src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy
author paulson <lp15@cam.ac.uk>
Fri Sep 18 16:27:37 2015 +0100 (2015-09-18)
changeset 61190 2bd401e364f9
parent 61104 3c2d4636cebc
child 61200 a5674da43c2b
permissions -rw-r--r--
Massive revisions, as a valid path must now be continously differentiable (C!)
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section \<open>Complex path integrals and Cauchy's integral theorem\<close>
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theory Cauchy_Integral_Thm
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imports Complex_Transcendental Weierstrass
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begin
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(*FIXME migrate into libraries*)
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lemma inj_mult_left [simp]: "inj (op* x) \<longleftrightarrow> x \<noteq> (0::'a::idom)"
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  by (metis injI mult_cancel_left the_inv_f_f zero_neq_one)
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lemma continuous_on_strong_cong:
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  "s = t \<Longrightarrow> (\<And>x. x \<in> t =simp=> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"
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  by (simp add: simp_implies_def)
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thm image_affinity_atLeastAtMost_div_diff
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lemma image_affinity_atLeastAtMost_div:
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  fixes c :: "'a::linordered_field"
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  shows "((\<lambda>x. x/m + c) ` {a..b}) = (if {a..b}={} then {}
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            else if 0 \<le> m then {a/m + c .. b/m + c}
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            else {b/m + c .. a/m + c})"
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  using image_affinity_atLeastAtMost [of "inverse m" c a b]
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  by (simp add: field_class.field_divide_inverse algebra_simps)
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thm continuous_on_closed_Un
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lemma continuous_on_open_Un:
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  "open s \<Longrightarrow> open t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
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  using continuous_on_open_Union [of "{s,t}"] by auto
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thm continuous_on_eq (*REPLACE*)
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lemma continuous_on_eq:
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  "\<lbrakk>continuous_on s f; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> continuous_on s g"
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  unfolding continuous_on_def tendsto_def eventually_at_topological
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  by simp
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thm vector_derivative_add_at
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lemma vector_derivative_mult_at [simp]:
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  fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
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  shows  "\<lbrakk>f differentiable at a; g differentiable at a\<rbrakk>
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   \<Longrightarrow> vector_derivative (\<lambda>x. f x * g x) (at a) = f a * vector_derivative g (at a) + vector_derivative f (at a) * g a"
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  by (simp add: vector_derivative_at has_vector_derivative_mult vector_derivative_works [symmetric])
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lemma vector_derivative_scaleR_at [simp]:
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    "\<lbrakk>f differentiable at a; g differentiable at a\<rbrakk>
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   \<Longrightarrow> vector_derivative (\<lambda>x. f x *\<^sub>R g x) (at a) = f a *\<^sub>R vector_derivative g (at a) + vector_derivative f (at a) *\<^sub>R g a"
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apply (rule vector_derivative_at)
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apply (rule has_vector_derivative_scaleR)
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apply (auto simp: vector_derivative_works has_vector_derivative_def has_field_derivative_def mult_commute_abs)
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done
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thm Derivative.vector_diff_chain_at
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lemma vector_derivative_chain_at:
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  assumes "f differentiable at x" "(g differentiable at (f x))"
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  shows "vector_derivative (g \<circ> f) (at x) =
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         vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"
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by (metis Derivative.vector_diff_chain_at vector_derivative_at vector_derivative_works assms)
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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_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 `finite s` `finite t` 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 dist_nz [symmetric])
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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 dist_nz [symmetric])
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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: split_if_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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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"
lp15@61190
   287
  by (auto simp: C1_differentiable_on_eq continuous_on_const)
lp15@61190
   288
lp15@61190
   289
lemma C1_differentiable_on_add [simp, derivative_intros]:
lp15@61190
   290
  "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x + g x) C1_differentiable_on s"
lp15@61190
   291
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
lp15@61190
   292
lp15@61190
   293
lemma C1_differentiable_on_minus [simp, derivative_intros]:
lp15@61190
   294
  "f C1_differentiable_on s \<Longrightarrow> (\<lambda>x. - f x) C1_differentiable_on s"
lp15@61190
   295
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
lp15@61190
   296
lp15@61190
   297
lemma C1_differentiable_on_diff [simp, derivative_intros]:
lp15@61190
   298
  "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x - g x) C1_differentiable_on s"
lp15@61190
   299
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
lp15@61190
   300
lp15@61190
   301
lemma C1_differentiable_on_mult [simp, derivative_intros]:
lp15@61190
   302
  fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
lp15@61190
   303
  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
   304
  unfolding C1_differentiable_on_eq
lp15@61190
   305
  by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within)
lp15@61190
   306
lp15@61190
   307
lemma C1_differentiable_on_scaleR [simp, derivative_intros]:
lp15@61190
   308
  "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
   309
  unfolding C1_differentiable_on_eq
lp15@61190
   310
  by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+
lp15@61190
   311
lp15@61190
   312
lp15@61190
   313
definition piecewise_C1_differentiable_on
lp15@61190
   314
           (infixr "piecewise'_C1'_differentiable'_on" 50)
lp15@61190
   315
  where "f piecewise_C1_differentiable_on i  \<equiv>
lp15@61190
   316
           continuous_on i f \<and>
lp15@61190
   317
           (\<exists>s. finite s \<and> (f C1_differentiable_on (i - s)))"
lp15@61190
   318
lp15@61190
   319
lemma C1_differentiable_imp_piecewise:
lp15@61190
   320
    "f C1_differentiable_on s \<Longrightarrow> f piecewise_C1_differentiable_on s"
lp15@61190
   321
  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within)
lp15@61190
   322
lp15@61190
   323
lemma piecewise_C1_imp_differentiable:
lp15@61190
   324
    "f piecewise_C1_differentiable_on i \<Longrightarrow> f piecewise_differentiable_on i"
lp15@61190
   325
  by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def
lp15@61190
   326
           C1_differentiable_on_def differentiable_def has_vector_derivative_def
lp15@61190
   327
           intro: has_derivative_at_within)
lp15@61190
   328
lp15@61190
   329
lemma piecewise_C1_differentiable_compose:
lp15@61190
   330
    "\<lbrakk>f piecewise_C1_differentiable_on s; g piecewise_C1_differentiable_on (f ` s);
lp15@61190
   331
      \<And>x. finite (s \<inter> f-`{x})\<rbrakk>
lp15@61190
   332
      \<Longrightarrow> (g o f) piecewise_C1_differentiable_on s"
lp15@61190
   333
  apply (simp add: piecewise_C1_differentiable_on_def, safe)
lp15@61190
   334
  apply (blast intro: continuous_on_compose2)
lp15@61190
   335
  apply (rename_tac A B)
lp15@61190
   336
  apply (rule_tac x="A \<union> (\<Union>x\<in>B. s \<inter> f-`{x})" in exI)
lp15@61190
   337
  apply (rule conjI, blast)
lp15@61190
   338
  apply (rule C1_differentiable_compose)
lp15@61190
   339
  apply (blast intro: C1_differentiable_on_subset)
lp15@61190
   340
  apply (blast intro: C1_differentiable_on_subset)
lp15@61190
   341
  by (simp add: Diff_Int_distrib2)
lp15@61190
   342
lp15@61190
   343
lemma piecewise_C1_differentiable_on_subset:
lp15@61190
   344
    "f piecewise_C1_differentiable_on s \<Longrightarrow> t \<le> s \<Longrightarrow> f piecewise_C1_differentiable_on t"
lp15@61190
   345
  by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset)
lp15@61190
   346
lp15@61190
   347
lemma C1_differentiable_imp_continuous_on:
lp15@61190
   348
  "f C1_differentiable_on s \<Longrightarrow> continuous_on s f"
lp15@61190
   349
  unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within
lp15@61190
   350
  using differentiable_at_withinI differentiable_imp_continuous_within by blast
lp15@61190
   351
lp15@61190
   352
lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}"
lp15@61190
   353
  unfolding C1_differentiable_on_def
lp15@61190
   354
  by auto
lp15@61190
   355
lp15@61190
   356
lemma piecewise_C1_differentiable_affine:
lp15@61190
   357
  fixes m::real
lp15@61190
   358
  assumes "f piecewise_C1_differentiable_on ((\<lambda>x. m * x + c) ` s)"
lp15@61190
   359
  shows "(f o (\<lambda>x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on s"
lp15@61190
   360
proof (cases "m = 0")
lp15@61190
   361
  case True
lp15@61190
   362
  then show ?thesis
lp15@61190
   363
    unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def continuous_on_const)
lp15@61190
   364
next
lp15@61190
   365
  case False
lp15@61190
   366
  show ?thesis
lp15@61190
   367
    apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise])
lp15@61190
   368
    apply (rule assms derivative_intros | simp add: False vimage_def)+
lp15@61190
   369
    using real_vector_affinity_eq [OF False, where c=c, unfolded scaleR_conv_of_real]
lp15@61190
   370
    apply simp
lp15@61190
   371
    done
lp15@61190
   372
qed
lp15@61190
   373
lp15@61190
   374
lemma piecewise_C1_differentiable_cases:
lp15@61190
   375
  fixes c::real
lp15@61190
   376
  assumes "f piecewise_C1_differentiable_on {a..c}"
lp15@61190
   377
          "g piecewise_C1_differentiable_on {c..b}"
lp15@61190
   378
           "a \<le> c" "c \<le> b" "f c = g c"
lp15@61190
   379
  shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}"
lp15@61190
   380
proof -
lp15@61190
   381
  obtain s t where st: "f C1_differentiable_on ({a..c} - s)"
lp15@61190
   382
                       "g C1_differentiable_on ({c..b} - t)"
lp15@61190
   383
                       "finite s" "finite t"
lp15@61190
   384
    using assms
lp15@61190
   385
    by (force simp: piecewise_C1_differentiable_on_def)
lp15@61190
   386
  then have f_diff: "f differentiable_on {a..<c} - s"
lp15@61190
   387
        and g_diff: "g differentiable_on {c<..b} - t"
lp15@61190
   388
    by (simp_all add: C1_differentiable_on_eq differentiable_at_withinI differentiable_on_def)
lp15@61190
   389
  have "continuous_on {a..c} f" "continuous_on {c..b} g"
lp15@61190
   390
    using assms piecewise_C1_differentiable_on_def by auto
lp15@61190
   391
  then have cab: "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
lp15@61190
   392
    using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
lp15@61190
   393
                               OF closed_real_atLeastAtMost [of c b],
lp15@61190
   394
                               of f g "\<lambda>x. x\<le>c"]  assms
lp15@61190
   395
    by (force simp: ivl_disj_un_two_touch)
lp15@61190
   396
  { fix x
lp15@61190
   397
    assume x: "x \<in> {a..b} - insert c (s \<union> t)"
lp15@61190
   398
    have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x" (is "?diff_fg")
lp15@61190
   399
    proof (cases x c rule: le_cases)
lp15@61190
   400
      case le show ?diff_fg
lp15@61190
   401
        apply (rule differentiable_transform_at [of "dist x c" _ f])
lp15@61190
   402
        using x dist_real_def le st by (auto simp: C1_differentiable_on_eq)
lp15@61190
   403
    next
lp15@61190
   404
      case ge show ?diff_fg
lp15@61190
   405
        apply (rule differentiable_transform_at [of "dist x c" _ g])
lp15@61190
   406
        using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq)
lp15@61190
   407
    qed
lp15@61190
   408
  }
lp15@61190
   409
  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
   410
    by auto
lp15@61190
   411
  moreover
lp15@61190
   412
  { assume fcon: "continuous_on ({a<..<c} - s) (\<lambda>x. vector_derivative f (at x))"
lp15@61190
   413
       and gcon: "continuous_on ({c<..<b} - t) (\<lambda>x. vector_derivative g (at x))"
lp15@61190
   414
    have "open ({a<..<c} - s)"  "open ({c<..<b} - t)"
lp15@61190
   415
      using st by (simp_all add: open_Diff finite_imp_closed)
lp15@61190
   416
    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
   417
      apply (rule continuous_on_eq [OF fcon])
lp15@61190
   418
      apply (simp add:)
lp15@61190
   419
      apply (rule vector_derivative_at [symmetric])
lp15@61190
   420
      apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_at)
lp15@61190
   421
      apply (simp_all add: dist_norm vector_derivative_works [symmetric])
lp15@61190
   422
      using f_diff
lp15@61190
   423
      by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_imp_le st(1))
lp15@61190
   424
    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
   425
      apply (rule continuous_on_eq [OF gcon])
lp15@61190
   426
      apply (simp add:)
lp15@61190
   427
      apply (rule vector_derivative_at [symmetric])
lp15@61190
   428
      apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_at)
lp15@61190
   429
      apply (simp_all add: dist_norm vector_derivative_works [symmetric])
lp15@61190
   430
      using g_diff
lp15@61190
   431
      by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_imp_le st(2))
lp15@61190
   432
    ultimately have "continuous_on ({a<..<b} - insert c (s \<union> t))
lp15@61190
   433
        (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
lp15@61190
   434
      apply (rule continuous_on_subset [OF continuous_on_open_Un], auto)
lp15@61190
   435
      done
lp15@61190
   436
  } note * = this
lp15@61190
   437
  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
   438
    using st
lp15@61190
   439
    by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *)
lp15@61190
   440
  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
   441
    apply (rule_tac x="{a,b,c} \<union> s \<union> t" in exI)
lp15@61190
   442
    using st  by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset)
lp15@61190
   443
  with cab show ?thesis
lp15@61190
   444
    by (simp add: piecewise_C1_differentiable_on_def)
lp15@61190
   445
qed
lp15@61190
   446
lp15@61190
   447
lemma piecewise_C1_differentiable_neg:
lp15@61190
   448
    "f piecewise_C1_differentiable_on s \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_C1_differentiable_on s"
lp15@61190
   449
  unfolding piecewise_C1_differentiable_on_def
lp15@61190
   450
  by (auto intro!: continuous_on_minus C1_differentiable_on_minus)
lp15@61190
   451
lp15@61190
   452
lemma piecewise_C1_differentiable_add:
lp15@61190
   453
  assumes "f piecewise_C1_differentiable_on i"
lp15@61190
   454
          "g piecewise_C1_differentiable_on i"
lp15@61190
   455
    shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i"
lp15@61190
   456
proof -
lp15@61190
   457
  obtain s t where st: "finite s" "finite t"
lp15@61190
   458
                       "f C1_differentiable_on (i-s)"
lp15@61190
   459
                       "g C1_differentiable_on (i-t)"
lp15@61190
   460
    using assms by (auto simp: piecewise_C1_differentiable_on_def)
lp15@61190
   461
  then have "finite (s \<union> t) \<and> (\<lambda>x. f x + g x) C1_differentiable_on i - (s \<union> t)"
lp15@61190
   462
    by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset)
lp15@61190
   463
  moreover have "continuous_on i f" "continuous_on i g"
lp15@61190
   464
    using assms piecewise_C1_differentiable_on_def by auto
lp15@61190
   465
  ultimately show ?thesis
lp15@61190
   466
    by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add)
lp15@61190
   467
qed
lp15@61190
   468
lp15@61190
   469
lemma piecewise_C1_differentiable_C1_diff:
lp15@61190
   470
    "\<lbrakk>f piecewise_C1_differentiable_on s;  g piecewise_C1_differentiable_on s\<rbrakk>
lp15@61190
   471
     \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_C1_differentiable_on s"
lp15@61190
   472
  unfolding diff_conv_add_uminus
lp15@61190
   473
  by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg)
lp15@61190
   474
lp15@61190
   475
lemma piecewise_C1_differentiable_D1:
lp15@61190
   476
  fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   477
  assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}"
lp15@61190
   478
    shows "g1 piecewise_C1_differentiable_on {0..1}"
lp15@61190
   479
proof -
lp15@61190
   480
  obtain s where "finite s"
lp15@61190
   481
             and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   482
             and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
lp15@61190
   483
    using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@61190
   484
  then have g1D: "g1 differentiable at x" if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
lp15@61190
   485
    apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))" in differentiable_transform_at)
lp15@61190
   486
    using that
lp15@61190
   487
    apply (simp_all add: dist_real_def joinpaths_def)
lp15@61190
   488
    apply (rule differentiable_chain_at derivative_intros | force)+
lp15@61190
   489
    done
lp15@61190
   490
  have [simp]: "vector_derivative (g1 \<circ> op * 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)"
lp15@61190
   491
               if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
lp15@61190
   492
    apply (subst vector_derivative_chain_at)
lp15@61190
   493
    using that
lp15@61190
   494
    apply (rule derivative_eq_intros g1D | simp)+
lp15@61190
   495
    done
lp15@61190
   496
  have "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   497
    using co12 by (rule continuous_on_subset) force
lp15@61190
   498
  then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 o op*2) (at x))"
lp15@61190
   499
    apply (rule continuous_on_eq [OF _ vector_derivative_at])
lp15@61190
   500
    apply (rule_tac f="g1 o op*2" and d="dist x (1/2)" in has_vector_derivative_transform_at)
lp15@61190
   501
    apply (simp_all add: dist_norm joinpaths_def vector_derivative_works [symmetric])
lp15@61190
   502
    apply (force intro: g1D differentiable_chain_at)
lp15@61190
   503
    done
lp15@61190
   504
  have "continuous_on ({0..1} - insert 1 (op * 2 ` s))
lp15@61190
   505
                      ((\<lambda>x. 1/2 * vector_derivative (g1 o op*2) (at x)) o op*(1/2))"
lp15@61190
   506
    apply (rule continuous_intros)+
lp15@61190
   507
    using coDhalf
lp15@61190
   508
    apply (simp add: scaleR_conv_of_real image_set_diff image_image)
lp15@61190
   509
    done
lp15@61190
   510
  then have con_g1: "continuous_on ({0..1} - insert 1 (op * 2 ` s)) (\<lambda>x. vector_derivative g1 (at x))"
lp15@61190
   511
    by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
lp15@61190
   512
  have "continuous_on {0..1} g1"
lp15@61190
   513
    using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast
lp15@61190
   514
  with `finite s` show ?thesis
lp15@61190
   515
    apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@61190
   516
    apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
lp15@61190
   517
    apply (simp add: g1D con_g1)
lp15@61190
   518
  done
lp15@61190
   519
qed
lp15@61190
   520
lp15@61190
   521
lemma piecewise_C1_differentiable_D2:
lp15@61190
   522
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   523
  assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" "pathfinish g1 = pathstart g2"
lp15@61190
   524
    shows "g2 piecewise_C1_differentiable_on {0..1}"
lp15@61190
   525
proof -
lp15@61190
   526
  obtain s where "finite s"
lp15@61190
   527
             and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   528
             and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
lp15@61190
   529
    using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@61190
   530
  then have g2D: "g2 differentiable at x" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)" for x
lp15@61190
   531
    apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\<lambda>x. (x+1)/2)" in differentiable_transform_at)
lp15@61190
   532
    using that
lp15@61190
   533
    apply (simp_all add: dist_real_def joinpaths_def)
lp15@61190
   534
    apply (auto simp: dist_real_def joinpaths_def field_simps)
lp15@61190
   535
    apply (rule differentiable_chain_at derivative_intros | force)+
lp15@61190
   536
    apply (drule_tac x= "(x + 1) / 2" in bspec, force simp: divide_simps)
lp15@61190
   537
    apply assumption
lp15@61190
   538
    done
lp15@61190
   539
  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
   540
               if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)" for x
lp15@61190
   541
    using that  by (auto simp: vector_derivative_chain_at divide_simps g2D)
lp15@61190
   542
  have "continuous_on ({1/2..1} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   543
    using co12 by (rule continuous_on_subset) force
lp15@61190
   544
  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
   545
    apply (rule continuous_on_eq [OF _ vector_derivative_at])
lp15@61190
   546
    apply (rule_tac f="g2 o (\<lambda>x. 2*x-1)" and d="dist (3/4) ((x+1)/2)" in has_vector_derivative_transform_at)
lp15@61190
   547
    apply (auto simp: dist_real_def field_simps joinpaths_def vector_derivative_works [symmetric]
lp15@61190
   548
                intro!: g2D differentiable_chain_at)
lp15@61190
   549
    done
lp15@61190
   550
  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
   551
    apply (simp add: image_set_diff inj_on_def image_image)
lp15@61190
   552
    apply (auto simp: image_affinity_atLeastAtMost_div add_divide_distrib)
lp15@61190
   553
    done
lp15@61190
   554
  have "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s))
lp15@61190
   555
                      ((\<lambda>x. 1/2 * vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x)) o (\<lambda>x. (x+1)/2))"
lp15@61190
   556
    by (rule continuous_intros | simp add:  coDhalf)+
lp15@61190
   557
  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
   558
    by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
lp15@61190
   559
  have "continuous_on {0..1} g2"
lp15@61190
   560
    using continuous_on_joinpaths_D2 assms piecewise_C1_differentiable_on_def by blast
lp15@61190
   561
  with `finite s` show ?thesis
lp15@61190
   562
    apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@61190
   563
    apply (rule_tac x="insert 0 ((\<lambda>x. 2 * x - 1) ` s)" in exI)
lp15@61190
   564
    apply (simp add: g2D con_g2)
lp15@61190
   565
  done
lp15@61190
   566
qed
lp15@60809
   567
lp15@60809
   568
subsection \<open>Valid paths, and their start and finish\<close>
lp15@60809
   569
lp15@60809
   570
lemma Diff_Un_eq: "A - (B \<union> C) = A - B - C"
lp15@60809
   571
  by blast
lp15@60809
   572
lp15@60809
   573
definition valid_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
lp15@61190
   574
  where "valid_path f \<equiv> f piecewise_C1_differentiable_on {0..1::real}"
lp15@60809
   575
lp15@60809
   576
definition closed_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
lp15@60809
   577
  where "closed_path g \<equiv> g 0 = g 1"
lp15@60809
   578
lp15@60809
   579
subsubsection\<open>In particular, all results for paths apply\<close>
lp15@60809
   580
lp15@60809
   581
lemma valid_path_imp_path: "valid_path g \<Longrightarrow> path g"
lp15@61190
   582
by (simp add: path_def piecewise_C1_differentiable_on_def valid_path_def)
lp15@60809
   583
lp15@60809
   584
lemma connected_valid_path_image: "valid_path g \<Longrightarrow> connected(path_image g)"
lp15@60809
   585
  by (metis connected_path_image valid_path_imp_path)
lp15@60809
   586
lp15@60809
   587
lemma compact_valid_path_image: "valid_path g \<Longrightarrow> compact(path_image g)"
lp15@60809
   588
  by (metis compact_path_image valid_path_imp_path)
lp15@60809
   589
lp15@60809
   590
lemma bounded_valid_path_image: "valid_path g \<Longrightarrow> bounded(path_image g)"
lp15@60809
   591
  by (metis bounded_path_image valid_path_imp_path)
lp15@60809
   592
lp15@60809
   593
lemma closed_valid_path_image: "valid_path g \<Longrightarrow> closed(path_image g)"
lp15@60809
   594
  by (metis closed_path_image valid_path_imp_path)
lp15@60809
   595
lp15@60809
   596
lp15@60809
   597
subsection\<open>Contour Integrals along a path\<close>
lp15@60809
   598
lp15@60809
   599
text\<open>This definition is for complex numbers only, and does not generalise to line integrals in a vector field\<close>
lp15@60809
   600
lp15@61190
   601
text\<open>piecewise differentiable function on [0,1]\<close>
lp15@60809
   602
lp15@60809
   603
definition has_path_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool"
lp15@60809
   604
           (infixr "has'_path'_integral" 50)
lp15@60809
   605
  where "(f has_path_integral i) g \<equiv>
lp15@60809
   606
           ((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1}))
lp15@60809
   607
            has_integral i) {0..1}"
lp15@60809
   608
lp15@60809
   609
definition path_integrable_on
lp15@60809
   610
           (infixr "path'_integrable'_on" 50)
lp15@60809
   611
  where "f path_integrable_on g \<equiv> \<exists>i. (f has_path_integral i) g"
lp15@60809
   612
lp15@60809
   613
definition path_integral
lp15@60809
   614
  where "path_integral g f \<equiv> @i. (f has_path_integral i) g"
lp15@60809
   615
lp15@60809
   616
lemma path_integral_unique: "(f has_path_integral i)  g \<Longrightarrow> path_integral g f = i"
lp15@60809
   617
  by (auto simp: path_integral_def has_path_integral_def integral_def [symmetric])
lp15@60809
   618
lp15@60809
   619
lemma has_path_integral_integral:
lp15@60809
   620
    "f path_integrable_on i \<Longrightarrow> (f has_path_integral (path_integral i f)) i"
lp15@60809
   621
  by (metis path_integral_unique path_integrable_on_def)
lp15@60809
   622
lp15@60809
   623
lemma has_path_integral_unique:
lp15@60809
   624
    "(f has_path_integral i) g \<Longrightarrow> (f has_path_integral j) g \<Longrightarrow> i = j"
lp15@60809
   625
  using has_integral_unique
lp15@60809
   626
  by (auto simp: has_path_integral_def)
lp15@60809
   627
lp15@60809
   628
lemma has_path_integral_integrable: "(f has_path_integral i) g \<Longrightarrow> f path_integrable_on g"
lp15@60809
   629
  using path_integrable_on_def by blast
lp15@60809
   630
lp15@60809
   631
(* Show that we can forget about the localized derivative.*)
lp15@60809
   632
lp15@60809
   633
lemma vector_derivative_within_interior:
lp15@60809
   634
     "\<lbrakk>x \<in> interior s; NO_MATCH UNIV s\<rbrakk>
lp15@60809
   635
      \<Longrightarrow> vector_derivative f (at x within s) = vector_derivative f (at x)"
lp15@60809
   636
  apply (simp add: vector_derivative_def has_vector_derivative_def has_derivative_def netlimit_within_interior)
lp15@60809
   637
  apply (subst lim_within_interior, auto)
lp15@60809
   638
  done
lp15@60809
   639
lp15@60809
   640
lemma has_integral_localized_vector_derivative:
lp15@60809
   641
    "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
lp15@60809
   642
     ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
lp15@60809
   643
proof -
lp15@60809
   644
  have "{a..b} - {a,b} = interior {a..b}"
lp15@60809
   645
    by (simp add: atLeastAtMost_diff_ends)
lp15@60809
   646
  show ?thesis
lp15@60809
   647
    apply (rule has_integral_spike_eq [of "{a,b}"])
lp15@60809
   648
    apply (auto simp: vector_derivative_within_interior)
lp15@60809
   649
    done
lp15@60809
   650
qed
lp15@60809
   651
lp15@60809
   652
lemma integrable_on_localized_vector_derivative:
lp15@60809
   653
    "(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow>
lp15@60809
   654
     (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
lp15@60809
   655
  by (simp add: integrable_on_def has_integral_localized_vector_derivative)
lp15@60809
   656
lp15@60809
   657
lemma has_path_integral:
lp15@60809
   658
     "(f has_path_integral i) g \<longleftrightarrow>
lp15@60809
   659
      ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
lp15@60809
   660
  by (simp add: has_integral_localized_vector_derivative has_path_integral_def)
lp15@60809
   661
lp15@60809
   662
lemma path_integrable_on:
lp15@60809
   663
     "f path_integrable_on g \<longleftrightarrow>
lp15@60809
   664
      (\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
lp15@60809
   665
  by (simp add: has_path_integral integrable_on_def path_integrable_on_def)
lp15@60809
   666
lp15@60809
   667
subsection\<open>Reversing a path\<close>
lp15@60809
   668
lp15@60809
   669
lemma valid_path_imp_reverse:
lp15@60809
   670
  assumes "valid_path g"
lp15@60809
   671
    shows "valid_path(reversepath g)"
lp15@60809
   672
proof -
lp15@61190
   673
  obtain s where "finite s" "g C1_differentiable_on ({0..1} - s)"
lp15@61190
   674
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
lp15@61190
   675
  then have "finite (op - 1 ` s)" "(reversepath g C1_differentiable_on ({0..1} - op - 1 ` s))"
lp15@60809
   676
    apply (auto simp: reversepath_def)
lp15@61190
   677
    apply (rule C1_differentiable_compose [of "\<lambda>x::real. 1-x" _ g, unfolded o_def])
lp15@61190
   678
    apply (auto simp: C1_differentiable_on_eq)
lp15@61190
   679
    apply (rule continuous_intros, force)
lp15@61190
   680
    apply (force elim!: continuous_on_subset)
lp15@61190
   681
    apply (simp add: finite_vimageI inj_on_def)
lp15@60809
   682
    done
lp15@60809
   683
  then show ?thesis using assms
lp15@61190
   684
    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def path_def [symmetric])
lp15@60809
   685
qed
lp15@60809
   686
lp15@60809
   687
lemma valid_path_reversepath: "valid_path(reversepath g) \<longleftrightarrow> valid_path g"
lp15@60809
   688
  using valid_path_imp_reverse by force
lp15@60809
   689
lp15@60809
   690
lemma has_path_integral_reversepath:
lp15@60809
   691
  assumes "valid_path g" "(f has_path_integral i) g"
lp15@60809
   692
    shows "(f has_path_integral (-i)) (reversepath g)"
lp15@60809
   693
proof -
lp15@60809
   694
  { fix s x
lp15@61190
   695
    assume xs: "g C1_differentiable_on ({0..1} - s)" "x \<notin> op - 1 ` s" "0 \<le> x" "x \<le> 1"
lp15@60809
   696
      have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
lp15@60809
   697
            - vector_derivative g (at (1 - x) within {0..1})"
lp15@60809
   698
      proof -
lp15@60809
   699
        obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
lp15@60809
   700
          using xs
lp15@61190
   701
          by (force simp: has_vector_derivative_def C1_differentiable_on_def)
lp15@60809
   702
        have "(g o (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
lp15@60809
   703
          apply (rule vector_diff_chain_within)
lp15@60809
   704
          apply (intro vector_diff_chain_within derivative_eq_intros | simp)+
lp15@60809
   705
          apply (rule has_vector_derivative_at_within [OF f'])
lp15@60809
   706
          done
lp15@60809
   707
        then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)"
lp15@60809
   708
          by (simp add: o_def)
lp15@60809
   709
        show ?thesis
lp15@60809
   710
          using xs
lp15@60809
   711
          by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
lp15@60809
   712
      qed
lp15@60809
   713
  } note * = this
lp15@60809
   714
  have 01: "{0..1::real} = cbox 0 1"
lp15@60809
   715
    by simp
lp15@60809
   716
  show ?thesis using assms
lp15@60809
   717
    apply (auto simp: has_path_integral_def)
lp15@60809
   718
    apply (drule has_integral_affinity01 [where m= "-1" and c=1])
lp15@61190
   719
    apply (auto simp: reversepath_def valid_path_def piecewise_C1_differentiable_on_def)
lp15@60809
   720
    apply (drule has_integral_neg)
lp15@60809
   721
    apply (rule_tac s = "(\<lambda>x. 1 - x) ` s" in has_integral_spike_finite)
lp15@60809
   722
    apply (auto simp: *)
lp15@60809
   723
    done
lp15@60809
   724
qed
lp15@60809
   725
lp15@60809
   726
lemma path_integrable_reversepath:
lp15@60809
   727
    "valid_path g \<Longrightarrow> f path_integrable_on g \<Longrightarrow> f path_integrable_on (reversepath g)"
lp15@60809
   728
  using has_path_integral_reversepath path_integrable_on_def by blast
lp15@60809
   729
lp15@60809
   730
lemma path_integrable_reversepath_eq:
lp15@60809
   731
    "valid_path g \<Longrightarrow> (f path_integrable_on (reversepath g) \<longleftrightarrow> f path_integrable_on g)"
lp15@60809
   732
  using path_integrable_reversepath valid_path_reversepath by fastforce
lp15@60809
   733
lp15@60809
   734
lemma path_integral_reversepath:
lp15@60809
   735
    "\<lbrakk>valid_path g; f path_integrable_on g\<rbrakk> \<Longrightarrow> path_integral (reversepath g) f = -(path_integral g f)"
lp15@60809
   736
  using has_path_integral_reversepath path_integrable_on_def path_integral_unique by blast
lp15@60809
   737
lp15@60809
   738
lp15@60809
   739
subsection\<open>Joining two paths together\<close>
lp15@60809
   740
lp15@60809
   741
lemma valid_path_join:
lp15@60809
   742
  assumes "valid_path g1" "valid_path g2" "pathfinish g1 = pathstart g2"
lp15@60809
   743
    shows "valid_path(g1 +++ g2)"
lp15@60809
   744
proof -
lp15@60809
   745
  have "g1 1 = g2 0"
lp15@60809
   746
    using assms by (auto simp: pathfinish_def pathstart_def)
lp15@61190
   747
  moreover have "(g1 o (\<lambda>x. 2*x)) piecewise_C1_differentiable_on {0..1/2}"
lp15@61190
   748
    apply (rule piecewise_C1_differentiable_compose)
lp15@60809
   749
    using assms
lp15@61190
   750
    apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths)
lp15@60809
   751
    apply (rule continuous_intros | simp)+
lp15@60809
   752
    apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
lp15@60809
   753
    done
lp15@61190
   754
  moreover have "(g2 o (\<lambda>x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}"
lp15@61190
   755
    apply (rule piecewise_C1_differentiable_compose)
lp15@61190
   756
    using assms unfolding valid_path_def piecewise_C1_differentiable_on_def
lp15@61190
   757
    by (auto intro!: continuous_intros finite_vimageI [where h = "(\<lambda>x. 2*x - 1)"] inj_onI
lp15@61190
   758
             simp: image_affinity_atLeastAtMost_diff continuous_on_joinpaths)
lp15@60809
   759
  ultimately show ?thesis
lp15@60809
   760
    apply (simp only: valid_path_def continuous_on_joinpaths joinpaths_def)
lp15@61190
   761
    apply (rule piecewise_C1_differentiable_cases)
lp15@60809
   762
    apply (auto simp: o_def)
lp15@60809
   763
    done
lp15@60809
   764
qed
lp15@60809
   765
lp15@61190
   766
lemma valid_path_join_D1:
lp15@61190
   767
  fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   768
  shows "valid_path (g1 +++ g2) \<Longrightarrow> valid_path g1"
lp15@61190
   769
  unfolding valid_path_def
lp15@61190
   770
  by (rule piecewise_C1_differentiable_D1)
lp15@60809
   771
lp15@61190
   772
lemma valid_path_join_D2:
lp15@61190
   773
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   774
  shows "\<lbrakk>valid_path (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> valid_path g2"
lp15@61190
   775
  unfolding valid_path_def
lp15@61190
   776
  by (rule piecewise_C1_differentiable_D2)
lp15@60809
   777
lp15@60809
   778
lemma valid_path_join_eq [simp]:
lp15@61190
   779
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   780
  shows "pathfinish g1 = pathstart g2 \<Longrightarrow> (valid_path(g1 +++ g2) \<longleftrightarrow> valid_path g1 \<and> valid_path g2)"
lp15@60809
   781
  using valid_path_join_D1 valid_path_join_D2 valid_path_join by blast
lp15@60809
   782
lp15@60809
   783
lemma has_path_integral_join:
lp15@60809
   784
  assumes "(f has_path_integral i1) g1" "(f has_path_integral i2) g2"
lp15@60809
   785
          "valid_path g1" "valid_path g2"
lp15@60809
   786
    shows "(f has_path_integral (i1 + i2)) (g1 +++ g2)"
lp15@60809
   787
proof -
lp15@60809
   788
  obtain s1 s2
lp15@60809
   789
    where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
lp15@60809
   790
      and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
lp15@60809
   791
    using assms
lp15@61190
   792
    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   793
  have 1: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
lp15@60809
   794
   and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
lp15@60809
   795
    using assms
lp15@60809
   796
    by (auto simp: has_path_integral)
lp15@60809
   797
  have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
lp15@60809
   798
   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
   799
    using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
lp15@60809
   800
          has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
lp15@60809
   801
    by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
lp15@60809
   802
  have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
lp15@60809
   803
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
lp15@60809
   804
            2 *\<^sub>R vector_derivative g1 (at (z*2))" for z
lp15@60809
   805
    apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\<bar>z - 1/2\<bar>" _ "(\<lambda>x. g1(2*x))"]])
lp15@60809
   806
    apply (simp_all add: dist_real_def abs_if split: split_if_asm)
lp15@60809
   807
    apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
lp15@60809
   808
    apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
lp15@60809
   809
    using s1
lp15@60809
   810
    apply (auto simp: algebra_simps vector_derivative_works)
lp15@60809
   811
    done
lp15@60809
   812
  have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
lp15@60809
   813
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
lp15@60809
   814
            2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z
lp15@60809
   815
    apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\<bar>z - 1/2\<bar>" _ "(\<lambda>x. g2 (2*x - 1))"]])
lp15@60809
   816
    apply (simp_all add: dist_real_def abs_if split: split_if_asm)
lp15@60809
   817
    apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
lp15@60809
   818
    apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
lp15@60809
   819
    using s2
lp15@60809
   820
    apply (auto simp: algebra_simps vector_derivative_works)
lp15@60809
   821
    done
lp15@60809
   822
  have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
lp15@60809
   823
    apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) (op*2 -` s1)"])
lp15@60809
   824
    using s1
lp15@60809
   825
    apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
lp15@60809
   826
    apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
lp15@60809
   827
    done
lp15@60809
   828
  moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
lp15@60809
   829
    apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
lp15@60809
   830
    using s2
lp15@60809
   831
    apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
lp15@60809
   832
    apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
lp15@60809
   833
    done
lp15@60809
   834
  ultimately
lp15@60809
   835
  show ?thesis
lp15@60809
   836
    apply (simp add: has_path_integral)
lp15@60809
   837
    apply (rule has_integral_combine [where c = "1/2"], auto)
lp15@60809
   838
    done
lp15@60809
   839
qed
lp15@60809
   840
lp15@60809
   841
lemma path_integrable_joinI:
lp15@60809
   842
  assumes "f path_integrable_on g1" "f path_integrable_on g2"
lp15@60809
   843
          "valid_path g1" "valid_path g2"
lp15@60809
   844
    shows "f path_integrable_on (g1 +++ g2)"
lp15@60809
   845
  using assms
lp15@60809
   846
  by (meson has_path_integral_join path_integrable_on_def)
lp15@60809
   847
lp15@60809
   848
lemma path_integrable_joinD1:
lp15@60809
   849
  assumes "f path_integrable_on (g1 +++ g2)" "valid_path g1"
lp15@60809
   850
    shows "f path_integrable_on g1"
lp15@60809
   851
proof -
lp15@60809
   852
  obtain s1
lp15@60809
   853
    where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
lp15@61190
   854
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   855
  have "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
lp15@60809
   856
    using assms
lp15@60809
   857
    apply (auto simp: path_integrable_on)
lp15@60809
   858
    apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
lp15@60809
   859
    apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
lp15@60809
   860
    done
lp15@60809
   861
  then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
lp15@61190
   862
    by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
lp15@60809
   863
  have g1: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow>
lp15@60809
   864
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
lp15@60809
   865
            2 *\<^sub>R vector_derivative g1 (at z)"  for z
lp15@60809
   866
    apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\<bar>(z-1)/2\<bar>" _ "(\<lambda>x. g1(2*x))"]])
lp15@60809
   867
    apply (simp_all add: field_simps dist_real_def abs_if split: split_if_asm)
lp15@60809
   868
    apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def])
lp15@60809
   869
    using s1
lp15@60809
   870
    apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
lp15@60809
   871
    done
lp15@60809
   872
  show ?thesis
lp15@60809
   873
    using s1
lp15@60809
   874
    apply (auto simp: path_integrable_on)
lp15@60809
   875
    apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
lp15@60809
   876
    apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
lp15@60809
   877
    done
lp15@60809
   878
qed
lp15@60809
   879
lp15@61190
   880
lemma path_integrable_joinD2:
lp15@60809
   881
  assumes "f path_integrable_on (g1 +++ g2)" "valid_path g2"
lp15@60809
   882
    shows "f path_integrable_on g2"
lp15@60809
   883
proof -
lp15@60809
   884
  obtain s2
lp15@60809
   885
    where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
lp15@61190
   886
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   887
  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
   888
    using assms
lp15@60809
   889
    apply (auto simp: path_integrable_on)
lp15@60809
   890
    apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
lp15@60809
   891
    apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
lp15@60809
   892
    apply (simp add: image_affinity_atLeastAtMost_diff)
lp15@60809
   893
    done
lp15@60809
   894
  then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2)))
lp15@60809
   895
                integrable_on {0..1}"
lp15@60809
   896
    by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
lp15@60809
   897
  have g2: "\<lbrakk>0 < z; z < 1; z \<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/2+1/2)) =
lp15@60809
   899
            2 *\<^sub>R vector_derivative g2 (at z)" for z
lp15@60809
   900
    apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\<bar>z/2\<bar>" _ "(\<lambda>x. g2(2*x-1))"]])
lp15@60809
   901
    apply (simp_all add: field_simps dist_real_def abs_if split: split_if_asm)
lp15@60809
   902
    apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def])
lp15@60809
   903
    using s2
lp15@60809
   904
    apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left
lp15@60809
   905
                      vector_derivative_works add_divide_distrib)
lp15@60809
   906
    done
lp15@60809
   907
  show ?thesis
lp15@60809
   908
    using s2
lp15@60809
   909
    apply (auto simp: path_integrable_on)
lp15@60809
   910
    apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
lp15@60809
   911
    apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
lp15@60809
   912
    done
lp15@60809
   913
qed
lp15@60809
   914
lp15@60809
   915
lemma path_integrable_join [simp]:
lp15@60809
   916
  shows
lp15@60809
   917
    "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
lp15@60809
   918
     \<Longrightarrow> f path_integrable_on (g1 +++ g2) \<longleftrightarrow> f path_integrable_on g1 \<and> f path_integrable_on g2"
lp15@60809
   919
using path_integrable_joinD1 path_integrable_joinD2 path_integrable_joinI by blast
lp15@60809
   920
lp15@60809
   921
lemma path_integral_join [simp]:
lp15@60809
   922
  shows
lp15@60809
   923
    "\<lbrakk>f path_integrable_on g1; f path_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
lp15@60809
   924
        \<Longrightarrow> path_integral (g1 +++ g2) f = path_integral g1 f + path_integral g2 f"
lp15@60809
   925
  by (simp add: has_path_integral_integral has_path_integral_join path_integral_unique)
lp15@60809
   926
lp15@60809
   927
lp15@60809
   928
subsection\<open>Shifting the starting point of a (closed) path\<close>
lp15@60809
   929
lp15@60809
   930
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
   931
  by (auto simp: shiftpath_def)
lp15@60809
   932
lp15@60809
   933
lemma valid_path_shiftpath [intro]:
lp15@60809
   934
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@60809
   935
    shows "valid_path(shiftpath a g)"
lp15@60809
   936
  using assms
lp15@60809
   937
  apply (auto simp: valid_path_def shiftpath_alt_def)
lp15@61190
   938
  apply (rule piecewise_C1_differentiable_cases)
lp15@60809
   939
  apply (auto simp: algebra_simps)
lp15@61190
   940
  apply (rule piecewise_C1_differentiable_affine [of g 1 a, simplified o_def scaleR_one])
lp15@61190
   941
  apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
lp15@61190
   942
  apply (rule piecewise_C1_differentiable_affine [of g 1 "a-1", simplified o_def scaleR_one algebra_simps])
lp15@61190
   943
  apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
lp15@60809
   944
  done
lp15@60809
   945
lp15@60809
   946
lemma has_path_integral_shiftpath:
lp15@60809
   947
  assumes f: "(f has_path_integral i) g" "valid_path g"
lp15@60809
   948
      and a: "a \<in> {0..1}"
lp15@60809
   949
    shows "(f has_path_integral i) (shiftpath a g)"
lp15@60809
   950
proof -
lp15@60809
   951
  obtain s
lp15@60809
   952
    where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
lp15@61190
   953
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   954
  have *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
lp15@60809
   955
    using assms by (auto simp: has_path_integral)
lp15@60809
   956
  then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) +
lp15@60809
   957
                    integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
lp15@60809
   958
    apply (rule has_integral_unique)
lp15@60809
   959
    apply (subst add.commute)
lp15@60809
   960
    apply (subst Integration.integral_combine)
lp15@60809
   961
    using assms * integral_unique by auto
lp15@60809
   962
  { fix x
lp15@60809
   963
    have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow>
lp15@60809
   964
         vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
lp15@60809
   965
      unfolding shiftpath_def
lp15@60809
   966
      apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "dist(1-a) x" _ "(\<lambda>x. g(a+x))"]])
lp15@60809
   967
        apply (auto simp: field_simps dist_real_def abs_if split: split_if_asm)
lp15@60809
   968
      apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one])
lp15@60809
   969
       apply (intro derivative_eq_intros | simp)+
lp15@60809
   970
      using g
lp15@60809
   971
       apply (drule_tac x="x+a" in bspec)
lp15@60809
   972
      using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
lp15@60809
   973
      done
lp15@60809
   974
  } note vd1 = this
lp15@60809
   975
  { fix x
lp15@60809
   976
    have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow>
lp15@60809
   977
          vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
lp15@60809
   978
      unfolding shiftpath_def
lp15@60809
   979
      apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "dist (1-a) x" _ "(\<lambda>x. g(a+x-1))"]])
lp15@60809
   980
        apply (auto simp: field_simps dist_real_def abs_if split: split_if_asm)
lp15@60809
   981
      apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one])
lp15@60809
   982
       apply (intro derivative_eq_intros | simp)+
lp15@60809
   983
      using g
lp15@60809
   984
      apply (drule_tac x="x+a-1" in bspec)
lp15@60809
   985
      using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
lp15@60809
   986
      done
lp15@60809
   987
  } note vd2 = this
lp15@60809
   988
  have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
lp15@60809
   989
    using * a   by (fastforce intro: integrable_subinterval_real)
lp15@60809
   990
  have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
lp15@60809
   991
    apply (rule integrable_subinterval_real)
lp15@60809
   992
    using * a by auto
lp15@60809
   993
  have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
lp15@60809
   994
        has_integral  integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {0..1 - a}"
lp15@60809
   995
    apply (rule has_integral_spike_finite
lp15@60809
   996
             [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
   997
      using s apply blast
lp15@60809
   998
     using a apply (auto simp: algebra_simps vd1)
lp15@60809
   999
     apply (force simp: shiftpath_def add.commute)
lp15@60809
  1000
    using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]]
lp15@60809
  1001
    apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute)
lp15@60809
  1002
    done
lp15@60809
  1003
  moreover
lp15@60809
  1004
  have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
lp15@60809
  1005
        has_integral  integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {1 - a..1}"
lp15@60809
  1006
    apply (rule has_integral_spike_finite
lp15@60809
  1007
             [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
  1008
      using s apply blast
lp15@60809
  1009
     using a apply (auto simp: algebra_simps vd2)
lp15@60809
  1010
     apply (force simp: shiftpath_def add.commute)
lp15@60809
  1011
    using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
lp15@60809
  1012
    apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified])
lp15@60809
  1013
    apply (simp add: algebra_simps)
lp15@60809
  1014
    done
lp15@60809
  1015
  ultimately show ?thesis
lp15@60809
  1016
    using a
lp15@60809
  1017
    by (auto simp: i has_path_integral intro: has_integral_combine [where c = "1-a"])
lp15@60809
  1018
qed
lp15@60809
  1019
lp15@60809
  1020
lemma has_path_integral_shiftpath_D:
lp15@60809
  1021
  assumes "(f has_path_integral i) (shiftpath a g)"
lp15@60809
  1022
          "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@60809
  1023
    shows "(f has_path_integral i) g"
lp15@60809
  1024
proof -
lp15@60809
  1025
  obtain s
lp15@60809
  1026
    where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
lp15@61190
  1027
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
  1028
  { fix x
lp15@60809
  1029
    assume x: "0 < x" "x < 1" "x \<notin> s"
lp15@60809
  1030
    then have gx: "g differentiable at x"
lp15@60809
  1031
      using g by auto
lp15@60809
  1032
    have "vector_derivative g (at x within {0..1}) =
lp15@60809
  1033
          vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
lp15@60809
  1034
      apply (rule vector_derivative_at_within_ivl
lp15@60809
  1035
                  [OF has_vector_derivative_transform_within_open
lp15@60809
  1036
                      [of "{0<..<1}-s" _ "(shiftpath (1 - a) (shiftpath a g))"]])
lp15@60809
  1037
      using s g assms x
lp15@60809
  1038
      apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
lp15@60809
  1039
                        vector_derivative_within_interior vector_derivative_works [symmetric])
lp15@60809
  1040
      apply (rule Derivative.differentiable_transform_at [of "min x (1-x)", OF _ _ gx])
lp15@60809
  1041
      apply (auto simp: dist_real_def shiftpath_shiftpath abs_if)
lp15@60809
  1042
      done
lp15@60809
  1043
  } note vd = this
lp15@60809
  1044
  have fi: "(f has_path_integral i) (shiftpath (1 - a) (shiftpath a g))"
lp15@60809
  1045
    using assms  by (auto intro!: has_path_integral_shiftpath)
lp15@60809
  1046
  show ?thesis
lp15@60809
  1047
    apply (simp add: has_path_integral_def)
lp15@60809
  1048
    apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _  fi [unfolded has_path_integral_def]])
lp15@60809
  1049
    using s assms vd
lp15@60809
  1050
    apply (auto simp: Path_Connected.shiftpath_shiftpath)
lp15@60809
  1051
    done
lp15@60809
  1052
qed
lp15@60809
  1053
lp15@60809
  1054
lemma has_path_integral_shiftpath_eq:
lp15@60809
  1055
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@60809
  1056
    shows "(f has_path_integral i) (shiftpath a g) \<longleftrightarrow> (f has_path_integral i) g"
lp15@60809
  1057
  using assms has_path_integral_shiftpath has_path_integral_shiftpath_D by blast
lp15@60809
  1058
lp15@60809
  1059
lemma path_integral_shiftpath:
lp15@60809
  1060
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@60809
  1061
    shows "path_integral (shiftpath a g) f = path_integral g f"
lp15@60809
  1062
   using assms by (simp add: path_integral_def has_path_integral_shiftpath_eq)
lp15@60809
  1063
lp15@60809
  1064
lp15@60809
  1065
subsection\<open>More about straight-line paths\<close>
lp15@60809
  1066
lp15@60809
  1067
lemma has_vector_derivative_linepath_within:
lp15@60809
  1068
    "(linepath a b has_vector_derivative (b - a)) (at x within s)"
lp15@60809
  1069
apply (simp add: linepath_def has_vector_derivative_def algebra_simps)
lp15@60809
  1070
apply (rule derivative_eq_intros | simp)+
lp15@60809
  1071
done
lp15@60809
  1072
lp15@60809
  1073
lemma vector_derivative_linepath_within:
lp15@60809
  1074
    "x \<in> {0..1} \<Longrightarrow> vector_derivative (linepath a b) (at x within {0..1}) = b - a"
lp15@60809
  1075
  apply (rule vector_derivative_within_closed_interval [of 0 "1::real", simplified])
lp15@60809
  1076
  apply (auto simp: has_vector_derivative_linepath_within)
lp15@60809
  1077
  done
lp15@60809
  1078
lp15@61190
  1079
lemma vector_derivative_linepath_at [simp]: "vector_derivative (linepath a b) (at x) = b - a"
lp15@60809
  1080
  by (simp add: has_vector_derivative_linepath_within vector_derivative_at)
lp15@60809
  1081
lp15@61190
  1082
lemma valid_path_linepath [iff]: "valid_path (linepath a b)"
lp15@61190
  1083
  apply (simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_on_linepath)
lp15@61190
  1084
  apply (rule_tac x="{}" in exI)
lp15@61190
  1085
  apply (simp add: differentiable_on_def differentiable_def)
lp15@61190
  1086
  using has_vector_derivative_def has_vector_derivative_linepath_within
lp15@61190
  1087
  apply (fastforce simp add: continuous_on_eq_continuous_within)
lp15@61190
  1088
  done
lp15@61190
  1089
lp15@60809
  1090
lemma has_path_integral_linepath:
lp15@60809
  1091
  shows "(f has_path_integral i) (linepath a b) \<longleftrightarrow>
lp15@60809
  1092
         ((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
lp15@60809
  1093
  by (simp add: has_path_integral vector_derivative_linepath_at)
lp15@60809
  1094
lp15@60809
  1095
lemma linepath_in_path:
lp15@60809
  1096
  shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b"
lp15@60809
  1097
  by (auto simp: segment linepath_def)
lp15@60809
  1098
lp15@60809
  1099
lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
lp15@60809
  1100
  by (auto simp: segment linepath_def)
lp15@60809
  1101
lp15@60809
  1102
lemma linepath_in_convex_hull:
lp15@60809
  1103
    fixes x::real
lp15@60809
  1104
    assumes a: "a \<in> convex hull s"
lp15@60809
  1105
        and b: "b \<in> convex hull s"
lp15@60809
  1106
        and x: "0\<le>x" "x\<le>1"
lp15@60809
  1107
       shows "linepath a b x \<in> convex hull s"
lp15@60809
  1108
  apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD])
lp15@60809
  1109
  using x
lp15@60809
  1110
  apply (auto simp: linepath_image_01 [symmetric])
lp15@60809
  1111
  done
lp15@60809
  1112
lp15@60809
  1113
lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
lp15@60809
  1114
  by (simp add: linepath_def)
lp15@60809
  1115
lp15@60809
  1116
lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
lp15@60809
  1117
  by (simp add: linepath_def)
lp15@60809
  1118
lp15@60809
  1119
lemma linepath_of_real: "(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)"
lp15@60809
  1120
  by (simp add: scaleR_conv_of_real linepath_def)
lp15@60809
  1121
lp15@60809
  1122
lemma of_real_linepath: "of_real (linepath a b x) = linepath (of_real a) (of_real b) x"
lp15@60809
  1123
  by (metis linepath_of_real mult.right_neutral of_real_def real_scaleR_def)
lp15@60809
  1124
lp15@60809
  1125
lemma has_path_integral_trivial [iff]: "(f has_path_integral 0) (linepath a a)"
lp15@60809
  1126
  by (simp add: has_path_integral_linepath)
lp15@60809
  1127
lp15@60809
  1128
lemma path_integral_trivial [simp]: "path_integral (linepath a a) f = 0"
lp15@60809
  1129
  using has_path_integral_trivial path_integral_unique by blast
lp15@60809
  1130
lp15@60809
  1131
lp15@60809
  1132
subsection\<open>Relation to subpath construction\<close>
lp15@60809
  1133
lp15@60809
  1134
lemma valid_path_subpath:
lp15@60809
  1135
  fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
lp15@60809
  1136
  assumes "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
lp15@60809
  1137
    shows "valid_path(subpath u v g)"
lp15@60809
  1138
proof (cases "v=u")
lp15@60809
  1139
  case True
lp15@60809
  1140
  then show ?thesis
lp15@61190
  1141
    unfolding valid_path_def subpath_def
lp15@61190
  1142
    by (force intro: C1_differentiable_on_const C1_differentiable_imp_piecewise)
lp15@60809
  1143
next
lp15@60809
  1144
  case False
lp15@61190
  1145
  have "(g o (\<lambda>x. ((v-u) * x + u))) piecewise_C1_differentiable_on {0..1}"
lp15@61190
  1146
    apply (rule piecewise_C1_differentiable_compose)
lp15@61190
  1147
    apply (simp add: C1_differentiable_imp_piecewise)
lp15@60809
  1148
     apply (simp add: image_affinity_atLeastAtMost)
lp15@60809
  1149
    using assms False
lp15@61190
  1150
    apply (auto simp: algebra_simps valid_path_def piecewise_C1_differentiable_on_subset)
lp15@60809
  1151
    apply (subst Int_commute)
lp15@60809
  1152
    apply (auto simp: inj_on_def algebra_simps crossproduct_eq finite_vimage_IntI)
lp15@60809
  1153
    done
lp15@60809
  1154
  then show ?thesis
lp15@60809
  1155
    by (auto simp: o_def valid_path_def subpath_def)
lp15@60809
  1156
qed
lp15@60809
  1157
lp15@60809
  1158
lemma has_path_integral_subpath_refl [iff]: "(f has_path_integral 0) (subpath u u g)"
lp15@60809
  1159
  by (simp add: has_path_integral subpath_def)
lp15@60809
  1160
lp15@60809
  1161
lemma path_integrable_subpath_refl [iff]: "f path_integrable_on (subpath u u g)"
lp15@60809
  1162
  using has_path_integral_subpath_refl path_integrable_on_def by blast
lp15@60809
  1163
lp15@60809
  1164
lemma path_integral_subpath_refl [simp]: "path_integral (subpath u u g) f = 0"
lp15@60809
  1165
  by (simp add: has_path_integral_subpath_refl path_integral_unique)
lp15@60809
  1166
lp15@60809
  1167
lemma has_path_integral_subpath:
lp15@60809
  1168
  assumes f: "f path_integrable_on g" and g: "valid_path g"
lp15@60809
  1169
      and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
lp15@60809
  1170
    shows "(f has_path_integral  integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x)))
lp15@60809
  1171
           (subpath u v g)"
lp15@60809
  1172
proof (cases "v=u")
lp15@60809
  1173
  case True
lp15@60809
  1174
  then show ?thesis
lp15@60809
  1175
    using f   by (simp add: path_integrable_on_def subpath_def has_path_integral)
lp15@60809
  1176
next
lp15@60809
  1177
  case False
lp15@60809
  1178
  obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s"
lp15@61190
  1179
    using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
lp15@60809
  1180
  have *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
lp15@60809
  1181
            has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t)))
lp15@60809
  1182
           {0..1}"
lp15@60809
  1183
    using f uv
lp15@60809
  1184
    apply (simp add: path_integrable_on subpath_def has_path_integral)
lp15@60809
  1185
    apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
lp15@60809
  1186
    apply (simp_all add: has_integral_integral)
lp15@60809
  1187
    apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
lp15@60809
  1188
    apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
lp15@60809
  1189
    apply (simp add: divide_simps False)
lp15@60809
  1190
    done
lp15@60809
  1191
  { fix x
lp15@60809
  1192
    have "x \<in> {0..1} \<Longrightarrow>
lp15@60809
  1193
           x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
lp15@60809
  1194
           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
  1195
      apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
lp15@60809
  1196
      apply (intro derivative_eq_intros | simp)+
lp15@60809
  1197
      apply (cut_tac s [of "(v - u) * x + u"])
lp15@60809
  1198
      using uv mult_left_le [of x "v-u"]
lp15@60809
  1199
      apply (auto simp:  vector_derivative_works)
lp15@60809
  1200
      done
lp15@60809
  1201
  } note vd = this
lp15@60809
  1202
  show ?thesis
lp15@60809
  1203
    apply (cut_tac has_integral_cmul [OF *, where c = "v-u"])
lp15@60809
  1204
    using fs assms
lp15@60809
  1205
    apply (simp add: False subpath_def has_path_integral)
lp15@60809
  1206
    apply (rule_tac s = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
lp15@60809
  1207
    apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real)
lp15@60809
  1208
    done
lp15@60809
  1209
qed
lp15@60809
  1210
lp15@60809
  1211
lemma path_integrable_subpath:
lp15@60809
  1212
  assumes "f path_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
lp15@60809
  1213
    shows "f path_integrable_on (subpath u v g)"
lp15@60809
  1214
  apply (cases u v rule: linorder_class.le_cases)
lp15@60809
  1215
   apply (metis path_integrable_on_def has_path_integral_subpath [OF assms])
lp15@60809
  1216
  apply (subst reversepath_subpath [symmetric])
lp15@60809
  1217
  apply (rule path_integrable_reversepath)
lp15@60809
  1218
   using assms apply (blast intro: valid_path_subpath)
lp15@60809
  1219
  apply (simp add: path_integrable_on_def)
lp15@60809
  1220
  using assms apply (blast intro: has_path_integral_subpath)
lp15@60809
  1221
  done
lp15@60809
  1222
lp15@60809
  1223
lemma has_integral_integrable_integral: "(f has_integral i) s \<longleftrightarrow> f integrable_on s \<and> integral s f = i"
lp15@60809
  1224
  by blast
lp15@60809
  1225
lp15@60809
  1226
lemma has_integral_path_integral_subpath:
lp15@60809
  1227
  assumes "f path_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
lp15@60809
  1228
    shows "(((\<lambda>x. f(g x) * vector_derivative g (at x)))
lp15@60809
  1229
            has_integral  path_integral (subpath u v g) f) {u..v}"
lp15@60809
  1230
  using assms
lp15@60809
  1231
  apply (auto simp: has_integral_integrable_integral)
lp15@60809
  1232
  apply (rule integrable_on_subcbox [where a=u and b=v and s = "{0..1}", simplified])
lp15@60809
  1233
  apply (auto simp: path_integral_unique [OF has_path_integral_subpath] path_integrable_on)
lp15@60809
  1234
  done
lp15@60809
  1235
lp15@60809
  1236
lemma path_integral_subpath_integral:
lp15@60809
  1237
  assumes "f path_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
lp15@60809
  1238
    shows "path_integral (subpath u v g) f =
lp15@60809
  1239
           integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))"
lp15@60809
  1240
  using assms has_path_integral_subpath path_integral_unique by blast
lp15@60809
  1241
lp15@60809
  1242
lemma path_integral_subpath_combine_less:
lp15@60809
  1243
  assumes "f path_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
lp15@60809
  1244
          "u<v" "v<w"
lp15@60809
  1245
    shows "path_integral (subpath u v g) f + path_integral (subpath v w g) f =
lp15@60809
  1246
           path_integral (subpath u w g) f"
lp15@60809
  1247
  using assms apply (auto simp: path_integral_subpath_integral)
lp15@60809
  1248
  apply (rule integral_combine, auto)
lp15@60809
  1249
  apply (rule integrable_on_subcbox [where a=u and b=w and s = "{0..1}", simplified])
lp15@60809
  1250
  apply (auto simp: path_integrable_on)
lp15@60809
  1251
  done
lp15@60809
  1252
lp15@60809
  1253
lemma path_integral_subpath_combine:
lp15@60809
  1254
  assumes "f path_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
lp15@60809
  1255
    shows "path_integral (subpath u v g) f + path_integral (subpath v w g) f =
lp15@60809
  1256
           path_integral (subpath u w g) f"
lp15@60809
  1257
proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w")
lp15@60809
  1258
  case True
lp15@60809
  1259
    have *: "subpath v u g = reversepath(subpath u v g) \<and>
lp15@60809
  1260
             subpath w u g = reversepath(subpath u w g) \<and>
lp15@60809
  1261
             subpath w v g = reversepath(subpath v w g)"
lp15@60809
  1262
      by (auto simp: reversepath_subpath)
lp15@60809
  1263
    have "u < v \<and> v < w \<or>
lp15@60809
  1264
          u < w \<and> w < v \<or>
lp15@60809
  1265
          v < u \<and> u < w \<or>
lp15@60809
  1266
          v < w \<and> w < u \<or>
lp15@60809
  1267
          w < u \<and> u < v \<or>
lp15@60809
  1268
          w < v \<and> v < u"
lp15@60809
  1269
      using True assms by linarith
lp15@60809
  1270
    with assms show ?thesis
lp15@60809
  1271
      using path_integral_subpath_combine_less [of f g u v w]
lp15@60809
  1272
            path_integral_subpath_combine_less [of f g u w v]
lp15@60809
  1273
            path_integral_subpath_combine_less [of f g v u w]
lp15@60809
  1274
            path_integral_subpath_combine_less [of f g v w u]
lp15@60809
  1275
            path_integral_subpath_combine_less [of f g w u v]
lp15@60809
  1276
            path_integral_subpath_combine_less [of f g w v u]
lp15@60809
  1277
      apply simp
lp15@60809
  1278
      apply (elim disjE)
lp15@60809
  1279
      apply (auto simp: * path_integral_reversepath path_integrable_subpath
lp15@60809
  1280
                   valid_path_reversepath valid_path_subpath algebra_simps)
lp15@60809
  1281
      done
lp15@60809
  1282
next
lp15@60809
  1283
  case False
lp15@60809
  1284
  then show ?thesis
lp15@60809
  1285
    apply (auto simp: path_integral_subpath_refl)
lp15@60809
  1286
    using assms
lp15@60809
  1287
    by (metis eq_neg_iff_add_eq_0 path_integrable_subpath path_integral_reversepath reversepath_subpath valid_path_subpath)
lp15@60809
  1288
qed
lp15@60809
  1289
lp15@60809
  1290
lemma path_integral_integral:
lp15@60809
  1291
  shows "path_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
lp15@60809
  1292
  by (simp add: path_integral_def integral_def has_path_integral)
lp15@60809
  1293
lp15@60809
  1294
lp15@60809
  1295
subsection\<open>Segments via convex hulls\<close>
lp15@60809
  1296
lp15@60809
  1297
lemma segments_subset_convex_hull:
lp15@60809
  1298
    "closed_segment a b \<subseteq> (convex hull {a,b,c})"
lp15@60809
  1299
    "closed_segment a c \<subseteq> (convex hull {a,b,c})"
lp15@60809
  1300
    "closed_segment b c \<subseteq> (convex hull {a,b,c})"
lp15@60809
  1301
    "closed_segment b a \<subseteq> (convex hull {a,b,c})"
lp15@60809
  1302
    "closed_segment c a \<subseteq> (convex hull {a,b,c})"
lp15@60809
  1303
    "closed_segment c b \<subseteq> (convex hull {a,b,c})"
lp15@60809
  1304
by (auto simp: segment_convex_hull linepath_of_real  elim!: rev_subsetD [OF _ hull_mono])
lp15@60809
  1305
lp15@60809
  1306
lemma midpoints_in_convex_hull:
lp15@60809
  1307
  assumes "x \<in> convex hull s" "y \<in> convex hull s"
lp15@60809
  1308
    shows "midpoint x y \<in> convex hull s"
lp15@60809
  1309
proof -
lp15@60809
  1310
  have "(1 - inverse(2)) *\<^sub>R x + inverse(2) *\<^sub>R y \<in> convex hull s"
lp15@60809
  1311
    apply (rule mem_convex)
lp15@60809
  1312
    using assms
lp15@60809
  1313
    apply (auto simp: convex_convex_hull)
lp15@60809
  1314
    done
lp15@60809
  1315
  then show ?thesis
lp15@60809
  1316
    by (simp add: midpoint_def algebra_simps)
lp15@60809
  1317
qed
lp15@60809
  1318
lp15@60809
  1319
lemma convex_hull_subset:
lp15@60809
  1320
    "s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
lp15@60809
  1321
  by (simp add: convex_convex_hull subset_hull)
lp15@60809
  1322
lp15@60809
  1323
lemma not_in_interior_convex_hull_3:
lp15@60809
  1324
  fixes a :: "complex"
lp15@60809
  1325
  shows "a \<notin> interior(convex hull {a,b,c})"
lp15@60809
  1326
        "b \<notin> interior(convex hull {a,b,c})"
lp15@60809
  1327
        "c \<notin> interior(convex hull {a,b,c})"
lp15@60809
  1328
  by (auto simp: card_insert_le_m1 not_in_interior_convex_hull)
lp15@60809
  1329
lp15@60809
  1330
lp15@60809
  1331
text\<open>Cauchy's theorem where there's a primitive\<close>
lp15@60809
  1332
lp15@60809
  1333
lemma path_integral_primitive_lemma:
lp15@60809
  1334
  fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
lp15@60809
  1335
  assumes "a \<le> b"
lp15@60809
  1336
      and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
lp15@60809
  1337
      and "g piecewise_differentiable_on {a..b}"  "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
lp15@60809
  1338
    shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b}))
lp15@60809
  1339
             has_integral (f(g b) - f(g a))) {a..b}"
lp15@60809
  1340
proof -
lp15@61190
  1341
  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
  1342
    using assms by (auto simp: piecewise_differentiable_on_def)
lp15@60809
  1343
  have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
lp15@60809
  1344
    apply (rule continuous_on_compose [OF cg, unfolded o_def])
lp15@60809
  1345
    using assms
lp15@60809
  1346
    apply (metis complex_differentiable_def complex_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
lp15@60809
  1347
    done
lp15@60809
  1348
  { fix x::real
lp15@60809
  1349
    assume a: "a < x" and b: "x < b" and xk: "x \<notin> k"
lp15@60809
  1350
    then have "g differentiable at x within {a..b}"
lp15@60809
  1351
      using k by (simp add: differentiable_at_withinI)
lp15@60809
  1352
    then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
lp15@60809
  1353
      by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
lp15@60809
  1354
    then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
lp15@60809
  1355
      by (simp add: has_vector_derivative_def scaleR_conv_of_real)
lp15@60809
  1356
    have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
lp15@60809
  1357
      using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
lp15@60809
  1358
    then have fdiff: "(f has_derivative op * (f' (g x))) (at (g x) within g ` {a..b})"
lp15@60809
  1359
      by (simp add: has_field_derivative_def)
lp15@60809
  1360
    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
  1361
      using diff_chain_within [OF gdiff fdiff]
lp15@60809
  1362
      by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
lp15@60809
  1363
  } note * = this
lp15@60809
  1364
  show ?thesis
lp15@60809
  1365
    apply (rule fundamental_theorem_of_calculus_interior_strong)
lp15@60809
  1366
    using k assms cfg *
lp15@60809
  1367
    apply (auto simp: at_within_closed_interval)
lp15@60809
  1368
    done
lp15@60809
  1369
qed
lp15@60809
  1370
lp15@60809
  1371
lemma path_integral_primitive:
lp15@60809
  1372
  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
lp15@60809
  1373
      and "valid_path g" "path_image g \<subseteq> s"
lp15@60809
  1374
    shows "(f' has_path_integral (f(pathfinish g) - f(pathstart g))) g"
lp15@60809
  1375
  using assms
lp15@60809
  1376
  apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_path_integral_def)
lp15@61190
  1377
  apply (auto intro!: piecewise_C1_imp_differentiable path_integral_primitive_lemma [of 0 1 s])
lp15@60809
  1378
  done
lp15@60809
  1379
lp15@60809
  1380
corollary Cauchy_theorem_primitive:
lp15@60809
  1381
  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
lp15@60809
  1382
      and "valid_path g"  "path_image g \<subseteq> s" "pathfinish g = pathstart g"
lp15@60809
  1383
    shows "(f' has_path_integral 0) g"
lp15@60809
  1384
  using assms
lp15@60809
  1385
  by (metis diff_self path_integral_primitive)
lp15@60809
  1386
lp15@60809
  1387
lp15@60809
  1388
text\<open>Existence of path integral for continuous function\<close>
lp15@60809
  1389
lemma path_integrable_continuous_linepath:
lp15@60809
  1390
  assumes "continuous_on (closed_segment a b) f"
lp15@60809
  1391
  shows "f path_integrable_on (linepath a b)"
lp15@60809
  1392
proof -
lp15@60809
  1393
  have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) o linepath a b)"
lp15@60809
  1394
    apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01)
lp15@60809
  1395
    apply (rule continuous_intros | simp add: assms)+
lp15@60809
  1396
    done
lp15@60809
  1397
  then show ?thesis
lp15@60809
  1398
    apply (simp add: path_integrable_on_def has_path_integral_def integrable_on_def [symmetric])
lp15@60809
  1399
    apply (rule integrable_continuous [of 0 "1::real", simplified])
lp15@60809
  1400
    apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"])
lp15@60809
  1401
    apply (auto simp: vector_derivative_linepath_within)
lp15@60809
  1402
    done
lp15@60809
  1403
qed
lp15@60809
  1404
lp15@60809
  1405
lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)"
lp15@60809
  1406
  by (rule has_derivative_imp_has_field_derivative)
lp15@60809
  1407
     (rule derivative_intros | simp)+
lp15@60809
  1408
lp15@60809
  1409
lemma path_integral_id [simp]: "path_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2"
lp15@60809
  1410
  apply (rule path_integral_unique)
lp15@60809
  1411
  using path_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"]
lp15@60809
  1412
  apply (auto simp: field_simps has_field_der_id)
lp15@60809
  1413
  done
lp15@60809
  1414
lp15@60809
  1415
lemma path_integrable_on_const [iff]: "(\<lambda>x. c) path_integrable_on (linepath a b)"
lp15@60809
  1416
  by (simp add: continuous_on_const path_integrable_continuous_linepath)
lp15@60809
  1417
lp15@60809
  1418
lemma path_integrable_on_id [iff]: "(\<lambda>x. x) path_integrable_on (linepath a b)"
lp15@60809
  1419
  by (simp add: continuous_on_id path_integrable_continuous_linepath)
lp15@60809
  1420
lp15@60809
  1421
lp15@60809
  1422
subsection\<open>Arithmetical combining theorems\<close>
lp15@60809
  1423
lp15@60809
  1424
lemma has_path_integral_neg:
lp15@60809
  1425
    "(f has_path_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_path_integral (-i)) g"
lp15@60809
  1426
  by (simp add: has_integral_neg has_path_integral_def)
lp15@60809
  1427
lp15@60809
  1428
lemma has_path_integral_add:
lp15@60809
  1429
    "\<lbrakk>(f1 has_path_integral i1) g; (f2 has_path_integral i2) g\<rbrakk>
lp15@60809
  1430
     \<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_path_integral (i1 + i2)) g"
lp15@60809
  1431
  by (simp add: has_integral_add has_path_integral_def algebra_simps)
lp15@60809
  1432
lp15@60809
  1433
lemma has_path_integral_diff:
lp15@61190
  1434
  "\<lbrakk>(f1 has_path_integral i1) g; (f2 has_path_integral i2) g\<rbrakk>
lp15@60809
  1435
         \<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_path_integral (i1 - i2)) g"
lp15@60809
  1436
  by (simp add: has_integral_sub has_path_integral_def algebra_simps)
lp15@60809
  1437
lp15@60809
  1438
lemma has_path_integral_lmul:
lp15@61190
  1439
  "(f has_path_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_path_integral (c*i)) g"
lp15@60809
  1440
apply (simp add: has_path_integral_def)
lp15@60809
  1441
apply (drule has_integral_mult_right)
lp15@60809
  1442
apply (simp add: algebra_simps)
lp15@60809
  1443
done
lp15@60809
  1444
lp15@60809
  1445
lemma has_path_integral_rmul:
lp15@61190
  1446
  "(f has_path_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_path_integral (i*c)) g"
lp15@60809
  1447
apply (drule has_path_integral_lmul)
lp15@60809
  1448
apply (simp add: mult.commute)
lp15@60809
  1449
done
lp15@60809
  1450
lp15@60809
  1451
lemma has_path_integral_div:
lp15@61190
  1452
  "(f has_path_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_path_integral (i/c)) g"
lp15@60809
  1453
  by (simp add: field_class.field_divide_inverse) (metis has_path_integral_rmul)
lp15@60809
  1454
lp15@60809
  1455
lemma has_path_integral_eq:
lp15@60809
  1456
    "\<lbrakk>(f has_path_integral y) p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> (g has_path_integral y) p"
lp15@60809
  1457
apply (simp add: path_image_def has_path_integral_def)
lp15@60809
  1458
by (metis (no_types, lifting) image_eqI has_integral_eq)
lp15@60809
  1459
lp15@60809
  1460
lemma has_path_integral_bound_linepath:
lp15@60809
  1461
  assumes "(f has_path_integral i) (linepath a b)"
lp15@60809
  1462
          "0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
lp15@60809
  1463
    shows "norm i \<le> B * norm(b - a)"
lp15@60809
  1464
proof -
lp15@60809
  1465
  { fix x::real
lp15@60809
  1466
    assume x: "0 \<le> x" "x \<le> 1"
lp15@60809
  1467
  have "norm (f (linepath a b x)) *
lp15@60809
  1468
        norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)"
lp15@60809
  1469
    by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x)
lp15@60809
  1470
  } note * = this
lp15@60809
  1471
  have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
lp15@60809
  1472
    apply (rule has_integral_bound
lp15@60809
  1473
       [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
lp15@60809
  1474
    using assms * unfolding has_path_integral_def
lp15@60809
  1475
    apply (auto simp: norm_mult)
lp15@60809
  1476
    done
lp15@60809
  1477
  then show ?thesis
lp15@60809
  1478
    by (auto simp: content_real)
lp15@60809
  1479
qed
lp15@60809
  1480
lp15@60809
  1481
(*UNUSED
lp15@60809
  1482
lemma has_path_integral_bound_linepath_strong:
lp15@60809
  1483
  fixes a :: real and f :: "complex \<Rightarrow> real"
lp15@60809
  1484
  assumes "(f has_path_integral i) (linepath a b)"
lp15@60809
  1485
          "finite k"
lp15@60809
  1486
          "0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B"
lp15@60809
  1487
    shows "norm i \<le> B*norm(b - a)"
lp15@60809
  1488
*)
lp15@60809
  1489
lp15@60809
  1490
lemma has_path_integral_const_linepath: "((\<lambda>x. c) has_path_integral c*(b - a))(linepath a b)"
lp15@60809
  1491
  unfolding has_path_integral_linepath
lp15@60809
  1492
  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
  1493
lp15@60809
  1494
lemma has_path_integral_0: "((\<lambda>x. 0) has_path_integral 0) g"
lp15@60809
  1495
  by (simp add: has_path_integral_def)
lp15@60809
  1496
lp15@60809
  1497
lemma has_path_integral_is_0:
lp15@60809
  1498
    "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_path_integral 0) g"
lp15@60809
  1499
  by (rule has_path_integral_eq [OF has_path_integral_0]) auto
lp15@60809
  1500
lp15@60809
  1501
lemma has_path_integral_setsum:
lp15@60809
  1502
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_path_integral i a) p\<rbrakk>
lp15@60809
  1503
     \<Longrightarrow> ((\<lambda>x. setsum (\<lambda>a. f a x) s) has_path_integral setsum i s) p"
lp15@60809
  1504
  by (induction s rule: finite_induct) (auto simp: has_path_integral_0 has_path_integral_add)
lp15@60809
  1505
lp15@60809
  1506
lp15@60809
  1507
subsection \<open>Operations on path integrals\<close>
lp15@60809
  1508
lp15@60809
  1509
lemma path_integral_const_linepath [simp]: "path_integral (linepath a b) (\<lambda>x. c) = c*(b - a)"
lp15@60809
  1510
  by (rule path_integral_unique [OF has_path_integral_const_linepath])
lp15@60809
  1511
lp15@60809
  1512
lemma path_integral_neg:
lp15@60809
  1513
    "f path_integrable_on g \<Longrightarrow> path_integral g (\<lambda>x. -(f x)) = -(path_integral g f)"
lp15@60809
  1514
  by (simp add: path_integral_unique has_path_integral_integral has_path_integral_neg)
lp15@60809
  1515
lp15@60809
  1516
lemma path_integral_add:
lp15@60809
  1517
    "f1 path_integrable_on g \<Longrightarrow> f2 path_integrable_on g \<Longrightarrow> path_integral g (\<lambda>x. f1 x + f2 x) =
lp15@60809
  1518
                path_integral g f1 + path_integral g f2"
lp15@60809
  1519
  by (simp add: path_integral_unique has_path_integral_integral has_path_integral_add)
lp15@60809
  1520
lp15@60809
  1521
lemma path_integral_diff:
lp15@60809
  1522
    "f1 path_integrable_on g \<Longrightarrow> f2 path_integrable_on g \<Longrightarrow> path_integral g (\<lambda>x. f1 x - f2 x) =
lp15@60809
  1523
                path_integral g f1 - path_integral g f2"
lp15@60809
  1524
  by (simp add: path_integral_unique has_path_integral_integral has_path_integral_diff)
lp15@60809
  1525
lp15@60809
  1526
lemma path_integral_lmul:
lp15@60809
  1527
  shows "f path_integrable_on g
lp15@60809
  1528
           \<Longrightarrow> path_integral g (\<lambda>x. c * f x) = c*path_integral g f"
lp15@60809
  1529
  by (simp add: path_integral_unique has_path_integral_integral has_path_integral_lmul)
lp15@60809
  1530
lp15@60809
  1531
lemma path_integral_rmul:
lp15@60809
  1532
  shows "f path_integrable_on g
lp15@60809
  1533
        \<Longrightarrow> path_integral g (\<lambda>x. f x * c) = path_integral g f * c"
lp15@60809
  1534
  by (simp add: path_integral_unique has_path_integral_integral has_path_integral_rmul)
lp15@60809
  1535
lp15@60809
  1536
lemma path_integral_div:
lp15@60809
  1537
  shows "f path_integrable_on g
lp15@60809
  1538
        \<Longrightarrow> path_integral g (\<lambda>x. f x / c) = path_integral g f / c"
lp15@60809
  1539
  by (simp add: path_integral_unique has_path_integral_integral has_path_integral_div)
lp15@60809
  1540
lp15@60809
  1541
lemma path_integral_eq:
lp15@60809
  1542
    "(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> path_integral p f = path_integral p g"
lp15@60809
  1543
  by (simp add: path_integral_def) (metis has_path_integral_eq)
lp15@60809
  1544
lp15@60809
  1545
lemma path_integral_eq_0:
lp15@60809
  1546
    "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> path_integral g f = 0"
lp15@60809
  1547
  by (simp add: has_path_integral_is_0 path_integral_unique)
lp15@60809
  1548
lp15@60809
  1549
lemma path_integral_bound_linepath:
lp15@60809
  1550
  shows
lp15@60809
  1551
    "\<lbrakk>f path_integrable_on (linepath a b);
lp15@60809
  1552
      0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
lp15@60809
  1553
     \<Longrightarrow> norm(path_integral (linepath a b) f) \<le> B*norm(b - a)"
lp15@60809
  1554
  apply (rule has_path_integral_bound_linepath [of f])
lp15@60809
  1555
  apply (auto simp: has_path_integral_integral)
lp15@60809
  1556
  done
lp15@60809
  1557
lp15@60809
  1558
lemma path_integral_0: "path_integral g (\<lambda>x. 0) = 0"
lp15@60809
  1559
  by (simp add: path_integral_unique has_path_integral_0)
lp15@60809
  1560
lp15@60809
  1561
lemma path_integral_setsum:
lp15@60809
  1562
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) path_integrable_on p\<rbrakk>
lp15@60809
  1563
     \<Longrightarrow> path_integral p (\<lambda>x. setsum (\<lambda>a. f a x) s) = setsum (\<lambda>a. path_integral p (f a)) s"
lp15@60809
  1564
  by (auto simp: path_integral_unique has_path_integral_setsum has_path_integral_integral)
lp15@60809
  1565
lp15@60809
  1566
lemma path_integrable_eq:
lp15@60809
  1567
    "\<lbrakk>f path_integrable_on p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g path_integrable_on p"
lp15@60809
  1568
  unfolding path_integrable_on_def
lp15@60809
  1569
  by (metis has_path_integral_eq)
lp15@60809
  1570
lp15@60809
  1571
lp15@60809
  1572
subsection \<open>Arithmetic theorems for path integrability\<close>
lp15@60809
  1573
lp15@60809
  1574
lemma path_integrable_neg:
lp15@60809
  1575
    "f path_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) path_integrable_on g"
lp15@60809
  1576
  using has_path_integral_neg path_integrable_on_def by blast
lp15@60809
  1577
lp15@60809
  1578
lemma path_integrable_add:
lp15@60809
  1579
    "\<lbrakk>f1 path_integrable_on g; f2 path_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x + f2 x) path_integrable_on g"
lp15@60809
  1580
  using has_path_integral_add path_integrable_on_def
lp15@60809
  1581
  by fastforce
lp15@60809
  1582
lp15@60809
  1583
lemma path_integrable_diff:
lp15@60809
  1584
    "\<lbrakk>f1 path_integrable_on g; f2 path_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x - f2 x) path_integrable_on g"
lp15@60809
  1585
  using has_path_integral_diff path_integrable_on_def
lp15@60809
  1586
  by fastforce
lp15@60809
  1587
lp15@60809
  1588
lemma path_integrable_lmul:
lp15@60809
  1589
    "f path_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) path_integrable_on g"
lp15@60809
  1590
  using has_path_integral_lmul path_integrable_on_def
lp15@60809
  1591
  by fastforce
lp15@60809
  1592
lp15@60809
  1593
lemma path_integrable_rmul:
lp15@60809
  1594
    "f path_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) path_integrable_on g"
lp15@60809
  1595
  using has_path_integral_rmul path_integrable_on_def
lp15@60809
  1596
  by fastforce
lp15@60809
  1597
lp15@60809
  1598
lemma path_integrable_div:
lp15@60809
  1599
    "f path_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) path_integrable_on g"
lp15@60809
  1600
  using has_path_integral_div path_integrable_on_def
lp15@60809
  1601
  by fastforce
lp15@60809
  1602
lp15@60809
  1603
lemma path_integrable_setsum:
lp15@60809
  1604
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) path_integrable_on p\<rbrakk>
lp15@60809
  1605
     \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) s) path_integrable_on p"
lp15@60809
  1606
   unfolding path_integrable_on_def
lp15@60809
  1607
   by (metis has_path_integral_setsum)
lp15@60809
  1608
lp15@60809
  1609
lp15@60809
  1610
subsection\<open>Reversing a path integral\<close>
lp15@60809
  1611
lp15@60809
  1612
lemma has_path_integral_reverse_linepath:
lp15@60809
  1613
    "(f has_path_integral i) (linepath a b)
lp15@60809
  1614
     \<Longrightarrow> (f has_path_integral (-i)) (linepath b a)"
lp15@60809
  1615
  using has_path_integral_reversepath valid_path_linepath by fastforce
lp15@60809
  1616
lp15@60809
  1617
lemma path_integral_reverse_linepath:
lp15@60809
  1618
    "continuous_on (closed_segment a b) f
lp15@60809
  1619
     \<Longrightarrow> path_integral (linepath a b) f = - (path_integral(linepath b a) f)"
lp15@60809
  1620
apply (rule path_integral_unique)
lp15@60809
  1621
apply (rule has_path_integral_reverse_linepath)
lp15@60809
  1622
by (simp add: closed_segment_commute path_integrable_continuous_linepath has_path_integral_integral)
lp15@60809
  1623
lp15@60809
  1624
lp15@60809
  1625
(* Splitting a path integral in a flat way.*)
lp15@60809
  1626
lp15@60809
  1627
lemma has_path_integral_split:
lp15@60809
  1628
  assumes f: "(f has_path_integral i) (linepath a c)" "(f has_path_integral j) (linepath c b)"
lp15@60809
  1629
      and k: "0 \<le> k" "k \<le> 1"
lp15@60809
  1630
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@60809
  1631
    shows "(f has_path_integral (i + j)) (linepath a b)"
lp15@60809
  1632
proof (cases "k = 0 \<or> k = 1")
lp15@60809
  1633
  case True
lp15@60809
  1634
  then show ?thesis
lp15@60809
  1635
    using assms
lp15@60809
  1636
    apply auto
lp15@60809
  1637
    apply (metis add.left_neutral has_path_integral_trivial has_path_integral_unique)
lp15@60809
  1638
    apply (metis add.right_neutral has_path_integral_trivial has_path_integral_unique)
lp15@60809
  1639
    done
lp15@60809
  1640
next
lp15@60809
  1641
  case False
lp15@60809
  1642
  then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1"
lp15@60809
  1643
    using assms apply auto
lp15@60809
  1644
    using of_real_eq_iff by fastforce
lp15@60809
  1645
  have c': "c = k *\<^sub>R (b - a) + a"
lp15@60809
  1646
    by (metis diff_add_cancel c)
lp15@60809
  1647
  have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)"
lp15@60809
  1648
    by (simp add: algebra_simps c')
lp15@60809
  1649
  { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
lp15@60809
  1650
    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
  1651
      using False
lp15@60809
  1652
      apply (simp add: c' algebra_simps)
lp15@60809
  1653
      apply (simp add: real_vector.scale_left_distrib [symmetric] divide_simps)
lp15@60809
  1654
      done
lp15@60809
  1655
    have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
lp15@60809
  1656
      using * k
lp15@60809
  1657
      apply -
lp15@60809
  1658
      apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse k" "0", simplified])
lp15@60809
  1659
      apply (simp_all add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c)
lp15@60809
  1660
      apply (drule Integration.has_integral_cmul [where c = "inverse k"])
lp15@60809
  1661
      apply (simp add: Integration.has_integral_cmul)
lp15@60809
  1662
      done
lp15@60809
  1663
  } note fi = this
lp15@60809
  1664
  { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}"
lp15@60809
  1665
    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
  1666
      using k
lp15@60809
  1667
      apply (simp add: c' field_simps)
lp15@60809
  1668
      apply (simp add: scaleR_conv_of_real divide_simps)
lp15@60809
  1669
      apply (simp add: field_simps)
lp15@60809
  1670
      done
lp15@60809
  1671
    have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
lp15@60809
  1672
      using * k
lp15@60809
  1673
      apply -
lp15@60809
  1674
      apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse(1 - k)" "-(k/(1 - k))", simplified])
lp15@60809
  1675
      apply (simp_all add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
lp15@60809
  1676
      apply (drule Integration.has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
lp15@60809
  1677
      apply (simp add: Integration.has_integral_cmul)
lp15@60809
  1678
      done
lp15@60809
  1679
  } note fj = this
lp15@60809
  1680
  show ?thesis
lp15@60809
  1681
    using f k
lp15@60809
  1682
    apply (simp add: has_path_integral_linepath)
lp15@60809
  1683
    apply (simp add: linepath_def)
lp15@60809
  1684
    apply (rule has_integral_combine [OF _ _ fi fj], simp_all)
lp15@60809
  1685
    done
lp15@60809
  1686
qed
lp15@60809
  1687
lp15@60809
  1688
lemma continuous_on_closed_segment_transform:
lp15@60809
  1689
  assumes f: "continuous_on (closed_segment a b) f"
lp15@60809
  1690
      and k: "0 \<le> k" "k \<le> 1"
lp15@60809
  1691
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@60809
  1692
    shows "continuous_on (closed_segment a c) f"
lp15@60809
  1693
proof -
lp15@60809
  1694
  have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
lp15@60809
  1695
    using c by (simp add: algebra_simps)
lp15@60809
  1696
  show "continuous_on (closed_segment a c) f"
lp15@60809
  1697
    apply (rule continuous_on_subset [OF f])
lp15@60809
  1698
    apply (simp add: segment_convex_hull)
lp15@60809
  1699
    apply (rule convex_hull_subset)
lp15@60809
  1700
    using assms
lp15@60809
  1701
    apply (auto simp: hull_inc c' Convex.mem_convex)
lp15@60809
  1702
    done
lp15@60809
  1703
qed
lp15@60809
  1704
lp15@60809
  1705
lemma path_integral_split:
lp15@60809
  1706
  assumes f: "continuous_on (closed_segment a b) f"
lp15@60809
  1707
      and k: "0 \<le> k" "k \<le> 1"
lp15@60809
  1708
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@60809
  1709
    shows "path_integral(linepath a b) f = path_integral(linepath a c) f + path_integral(linepath c b) f"
lp15@60809
  1710
proof -
lp15@60809
  1711
  have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
lp15@60809
  1712
    using c by (simp add: algebra_simps)
lp15@60809
  1713
  have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
lp15@60809
  1714
    apply (rule_tac [!] continuous_on_subset [OF f])
lp15@60809
  1715
    apply (simp_all add: segment_convex_hull)
lp15@60809
  1716
    apply (rule_tac [!] convex_hull_subset)
lp15@60809
  1717
    using assms
lp15@60809
  1718
    apply (auto simp: hull_inc c' Convex.mem_convex)
lp15@60809
  1719
    done
lp15@60809
  1720
  show ?thesis
lp15@60809
  1721
    apply (rule path_integral_unique)
lp15@60809
  1722
    apply (rule has_path_integral_split [OF has_path_integral_integral has_path_integral_integral k c])
lp15@60809
  1723
    apply (rule path_integrable_continuous_linepath *)+
lp15@60809
  1724
    done
lp15@60809
  1725
qed
lp15@60809
  1726
lp15@60809
  1727
lemma path_integral_split_linepath:
lp15@60809
  1728
  assumes f: "continuous_on (closed_segment a b) f"
lp15@60809
  1729
      and c: "c \<in> closed_segment a b"
lp15@60809
  1730
    shows "path_integral(linepath a b) f = path_integral(linepath a c) f + path_integral(linepath c b) f"
lp15@60809
  1731
  using c
lp15@60809
  1732
  by (auto simp: closed_segment_def algebra_simps intro!: path_integral_split [OF f])
lp15@60809
  1733
lp15@60809
  1734
(* The special case of midpoints used in the main quadrisection.*)
lp15@60809
  1735
lp15@60809
  1736
lemma has_path_integral_midpoint:
lp15@60809
  1737
  assumes "(f has_path_integral i) (linepath a (midpoint a b))"
lp15@60809
  1738
          "(f has_path_integral j) (linepath (midpoint a b) b)"
lp15@60809
  1739
    shows "(f has_path_integral (i + j)) (linepath a b)"
lp15@60809
  1740
  apply (rule has_path_integral_split [where c = "midpoint a b" and k = "1/2"])
lp15@60809
  1741
  using assms
lp15@60809
  1742
  apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
lp15@60809
  1743
  done
lp15@60809
  1744
lp15@60809
  1745
lemma path_integral_midpoint:
lp15@60809
  1746
   "continuous_on (closed_segment a b) f
lp15@60809
  1747
    \<Longrightarrow> path_integral (linepath a b) f =
lp15@60809
  1748
        path_integral (linepath a (midpoint a b)) f + path_integral (linepath (midpoint a b) b) f"
lp15@60809
  1749
  apply (rule path_integral_split [where c = "midpoint a b" and k = "1/2"])
lp15@60809
  1750
  using assms
lp15@60809
  1751
  apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
lp15@60809
  1752
  done
lp15@60809
  1753
lp15@60809
  1754
lp15@60809
  1755
text\<open>A couple of special case lemmas that are useful below\<close>
lp15@60809
  1756
lp15@60809
  1757
lemma triangle_linear_has_chain_integral:
lp15@60809
  1758
    "((\<lambda>x. m*x + d) has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  1759
  apply (rule Cauchy_theorem_primitive [of UNIV "\<lambda>x. m/2 * x^2 + d*x"])
lp15@60809
  1760
  apply (auto intro!: derivative_eq_intros)
lp15@60809
  1761
  done
lp15@60809
  1762
lp15@60809
  1763
lemma has_chain_integral_chain_integral3:
lp15@60809
  1764
     "(f has_path_integral i) (linepath a b +++ linepath b c +++ linepath c d)
lp15@60809
  1765
      \<Longrightarrow> path_integral (linepath a b) f + path_integral (linepath b c) f + path_integral (linepath c d) f = i"
lp15@60809
  1766
  apply (subst path_integral_unique [symmetric], assumption)
lp15@60809
  1767
  apply (drule has_path_integral_integrable)
lp15@60809
  1768
  apply (simp add: valid_path_join)
lp15@60809
  1769
  done
lp15@60809
  1770
lp15@60809
  1771
subsection\<open>Reversing the order in a double path integral\<close>
lp15@60809
  1772
lp15@60809
  1773
text\<open>The condition is stronger than needed but it's often true in typical situations\<close>
lp15@60809
  1774
lp15@60809
  1775
lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b"
lp15@60809
  1776
  by (auto simp: cbox_Pair_eq)
lp15@60809
  1777
lp15@60809
  1778
lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d"
lp15@60809
  1779
  by (auto simp: cbox_Pair_eq)
lp15@60809
  1780
lp15@60809
  1781
lemma path_integral_swap:
lp15@60809
  1782
  assumes fcon:  "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)"
lp15@60809
  1783
      and vp:    "valid_path g" "valid_path h"
lp15@60809
  1784
      and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
lp15@60809
  1785
      and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
lp15@60809
  1786
  shows "path_integral g (\<lambda>w. path_integral h (f w)) =
lp15@60809
  1787
         path_integral h (\<lambda>z. path_integral g (\<lambda>w. f w z))"
lp15@60809
  1788
proof -
lp15@60809
  1789
  have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h"
lp15@61190
  1790
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
lp15@60809
  1791
  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
  1792
    by (rule ext) simp
lp15@60809
  1793
  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
  1794
    by (rule ext) simp
lp15@60809
  1795
  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
  1796
    by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
lp15@60809
  1797
  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
  1798
    by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
lp15@60809
  1799
  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
  1800
    apply (rule integrable_continuous_real)
lp15@60809
  1801
    apply (rule continuous_on_mult [OF _ gvcon])
lp15@60809
  1802
    apply (subst fgh2)
lp15@60809
  1803
    apply (rule fcon_im2 gcon continuous_intros | simp)+
lp15@60809
  1804
    done
lp15@60809
  1805
  have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) o fst"
lp15@60809
  1806
    by auto
lp15@60809
  1807
  then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))"
lp15@60809
  1808
    apply (rule ssubst)
lp15@60809
  1809
    apply (rule continuous_intros | simp add: gvcon)+
lp15@60809
  1810
    done
lp15@60809
  1811
  have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) o snd"
lp15@60809
  1812
    by auto
lp15@60809
  1813
  then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
lp15@60809
  1814
    apply (rule ssubst)
lp15@60809
  1815
    apply (rule continuous_intros | simp add: hvcon)+
lp15@60809
  1816
    done
lp15@60809
  1817
  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
  1818
    by auto
lp15@60809
  1819
  then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
lp15@60809
  1820
    apply (rule ssubst)
lp15@60809
  1821
    apply (rule gcon hcon continuous_intros | simp)+
lp15@60809
  1822
    apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
lp15@60809
  1823
    done
lp15@60809
  1824
  have "integral {0..1} (\<lambda>x. path_integral h (f (g x)) * vector_derivative g (at x)) =
lp15@60809
  1825
        integral {0..1} (\<lambda>x. path_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))"
lp15@60809
  1826
    apply (rule integral_cong [OF path_integral_rmul [symmetric]])
lp15@60809
  1827
    apply (clarsimp simp: path_integrable_on)
lp15@60809
  1828
    apply (rule integrable_continuous_real)
lp15@60809
  1829
    apply (rule continuous_on_mult [OF _ hvcon])
lp15@60809
  1830
    apply (subst fgh1)
lp15@60809
  1831
    apply (rule fcon_im1 hcon continuous_intros | simp)+
lp15@60809
  1832
    done
lp15@60809
  1833
  also have "... = integral {0..1}
lp15@60809
  1834
                     (\<lambda>y. path_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))"
lp15@60809
  1835
    apply (simp add: path_integral_integral)
lp15@60809
  1836
    apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
lp15@60809
  1837
    apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
lp15@60809
  1838
    apply (simp add: algebra_simps)
lp15@60809
  1839
    done
lp15@60809
  1840
  also have "... = path_integral h (\<lambda>z. path_integral g (\<lambda>w. f w z))"
lp15@60809
  1841
    apply (simp add: path_integral_integral)
lp15@60809
  1842
    apply (rule integral_cong)
lp15@60809
  1843
    apply (subst integral_mult_left [symmetric])
lp15@60809
  1844
    apply (blast intro: vdg)
lp15@60809
  1845
    apply (simp add: algebra_simps)
lp15@60809
  1846
    done
lp15@60809
  1847
  finally show ?thesis
lp15@60809
  1848
    by (simp add: path_integral_integral)
lp15@60809
  1849
qed
lp15@60809
  1850
lp15@60809
  1851
lp15@60809
  1852
subsection\<open>The key quadrisection step\<close>
lp15@60809
  1853
lp15@60809
  1854
lemma norm_sum_half:
lp15@60809
  1855
  assumes "norm(a + b) >= e"
lp15@60809
  1856
    shows "norm a >= e/2 \<or> norm b >= e/2"
lp15@60809
  1857
proof -
lp15@60809
  1858
  have "e \<le> norm (- a - b)"
lp15@60809
  1859
    by (simp add: add.commute assms norm_minus_commute)
lp15@60809
  1860
  thus ?thesis
lp15@60809
  1861
    using norm_triangle_ineq4 order_trans by fastforce
lp15@60809
  1862
qed
lp15@60809
  1863
lp15@60809
  1864
lemma norm_sum_lemma:
lp15@60809
  1865
  assumes "e \<le> norm (a + b + c + d)"
lp15@60809
  1866
    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
  1867
proof -
lp15@60809
  1868
  have "e \<le> norm ((a + b) + (c + d))" using assms
lp15@60809
  1869
    by (simp add: algebra_simps)
lp15@60809
  1870
  then show ?thesis
lp15@60809
  1871
    by (auto dest!: norm_sum_half)
lp15@60809
  1872
qed
lp15@60809
  1873
lp15@60809
  1874
lemma Cauchy_theorem_quadrisection:
lp15@60809
  1875
  assumes f: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  1876
      and dist: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
lp15@60809
  1877
      and e: "e * K^2 \<le>
lp15@60809
  1878
              norm (path_integral(linepath a b) f + path_integral(linepath b c) f + path_integral(linepath c a) f)"
lp15@60809
  1879
  shows "\<exists>a' b' c'.
lp15@60809
  1880
           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
  1881
           dist a' b' \<le> K/2  \<and>  dist b' c' \<le> K/2  \<and>  dist c' a' \<le> K/2  \<and>
lp15@60809
  1882
           e * (K/2)^2 \<le> norm(path_integral(linepath a' b') f + path_integral(linepath b' c') f + path_integral(linepath c' a') f)"
lp15@60809
  1883
proof -
lp15@60809
  1884
  note divide_le_eq_numeral1 [simp del]
lp15@60809
  1885
  def a' \<equiv> "midpoint b c"
lp15@60809
  1886
  def b' \<equiv> "midpoint c a"
lp15@60809
  1887
  def c' \<equiv> "midpoint a b"
lp15@60809
  1888
  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
  1889
    using f continuous_on_subset segments_subset_convex_hull by metis+
lp15@60809
  1890
  have fcont': "continuous_on (closed_segment c' b') f"
lp15@60809
  1891
               "continuous_on (closed_segment a' c') f"
lp15@60809
  1892
               "continuous_on (closed_segment b' a') f"
lp15@60809
  1893
    unfolding a'_def b'_def c'_def
lp15@60809
  1894
    apply (rule continuous_on_subset [OF f],
lp15@60809
  1895
           metis midpoints_in_convex_hull convex_hull_subset hull_subset insert_subset segment_convex_hull)+
lp15@60809
  1896
    done
lp15@60809
  1897
  let ?pathint = "\<lambda>x y. path_integral(linepath x y) f"
lp15@60809
  1898
  have *: "?pathint a b + ?pathint b c + ?pathint c a =
lp15@60809
  1899
          (?pathint a c' + ?pathint c' b' + ?pathint b' a) +
lp15@60809
  1900
          (?pathint a' c' + ?pathint c' b + ?pathint b a') +
lp15@60809
  1901
          (?pathint a' c + ?pathint c b' + ?pathint b' a') +
lp15@60809
  1902
          (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
lp15@60809
  1903
    apply (simp add: fcont' path_integral_reverse_linepath)
lp15@60809
  1904
    apply (simp add: a'_def b'_def c'_def path_integral_midpoint fabc)
lp15@60809
  1905
    done
lp15@60809
  1906
  have [simp]: "\<And>x y. cmod (x * 2 - y * 2) = cmod (x - y) * 2"
lp15@60809
  1907
    by (metis left_diff_distrib mult.commute norm_mult_numeral1)
lp15@60809
  1908
  have [simp]: "\<And>x y. cmod (x - y) = cmod (y - x)"
lp15@60809
  1909
    by (simp add: norm_minus_commute)
lp15@60809
  1910
  consider "e * K\<^sup>2 / 4 \<le> cmod (?pathint a c' + ?pathint c' b' + ?pathint b' a)" |
lp15@60809
  1911
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c' + ?pathint c' b + ?pathint b a')" |
lp15@60809
  1912
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c + ?pathint c b' + ?pathint b' a')" |
lp15@60809
  1913
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
lp15@60809
  1914
    using assms
lp15@60809
  1915
    apply (simp only: *)
lp15@60809
  1916
    apply (blast intro: that dest!: norm_sum_lemma)
lp15@60809
  1917
    done
lp15@60809
  1918
  then show ?thesis
lp15@60809
  1919
  proof cases
lp15@60809
  1920
    case 1 then show ?thesis
lp15@60809
  1921
      apply (rule_tac x=a in exI)
lp15@60809
  1922
      apply (rule exI [where x=c'])
lp15@60809
  1923
      apply (rule exI [where x=b'])
lp15@60809
  1924
      using assms
lp15@60809
  1925
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  1926
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  1927
      done
lp15@60809
  1928
  next
lp15@60809
  1929
    case 2 then show ?thesis
lp15@60809
  1930
      apply (rule_tac x=a' in exI)
lp15@60809
  1931
      apply (rule exI [where x=c'])
lp15@60809
  1932
      apply (rule exI [where x=b])
lp15@60809
  1933
      using assms
lp15@60809
  1934
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  1935
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  1936
      done
lp15@60809
  1937
  next
lp15@60809
  1938
    case 3 then show ?thesis
lp15@60809
  1939
      apply (rule_tac x=a' in exI)
lp15@60809
  1940
      apply (rule exI [where x=c])
lp15@60809
  1941
      apply (rule exI [where x=b'])
lp15@60809
  1942
      using assms
lp15@60809
  1943
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  1944
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  1945
      done
lp15@60809
  1946
  next
lp15@60809
  1947
    case 4 then show ?thesis
lp15@60809
  1948
      apply (rule_tac x=a' in exI)
lp15@60809
  1949
      apply (rule exI [where x=b'])
lp15@60809
  1950
      apply (rule exI [where x=c'])
lp15@60809
  1951
      using assms
lp15@60809
  1952
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  1953
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  1954
      done
lp15@60809
  1955
  qed
lp15@60809
  1956
qed
lp15@60809
  1957
lp15@60809
  1958
subsection\<open>Cauchy's theorem for triangles\<close>
lp15@60809
  1959
lp15@60809
  1960
lemma triangle_points_closer:
lp15@60809
  1961
  fixes a::complex
lp15@60809
  1962
  shows "\<lbrakk>x \<in> convex hull {a,b,c};  y \<in> convex hull {a,b,c}\<rbrakk>
lp15@60809
  1963
         \<Longrightarrow> norm(x - y) \<le> norm(a - b) \<or>
lp15@60809
  1964
             norm(x - y) \<le> norm(b - c) \<or>
lp15@60809
  1965
             norm(x - y) \<le> norm(c - a)"
lp15@60809
  1966
  using simplex_extremal_le [of "{a,b,c}"]
lp15@60809
  1967
  by (auto simp: norm_minus_commute)
lp15@60809
  1968
lp15@60809
  1969
lemma holomorphic_point_small_triangle:
lp15@60809
  1970
  assumes x: "x \<in> s"
lp15@60809
  1971
      and f: "continuous_on s f"
lp15@60809
  1972
      and cd: "f complex_differentiable (at x within s)"
lp15@60809
  1973
      and e: "0 < e"
lp15@60809
  1974
    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
  1975
              x \<in> convex hull {a,b,c} \<and> convex hull {a,b,c} \<subseteq> s
lp15@60809
  1976
              \<longrightarrow> norm(path_integral(linepath a b) f + path_integral(linepath b c) f +
lp15@60809
  1977
                       path_integral(linepath c a) f)
lp15@60809
  1978
                  \<le> e*(dist a b + dist b c + dist c a)^2"
lp15@60809
  1979
           (is "\<exists>k>0. \<forall>a b c. _ \<longrightarrow> ?normle a b c")
lp15@60809
  1980
proof -
lp15@60809
  1981
  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
  1982
                     \<Longrightarrow> a \<le> e*(x + y + z)^2"
lp15@60809
  1983
    by (simp add: algebra_simps power2_eq_square)
lp15@60809
  1984
  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
  1985
             for x::real and a b c
lp15@60809
  1986
    by linarith
lp15@60809
  1987
  have fabc: "f path_integrable_on linepath a b" "f path_integrable_on linepath b c" "f path_integrable_on linepath c a"
lp15@60809
  1988
              if "convex hull {a, b, c} \<subseteq> s" for a b c
lp15@60809
  1989
    using segments_subset_convex_hull that
lp15@60809
  1990
    by (metis continuous_on_subset f path_integrable_continuous_linepath)+
lp15@60809
  1991
  note path_bound = has_path_integral_bound_linepath [simplified norm_minus_commute, OF has_path_integral_integral]
lp15@60809
  1992
  { fix f' a b c d
lp15@60809
  1993
    assume d: "0 < d"
lp15@60809
  1994
       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
  1995
       and le: "cmod (a - b) \<le> d" "cmod (b - c) \<le> d" "cmod (c - a) \<le> d"
lp15@60809
  1996
       and xc: "x \<in> convex hull {a, b, c}"
lp15@60809
  1997
       and s: "convex hull {a, b, c} \<subseteq> s"
lp15@60809
  1998
    have pa: "path_integral (linepath a b) f + path_integral (linepath b c) f + path_integral (linepath c a) f =
lp15@60809
  1999
              path_integral (linepath a b) (\<lambda>y. f y - f x - f'*(y - x)) +
lp15@60809
  2000
              path_integral (linepath b c) (\<lambda>y. f y - f x - f'*(y - x)) +
lp15@60809
  2001
              path_integral (linepath c a) (\<lambda>y. f y - f x - f'*(y - x))"
lp15@60809
  2002
      apply (simp add: path_integral_diff path_integral_lmul path_integrable_lmul path_integrable_diff fabc [OF s])
lp15@60809
  2003
      apply (simp add: field_simps)
lp15@60809
  2004
      done
lp15@60809
  2005
    { fix y
lp15@60809
  2006
      assume yc: "y \<in> convex hull {a,b,c}"
lp15@60809
  2007
      have "cmod (f y - f x - f' * (y - x)) \<le> e*norm(y - x)"
lp15@60809
  2008
        apply (rule f')
lp15@60809
  2009
        apply (metis triangle_points_closer [OF xc yc] le norm_minus_commute order_trans)
lp15@60809
  2010
        using s yc by blast
lp15@60809
  2011
      also have "... \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))"
lp15@60809
  2012
        by (simp add: yc e xc disj_le [OF triangle_points_closer])
lp15@60809
  2013
      finally have "cmod (f y - f x - f' * (y - x)) \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" .
lp15@60809
  2014
    } note cm_le = this
lp15@60809
  2015
    have "?normle a b c"
lp15@60809
  2016
      apply (simp add: dist_norm pa)
lp15@60809
  2017
      apply (rule le_of_3)
lp15@60809
  2018
      using f' xc s e
lp15@60809
  2019
      apply simp_all
lp15@60809
  2020
      apply (intro norm_triangle_le add_mono path_bound)
lp15@60809
  2021
      apply (simp_all add: path_integral_diff path_integral_lmul path_integrable_lmul path_integrable_diff fabc)
lp15@60809
  2022
      apply (blast intro: cm_le elim: dest: segments_subset_convex_hull [THEN subsetD])+
lp15@60809
  2023
      done
lp15@60809
  2024
  } note * = this
lp15@60809
  2025
  show ?thesis
lp15@60809
  2026
    using cd e
lp15@60809
  2027
    apply (simp add: complex_differentiable_def has_field_derivative_def has_derivative_within_alt approachable_lt_le2 Ball_def)
lp15@60809
  2028
    apply (clarify dest!: spec mp)
lp15@60809
  2029
    using *
lp15@60809
  2030
    apply (simp add: dist_norm, blast)
lp15@60809
  2031
    done
lp15@60809
  2032
qed
lp15@60809
  2033
lp15@60809
  2034
lp15@60809
  2035
(* Hence the most basic theorem for a triangle.*)
lp15@60809
  2036
locale Chain =
lp15@60809
  2037
  fixes x0 At Follows
lp15@60809
  2038
  assumes At0: "At x0 0"
lp15@60809
  2039
      and AtSuc: "\<And>x n. At x n \<Longrightarrow> \<exists>x'. At x' (Suc n) \<and> Follows x' x"
lp15@60809
  2040
begin
lp15@60809
  2041
  primrec f where
lp15@60809
  2042
    "f 0 = x0"
lp15@60809
  2043
  | "f (Suc n) = (SOME x. At x (Suc n) \<and> Follows x (f n))"
lp15@60809
  2044
lp15@60809
  2045
  lemma At: "At (f n) n"
lp15@60809
  2046
  proof (induct n)
lp15@60809
  2047
    case 0 show ?case
lp15@60809
  2048
      by (simp add: At0)
lp15@60809
  2049
  next
lp15@60809
  2050
    case (Suc n) show ?case
lp15@60809
  2051
      by (metis (no_types, lifting) AtSuc [OF Suc] f.simps(2) someI_ex)
lp15@60809
  2052
  qed
lp15@60809
  2053
lp15@60809
  2054
  lemma Follows: "Follows (f(Suc n)) (f n)"
lp15@60809
  2055
    by (metis (no_types, lifting) AtSuc [OF At [of n]] f.simps(2) someI_ex)
lp15@60809
  2056
lp15@60809
  2057
  declare f.simps(2) [simp del]
lp15@60809
  2058
end
lp15@60809
  2059
lp15@60809
  2060
lemma Chain3:
lp15@60809
  2061
  assumes At0: "At x0 y0 z0 0"
lp15@60809
  2062
      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
  2063
  obtains f g h where
lp15@60809
  2064
    "f 0 = x0" "g 0 = y0" "h 0 = z0"
lp15@60809
  2065
                      "\<And>n. At (f n) (g n) (h n) n"
lp15@60809
  2066
                       "\<And>n. Follows (f(Suc n)) (g(Suc n)) (h(Suc n)) (f n) (g n) (h n)"
lp15@60809
  2067
proof -
lp15@60809
  2068
  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
  2069
    apply unfold_locales
lp15@60809
  2070
    using At0 AtSuc by auto
lp15@60809
  2071
  show ?thesis
lp15@60809
  2072
  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
  2073
  apply simp_all
lp15@60809
  2074
  using three.At three.Follows
lp15@60809
  2075
  apply (simp_all add: split_beta')
lp15@60809
  2076
  done
lp15@60809
  2077
qed
lp15@60809
  2078
lp15@60809
  2079
lemma Cauchy_theorem_triangle:
lp15@60809
  2080
  assumes "f holomorphic_on (convex hull {a,b,c})"
lp15@60809
  2081
    shows "(f has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2082
proof -
lp15@60809
  2083
  have contf: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  2084
    by (metis assms holomorphic_on_imp_continuous_on)
lp15@60809
  2085
  let ?pathint = "\<lambda>x y. path_integral(linepath x y) f"
lp15@60809
  2086
  { fix y::complex
lp15@60809
  2087
    assume fy: "(f has_path_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2088
       and ynz: "y \<noteq> 0"
lp15@60809
  2089
    def K \<equiv> "1 + max (dist a b) (max (dist b c) (dist c a))"
lp15@60809
  2090
    def e \<equiv> "norm y / K^2"
lp15@60809
  2091
    have K1: "K \<ge> 1"  by (simp add: K_def max.coboundedI1)
lp15@60809
  2092
    then have K: "K > 0" by linarith
lp15@60809
  2093
    have [iff]: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
lp15@60809
  2094
      by (simp_all add: K_def)
lp15@60809
  2095
    have e: "e > 0"
lp15@60809
  2096
      unfolding e_def using ynz K1 by simp
lp15@60809
  2097
    def At \<equiv> "\<lambda>x y z n. convex hull {x,y,z} \<subseteq> convex hull {a,b,c} \<and>
lp15@60809
  2098
                         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
  2099
                         norm(?pathint x y + ?pathint y z + ?pathint z x) \<ge> e*(K/2^n)^2"
lp15@60809
  2100
    have At0: "At a b c 0"
lp15@60809
  2101
      using fy
lp15@60809
  2102
      by (simp add: At_def e_def has_chain_integral_chain_integral3)
lp15@60809
  2103
    { fix x y z n
lp15@60809
  2104
      assume At: "At x y z n"
lp15@60809
  2105
      then have contf': "continuous_on (convex hull {x,y,z}) f"
lp15@60809
  2106
        using contf At_def continuous_on_subset by blast
lp15@60809
  2107
      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
  2108
        using At
lp15@60809
  2109
        apply (simp add: At_def)
lp15@60809
  2110
        using  Cauchy_theorem_quadrisection [OF contf', of "K/2^n" e]
lp15@60809
  2111
        apply clarsimp
lp15@60809
  2112
        apply (rule_tac x="a'" in exI)
lp15@60809
  2113
        apply (rule_tac x="b'" in exI)
lp15@60809
  2114
        apply (rule_tac x="c'" in exI)
lp15@60809
  2115
        apply (simp add: algebra_simps)
lp15@60809
  2116
        apply (meson convex_hull_subset empty_subsetI insert_subset subsetCE)
lp15@60809
  2117
        done
lp15@60809
  2118
    } note AtSuc = this
lp15@60809
  2119
    obtain fa fb fc
lp15@60809
  2120
      where f0 [simp]: "fa 0 = a" "fb 0 = b" "fc 0 = c"
lp15@60809
  2121
        and cosb: "\<And>n. convex hull {fa n, fb n, fc n} \<subseteq> convex hull {a,b,c}"
lp15@60809
  2122
        and dist: "\<And>n. dist (fa n) (fb n) \<le> K/2^n"
lp15@60809
  2123
                  "\<And>n. dist (fb n) (fc n) \<le> K/2^n"
lp15@60809
  2124
                  "\<And>n. dist (fc n) (fa n) \<le> K/2^n"
lp15@60809
  2125
        and no: "\<And>n. norm(?pathint (fa n) (fb n) +
lp15@60809
  2126
                           ?pathint (fb n) (fc n) +
lp15@60809
  2127
                           ?pathint (fc n) (fa n)) \<ge> e * (K/2^n)^2"
lp15@60809
  2128
        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
  2129
      apply (rule Chain3 [of At, OF At0 AtSuc])
lp15@60809
  2130
      apply (auto simp: At_def)
lp15@60809
  2131
      done
lp15@60809
  2132
    have "\<exists>x. \<forall>n. x \<in> convex hull {fa n, fb n, fc n}"
lp15@60809
  2133
      apply (rule bounded_closed_nest)
lp15@60809
  2134
      apply (simp_all add: compact_imp_closed finite_imp_compact_convex_hull finite_imp_bounded_convex_hull)
lp15@60809
  2135
      apply (rule allI)
lp15@60809
  2136
      apply (rule transitive_stepwise_le)
lp15@60809
  2137
      apply (auto simp: conv_le)
lp15@60809
  2138
      done
lp15@60809
  2139
    then obtain x where x: "\<And>n. x \<in> convex hull {fa n, fb n, fc n}" by auto
lp15@60809
  2140
    then have xin: "x \<in> convex hull {a,b,c}"
lp15@60809
  2141
      using assms f0 by blast
lp15@60809
  2142
    then have fx: "f complex_differentiable at x within (convex hull {a,b,c})"
lp15@60809
  2143
      using assms holomorphic_on_def by blast
lp15@60809
  2144
    { fix k n
lp15@60809
  2145
      assume k: "0 < k"
lp15@60809
  2146
         and le:
lp15@60809
  2147
            "\<And>x' y' z'.
lp15@60809
  2148
               \<lbrakk>dist x' y' \<le> k; dist y' z' \<le> k; dist z' x' \<le> k;
lp15@60809
  2149
                x \<in> convex hull {x',y',z'};
lp15@60809
  2150
                convex hull {x',y',z'} \<subseteq> convex hull {a,b,c}\<rbrakk>
lp15@60809
  2151
               \<Longrightarrow>
lp15@60809
  2152
               cmod (?pathint x' y' + ?pathint y' z' + ?pathint z' x') * 10
lp15@60809
  2153
                     \<le> e * (dist x' y' + dist y' z' + dist z' x')\<^sup>2"
lp15@60809
  2154
         and Kk: "K / k < 2 ^ n"
lp15@60809
  2155
      have "K / 2 ^ n < k" using Kk k
lp15@60809
  2156
        by (auto simp: field_simps)
lp15@60809
  2157
      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
  2158
        using dist [of n]  k
lp15@60809
  2159
        by linarith+
lp15@60809
  2160
      have dle: "(dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2
lp15@60809
  2161
               \<le> (3 * K / 2 ^ n)\<^sup>2"
lp15@60809
  2162
        using dist [of n] e K
lp15@60809
  2163
        by (simp add: abs_le_square_iff [symmetric])
lp15@60809
  2164
      have less10: "\<And>x y::real. 0 < x \<Longrightarrow> y \<le> 9*x \<Longrightarrow> y < x*10"
lp15@60809
  2165
        by linarith
lp15@60809
  2166
      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
  2167
        using ynz dle e mult_le_cancel_left_pos by blast
lp15@60809
  2168
      also have "... <
lp15@60809
  2169
          cmod (?pathint (fa n) (fb n) + ?pathint (fb n) (fc n) + ?pathint (fc n) (fa n)) * 10"
lp15@60809
  2170
        using no [of n] e K
lp15@60809
  2171
        apply (simp add: e_def field_simps)
lp15@60809
  2172
        apply (simp only: zero_less_norm_iff [symmetric])
lp15@60809
  2173
        done
lp15@60809
  2174
      finally have False
lp15@60809
  2175
        using le [OF DD x cosb] by auto
lp15@60809
  2176
    } then
lp15@60809
  2177
    have ?thesis
lp15@60809
  2178
      using holomorphic_point_small_triangle [OF xin contf fx, of "e/10"] e
lp15@60809
  2179
      apply clarsimp
lp15@60809
  2180
      apply (rule_tac x1="K/k" in exE [OF real_arch_pow2], blast)
lp15@60809
  2181
      done
lp15@60809
  2182
  }
lp15@60809
  2183
  moreover have "f path_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2184
    by simp (meson contf continuous_on_subset path_integrable_continuous_linepath segments_subset_convex_hull(1)
lp15@60809
  2185
                   segments_subset_convex_hull(3) segments_subset_convex_hull(5))
lp15@60809
  2186
  ultimately show ?thesis
lp15@60809
  2187
    using has_path_integral_integral by fastforce
lp15@60809
  2188
qed
lp15@60809
  2189
lp15@60809
  2190
lp15@60809
  2191
subsection\<open>Version needing function holomorphic in interior only\<close>
lp15@60809
  2192
lp15@60809
  2193
lemma Cauchy_theorem_flat_lemma:
lp15@60809
  2194
  assumes f: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  2195
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@60809
  2196
      and k: "0 \<le> k"
lp15@60809
  2197
    shows "path_integral (linepath a b) f + path_integral (linepath b c) f +
lp15@60809
  2198
          path_integral (linepath c a) f = 0"
lp15@60809
  2199
proof -
lp15@60809
  2200
  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
  2201
    using f continuous_on_subset segments_subset_convex_hull by metis+
lp15@60809
  2202
  show ?thesis
lp15@60809
  2203
  proof (cases "k \<le> 1")
lp15@60809
  2204
    case True show ?thesis
lp15@60809
  2205
      by (simp add: path_integral_split [OF fabc(1) k True c] path_integral_reverse_linepath fabc)
lp15@60809
  2206
  next
lp15@60809
  2207
    case False then show ?thesis
lp15@60809
  2208
      using fabc c
lp15@60809
  2209
      apply (subst path_integral_split [of a c f "1/k" b, symmetric])
lp15@60809
  2210
      apply (metis closed_segment_commute fabc(3))
lp15@60809
  2211
      apply (auto simp: k path_integral_reverse_linepath)
lp15@60809
  2212
      done
lp15@60809
  2213
  qed
lp15@60809
  2214
qed
lp15@60809
  2215
lp15@60809
  2216
lemma Cauchy_theorem_flat:
lp15@60809
  2217
  assumes f: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  2218
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@60809
  2219
    shows "path_integral (linepath a b) f +
lp15@60809
  2220
           path_integral (linepath b c) f +
lp15@60809
  2221
           path_integral (linepath c a) f = 0"
lp15@60809
  2222
proof (cases "0 \<le> k")
lp15@60809
  2223
  case True with assms show ?thesis
lp15@60809
  2224
    by (blast intro: Cauchy_theorem_flat_lemma)
lp15@60809
  2225
next
lp15@60809
  2226
  case False
lp15@60809
  2227
  have "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
lp15@60809
  2228
    using f continuous_on_subset segments_subset_convex_hull by metis+
lp15@60809
  2229
  moreover have "path_integral (linepath b a) f + path_integral (linepath a c) f +
lp15@60809
  2230
        path_integral (linepath c b) f = 0"
lp15@60809
  2231
    apply (rule Cauchy_theorem_flat_lemma [of b a c f "1-k"])
lp15@60809
  2232
    using False c
lp15@60809
  2233
    apply (auto simp: f insert_commute scaleR_conv_of_real algebra_simps)
lp15@60809
  2234
    done
lp15@60809
  2235
  ultimately show ?thesis
lp15@60809
  2236
    apply (auto simp: path_integral_reverse_linepath)
lp15@60809
  2237
    using add_eq_0_iff by force
lp15@60809
  2238
qed
lp15@60809
  2239
lp15@60809
  2240
lp15@60809
  2241
lemma Cauchy_theorem_triangle_interior:
lp15@60809
  2242
  assumes contf: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  2243
      and holf:  "f holomorphic_on interior (convex hull {a,b,c})"
lp15@60809
  2244
     shows "(f has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2245
proof -
lp15@60809
  2246
  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
  2247
    using contf continuous_on_subset segments_subset_convex_hull by metis+
lp15@60809
  2248
  have "bounded (f ` (convex hull {a,b,c}))"
lp15@60809
  2249
    by (simp add: compact_continuous_image compact_convex_hull compact_imp_bounded contf)
lp15@60809
  2250
  then obtain B where "0 < B" and Bnf: "\<And>x. x \<in> convex hull {a,b,c} \<Longrightarrow> norm (f x) \<le> B"
lp15@60809
  2251
     by (auto simp: dest!: bounded_pos [THEN iffD1])
lp15@60809
  2252
  have "bounded (convex hull {a,b,c})"
lp15@60809
  2253
    by (simp add: bounded_convex_hull)
lp15@60809
  2254
  then obtain C where C: "0 < C" and Cno: "\<And>y. y \<in> convex hull {a,b,c} \<Longrightarrow> norm y < C"
lp15@60809
  2255
    using bounded_pos_less by blast
lp15@60809
  2256
  then have diff_2C: "norm(x - y) \<le> 2*C"
lp15@60809
  2257
           if x: "x \<in> convex hull {a, b, c}" and y: "y \<in> convex hull {a, b, c}" for x y
lp15@60809
  2258
  proof -
lp15@60809
  2259
    have "cmod x \<le> C"
lp15@60809
  2260
      using x by (meson Cno not_le not_less_iff_gr_or_eq)
lp15@60809
  2261
    hence "cmod (x - y) \<le> C + C"
lp15@60809
  2262
      using y by (meson Cno add_mono_thms_linordered_field(4) less_eq_real_def norm_triangle_ineq4 order_trans)
lp15@60809
  2263
    thus "cmod (x - y) \<le> 2 * C"
lp15@60809
  2264
      by (metis mult_2)
lp15@60809
  2265
  qed
lp15@60809
  2266
  have contf': "continuous_on (convex hull {b,a,c}) f"
lp15@60809
  2267
    using contf by (simp add: insert_commute)
lp15@60809
  2268
  { fix y::complex
lp15@60809
  2269
    assume fy: "(f has_path_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2270
       and ynz: "y \<noteq> 0"
lp15@60809
  2271
    have pi_eq_y: "path_integral (linepath a b) f + path_integral (linepath b c) f + path_integral (linepath c a) f = y"
lp15@60809
  2272
      by (rule has_chain_integral_chain_integral3 [OF fy])
lp15@60809
  2273
    have ?thesis
lp15@60809
  2274
    proof (cases "c=a \<or> a=b \<or> b=c")
lp15@60809
  2275
      case True then show ?t