src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy
 author paulson Tue Oct 13 12:42:08 2015 +0100 (2015-10-13) changeset 61426 d53db136e8fd parent 61284 2314c2f62eb1 child 61518 ff12606337e9 permissions -rw-r--r--
new material on path_component_sets, inside, outside, etc. And more default simprules
 lp15@60809 ` 1` ```section \Complex path integrals and Cauchy's integral theorem\ ``` lp15@60809 ` 2` lp15@60809 ` 3` ```theory Cauchy_Integral_Thm ``` paulson@61104 ` 4` ```imports Complex_Transcendental Weierstrass ``` lp15@60809 ` 5` ```begin ``` lp15@60809 ` 6` lp15@61190 ` 7` ```subsection \Piecewise differentiable functions\ ``` lp15@60809 ` 8` lp15@60809 ` 9` ```definition piecewise_differentiable_on ``` lp15@60809 ` 10` ``` (infixr "piecewise'_differentiable'_on" 50) ``` lp15@60809 ` 11` ``` where "f piecewise_differentiable_on i \ ``` lp15@60809 ` 12` ``` continuous_on i f \ ``` lp15@61190 ` 13` ``` (\s. finite s \ (\x \ i - s. f differentiable (at x within i)))" ``` lp15@60809 ` 14` lp15@60809 ` 15` ```lemma piecewise_differentiable_on_imp_continuous_on: ``` lp15@60809 ` 16` ``` "f piecewise_differentiable_on s \ continuous_on s f" ``` lp15@60809 ` 17` ```by (simp add: piecewise_differentiable_on_def) ``` lp15@60809 ` 18` lp15@60809 ` 19` ```lemma piecewise_differentiable_on_subset: ``` lp15@60809 ` 20` ``` "f piecewise_differentiable_on s \ t \ s \ f piecewise_differentiable_on t" ``` lp15@60809 ` 21` ``` using continuous_on_subset ``` lp15@61190 ` 22` ``` unfolding piecewise_differentiable_on_def ``` lp15@61190 ` 23` ``` apply safe ``` lp15@61190 ` 24` ``` apply (blast intro: elim: continuous_on_subset) ``` lp15@61190 ` 25` ``` by (meson Diff_iff differentiable_within_subset subsetCE) ``` lp15@60809 ` 26` lp15@60809 ` 27` ```lemma differentiable_on_imp_piecewise_differentiable: ``` lp15@60809 ` 28` ``` fixes a:: "'a::{linorder_topology,real_normed_vector}" ``` lp15@60809 ` 29` ``` shows "f differentiable_on {a..b} \ f piecewise_differentiable_on {a..b}" ``` lp15@60809 ` 30` ``` apply (simp add: piecewise_differentiable_on_def differentiable_imp_continuous_on) ``` lp15@61190 ` 31` ``` apply (rule_tac x="{a,b}" in exI, simp add: differentiable_on_def) ``` lp15@61190 ` 32` ``` done ``` lp15@60809 ` 33` lp15@60809 ` 34` ```lemma differentiable_imp_piecewise_differentiable: ``` lp15@61190 ` 35` ``` "(\x. x \ s \ f differentiable (at x within s)) ``` lp15@60809 ` 36` ``` \ f piecewise_differentiable_on s" ``` lp15@61190 ` 37` ```by (auto simp: piecewise_differentiable_on_def differentiable_imp_continuous_on differentiable_on_def ``` lp15@61190 ` 38` ``` intro: differentiable_within_subset) ``` lp15@60809 ` 39` paulson@61204 ` 40` ```lemma piecewise_differentiable_const [iff]: "(\x. z) piecewise_differentiable_on s" ``` paulson@61204 ` 41` ``` by (simp add: differentiable_imp_piecewise_differentiable) ``` paulson@61204 ` 42` lp15@60809 ` 43` ```lemma piecewise_differentiable_compose: ``` lp15@60809 ` 44` ``` "\f piecewise_differentiable_on s; g piecewise_differentiable_on (f ` s); ``` lp15@60809 ` 45` ``` \x. finite (s \ f-`{x})\ ``` lp15@60809 ` 46` ``` \ (g o f) piecewise_differentiable_on s" ``` lp15@60809 ` 47` ``` apply (simp add: piecewise_differentiable_on_def, safe) ``` lp15@60809 ` 48` ``` apply (blast intro: continuous_on_compose2) ``` lp15@60809 ` 49` ``` apply (rename_tac A B) ``` lp15@60809 ` 50` ``` apply (rule_tac x="A \ (\x\B. s \ f-`{x})" in exI) ``` lp15@61190 ` 51` ``` apply (blast intro: differentiable_chain_within) ``` lp15@61190 ` 52` ``` done ``` lp15@60809 ` 53` lp15@60809 ` 54` ```lemma piecewise_differentiable_affine: ``` lp15@60809 ` 55` ``` fixes m::real ``` lp15@60809 ` 56` ``` assumes "f piecewise_differentiable_on ((\x. m *\<^sub>R x + c) ` s)" ``` lp15@60809 ` 57` ``` shows "(f o (\x. m *\<^sub>R x + c)) piecewise_differentiable_on s" ``` lp15@60809 ` 58` ```proof (cases "m = 0") ``` lp15@60809 ` 59` ``` case True ``` lp15@60809 ` 60` ``` then show ?thesis ``` lp15@60809 ` 61` ``` unfolding o_def ``` lp15@60809 ` 62` ``` by (force intro: differentiable_imp_piecewise_differentiable differentiable_const) ``` lp15@60809 ` 63` ```next ``` lp15@60809 ` 64` ``` case False ``` lp15@60809 ` 65` ``` show ?thesis ``` lp15@60809 ` 66` ``` apply (rule piecewise_differentiable_compose [OF differentiable_imp_piecewise_differentiable]) ``` lp15@60809 ` 67` ``` apply (rule assms derivative_intros | simp add: False vimage_def real_vector_affinity_eq)+ ``` lp15@60809 ` 68` ``` done ``` lp15@60809 ` 69` ```qed ``` lp15@60809 ` 70` lp15@60809 ` 71` ```lemma piecewise_differentiable_cases: ``` lp15@60809 ` 72` ``` fixes c::real ``` lp15@60809 ` 73` ``` assumes "f piecewise_differentiable_on {a..c}" ``` lp15@60809 ` 74` ``` "g piecewise_differentiable_on {c..b}" ``` lp15@60809 ` 75` ``` "a \ c" "c \ b" "f c = g c" ``` lp15@60809 ` 76` ``` shows "(\x. if x \ c then f x else g x) piecewise_differentiable_on {a..b}" ``` lp15@60809 ` 77` ```proof - ``` lp15@60809 ` 78` ``` obtain s t where st: "finite s" "finite t" ``` lp15@61190 ` 79` ``` "\x\{a..c} - s. f differentiable at x within {a..c}" ``` lp15@61190 ` 80` ``` "\x\{c..b} - t. g differentiable at x within {c..b}" ``` lp15@60809 ` 81` ``` using assms ``` lp15@60809 ` 82` ``` by (auto simp: piecewise_differentiable_on_def) ``` lp15@61190 ` 83` ``` have finabc: "finite ({a,b,c} \ (s \ t))" ``` wenzelm@61222 ` 84` ``` by (metis \finite s\ \finite t\ finite_Un finite_insert finite.emptyI) ``` lp15@60809 ` 85` ``` have "continuous_on {a..c} f" "continuous_on {c..b} g" ``` lp15@60809 ` 86` ``` using assms piecewise_differentiable_on_def by auto ``` lp15@60809 ` 87` ``` then have "continuous_on {a..b} (\x. if x \ c then f x else g x)" ``` lp15@60809 ` 88` ``` using continuous_on_cases [OF closed_real_atLeastAtMost [of a c], ``` lp15@60809 ` 89` ``` OF closed_real_atLeastAtMost [of c b], ``` lp15@60809 ` 90` ``` of f g "\x. x\c"] assms ``` lp15@60809 ` 91` ``` by (force simp: ivl_disj_un_two_touch) ``` lp15@60809 ` 92` ``` moreover ``` lp15@60809 ` 93` ``` { fix x ``` lp15@61190 ` 94` ``` assume x: "x \ {a..b} - ({a,b,c} \ (s \ t))" ``` lp15@61190 ` 95` ``` have "(\x. if x \ c then f x else g x) differentiable at x within {a..b}" (is "?diff_fg") ``` lp15@60809 ` 96` ``` proof (cases x c rule: le_cases) ``` lp15@60809 ` 97` ``` case le show ?diff_fg ``` lp15@61190 ` 98` ``` apply (rule differentiable_transform_within [where d = "dist x c" and f = f]) ``` lp15@61190 ` 99` ``` using x le st ``` lp15@61190 ` 100` ``` apply (simp_all add: dist_real_def dist_nz [symmetric]) ``` lp15@61190 ` 101` ``` apply (rule differentiable_at_withinI) ``` lp15@61190 ` 102` ``` apply (rule differentiable_within_open [where s = "{a<..s. finite s \ ``` lp15@61190 ` 119` ``` (\x\{a..b} - s. (\x. if x \ c then f x else g x) differentiable at x within {a..b})" ``` lp15@61190 ` 120` ``` by (meson finabc) ``` lp15@60809 ` 121` ``` ultimately show ?thesis ``` lp15@60809 ` 122` ``` by (simp add: piecewise_differentiable_on_def) ``` lp15@60809 ` 123` ```qed ``` lp15@60809 ` 124` lp15@60809 ` 125` ```lemma piecewise_differentiable_neg: ``` lp15@60809 ` 126` ``` "f piecewise_differentiable_on s \ (\x. -(f x)) piecewise_differentiable_on s" ``` lp15@60809 ` 127` ``` by (auto simp: piecewise_differentiable_on_def continuous_on_minus) ``` lp15@60809 ` 128` lp15@60809 ` 129` ```lemma piecewise_differentiable_add: ``` lp15@60809 ` 130` ``` assumes "f piecewise_differentiable_on i" ``` lp15@60809 ` 131` ``` "g piecewise_differentiable_on i" ``` lp15@60809 ` 132` ``` shows "(\x. f x + g x) piecewise_differentiable_on i" ``` lp15@60809 ` 133` ```proof - ``` lp15@60809 ` 134` ``` obtain s t where st: "finite s" "finite t" ``` lp15@61190 ` 135` ``` "\x\i - s. f differentiable at x within i" ``` lp15@61190 ` 136` ``` "\x\i - t. g differentiable at x within i" ``` lp15@60809 ` 137` ``` using assms by (auto simp: piecewise_differentiable_on_def) ``` lp15@61190 ` 138` ``` then have "finite (s \ t) \ (\x\i - (s \ t). (\x. f x + g x) differentiable at x within i)" ``` lp15@60809 ` 139` ``` by auto ``` lp15@60809 ` 140` ``` moreover have "continuous_on i f" "continuous_on i g" ``` lp15@60809 ` 141` ``` using assms piecewise_differentiable_on_def by auto ``` lp15@60809 ` 142` ``` ultimately show ?thesis ``` lp15@60809 ` 143` ``` by (auto simp: piecewise_differentiable_on_def continuous_on_add) ``` lp15@60809 ` 144` ```qed ``` lp15@60809 ` 145` lp15@60809 ` 146` ```lemma piecewise_differentiable_diff: ``` lp15@60809 ` 147` ``` "\f piecewise_differentiable_on s; g piecewise_differentiable_on s\ ``` lp15@60809 ` 148` ``` \ (\x. f x - g x) piecewise_differentiable_on s" ``` lp15@60809 ` 149` ``` unfolding diff_conv_add_uminus ``` lp15@60809 ` 150` ``` by (metis piecewise_differentiable_add piecewise_differentiable_neg) ``` lp15@60809 ` 151` lp15@61190 ` 152` ```lemma continuous_on_joinpaths_D1: ``` lp15@61190 ` 153` ``` "continuous_on {0..1} (g1 +++ g2) \ continuous_on {0..1} g1" ``` lp15@61190 ` 154` ``` apply (rule continuous_on_eq [of _ "(g1 +++ g2) o (op*(inverse 2))"]) ``` lp15@61190 ` 155` ``` apply (rule continuous_intros | simp)+ ``` lp15@61190 ` 156` ``` apply (auto elim!: continuous_on_subset simp: joinpaths_def) ``` lp15@61190 ` 157` ``` done ``` lp15@61190 ` 158` lp15@61190 ` 159` ```lemma continuous_on_joinpaths_D2: ``` lp15@61190 ` 160` ``` "\continuous_on {0..1} (g1 +++ g2); pathfinish g1 = pathstart g2\ \ continuous_on {0..1} g2" ``` lp15@61190 ` 161` ``` apply (rule continuous_on_eq [of _ "(g1 +++ g2) o (\x. inverse 2*x + 1/2)"]) ``` lp15@61190 ` 162` ``` apply (rule continuous_intros | simp)+ ``` lp15@61190 ` 163` ``` apply (auto elim!: continuous_on_subset simp add: joinpaths_def pathfinish_def pathstart_def Ball_def) ``` lp15@61190 ` 164` ``` done ``` lp15@61190 ` 165` lp15@61190 ` 166` ```lemma piecewise_differentiable_D1: ``` lp15@61190 ` 167` ``` "(g1 +++ g2) piecewise_differentiable_on {0..1} \ g1 piecewise_differentiable_on {0..1}" ``` lp15@61190 ` 168` ``` apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D1) ``` lp15@61190 ` 169` ``` apply (rule_tac x="insert 1 ((op*2)`s)" in exI) ``` lp15@61190 ` 170` ``` apply simp ``` lp15@61190 ` 171` ``` apply (intro ballI) ``` lp15@61190 ` 172` ``` apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))" ``` lp15@61190 ` 173` ``` in differentiable_transform_within) ``` lp15@61190 ` 174` ``` apply (auto simp: dist_real_def joinpaths_def) ``` lp15@61190 ` 175` ``` apply (rule differentiable_chain_within derivative_intros | simp)+ ``` lp15@61190 ` 176` ``` apply (rule differentiable_subset) ``` lp15@61190 ` 177` ``` apply (force simp:)+ ``` lp15@61190 ` 178` ``` done ``` lp15@61190 ` 179` lp15@61190 ` 180` ```lemma piecewise_differentiable_D2: ``` lp15@61190 ` 181` ``` "\(g1 +++ g2) piecewise_differentiable_on {0..1}; pathfinish g1 = pathstart g2\ ``` lp15@61190 ` 182` ``` \ g2 piecewise_differentiable_on {0..1}" ``` lp15@61190 ` 183` ``` apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D2) ``` lp15@61190 ` 184` ``` apply (rule_tac x="insert 0 ((\x. 2*x-1)`s)" in exI) ``` lp15@61190 ` 185` ``` apply simp ``` lp15@61190 ` 186` ``` apply (intro ballI) ``` lp15@61190 ` 187` ``` apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\x. (x+1)/2)" ``` lp15@61190 ` 188` ``` in differentiable_transform_within) ``` lp15@61190 ` 189` ``` apply (auto simp: dist_real_def joinpaths_def abs_if field_simps split: split_if_asm) ``` lp15@61190 ` 190` ``` apply (rule differentiable_chain_within derivative_intros | simp)+ ``` lp15@61190 ` 191` ``` apply (rule differentiable_subset) ``` lp15@61190 ` 192` ``` apply (force simp: divide_simps)+ ``` lp15@61190 ` 193` ``` done ``` lp15@61190 ` 194` lp15@61190 ` 195` lp15@61190 ` 196` ```subsubsection\The concept of continuously differentiable\ ``` lp15@61190 ` 197` lp15@61190 ` 198` ```definition C1_differentiable_on :: "(real \ 'a::real_normed_vector) \ real set \ bool" ``` lp15@61190 ` 199` ``` (infix "C1'_differentiable'_on" 50) ``` lp15@61190 ` 200` ``` where ``` lp15@61190 ` 201` ``` "f C1_differentiable_on s \ ``` lp15@61190 ` 202` ``` (\D. (\x \ s. (f has_vector_derivative (D x)) (at x)) \ continuous_on s D)" ``` lp15@61190 ` 203` lp15@61190 ` 204` ```lemma C1_differentiable_on_eq: ``` lp15@61190 ` 205` ``` "f C1_differentiable_on s \ ``` lp15@61190 ` 206` ``` (\x \ s. f differentiable at x) \ continuous_on s (\x. vector_derivative f (at x))" ``` lp15@61190 ` 207` ``` unfolding C1_differentiable_on_def ``` lp15@61190 ` 208` ``` apply safe ``` lp15@61190 ` 209` ``` using differentiable_def has_vector_derivative_def apply blast ``` lp15@61190 ` 210` ``` apply (erule continuous_on_eq) ``` lp15@61190 ` 211` ``` using vector_derivative_at apply fastforce ``` lp15@61190 ` 212` ``` using vector_derivative_works apply fastforce ``` lp15@61190 ` 213` ``` done ``` lp15@61190 ` 214` lp15@61190 ` 215` ```lemma C1_differentiable_on_subset: ``` lp15@61190 ` 216` ``` "f C1_differentiable_on t \ s \ t \ f C1_differentiable_on s" ``` lp15@61190 ` 217` ``` unfolding C1_differentiable_on_def continuous_on_eq_continuous_within ``` lp15@61190 ` 218` ``` by (blast intro: continuous_within_subset) ``` lp15@61190 ` 219` lp15@61190 ` 220` ```lemma C1_differentiable_compose: ``` lp15@61190 ` 221` ``` "\f C1_differentiable_on s; g C1_differentiable_on (f ` s); ``` lp15@61190 ` 222` ``` \x. finite (s \ f-`{x})\ ``` lp15@61190 ` 223` ``` \ (g o f) C1_differentiable_on s" ``` lp15@61190 ` 224` ``` apply (simp add: C1_differentiable_on_eq, safe) ``` lp15@61190 ` 225` ``` using differentiable_chain_at apply blast ``` lp15@61190 ` 226` ``` apply (rule continuous_on_eq [of _ "\x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"]) ``` lp15@61190 ` 227` ``` apply (rule Limits.continuous_on_scaleR, assumption) ``` lp15@61190 ` 228` ``` apply (metis (mono_tags, lifting) continuous_on_eq continuous_at_imp_continuous_on continuous_on_compose differentiable_imp_continuous_within o_def) ``` lp15@61190 ` 229` ``` by (simp add: vector_derivative_chain_at) ``` lp15@61190 ` 230` lp15@61190 ` 231` ```lemma C1_diff_imp_diff: "f C1_differentiable_on s \ f differentiable_on s" ``` lp15@61190 ` 232` ``` by (simp add: C1_differentiable_on_eq differentiable_at_imp_differentiable_on) ``` lp15@61190 ` 233` lp15@61190 ` 234` ```lemma C1_differentiable_on_ident [simp, derivative_intros]: "(\x. x) C1_differentiable_on s" ``` lp15@61190 ` 235` ``` by (auto simp: C1_differentiable_on_eq continuous_on_const) ``` lp15@61190 ` 236` lp15@61190 ` 237` ```lemma C1_differentiable_on_const [simp, derivative_intros]: "(\z. a) C1_differentiable_on s" ``` lp15@61190 ` 238` ``` by (auto simp: C1_differentiable_on_eq continuous_on_const) ``` lp15@61190 ` 239` lp15@61190 ` 240` ```lemma C1_differentiable_on_add [simp, derivative_intros]: ``` lp15@61190 ` 241` ``` "f C1_differentiable_on s \ g C1_differentiable_on s \ (\x. f x + g x) C1_differentiable_on s" ``` lp15@61190 ` 242` ``` unfolding C1_differentiable_on_eq by (auto intro: continuous_intros) ``` lp15@61190 ` 243` lp15@61190 ` 244` ```lemma C1_differentiable_on_minus [simp, derivative_intros]: ``` lp15@61190 ` 245` ``` "f C1_differentiable_on s \ (\x. - f x) C1_differentiable_on s" ``` lp15@61190 ` 246` ``` unfolding C1_differentiable_on_eq by (auto intro: continuous_intros) ``` lp15@61190 ` 247` lp15@61190 ` 248` ```lemma C1_differentiable_on_diff [simp, derivative_intros]: ``` lp15@61190 ` 249` ``` "f C1_differentiable_on s \ g C1_differentiable_on s \ (\x. f x - g x) C1_differentiable_on s" ``` lp15@61190 ` 250` ``` unfolding C1_differentiable_on_eq by (auto intro: continuous_intros) ``` lp15@61190 ` 251` lp15@61190 ` 252` ```lemma C1_differentiable_on_mult [simp, derivative_intros]: ``` lp15@61190 ` 253` ``` fixes f g :: "real \ 'a :: real_normed_algebra" ``` lp15@61190 ` 254` ``` shows "f C1_differentiable_on s \ g C1_differentiable_on s \ (\x. f x * g x) C1_differentiable_on s" ``` lp15@61190 ` 255` ``` unfolding C1_differentiable_on_eq ``` lp15@61190 ` 256` ``` by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within) ``` lp15@61190 ` 257` lp15@61190 ` 258` ```lemma C1_differentiable_on_scaleR [simp, derivative_intros]: ``` lp15@61190 ` 259` ``` "f C1_differentiable_on s \ g C1_differentiable_on s \ (\x. f x *\<^sub>R g x) C1_differentiable_on s" ``` lp15@61190 ` 260` ``` unfolding C1_differentiable_on_eq ``` lp15@61190 ` 261` ``` by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+ ``` lp15@61190 ` 262` lp15@61190 ` 263` lp15@61190 ` 264` ```definition piecewise_C1_differentiable_on ``` lp15@61190 ` 265` ``` (infixr "piecewise'_C1'_differentiable'_on" 50) ``` lp15@61190 ` 266` ``` where "f piecewise_C1_differentiable_on i \ ``` lp15@61190 ` 267` ``` continuous_on i f \ ``` lp15@61190 ` 268` ``` (\s. finite s \ (f C1_differentiable_on (i - s)))" ``` lp15@61190 ` 269` lp15@61190 ` 270` ```lemma C1_differentiable_imp_piecewise: ``` lp15@61190 ` 271` ``` "f C1_differentiable_on s \ f piecewise_C1_differentiable_on s" ``` lp15@61190 ` 272` ``` by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within) ``` lp15@61190 ` 273` lp15@61190 ` 274` ```lemma piecewise_C1_imp_differentiable: ``` lp15@61190 ` 275` ``` "f piecewise_C1_differentiable_on i \ f piecewise_differentiable_on i" ``` lp15@61190 ` 276` ``` by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def ``` lp15@61190 ` 277` ``` C1_differentiable_on_def differentiable_def has_vector_derivative_def ``` lp15@61190 ` 278` ``` intro: has_derivative_at_within) ``` lp15@61190 ` 279` lp15@61190 ` 280` ```lemma piecewise_C1_differentiable_compose: ``` lp15@61190 ` 281` ``` "\f piecewise_C1_differentiable_on s; g piecewise_C1_differentiable_on (f ` s); ``` lp15@61190 ` 282` ``` \x. finite (s \ f-`{x})\ ``` lp15@61190 ` 283` ``` \ (g o f) piecewise_C1_differentiable_on s" ``` lp15@61190 ` 284` ``` apply (simp add: piecewise_C1_differentiable_on_def, safe) ``` lp15@61190 ` 285` ``` apply (blast intro: continuous_on_compose2) ``` lp15@61190 ` 286` ``` apply (rename_tac A B) ``` lp15@61190 ` 287` ``` apply (rule_tac x="A \ (\x\B. s \ f-`{x})" in exI) ``` lp15@61190 ` 288` ``` apply (rule conjI, blast) ``` lp15@61190 ` 289` ``` apply (rule C1_differentiable_compose) ``` lp15@61190 ` 290` ``` apply (blast intro: C1_differentiable_on_subset) ``` lp15@61190 ` 291` ``` apply (blast intro: C1_differentiable_on_subset) ``` lp15@61190 ` 292` ``` by (simp add: Diff_Int_distrib2) ``` lp15@61190 ` 293` lp15@61190 ` 294` ```lemma piecewise_C1_differentiable_on_subset: ``` lp15@61190 ` 295` ``` "f piecewise_C1_differentiable_on s \ t \ s \ f piecewise_C1_differentiable_on t" ``` lp15@61190 ` 296` ``` by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset) ``` lp15@61190 ` 297` lp15@61190 ` 298` ```lemma C1_differentiable_imp_continuous_on: ``` lp15@61190 ` 299` ``` "f C1_differentiable_on s \ continuous_on s f" ``` lp15@61190 ` 300` ``` unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within ``` lp15@61190 ` 301` ``` using differentiable_at_withinI differentiable_imp_continuous_within by blast ``` lp15@61190 ` 302` lp15@61190 ` 303` ```lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}" ``` lp15@61190 ` 304` ``` unfolding C1_differentiable_on_def ``` lp15@61190 ` 305` ``` by auto ``` lp15@61190 ` 306` lp15@61190 ` 307` ```lemma piecewise_C1_differentiable_affine: ``` lp15@61190 ` 308` ``` fixes m::real ``` lp15@61190 ` 309` ``` assumes "f piecewise_C1_differentiable_on ((\x. m * x + c) ` s)" ``` lp15@61190 ` 310` ``` shows "(f o (\x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on s" ``` lp15@61190 ` 311` ```proof (cases "m = 0") ``` lp15@61190 ` 312` ``` case True ``` lp15@61190 ` 313` ``` then show ?thesis ``` lp15@61190 ` 314` ``` unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def continuous_on_const) ``` lp15@61190 ` 315` ```next ``` lp15@61190 ` 316` ``` case False ``` lp15@61190 ` 317` ``` show ?thesis ``` lp15@61190 ` 318` ``` apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise]) ``` lp15@61190 ` 319` ``` apply (rule assms derivative_intros | simp add: False vimage_def)+ ``` lp15@61190 ` 320` ``` using real_vector_affinity_eq [OF False, where c=c, unfolded scaleR_conv_of_real] ``` lp15@61190 ` 321` ``` apply simp ``` lp15@61190 ` 322` ``` done ``` lp15@61190 ` 323` ```qed ``` lp15@61190 ` 324` lp15@61190 ` 325` ```lemma piecewise_C1_differentiable_cases: ``` lp15@61190 ` 326` ``` fixes c::real ``` lp15@61190 ` 327` ``` assumes "f piecewise_C1_differentiable_on {a..c}" ``` lp15@61190 ` 328` ``` "g piecewise_C1_differentiable_on {c..b}" ``` lp15@61190 ` 329` ``` "a \ c" "c \ b" "f c = g c" ``` lp15@61190 ` 330` ``` shows "(\x. if x \ c then f x else g x) piecewise_C1_differentiable_on {a..b}" ``` lp15@61190 ` 331` ```proof - ``` lp15@61190 ` 332` ``` obtain s t where st: "f C1_differentiable_on ({a..c} - s)" ``` lp15@61190 ` 333` ``` "g C1_differentiable_on ({c..b} - t)" ``` lp15@61190 ` 334` ``` "finite s" "finite t" ``` lp15@61190 ` 335` ``` using assms ``` lp15@61190 ` 336` ``` by (force simp: piecewise_C1_differentiable_on_def) ``` lp15@61190 ` 337` ``` then have f_diff: "f differentiable_on {a..x. if x \ c then f x else g x)" ``` lp15@61190 ` 343` ``` using continuous_on_cases [OF closed_real_atLeastAtMost [of a c], ``` lp15@61190 ` 344` ``` OF closed_real_atLeastAtMost [of c b], ``` lp15@61190 ` 345` ``` of f g "\x. x\c"] assms ``` lp15@61190 ` 346` ``` by (force simp: ivl_disj_un_two_touch) ``` lp15@61190 ` 347` ``` { fix x ``` lp15@61190 ` 348` ``` assume x: "x \ {a..b} - insert c (s \ t)" ``` lp15@61190 ` 349` ``` have "(\x. if x \ c then f x else g x) differentiable at x" (is "?diff_fg") ``` lp15@61190 ` 350` ``` proof (cases x c rule: le_cases) ``` lp15@61190 ` 351` ``` case le show ?diff_fg ``` lp15@61190 ` 352` ``` apply (rule differentiable_transform_at [of "dist x c" _ f]) ``` lp15@61190 ` 353` ``` using x dist_real_def le st by (auto simp: C1_differentiable_on_eq) ``` lp15@61190 ` 354` ``` next ``` lp15@61190 ` 355` ``` case ge show ?diff_fg ``` lp15@61190 ` 356` ``` apply (rule differentiable_transform_at [of "dist x c" _ g]) ``` lp15@61190 ` 357` ``` using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq) ``` lp15@61190 ` 358` ``` qed ``` lp15@61190 ` 359` ``` } ``` lp15@61190 ` 360` ``` then have "(\x \ {a..b} - insert c (s \ t). (\x. if x \ c then f x else g x) differentiable at x)" ``` lp15@61190 ` 361` ``` by auto ``` lp15@61190 ` 362` ``` moreover ``` lp15@61190 ` 363` ``` { assume fcon: "continuous_on ({a<..x. vector_derivative f (at x))" ``` lp15@61190 ` 364` ``` and gcon: "continuous_on ({c<..x. vector_derivative g (at x))" ``` lp15@61190 ` 365` ``` have "open ({a<..x. vector_derivative (\x. if x \ c then f x else g x) (at x))" ``` lp15@61190 ` 368` ``` apply (rule continuous_on_eq [OF fcon]) ``` lp15@61190 ` 369` ``` apply (simp add:) ``` lp15@61190 ` 370` ``` apply (rule vector_derivative_at [symmetric]) ``` lp15@61190 ` 371` ``` apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_at) ``` lp15@61190 ` 372` ``` apply (simp_all add: dist_norm vector_derivative_works [symmetric]) ``` lp15@61190 ` 373` ``` using f_diff ``` lp15@61190 ` 374` ``` by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_imp_le st(1)) ``` lp15@61190 ` 375` ``` moreover have "continuous_on ({c<..x. vector_derivative (\x. if x \ c then f x else g x) (at x))" ``` lp15@61190 ` 376` ``` apply (rule continuous_on_eq [OF gcon]) ``` lp15@61190 ` 377` ``` apply (simp add:) ``` lp15@61190 ` 378` ``` apply (rule vector_derivative_at [symmetric]) ``` lp15@61190 ` 379` ``` apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_at) ``` lp15@61190 ` 380` ``` apply (simp_all add: dist_norm vector_derivative_works [symmetric]) ``` lp15@61190 ` 381` ``` using g_diff ``` lp15@61190 ` 382` ``` by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_imp_le st(2)) ``` lp15@61190 ` 383` ``` ultimately have "continuous_on ({a<.. t)) ``` lp15@61190 ` 384` ``` (\x. vector_derivative (\x. if x \ c then f x else g x) (at x))" ``` lp15@61190 ` 385` ``` apply (rule continuous_on_subset [OF continuous_on_open_Un], auto) ``` lp15@61190 ` 386` ``` done ``` lp15@61190 ` 387` ``` } note * = this ``` lp15@61190 ` 388` ``` have "continuous_on ({a<.. t)) (\x. vector_derivative (\x. if x \ c then f x else g x) (at x))" ``` lp15@61190 ` 389` ``` using st ``` lp15@61190 ` 390` ``` by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *) ``` lp15@61190 ` 391` ``` ultimately have "\s. finite s \ ((\x. if x \ c then f x else g x) C1_differentiable_on {a..b} - s)" ``` lp15@61190 ` 392` ``` apply (rule_tac x="{a,b,c} \ s \ t" in exI) ``` lp15@61190 ` 393` ``` using st by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset) ``` lp15@61190 ` 394` ``` with cab show ?thesis ``` lp15@61190 ` 395` ``` by (simp add: piecewise_C1_differentiable_on_def) ``` lp15@61190 ` 396` ```qed ``` lp15@61190 ` 397` lp15@61190 ` 398` ```lemma piecewise_C1_differentiable_neg: ``` lp15@61190 ` 399` ``` "f piecewise_C1_differentiable_on s \ (\x. -(f x)) piecewise_C1_differentiable_on s" ``` lp15@61190 ` 400` ``` unfolding piecewise_C1_differentiable_on_def ``` lp15@61190 ` 401` ``` by (auto intro!: continuous_on_minus C1_differentiable_on_minus) ``` lp15@61190 ` 402` lp15@61190 ` 403` ```lemma piecewise_C1_differentiable_add: ``` lp15@61190 ` 404` ``` assumes "f piecewise_C1_differentiable_on i" ``` lp15@61190 ` 405` ``` "g piecewise_C1_differentiable_on i" ``` lp15@61190 ` 406` ``` shows "(\x. f x + g x) piecewise_C1_differentiable_on i" ``` lp15@61190 ` 407` ```proof - ``` lp15@61190 ` 408` ``` obtain s t where st: "finite s" "finite t" ``` lp15@61190 ` 409` ``` "f C1_differentiable_on (i-s)" ``` lp15@61190 ` 410` ``` "g C1_differentiable_on (i-t)" ``` lp15@61190 ` 411` ``` using assms by (auto simp: piecewise_C1_differentiable_on_def) ``` lp15@61190 ` 412` ``` then have "finite (s \ t) \ (\x. f x + g x) C1_differentiable_on i - (s \ t)" ``` lp15@61190 ` 413` ``` by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset) ``` lp15@61190 ` 414` ``` moreover have "continuous_on i f" "continuous_on i g" ``` lp15@61190 ` 415` ``` using assms piecewise_C1_differentiable_on_def by auto ``` lp15@61190 ` 416` ``` ultimately show ?thesis ``` lp15@61190 ` 417` ``` by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add) ``` lp15@61190 ` 418` ```qed ``` lp15@61190 ` 419` paulson@61204 ` 420` ```lemma piecewise_C1_differentiable_diff: ``` lp15@61190 ` 421` ``` "\f piecewise_C1_differentiable_on s; g piecewise_C1_differentiable_on s\ ``` lp15@61190 ` 422` ``` \ (\x. f x - g x) piecewise_C1_differentiable_on s" ``` lp15@61190 ` 423` ``` unfolding diff_conv_add_uminus ``` lp15@61190 ` 424` ``` by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg) ``` lp15@61190 ` 425` lp15@61190 ` 426` ```lemma piecewise_C1_differentiable_D1: ``` lp15@61190 ` 427` ``` fixes g1 :: "real \ 'a::real_normed_field" ``` lp15@61190 ` 428` ``` assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" ``` lp15@61190 ` 429` ``` shows "g1 piecewise_C1_differentiable_on {0..1}" ``` lp15@61190 ` 430` ```proof - ``` lp15@61190 ` 431` ``` obtain s where "finite s" ``` lp15@61190 ` 432` ``` and co12: "continuous_on ({0..1} - s) (\x. vector_derivative (g1 +++ g2) (at x))" ``` lp15@61190 ` 433` ``` and g12D: "\x\{0..1} - s. g1 +++ g2 differentiable at x" ``` lp15@61190 ` 434` ``` using assms by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq) ``` lp15@61190 ` 435` ``` then have g1D: "g1 differentiable at x" if "x \ {0..1} - insert 1 (op * 2 ` s)" for x ``` lp15@61190 ` 436` ``` apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))" in differentiable_transform_at) ``` lp15@61190 ` 437` ``` using that ``` lp15@61190 ` 438` ``` apply (simp_all add: dist_real_def joinpaths_def) ``` lp15@61190 ` 439` ``` apply (rule differentiable_chain_at derivative_intros | force)+ ``` lp15@61190 ` 440` ``` done ``` lp15@61190 ` 441` ``` have [simp]: "vector_derivative (g1 \ op * 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)" ``` lp15@61190 ` 442` ``` if "x \ {0..1} - insert 1 (op * 2 ` s)" for x ``` lp15@61190 ` 443` ``` apply (subst vector_derivative_chain_at) ``` lp15@61190 ` 444` ``` using that ``` lp15@61190 ` 445` ``` apply (rule derivative_eq_intros g1D | simp)+ ``` lp15@61190 ` 446` ``` done ``` lp15@61190 ` 447` ``` have "continuous_on ({0..1/2} - insert (1/2) s) (\x. vector_derivative (g1 +++ g2) (at x))" ``` lp15@61190 ` 448` ``` using co12 by (rule continuous_on_subset) force ``` lp15@61190 ` 449` ``` then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) s) (\x. vector_derivative (g1 o op*2) (at x))" ``` lp15@61190 ` 450` ``` apply (rule continuous_on_eq [OF _ vector_derivative_at]) ``` lp15@61190 ` 451` ``` apply (rule_tac f="g1 o op*2" and d="dist x (1/2)" in has_vector_derivative_transform_at) ``` lp15@61190 ` 452` ``` apply (simp_all add: dist_norm joinpaths_def vector_derivative_works [symmetric]) ``` lp15@61190 ` 453` ``` apply (force intro: g1D differentiable_chain_at) ``` lp15@61190 ` 454` ``` done ``` lp15@61190 ` 455` ``` have "continuous_on ({0..1} - insert 1 (op * 2 ` s)) ``` lp15@61190 ` 456` ``` ((\x. 1/2 * vector_derivative (g1 o op*2) (at x)) o op*(1/2))" ``` lp15@61190 ` 457` ``` apply (rule continuous_intros)+ ``` lp15@61190 ` 458` ``` using coDhalf ``` lp15@61190 ` 459` ``` apply (simp add: scaleR_conv_of_real image_set_diff image_image) ``` lp15@61190 ` 460` ``` done ``` lp15@61190 ` 461` ``` then have con_g1: "continuous_on ({0..1} - insert 1 (op * 2 ` s)) (\x. vector_derivative g1 (at x))" ``` lp15@61190 ` 462` ``` by (rule continuous_on_eq) (simp add: scaleR_conv_of_real) ``` lp15@61190 ` 463` ``` have "continuous_on {0..1} g1" ``` lp15@61190 ` 464` ``` using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast ``` wenzelm@61222 ` 465` ``` with \finite s\ show ?thesis ``` lp15@61190 ` 466` ``` apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq) ``` lp15@61190 ` 467` ``` apply (rule_tac x="insert 1 ((op*2)`s)" in exI) ``` lp15@61190 ` 468` ``` apply (simp add: g1D con_g1) ``` lp15@61190 ` 469` ``` done ``` lp15@61190 ` 470` ```qed ``` lp15@61190 ` 471` lp15@61190 ` 472` ```lemma piecewise_C1_differentiable_D2: ``` lp15@61190 ` 473` ``` fixes g2 :: "real \ 'a::real_normed_field" ``` lp15@61190 ` 474` ``` assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" "pathfinish g1 = pathstart g2" ``` lp15@61190 ` 475` ``` shows "g2 piecewise_C1_differentiable_on {0..1}" ``` lp15@61190 ` 476` ```proof - ``` lp15@61190 ` 477` ``` obtain s where "finite s" ``` lp15@61190 ` 478` ``` and co12: "continuous_on ({0..1} - s) (\x. vector_derivative (g1 +++ g2) (at x))" ``` lp15@61190 ` 479` ``` and g12D: "\x\{0..1} - s. g1 +++ g2 differentiable at x" ``` lp15@61190 ` 480` ``` using assms by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq) ``` lp15@61190 ` 481` ``` then have g2D: "g2 differentiable at x" if "x \ {0..1} - insert 0 ((\x. 2*x-1) ` s)" for x ``` lp15@61190 ` 482` ``` apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\x. (x+1)/2)" in differentiable_transform_at) ``` lp15@61190 ` 483` ``` using that ``` lp15@61190 ` 484` ``` apply (simp_all add: dist_real_def joinpaths_def) ``` lp15@61190 ` 485` ``` apply (auto simp: dist_real_def joinpaths_def field_simps) ``` lp15@61190 ` 486` ``` apply (rule differentiable_chain_at derivative_intros | force)+ ``` lp15@61190 ` 487` ``` apply (drule_tac x= "(x + 1) / 2" in bspec, force simp: divide_simps) ``` lp15@61190 ` 488` ``` apply assumption ``` lp15@61190 ` 489` ``` done ``` lp15@61190 ` 490` ``` have [simp]: "vector_derivative (g2 \ (\x. 2*x-1)) (at ((x+1)/2)) = 2 *\<^sub>R vector_derivative g2 (at x)" ``` lp15@61190 ` 491` ``` if "x \ {0..1} - insert 0 ((\x. 2*x-1) ` s)" for x ``` lp15@61190 ` 492` ``` using that by (auto simp: vector_derivative_chain_at divide_simps g2D) ``` lp15@61190 ` 493` ``` have "continuous_on ({1/2..1} - insert (1/2) s) (\x. vector_derivative (g1 +++ g2) (at x))" ``` lp15@61190 ` 494` ``` using co12 by (rule continuous_on_subset) force ``` lp15@61190 ` 495` ``` then have coDhalf: "continuous_on ({1/2..1} - insert (1/2) s) (\x. vector_derivative (g2 o (\x. 2*x-1)) (at x))" ``` lp15@61190 ` 496` ``` apply (rule continuous_on_eq [OF _ vector_derivative_at]) ``` lp15@61190 ` 497` ``` apply (rule_tac f="g2 o (\x. 2*x-1)" and d="dist (3/4) ((x+1)/2)" in has_vector_derivative_transform_at) ``` lp15@61190 ` 498` ``` apply (auto simp: dist_real_def field_simps joinpaths_def vector_derivative_works [symmetric] ``` lp15@61190 ` 499` ``` intro!: g2D differentiable_chain_at) ``` lp15@61190 ` 500` ``` done ``` lp15@61190 ` 501` ``` have [simp]: "((\x. (x + 1) / 2) ` ({0..1} - insert 0 ((\x. 2 * x - 1) ` s))) = ({1/2..1} - insert (1/2) s)" ``` lp15@61190 ` 502` ``` apply (simp add: image_set_diff inj_on_def image_image) ``` lp15@61190 ` 503` ``` apply (auto simp: image_affinity_atLeastAtMost_div add_divide_distrib) ``` lp15@61190 ` 504` ``` done ``` lp15@61190 ` 505` ``` have "continuous_on ({0..1} - insert 0 ((\x. 2*x-1) ` s)) ``` lp15@61190 ` 506` ``` ((\x. 1/2 * vector_derivative (g2 \ (\x. 2*x-1)) (at x)) o (\x. (x+1)/2))" ``` lp15@61190 ` 507` ``` by (rule continuous_intros | simp add: coDhalf)+ ``` lp15@61190 ` 508` ``` then have con_g2: "continuous_on ({0..1} - insert 0 ((\x. 2*x-1) ` s)) (\x. vector_derivative g2 (at x))" ``` lp15@61190 ` 509` ``` by (rule continuous_on_eq) (simp add: scaleR_conv_of_real) ``` lp15@61190 ` 510` ``` have "continuous_on {0..1} g2" ``` lp15@61190 ` 511` ``` using continuous_on_joinpaths_D2 assms piecewise_C1_differentiable_on_def by blast ``` wenzelm@61222 ` 512` ``` with \finite s\ show ?thesis ``` lp15@61190 ` 513` ``` apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq) ``` lp15@61190 ` 514` ``` apply (rule_tac x="insert 0 ((\x. 2 * x - 1) ` s)" in exI) ``` lp15@61190 ` 515` ``` apply (simp add: g2D con_g2) ``` lp15@61190 ` 516` ``` done ``` lp15@61190 ` 517` ```qed ``` lp15@60809 ` 518` lp15@60809 ` 519` ```subsection \Valid paths, and their start and finish\ ``` lp15@60809 ` 520` lp15@60809 ` 521` ```lemma Diff_Un_eq: "A - (B \ C) = A - B - C" ``` lp15@60809 ` 522` ``` by blast ``` lp15@60809 ` 523` lp15@60809 ` 524` ```definition valid_path :: "(real \ 'a :: real_normed_vector) \ bool" ``` lp15@61190 ` 525` ``` where "valid_path f \ f piecewise_C1_differentiable_on {0..1::real}" ``` lp15@60809 ` 526` lp15@60809 ` 527` ```definition closed_path :: "(real \ 'a :: real_normed_vector) \ bool" ``` lp15@60809 ` 528` ``` where "closed_path g \ g 0 = g 1" ``` lp15@60809 ` 529` lp15@60809 ` 530` ```subsubsection\In particular, all results for paths apply\ ``` lp15@60809 ` 531` lp15@60809 ` 532` ```lemma valid_path_imp_path: "valid_path g \ path g" ``` lp15@61190 ` 533` ```by (simp add: path_def piecewise_C1_differentiable_on_def valid_path_def) ``` lp15@60809 ` 534` lp15@60809 ` 535` ```lemma connected_valid_path_image: "valid_path g \ connected(path_image g)" ``` lp15@60809 ` 536` ``` by (metis connected_path_image valid_path_imp_path) ``` lp15@60809 ` 537` lp15@60809 ` 538` ```lemma compact_valid_path_image: "valid_path g \ compact(path_image g)" ``` lp15@60809 ` 539` ``` by (metis compact_path_image valid_path_imp_path) ``` lp15@60809 ` 540` lp15@60809 ` 541` ```lemma bounded_valid_path_image: "valid_path g \ bounded(path_image g)" ``` lp15@60809 ` 542` ``` by (metis bounded_path_image valid_path_imp_path) ``` lp15@60809 ` 543` lp15@60809 ` 544` ```lemma closed_valid_path_image: "valid_path g \ closed(path_image g)" ``` lp15@60809 ` 545` ``` by (metis closed_path_image valid_path_imp_path) ``` lp15@60809 ` 546` lp15@60809 ` 547` lp15@60809 ` 548` ```subsection\Contour Integrals along a path\ ``` lp15@60809 ` 549` lp15@60809 ` 550` ```text\This definition is for complex numbers only, and does not generalise to line integrals in a vector field\ ``` lp15@60809 ` 551` lp15@61190 ` 552` ```text\piecewise differentiable function on [0,1]\ ``` lp15@60809 ` 553` lp15@60809 ` 554` ```definition has_path_integral :: "(complex \ complex) \ complex \ (real \ complex) \ bool" ``` lp15@60809 ` 555` ``` (infixr "has'_path'_integral" 50) ``` lp15@60809 ` 556` ``` where "(f has_path_integral i) g \ ``` lp15@60809 ` 557` ``` ((\x. f(g x) * vector_derivative g (at x within {0..1})) ``` lp15@60809 ` 558` ``` has_integral i) {0..1}" ``` lp15@60809 ` 559` lp15@60809 ` 560` ```definition path_integrable_on ``` lp15@60809 ` 561` ``` (infixr "path'_integrable'_on" 50) ``` lp15@60809 ` 562` ``` where "f path_integrable_on g \ \i. (f has_path_integral i) g" ``` lp15@60809 ` 563` lp15@60809 ` 564` ```definition path_integral ``` lp15@60809 ` 565` ``` where "path_integral g f \ @i. (f has_path_integral i) g" ``` lp15@60809 ` 566` lp15@60809 ` 567` ```lemma path_integral_unique: "(f has_path_integral i) g \ path_integral g f = i" ``` lp15@60809 ` 568` ``` by (auto simp: path_integral_def has_path_integral_def integral_def [symmetric]) ``` lp15@60809 ` 569` lp15@60809 ` 570` ```lemma has_path_integral_integral: ``` lp15@60809 ` 571` ``` "f path_integrable_on i \ (f has_path_integral (path_integral i f)) i" ``` lp15@60809 ` 572` ``` by (metis path_integral_unique path_integrable_on_def) ``` lp15@60809 ` 573` lp15@60809 ` 574` ```lemma has_path_integral_unique: ``` lp15@60809 ` 575` ``` "(f has_path_integral i) g \ (f has_path_integral j) g \ i = j" ``` lp15@60809 ` 576` ``` using has_integral_unique ``` lp15@60809 ` 577` ``` by (auto simp: has_path_integral_def) ``` lp15@60809 ` 578` lp15@60809 ` 579` ```lemma has_path_integral_integrable: "(f has_path_integral i) g \ f path_integrable_on g" ``` lp15@60809 ` 580` ``` using path_integrable_on_def by blast ``` lp15@60809 ` 581` lp15@60809 ` 582` ```(* Show that we can forget about the localized derivative.*) ``` lp15@60809 ` 583` lp15@60809 ` 584` ```lemma vector_derivative_within_interior: ``` lp15@60809 ` 585` ``` "\x \ interior s; NO_MATCH UNIV s\ ``` lp15@60809 ` 586` ``` \ vector_derivative f (at x within s) = vector_derivative f (at x)" ``` lp15@60809 ` 587` ``` apply (simp add: vector_derivative_def has_vector_derivative_def has_derivative_def netlimit_within_interior) ``` lp15@60809 ` 588` ``` apply (subst lim_within_interior, auto) ``` lp15@60809 ` 589` ``` done ``` lp15@60809 ` 590` lp15@60809 ` 591` ```lemma has_integral_localized_vector_derivative: ``` lp15@60809 ` 592` ``` "((\x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \ ``` lp15@60809 ` 593` ``` ((\x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}" ``` lp15@60809 ` 594` ```proof - ``` lp15@60809 ` 595` ``` have "{a..b} - {a,b} = interior {a..b}" ``` lp15@60809 ` 596` ``` by (simp add: atLeastAtMost_diff_ends) ``` lp15@60809 ` 597` ``` show ?thesis ``` lp15@60809 ` 598` ``` apply (rule has_integral_spike_eq [of "{a,b}"]) ``` lp15@60809 ` 599` ``` apply (auto simp: vector_derivative_within_interior) ``` lp15@60809 ` 600` ``` done ``` lp15@60809 ` 601` ```qed ``` lp15@60809 ` 602` lp15@60809 ` 603` ```lemma integrable_on_localized_vector_derivative: ``` lp15@60809 ` 604` ``` "(\x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \ ``` lp15@60809 ` 605` ``` (\x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}" ``` lp15@60809 ` 606` ``` by (simp add: integrable_on_def has_integral_localized_vector_derivative) ``` lp15@60809 ` 607` lp15@60809 ` 608` ```lemma has_path_integral: ``` lp15@60809 ` 609` ``` "(f has_path_integral i) g \ ``` lp15@60809 ` 610` ``` ((\x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}" ``` lp15@60809 ` 611` ``` by (simp add: has_integral_localized_vector_derivative has_path_integral_def) ``` lp15@60809 ` 612` lp15@60809 ` 613` ```lemma path_integrable_on: ``` lp15@60809 ` 614` ``` "f path_integrable_on g \ ``` lp15@60809 ` 615` ``` (\t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}" ``` lp15@60809 ` 616` ``` by (simp add: has_path_integral integrable_on_def path_integrable_on_def) ``` lp15@60809 ` 617` lp15@60809 ` 618` ```subsection\Reversing a path\ ``` lp15@60809 ` 619` lp15@60809 ` 620` ```lemma valid_path_imp_reverse: ``` lp15@60809 ` 621` ``` assumes "valid_path g" ``` lp15@60809 ` 622` ``` shows "valid_path(reversepath g)" ``` lp15@60809 ` 623` ```proof - ``` lp15@61190 ` 624` ``` obtain s where "finite s" "g C1_differentiable_on ({0..1} - s)" ``` lp15@61190 ` 625` ``` using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def) ``` lp15@61190 ` 626` ``` then have "finite (op - 1 ` s)" "(reversepath g C1_differentiable_on ({0..1} - op - 1 ` s))" ``` lp15@60809 ` 627` ``` apply (auto simp: reversepath_def) ``` lp15@61190 ` 628` ``` apply (rule C1_differentiable_compose [of "\x::real. 1-x" _ g, unfolded o_def]) ``` lp15@61190 ` 629` ``` apply (auto simp: C1_differentiable_on_eq) ``` lp15@61190 ` 630` ``` apply (rule continuous_intros, force) ``` lp15@61190 ` 631` ``` apply (force elim!: continuous_on_subset) ``` lp15@61190 ` 632` ``` apply (simp add: finite_vimageI inj_on_def) ``` lp15@60809 ` 633` ``` done ``` lp15@60809 ` 634` ``` then show ?thesis using assms ``` lp15@61190 ` 635` ``` by (auto simp: valid_path_def piecewise_C1_differentiable_on_def path_def [symmetric]) ``` lp15@60809 ` 636` ```qed ``` lp15@60809 ` 637` lp15@60809 ` 638` ```lemma valid_path_reversepath: "valid_path(reversepath g) \ valid_path g" ``` lp15@60809 ` 639` ``` using valid_path_imp_reverse by force ``` lp15@60809 ` 640` lp15@60809 ` 641` ```lemma has_path_integral_reversepath: ``` lp15@60809 ` 642` ``` assumes "valid_path g" "(f has_path_integral i) g" ``` lp15@60809 ` 643` ``` shows "(f has_path_integral (-i)) (reversepath g)" ``` lp15@60809 ` 644` ```proof - ``` lp15@60809 ` 645` ``` { fix s x ``` lp15@61190 ` 646` ``` assume xs: "g C1_differentiable_on ({0..1} - s)" "x \ op - 1 ` s" "0 \ x" "x \ 1" ``` lp15@60809 ` 647` ``` have "vector_derivative (\x. g (1 - x)) (at x within {0..1}) = ``` lp15@60809 ` 648` ``` - vector_derivative g (at (1 - x) within {0..1})" ``` lp15@60809 ` 649` ``` proof - ``` lp15@60809 ` 650` ``` obtain f' where f': "(g has_vector_derivative f') (at (1 - x))" ``` lp15@60809 ` 651` ``` using xs ``` lp15@61190 ` 652` ``` by (force simp: has_vector_derivative_def C1_differentiable_on_def) ``` lp15@60809 ` 653` ``` have "(g o (\x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)" ``` lp15@60809 ` 654` ``` apply (rule vector_diff_chain_within) ``` lp15@60809 ` 655` ``` apply (intro vector_diff_chain_within derivative_eq_intros | simp)+ ``` lp15@60809 ` 656` ``` apply (rule has_vector_derivative_at_within [OF f']) ``` lp15@60809 ` 657` ``` done ``` lp15@60809 ` 658` ``` then have mf': "((\x. g (1 - x)) has_vector_derivative -f') (at x)" ``` lp15@60809 ` 659` ``` by (simp add: o_def) ``` lp15@60809 ` 660` ``` show ?thesis ``` lp15@60809 ` 661` ``` using xs ``` lp15@60809 ` 662` ``` by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f']) ``` lp15@60809 ` 663` ``` qed ``` lp15@60809 ` 664` ``` } note * = this ``` lp15@60809 ` 665` ``` have 01: "{0..1::real} = cbox 0 1" ``` lp15@60809 ` 666` ``` by simp ``` lp15@60809 ` 667` ``` show ?thesis using assms ``` lp15@60809 ` 668` ``` apply (auto simp: has_path_integral_def) ``` lp15@60809 ` 669` ``` apply (drule has_integral_affinity01 [where m= "-1" and c=1]) ``` lp15@61190 ` 670` ``` apply (auto simp: reversepath_def valid_path_def piecewise_C1_differentiable_on_def) ``` lp15@60809 ` 671` ``` apply (drule has_integral_neg) ``` lp15@60809 ` 672` ``` apply (rule_tac s = "(\x. 1 - x) ` s" in has_integral_spike_finite) ``` lp15@60809 ` 673` ``` apply (auto simp: *) ``` lp15@60809 ` 674` ``` done ``` lp15@60809 ` 675` ```qed ``` lp15@60809 ` 676` lp15@60809 ` 677` ```lemma path_integrable_reversepath: ``` lp15@60809 ` 678` ``` "valid_path g \ f path_integrable_on g \ f path_integrable_on (reversepath g)" ``` lp15@60809 ` 679` ``` using has_path_integral_reversepath path_integrable_on_def by blast ``` lp15@60809 ` 680` lp15@60809 ` 681` ```lemma path_integrable_reversepath_eq: ``` lp15@60809 ` 682` ``` "valid_path g \ (f path_integrable_on (reversepath g) \ f path_integrable_on g)" ``` lp15@60809 ` 683` ``` using path_integrable_reversepath valid_path_reversepath by fastforce ``` lp15@60809 ` 684` lp15@60809 ` 685` ```lemma path_integral_reversepath: ``` lp15@60809 ` 686` ``` "\valid_path g; f path_integrable_on g\ \ path_integral (reversepath g) f = -(path_integral g f)" ``` lp15@60809 ` 687` ``` using has_path_integral_reversepath path_integrable_on_def path_integral_unique by blast ``` lp15@60809 ` 688` lp15@60809 ` 689` lp15@60809 ` 690` ```subsection\Joining two paths together\ ``` lp15@60809 ` 691` lp15@60809 ` 692` ```lemma valid_path_join: ``` lp15@60809 ` 693` ``` assumes "valid_path g1" "valid_path g2" "pathfinish g1 = pathstart g2" ``` lp15@60809 ` 694` ``` shows "valid_path(g1 +++ g2)" ``` lp15@60809 ` 695` ```proof - ``` lp15@60809 ` 696` ``` have "g1 1 = g2 0" ``` lp15@60809 ` 697` ``` using assms by (auto simp: pathfinish_def pathstart_def) ``` lp15@61190 ` 698` ``` moreover have "(g1 o (\x. 2*x)) piecewise_C1_differentiable_on {0..1/2}" ``` lp15@61190 ` 699` ``` apply (rule piecewise_C1_differentiable_compose) ``` lp15@60809 ` 700` ``` using assms ``` lp15@61190 ` 701` ``` apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths) ``` lp15@60809 ` 702` ``` apply (rule continuous_intros | simp)+ ``` lp15@60809 ` 703` ``` apply (force intro: finite_vimageI [where h = "op*2"] inj_onI) ``` lp15@60809 ` 704` ``` done ``` lp15@61190 ` 705` ``` moreover have "(g2 o (\x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}" ``` lp15@61190 ` 706` ``` apply (rule piecewise_C1_differentiable_compose) ``` lp15@61190 ` 707` ``` using assms unfolding valid_path_def piecewise_C1_differentiable_on_def ``` lp15@61190 ` 708` ``` by (auto intro!: continuous_intros finite_vimageI [where h = "(\x. 2*x - 1)"] inj_onI ``` lp15@61190 ` 709` ``` simp: image_affinity_atLeastAtMost_diff continuous_on_joinpaths) ``` lp15@60809 ` 710` ``` ultimately show ?thesis ``` lp15@60809 ` 711` ``` apply (simp only: valid_path_def continuous_on_joinpaths joinpaths_def) ``` lp15@61190 ` 712` ``` apply (rule piecewise_C1_differentiable_cases) ``` lp15@60809 ` 713` ``` apply (auto simp: o_def) ``` lp15@60809 ` 714` ``` done ``` lp15@60809 ` 715` ```qed ``` lp15@60809 ` 716` lp15@61190 ` 717` ```lemma valid_path_join_D1: ``` lp15@61190 ` 718` ``` fixes g1 :: "real \ 'a::real_normed_field" ``` lp15@61190 ` 719` ``` shows "valid_path (g1 +++ g2) \ valid_path g1" ``` lp15@61190 ` 720` ``` unfolding valid_path_def ``` lp15@61190 ` 721` ``` by (rule piecewise_C1_differentiable_D1) ``` lp15@60809 ` 722` lp15@61190 ` 723` ```lemma valid_path_join_D2: ``` lp15@61190 ` 724` ``` fixes g2 :: "real \ 'a::real_normed_field" ``` lp15@61190 ` 725` ``` shows "\valid_path (g1 +++ g2); pathfinish g1 = pathstart g2\ \ valid_path g2" ``` lp15@61190 ` 726` ``` unfolding valid_path_def ``` lp15@61190 ` 727` ``` by (rule piecewise_C1_differentiable_D2) ``` lp15@60809 ` 728` lp15@60809 ` 729` ```lemma valid_path_join_eq [simp]: ``` lp15@61190 ` 730` ``` fixes g2 :: "real \ 'a::real_normed_field" ``` lp15@61190 ` 731` ``` shows "pathfinish g1 = pathstart g2 \ (valid_path(g1 +++ g2) \ valid_path g1 \ valid_path g2)" ``` lp15@60809 ` 732` ``` using valid_path_join_D1 valid_path_join_D2 valid_path_join by blast ``` lp15@60809 ` 733` lp15@60809 ` 734` ```lemma has_path_integral_join: ``` lp15@60809 ` 735` ``` assumes "(f has_path_integral i1) g1" "(f has_path_integral i2) g2" ``` lp15@60809 ` 736` ``` "valid_path g1" "valid_path g2" ``` lp15@60809 ` 737` ``` shows "(f has_path_integral (i1 + i2)) (g1 +++ g2)" ``` lp15@60809 ` 738` ```proof - ``` lp15@60809 ` 739` ``` obtain s1 s2 ``` lp15@60809 ` 740` ``` where s1: "finite s1" "\x\{0..1} - s1. g1 differentiable at x" ``` lp15@60809 ` 741` ``` and s2: "finite s2" "\x\{0..1} - s2. g2 differentiable at x" ``` lp15@60809 ` 742` ``` using assms ``` lp15@61190 ` 743` ``` by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq) ``` lp15@60809 ` 744` ``` have 1: "((\x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}" ``` lp15@60809 ` 745` ``` and 2: "((\x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}" ``` lp15@60809 ` 746` ``` using assms ``` lp15@60809 ` 747` ``` by (auto simp: has_path_integral) ``` lp15@60809 ` 748` ``` have i1: "((\x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}" ``` lp15@60809 ` 749` ``` and i2: "((\x. (2*f (g2 (2*x - 1))) * vector_derivative g2 (at (2*x - 1))) has_integral i2) {1/2..1}" ``` lp15@60809 ` 750` ``` using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]] ``` lp15@60809 ` 751` ``` has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]] ``` lp15@60809 ` 752` ``` by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac) ``` lp15@60809 ` 753` ``` have g1: "\0 \ z; z*2 < 1; z*2 \ s1\ \ ``` lp15@60809 ` 754` ``` vector_derivative (\x. if x*2 \ 1 then g1 (2*x) else g2 (2*x - 1)) (at z) = ``` lp15@60809 ` 755` ``` 2 *\<^sub>R vector_derivative g1 (at (z*2))" for z ``` lp15@60809 ` 756` ``` apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\z - 1/2\" _ "(\x. g1(2*x))"]]) ``` lp15@60809 ` 757` ``` apply (simp_all add: dist_real_def abs_if split: split_if_asm) ``` lp15@60809 ` 758` ``` apply (rule vector_diff_chain_at [of "\x. 2*x" 2 _ g1, simplified o_def]) ``` lp15@60809 ` 759` ``` apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left) ``` lp15@60809 ` 760` ``` using s1 ``` lp15@60809 ` 761` ``` apply (auto simp: algebra_simps vector_derivative_works) ``` lp15@60809 ` 762` ``` done ``` lp15@60809 ` 763` ``` have g2: "\1 < z*2; z \ 1; z*2 - 1 \ s2\ \ ``` lp15@60809 ` 764` ``` vector_derivative (\x. if x*2 \ 1 then g1 (2*x) else g2 (2*x - 1)) (at z) = ``` lp15@60809 ` 765` ``` 2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z ``` lp15@60809 ` 766` ``` apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\z - 1/2\" _ "(\x. g2 (2*x - 1))"]]) ``` lp15@60809 ` 767` ``` apply (simp_all add: dist_real_def abs_if split: split_if_asm) ``` lp15@60809 ` 768` ``` apply (rule vector_diff_chain_at [of "\x. 2*x - 1" 2 _ g2, simplified o_def]) ``` lp15@60809 ` 769` ``` apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left) ``` lp15@60809 ` 770` ``` using s2 ``` lp15@60809 ` 771` ``` apply (auto simp: algebra_simps vector_derivative_works) ``` lp15@60809 ` 772` ``` done ``` lp15@60809 ` 773` ``` have "((\x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}" ``` lp15@60809 ` 774` ``` apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) (op*2 -` s1)"]) ``` lp15@60809 ` 775` ``` using s1 ``` lp15@60809 ` 776` ``` apply (force intro: finite_vimageI [where h = "op*2"] inj_onI) ``` lp15@60809 ` 777` ``` apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1) ``` lp15@60809 ` 778` ``` done ``` lp15@60809 ` 779` ``` moreover have "((\x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}" ``` lp15@60809 ` 780` ``` apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\x. 2*x-1) -` s2)"]) ``` lp15@60809 ` 781` ``` using s2 ``` lp15@60809 ` 782` ``` apply (force intro: finite_vimageI [where h = "\x. 2*x-1"] inj_onI) ``` lp15@60809 ` 783` ``` apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2) ``` lp15@60809 ` 784` ``` done ``` lp15@60809 ` 785` ``` ultimately ``` lp15@60809 ` 786` ``` show ?thesis ``` lp15@60809 ` 787` ``` apply (simp add: has_path_integral) ``` lp15@60809 ` 788` ``` apply (rule has_integral_combine [where c = "1/2"], auto) ``` lp15@60809 ` 789` ``` done ``` lp15@60809 ` 790` ```qed ``` lp15@60809 ` 791` lp15@60809 ` 792` ```lemma path_integrable_joinI: ``` lp15@60809 ` 793` ``` assumes "f path_integrable_on g1" "f path_integrable_on g2" ``` lp15@60809 ` 794` ``` "valid_path g1" "valid_path g2" ``` lp15@60809 ` 795` ``` shows "f path_integrable_on (g1 +++ g2)" ``` lp15@60809 ` 796` ``` using assms ``` lp15@60809 ` 797` ``` by (meson has_path_integral_join path_integrable_on_def) ``` lp15@60809 ` 798` lp15@60809 ` 799` ```lemma path_integrable_joinD1: ``` lp15@60809 ` 800` ``` assumes "f path_integrable_on (g1 +++ g2)" "valid_path g1" ``` lp15@60809 ` 801` ``` shows "f path_integrable_on g1" ``` lp15@60809 ` 802` ```proof - ``` lp15@60809 ` 803` ``` obtain s1 ``` lp15@60809 ` 804` ``` where s1: "finite s1" "\x\{0..1} - s1. g1 differentiable at x" ``` lp15@61190 ` 805` ``` using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq) ``` lp15@60809 ` 806` ``` have "(\x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}" ``` lp15@60809 ` 807` ``` using assms ``` lp15@60809 ` 808` ``` apply (auto simp: path_integrable_on) ``` lp15@60809 ` 809` ``` apply (drule integrable_on_subcbox [where a=0 and b="1/2"]) ``` lp15@60809 ` 810` ``` apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified]) ``` lp15@60809 ` 811` ``` done ``` lp15@60809 ` 812` ``` then have *: "(\x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}" ``` lp15@61190 ` 813` ``` by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real) ``` lp15@60809 ` 814` ``` have g1: "\0 < z; z < 1; z \ s1\ \ ``` lp15@60809 ` 815` ``` vector_derivative (\x. if x*2 \ 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) = ``` lp15@60809 ` 816` ``` 2 *\<^sub>R vector_derivative g1 (at z)" for z ``` lp15@60809 ` 817` ``` apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\(z-1)/2\" _ "(\x. g1(2*x))"]]) ``` lp15@60809 ` 818` ``` apply (simp_all add: field_simps dist_real_def abs_if split: split_if_asm) ``` lp15@60809 ` 819` ``` apply (rule vector_diff_chain_at [of "\x. x*2" 2 _ g1, simplified o_def]) ``` lp15@60809 ` 820` ``` using s1 ``` lp15@60809 ` 821` ``` apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left) ``` lp15@60809 ` 822` ``` done ``` lp15@60809 ` 823` ``` show ?thesis ``` lp15@60809 ` 824` ``` using s1 ``` lp15@60809 ` 825` ``` apply (auto simp: path_integrable_on) ``` lp15@60809 ` 826` ``` apply (rule integrable_spike_finite [of "{0,1} \ s1", OF _ _ *]) ``` lp15@60809 ` 827` ``` apply (auto simp: joinpaths_def scaleR_conv_of_real g1) ``` lp15@60809 ` 828` ``` done ``` lp15@60809 ` 829` ```qed ``` lp15@60809 ` 830` lp15@61190 ` 831` ```lemma path_integrable_joinD2: ``` lp15@60809 ` 832` ``` assumes "f path_integrable_on (g1 +++ g2)" "valid_path g2" ``` lp15@60809 ` 833` ``` shows "f path_integrable_on g2" ``` lp15@60809 ` 834` ```proof - ``` lp15@60809 ` 835` ``` obtain s2 ``` lp15@60809 ` 836` ``` where s2: "finite s2" "\x\{0..1} - s2. g2 differentiable at x" ``` lp15@61190 ` 837` ``` using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq) ``` lp15@60809 ` 838` ``` have "(\x. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}" ``` lp15@60809 ` 839` ``` using assms ``` lp15@60809 ` 840` ``` apply (auto simp: path_integrable_on) ``` lp15@60809 ` 841` ``` apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto) ``` lp15@60809 ` 842` ``` apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified]) ``` lp15@60809 ` 843` ``` apply (simp add: image_affinity_atLeastAtMost_diff) ``` lp15@60809 ` 844` ``` done ``` lp15@60809 ` 845` ``` then have *: "(\x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) ``` lp15@60809 ` 846` ``` integrable_on {0..1}" ``` lp15@60809 ` 847` ``` by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real) ``` lp15@60809 ` 848` ``` have g2: "\0 < z; z < 1; z \ s2\ \ ``` lp15@60809 ` 849` ``` vector_derivative (\x. if x*2 \ 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) = ``` lp15@60809 ` 850` ``` 2 *\<^sub>R vector_derivative g2 (at z)" for z ``` lp15@60809 ` 851` ``` apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\z/2\" _ "(\x. g2(2*x-1))"]]) ``` lp15@60809 ` 852` ``` apply (simp_all add: field_simps dist_real_def abs_if split: split_if_asm) ``` lp15@60809 ` 853` ``` apply (rule vector_diff_chain_at [of "\x. x*2-1" 2 _ g2, simplified o_def]) ``` lp15@60809 ` 854` ``` using s2 ``` lp15@60809 ` 855` ``` apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left ``` lp15@60809 ` 856` ``` vector_derivative_works add_divide_distrib) ``` lp15@60809 ` 857` ``` done ``` lp15@60809 ` 858` ``` show ?thesis ``` lp15@60809 ` 859` ``` using s2 ``` lp15@60809 ` 860` ``` apply (auto simp: path_integrable_on) ``` lp15@60809 ` 861` ``` apply (rule integrable_spike_finite [of "{0,1} \ s2", OF _ _ *]) ``` lp15@60809 ` 862` ``` apply (auto simp: joinpaths_def scaleR_conv_of_real g2) ``` lp15@60809 ` 863` ``` done ``` lp15@60809 ` 864` ```qed ``` lp15@60809 ` 865` lp15@60809 ` 866` ```lemma path_integrable_join [simp]: ``` lp15@60809 ` 867` ``` shows ``` lp15@60809 ` 868` ``` "\valid_path g1; valid_path g2\ ``` lp15@60809 ` 869` ``` \ f path_integrable_on (g1 +++ g2) \ f path_integrable_on g1 \ f path_integrable_on g2" ``` lp15@60809 ` 870` ```using path_integrable_joinD1 path_integrable_joinD2 path_integrable_joinI by blast ``` lp15@60809 ` 871` lp15@60809 ` 872` ```lemma path_integral_join [simp]: ``` lp15@60809 ` 873` ``` shows ``` lp15@60809 ` 874` ``` "\f path_integrable_on g1; f path_integrable_on g2; valid_path g1; valid_path g2\ ``` lp15@60809 ` 875` ``` \ path_integral (g1 +++ g2) f = path_integral g1 f + path_integral g2 f" ``` lp15@60809 ` 876` ``` by (simp add: has_path_integral_integral has_path_integral_join path_integral_unique) ``` lp15@60809 ` 877` lp15@60809 ` 878` lp15@60809 ` 879` ```subsection\Shifting the starting point of a (closed) path\ ``` lp15@60809 ` 880` lp15@60809 ` 881` ```lemma shiftpath_alt_def: "shiftpath a f = (\x. if x \ 1-a then f (a + x) else f (a + x - 1))" ``` lp15@60809 ` 882` ``` by (auto simp: shiftpath_def) ``` lp15@60809 ` 883` lp15@60809 ` 884` ```lemma valid_path_shiftpath [intro]: ``` lp15@60809 ` 885` ``` assumes "valid_path g" "pathfinish g = pathstart g" "a \ {0..1}" ``` lp15@60809 ` 886` ``` shows "valid_path(shiftpath a g)" ``` lp15@60809 ` 887` ``` using assms ``` lp15@60809 ` 888` ``` apply (auto simp: valid_path_def shiftpath_alt_def) ``` lp15@61190 ` 889` ``` apply (rule piecewise_C1_differentiable_cases) ``` lp15@60809 ` 890` ``` apply (auto simp: algebra_simps) ``` lp15@61190 ` 891` ``` apply (rule piecewise_C1_differentiable_affine [of g 1 a, simplified o_def scaleR_one]) ``` lp15@61190 ` 892` ``` apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset) ``` lp15@61190 ` 893` ``` apply (rule piecewise_C1_differentiable_affine [of g 1 "a-1", simplified o_def scaleR_one algebra_simps]) ``` lp15@61190 ` 894` ``` apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset) ``` lp15@60809 ` 895` ``` done ``` lp15@60809 ` 896` lp15@60809 ` 897` ```lemma has_path_integral_shiftpath: ``` lp15@60809 ` 898` ``` assumes f: "(f has_path_integral i) g" "valid_path g" ``` lp15@60809 ` 899` ``` and a: "a \ {0..1}" ``` lp15@60809 ` 900` ``` shows "(f has_path_integral i) (shiftpath a g)" ``` lp15@60809 ` 901` ```proof - ``` lp15@60809 ` 902` ``` obtain s ``` lp15@60809 ` 903` ``` where s: "finite s" and g: "\x\{0..1} - s. g differentiable at x" ``` lp15@61190 ` 904` ``` using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq) ``` lp15@60809 ` 905` ``` have *: "((\x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}" ``` lp15@60809 ` 906` ``` using assms by (auto simp: has_path_integral) ``` lp15@60809 ` 907` ``` then have i: "i = integral {a..1} (\x. f (g x) * vector_derivative g (at x)) + ``` lp15@60809 ` 908` ``` integral {0..a} (\x. f (g x) * vector_derivative g (at x))" ``` lp15@60809 ` 909` ``` apply (rule has_integral_unique) ``` lp15@60809 ` 910` ``` apply (subst add.commute) ``` lp15@60809 ` 911` ``` apply (subst Integration.integral_combine) ``` lp15@60809 ` 912` ``` using assms * integral_unique by auto ``` lp15@60809 ` 913` ``` { fix x ``` lp15@60809 ` 914` ``` have "0 \ x \ x + a < 1 \ x \ (\x. x - a) ` s \ ``` lp15@60809 ` 915` ``` vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))" ``` lp15@60809 ` 916` ``` unfolding shiftpath_def ``` lp15@60809 ` 917` ``` apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "dist(1-a) x" _ "(\x. g(a+x))"]]) ``` lp15@60809 ` 918` ``` apply (auto simp: field_simps dist_real_def abs_if split: split_if_asm) ``` lp15@60809 ` 919` ``` apply (rule vector_diff_chain_at [of "\x. x+a" 1 _ g, simplified o_def scaleR_one]) ``` lp15@60809 ` 920` ``` apply (intro derivative_eq_intros | simp)+ ``` lp15@60809 ` 921` ``` using g ``` lp15@60809 ` 922` ``` apply (drule_tac x="x+a" in bspec) ``` lp15@60809 ` 923` ``` using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute) ``` lp15@60809 ` 924` ``` done ``` lp15@60809 ` 925` ``` } note vd1 = this ``` lp15@60809 ` 926` ``` { fix x ``` lp15@60809 ` 927` ``` have "1 < x + a \ x \ 1 \ x \ (\x. x - a + 1) ` s \ ``` lp15@60809 ` 928` ``` vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))" ``` lp15@60809 ` 929` ``` unfolding shiftpath_def ``` lp15@60809 ` 930` ``` apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "dist (1-a) x" _ "(\x. g(a+x-1))"]]) ``` lp15@60809 ` 931` ``` apply (auto simp: field_simps dist_real_def abs_if split: split_if_asm) ``` lp15@60809 ` 932` ``` apply (rule vector_diff_chain_at [of "\x. x+a-1" 1 _ g, simplified o_def scaleR_one]) ``` lp15@60809 ` 933` ``` apply (intro derivative_eq_intros | simp)+ ``` lp15@60809 ` 934` ``` using g ``` lp15@60809 ` 935` ``` apply (drule_tac x="x+a-1" in bspec) ``` lp15@60809 ` 936` ``` using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute) ``` lp15@60809 ` 937` ``` done ``` lp15@60809 ` 938` ``` } note vd2 = this ``` lp15@60809 ` 939` ``` have va1: "(\x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})" ``` lp15@60809 ` 940` ``` using * a by (fastforce intro: integrable_subinterval_real) ``` lp15@60809 ` 941` ``` have v0a: "(\x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})" ``` lp15@60809 ` 942` ``` apply (rule integrable_subinterval_real) ``` lp15@60809 ` 943` ``` using * a by auto ``` lp15@60809 ` 944` ``` have "((\x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x)) ``` lp15@60809 ` 945` ``` has_integral integral {a..1} (\x. f (g x) * vector_derivative g (at x))) {0..1 - a}" ``` lp15@60809 ` 946` ``` apply (rule has_integral_spike_finite ``` lp15@60809 ` 947` ``` [where s = "{1-a} \ (\x. x-a) ` s" and f = "\x. f(g(a+x)) * vector_derivative g (at(a+x))"]) ``` lp15@60809 ` 948` ``` using s apply blast ``` lp15@60809 ` 949` ``` using a apply (auto simp: algebra_simps vd1) ``` lp15@60809 ` 950` ``` apply (force simp: shiftpath_def add.commute) ``` lp15@60809 ` 951` ``` using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]] ``` lp15@60809 ` 952` ``` apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute) ``` lp15@60809 ` 953` ``` done ``` lp15@60809 ` 954` ``` moreover ``` lp15@60809 ` 955` ``` have "((\x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x)) ``` lp15@60809 ` 956` ``` has_integral integral {0..a} (\x. f (g x) * vector_derivative g (at x))) {1 - a..1}" ``` lp15@60809 ` 957` ``` apply (rule has_integral_spike_finite ``` lp15@60809 ` 958` ``` [where s = "{1-a} \ (\x. x-a+1) ` s" and f = "\x. f(g(a+x-1)) * vector_derivative g (at(a+x-1))"]) ``` lp15@60809 ` 959` ``` using s apply blast ``` lp15@60809 ` 960` ``` using a apply (auto simp: algebra_simps vd2) ``` lp15@60809 ` 961` ``` apply (force simp: shiftpath_def add.commute) ``` lp15@60809 ` 962` ``` using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]] ``` lp15@60809 ` 963` ``` apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified]) ``` lp15@60809 ` 964` ``` apply (simp add: algebra_simps) ``` lp15@60809 ` 965` ``` done ``` lp15@60809 ` 966` ``` ultimately show ?thesis ``` lp15@60809 ` 967` ``` using a ``` lp15@60809 ` 968` ``` by (auto simp: i has_path_integral intro: has_integral_combine [where c = "1-a"]) ``` lp15@60809 ` 969` ```qed ``` lp15@60809 ` 970` lp15@60809 ` 971` ```lemma has_path_integral_shiftpath_D: ``` lp15@60809 ` 972` ``` assumes "(f has_path_integral i) (shiftpath a g)" ``` lp15@60809 ` 973` ``` "valid_path g" "pathfinish g = pathstart g" "a \ {0..1}" ``` lp15@60809 ` 974` ``` shows "(f has_path_integral i) g" ``` lp15@60809 ` 975` ```proof - ``` lp15@60809 ` 976` ``` obtain s ``` lp15@60809 ` 977` ``` where s: "finite s" and g: "\x\{0..1} - s. g differentiable at x" ``` lp15@61190 ` 978` ``` using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq) ``` lp15@60809 ` 979` ``` { fix x ``` lp15@60809 ` 980` ``` assume x: "0 < x" "x < 1" "x \ s" ``` lp15@60809 ` 981` ``` then have gx: "g differentiable at x" ``` lp15@60809 ` 982` ``` using g by auto ``` lp15@60809 ` 983` ``` have "vector_derivative g (at x within {0..1}) = ``` lp15@60809 ` 984` ``` vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})" ``` lp15@60809 ` 985` ``` apply (rule vector_derivative_at_within_ivl ``` lp15@60809 ` 986` ``` [OF has_vector_derivative_transform_within_open ``` lp15@60809 ` 987` ``` [of "{0<..<1}-s" _ "(shiftpath (1 - a) (shiftpath a g))"]]) ``` lp15@60809 ` 988` ``` using s g assms x ``` lp15@60809 ` 989` ``` apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath ``` lp15@60809 ` 990` ``` vector_derivative_within_interior vector_derivative_works [symmetric]) ``` lp15@60809 ` 991` ``` apply (rule Derivative.differentiable_transform_at [of "min x (1-x)", OF _ _ gx]) ``` lp15@60809 ` 992` ``` apply (auto simp: dist_real_def shiftpath_shiftpath abs_if) ``` lp15@60809 ` 993` ``` done ``` lp15@60809 ` 994` ``` } note vd = this ``` lp15@60809 ` 995` ``` have fi: "(f has_path_integral i) (shiftpath (1 - a) (shiftpath a g))" ``` lp15@60809 ` 996` ``` using assms by (auto intro!: has_path_integral_shiftpath) ``` lp15@60809 ` 997` ``` show ?thesis ``` lp15@60809 ` 998` ``` apply (simp add: has_path_integral_def) ``` lp15@60809 ` 999` ``` apply (rule has_integral_spike_finite [of "{0,1} \ s", OF _ _ fi [unfolded has_path_integral_def]]) ``` lp15@60809 ` 1000` ``` using s assms vd ``` lp15@60809 ` 1001` ``` apply (auto simp: Path_Connected.shiftpath_shiftpath) ``` lp15@60809 ` 1002` ``` done ``` lp15@60809 ` 1003` ```qed ``` lp15@60809 ` 1004` lp15@60809 ` 1005` ```lemma has_path_integral_shiftpath_eq: ``` lp15@60809 ` 1006` ``` assumes "valid_path g" "pathfinish g = pathstart g" "a \ {0..1}" ``` lp15@60809 ` 1007` ``` shows "(f has_path_integral i) (shiftpath a g) \ (f has_path_integral i) g" ``` lp15@60809 ` 1008` ``` using assms has_path_integral_shiftpath has_path_integral_shiftpath_D by blast ``` lp15@60809 ` 1009` lp15@60809 ` 1010` ```lemma path_integral_shiftpath: ``` lp15@60809 ` 1011` ``` assumes "valid_path g" "pathfinish g = pathstart g" "a \ {0..1}" ``` lp15@60809 ` 1012` ``` shows "path_integral (shiftpath a g) f = path_integral g f" ``` lp15@60809 ` 1013` ``` using assms by (simp add: path_integral_def has_path_integral_shiftpath_eq) ``` lp15@60809 ` 1014` lp15@60809 ` 1015` lp15@60809 ` 1016` ```subsection\More about straight-line paths\ ``` lp15@60809 ` 1017` lp15@60809 ` 1018` ```lemma has_vector_derivative_linepath_within: ``` lp15@60809 ` 1019` ``` "(linepath a b has_vector_derivative (b - a)) (at x within s)" ``` lp15@60809 ` 1020` ```apply (simp add: linepath_def has_vector_derivative_def algebra_simps) ``` lp15@60809 ` 1021` ```apply (rule derivative_eq_intros | simp)+ ``` lp15@60809 ` 1022` ```done ``` lp15@60809 ` 1023` lp15@60809 ` 1024` ```lemma vector_derivative_linepath_within: ``` lp15@60809 ` 1025` ``` "x \ {0..1} \ vector_derivative (linepath a b) (at x within {0..1}) = b - a" ``` lp15@60809 ` 1026` ``` apply (rule vector_derivative_within_closed_interval [of 0 "1::real", simplified]) ``` lp15@60809 ` 1027` ``` apply (auto simp: has_vector_derivative_linepath_within) ``` lp15@60809 ` 1028` ``` done ``` lp15@60809 ` 1029` lp15@61190 ` 1030` ```lemma vector_derivative_linepath_at [simp]: "vector_derivative (linepath a b) (at x) = b - a" ``` lp15@60809 ` 1031` ``` by (simp add: has_vector_derivative_linepath_within vector_derivative_at) ``` lp15@60809 ` 1032` lp15@61190 ` 1033` ```lemma valid_path_linepath [iff]: "valid_path (linepath a b)" ``` lp15@61190 ` 1034` ``` apply (simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_on_linepath) ``` lp15@61190 ` 1035` ``` apply (rule_tac x="{}" in exI) ``` lp15@61190 ` 1036` ``` apply (simp add: differentiable_on_def differentiable_def) ``` lp15@61190 ` 1037` ``` using has_vector_derivative_def has_vector_derivative_linepath_within ``` lp15@61190 ` 1038` ``` apply (fastforce simp add: continuous_on_eq_continuous_within) ``` lp15@61190 ` 1039` ``` done ``` lp15@61190 ` 1040` lp15@60809 ` 1041` ```lemma has_path_integral_linepath: ``` lp15@60809 ` 1042` ``` shows "(f has_path_integral i) (linepath a b) \ ``` lp15@60809 ` 1043` ``` ((\x. f(linepath a b x) * (b - a)) has_integral i) {0..1}" ``` lp15@60809 ` 1044` ``` by (simp add: has_path_integral vector_derivative_linepath_at) ``` lp15@60809 ` 1045` lp15@60809 ` 1046` ```lemma linepath_in_path: ``` lp15@60809 ` 1047` ``` shows "x \ {0..1} \ linepath a b x \ closed_segment a b" ``` lp15@60809 ` 1048` ``` by (auto simp: segment linepath_def) ``` lp15@60809 ` 1049` lp15@60809 ` 1050` ```lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b" ``` lp15@60809 ` 1051` ``` by (auto simp: segment linepath_def) ``` lp15@60809 ` 1052` lp15@60809 ` 1053` ```lemma linepath_in_convex_hull: ``` lp15@60809 ` 1054` ``` fixes x::real ``` lp15@60809 ` 1055` ``` assumes a: "a \ convex hull s" ``` lp15@60809 ` 1056` ``` and b: "b \ convex hull s" ``` lp15@60809 ` 1057` ``` and x: "0\x" "x\1" ``` lp15@60809 ` 1058` ``` shows "linepath a b x \ convex hull s" ``` lp15@60809 ` 1059` ``` apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD]) ``` lp15@60809 ` 1060` ``` using x ``` lp15@60809 ` 1061` ``` apply (auto simp: linepath_image_01 [symmetric]) ``` lp15@60809 ` 1062` ``` done ``` lp15@60809 ` 1063` lp15@60809 ` 1064` ```lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b" ``` lp15@60809 ` 1065` ``` by (simp add: linepath_def) ``` lp15@60809 ` 1066` lp15@60809 ` 1067` ```lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0" ``` lp15@60809 ` 1068` ``` by (simp add: linepath_def) ``` lp15@60809 ` 1069` lp15@60809 ` 1070` ```lemma linepath_of_real: "(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)" ``` lp15@60809 ` 1071` ``` by (simp add: scaleR_conv_of_real linepath_def) ``` lp15@60809 ` 1072` lp15@60809 ` 1073` ```lemma of_real_linepath: "of_real (linepath a b x) = linepath (of_real a) (of_real b) x" ``` lp15@60809 ` 1074` ``` by (metis linepath_of_real mult.right_neutral of_real_def real_scaleR_def) ``` lp15@60809 ` 1075` lp15@60809 ` 1076` ```lemma has_path_integral_trivial [iff]: "(f has_path_integral 0) (linepath a a)" ``` lp15@60809 ` 1077` ``` by (simp add: has_path_integral_linepath) ``` lp15@60809 ` 1078` lp15@60809 ` 1079` ```lemma path_integral_trivial [simp]: "path_integral (linepath a a) f = 0" ``` lp15@60809 ` 1080` ``` using has_path_integral_trivial path_integral_unique by blast ``` lp15@60809 ` 1081` lp15@60809 ` 1082` lp15@60809 ` 1083` ```subsection\Relation to subpath construction\ ``` lp15@60809 ` 1084` lp15@60809 ` 1085` ```lemma valid_path_subpath: ``` lp15@60809 ` 1086` ``` fixes g :: "real \ 'a :: real_normed_vector" ``` lp15@60809 ` 1087` ``` assumes "valid_path g" "u \ {0..1}" "v \ {0..1}" ``` lp15@60809 ` 1088` ``` shows "valid_path(subpath u v g)" ``` lp15@60809 ` 1089` ```proof (cases "v=u") ``` lp15@60809 ` 1090` ``` case True ``` lp15@60809 ` 1091` ``` then show ?thesis ``` lp15@61190 ` 1092` ``` unfolding valid_path_def subpath_def ``` lp15@61190 ` 1093` ``` by (force intro: C1_differentiable_on_const C1_differentiable_imp_piecewise) ``` lp15@60809 ` 1094` ```next ``` lp15@60809 ` 1095` ``` case False ``` lp15@61190 ` 1096` ``` have "(g o (\x. ((v-u) * x + u))) piecewise_C1_differentiable_on {0..1}" ``` lp15@61190 ` 1097` ``` apply (rule piecewise_C1_differentiable_compose) ``` lp15@61190 ` 1098` ``` apply (simp add: C1_differentiable_imp_piecewise) ``` lp15@60809 ` 1099` ``` apply (simp add: image_affinity_atLeastAtMost) ``` lp15@60809 ` 1100` ``` using assms False ``` lp15@61190 ` 1101` ``` apply (auto simp: algebra_simps valid_path_def piecewise_C1_differentiable_on_subset) ``` lp15@60809 ` 1102` ``` apply (subst Int_commute) ``` lp15@60809 ` 1103` ``` apply (auto simp: inj_on_def algebra_simps crossproduct_eq finite_vimage_IntI) ``` lp15@60809 ` 1104` ``` done ``` lp15@60809 ` 1105` ``` then show ?thesis ``` lp15@60809 ` 1106` ``` by (auto simp: o_def valid_path_def subpath_def) ``` lp15@60809 ` 1107` ```qed ``` lp15@60809 ` 1108` lp15@60809 ` 1109` ```lemma has_path_integral_subpath_refl [iff]: "(f has_path_integral 0) (subpath u u g)" ``` lp15@60809 ` 1110` ``` by (simp add: has_path_integral subpath_def) ``` lp15@60809 ` 1111` lp15@60809 ` 1112` ```lemma path_integrable_subpath_refl [iff]: "f path_integrable_on (subpath u u g)" ``` lp15@60809 ` 1113` ``` using has_path_integral_subpath_refl path_integrable_on_def by blast ``` lp15@60809 ` 1114` lp15@60809 ` 1115` ```lemma path_integral_subpath_refl [simp]: "path_integral (subpath u u g) f = 0" ``` lp15@60809 ` 1116` ``` by (simp add: has_path_integral_subpath_refl path_integral_unique) ``` lp15@60809 ` 1117` lp15@60809 ` 1118` ```lemma has_path_integral_subpath: ``` lp15@60809 ` 1119` ``` assumes f: "f path_integrable_on g" and g: "valid_path g" ``` lp15@60809 ` 1120` ``` and uv: "u \ {0..1}" "v \ {0..1}" "u \ v" ``` lp15@60809 ` 1121` ``` shows "(f has_path_integral integral {u..v} (\x. f(g x) * vector_derivative g (at x))) ``` lp15@60809 ` 1122` ``` (subpath u v g)" ``` lp15@60809 ` 1123` ```proof (cases "v=u") ``` lp15@60809 ` 1124` ``` case True ``` lp15@60809 ` 1125` ``` then show ?thesis ``` lp15@60809 ` 1126` ``` using f by (simp add: path_integrable_on_def subpath_def has_path_integral) ``` lp15@60809 ` 1127` ```next ``` lp15@60809 ` 1128` ``` case False ``` lp15@60809 ` 1129` ``` obtain s where s: "\x. x \ {0..1} - s \ g differentiable at x" and fs: "finite s" ``` lp15@61190 ` 1130` ``` using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast ``` lp15@60809 ` 1131` ``` have *: "((\x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u))) ``` lp15@60809 ` 1132` ``` has_integral (1 / (v - u)) * integral {u..v} (\t. f (g t) * vector_derivative g (at t))) ``` lp15@60809 ` 1133` ``` {0..1}" ``` lp15@60809 ` 1134` ``` using f uv ``` lp15@60809 ` 1135` ``` apply (simp add: path_integrable_on subpath_def has_path_integral) ``` lp15@60809 ` 1136` ``` apply (drule integrable_on_subcbox [where a=u and b=v, simplified]) ``` lp15@60809 ` 1137` ``` apply (simp_all add: has_integral_integral) ``` lp15@60809 ` 1138` ``` apply (drule has_integral_affinity [where m="v-u" and c=u, simplified]) ``` lp15@60809 ` 1139` ``` apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real) ``` lp15@60809 ` 1140` ``` apply (simp add: divide_simps False) ``` lp15@60809 ` 1141` ``` done ``` lp15@60809 ` 1142` ``` { fix x ``` lp15@60809 ` 1143` ``` have "x \ {0..1} \ ``` lp15@60809 ` 1144` ``` x \ (\t. (v-u) *\<^sub>R t + u) -` s \ ``` lp15@60809 ` 1145` ``` vector_derivative (\x. g ((v-u) * x + u)) (at x) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u))" ``` lp15@60809 ` 1146` ``` apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]]) ``` lp15@60809 ` 1147` ``` apply (intro derivative_eq_intros | simp)+ ``` lp15@60809 ` 1148` ``` apply (cut_tac s [of "(v - u) * x + u"]) ``` lp15@60809 ` 1149` ``` using uv mult_left_le [of x "v-u"] ``` lp15@60809 ` 1150` ``` apply (auto simp: vector_derivative_works) ``` lp15@60809 ` 1151` ``` done ``` lp15@60809 ` 1152` ``` } note vd = this ``` lp15@60809 ` 1153` ``` show ?thesis ``` lp15@60809 ` 1154` ``` apply (cut_tac has_integral_cmul [OF *, where c = "v-u"]) ``` lp15@60809 ` 1155` ``` using fs assms ``` lp15@60809 ` 1156` ``` apply (simp add: False subpath_def has_path_integral) ``` lp15@60809 ` 1157` ``` apply (rule_tac s = "(\t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite) ``` lp15@60809 ` 1158` ``` apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real) ``` lp15@60809 ` 1159` ``` done ``` lp15@60809 ` 1160` ```qed ``` lp15@60809 ` 1161` lp15@60809 ` 1162` ```lemma path_integrable_subpath: ``` lp15@60809 ` 1163` ``` assumes "f path_integrable_on g" "valid_path g" "u \ {0..1}" "v \ {0..1}" ``` lp15@60809 ` 1164` ``` shows "f path_integrable_on (subpath u v g)" ``` lp15@60809 ` 1165` ``` apply (cases u v rule: linorder_class.le_cases) ``` lp15@60809 ` 1166` ``` apply (metis path_integrable_on_def has_path_integral_subpath [OF assms]) ``` lp15@60809 ` 1167` ``` apply (subst reversepath_subpath [symmetric]) ``` lp15@60809 ` 1168` ``` apply (rule path_integrable_reversepath) ``` lp15@60809 ` 1169` ``` using assms apply (blast intro: valid_path_subpath) ``` lp15@60809 ` 1170` ``` apply (simp add: path_integrable_on_def) ``` lp15@60809 ` 1171` ``` using assms apply (blast intro: has_path_integral_subpath) ``` lp15@60809 ` 1172` ``` done ``` lp15@60809 ` 1173` lp15@60809 ` 1174` ```lemma has_integral_integrable_integral: "(f has_integral i) s \ f integrable_on s \ integral s f = i" ``` lp15@60809 ` 1175` ``` by blast ``` lp15@60809 ` 1176` lp15@60809 ` 1177` ```lemma has_integral_path_integral_subpath: ``` lp15@60809 ` 1178` ``` assumes "f path_integrable_on g" "valid_path g" "u \ {0..1}" "v \ {0..1}" "u \ v" ``` lp15@60809 ` 1179` ``` shows "(((\x. f(g x) * vector_derivative g (at x))) ``` lp15@60809 ` 1180` ``` has_integral path_integral (subpath u v g) f) {u..v}" ``` lp15@60809 ` 1181` ``` using assms ``` lp15@60809 ` 1182` ``` apply (auto simp: has_integral_integrable_integral) ``` lp15@60809 ` 1183` ``` apply (rule integrable_on_subcbox [where a=u and b=v and s = "{0..1}", simplified]) ``` lp15@60809 ` 1184` ``` apply (auto simp: path_integral_unique [OF has_path_integral_subpath] path_integrable_on) ``` lp15@60809 ` 1185` ``` done ``` lp15@60809 ` 1186` lp15@60809 ` 1187` ```lemma path_integral_subpath_integral: ``` lp15@60809 ` 1188` ``` assumes "f path_integrable_on g" "valid_path g" "u \ {0..1}" "v \ {0..1}" "u \ v" ``` lp15@60809 ` 1189` ``` shows "path_integral (subpath u v g) f = ``` lp15@60809 ` 1190` ``` integral {u..v} (\x. f(g x) * vector_derivative g (at x))" ``` lp15@60809 ` 1191` ``` using assms has_path_integral_subpath path_integral_unique by blast ``` lp15@60809 ` 1192` lp15@60809 ` 1193` ```lemma path_integral_subpath_combine_less: ``` lp15@60809 ` 1194` ``` assumes "f path_integrable_on g" "valid_path g" "u \ {0..1}" "v \ {0..1}" "w \ {0..1}" ``` lp15@60809 ` 1195` ``` "u {0..1}" "v \ {0..1}" "w \ {0..1}" ``` lp15@60809 ` 1206` ``` shows "path_integral (subpath u v g) f + path_integral (subpath v w g) f = ``` lp15@60809 ` 1207` ``` path_integral (subpath u w g) f" ``` lp15@60809 ` 1208` ```proof (cases "u\v \ v\w \ u\w") ``` lp15@60809 ` 1209` ``` case True ``` lp15@60809 ` 1210` ``` have *: "subpath v u g = reversepath(subpath u v g) \ ``` lp15@60809 ` 1211` ``` subpath w u g = reversepath(subpath u w g) \ ``` lp15@60809 ` 1212` ``` subpath w v g = reversepath(subpath v w g)" ``` lp15@60809 ` 1213` ``` by (auto simp: reversepath_subpath) ``` lp15@60809 ` 1214` ``` have "u < v \ v < w \ ``` lp15@60809 ` 1215` ``` u < w \ w < v \ ``` lp15@60809 ` 1216` ``` v < u \ u < w \ ``` lp15@60809 ` 1217` ``` v < w \ w < u \ ``` lp15@60809 ` 1218` ``` w < u \ u < v \ ``` lp15@60809 ` 1219` ``` w < v \ v < u" ``` lp15@60809 ` 1220` ``` using True assms by linarith ``` lp15@60809 ` 1221` ``` with assms show ?thesis ``` lp15@60809 ` 1222` ``` using path_integral_subpath_combine_less [of f g u v w] ``` lp15@60809 ` 1223` ``` path_integral_subpath_combine_less [of f g u w v] ``` lp15@60809 ` 1224` ``` path_integral_subpath_combine_less [of f g v u w] ``` lp15@60809 ` 1225` ``` path_integral_subpath_combine_less [of f g v w u] ``` lp15@60809 ` 1226` ``` path_integral_subpath_combine_less [of f g w u v] ``` lp15@60809 ` 1227` ``` path_integral_subpath_combine_less [of f g w v u] ``` lp15@60809 ` 1228` ``` apply simp ``` lp15@60809 ` 1229` ``` apply (elim disjE) ``` lp15@60809 ` 1230` ``` apply (auto simp: * path_integral_reversepath path_integrable_subpath ``` lp15@60809 ` 1231` ``` valid_path_reversepath valid_path_subpath algebra_simps) ``` lp15@60809 ` 1232` ``` done ``` lp15@60809 ` 1233` ```next ``` lp15@60809 ` 1234` ``` case False ``` lp15@60809 ` 1235` ``` then show ?thesis ``` lp15@60809 ` 1236` ``` apply (auto simp: path_integral_subpath_refl) ``` lp15@60809 ` 1237` ``` using assms ``` lp15@60809 ` 1238` ``` by (metis eq_neg_iff_add_eq_0 path_integrable_subpath path_integral_reversepath reversepath_subpath valid_path_subpath) ``` lp15@60809 ` 1239` ```qed ``` lp15@60809 ` 1240` lp15@60809 ` 1241` ```lemma path_integral_integral: ``` lp15@60809 ` 1242` ``` shows "path_integral g f = integral {0..1} (\x. f (g x) * vector_derivative g (at x))" ``` lp15@60809 ` 1243` ``` by (simp add: path_integral_def integral_def has_path_integral) ``` lp15@60809 ` 1244` lp15@60809 ` 1245` lp15@60809 ` 1246` ```subsection\Segments via convex hulls\ ``` lp15@60809 ` 1247` lp15@60809 ` 1248` ```lemma segments_subset_convex_hull: ``` lp15@60809 ` 1249` ``` "closed_segment a b \ (convex hull {a,b,c})" ``` lp15@60809 ` 1250` ``` "closed_segment a c \ (convex hull {a,b,c})" ``` lp15@60809 ` 1251` ``` "closed_segment b c \ (convex hull {a,b,c})" ``` lp15@60809 ` 1252` ``` "closed_segment b a \ (convex hull {a,b,c})" ``` lp15@60809 ` 1253` ``` "closed_segment c a \ (convex hull {a,b,c})" ``` lp15@60809 ` 1254` ``` "closed_segment c b \ (convex hull {a,b,c})" ``` lp15@60809 ` 1255` ```by (auto simp: segment_convex_hull linepath_of_real elim!: rev_subsetD [OF _ hull_mono]) ``` lp15@60809 ` 1256` lp15@60809 ` 1257` ```lemma midpoints_in_convex_hull: ``` lp15@60809 ` 1258` ``` assumes "x \ convex hull s" "y \ convex hull s" ``` lp15@60809 ` 1259` ``` shows "midpoint x y \ convex hull s" ``` lp15@60809 ` 1260` ```proof - ``` lp15@60809 ` 1261` ``` have "(1 - inverse(2)) *\<^sub>R x + inverse(2) *\<^sub>R y \ convex hull s" ``` lp15@61426 ` 1262` ``` apply (rule convexD_alt) ``` lp15@60809 ` 1263` ``` using assms ``` lp15@60809 ` 1264` ``` apply (auto simp: convex_convex_hull) ``` lp15@60809 ` 1265` ``` done ``` lp15@60809 ` 1266` ``` then show ?thesis ``` lp15@60809 ` 1267` ``` by (simp add: midpoint_def algebra_simps) ``` lp15@60809 ` 1268` ```qed ``` lp15@60809 ` 1269` lp15@60809 ` 1270` ```lemma convex_hull_subset: ``` lp15@60809 ` 1271` ``` "s \ convex hull t \ convex hull s \ convex hull t" ``` lp15@60809 ` 1272` ``` by (simp add: convex_convex_hull subset_hull) ``` lp15@60809 ` 1273` lp15@60809 ` 1274` ```lemma not_in_interior_convex_hull_3: ``` lp15@60809 ` 1275` ``` fixes a :: "complex" ``` lp15@60809 ` 1276` ``` shows "a \ interior(convex hull {a,b,c})" ``` lp15@60809 ` 1277` ``` "b \ interior(convex hull {a,b,c})" ``` lp15@60809 ` 1278` ``` "c \ interior(convex hull {a,b,c})" ``` lp15@60809 ` 1279` ``` by (auto simp: card_insert_le_m1 not_in_interior_convex_hull) ``` lp15@60809 ` 1280` lp15@60809 ` 1281` lp15@60809 ` 1282` ```text\Cauchy's theorem where there's a primitive\ ``` lp15@60809 ` 1283` lp15@60809 ` 1284` ```lemma path_integral_primitive_lemma: ``` lp15@60809 ` 1285` ``` fixes f :: "complex \ complex" and g :: "real \ complex" ``` lp15@60809 ` 1286` ``` assumes "a \ b" ``` lp15@60809 ` 1287` ``` and "\x. x \ s \ (f has_field_derivative f' x) (at x within s)" ``` lp15@60809 ` 1288` ``` and "g piecewise_differentiable_on {a..b}" "\x. x \ {a..b} \ g x \ s" ``` lp15@60809 ` 1289` ``` shows "((\x. f'(g x) * vector_derivative g (at x within {a..b})) ``` lp15@60809 ` 1290` ``` has_integral (f(g b) - f(g a))) {a..b}" ``` lp15@60809 ` 1291` ```proof - ``` lp15@61190 ` 1292` ``` obtain k where k: "finite k" "\x\{a..b} - k. g differentiable (at x within {a..b})" and cg: "continuous_on {a..b} g" ``` lp15@60809 ` 1293` ``` using assms by (auto simp: piecewise_differentiable_on_def) ``` lp15@60809 ` 1294` ``` have cfg: "continuous_on {a..b} (\x. f (g x))" ``` lp15@60809 ` 1295` ``` apply (rule continuous_on_compose [OF cg, unfolded o_def]) ``` lp15@60809 ` 1296` ``` using assms ``` lp15@60809 ` 1297` ``` apply (metis complex_differentiable_def complex_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff) ``` lp15@60809 ` 1298` ``` done ``` lp15@60809 ` 1299` ``` { fix x::real ``` lp15@60809 ` 1300` ``` assume a: "a < x" and b: "x < b" and xk: "x \ k" ``` lp15@60809 ` 1301` ``` then have "g differentiable at x within {a..b}" ``` lp15@60809 ` 1302` ``` using k by (simp add: differentiable_at_withinI) ``` lp15@60809 ` 1303` ``` then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})" ``` lp15@60809 ` 1304` ``` by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real) ``` lp15@60809 ` 1305` ``` then have gdiff: "(g has_derivative (\u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})" ``` lp15@60809 ` 1306` ``` by (simp add: has_vector_derivative_def scaleR_conv_of_real) ``` lp15@60809 ` 1307` ``` have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})" ``` lp15@60809 ` 1308` ``` using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def) ``` lp15@60809 ` 1309` ``` then have fdiff: "(f has_derivative op * (f' (g x))) (at (g x) within g ` {a..b})" ``` lp15@60809 ` 1310` ``` by (simp add: has_field_derivative_def) ``` lp15@60809 ` 1311` ``` have "((\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 ` 1312` ``` using diff_chain_within [OF gdiff fdiff] ``` lp15@60809 ` 1313` ``` by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac) ``` lp15@60809 ` 1314` ``` } note * = this ``` lp15@60809 ` 1315` ``` show ?thesis ``` lp15@60809 ` 1316` ``` apply (rule fundamental_theorem_of_calculus_interior_strong) ``` lp15@60809 ` 1317` ``` using k assms cfg * ``` lp15@60809 ` 1318` ``` apply (auto simp: at_within_closed_interval) ``` lp15@60809 ` 1319` ``` done ``` lp15@60809 ` 1320` ```qed ``` lp15@60809 ` 1321` lp15@60809 ` 1322` ```lemma path_integral_primitive: ``` lp15@60809 ` 1323` ``` assumes "\x. x \ s \ (f has_field_derivative f' x) (at x within s)" ``` lp15@60809 ` 1324` ``` and "valid_path g" "path_image g \ s" ``` lp15@60809 ` 1325` ``` shows "(f' has_path_integral (f(pathfinish g) - f(pathstart g))) g" ``` lp15@60809 ` 1326` ``` using assms ``` lp15@60809 ` 1327` ``` apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_path_integral_def) ``` lp15@61190 ` 1328` ``` apply (auto intro!: piecewise_C1_imp_differentiable path_integral_primitive_lemma [of 0 1 s]) ``` lp15@60809 ` 1329` ``` done ``` lp15@60809 ` 1330` lp15@60809 ` 1331` ```corollary Cauchy_theorem_primitive: ``` lp15@60809 ` 1332` ``` assumes "\x. x \ s \ (f has_field_derivative f' x) (at x within s)" ``` lp15@60809 ` 1333` ``` and "valid_path g" "path_image g \ s" "pathfinish g = pathstart g" ``` lp15@60809 ` 1334` ``` shows "(f' has_path_integral 0) g" ``` lp15@60809 ` 1335` ``` using assms ``` lp15@60809 ` 1336` ``` by (metis diff_self path_integral_primitive) ``` lp15@60809 ` 1337` lp15@60809 ` 1338` lp15@60809 ` 1339` ```text\Existence of path integral for continuous function\ ``` lp15@60809 ` 1340` ```lemma path_integrable_continuous_linepath: ``` lp15@60809 ` 1341` ``` assumes "continuous_on (closed_segment a b) f" ``` lp15@60809 ` 1342` ``` shows "f path_integrable_on (linepath a b)" ``` lp15@60809 ` 1343` ```proof - ``` lp15@60809 ` 1344` ``` have "continuous_on {0..1} ((\x. f x * (b - a)) o linepath a b)" ``` lp15@60809 ` 1345` ``` apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01) ``` lp15@60809 ` 1346` ``` apply (rule continuous_intros | simp add: assms)+ ``` lp15@60809 ` 1347` ``` done ``` lp15@60809 ` 1348` ``` then show ?thesis ``` lp15@60809 ` 1349` ``` apply (simp add: path_integrable_on_def has_path_integral_def integrable_on_def [symmetric]) ``` lp15@60809 ` 1350` ``` apply (rule integrable_continuous [of 0 "1::real", simplified]) ``` lp15@60809 ` 1351` ``` apply (rule continuous_on_eq [where f = "\x. f(linepath a b x)*(b - a)"]) ``` lp15@60809 ` 1352` ``` apply (auto simp: vector_derivative_linepath_within) ``` lp15@60809 ` 1353` ``` done ``` lp15@60809 ` 1354` ```qed ``` lp15@60809 ` 1355` lp15@60809 ` 1356` ```lemma has_field_der_id: "((\x. x\<^sup>2 / 2) has_field_derivative x) (at x)" ``` lp15@60809 ` 1357` ``` by (rule has_derivative_imp_has_field_derivative) ``` lp15@60809 ` 1358` ``` (rule derivative_intros | simp)+ ``` lp15@60809 ` 1359` lp15@60809 ` 1360` ```lemma path_integral_id [simp]: "path_integral (linepath a b) (\y. y) = (b^2 - a^2)/2" ``` lp15@60809 ` 1361` ``` apply (rule path_integral_unique) ``` lp15@60809 ` 1362` ``` using path_integral_primitive [of UNIV "\x. x^2/2" "\x. x" "linepath a b"] ``` lp15@60809 ` 1363` ``` apply (auto simp: field_simps has_field_der_id) ``` lp15@60809 ` 1364` ``` done ``` lp15@60809 ` 1365` lp15@60809 ` 1366` ```lemma path_integrable_on_const [iff]: "(\x. c) path_integrable_on (linepath a b)" ``` lp15@60809 ` 1367` ``` by (simp add: continuous_on_const path_integrable_continuous_linepath) ``` lp15@60809 ` 1368` lp15@60809 ` 1369` ```lemma path_integrable_on_id [iff]: "(\x. x) path_integrable_on (linepath a b)" ``` lp15@60809 ` 1370` ``` by (simp add: continuous_on_id path_integrable_continuous_linepath) ``` lp15@60809 ` 1371` lp15@60809 ` 1372` lp15@60809 ` 1373` ```subsection\Arithmetical combining theorems\ ``` lp15@60809 ` 1374` lp15@60809 ` 1375` ```lemma has_path_integral_neg: ``` lp15@60809 ` 1376` ``` "(f has_path_integral i) g \ ((\x. -(f x)) has_path_integral (-i)) g" ``` lp15@60809 ` 1377` ``` by (simp add: has_integral_neg has_path_integral_def) ``` lp15@60809 ` 1378` lp15@60809 ` 1379` ```lemma has_path_integral_add: ``` lp15@60809 ` 1380` ``` "\(f1 has_path_integral i1) g; (f2 has_path_integral i2) g\ ``` lp15@60809 ` 1381` ``` \ ((\x. f1 x + f2 x) has_path_integral (i1 + i2)) g" ``` lp15@60809 ` 1382` ``` by (simp add: has_integral_add has_path_integral_def algebra_simps) ``` lp15@60809 ` 1383` lp15@60809 ` 1384` ```lemma has_path_integral_diff: ``` lp15@61190 ` 1385` ``` "\(f1 has_path_integral i1) g; (f2 has_path_integral i2) g\ ``` lp15@60809 ` 1386` ``` \ ((\x. f1 x - f2 x) has_path_integral (i1 - i2)) g" ``` lp15@60809 ` 1387` ``` by (simp add: has_integral_sub has_path_integral_def algebra_simps) ``` lp15@60809 ` 1388` lp15@60809 ` 1389` ```lemma has_path_integral_lmul: ``` lp15@61190 ` 1390` ``` "(f has_path_integral i) g \ ((\x. c * (f x)) has_path_integral (c*i)) g" ``` lp15@60809 ` 1391` ```apply (simp add: has_path_integral_def) ``` lp15@60809 ` 1392` ```apply (drule has_integral_mult_right) ``` lp15@60809 ` 1393` ```apply (simp add: algebra_simps) ``` lp15@60809 ` 1394` ```done ``` lp15@60809 ` 1395` lp15@60809 ` 1396` ```lemma has_path_integral_rmul: ``` lp15@61190 ` 1397` ``` "(f has_path_integral i) g \ ((\x. (f x) * c) has_path_integral (i*c)) g" ``` lp15@60809 ` 1398` ```apply (drule has_path_integral_lmul) ``` lp15@60809 ` 1399` ```apply (simp add: mult.commute) ``` lp15@60809 ` 1400` ```done ``` lp15@60809 ` 1401` lp15@60809 ` 1402` ```lemma has_path_integral_div: ``` lp15@61190 ` 1403` ``` "(f has_path_integral i) g \ ((\x. f x/c) has_path_integral (i/c)) g" ``` lp15@60809 ` 1404` ``` by (simp add: field_class.field_divide_inverse) (metis has_path_integral_rmul) ``` lp15@60809 ` 1405` lp15@60809 ` 1406` ```lemma has_path_integral_eq: ``` lp15@60809 ` 1407` ``` "\(f has_path_integral y) p; \x. x \ path_image p \ f x = g x\ \ (g has_path_integral y) p" ``` lp15@60809 ` 1408` ```apply (simp add: path_image_def has_path_integral_def) ``` lp15@60809 ` 1409` ```by (metis (no_types, lifting) image_eqI has_integral_eq) ``` lp15@60809 ` 1410` lp15@60809 ` 1411` ```lemma has_path_integral_bound_linepath: ``` lp15@60809 ` 1412` ``` assumes "(f has_path_integral i) (linepath a b)" ``` lp15@60809 ` 1413` ``` "0 \ B" "\x. x \ closed_segment a b \ norm(f x) \ B" ``` lp15@60809 ` 1414` ``` shows "norm i \ B * norm(b - a)" ``` lp15@60809 ` 1415` ```proof - ``` lp15@60809 ` 1416` ``` { fix x::real ``` lp15@60809 ` 1417` ``` assume x: "0 \ x" "x \ 1" ``` lp15@60809 ` 1418` ``` have "norm (f (linepath a b x)) * ``` lp15@60809 ` 1419` ``` norm (vector_derivative (linepath a b) (at x within {0..1})) \ B * norm (b - a)" ``` lp15@60809 ` 1420` ``` by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x) ``` lp15@60809 ` 1421` ``` } note * = this ``` lp15@60809 ` 1422` ``` have "norm i \ (B * norm (b - a)) * content (cbox 0 (1::real))" ``` lp15@60809 ` 1423` ``` apply (rule has_integral_bound ``` lp15@60809 ` 1424` ``` [of _ "\x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"]) ``` lp15@60809 ` 1425` ``` using assms * unfolding has_path_integral_def ``` lp15@60809 ` 1426` ``` apply (auto simp: norm_mult) ``` lp15@60809 ` 1427` ``` done ``` lp15@60809 ` 1428` ``` then show ?thesis ``` lp15@60809 ` 1429` ``` by (auto simp: content_real) ``` lp15@60809 ` 1430` ```qed ``` lp15@60809 ` 1431` lp15@60809 ` 1432` ```(*UNUSED ``` lp15@60809 ` 1433` ```lemma has_path_integral_bound_linepath_strong: ``` lp15@60809 ` 1434` ``` fixes a :: real and f :: "complex \ real" ``` lp15@60809 ` 1435` ``` assumes "(f has_path_integral i) (linepath a b)" ``` lp15@60809 ` 1436` ``` "finite k" ``` lp15@60809 ` 1437` ``` "0 \ B" "\x::real. x \ closed_segment a b - k \ norm(f x) \ B" ``` lp15@60809 ` 1438` ``` shows "norm i \ B*norm(b - a)" ``` lp15@60809 ` 1439` ```*) ``` lp15@60809 ` 1440` lp15@60809 ` 1441` ```lemma has_path_integral_const_linepath: "((\x. c) has_path_integral c*(b - a))(linepath a b)" ``` lp15@60809 ` 1442` ``` unfolding has_path_integral_linepath ``` lp15@60809 ` 1443` ``` 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 ` 1444` lp15@60809 ` 1445` ```lemma has_path_integral_0: "((\x. 0) has_path_integral 0) g" ``` lp15@60809 ` 1446` ``` by (simp add: has_path_integral_def) ``` lp15@60809 ` 1447` lp15@60809 ` 1448` ```lemma has_path_integral_is_0: ``` lp15@60809 ` 1449` ``` "(\z. z \ path_image g \ f z = 0) \ (f has_path_integral 0) g" ``` lp15@60809 ` 1450` ``` by (rule has_path_integral_eq [OF has_path_integral_0]) auto ``` lp15@60809 ` 1451` lp15@60809 ` 1452` ```lemma has_path_integral_setsum: ``` lp15@60809 ` 1453` ``` "\finite s; \a. a \ s \ (f a has_path_integral i a) p\ ``` lp15@60809 ` 1454` ``` \ ((\x. setsum (\a. f a x) s) has_path_integral setsum i s) p" ``` lp15@60809 ` 1455` ``` by (induction s rule: finite_induct) (auto simp: has_path_integral_0 has_path_integral_add) ``` lp15@60809 ` 1456` lp15@60809 ` 1457` lp15@60809 ` 1458` ```subsection \Operations on path integrals\ ``` lp15@60809 ` 1459` lp15@60809 ` 1460` ```lemma path_integral_const_linepath [simp]: "path_integral (linepath a b) (\x. c) = c*(b - a)" ``` lp15@60809 ` 1461` ``` by (rule path_integral_unique [OF has_path_integral_const_linepath]) ``` lp15@60809 ` 1462` lp15@60809 ` 1463` ```lemma path_integral_neg: ``` lp15@60809 ` 1464` ``` "f path_integrable_on g \ path_integral g (\x. -(f x)) = -(path_integral g f)" ``` lp15@60809 ` 1465` ``` by (simp add: path_integral_unique has_path_integral_integral has_path_integral_neg) ``` lp15@60809 ` 1466` lp15@60809 ` 1467` ```lemma path_integral_add: ``` lp15@60809 ` 1468` ``` "f1 path_integrable_on g \ f2 path_integrable_on g \ path_integral g (\x. f1 x + f2 x) = ``` lp15@60809 ` 1469` ``` path_integral g f1 + path_integral g f2" ``` lp15@60809 ` 1470` ``` by (simp add: path_integral_unique has_path_integral_integral has_path_integral_add) ``` lp15@60809 ` 1471` lp15@60809 ` 1472` ```lemma path_integral_diff: ``` lp15@60809 ` 1473` ``` "f1 path_integrable_on g \ f2 path_integrable_on g \ path_integral g (\x. f1 x - f2 x) = ``` lp15@60809 ` 1474` ``` path_integral g f1 - path_integral g f2" ``` lp15@60809 ` 1475` ``` by (simp add: path_integral_unique has_path_integral_integral has_path_integral_diff) ``` lp15@60809 ` 1476` lp15@60809 ` 1477` ```lemma path_integral_lmul: ``` lp15@60809 ` 1478` ``` shows "f path_integrable_on g ``` lp15@60809 ` 1479` ``` \ path_integral g (\x. c * f x) = c*path_integral g f" ``` lp15@60809 ` 1480` ``` by (simp add: path_integral_unique has_path_integral_integral has_path_integral_lmul) ``` lp15@60809 ` 1481` lp15@60809 ` 1482` ```lemma path_integral_rmul: ``` lp15@60809 ` 1483` ``` shows "f path_integrable_on g ``` lp15@60809 ` 1484` ``` \ path_integral g (\x. f x * c) = path_integral g f * c" ``` lp15@60809 ` 1485` ``` by (simp add: path_integral_unique has_path_integral_integral has_path_integral_rmul) ``` lp15@60809 ` 1486` lp15@60809 ` 1487` ```lemma path_integral_div: ``` lp15@60809 ` 1488` ``` shows "f path_integrable_on g ``` lp15@60809 ` 1489` ``` \ path_integral g (\x. f x / c) = path_integral g f / c" ``` lp15@60809 ` 1490` ``` by (simp add: path_integral_unique has_path_integral_integral has_path_integral_div) ``` lp15@60809 ` 1491` lp15@60809 ` 1492` ```lemma path_integral_eq: ``` lp15@60809 ` 1493` ``` "(\x. x \ path_image p \ f x = g x) \ path_integral p f = path_integral p g" ``` lp15@60809 ` 1494` ``` by (simp add: path_integral_def) (metis has_path_integral_eq) ``` lp15@60809 ` 1495` lp15@60809 ` 1496` ```lemma path_integral_eq_0: ``` lp15@60809 ` 1497` ``` "(\z. z \ path_image g \ f z = 0) \ path_integral g f = 0" ``` lp15@60809 ` 1498` ``` by (simp add: has_path_integral_is_0 path_integral_unique) ``` lp15@60809 ` 1499` lp15@60809 ` 1500` ```lemma path_integral_bound_linepath: ``` lp15@60809 ` 1501` ``` shows ``` lp15@60809 ` 1502` ``` "\f path_integrable_on (linepath a b); ``` lp15@60809 ` 1503` ``` 0 \ B; \x. x \ closed_segment a b \ norm(f x) \ B\ ``` lp15@60809 ` 1504` ``` \ norm(path_integral (linepath a b) f) \ B*norm(b - a)" ``` lp15@60809 ` 1505` ``` apply (rule has_path_integral_bound_linepath [of f]) ``` lp15@60809 ` 1506` ``` apply (auto simp: has_path_integral_integral) ``` lp15@60809 ` 1507` ``` done ``` lp15@60809 ` 1508` lp15@60809 ` 1509` ```lemma path_integral_0: "path_integral g (\x. 0) = 0" ``` lp15@60809 ` 1510` ``` by (simp add: path_integral_unique has_path_integral_0) ``` lp15@60809 ` 1511` lp15@60809 ` 1512` ```lemma path_integral_setsum: ``` lp15@60809 ` 1513` ``` "\finite s; \a. a \ s \ (f a) path_integrable_on p\ ``` lp15@60809 ` 1514` ``` \ path_integral p (\x. setsum (\a. f a x) s) = setsum (\a. path_integral p (f a)) s" ``` lp15@60809 ` 1515` ``` by (auto simp: path_integral_unique has_path_integral_setsum has_path_integral_integral) ``` lp15@60809 ` 1516` lp15@60809 ` 1517` ```lemma path_integrable_eq: ``` lp15@60809 ` 1518` ``` "\f path_integrable_on p; \x. x \ path_image p \ f x = g x\ \ g path_integrable_on p" ``` lp15@60809 ` 1519` ``` unfolding path_integrable_on_def ``` lp15@60809 ` 1520` ``` by (metis has_path_integral_eq) ``` lp15@60809 ` 1521` lp15@60809 ` 1522` lp15@60809 ` 1523` ```subsection \Arithmetic theorems for path integrability\ ``` lp15@60809 ` 1524` lp15@60809 ` 1525` ```lemma path_integrable_neg: ``` lp15@60809 ` 1526` ``` "f path_integrable_on g \ (\x. -(f x)) path_integrable_on g" ``` lp15@60809 ` 1527` ``` using has_path_integral_neg path_integrable_on_def by blast ``` lp15@60809 ` 1528` lp15@60809 ` 1529` ```lemma path_integrable_add: ``` lp15@60809 ` 1530` ``` "\f1 path_integrable_on g; f2 path_integrable_on g\ \ (\x. f1 x + f2 x) path_integrable_on g" ``` lp15@60809 ` 1531` ``` using has_path_integral_add path_integrable_on_def ``` lp15@60809 ` 1532` ``` by fastforce ``` lp15@60809 ` 1533` lp15@60809 ` 1534` ```lemma path_integrable_diff: ``` lp15@60809 ` 1535` ``` "\f1 path_integrable_on g; f2 path_integrable_on g\ \ (\x. f1 x - f2 x) path_integrable_on g" ``` lp15@60809 ` 1536` ``` using has_path_integral_diff path_integrable_on_def ``` lp15@60809 ` 1537` ``` by fastforce ``` lp15@60809 ` 1538` lp15@60809 ` 1539` ```lemma path_integrable_lmul: ``` lp15@60809 ` 1540` ``` "f path_integrable_on g \ (\x. c * f x) path_integrable_on g" ``` lp15@60809 ` 1541` ``` using has_path_integral_lmul path_integrable_on_def ``` lp15@60809 ` 1542` ``` by fastforce ``` lp15@60809 ` 1543` lp15@60809 ` 1544` ```lemma path_integrable_rmul: ``` lp15@60809 ` 1545` ``` "f path_integrable_on g \ (\x. f x * c) path_integrable_on g" ``` lp15@60809 ` 1546` ``` using has_path_integral_rmul path_integrable_on_def ``` lp15@60809 ` 1547` ``` by fastforce ``` lp15@60809 ` 1548` lp15@60809 ` 1549` ```lemma path_integrable_div: ``` lp15@60809 ` 1550` ``` "f path_integrable_on g \ (\x. f x / c) path_integrable_on g" ``` lp15@60809 ` 1551` ``` using has_path_integral_div path_integrable_on_def ``` lp15@60809 ` 1552` ``` by fastforce ``` lp15@60809 ` 1553` lp15@60809 ` 1554` ```lemma path_integrable_setsum: ``` lp15@60809 ` 1555` ``` "\finite s; \a. a \ s \ (f a) path_integrable_on p\ ``` lp15@60809 ` 1556` ``` \ (\x. setsum (\a. f a x) s) path_integrable_on p" ``` lp15@60809 ` 1557` ``` unfolding path_integrable_on_def ``` lp15@60809 ` 1558` ``` by (metis has_path_integral_setsum) ``` lp15@60809 ` 1559` lp15@60809 ` 1560` lp15@60809 ` 1561` ```subsection\Reversing a path integral\ ``` lp15@60809 ` 1562` lp15@60809 ` 1563` ```lemma has_path_integral_reverse_linepath: ``` lp15@60809 ` 1564` ``` "(f has_path_integral i) (linepath a b) ``` lp15@60809 ` 1565` ``` \ (f has_path_integral (-i)) (linepath b a)" ``` lp15@60809 ` 1566` ``` using has_path_integral_reversepath valid_path_linepath by fastforce ``` lp15@60809 ` 1567` lp15@60809 ` 1568` ```lemma path_integral_reverse_linepath: ``` lp15@60809 ` 1569` ``` "continuous_on (closed_segment a b) f ``` lp15@60809 ` 1570` ``` \ path_integral (linepath a b) f = - (path_integral(linepath b a) f)" ``` lp15@60809 ` 1571` ```apply (rule path_integral_unique) ``` lp15@60809 ` 1572` ```apply (rule has_path_integral_reverse_linepath) ``` lp15@60809 ` 1573` ```by (simp add: closed_segment_commute path_integrable_continuous_linepath has_path_integral_integral) ``` lp15@60809 ` 1574` lp15@60809 ` 1575` lp15@60809 ` 1576` ```(* Splitting a path integral in a flat way.*) ``` lp15@60809 ` 1577` lp15@60809 ` 1578` ```lemma has_path_integral_split: ``` lp15@60809 ` 1579` ``` assumes f: "(f has_path_integral i) (linepath a c)" "(f has_path_integral j) (linepath c b)" ``` lp15@60809 ` 1580` ``` and k: "0 \ k" "k \ 1" ``` lp15@60809 ` 1581` ``` and c: "c - a = k *\<^sub>R (b - a)" ``` lp15@60809 ` 1582` ``` shows "(f has_path_integral (i + j)) (linepath a b)" ``` lp15@60809 ` 1583` ```proof (cases "k = 0 \ k = 1") ``` lp15@60809 ` 1584` ``` case True ``` lp15@60809 ` 1585` ``` then show ?thesis ``` lp15@60809 ` 1586` ``` using assms ``` lp15@60809 ` 1587` ``` apply auto ``` lp15@60809 ` 1588` ``` apply (metis add.left_neutral has_path_integral_trivial has_path_integral_unique) ``` lp15@60809 ` 1589` ``` apply (metis add.right_neutral has_path_integral_trivial has_path_integral_unique) ``` lp15@60809 ` 1590` ``` done ``` lp15@60809 ` 1591` ```next ``` lp15@60809 ` 1592` ``` case False ``` lp15@60809 ` 1593` ``` then have k: "0 < k" "k < 1" "complex_of_real k \ 1" ``` lp15@60809 ` 1594` ``` using assms apply auto ``` lp15@60809 ` 1595` ``` using of_real_eq_iff by fastforce ``` lp15@60809 ` 1596` ``` have c': "c = k *\<^sub>R (b - a) + a" ``` lp15@60809 ` 1597` ``` by (metis diff_add_cancel c) ``` lp15@60809 ` 1598` ``` have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)" ``` lp15@60809 ` 1599` ``` by (simp add: algebra_simps c') ``` lp15@60809 ` 1600` ``` { assume *: "((\x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}" ``` lp15@60809 ` 1601` ``` have **: "\x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b" ``` lp15@60809 ` 1602` ``` using False ``` lp15@60809 ` 1603` ``` apply (simp add: c' algebra_simps) ``` lp15@60809 ` 1604` ``` apply (simp add: real_vector.scale_left_distrib [symmetric] divide_simps) ``` lp15@60809 ` 1605` ``` done ``` lp15@60809 ` 1606` ``` have "((\x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}" ``` lp15@60809 ` 1607` ``` using * k ``` lp15@60809 ` 1608` ``` apply - ``` lp15@60809 ` 1609` ``` apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse k" "0", simplified]) ``` lp15@60809 ` 1610` ``` apply (simp_all add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c) ``` lp15@60809 ` 1611` ``` apply (drule Integration.has_integral_cmul [where c = "inverse k"]) ``` lp15@60809 ` 1612` ``` apply (simp add: Integration.has_integral_cmul) ``` lp15@60809 ` 1613` ``` done ``` lp15@60809 ` 1614` ``` } note fi = this ``` lp15@60809 ` 1615` ``` { assume *: "((\x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}" ``` lp15@60809 ` 1616` ``` have **: "\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 ` 1617` ``` using k ``` lp15@60809 ` 1618` ``` apply (simp add: c' field_simps) ``` lp15@60809 ` 1619` ``` apply (simp add: scaleR_conv_of_real divide_simps) ``` lp15@60809 ` 1620` ``` apply (simp add: field_simps) ``` lp15@60809 ` 1621` ``` done ``` lp15@60809 ` 1622` ``` have "((\x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}" ``` lp15@60809 ` 1623` ``` using * k ``` lp15@60809 ` 1624` ``` apply - ``` lp15@60809 ` 1625` ``` apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse(1 - k)" "-(k/(1 - k))", simplified]) ``` lp15@60809 ` 1626` ``` apply (simp_all add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc) ``` lp15@60809 ` 1627` ``` apply (drule Integration.has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"]) ``` lp15@60809 ` 1628` ``` apply (simp add: Integration.has_integral_cmul) ``` lp15@60809 ` 1629` ``` done ``` lp15@60809 ` 1630` ``` } note fj = this ``` lp15@60809 ` 1631` ``` show ?thesis ``` lp15@60809 ` 1632` ``` using f k ``` lp15@60809 ` 1633` ``` apply (simp add: has_path_integral_linepath) ``` lp15@60809 ` 1634` ``` apply (simp add: linepath_def) ``` lp15@60809 ` 1635` ``` apply (rule has_integral_combine [OF _ _ fi fj], simp_all) ``` lp15@60809 ` 1636` ``` done ``` lp15@60809 ` 1637` ```qed ``` lp15@60809 ` 1638` lp15@60809 ` 1639` ```lemma continuous_on_closed_segment_transform: ``` lp15@60809 ` 1640` ``` assumes f: "continuous_on (closed_segment a b) f" ``` lp15@60809 ` 1641` ``` and k: "0 \ k" "k \ 1" ``` lp15@60809 ` 1642` ``` and c: "c - a = k *\<^sub>R (b - a)" ``` lp15@60809 ` 1643` ``` shows "continuous_on (closed_segment a c) f" ``` lp15@60809 ` 1644` ```proof - ``` lp15@60809 ` 1645` ``` have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b" ``` lp15@60809 ` 1646` ``` using c by (simp add: algebra_simps) ``` lp15@60809 ` 1647` ``` show "continuous_on (closed_segment a c) f" ``` lp15@60809 ` 1648` ``` apply (rule continuous_on_subset [OF f]) ``` lp15@60809 ` 1649` ``` apply (simp add: segment_convex_hull) ``` lp15@60809 ` 1650` ``` apply (rule convex_hull_subset) ``` lp15@60809 ` 1651` ``` using assms ``` lp15@61426 ` 1652` ``` apply (auto simp: hull_inc c' Convex.convexD_alt) ``` lp15@60809 ` 1653` ``` done ``` lp15@60809 ` 1654` ```qed ``` lp15@60809 ` 1655` lp15@60809 ` 1656` ```lemma path_integral_split: ``` lp15@60809 ` 1657` ``` assumes f: "continuous_on (closed_segment a b) f" ``` lp15@60809 ` 1658` ``` and k: "0 \ k" "k \ 1" ``` lp15@60809 ` 1659` ``` and c: "c - a = k *\<^sub>R (b - a)" ``` lp15@60809 ` 1660` ``` shows "path_integral(linepath a b) f = path_integral(linepath a c) f + path_integral(linepath c b) f" ``` lp15@60809 ` 1661` ```proof - ``` lp15@60809 ` 1662` ``` have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b" ``` lp15@60809 ` 1663` ``` using c by (simp add: algebra_simps) ``` lp15@60809 ` 1664` ``` have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f" ``` lp15@60809 ` 1665` ``` apply (rule_tac [!] continuous_on_subset [OF f]) ``` lp15@60809 ` 1666` ``` apply (simp_all add: segment_convex_hull) ``` lp15@60809 ` 1667` ``` apply (rule_tac [!] convex_hull_subset) ``` lp15@60809 ` 1668` ``` using assms ``` lp15@61426 ` 1669` ``` apply (auto simp: hull_inc c' Convex.convexD_alt) ``` lp15@60809 ` 1670` ``` done ``` lp15@60809 ` 1671` ``` show ?thesis ``` lp15@60809 ` 1672` ``` apply (rule path_integral_unique) ``` lp15@60809 ` 1673` ``` apply (rule has_path_integral_split [OF has_path_integral_integral has_path_integral_integral k c]) ``` lp15@60809 ` 1674` ``` apply (rule path_integrable_continuous_linepath *)+ ``` lp15@60809 ` 1675` ``` done ``` lp15@60809 ` 1676` ```qed ``` lp15@60809 ` 1677` lp15@60809 ` 1678` ```lemma path_integral_split_linepath: ``` lp15@60809 ` 1679` ``` assumes f: "continuous_on (closed_segment a b) f" ``` lp15@60809 ` 1680` ``` and c: "c \ closed_segment a b" ``` lp15@60809 ` 1681` ``` shows "path_integral(linepath a b) f = path_integral(linepath a c) f + path_integral(linepath c b) f" ``` lp15@60809 ` 1682` ``` using c ``` lp15@60809 ` 1683` ``` by (auto simp: closed_segment_def algebra_simps intro!: path_integral_split [OF f]) ``` lp15@60809 ` 1684` lp15@60809 ` 1685` ```(* The special case of midpoints used in the main quadrisection.*) ``` lp15@60809 ` 1686` lp15@60809 ` 1687` ```lemma has_path_integral_midpoint: ``` lp15@60809 ` 1688` ``` assumes "(f has_path_integral i) (linepath a (midpoint a b))" ``` lp15@60809 ` 1689` ``` "(f has_path_integral j) (linepath (midpoint a b) b)" ``` lp15@60809 ` 1690` ``` shows "(f has_path_integral (i + j)) (linepath a b)" ``` lp15@60809 ` 1691` ``` apply (rule has_path_integral_split [where c = "midpoint a b" and k = "1/2"]) ``` lp15@60809 ` 1692` ``` using assms ``` lp15@60809 ` 1693` ``` apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real) ``` lp15@60809 ` 1694` ``` done ``` lp15@60809 ` 1695` lp15@60809 ` 1696` ```lemma path_integral_midpoint: ``` lp15@60809 ` 1697` ``` "continuous_on (closed_segment a b) f ``` lp15@60809 ` 1698` ``` \ path_integral (linepath a b) f = ``` lp15@60809 ` 1699` ``` path_integral (linepath a (midpoint a b)) f + path_integral (linepath (midpoint a b) b) f" ``` lp15@60809 ` 1700` ``` apply (rule path_integral_split [where c = "midpoint a b" and k = "1/2"]) ``` lp15@60809 ` 1701` ``` using assms ``` lp15@60809 ` 1702` ``` apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real) ``` lp15@60809 ` 1703` ``` done ``` lp15@60809 ` 1704` lp15@60809 ` 1705` lp15@60809 ` 1706` ```text\A couple of special case lemmas that are useful below\ ``` lp15@60809 ` 1707` lp15@60809 ` 1708` ```lemma triangle_linear_has_chain_integral: ``` lp15@60809 ` 1709` ``` "((\x. m*x + d) has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)" ``` lp15@60809 ` 1710` ``` apply (rule Cauchy_theorem_primitive [of UNIV "\x. m/2 * x^2 + d*x"]) ``` lp15@60809 ` 1711` ``` apply (auto intro!: derivative_eq_intros) ``` lp15@60809 ` 1712` ``` done ``` lp15@60809 ` 1713` lp15@60809 ` 1714` ```lemma has_chain_integral_chain_integral3: ``` lp15@60809 ` 1715` ``` "(f has_path_integral i) (linepath a b +++ linepath b c +++ linepath c d) ``` lp15@60809 ` 1716` ``` \ path_integral (linepath a b) f + path_integral (linepath b c) f + path_integral (linepath c d) f = i" ``` lp15@60809 ` 1717` ``` apply (subst path_integral_unique [symmetric], assumption) ``` lp15@60809 ` 1718` ``` apply (drule has_path_integral_integrable) ``` lp15@60809 ` 1719` ``` apply (simp add: valid_path_join) ``` lp15@60809 ` 1720` ``` done ``` lp15@60809 ` 1721` lp15@60809 ` 1722` ```subsection\Reversing the order in a double path integral\ ``` lp15@60809 ` 1723` lp15@60809 ` 1724` ```text\The condition is stronger than needed but it's often true in typical situations\ ``` lp15@60809 ` 1725` lp15@60809 ` 1726` ```lemma fst_im_cbox [simp]: "cbox c d \ {} \ (fst ` cbox (a,c) (b,d)) = cbox a b" ``` lp15@60809 ` 1727` ``` by (auto simp: cbox_Pair_eq) ``` lp15@60809 ` 1728` lp15@60809 ` 1729` ```lemma snd_im_cbox [simp]: "cbox a b \ {} \ (snd ` cbox (a,c) (b,d)) = cbox c d" ``` lp15@60809 ` 1730` ``` by (auto simp: cbox_Pair_eq) ``` lp15@60809 ` 1731` lp15@60809 ` 1732` ```lemma path_integral_swap: ``` lp15@60809 ` 1733` ``` assumes fcon: "continuous_on (path_image g \ path_image h) (\(y1,y2). f y1 y2)" ``` lp15@60809 ` 1734` ``` and vp: "valid_path g" "valid_path h" ``` lp15@60809 ` 1735` ``` and gvcon: "continuous_on {0..1} (\t. vector_derivative g (at t))" ``` lp15@60809 ` 1736` ``` and hvcon: "continuous_on {0..1} (\t. vector_derivative h (at t))" ``` lp15@60809 ` 1737` ``` shows "path_integral g (\w. path_integral h (f w)) = ``` lp15@60809 ` 1738` ``` path_integral h (\z. path_integral g (\w. f w z))" ``` lp15@60809 ` 1739` ```proof - ``` lp15@60809 ` 1740` ``` have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h" ``` lp15@61190 ` 1741` ``` using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def) ``` lp15@60809 ` 1742` ``` have fgh1: "\x. (\t. f (g x) (h t)) = (\(y1,y2). f y1 y2) o (\t. (g x, h t))" ``` lp15@60809 ` 1743` ``` by (rule ext) simp ``` lp15@60809 ` 1744` ``` have fgh2: "\x. (\t. f (g t) (h x)) = (\(y1,y2). f y1 y2) o (\t. (g t, h x))" ``` lp15@60809 ` 1745` ``` by (rule ext) simp ``` lp15@60809 ` 1746` ``` have fcon_im1: "\x. 0 \ x \ x \ 1 \ continuous_on ((\t. (g x, h t)) ` {0..1}) (\(x, y). f x y)" ``` lp15@60809 ` 1747` ``` by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def) ``` lp15@60809 ` 1748` ``` have fcon_im2: "\x. 0 \ x \ x \ 1 \ continuous_on ((\t. (g t, h x)) ` {0..1}) (\(x, y). f x y)" ``` lp15@60809 ` 1749` ``` by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def) ``` lp15@60809 ` 1750` ``` have vdg: "\y. y \ {0..1} \ (\x. f (g x) (h y) * vector_derivative g (at x)) integrable_on {0..1}" ``` lp15@60809 ` 1751` ``` apply (rule integrable_continuous_real) ``` lp15@60809 ` 1752` ``` apply (rule continuous_on_mult [OF _ gvcon]) ``` lp15@60809 ` 1753` ``` apply (subst fgh2) ``` lp15@60809 ` 1754` ``` apply (rule fcon_im2 gcon continuous_intros | simp)+ ``` lp15@60809 ` 1755` ``` done ``` lp15@60809 ` 1756` ``` have "(\z. vector_derivative g (at (fst z))) = (\x. vector_derivative g (at x)) o fst" ``` lp15@60809 ` 1757` ``` by auto ``` lp15@60809 ` 1758` ``` then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\x. vector_derivative g (at (fst x)))" ``` lp15@60809 ` 1759` ``` apply (rule ssubst) ``` lp15@60809 ` 1760` ``` apply (rule continuous_intros | simp add: gvcon)+ ``` lp15@60809 ` 1761` ``` done ``` lp15@60809 ` 1762` ``` have "(\z. vector_derivative h (at (snd z))) = (\x. vector_derivative h (at x)) o snd" ``` lp15@60809 ` 1763` ``` by auto ``` lp15@60809 ` 1764` ``` then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\x. vector_derivative h (at (snd x)))" ``` lp15@60809 ` 1765` ``` apply (rule ssubst) ``` lp15@60809 ` 1766` ``` apply (rule continuous_intros | simp add: hvcon)+ ``` lp15@60809 ` 1767` ``` done ``` lp15@60809 ` 1768` ``` have "(\x. f (g (fst x)) (h (snd x))) = (\(y1,y2). f y1 y2) o (\w. ((g o fst) w, (h o snd) w))" ``` lp15@60809 ` 1769` ``` by auto ``` lp15@60809 ` 1770` ``` then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\x. f (g (fst x)) (h (snd x)))" ``` lp15@60809 ` 1771` ``` apply (rule ssubst) ``` lp15@60809 ` 1772` ``` apply (rule gcon hcon continuous_intros | simp)+ ``` lp15@60809 ` 1773` ``` apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon]) ``` lp15@60809 ` 1774` ``` done ``` lp15@60809 ` 1775` ``` have "integral {0..1} (\x. path_integral h (f (g x)) * vector_derivative g (at x)) = ``` lp15@60809 ` 1776` ``` integral {0..1} (\x. path_integral h (\y. f (g x) y * vector_derivative g (at x)))" ``` lp15@60809 ` 1777` ``` apply (rule integral_cong [OF path_integral_rmul [symmetric]]) ``` lp15@60809 ` 1778` ``` apply (clarsimp simp: path_integrable_on) ``` lp15@60809 ` 1779` ``` apply (rule integrable_continuous_real) ``` lp15@60809 ` 1780` ``` apply (rule continuous_on_mult [OF _ hvcon]) ``` lp15@60809 ` 1781` ``` apply (subst fgh1) ``` lp15@60809 ` 1782` ``` apply (rule fcon_im1 hcon continuous_intros | simp)+ ``` lp15@60809 ` 1783` ``` done ``` lp15@60809 ` 1784` ``` also have "... = integral {0..1} ``` lp15@60809 ` 1785` ``` (\y. path_integral g (\x. f x (h y) * vector_derivative h (at y)))" ``` lp15@60809 ` 1786` ``` apply (simp add: path_integral_integral) ``` lp15@60809 ` 1787` ``` apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified]) ``` lp15@60809 ` 1788` ``` apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+ ``` lp15@60809 ` 1789` ``` apply (simp add: algebra_simps) ``` lp15@60809 ` 1790` ``` done ``` lp15@60809 ` 1791` ``` also have "... = path_integral h (\z. path_integral g (\w. f w z))" ``` lp15@60809 ` 1792` ``` apply (simp add: path_integral_integral) ``` lp15@60809 ` 1793` ``` apply (rule integral_cong) ``` lp15@60809 ` 1794` ``` apply (subst integral_mult_left [symmetric]) ``` lp15@60809 ` 1795` ``` apply (blast intro: vdg) ``` lp15@60809 ` 1796` ``` apply (simp add: algebra_simps) ``` lp15@60809 ` 1797` ``` done ``` lp15@60809 ` 1798` ``` finally show ?thesis ``` lp15@60809 ` 1799` ``` by (simp add: path_integral_integral) ``` lp15@60809 ` 1800` ```qed ``` lp15@60809 ` 1801` lp15@60809 ` 1802` lp15@60809 ` 1803` ```subsection\The key quadrisection step\ ``` lp15@60809 ` 1804` lp15@60809 ` 1805` ```lemma norm_sum_half: ``` lp15@60809 ` 1806` ``` assumes "norm(a + b) >= e" ``` lp15@60809 ` 1807` ``` shows "norm a >= e/2 \ norm b >= e/2" ``` lp15@60809 ` 1808` ```proof - ``` lp15@60809 ` 1809` ``` have "e \ norm (- a - b)" ``` lp15@60809 ` 1810` ``` by (simp add: add.commute assms norm_minus_commute) ``` lp15@60809 ` 1811` ``` thus ?thesis ``` lp15@60809 ` 1812` ``` using norm_triangle_ineq4 order_trans by fastforce ``` lp15@60809 ` 1813` ```qed ``` lp15@60809 ` 1814` lp15@60809 ` 1815` ```lemma norm_sum_lemma: ``` lp15@60809 ` 1816` ``` assumes "e \ norm (a + b + c + d)" ``` lp15@60809 ` 1817` ``` shows "e / 4 \ norm a \ e / 4 \ norm b \ e / 4 \ norm c \ e / 4 \ norm d" ``` lp15@60809 ` 1818` ```proof - ``` lp15@60809 ` 1819` ``` have "e \ norm ((a + b) + (c + d))" using assms ``` lp15@60809 ` 1820` ``` by (simp add: algebra_simps) ``` lp15@60809 ` 1821` ``` then show ?thesis ``` lp15@60809 ` 1822` ``` by (auto dest!: norm_sum_half) ``` lp15@60809 ` 1823` ```qed ``` lp15@60809 ` 1824` lp15@60809 ` 1825` ```lemma Cauchy_theorem_quadrisection: ``` lp15@60809 ` 1826` ``` assumes f: "continuous_on (convex hull {a,b,c}) f" ``` lp15@60809 ` 1827` ``` and dist: "dist a b \ K" "dist b c \ K" "dist c a \ K" ``` lp15@60809 ` 1828` ``` and e: "e * K^2 \ ``` lp15@60809 ` 1829` ``` norm (path_integral(linepath a b) f + path_integral(linepath b c) f + path_integral(linepath c a) f)" ``` lp15@60809 ` 1830` ``` shows "\a' b' c'. ``` lp15@60809 ` 1831` ``` a' \ convex hull {a,b,c} \ b' \ convex hull {a,b,c} \ c' \ convex hull {a,b,c} \ ``` lp15@60809 ` 1832` ``` dist a' b' \ K/2 \ dist b' c' \ K/2 \ dist c' a' \ K/2 \ ``` lp15@60809 ` 1833` ``` e * (K/2)^2 \ norm(path_integral(linepath a' b') f + path_integral(linepath b' c') f + path_integral(linepath c' a') f)" ``` lp15@60809 ` 1834` ```proof - ``` lp15@60809 ` 1835` ``` note divide_le_eq_numeral1 [simp del] ``` lp15@60809 ` 1836` ``` def a' \ "midpoint b c" ``` lp15@60809 ` 1837` ``` def b' \ "midpoint c a" ``` lp15@60809 ` 1838` ``` def c' \ "midpoint a b" ``` lp15@60809 ` 1839` ``` 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 ` 1840` ``` using f continuous_on_subset segments_subset_convex_hull by metis+ ``` lp15@60809 ` 1841` ``` have fcont': "continuous_on (closed_segment c' b') f" ``` lp15@60809 ` 1842` ``` "continuous_on (closed_segment a' c') f" ``` lp15@60809 ` 1843` ``` "continuous_on (closed_segment b' a') f" ``` lp15@60809 ` 1844` ``` unfolding a'_def b'_def c'_def ``` lp15@60809 ` 1845` ``` apply (rule continuous_on_subset [OF f], ``` lp15@60809 ` 1846` ``` metis midpoints_in_convex_hull convex_hull_subset hull_subset insert_subset segment_convex_hull)+ ``` lp15@60809 ` 1847` ``` done ``` lp15@60809 ` 1848` ``` let ?pathint = "\x y. path_integral(linepath x y) f" ``` lp15@60809 ` 1849` ``` have *: "?pathint a b + ?pathint b c + ?pathint c a = ``` lp15@60809 ` 1850` ``` (?pathint a c' + ?pathint c' b' + ?pathint b' a) + ``` lp15@60809 ` 1851` ``` (?pathint a' c' + ?pathint c' b + ?pathint b a') + ``` lp15@60809 ` 1852` ``` (?pathint a' c + ?pathint c b' + ?pathint b' a') + ``` lp15@60809 ` 1853` ``` (?pathint a' b' + ?pathint b' c' + ?pathint c' a')" ``` lp15@60809 ` 1854` ``` apply (simp add: fcont' path_integral_reverse_linepath) ``` lp15@60809 ` 1855` ``` apply (simp add: a'_def b'_def c'_def path_integral_midpoint fabc) ``` lp15@60809 ` 1856` ``` done ``` lp15@60809 ` 1857` ``` have [simp]: "\x y. cmod (x * 2 - y * 2) = cmod (x - y) * 2" ``` lp15@60809 ` 1858` ``` by (metis left_diff_distrib mult.commute norm_mult_numeral1) ``` lp15@60809 ` 1859` ``` have [simp]: "\x y. cmod (x - y) = cmod (y - x)" ``` lp15@60809 ` 1860` ``` by (simp add: norm_minus_commute) ``` lp15@60809 ` 1861` ``` consider "e * K\<^sup>2 / 4 \ cmod (?pathint a c' + ?pathint c' b' + ?pathint b' a)" | ``` lp15@60809 ` 1862` ``` "e * K\<^sup>2 / 4 \ cmod (?pathint a' c' + ?pathint c' b + ?pathint b a')" | ``` lp15@60809 ` 1863` ``` "e * K\<^sup>2 / 4 \ cmod (?pathint a' c + ?pathint c b' + ?pathint b' a')" | ``` lp15@60809 ` 1864` ``` "e * K\<^sup>2 / 4 \ cmod (?pathint a' b' + ?pathint b' c' + ?pathint c' a')" ``` lp15@60809 ` 1865` ``` using assms ``` lp15@60809 ` 1866` ``` apply (simp only: *) ``` lp15@60809 ` 1867` ``` apply (blast intro: that dest!: norm_sum_lemma) ``` lp15@60809 ` 1868` ``` done ``` lp15@60809 ` 1869` ``` then show ?thesis ``` lp15@60809 ` 1870` ``` proof cases ``` lp15@60809 ` 1871` ``` case 1 then show ?thesis ``` lp15@60809 ` 1872` ``` apply (rule_tac x=a in exI) ``` lp15@60809 ` 1873` ``` apply (rule exI [where x=c']) ``` lp15@60809 ` 1874` ``` apply (rule exI [where x=b']) ``` lp15@60809 ` 1875` ``` using assms ``` lp15@60809 ` 1876` ``` apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD]) ``` lp15@60809 ` 1877` ``` apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps) ``` lp15@60809 ` 1878` ``` done ``` lp15@60809 ` 1879` ``` next ``` lp15@60809 ` 1880` ``` case 2 then show ?thesis ``` lp15@60809 ` 1881` ``` apply (rule_tac x=a' in exI) ``` lp15@60809 ` 1882` ``` apply (rule exI [where x=c']) ``` lp15@60809 ` 1883` ``` apply (rule exI [where x=b]) ``` lp15@60809 ` 1884` ``` using assms ``` lp15@60809 ` 1885` ``` apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD]) ``` lp15@60809 ` 1886` ``` apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps) ``` lp15@60809 ` 1887` ``` done ``` lp15@60809 ` 1888` ``` next ``` lp15@60809 ` 1889` ``` case 3 then show ?thesis ``` lp15@60809 ` 1890` ``` apply (rule_tac x=a' in exI) ``` lp15@60809 ` 1891` ``` apply (rule exI [where x=c]) ``` lp15@60809 ` 1892` ``` apply (rule exI [where x=b']) ``` lp15@60809 ` 1893` ``` using assms ``` lp15@60809 ` 1894` ``` apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD]) ``` lp15@60809 ` 1895` ``` apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps) ``` lp15@60809 ` 1896` ``` done ``` lp15@60809 ` 1897` ``` next ``` lp15@60809 ` 1898` ``` case 4 then show ?thesis ``` lp15@60809 ` 1899` ``` apply (rule_tac x=a' in exI) ``` lp15@60809 ` 1900` ``` apply (rule exI [where x=b']) ``` lp15@60809 ` 1901` ``` apply (rule exI [where x=c']) ``` lp15@60809 ` 1902` ``` using assms ``` lp15@60809 ` 1903` ``` apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD]) ``` lp15@60809 ` 1904` ``` apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps) ``` lp15@60809 ` 1905` ``` done ``` lp15@60809 ` 1906` ``` qed ``` lp15@60809 ` 1907` ```qed ``` lp15@60809 ` 1908` lp15@60809 ` 1909` ```subsection\Cauchy's theorem for triangles\ ``` lp15@60809 ` 1910` lp15@60809 ` 1911` ```lemma triangle_points_closer: ``` lp15@60809 ` 1912` ``` fixes a::complex ``` lp15@60809 ` 1913` ``` shows "\x \ convex hull {a,b,c}; y \ convex hull {a,b,c}\ ``` lp15@60809 ` 1914` ``` \ norm(x - y) \ norm(a - b) \ ``` lp15@60809 ` 1915` ``` norm(x - y) \ norm(b - c) \ ``` lp15@60809 ` 1916` ``` norm(x - y) \ norm(c - a)" ``` lp15@60809 ` 1917` ``` using simplex_extremal_le [of "{a,b,c}"] ``` lp15@60809 ` 1918` ``` by (auto simp: norm_minus_commute) ``` lp15@60809 ` 1919` lp15@60809 ` 1920` ```lemma holomorphic_point_small_triangle: ``` lp15@60809 ` 1921` ``` assumes x: "x \ s" ``` lp15@60809 ` 1922` ``` and f: "continuous_on s f" ``` lp15@60809 ` 1923` ``` and cd: "f complex_differentiable (at x within s)" ``` lp15@60809 ` 1924` ``` and e: "0 < e" ``` lp15@60809 ` 1925` ``` shows "\k>0. \a b c. dist a b \ k \ dist b c \ k \ dist c a \ k \ ``` lp15@60809 ` 1926` ``` x \ convex hull {a,b,c} \ convex hull {a,b,c} \ s ``` lp15@60809 ` 1927` ``` \ norm(path_integral(linepath a b) f + path_integral(linepath b c) f + ``` lp15@60809 ` 1928` ``` path_integral(linepath c a) f) ``` lp15@60809 ` 1929` ``` \ e*(dist a b + dist b c + dist c a)^2" ``` lp15@60809 ` 1930` ``` (is "\k>0. \a b c. _ \ ?normle a b c") ``` lp15@60809 ` 1931` ```proof - ``` lp15@60809 ` 1932` ``` have le_of_3: "\a x y z. \0 \ x*y; 0 \ x*z; 0 \ y*z; a \ (e*(x + y + z))*x + (e*(x + y + z))*y + (e*(x + y + z))*z\ ``` lp15@60809 ` 1933` ``` \ a \ e*(x + y + z)^2" ``` lp15@60809 ` 1934` ``` by (simp add: algebra_simps power2_eq_square) ``` lp15@60809 ` 1935` ``` have disj_le: "\x \ a \ x \ b \ x \ c; 0 \ a; 0 \ b; 0 \ c\ \ x \ a + b + c" ``` lp15@60809 ` 1936` ``` for x::real and a b c ``` lp15@60809 ` 1937` ``` by linarith ``` lp15@60809 ` 1938` ``` 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 ` 1939` ``` if "convex hull {a, b, c} \ s" for a b c ``` lp15@60809 ` 1940` ``` using segments_subset_convex_hull that ``` lp15@60809 ` 1941` ``` by (metis continuous_on_subset f path_integrable_continuous_linepath)+ ``` lp15@60809 ` 1942` ``` note path_bound = has_path_integral_bound_linepath [simplified norm_minus_commute, OF has_path_integral_integral] ``` lp15@60809 ` 1943` ``` { fix f' a b c d ``` lp15@60809 ` 1944` ``` assume d: "0 < d" ``` lp15@60809 ` 1945` ``` and f': "\y. \cmod (y - x) \ d; y \ s\ \ cmod (f y - f x - f' * (y - x)) \ e * cmod (y - x)" ``` lp15@60809 ` 1946` ``` and le: "cmod (a - b) \ d" "cmod (b - c) \ d" "cmod (c - a) \ d" ``` lp15@60809 ` 1947` ``` and xc: "x \ convex hull {a, b, c}" ``` lp15@60809 ` 1948` ``` and s: "convex hull {a, b, c} \ s" ``` lp15@60809 ` 1949` ``` have pa: "path_integral (linepath a b) f + path_integral (linepath b c) f + path_integral (linepath c a) f = ``` lp15@60809 ` 1950` ``` path_integral (linepath a b) (\y. f y - f x - f'*(y - x)) + ``` lp15@60809 ` 1951` ``` path_integral (linepath b c) (\y. f y - f x - f'*(y - x)) + ``` lp15@60809 ` 1952` ``` path_integral (linepath c a) (\y. f y - f x - f'*(y - x))" ``` lp15@60809 ` 1953` ``` apply (simp add: path_integral_diff path_integral_lmul path_integrable_lmul path_integrable_diff fabc [OF s]) ``` lp15@60809 ` 1954` ``` apply (simp add: field_simps) ``` lp15@60809 ` 1955` ``` done ``` lp15@60809 ` 1956` ``` { fix y ``` lp15@60809 ` 1957` ``` assume yc: "y \ convex hull {a,b,c}" ``` lp15@60809 ` 1958` ``` have "cmod (f y - f x - f' * (y - x)) \ e*norm(y - x)" ``` lp15@60809 ` 1959` ``` apply (rule f') ``` lp15@60809 ` 1960` ``` apply (metis triangle_points_closer [OF xc yc] le norm_minus_commute order_trans) ``` lp15@60809 ` 1961` ``` using s yc by blast ``` lp15@60809 ` 1962` ``` also have "... \ e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" ``` lp15@60809 ` 1963` ``` by (simp add: yc e xc disj_le [OF triangle_points_closer]) ``` lp15@60809 ` 1964` ``` finally have "cmod (f y - f x - f' * (y - x)) \ e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" . ``` lp15@60809 ` 1965` ``` } note cm_le = this ``` lp15@60809 ` 1966` ``` have "?normle a b c" ``` lp15@60809 ` 1967` ``` apply (simp add: dist_norm pa) ``` lp15@60809 ` 1968` ``` apply (rule le_of_3) ``` lp15@60809 ` 1969` ``` using f' xc s e ``` lp15@60809 ` 1970` ``` apply simp_all ``` lp15@60809 ` 1971` ``` apply (intro norm_triangle_le add_mono path_bound) ``` lp15@60809 ` 1972` ``` apply (simp_all add: path_integral_diff path_integral_lmul path_integrable_lmul path_integrable_diff fabc) ``` lp15@60809 ` 1973` ``` apply (blast intro: cm_le elim: dest: segments_subset_convex_hull [THEN subsetD])+ ``` lp15@60809 ` 1974` ``` done ``` lp15@60809 ` 1975` ``` } note * = this ``` lp15@60809 ` 1976` ``` show ?thesis ``` lp15@60809 ` 1977` ``` using cd e ``` lp15@60809 ` 1978` ``` apply (simp add: complex_differentiable_def has_field_derivative_def has_derivative_within_alt approachable_lt_le2 Ball_def) ``` lp15@60809 ` 1979` ``` apply (clarify dest!: spec mp) ``` lp15@60809 ` 1980` ``` using * ``` lp15@60809 ` 1981` ``` apply (simp add: dist_norm, blast) ``` lp15@60809 ` 1982` ``` done ``` lp15@60809 ` 1983` ```qed ``` lp15@60809 ` 1984` lp15@60809 ` 1985` lp15@60809 ` 1986` ```(* Hence the most basic theorem for a triangle.*) ``` lp15@60809 ` 1987` ```locale Chain = ``` lp15@60809 ` 1988` ``` fixes x0 At Follows ``` lp15@60809 ` 1989` ``` assumes At0: "At x0 0" ``` lp15@60809 ` 1990` ``` and AtSuc: "\x n. At x n \ \x'. At x' (Suc n) \ Follows x' x" ``` lp15@60809 ` 1991` ```begin ``` lp15@60809 ` 1992` ``` primrec f where ``` lp15@60809 ` 1993` ``` "f 0 = x0" ``` lp15@60809 ` 1994` ``` | "f (Suc n) = (SOME x. At x (Suc n) \ Follows x (f n))" ``` lp15@60809 ` 1995` lp15@60809 ` 1996` ``` lemma At: "At (f n) n" ``` lp15@60809 ` 1997` ``` proof (induct n) ``` lp15@60809 ` 1998` ``` case 0 show ?case ``` lp15@60809 ` 1999` ``` by (simp add: At0) ``` lp15@60809 ` 2000` ``` next ``` lp15@60809 ` 2001` ``` case (Suc n) show ?case ``` lp15@60809 ` 2002` ``` by (metis (no_types, lifting) AtSuc [OF Suc] f.simps(2) someI_ex) ``` lp15@60809 ` 2003` ``` qed ``` lp15@60809 ` 2004` lp15@60809 ` 2005` ``` lemma Follows: "Follows (f(Suc n)) (f n)" ``` lp15@60809 ` 2006` ``` by (metis (no_types, lifting) AtSuc [OF At [of n]] f.simps(2) someI_ex) ``` lp15@60809 ` 2007` lp15@60809 ` 2008` ``` declare f.simps(2) [simp del] ``` lp15@60809 ` 2009` ```end ``` lp15@60809 ` 2010` lp15@60809 ` 2011` ```lemma Chain3: ``` lp15@60809 ` 2012` ``` assumes At0: "At x0 y0 z0 0" ``` lp15@60809 ` 2013` ``` and AtSuc: "\x y z n. At x y z n \ \x' y' z'. At x' y' z' (Suc n) \ Follows x' y' z' x y z" ``` lp15@60809 ` 2014` ``` obtains f g h where ``` lp15@60809 ` 2015` ``` "f 0 = x0" "g 0 = y0" "h 0 = z0" ``` lp15@60809 ` 2016` ``` "\n. At (f n) (g n) (h n) n" ``` lp15@60809 ` 2017` ``` "\n. Follows (f(Suc n)) (g(Suc n)) (h(Suc n)) (f n) (g n) (h n)" ``` lp15@60809 ` 2018` ```proof - ``` lp15@60809 ` 2019` ``` interpret three: Chain "(x0,y0,z0)" "\(x,y,z). At x y z" "\(x',y',z'). \(x,y,z). Follows x' y' z' x y z" ``` lp15@60809 ` 2020` ``` apply unfold_locales ``` lp15@60809 ` 2021` ``` using At0 AtSuc by auto ``` lp15@60809 ` 2022` ``` show ?thesis ``` lp15@60809 ` 2023` ``` apply (rule that [of "\n. fst (three.f n)" "\n. fst (snd (three.f n))" "\n. snd (snd (three.f n))"]) ``` lp15@60809 ` 2024` ``` apply simp_all ``` lp15@60809 ` 2025` ``` using three.At three.Follows ``` lp15@60809 ` 2026` ``` apply (simp_all add: split_beta') ``` lp15@60809 ` 2027` ``` done ``` lp15@60809 ` 2028` ```qed ``` lp15@60809 ` 2029` lp15@60809 ` 2030` ```lemma Cauchy_theorem_triangle: ``` lp15@60809 ` 2031` ``` assumes "f holomorphic_on (convex hull {a,b,c})" ``` lp15@60809 ` 2032` ``` shows "(f has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)" ``` lp15@60809 ` 2033` ```proof - ``` lp15@60809 ` 2034` ``` have contf: "continuous_on (convex hull {a,b,c}) f" ``` lp15@60809 ` 2035` ``` by (metis assms holomorphic_on_imp_continuous_on) ``` lp15@60809 ` 2036` ``` let ?pathint = "\x y. path_integral(linepath x y) f" ``` lp15@60809 ` 2037` ``` { fix y::complex ``` lp15@60809 ` 2038` ``` assume fy: "(f has_path_integral y) (linepath a b +++ linepath b c +++ linepath c a)" ``` lp15@60809 ` 2039` ``` and ynz: "y \ 0" ``` lp15@60809 ` 2040` ``` def K \ "1 + max (dist a b) (max (dist b c) (dist c a))" ``` lp15@60809 ` 2041` ``` def e \ "norm y / K^2" ``` lp15@60809 ` 2042` ``` have K1: "K \ 1" by (simp add: K_def max.coboundedI1) ``` lp15@60809 ` 2043` ``` then have K: "K > 0" by linarith ``` lp15@60809 ` 2044` ``` have [iff]: "dist a b \ K" "dist b c \ K" "dist c a \ K" ``` lp15@60809 ` 2045` ``` by (simp_all add: K_def) ``` lp15@60809 ` 2046` ``` have e: "e > 0" ``` lp15@60809 ` 2047` ``` unfolding e_def using ynz K1 by simp ``` lp15@60809 ` 2048` ``` def At \ "\x y z n. convex hull {x,y,z} \ convex hull {a,b,c} \ ``` lp15@60809 ` 2049` ``` dist x y \ K/2^n \ dist y z \ K/2^n \ dist z x \ K/2^n \ ``` lp15@60809 ` 2050` ``` norm(?pathint x y + ?pathint y z + ?pathint z x) \ e*(K/2^n)^2" ``` lp15@60809 ` 2051` ``` have At0: "At a b c 0" ``` lp15@60809 ` 2052` ``` using fy ``` lp15@60809 ` 2053` ``` by (simp add: At_def e_def has_chain_integral_chain_integral3) ``` lp15@60809 ` 2054` ``` { fix x y z n ``` lp15@60809 ` 2055` ``` assume At: "At x y z n" ``` lp15@60809 ` 2056` ``` then have contf': "continuous_on (convex hull {x,y,z}) f" ``` lp15@60809 ` 2057` ``` using contf At_def continuous_on_subset by blast ``` lp15@60809 ` 2058` ``` have "\x' y' z'. At x' y' z' (Suc n) \ convex hull {x',y',z'} \ convex hull {x,y,z}" ``` lp15@60809 ` 2059` ``` using At ``` lp15@60809 ` 2060` ``` apply (simp add: At_def) ``` lp15@60809 ` 2061` ``` using Cauchy_theorem_quadrisection [OF contf', of "K/2^n" e] ``` lp15@60809 ` 2062` ``` apply clarsimp ``` lp15@60809 ` 2063` ``` apply (rule_tac x="a'" in exI) ``` lp15@60809 ` 2064` ``` apply (rule_tac x="b'" in exI) ``` lp15@60809 ` 2065` ``` apply (rule_tac x="c'" in exI) ``` lp15@60809 ` 2066` ``` apply (simp add: algebra_simps) ``` lp15@60809 ` 2067` ``` apply (meson convex_hull_subset empty_subsetI insert_subset subsetCE) ``` lp15@60809 ` 2068` ``` done ``` lp15@60809 ` 2069` ``` } note AtSuc = this ``` lp15@60809 ` 2070` ``` obtain fa fb fc ``` lp15@60809 ` 2071` ``` where f0 [simp]: "fa 0 = a" "fb 0 = b" "fc 0 = c" ``` lp15@60809 ` 2072` ``` and cosb: "\n. convex hull {fa n, fb n, fc n} \ convex hull {a,b,c}" ``` lp15@60809 ` 2073` ``` and dist: "\n. dist (fa n) (fb n) \ K/2^n" ``` lp15@60809 ` 2074` ``` "\n. dist (fb n) (fc n) \ K/2^n" ``` lp15@60809 ` 2075` ``` "\n. dist (fc n) (fa n) \ K/2^n" ``` lp15@60809 ` 2076` ``` and no: "\n. norm(?pathint (fa n) (fb n) + ``` lp15@60809 ` 2077` ``` ?pathint (fb n) (fc n) + ``` lp15@60809 ` 2078` ``` ?pathint (fc n) (fa n)) \ e * (K/2^n)^2" ``` lp15@60809 ` 2079` ``` and conv_le: "\n. convex hull {fa(Suc n), fb(Suc n), fc(Suc n)} \ convex hull {fa n, fb n, fc n}" ``` lp15@60809 ` 2080` ``` apply (rule Chain3 [of At, OF At0 AtSuc]) ``` lp15@60809 ` 2081` ``` apply (auto simp: At_def) ``` lp15@60809 ` 2082` ``` done ``` lp15@60809 ` 2083` ``` have "\x. \n. x \ convex hull {fa n, fb n, fc n}" ``` lp15@60809 ` 2084` ``` apply (rule bounded_closed_nest) ``` lp15@60809 ` 2085` ``` apply (simp_all add: compact_imp_closed finite_imp_compact_convex_hull finite_imp_bounded_convex_hull) ``` lp15@60809 ` 2086` ``` apply (rule allI) ``` lp15@60809 ` 2087` ``` apply (rule transitive_stepwise_le) ``` lp15@60809 ` 2088` ``` apply (auto simp: conv_le) ``` lp15@60809 ` 2089` ``` done ``` lp15@60809 ` 2090` ``` then obtain x where x: "\n. x \ convex hull {fa n, fb n, fc n}" by auto ``` lp15@60809 ` 2091` ``` then have xin: "x \ convex hull {a,b,c}" ``` lp15@60809 ` 2092` ``` using assms f0 by blast ``` lp15@60809 ` 2093` ``` then have fx: "f complex_differentiable at x within (convex hull {a,b,c})" ``` lp15@60809 ` 2094` ``` using assms holomorphic_on_def by blast ``` lp15@60809 ` 2095` ``` { fix k n ``` lp15@60809 ` 2096` ``` assume k: "0 < k" ``` lp15@60809 ` 2097` ``` and le: ``` lp15@60809 ` 2098` ``` "\x' y' z'. ``` lp15@60809 ` 2099` ``` \dist x' y' \ k; dist y' z' \ k; dist z' x' \ k; ``` lp15@60809 ` 2100` ``` x \ convex hull {x',y',z'}; ``` lp15@60809 ` 2101` ``` convex hull {x',y',z'} \ convex hull {a,b,c}\ ``` lp15@60809 ` 2102` ``` \ ``` lp15@60809 ` 2103` ``` cmod (?pathint x' y' + ?pathint y' z' + ?pathint z' x') * 10 ``` lp15@60809 ` 2104` ``` \ e * (dist x' y' + dist y' z' + dist z' x')\<^sup>2" ``` lp15@60809 ` 2105` ``` and Kk: "K / k < 2 ^ n" ``` lp15@60809 ` 2106` ``` have "K / 2 ^ n < k" using Kk k ``` lp15@60809 ` 2107` ``` by (auto simp: field_simps) ``` lp15@60809 ` 2108` ``` then have DD: "dist (fa n) (fb n) \ k" "dist (fb n) (fc n) \ k" "dist (fc n) (fa n) \ k" ``` lp15@60809 ` 2109` ``` using dist [of n] k ``` lp15@60809 ` 2110` ``` by linarith+ ``` lp15@60809 ` 2111` ``` have dle: "(dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2 ``` lp15@60809 ` 2112` ``` \ (3 * K / 2 ^ n)\<^sup>2" ``` lp15@60809 ` 2113` ``` using dist [of n] e K ``` lp15@60809 ` 2114` ``` by (simp add: abs_le_square_iff [symmetric]) ``` lp15@60809 ` 2115` ``` have less10: "\x y::real. 0 < x \ y \ 9*x \ y < x*10" ``` lp15@60809 ` 2116` ``` by linarith ``` lp15@60809 ` 2117` ``` have "e * (dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2 \ e * (3 * K / 2 ^ n)\<^sup>2" ``` lp15@60809 ` 2118` ``` using ynz dle e mult_le_cancel_left_pos by blast ``` lp15@60809 ` 2119` ``` also have "... < ``` lp15@60809 ` 2120` ``` cmod (?pathint (fa n) (fb n) + ?pathint (fb n) (fc n) + ?pathint (fc n) (fa n)) * 10" ``` lp15@60809 ` 2121` ``` using no [of n] e K ``` lp15@60809 ` 2122` ``` apply (simp add: e_def field_simps) ``` lp15@60809 ` 2123` ``` apply (simp only: zero_less_norm_iff [symmetric]) ``` lp15@60809 ` 2124` ``` done ``` lp15@60809 ` 2125` ``` finally have False ``` lp15@60809 ` 2126` ``` using le [OF DD x cosb] by auto ``` lp15@60809 ` 2127` ``` } then ``` lp15@60809 ` 2128` ``` have ?thesis ``` lp15@60809 ` 2129` ``` using holomorphic_point_small_triangle [OF xin contf fx, of "e/10"] e ``` lp15@60809 ` 2130` ``` apply clarsimp ``` lp15@60809 ` 2131` ``` apply (rule_tac x1="K/k" in exE [OF real_arch_pow2], blast) ``` lp15@60809 ` 2132` ``` done ``` lp15@60809 ` 2133` ``` } ``` lp15@60809 ` 2134` ``` moreover have "f path_integrable_on (linepath a b +++ linepath b c +++ linepath c a)" ``` lp15@60809 ` 2135` ``` by simp (meson contf continuous_on_subset path_integrable_continuous_linepath segments_subset_convex_hull(1) ``` lp15@60809 ` 2136` ``` segments_subset_convex_hull(3) segments_subset_convex_hull(5)) ``` lp15@60809 ` 2137` ``` ultimately show ?thesis ``` lp15@60809 ` 2138` ``` using has_path_integral_integral by fastforce ``` lp15@60809 ` 2139` ```qed ``` lp15@60809 ` 2140` lp15@60809 ` 2141` lp15@60809 ` 2142` ```subsection\Version needing function holomorphic in interior only\ ``` lp15@60809 ` 2143` lp15@60809 ` 2144` ```lemma Cauchy_theorem_flat_lemma: ``` lp15@60809 ` 2145` ``` assumes f: "continuous_on (convex hull {a,b,c}) f" ``` lp15@60809 ` 2146` ``` and c: "c - a = k *\<^sub>R (b - a)" ``` lp15@60809 ` 2147` ``` and k: "0 \ k" ``` lp15@60809 ` 2148` ``` shows "path_integral (linepath a b) f + path_integral (linepath b c) f + ``` lp15@60809 ` 2149` ``` path_integral (linepath c a) f = 0" ``` lp15@60809 ` 2150` ```proof - ``` lp15@60809 ` 2151` ``` 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 ` 2152` ``` using f continuous_on_subset segments_subset_convex_hull by metis+ ``` lp15@60809 ` 2153` ``` show ?thesis ``` lp15@60809 ` 2154` ``` proof (cases "k \ 1") ``` lp15@60809 ` 2155` ``` case True show ?thesis ``` lp15@60809 ` 2156` ``` by (simp add: path_integral_split [OF fabc(1) k True c] path_integral_reverse_linepath fabc) ``` lp15@60809 ` 2157` ``` next ``` lp15@60809 ` 2158` ``` case False then show ?thesis ``` lp15@60809 ` 2159` ``` using fabc c ``` lp15@60809 ` 2160` ``` apply (subst path_integral_split [of a c f "1/k" b, symmetric]) ``` lp15@60809 ` 2161` ``` apply (metis closed_segment_commute fabc(3)) ``` lp15@60809 ` 2162` ``` apply (auto simp: k path_integral_reverse_linepath) ``` lp15@60809 ` 2163` ``` done ``` lp15@60809 ` 2164` ``` qed ``` lp15@60809 ` 2165` ```qed ``` lp15@60809 ` 2166` lp15@60809 ` 2167` ```lemma Cauchy_theorem_flat: ``` lp15@60809 ` 2168` ``` assumes f: "continuous_on (convex hull {a,b,c}) f" ``` lp15@60809 ` 2169` ``` and c: "c - a = k *\<^sub>R (b - a)" ``` lp15@60809 ` 2170` ``` shows "path_integral (linepath a b) f + ``` lp15@60809 ` 2171` ``` path_integral (linepath b c) f + ``` lp15@60809 ` 2172` ``` path_integral (linepath c a) f = 0" ``` lp15@60809 ` 2173` ```proof (cases "0 \ k") ``` lp15@60809 ` 2174` ``` case True with assms show ?thesis ``` lp15@60809 ` 2175` ``` by (blast intro: Cauchy_theorem_flat_lemma) ``` lp15@60809 ` 2176` ```next ``` lp15@60809 ` 2177` ``` case False ``` lp15@60809 ` 2178` ``` have "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f" ``` lp15@60809 ` 2179` ``` using f continuous_on_subset segments_subset_convex_hull by metis+ ``` lp15@60809 ` 2180` ``` moreover have "path_integral (linepath b a) f + path_integral (linepath a c) f + ``` lp15@60809 ` 2181` ``` path_integral (linepath c b) f = 0" ``` lp15@60809 ` 2182` ``` apply (rule Cauchy_theorem_flat_lemma [of b a c f "1-k"]) ``` lp15@60809 ` 2183` ``` using False c ``` lp15@60809 ` 2184` ``` apply (auto simp: f insert_commute scaleR_conv_of_real algebra_simps) ``` lp15@60809 ` 2185` ``` done ``` lp15@60809 ` 2186` ``` ultimately show ?thesis ``` lp15@60809 ` 2187` ``` apply (auto simp: path_integral_reverse_linepath) ``` lp15@60809 ` 2188` ``` using add_eq_0_iff by force ``` lp15@60809 ` 2189` ```qed ``` lp15@60809 ` 2190` lp15@60809 ` 2191` lp15@60809 ` 2192` ```lemma Cauchy_theorem_triangle_interior: ``` lp15@60809 ` 2193` ``` assumes contf: "continuous_on (convex hull {a,b,c}) f" ``` lp15@60809 ` 2194` ``` and holf: "f holomorphic_on interior (convex hull {a,b,c})" ``` lp15@60809 ` 2195` ``` shows "(f has_path_integral 0) (linepath a b +++ linepath b c +++ linepath c a)" ``` lp15@60809 ` 2196` ```proof - ``` lp15@60809 ` 2197` ``` 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 ` 2198` ``` using contf continuous_on_subset segments_subset_convex_hull by metis+ ``` lp15@60809 ` 2199` ``` have "bounded (f ` (convex hull {a,b,c}))" ``` lp15@60809 ` 2200` ``` by (simp add: compact_continuous_image compact_convex_hull compact_imp_bounded contf) ``` lp15@60809 ` 2201` ``` then obtain B where "0 < B" and Bnf: "\x. x \ convex hull {a,b,c} \ norm (f x) \ B" ``` lp15@60809 ` 2202` ``` by (auto simp: dest!: bounded_pos [THEN iffD1]) ``` lp15@60809 ` 2203` ``` have "bounded (convex hull {a,b,c})" ``` lp15@60809 ` 2204` ``` by (simp add: bounded_convex_hull) ``` lp15@60809 ` 2205` ``` then obtain C where C: "0 < C" and Cno: "\y. y \ convex hull {a,b,c} \ norm y < C" ``` lp15@60809 ` 2206` ``` using bounded_pos_less by blast ``` lp15@60809 ` 2207` ``` then have diff_2C: "norm(x - y) \ 2*C" ``` lp15@60809 ` 2208` ``` if x: "x \ convex hull {a, b, c}" and y: "y \ convex hull {a, b, c}" for x y ``` lp15@60809 ` 2209` ``` proof - ``` lp15@60809 ` 2210` ``` have "cmod x \ C" ``` lp15@60809 ` 2211` ``` using x by (meson Cno not_le not_less_iff_gr_or_eq) ``` lp15@60809 ` 2212` ``` hence "cmod (x - y) \ C + C" ``` lp15@60809 ` 2213` ``` using y by (meson Cno add_mono_thms_linordered_field(4) less_eq_real_def norm_triangle_ineq4 order_trans) ``` lp15@60809 ` 2214` ``` thus "cmod (x - y) \ 2 * C" ``` lp15@60809 ` 2215` ``` by (metis mult_2) ``` lp15@60809 ` 2216` ``` qed ``` lp15@60809 ` 2217` ``` have contf': "continuous_on (convex hull {b,a,c}) f" ``` lp15@60809 ` 2218` ``` using contf by (simp add: insert_commute) ``` lp15@60809 ` 2219` ``` { fix y::complex ``` lp15@60809 ` 2220` ``` assume fy: "(f has_path_integral y) (linepath a b +++ linepath b c +++ linepath c a)" ``` lp15@60809 ` 2221` ``` and ynz: "y \ 0" ``` lp15@60809 ` 2222` ``` 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 ` 2223` ``` by (rule has_chain_integral_chain_integral3 [OF fy]) ``` lp15@60809 ` 2224` ``` have ?thesis ``` lp15@60809 ` 2225` ``` proof (cases "c=a \ a=b \ b=c") ``` lp15@60809 ` 2226` ``` case True then show ?thesis ``` lp15@60809 ` 2227` ``` using Cauchy_theorem_flat [OF contf, of 0] ``` lp15@60809 ` 2228` ``` using has_chain_integral_chain_integral3 [OF fy] ynz ``` lp15@60809 ` 2229` ``` by (force simp: fabc path_integral_reverse_linepath) ``` lp15@60809 ` 2230` ``` next ``` lp15@60809 ` 2231` ``` case False ``` lp15@60809 ` 2232` ``` then have car3: "card {a, b, c} = Suc (DIM(complex))" ``` lp15@60809 ` 2233` ``` by auto ``` lp15@60809 ` 2234` ``` { assume "interior(convex hull {a,b,c}) = {}" ``` lp15@60809 ` 2235` ``` then have "collinear{a,b,c}" ``` lp15@60809 ` 2236` ``` using interior_convex_hull_eq_empty [OF car3] ``` lp15@60809 ` 2237` ``` by (simp add: collinear_3_eq_affine_dependent) ``` lp15@60809 ` 2238` ``` then have "False" ``` lp15@60809 ` 2239` ``` using False ``` lp15@60809 ` 2240` ``` apply (clarsimp simp add: collinear_3 collinear_lemma) ``` lp15@60809 ` 2241` ``` apply (drule Cauchy_theorem_flat [OF contf']) ``` lp15@60809 ` 2242` ``` using pi_eq_y ynz ``` lp15@60809 ` 2243` ``` apply (simp add: fabc add_eq_0_iff path_integral_reverse_linepath) ``` lp15@60809 ` 2244` ``` done ``` lp15@60809 ` 2245` ``` } ``` lp15@60809 ` 2246` ``` then obtain d where d: "d \ interior (convex hull {a, b, c})" ``` lp15@60809 ` 2247` ``` by blast ``` lp15@60809 ` 2248` ``` { fix d1 ``` lp15@60809 ` 2249` ``` assume d1_pos: "0 < d1" ``` lp15@60809 ` 2250` ``` and d1: "\x x'. \x\convex hull {a, b, c}; x'\convex hull {a, b, c}; cmod (x' - x) < d1\ ``` lp15@60809 ` 2251` ``` \ cmod (f x' - f x) < cmod y / (24 * C)" ``` lp15@60809 ` 2252` ``` def e \ "min 1 (min (d1/(4*C)) ((norm y / 24 / C) / B))" ``` lp15@60809 ` 2253` ``` def shrink \ "\x. x - e *\<^sub>R (x - d)" ``` lp15@60809 ` 2254` ``` let ?pathint = "\x y. path_integral(linepath x y) f" ``` lp15@60809 ` 2255` ``` have e: "0 < e" "e \ 1" "e \ d1 / (4 * C)" "e \ cmod y / 24 / C / B" ``` wenzelm@61222 ` 2256` ``` using d1_pos \C>0\ \B>0\ ynz by (simp_all add: e_def) ``` lp15@60809 ` 2257` ``` then have eCB: "24 * e * C * B \ cmod y" ``` wenzelm@61222 ` 2258` ``` using \C>0\ \B>0\ by (simp add: field_simps) ``` lp15@60809 ` 2259` ``` have e_le_d1: "e * (4 * C) \ d1" ``` wenzelm@61222 ` 2260` ``` using e \C>0\ by (simp add: field_simps) ``` lp15@60809 ` 2261` ``` have "shrink a \ interior(convex hull {a,b,c})" ``` lp15@60809 ` 2262` ``` "shrink b \ interior(convex hull {a,b,c})" ``` lp15@60809 ` 2263` ``` "shrink c \ interior(convex hull {a,b,c})" ``` lp15@60809 ` 2264` ``` using d e by (auto simp: hull_inc mem_interior_convex_shrink shrink_def) ``` lp15@60809 ` 2265` ``` then have fhp0: "(f has_path_integral 0) ``` lp15@60809 ` 2266` ``` (linepath (shrink a) (shrink b) +++ linepath (shrink b) (shrink c) +++ linepath (shrink c) (shrink a))" ``` lp15@60809 ` 2267` ``` by (simp add: Cauchy_theorem_triangle holomorphic_on_subset [OF holf] hull_minimal convex_interior) ``` lp15@60809 ` 2268` ``` then have f_0_shrink: "?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a) = 0" ``` lp15@60809 ` 2269` ``` by (simp add: has_chain_integral_chain_integral3) ``` lp15@60809 ` 2270` ``` have fpi_abc: "f path_integrable_on linepath (shrink a) (shrink b)" ``` lp15@60809 ` 2271` ``` "f path_integrable_on linepath (shrink b) (shrink c)" ``` lp15@60809 ` 2272` ``` "f path_integrable_on linepath (shrink c) (shrink a)" ``` lp15@60809 ` 2273` ``` using fhp0 by (auto simp: valid_path_join dest: has_path_integral_integrable) ``` lp15@60809 ` 2274` ``` have cmod_shr: "\x y. cmod (shrink y - shrink x - (y - x)) = e * cmod (x - y)" ``` lp15@60809 ` 2275` ``` using e by (simp add: shrink_def real_vector.scale_right_diff_distrib [symmetric]) ``` lp15@60809 ` 2276` ``` have sh_eq: "\a b d::complex. (b - e *\<^sub>R (b - d)) - (a - e *\<^sub>R (a - d)) - (b - a) = e *\<^sub>R (a - b)" ```