insert_subset segment_convex_hull)+
lp15@60809
  1955
    done
lp15@61738
  1956
  let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
lp15@60809
  1957
  have *: "?pathint a b + ?pathint b c + ?pathint c a =
lp15@60809
  1958
          (?pathint a c' + ?pathint c' b' + ?pathint b' a) +
lp15@60809
  1959
          (?pathint a' c' + ?pathint c' b + ?pathint b a') +
lp15@60809
  1960
          (?pathint a' c + ?pathint c b' + ?pathint b' a') +
lp15@60809
  1961
          (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
lp15@61738
  1962
    apply (simp add: fcont' contour_integral_reverse_linepath)
lp15@61738
  1963
    apply (simp add: a'_def b'_def c'_def contour_integral_midpoint fabc)
lp15@60809
  1964
    done
lp15@60809
  1965
  have [simp]: "\<And>x y. cmod (x * 2 - y * 2) = cmod (x - y) * 2"
lp15@60809
  1966
    by (metis left_diff_distrib mult.commute norm_mult_numeral1)
lp15@60809
  1967
  have [simp]: "\<And>x y. cmod (x - y) = cmod (y - x)"
lp15@60809
  1968
    by (simp add: norm_minus_commute)
lp15@60809
  1969
  consider "e * K\<^sup>2 / 4 \<le> cmod (?pathint a c' + ?pathint c' b' + ?pathint b' a)" |
lp15@60809
  1970
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c' + ?pathint c' b + ?pathint b a')" |
lp15@60809
  1971
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c + ?pathint c b' + ?pathint b' a')" |
lp15@60809
  1972
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
lp15@60809
  1973
    using assms
lp15@60809
  1974
    apply (simp only: *)
lp15@60809
  1975
    apply (blast intro: that dest!: norm_sum_lemma)
lp15@60809
  1976
    done
lp15@60809
  1977
  then show ?thesis
lp15@60809
  1978
  proof cases
lp15@60809
  1979
    case 1 then show ?thesis
lp15@60809
  1980
      apply (rule_tac x=a in exI)
lp15@60809
  1981
      apply (rule exI [where x=c'])
lp15@60809
  1982
      apply (rule exI [where x=b'])
lp15@60809
  1983
      using assms
lp15@60809
  1984
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  1985
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  1986
      done
lp15@60809
  1987
  next
lp15@60809
  1988
    case 2 then show ?thesis
lp15@60809
  1989
      apply (rule_tac x=a' in exI)
lp15@60809
  1990
      apply (rule exI [where x=c'])
lp15@60809
  1991
      apply (rule exI [where x=b])
lp15@60809
  1992
      using assms
lp15@60809
  1993
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  1994
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  1995
      done
lp15@60809
  1996
  next
lp15@60809
  1997
    case 3 then show ?thesis
lp15@60809
  1998
      apply (rule_tac x=a' in exI)
lp15@60809
  1999
      apply (rule exI [where x=c])
lp15@60809
  2000
      apply (rule exI [where x=b'])
lp15@60809
  2001
      using assms
lp15@60809
  2002
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  2003
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  2004
      done
lp15@60809
  2005
  next
lp15@60809
  2006
    case 4 then show ?thesis
lp15@60809
  2007
      apply (rule_tac x=a' in exI)
lp15@60809
  2008
      apply (rule exI [where x=b'])
lp15@60809
  2009
      apply (rule exI [where x=c'])
lp15@60809
  2010
      using assms
lp15@60809
  2011
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  2012
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  2013
      done
lp15@60809
  2014
  qed
lp15@60809
  2015
qed
lp15@60809
  2016
lp15@60809
  2017
subsection\<open>Cauchy's theorem for triangles\<close>
lp15@60809
  2018
lp15@60809
  2019
lemma triangle_points_closer:
lp15@60809
  2020
  fixes a::complex
lp15@60809
  2021
  shows "\<lbrakk>x \<in> convex hull {a,b,c};  y \<in> convex hull {a,b,c}\<rbrakk>
lp15@60809
  2022
         \<Longrightarrow> norm(x - y) \<le> norm(a - b) \<or>
lp15@60809
  2023
             norm(x - y) \<le> norm(b - c) \<or>
lp15@60809
  2024
             norm(x - y) \<le> norm(c - a)"
lp15@60809
  2025
  using simplex_extremal_le [of "{a,b,c}"]
lp15@60809
  2026
  by (auto simp: norm_minus_commute)
lp15@60809
  2027
lp15@60809
  2028
lemma holomorphic_point_small_triangle:
lp15@60809
  2029
  assumes x: "x \<in> s"
lp15@60809
  2030
      and f: "continuous_on s f"
lp15@62534
  2031
      and cd: "f field_differentiable (at x within s)"
lp15@60809
  2032
      and e: "0 < e"
lp15@60809
  2033
    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@60809
  2034
              x \<in> convex hull {a,b,c} \<and> convex hull {a,b,c} \<subseteq> s
lp15@61738
  2035
              \<longrightarrow> norm(contour_integral(linepath a b) f + contour_integral(linepath b c) f +
lp15@61738
  2036
                       contour_integral(linepath c a) f)
lp15@60809
  2037
                  \<le> e*(dist a b + dist b c + dist c a)^2"
lp15@60809
  2038
           (is "\<exists>k>0. \<forall>a b c. _ \<longrightarrow> ?normle a b c")
lp15@60809
  2039
proof -
lp15@60809
  2040
  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
  2041
                     \<Longrightarrow> a \<le> e*(x + y + z)^2"
lp15@60809
  2042
    by (simp add: algebra_simps power2_eq_square)
lp15@60809
  2043
  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
  2044
             for x::real and a b c
lp15@60809
  2045
    by linarith
lp15@61738
  2046
  have fabc: "f contour_integrable_on linepath a b" "f contour_integrable_on linepath b c" "f contour_integrable_on linepath c a"
lp15@60809
  2047
              if "convex hull {a, b, c} \<subseteq> s" for a b c
lp15@60809
  2048
    using segments_subset_convex_hull that
lp15@61738
  2049
    by (metis continuous_on_subset f contour_integrable_continuous_linepath)+
lp15@61738
  2050
  note path_bound = has_contour_integral_bound_linepath [simplified norm_minus_commute, OF has_contour_integral_integral]
lp15@60809
  2051
  { fix f' a b c d
lp15@60809
  2052
    assume d: "0 < d"
lp15@60809
  2053
       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
  2054
       and le: "cmod (a - b) \<le> d" "cmod (b - c) \<le> d" "cmod (c - a) \<le> d"
lp15@60809
  2055
       and xc: "x \<in> convex hull {a, b, c}"
lp15@60809
  2056
       and s: "convex hull {a, b, c} \<subseteq> s"
lp15@61738
  2057
    have pa: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f =
lp15@61738
  2058
              contour_integral (linepath a b) (\<lambda>y. f y - f x - f'*(y - x)) +
lp15@61738
  2059
              contour_integral (linepath b c) (\<lambda>y. f y - f x - f'*(y - x)) +
lp15@61738
  2060
              contour_integral (linepath c a) (\<lambda>y. f y - f x - f'*(y - x))"
lp15@61738
  2061
      apply (simp add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc [OF s])
lp15@60809
  2062
      apply (simp add: field_simps)
lp15@60809
  2063
      done
lp15@60809
  2064
    { fix y
lp15@60809
  2065
      assume yc: "y \<in> convex hull {a,b,c}"
lp15@60809
  2066
      have "cmod (f y - f x - f' * (y - x)) \<le> e*norm(y - x)"
lp15@60809
  2067
        apply (rule f')
lp15@60809
  2068
        apply (metis triangle_points_closer [OF xc yc] le norm_minus_commute order_trans)
lp15@60809
  2069
        using s yc by blast
lp15@60809
  2070
      also have "... \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))"
lp15@60809
  2071
        by (simp add: yc e xc disj_le [OF triangle_points_closer])
lp15@60809
  2072
      finally have "cmod (f y - f x - f' * (y - x)) \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" .
lp15@60809
  2073
    } note cm_le = this
lp15@60809
  2074
    have "?normle a b c"
lp15@60809
  2075
      apply (simp add: dist_norm pa)
lp15@60809
  2076
      apply (rule le_of_3)
lp15@60809
  2077
      using f' xc s e
lp15@60809
  2078
      apply simp_all
lp15@60809
  2079
      apply (intro norm_triangle_le add_mono path_bound)
lp15@61738
  2080
      apply (simp_all add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc)
lp15@60809
  2081
      apply (blast intro: cm_le elim: dest: segments_subset_convex_hull [THEN subsetD])+
lp15@60809
  2082
      done
lp15@60809
  2083
  } note * = this
lp15@60809
  2084
  show ?thesis
lp15@60809
  2085
    using cd e
lp15@62534
  2086
    apply (simp add: field_differentiable_def has_field_derivative_def has_derivative_within_alt approachable_lt_le2 Ball_def)
lp15@60809
  2087
    apply (clarify dest!: spec mp)
lp15@60809
  2088
    using *
lp15@60809
  2089
    apply (simp add: dist_norm, blast)
lp15@60809
  2090
    done
lp15@60809
  2091
qed
lp15@60809
  2092
lp15@60809
  2093
lp15@60809
  2094
(* Hence the most basic theorem for a triangle.*)
lp15@60809
  2095
locale Chain =
lp15@60809
  2096
  fixes x0 At Follows
lp15@60809
  2097
  assumes At0: "At x0 0"
lp15@60809
  2098
      and AtSuc: "\<And>x n. At x n \<Longrightarrow> \<exists>x'. At x' (Suc n) \<and> Follows x' x"
lp15@60809
  2099
begin
lp15@60809
  2100
  primrec f where
lp15@60809
  2101
    "f 0 = x0"
lp15@60809
  2102
  | "f (Suc n) = (SOME x. At x (Suc n) \<and> Follows x (f n))"
lp15@60809
  2103
lp15@60809
  2104
  lemma At: "At (f n) n"
lp15@60809
  2105
  proof (induct n)
lp15@60809
  2106
    case 0 show ?case
lp15@60809
  2107
      by (simp add: At0)
lp15@60809
  2108
  next
lp15@60809
  2109
    case (Suc n) show ?case
lp15@60809
  2110
      by (metis (no_types, lifting) AtSuc [OF Suc] f.simps(2) someI_ex)
lp15@60809
  2111
  qed
lp15@60809
  2112
lp15@60809
  2113
  lemma Follows: "Follows (f(Suc n)) (f n)"
lp15@60809
  2114
    by (metis (no_types, lifting) AtSuc [OF At [of n]] f.simps(2) someI_ex)
lp15@60809
  2115
lp15@60809
  2116
  declare f.simps(2) [simp del]
lp15@60809
  2117
end
lp15@60809
  2118
lp15@60809
  2119
lemma Chain3:
lp15@60809
  2120
  assumes At0: "At x0 y0 z0 0"
lp15@60809
  2121
      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
  2122
  obtains f g h where
lp15@60809
  2123
    "f 0 = x0" "g 0 = y0" "h 0 = z0"
lp15@60809
  2124
                      "\<And>n. At (f n) (g n) (h n) n"
lp15@60809
  2125
                       "\<And>n. Follows (f(Suc n)) (g(Suc n)) (h(Suc n)) (f n) (g n) (h n)"
lp15@60809
  2126
proof -
lp15@60809
  2127
  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
  2128
    apply unfold_locales
lp15@60809
  2129
    using At0 AtSuc by auto
lp15@60809
  2130
  show ?thesis
lp15@60809
  2131
  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@60809
  2132
  apply simp_all
lp15@60809
  2133
  using three.At three.Follows
lp15@60809
  2134
  apply (simp_all add: split_beta')
lp15@60809
  2135
  done
lp15@60809
  2136
qed
lp15@60809
  2137
lp15@60809
  2138
lemma Cauchy_theorem_triangle:
lp15@60809
  2139
  assumes "f holomorphic_on (convex hull {a,b,c})"
lp15@61738
  2140
    shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2141
proof -
lp15@60809
  2142
  have contf: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  2143
    by (metis assms holomorphic_on_imp_continuous_on)
lp15@61738
  2144
  let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
lp15@60809
  2145
  { fix y::complex
lp15@61738
  2146
    assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2147
       and ynz: "y \<noteq> 0"
lp15@60809
  2148
    def K \<equiv> "1 + max (dist a b) (max (dist b c) (dist c a))"
lp15@60809
  2149
    def e \<equiv> "norm y / K^2"
lp15@60809
  2150
    have K1: "K \<ge> 1"  by (simp add: K_def max.coboundedI1)
lp15@60809
  2151
    then have K: "K > 0" by linarith
lp15@60809
  2152
    have [iff]: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
lp15@60809
  2153
      by (simp_all add: K_def)
lp15@60809
  2154
    have e: "e > 0"
lp15@60809
  2155
      unfolding e_def using ynz K1 by simp
lp15@60809
  2156
    def At \<equiv> "\<lambda>x y z n. convex hull {x,y,z} \<subseteq> convex hull {a,b,c} \<and>
lp15@60809
  2157
                         dist x y \<le> K/2^n \<and> dist y z \<le> K/2^n \<and> dist z x \<le> K/2^n \<and>
lp15@60809
  2158
                         norm(?pathint x y + ?pathint y z + ?pathint z x) \<ge> e*(K/2^n)^2"
lp15@60809
  2159
    have At0: "At a b c 0"
lp15@60809
  2160
      using fy
lp15@60809
  2161
      by (simp add: At_def e_def has_chain_integral_chain_integral3)
lp15@60809
  2162
    { fix x y z n
lp15@60809
  2163
      assume At: "At x y z n"
lp15@60809
  2164
      then have contf': "continuous_on (convex hull {x,y,z}) f"
lp15@60809
  2165
        using contf At_def continuous_on_subset by blast
lp15@60809
  2166
      have "\<exists>x' y' z'. At x' y' z' (Suc n) \<and> convex hull {x',y',z'} \<subseteq> convex hull {x,y,z}"
lp15@60809
  2167
        using At
lp15@60809
  2168
        apply (simp add: At_def)
lp15@60809
  2169
        using  Cauchy_theorem_quadrisection [OF contf', of "K/2^n" e]
lp15@60809
  2170
        apply clarsimp
lp15@60809
  2171
        apply (rule_tac x="a'" in exI)
lp15@60809
  2172
        apply (rule_tac x="b'" in exI)
lp15@60809
  2173
        apply (rule_tac x="c'" in exI)
lp15@60809
  2174
        apply (simp add: algebra_simps)
lp15@60809
  2175
        apply (meson convex_hull_subset empty_subsetI insert_subset subsetCE)
lp15@60809
  2176
        done
lp15@60809
  2177
    } note AtSuc = this
lp15@60809
  2178
    obtain fa fb fc
lp15@60809
  2179
      where f0 [simp]: "fa 0 = a" "fb 0 = b" "fc 0 = c"
lp15@60809
  2180
        and cosb: "\<And>n. convex hull {fa n, fb n, fc n} \<subseteq> convex hull {a,b,c}"
lp15@60809
  2181
        and dist: "\<And>n. dist (fa n) (fb n) \<le> K/2^n"
lp15@60809
  2182
                  "\<And>n. dist (fb n) (fc n) \<le> K/2^n"
lp15@60809
  2183
                  "\<And>n. dist (fc n) (fa n) \<le> K/2^n"
lp15@60809
  2184
        and no: "\<And>n. norm(?pathint (fa n) (fb n) +
lp15@60809
  2185
                           ?pathint (fb n) (fc n) +
lp15@60809
  2186
                           ?pathint (fc n) (fa n)) \<ge> e * (K/2^n)^2"
lp15@60809
  2187
        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}"
lp15@60809
  2188
      apply (rule Chain3 [of At, OF At0 AtSuc])
lp15@60809
  2189
      apply (auto simp: At_def)
lp15@60809
  2190
      done
lp15@60809
  2191
    have "\<exists>x. \<forall>n. x \<in> convex hull {fa n, fb n, fc n}"
lp15@60809
  2192
      apply (rule bounded_closed_nest)
lp15@60809
  2193
      apply (simp_all add: compact_imp_closed finite_imp_compact_convex_hull finite_imp_bounded_convex_hull)
lp15@60809
  2194
      apply (rule allI)
lp15@60809
  2195
      apply (rule transitive_stepwise_le)
lp15@60809
  2196
      apply (auto simp: conv_le)
lp15@60809
  2197
      done
lp15@60809
  2198
    then obtain x where x: "\<And>n. x \<in> convex hull {fa n, fb n, fc n}" by auto
lp15@60809
  2199
    then have xin: "x \<in> convex hull {a,b,c}"
lp15@60809
  2200
      using assms f0 by blast
lp15@62534
  2201
    then have fx: "f field_differentiable at x within (convex hull {a,b,c})"
lp15@60809
  2202
      using assms holomorphic_on_def by blast
lp15@60809
  2203
    { fix k n
lp15@60809
  2204
      assume k: "0 < k"
lp15@60809
  2205
         and le:
lp15@60809
  2206
            "\<And>x' y' z'.
lp15@60809
  2207
               \<lbrakk>dist x' y' \<le> k; dist y' z' \<le> k; dist z' x' \<le> k;
lp15@60809
  2208
                x \<in> convex hull {x',y',z'};
lp15@60809
  2209
                convex hull {x',y',z'} \<subseteq> convex hull {a,b,c}\<rbrakk>
lp15@60809
  2210
               \<Longrightarrow>
lp15@60809
  2211
               cmod (?pathint x' y' + ?pathint y' z' + ?pathint z' x') * 10
lp15@60809
  2212
                     \<le> e * (dist x' y' + dist y' z' + dist z' x')\<^sup>2"
lp15@60809
  2213
         and Kk: "K / k < 2 ^ n"
lp15@60809
  2214
      have "K / 2 ^ n < k" using Kk k
lp15@60809
  2215
        by (auto simp: field_simps)
lp15@60809
  2216
      then have DD: "dist (fa n) (fb n) \<le> k" "dist (fb n) (fc n) \<le> k" "dist (fc n) (fa n) \<le> k"
lp15@60809
  2217
        using dist [of n]  k
lp15@60809
  2218
        by linarith+
lp15@60809
  2219
      have dle: "(dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2
lp15@60809
  2220
               \<le> (3 * K / 2 ^ n)\<^sup>2"
lp15@60809
  2221
        using dist [of n] e K
lp15@60809
  2222
        by (simp add: abs_le_square_iff [symmetric])
lp15@60809
  2223
      have less10: "\<And>x y::real. 0 < x \<Longrightarrow> y \<le> 9*x \<Longrightarrow> y < x*10"
lp15@60809
  2224
        by linarith
lp15@60809
  2225
      have "e * (dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2 \<le> e * (3 * K / 2 ^ n)\<^sup>2"
lp15@60809
  2226
        using ynz dle e mult_le_cancel_left_pos by blast