src/HOL/Deriv.thy
 author hoelzl Mon Dec 03 18:19:07 2012 +0100 (2012-12-03) changeset 50327 bbea2e82871c parent 47108 2a1953f0d20d child 50328 25b1e8686ce0 permissions -rw-r--r--
 huffman@21164 ` 1` ```(* Title : Deriv.thy ``` huffman@21164 ` 2` ``` Author : Jacques D. Fleuriot ``` huffman@21164 ` 3` ``` Copyright : 1998 University of Cambridge ``` huffman@21164 ` 4` ``` Conversion to Isar and new proofs by Lawrence C Paulson, 2004 ``` huffman@21164 ` 5` ``` GMVT by Benjamin Porter, 2005 ``` huffman@21164 ` 6` ```*) ``` huffman@21164 ` 7` huffman@21164 ` 8` ```header{* Differentiation *} ``` huffman@21164 ` 9` huffman@21164 ` 10` ```theory Deriv ``` huffman@29987 ` 11` ```imports Lim ``` huffman@21164 ` 12` ```begin ``` huffman@21164 ` 13` huffman@22984 ` 14` ```text{*Standard Definitions*} ``` huffman@21164 ` 15` huffman@21164 ` 16` ```definition ``` huffman@21784 ` 17` ``` deriv :: "['a::real_normed_field \ 'a, 'a, 'a] \ bool" ``` huffman@21164 ` 18` ``` --{*Differentiation: D is derivative of function f at x*} ``` wenzelm@21404 ` 19` ``` ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where ``` huffman@21784 ` 20` ``` "DERIV f x :> D = ((%h. (f(x + h) - f x) / h) -- 0 --> D)" ``` huffman@21164 ` 21` huffman@21164 ` 22` ```primrec ``` haftmann@34941 ` 23` ``` Bolzano_bisect :: "(real \ real \ bool) \ real \ real \ nat \ real \ real" where ``` haftmann@34941 ` 24` ``` "Bolzano_bisect P a b 0 = (a, b)" ``` haftmann@34941 ` 25` ``` | "Bolzano_bisect P a b (Suc n) = ``` haftmann@34941 ` 26` ``` (let (x, y) = Bolzano_bisect P a b n ``` haftmann@34941 ` 27` ``` in if P (x, (x+y) / 2) then ((x+y)/2, y) ``` haftmann@34941 ` 28` ``` else (x, (x+y)/2))" ``` huffman@21164 ` 29` huffman@21164 ` 30` huffman@21164 ` 31` ```subsection {* Derivatives *} ``` huffman@21164 ` 32` huffman@21784 ` 33` ```lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)" ``` huffman@21164 ` 34` ```by (simp add: deriv_def) ``` huffman@21164 ` 35` huffman@21784 ` 36` ```lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --> D" ``` huffman@21164 ` 37` ```by (simp add: deriv_def) ``` huffman@21164 ` 38` huffman@21164 ` 39` ```lemma DERIV_const [simp]: "DERIV (\x. k) x :> 0" ``` huffman@44568 ` 40` ``` by (simp add: deriv_def tendsto_const) ``` huffman@21164 ` 41` huffman@23069 ` 42` ```lemma DERIV_ident [simp]: "DERIV (\x. x) x :> 1" ``` huffman@44568 ` 43` ``` by (simp add: deriv_def tendsto_const cong: LIM_cong) ``` huffman@21164 ` 44` huffman@21164 ` 45` ```lemma DERIV_add: ``` huffman@21164 ` 46` ``` "\DERIV f x :> D; DERIV g x :> E\ \ DERIV (\x. f x + g x) x :> D + E" ``` huffman@44568 ` 47` ``` by (simp only: deriv_def add_diff_add add_divide_distrib tendsto_add) ``` huffman@21164 ` 48` huffman@21164 ` 49` ```lemma DERIV_minus: ``` huffman@21164 ` 50` ``` "DERIV f x :> D \ DERIV (\x. - f x) x :> - D" ``` huffman@44568 ` 51` ``` by (simp only: deriv_def minus_diff_minus divide_minus_left tendsto_minus) ``` huffman@21164 ` 52` huffman@21164 ` 53` ```lemma DERIV_diff: ``` huffman@21164 ` 54` ``` "\DERIV f x :> D; DERIV g x :> E\ \ DERIV (\x. f x - g x) x :> D - E" ``` haftmann@37887 ` 55` ```by (simp only: diff_minus DERIV_add DERIV_minus) ``` huffman@21164 ` 56` huffman@21164 ` 57` ```lemma DERIV_add_minus: ``` huffman@21164 ` 58` ``` "\DERIV f x :> D; DERIV g x :> E\ \ DERIV (\x. f x + - g x) x :> D + - E" ``` huffman@21164 ` 59` ```by (simp only: DERIV_add DERIV_minus) ``` huffman@21164 ` 60` huffman@21164 ` 61` ```lemma DERIV_isCont: "DERIV f x :> D \ isCont f x" ``` huffman@21164 ` 62` ```proof (unfold isCont_iff) ``` huffman@21164 ` 63` ``` assume "DERIV f x :> D" ``` huffman@21784 ` 64` ``` hence "(\h. (f(x+h) - f(x)) / h) -- 0 --> D" ``` huffman@21164 ` 65` ``` by (rule DERIV_D) ``` huffman@21784 ` 66` ``` hence "(\h. (f(x+h) - f(x)) / h * h) -- 0 --> D * 0" ``` huffman@44568 ` 67` ``` by (intro tendsto_mult tendsto_ident_at) ``` huffman@21784 ` 68` ``` hence "(\h. (f(x+h) - f(x)) * (h / h)) -- 0 --> 0" ``` huffman@21784 ` 69` ``` by simp ``` huffman@21784 ` 70` ``` hence "(\h. f(x+h) - f(x)) -- 0 --> 0" ``` nipkow@23398 ` 71` ``` by (simp cong: LIM_cong) ``` huffman@21164 ` 72` ``` thus "(\h. f(x+h)) -- 0 --> f(x)" ``` huffman@31338 ` 73` ``` by (simp add: LIM_def dist_norm) ``` huffman@21164 ` 74` ```qed ``` huffman@21164 ` 75` huffman@21164 ` 76` ```lemma DERIV_mult_lemma: ``` huffman@21784 ` 77` ``` fixes a b c d :: "'a::real_field" ``` huffman@21784 ` 78` ``` shows "(a * b - c * d) / h = a * ((b - d) / h) + ((a - c) / h) * d" ``` nipkow@23477 ` 79` ```by (simp add: diff_minus add_divide_distrib [symmetric] ring_distribs) ``` huffman@21164 ` 80` huffman@21164 ` 81` ```lemma DERIV_mult': ``` huffman@21164 ` 82` ``` assumes f: "DERIV f x :> D" ``` huffman@21164 ` 83` ``` assumes g: "DERIV g x :> E" ``` huffman@21164 ` 84` ``` shows "DERIV (\x. f x * g x) x :> f x * E + D * g x" ``` huffman@21164 ` 85` ```proof (unfold deriv_def) ``` huffman@21164 ` 86` ``` from f have "isCont f x" ``` huffman@21164 ` 87` ``` by (rule DERIV_isCont) ``` huffman@21164 ` 88` ``` hence "(\h. f(x+h)) -- 0 --> f x" ``` huffman@21164 ` 89` ``` by (simp only: isCont_iff) ``` huffman@21784 ` 90` ``` hence "(\h. f(x+h) * ((g(x+h) - g x) / h) + ``` huffman@21784 ` 91` ``` ((f(x+h) - f x) / h) * g x) ``` huffman@21164 ` 92` ``` -- 0 --> f x * E + D * g x" ``` huffman@44568 ` 93` ``` by (intro tendsto_intros DERIV_D f g) ``` huffman@21784 ` 94` ``` thus "(\h. (f(x+h) * g(x+h) - f x * g x) / h) ``` huffman@21164 ` 95` ``` -- 0 --> f x * E + D * g x" ``` huffman@21164 ` 96` ``` by (simp only: DERIV_mult_lemma) ``` huffman@21164 ` 97` ```qed ``` huffman@21164 ` 98` huffman@21164 ` 99` ```lemma DERIV_mult: ``` huffman@21164 ` 100` ``` "[| DERIV f x :> Da; DERIV g x :> Db |] ``` huffman@21164 ` 101` ``` ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))" ``` huffman@21164 ` 102` ```by (drule (1) DERIV_mult', simp only: mult_commute add_commute) ``` huffman@21164 ` 103` huffman@21164 ` 104` ```lemma DERIV_unique: ``` huffman@21164 ` 105` ``` "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E" ``` huffman@21164 ` 106` ```apply (simp add: deriv_def) ``` huffman@21164 ` 107` ```apply (blast intro: LIM_unique) ``` huffman@21164 ` 108` ```done ``` huffman@21164 ` 109` huffman@21164 ` 110` ```text{*Differentiation of finite sum*} ``` huffman@21164 ` 111` hoelzl@31880 ` 112` ```lemma DERIV_setsum: ``` hoelzl@31880 ` 113` ``` assumes "finite S" ``` hoelzl@31880 ` 114` ``` and "\ n. n \ S \ DERIV (%x. f x n) x :> (f' x n)" ``` hoelzl@31880 ` 115` ``` shows "DERIV (%x. setsum (f x) S) x :> setsum (f' x) S" ``` hoelzl@31880 ` 116` ``` using assms by induct (auto intro!: DERIV_add) ``` hoelzl@31880 ` 117` huffman@21164 ` 118` ```lemma DERIV_sumr [rule_format (no_asm)]: ``` huffman@21164 ` 119` ``` "(\r. m \ r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x)) ``` huffman@21164 ` 120` ``` --> DERIV (%x. \n=m.. (\r=m.. 'a" shows ``` huffman@21784 ` 127` ``` "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) = ``` huffman@21164 ` 128` ``` ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)" ``` huffman@21164 ` 129` ```apply (rule iffI) ``` huffman@21164 ` 130` ```apply (drule_tac k="- a" in LIM_offset) ``` huffman@21164 ` 131` ```apply (simp add: diff_minus) ``` huffman@21164 ` 132` ```apply (drule_tac k="a" in LIM_offset) ``` huffman@21164 ` 133` ```apply (simp add: add_commute) ``` huffman@21164 ` 134` ```done ``` huffman@21164 ` 135` huffman@21784 ` 136` ```lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)" ``` huffman@21784 ` 137` ```by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff) ``` huffman@21164 ` 138` huffman@21164 ` 139` ```lemma DERIV_inverse_lemma: ``` huffman@21784 ` 140` ``` "\a \ 0; b \ (0::'a::real_normed_field)\ ``` huffman@21784 ` 141` ``` \ (inverse a - inverse b) / h ``` huffman@21784 ` 142` ``` = - (inverse a * ((a - b) / h) * inverse b)" ``` huffman@21164 ` 143` ```by (simp add: inverse_diff_inverse) ``` huffman@21164 ` 144` huffman@21164 ` 145` ```lemma DERIV_inverse': ``` huffman@21164 ` 146` ``` assumes der: "DERIV f x :> D" ``` huffman@21164 ` 147` ``` assumes neq: "f x \ 0" ``` huffman@21164 ` 148` ``` shows "DERIV (\x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))" ``` huffman@21164 ` 149` ``` (is "DERIV _ _ :> ?E") ``` huffman@21164 ` 150` ```proof (unfold DERIV_iff2) ``` huffman@21164 ` 151` ``` from der have lim_f: "f -- x --> f x" ``` huffman@21164 ` 152` ``` by (rule DERIV_isCont [unfolded isCont_def]) ``` huffman@21164 ` 153` huffman@21164 ` 154` ``` from neq have "0 < norm (f x)" by simp ``` huffman@21164 ` 155` ``` with LIM_D [OF lim_f] obtain s ``` huffman@21164 ` 156` ``` where s: "0 < s" ``` huffman@21164 ` 157` ``` and less_fx: "\z. \z \ x; norm (z - x) < s\ ``` huffman@21164 ` 158` ``` \ norm (f z - f x) < norm (f x)" ``` huffman@21164 ` 159` ``` by fast ``` huffman@21164 ` 160` huffman@21784 ` 161` ``` show "(\z. (inverse (f z) - inverse (f x)) / (z - x)) -- x --> ?E" ``` huffman@21164 ` 162` ``` proof (rule LIM_equal2 [OF s]) ``` huffman@21784 ` 163` ``` fix z ``` huffman@21164 ` 164` ``` assume "z \ x" "norm (z - x) < s" ``` huffman@21164 ` 165` ``` hence "norm (f z - f x) < norm (f x)" by (rule less_fx) ``` huffman@21164 ` 166` ``` hence "f z \ 0" by auto ``` huffman@21784 ` 167` ``` thus "(inverse (f z) - inverse (f x)) / (z - x) = ``` huffman@21784 ` 168` ``` - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x))" ``` huffman@21164 ` 169` ``` using neq by (rule DERIV_inverse_lemma) ``` huffman@21164 ` 170` ``` next ``` huffman@21784 ` 171` ``` from der have "(\z. (f z - f x) / (z - x)) -- x --> D" ``` huffman@21164 ` 172` ``` by (unfold DERIV_iff2) ``` huffman@21784 ` 173` ``` thus "(\z. - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x))) ``` huffman@21164 ` 174` ``` -- x --> ?E" ``` huffman@44568 ` 175` ``` by (intro tendsto_intros lim_f neq) ``` huffman@21164 ` 176` ``` qed ``` huffman@21164 ` 177` ```qed ``` huffman@21164 ` 178` huffman@21164 ` 179` ```lemma DERIV_divide: ``` huffman@21784 ` 180` ``` "\DERIV f x :> D; DERIV g x :> E; g x \ 0\ ``` huffman@21784 ` 181` ``` \ DERIV (\x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)" ``` huffman@21164 ` 182` ```apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) + ``` huffman@21164 ` 183` ``` D * inverse (g x) = (D * g x - f x * E) / (g x * g x)") ``` huffman@21164 ` 184` ```apply (erule subst) ``` huffman@21164 ` 185` ```apply (unfold divide_inverse) ``` huffman@21164 ` 186` ```apply (erule DERIV_mult') ``` huffman@21164 ` 187` ```apply (erule (1) DERIV_inverse') ``` nipkow@23477 ` 188` ```apply (simp add: ring_distribs nonzero_inverse_mult_distrib) ``` huffman@21164 ` 189` ```done ``` huffman@21164 ` 190` huffman@21164 ` 191` ```lemma DERIV_power_Suc: ``` haftmann@31017 ` 192` ``` fixes f :: "'a \ 'a::{real_normed_field}" ``` huffman@21164 ` 193` ``` assumes f: "DERIV f x :> D" ``` huffman@23431 ` 194` ``` shows "DERIV (\x. f x ^ Suc n) x :> (1 + of_nat n) * (D * f x ^ n)" ``` huffman@21164 ` 195` ```proof (induct n) ``` huffman@21164 ` 196` ```case 0 ``` huffman@30273 ` 197` ``` show ?case by (simp add: f) ``` huffman@21164 ` 198` ```case (Suc k) ``` huffman@21164 ` 199` ``` from DERIV_mult' [OF f Suc] show ?case ``` nipkow@23477 ` 200` ``` apply (simp only: of_nat_Suc ring_distribs mult_1_left) ``` nipkow@29667 ` 201` ``` apply (simp only: power_Suc algebra_simps) ``` huffman@21164 ` 202` ``` done ``` huffman@21164 ` 203` ```qed ``` huffman@21164 ` 204` huffman@21164 ` 205` ```lemma DERIV_power: ``` haftmann@31017 ` 206` ``` fixes f :: "'a \ 'a::{real_normed_field}" ``` huffman@21164 ` 207` ``` assumes f: "DERIV f x :> D" ``` huffman@21784 ` 208` ``` shows "DERIV (\x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))" ``` huffman@30273 ` 209` ```by (cases "n", simp, simp add: DERIV_power_Suc f del: power_Suc) ``` huffman@21164 ` 210` huffman@29975 ` 211` ```text {* Caratheodory formulation of derivative at a point *} ``` huffman@21164 ` 212` huffman@21164 ` 213` ```lemma CARAT_DERIV: ``` huffman@21164 ` 214` ``` "(DERIV f x :> l) = ``` huffman@21784 ` 215` ``` (\g. (\z. f z - f x = g z * (z-x)) & isCont g x & g x = l)" ``` huffman@21164 ` 216` ``` (is "?lhs = ?rhs") ``` huffman@21164 ` 217` ```proof ``` huffman@21164 ` 218` ``` assume der: "DERIV f x :> l" ``` huffman@21784 ` 219` ``` show "\g. (\z. f z - f x = g z * (z-x)) \ isCont g x \ g x = l" ``` huffman@21164 ` 220` ``` proof (intro exI conjI) ``` huffman@21784 ` 221` ``` let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))" ``` nipkow@23413 ` 222` ``` show "\z. f z - f x = ?g z * (z-x)" by simp ``` huffman@21164 ` 223` ``` show "isCont ?g x" using der ``` huffman@21164 ` 224` ``` by (simp add: isCont_iff DERIV_iff diff_minus ``` huffman@21164 ` 225` ``` cong: LIM_equal [rule_format]) ``` huffman@21164 ` 226` ``` show "?g x = l" by simp ``` huffman@21164 ` 227` ``` qed ``` huffman@21164 ` 228` ```next ``` huffman@21164 ` 229` ``` assume "?rhs" ``` huffman@21164 ` 230` ``` then obtain g where ``` huffman@21784 ` 231` ``` "(\z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast ``` huffman@21164 ` 232` ``` thus "(DERIV f x :> l)" ``` nipkow@23413 ` 233` ``` by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong) ``` huffman@21164 ` 234` ```qed ``` huffman@21164 ` 235` huffman@21164 ` 236` ```lemma DERIV_chain': ``` huffman@21164 ` 237` ``` assumes f: "DERIV f x :> D" ``` huffman@21164 ` 238` ``` assumes g: "DERIV g (f x) :> E" ``` huffman@21784 ` 239` ``` shows "DERIV (\x. g (f x)) x :> E * D" ``` huffman@21164 ` 240` ```proof (unfold DERIV_iff2) ``` huffman@21784 ` 241` ``` obtain d where d: "\y. g y - g (f x) = d y * (y - f x)" ``` huffman@21164 ` 242` ``` and cont_d: "isCont d (f x)" and dfx: "d (f x) = E" ``` huffman@21164 ` 243` ``` using CARAT_DERIV [THEN iffD1, OF g] by fast ``` huffman@21164 ` 244` ``` from f have "f -- x --> f x" ``` huffman@21164 ` 245` ``` by (rule DERIV_isCont [unfolded isCont_def]) ``` huffman@21164 ` 246` ``` with cont_d have "(\z. d (f z)) -- x --> d (f x)" ``` huffman@44568 ` 247` ``` by (rule isCont_tendsto_compose) ``` huffman@21784 ` 248` ``` hence "(\z. d (f z) * ((f z - f x) / (z - x))) ``` huffman@21784 ` 249` ``` -- x --> d (f x) * D" ``` huffman@44568 ` 250` ``` by (rule tendsto_mult [OF _ f [unfolded DERIV_iff2]]) ``` huffman@21784 ` 251` ``` thus "(\z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D" ``` huffman@35216 ` 252` ``` by (simp add: d dfx) ``` huffman@21164 ` 253` ```qed ``` huffman@21164 ` 254` wenzelm@31899 ` 255` ```text {* ``` wenzelm@31899 ` 256` ``` Let's do the standard proof, though theorem ``` wenzelm@31899 ` 257` ``` @{text "LIM_mult2"} follows from a NS proof ``` wenzelm@31899 ` 258` ```*} ``` huffman@21164 ` 259` huffman@21164 ` 260` ```lemma DERIV_cmult: ``` huffman@21164 ` 261` ``` "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D" ``` huffman@21164 ` 262` ```by (drule DERIV_mult' [OF DERIV_const], simp) ``` huffman@21164 ` 263` paulson@33654 ` 264` ```lemma DERIV_cdivide: "DERIV f x :> D ==> DERIV (%x. f x / c) x :> D / c" ``` paulson@33654 ` 265` ``` apply (subgoal_tac "DERIV (%x. (1 / c) * f x) x :> (1 / c) * D", force) ``` paulson@33654 ` 266` ``` apply (erule DERIV_cmult) ``` paulson@33654 ` 267` ``` done ``` paulson@33654 ` 268` wenzelm@31899 ` 269` ```text {* Standard version *} ``` huffman@21164 ` 270` ```lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db" ``` huffman@35216 ` 271` ```by (drule (1) DERIV_chain', simp add: o_def mult_commute) ``` huffman@21164 ` 272` huffman@21164 ` 273` ```lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db" ``` huffman@21164 ` 274` ```by (auto dest: DERIV_chain simp add: o_def) ``` huffman@21164 ` 275` wenzelm@31899 ` 276` ```text {* Derivative of linear multiplication *} ``` huffman@21164 ` 277` ```lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c" ``` huffman@23069 ` 278` ```by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp) ``` huffman@21164 ` 279` huffman@21164 ` 280` ```lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))" ``` huffman@23069 ` 281` ```apply (cut_tac DERIV_power [OF DERIV_ident]) ``` huffman@35216 ` 282` ```apply (simp add: real_of_nat_def) ``` huffman@21164 ` 283` ```done ``` huffman@21164 ` 284` wenzelm@31899 ` 285` ```text {* Power of @{text "-1"} *} ``` huffman@21164 ` 286` huffman@21784 ` 287` ```lemma DERIV_inverse: ``` haftmann@31017 ` 288` ``` fixes x :: "'a::{real_normed_field}" ``` huffman@21784 ` 289` ``` shows "x \ 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))" ``` huffman@30273 ` 290` ```by (drule DERIV_inverse' [OF DERIV_ident]) simp ``` huffman@21164 ` 291` wenzelm@31899 ` 292` ```text {* Derivative of inverse *} ``` huffman@21784 ` 293` ```lemma DERIV_inverse_fun: ``` haftmann@31017 ` 294` ``` fixes x :: "'a::{real_normed_field}" ``` huffman@21784 ` 295` ``` shows "[| DERIV f x :> d; f(x) \ 0 |] ``` huffman@21784 ` 296` ``` ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))" ``` huffman@30273 ` 297` ```by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib) ``` huffman@21164 ` 298` wenzelm@31899 ` 299` ```text {* Derivative of quotient *} ``` huffman@21784 ` 300` ```lemma DERIV_quotient: ``` haftmann@31017 ` 301` ``` fixes x :: "'a::{real_normed_field}" ``` huffman@21784 ` 302` ``` shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \ 0 |] ``` huffman@21784 ` 303` ``` ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))" ``` huffman@30273 ` 304` ```by (drule (2) DERIV_divide) (simp add: mult_commute) ``` huffman@21164 ` 305` wenzelm@31899 ` 306` ```text {* @{text "DERIV_intros"} *} ``` wenzelm@31899 ` 307` ```ML {* ``` wenzelm@31902 ` 308` ```structure Deriv_Intros = Named_Thms ``` wenzelm@31899 ` 309` ```( ``` wenzelm@45294 ` 310` ``` val name = @{binding DERIV_intros} ``` wenzelm@31899 ` 311` ``` val description = "DERIV introduction rules" ``` wenzelm@31899 ` 312` ```) ``` wenzelm@31899 ` 313` ```*} ``` hoelzl@31880 ` 314` wenzelm@31902 ` 315` ```setup Deriv_Intros.setup ``` hoelzl@31880 ` 316` hoelzl@31880 ` 317` ```lemma DERIV_cong: "\ DERIV f x :> X ; X = Y \ \ DERIV f x :> Y" ``` hoelzl@31880 ` 318` ``` by simp ``` hoelzl@31880 ` 319` hoelzl@31880 ` 320` ```declare ``` hoelzl@31880 ` 321` ``` DERIV_const[THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 322` ``` DERIV_ident[THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 323` ``` DERIV_add[THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 324` ``` DERIV_minus[THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 325` ``` DERIV_mult[THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 326` ``` DERIV_diff[THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 327` ``` DERIV_inverse'[THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 328` ``` DERIV_divide[THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 329` ``` DERIV_power[where 'a=real, THEN DERIV_cong, ``` hoelzl@31880 ` 330` ``` unfolded real_of_nat_def[symmetric], DERIV_intros] ``` hoelzl@31880 ` 331` ``` DERIV_setsum[THEN DERIV_cong, DERIV_intros] ``` huffman@22984 ` 332` wenzelm@31899 ` 333` huffman@22984 ` 334` ```subsection {* Differentiability predicate *} ``` huffman@21164 ` 335` huffman@29169 ` 336` ```definition ``` huffman@29169 ` 337` ``` differentiable :: "['a::real_normed_field \ 'a, 'a] \ bool" ``` huffman@29169 ` 338` ``` (infixl "differentiable" 60) where ``` huffman@29169 ` 339` ``` "f differentiable x = (\D. DERIV f x :> D)" ``` huffman@29169 ` 340` huffman@29169 ` 341` ```lemma differentiableE [elim?]: ``` huffman@29169 ` 342` ``` assumes "f differentiable x" ``` huffman@29169 ` 343` ``` obtains df where "DERIV f x :> df" ``` wenzelm@41550 ` 344` ``` using assms unfolding differentiable_def .. ``` huffman@29169 ` 345` huffman@21164 ` 346` ```lemma differentiableD: "f differentiable x ==> \D. DERIV f x :> D" ``` huffman@21164 ` 347` ```by (simp add: differentiable_def) ``` huffman@21164 ` 348` huffman@21164 ` 349` ```lemma differentiableI: "DERIV f x :> D ==> f differentiable x" ``` huffman@21164 ` 350` ```by (force simp add: differentiable_def) ``` huffman@21164 ` 351` huffman@29169 ` 352` ```lemma differentiable_ident [simp]: "(\x. x) differentiable x" ``` huffman@29169 ` 353` ``` by (rule DERIV_ident [THEN differentiableI]) ``` huffman@29169 ` 354` huffman@29169 ` 355` ```lemma differentiable_const [simp]: "(\z. a) differentiable x" ``` huffman@29169 ` 356` ``` by (rule DERIV_const [THEN differentiableI]) ``` huffman@21164 ` 357` huffman@29169 ` 358` ```lemma differentiable_compose: ``` huffman@29169 ` 359` ``` assumes f: "f differentiable (g x)" ``` huffman@29169 ` 360` ``` assumes g: "g differentiable x" ``` huffman@29169 ` 361` ``` shows "(\x. f (g x)) differentiable x" ``` huffman@29169 ` 362` ```proof - ``` huffman@29169 ` 363` ``` from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" .. ``` huffman@29169 ` 364` ``` moreover ``` huffman@29169 ` 365` ``` from `g differentiable x` obtain dg where "DERIV g x :> dg" .. ``` huffman@29169 ` 366` ``` ultimately ``` huffman@29169 ` 367` ``` have "DERIV (\x. f (g x)) x :> df * dg" by (rule DERIV_chain2) ``` huffman@29169 ` 368` ``` thus ?thesis by (rule differentiableI) ``` huffman@29169 ` 369` ```qed ``` huffman@29169 ` 370` huffman@29169 ` 371` ```lemma differentiable_sum [simp]: ``` huffman@21164 ` 372` ``` assumes "f differentiable x" ``` huffman@21164 ` 373` ``` and "g differentiable x" ``` huffman@21164 ` 374` ``` shows "(\x. f x + g x) differentiable x" ``` huffman@21164 ` 375` ```proof - ``` huffman@29169 ` 376` ``` from `f differentiable x` obtain df where "DERIV f x :> df" .. ``` huffman@29169 ` 377` ``` moreover ``` huffman@29169 ` 378` ``` from `g differentiable x` obtain dg where "DERIV g x :> dg" .. ``` huffman@29169 ` 379` ``` ultimately ``` huffman@29169 ` 380` ``` have "DERIV (\x. f x + g x) x :> df + dg" by (rule DERIV_add) ``` huffman@29169 ` 381` ``` thus ?thesis by (rule differentiableI) ``` huffman@29169 ` 382` ```qed ``` huffman@29169 ` 383` huffman@29169 ` 384` ```lemma differentiable_minus [simp]: ``` huffman@29169 ` 385` ``` assumes "f differentiable x" ``` huffman@29169 ` 386` ``` shows "(\x. - f x) differentiable x" ``` huffman@29169 ` 387` ```proof - ``` huffman@29169 ` 388` ``` from `f differentiable x` obtain df where "DERIV f x :> df" .. ``` huffman@29169 ` 389` ``` hence "DERIV (\x. - f x) x :> - df" by (rule DERIV_minus) ``` huffman@29169 ` 390` ``` thus ?thesis by (rule differentiableI) ``` huffman@21164 ` 391` ```qed ``` huffman@21164 ` 392` huffman@29169 ` 393` ```lemma differentiable_diff [simp]: ``` huffman@21164 ` 394` ``` assumes "f differentiable x" ``` huffman@29169 ` 395` ``` assumes "g differentiable x" ``` huffman@21164 ` 396` ``` shows "(\x. f x - g x) differentiable x" ``` wenzelm@41550 ` 397` ``` unfolding diff_minus using assms by simp ``` huffman@29169 ` 398` huffman@29169 ` 399` ```lemma differentiable_mult [simp]: ``` huffman@29169 ` 400` ``` assumes "f differentiable x" ``` huffman@29169 ` 401` ``` assumes "g differentiable x" ``` huffman@29169 ` 402` ``` shows "(\x. f x * g x) differentiable x" ``` huffman@21164 ` 403` ```proof - ``` huffman@29169 ` 404` ``` from `f differentiable x` obtain df where "DERIV f x :> df" .. ``` huffman@21164 ` 405` ``` moreover ``` huffman@29169 ` 406` ``` from `g differentiable x` obtain dg where "DERIV g x :> dg" .. ``` huffman@29169 ` 407` ``` ultimately ``` huffman@29169 ` 408` ``` have "DERIV (\x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult) ``` huffman@29169 ` 409` ``` thus ?thesis by (rule differentiableI) ``` huffman@21164 ` 410` ```qed ``` huffman@21164 ` 411` huffman@29169 ` 412` ```lemma differentiable_inverse [simp]: ``` huffman@29169 ` 413` ``` assumes "f differentiable x" and "f x \ 0" ``` huffman@29169 ` 414` ``` shows "(\x. inverse (f x)) differentiable x" ``` huffman@21164 ` 415` ```proof - ``` huffman@29169 ` 416` ``` from `f differentiable x` obtain df where "DERIV f x :> df" .. ``` huffman@29169 ` 417` ``` hence "DERIV (\x. inverse (f x)) x :> - (inverse (f x) * df * inverse (f x))" ``` huffman@29169 ` 418` ``` using `f x \ 0` by (rule DERIV_inverse') ``` huffman@29169 ` 419` ``` thus ?thesis by (rule differentiableI) ``` huffman@21164 ` 420` ```qed ``` huffman@21164 ` 421` huffman@29169 ` 422` ```lemma differentiable_divide [simp]: ``` huffman@29169 ` 423` ``` assumes "f differentiable x" ``` huffman@29169 ` 424` ``` assumes "g differentiable x" and "g x \ 0" ``` huffman@29169 ` 425` ``` shows "(\x. f x / g x) differentiable x" ``` wenzelm@41550 ` 426` ``` unfolding divide_inverse using assms by simp ``` huffman@29169 ` 427` huffman@29169 ` 428` ```lemma differentiable_power [simp]: ``` haftmann@31017 ` 429` ``` fixes f :: "'a::{real_normed_field} \ 'a" ``` huffman@29169 ` 430` ``` assumes "f differentiable x" ``` huffman@29169 ` 431` ``` shows "(\x. f x ^ n) differentiable x" ``` wenzelm@41550 ` 432` ``` apply (induct n) ``` wenzelm@41550 ` 433` ``` apply simp ``` wenzelm@41550 ` 434` ``` apply (simp add: assms) ``` wenzelm@41550 ` 435` ``` done ``` huffman@29169 ` 436` huffman@22984 ` 437` huffman@21164 ` 438` ```subsection {* Nested Intervals and Bisection *} ``` huffman@21164 ` 439` huffman@21164 ` 440` ```text{*Lemmas about nested intervals and proof by bisection (cf.Harrison). ``` huffman@21164 ` 441` ``` All considerably tidied by lcp.*} ``` huffman@21164 ` 442` huffman@21164 ` 443` ```lemma lemma_f_mono_add [rule_format (no_asm)]: "(\n. (f::nat=>real) n \ f (Suc n)) --> f m \ f(m + no)" ``` huffman@21164 ` 444` ```apply (induct "no") ``` huffman@21164 ` 445` ```apply (auto intro: order_trans) ``` huffman@21164 ` 446` ```done ``` huffman@21164 ` 447` huffman@21164 ` 448` ```lemma f_inc_g_dec_Beq_f: "[| \n. f(n) \ f(Suc n); ``` huffman@21164 ` 449` ``` \n. g(Suc n) \ g(n); ``` huffman@21164 ` 450` ``` \n. f(n) \ g(n) |] ``` huffman@21164 ` 451` ``` ==> Bseq (f :: nat \ real)" ``` huffman@21164 ` 452` ```apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI) ``` huffman@44921 ` 453` ```apply (rule conjI) ``` huffman@21164 ` 454` ```apply (induct_tac "n") ``` huffman@21164 ` 455` ```apply (auto intro: order_trans) ``` huffman@44921 ` 456` ```apply (rule_tac y = "g n" in order_trans) ``` huffman@44921 ` 457` ```apply (induct_tac [2] "n") ``` huffman@21164 ` 458` ```apply (auto intro: order_trans) ``` huffman@21164 ` 459` ```done ``` huffman@21164 ` 460` huffman@21164 ` 461` ```lemma f_inc_g_dec_Beq_g: "[| \n. f(n) \ f(Suc n); ``` huffman@21164 ` 462` ``` \n. g(Suc n) \ g(n); ``` huffman@21164 ` 463` ``` \n. f(n) \ g(n) |] ``` huffman@21164 ` 464` ``` ==> Bseq (g :: nat \ real)" ``` huffman@21164 ` 465` ```apply (subst Bseq_minus_iff [symmetric]) ``` huffman@21164 ` 466` ```apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f) ``` huffman@21164 ` 467` ```apply auto ``` huffman@21164 ` 468` ```done ``` huffman@21164 ` 469` huffman@21164 ` 470` ```lemma f_inc_imp_le_lim: ``` huffman@21164 ` 471` ``` fixes f :: "nat \ real" ``` huffman@21164 ` 472` ``` shows "\\n. f n \ f (Suc n); convergent f\ \ f n \ lim f" ``` huffman@44921 ` 473` ``` by (rule incseq_le, simp add: incseq_SucI, simp add: convergent_LIMSEQ_iff) ``` huffman@21164 ` 474` huffman@31404 ` 475` ```lemma lim_uminus: ``` huffman@31404 ` 476` ``` fixes g :: "nat \ 'a::real_normed_vector" ``` huffman@31404 ` 477` ``` shows "convergent g ==> lim (%x. - g x) = - (lim g)" ``` huffman@44568 ` 478` ```apply (rule tendsto_minus [THEN limI]) ``` huffman@21164 ` 479` ```apply (simp add: convergent_LIMSEQ_iff) ``` huffman@21164 ` 480` ```done ``` huffman@21164 ` 481` huffman@21164 ` 482` ```lemma g_dec_imp_lim_le: ``` huffman@21164 ` 483` ``` fixes g :: "nat \ real" ``` huffman@21164 ` 484` ``` shows "\\n. g (Suc n) \ g(n); convergent g\ \ lim g \ g n" ``` huffman@44921 ` 485` ``` by (rule decseq_le, simp add: decseq_SucI, simp add: convergent_LIMSEQ_iff) ``` huffman@21164 ` 486` huffman@21164 ` 487` ```lemma lemma_nest: "[| \n. f(n) \ f(Suc n); ``` huffman@21164 ` 488` ``` \n. g(Suc n) \ g(n); ``` huffman@21164 ` 489` ``` \n. f(n) \ g(n) |] ``` huffman@21164 ` 490` ``` ==> \l m :: real. l \ m & ((\n. f(n) \ l) & f ----> l) & ``` huffman@21164 ` 491` ``` ((\n. m \ g(n)) & g ----> m)" ``` huffman@21164 ` 492` ```apply (subgoal_tac "monoseq f & monoseq g") ``` huffman@21164 ` 493` ```prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc) ``` huffman@21164 ` 494` ```apply (subgoal_tac "Bseq f & Bseq g") ``` huffman@21164 ` 495` ```prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g) ``` huffman@21164 ` 496` ```apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff) ``` huffman@21164 ` 497` ```apply (rule_tac x = "lim f" in exI) ``` huffman@21164 ` 498` ```apply (rule_tac x = "lim g" in exI) ``` huffman@21164 ` 499` ```apply (auto intro: LIMSEQ_le) ``` huffman@21164 ` 500` ```apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff) ``` huffman@21164 ` 501` ```done ``` huffman@21164 ` 502` huffman@21164 ` 503` ```lemma lemma_nest_unique: "[| \n. f(n) \ f(Suc n); ``` huffman@21164 ` 504` ``` \n. g(Suc n) \ g(n); ``` huffman@21164 ` 505` ``` \n. f(n) \ g(n); ``` huffman@21164 ` 506` ``` (%n. f(n) - g(n)) ----> 0 |] ``` huffman@21164 ` 507` ``` ==> \l::real. ((\n. f(n) \ l) & f ----> l) & ``` huffman@21164 ` 508` ``` ((\n. l \ g(n)) & g ----> l)" ``` huffman@21164 ` 509` ```apply (drule lemma_nest, auto) ``` huffman@21164 ` 510` ```apply (subgoal_tac "l = m") ``` huffman@44568 ` 511` ```apply (drule_tac [2] f = f in tendsto_diff) ``` huffman@21164 ` 512` ```apply (auto intro: LIMSEQ_unique) ``` huffman@21164 ` 513` ```done ``` huffman@21164 ` 514` huffman@21164 ` 515` ```text{*The universal quantifiers below are required for the declaration ``` huffman@21164 ` 516` ``` of @{text Bolzano_nest_unique} below.*} ``` huffman@21164 ` 517` huffman@21164 ` 518` ```lemma Bolzano_bisect_le: ``` huffman@21164 ` 519` ``` "a \ b ==> \n. fst (Bolzano_bisect P a b n) \ snd (Bolzano_bisect P a b n)" ``` huffman@21164 ` 520` ```apply (rule allI) ``` huffman@21164 ` 521` ```apply (induct_tac "n") ``` huffman@21164 ` 522` ```apply (auto simp add: Let_def split_def) ``` huffman@21164 ` 523` ```done ``` huffman@21164 ` 524` huffman@21164 ` 525` ```lemma Bolzano_bisect_fst_le_Suc: "a \ b ==> ``` huffman@21164 ` 526` ``` \n. fst(Bolzano_bisect P a b n) \ fst (Bolzano_bisect P a b (Suc n))" ``` huffman@21164 ` 527` ```apply (rule allI) ``` huffman@21164 ` 528` ```apply (induct_tac "n") ``` huffman@21164 ` 529` ```apply (auto simp add: Bolzano_bisect_le Let_def split_def) ``` huffman@21164 ` 530` ```done ``` huffman@21164 ` 531` huffman@21164 ` 532` ```lemma Bolzano_bisect_Suc_le_snd: "a \ b ==> ``` huffman@21164 ` 533` ``` \n. snd(Bolzano_bisect P a b (Suc n)) \ snd (Bolzano_bisect P a b n)" ``` huffman@21164 ` 534` ```apply (rule allI) ``` huffman@21164 ` 535` ```apply (induct_tac "n") ``` huffman@21164 ` 536` ```apply (auto simp add: Bolzano_bisect_le Let_def split_def) ``` huffman@21164 ` 537` ```done ``` huffman@21164 ` 538` huffman@21164 ` 539` ```lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)" ``` huffman@21164 ` 540` ```apply (auto) ``` huffman@21164 ` 541` ```apply (drule_tac f = "%u. (1/2) *u" in arg_cong) ``` huffman@21164 ` 542` ```apply (simp) ``` huffman@21164 ` 543` ```done ``` huffman@21164 ` 544` huffman@21164 ` 545` ```lemma Bolzano_bisect_diff: ``` huffman@21164 ` 546` ``` "a \ b ==> ``` huffman@21164 ` 547` ``` snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) = ``` huffman@21164 ` 548` ``` (b-a) / (2 ^ n)" ``` huffman@21164 ` 549` ```apply (induct "n") ``` huffman@21164 ` 550` ```apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def) ``` huffman@21164 ` 551` ```done ``` huffman@21164 ` 552` huffman@21164 ` 553` ```lemmas Bolzano_nest_unique = ``` huffman@21164 ` 554` ``` lemma_nest_unique ``` huffman@21164 ` 555` ``` [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le] ``` huffman@21164 ` 556` huffman@21164 ` 557` huffman@21164 ` 558` ```lemma not_P_Bolzano_bisect: ``` huffman@21164 ` 559` ``` assumes P: "!!a b c. [| P(a,b); P(b,c); a \ b; b \ c|] ==> P(a,c)" ``` huffman@21164 ` 560` ``` and notP: "~ P(a,b)" ``` huffman@21164 ` 561` ``` and le: "a \ b" ``` huffman@21164 ` 562` ``` shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))" ``` huffman@21164 ` 563` ```proof (induct n) ``` huffman@23441 ` 564` ``` case 0 show ?case using notP by simp ``` huffman@21164 ` 565` ``` next ``` huffman@21164 ` 566` ``` case (Suc n) ``` huffman@21164 ` 567` ``` thus ?case ``` huffman@21164 ` 568` ``` by (auto simp del: surjective_pairing [symmetric] ``` huffman@21164 ` 569` ``` simp add: Let_def split_def Bolzano_bisect_le [OF le] ``` huffman@21164 ` 570` ``` P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"]) ``` huffman@21164 ` 571` ```qed ``` huffman@21164 ` 572` huffman@21164 ` 573` ```(*Now we re-package P_prem as a formula*) ``` huffman@21164 ` 574` ```lemma not_P_Bolzano_bisect': ``` huffman@21164 ` 575` ``` "[| \a b c. P(a,b) & P(b,c) & a \ b & b \ c --> P(a,c); ``` huffman@21164 ` 576` ``` ~ P(a,b); a \ b |] ==> ``` huffman@21164 ` 577` ``` \n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))" ``` huffman@21164 ` 578` ```by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE]) ``` huffman@21164 ` 579` huffman@21164 ` 580` huffman@21164 ` 581` huffman@21164 ` 582` ```lemma lemma_BOLZANO: ``` huffman@21164 ` 583` ``` "[| \a b c. P(a,b) & P(b,c) & a \ b & b \ c --> P(a,c); ``` huffman@21164 ` 584` ``` \x. \d::real. 0 < d & ``` huffman@21164 ` 585` ``` (\a b. a \ x & x \ b & (b-a) < d --> P(a,b)); ``` huffman@21164 ` 586` ``` a \ b |] ``` huffman@21164 ` 587` ``` ==> P(a,b)" ``` wenzelm@45600 ` 588` ```apply (rule Bolzano_nest_unique [where P=P, THEN exE], assumption+) ``` huffman@44568 ` 589` ```apply (rule tendsto_minus_cancel) ``` huffman@21164 ` 590` ```apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero) ``` huffman@21164 ` 591` ```apply (rule ccontr) ``` huffman@21164 ` 592` ```apply (drule not_P_Bolzano_bisect', assumption+) ``` huffman@21164 ` 593` ```apply (rename_tac "l") ``` huffman@21164 ` 594` ```apply (drule_tac x = l in spec, clarify) ``` huffman@31336 ` 595` ```apply (simp add: LIMSEQ_iff) ``` huffman@21164 ` 596` ```apply (drule_tac P = "%r. 0 ?Q r" and x = "d/2" in spec) ``` huffman@21164 ` 597` ```apply (drule_tac P = "%r. 0 ?Q r" and x = "d/2" in spec) ``` huffman@21164 ` 598` ```apply (drule real_less_half_sum, auto) ``` huffman@21164 ` 599` ```apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec) ``` huffman@21164 ` 600` ```apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec) ``` huffman@21164 ` 601` ```apply safe ``` huffman@21164 ` 602` ```apply (simp_all (no_asm_simp)) ``` huffman@21164 ` 603` ```apply (rule_tac y = "abs (fst (Bolzano_bisect P a b (no + noa)) - l) + abs (snd (Bolzano_bisect P a b (no + noa)) - l)" in order_le_less_trans) ``` huffman@21164 ` 604` ```apply (simp (no_asm_simp) add: abs_if) ``` huffman@21164 ` 605` ```apply (rule real_sum_of_halves [THEN subst]) ``` huffman@21164 ` 606` ```apply (rule add_strict_mono) ``` huffman@21164 ` 607` ```apply (simp_all add: diff_minus [symmetric]) ``` huffman@21164 ` 608` ```done ``` huffman@21164 ` 609` huffman@21164 ` 610` huffman@21164 ` 611` ```lemma lemma_BOLZANO2: "((\a b c. (a \ b & b \ c & P(a,b) & P(b,c)) --> P(a,c)) & ``` huffman@21164 ` 612` ``` (\x. \d::real. 0 < d & ``` huffman@21164 ` 613` ``` (\a b. a \ x & x \ b & (b-a) < d --> P(a,b)))) ``` huffman@21164 ` 614` ``` --> (\a b. a \ b --> P(a,b))" ``` huffman@21164 ` 615` ```apply clarify ``` huffman@21164 ` 616` ```apply (blast intro: lemma_BOLZANO) ``` huffman@21164 ` 617` ```done ``` huffman@21164 ` 618` huffman@21164 ` 619` huffman@21164 ` 620` ```subsection {* Intermediate Value Theorem *} ``` huffman@21164 ` 621` huffman@21164 ` 622` ```text {*Prove Contrapositive by Bisection*} ``` huffman@21164 ` 623` huffman@21164 ` 624` ```lemma IVT: "[| f(a::real) \ (y::real); y \ f(b); ``` huffman@21164 ` 625` ``` a \ b; ``` huffman@21164 ` 626` ``` (\x. a \ x & x \ b --> isCont f x) |] ``` huffman@21164 ` 627` ``` ==> \x. a \ x & x \ b & f(x) = y" ``` huffman@21164 ` 628` ```apply (rule contrapos_pp, assumption) ``` huffman@21164 ` 629` ```apply (cut_tac P = "% (u,v) . a \ u & u \ v & v \ b --> ~ (f (u) \ y & y \ f (v))" in lemma_BOLZANO2) ``` huffman@21164 ` 630` ```apply safe ``` huffman@21164 ` 631` ```apply simp_all ``` huffman@31338 ` 632` ```apply (simp add: isCont_iff LIM_eq) ``` huffman@21164 ` 633` ```apply (rule ccontr) ``` huffman@21164 ` 634` ```apply (subgoal_tac "a \ x & x \ b") ``` huffman@21164 ` 635` ``` prefer 2 ``` huffman@21164 ` 636` ``` apply simp ``` huffman@21164 ` 637` ``` apply (drule_tac P = "%d. 0 ?P d" and x = 1 in spec, arith) ``` huffman@21164 ` 638` ```apply (drule_tac x = x in spec)+ ``` huffman@21164 ` 639` ```apply simp ``` huffman@21164 ` 640` ```apply (drule_tac P = "%r. ?P r --> (\s>0. ?Q r s) " and x = "\y - f x\" in spec) ``` huffman@21164 ` 641` ```apply safe ``` huffman@21164 ` 642` ```apply simp ``` huffman@21164 ` 643` ```apply (drule_tac x = s in spec, clarify) ``` huffman@21164 ` 644` ```apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe) ``` huffman@21164 ` 645` ```apply (drule_tac x = "ba-x" in spec) ``` huffman@21164 ` 646` ```apply (simp_all add: abs_if) ``` huffman@21164 ` 647` ```apply (drule_tac x = "aa-x" in spec) ``` huffman@21164 ` 648` ```apply (case_tac "x \ aa", simp_all) ``` huffman@21164 ` 649` ```done ``` huffman@21164 ` 650` huffman@21164 ` 651` ```lemma IVT2: "[| f(b::real) \ (y::real); y \ f(a); ``` huffman@21164 ` 652` ``` a \ b; ``` huffman@21164 ` 653` ``` (\x. a \ x & x \ b --> isCont f x) ``` huffman@21164 ` 654` ``` |] ==> \x. a \ x & x \ b & f(x) = y" ``` huffman@21164 ` 655` ```apply (subgoal_tac "- f a \ -y & -y \ - f b", clarify) ``` huffman@21164 ` 656` ```apply (drule IVT [where f = "%x. - f x"], assumption) ``` huffman@44233 ` 657` ```apply simp_all ``` huffman@21164 ` 658` ```done ``` huffman@21164 ` 659` huffman@21164 ` 660` ```(*HOL style here: object-level formulations*) ``` huffman@21164 ` 661` ```lemma IVT_objl: "(f(a::real) \ (y::real) & y \ f(b) & a \ b & ``` huffman@21164 ` 662` ``` (\x. a \ x & x \ b --> isCont f x)) ``` huffman@21164 ` 663` ``` --> (\x. a \ x & x \ b & f(x) = y)" ``` huffman@21164 ` 664` ```apply (blast intro: IVT) ``` huffman@21164 ` 665` ```done ``` huffman@21164 ` 666` huffman@21164 ` 667` ```lemma IVT2_objl: "(f(b::real) \ (y::real) & y \ f(a) & a \ b & ``` huffman@21164 ` 668` ``` (\x. a \ x & x \ b --> isCont f x)) ``` huffman@21164 ` 669` ``` --> (\x. a \ x & x \ b & f(x) = y)" ``` huffman@21164 ` 670` ```apply (blast intro: IVT2) ``` huffman@21164 ` 671` ```done ``` huffman@21164 ` 672` huffman@29975 ` 673` huffman@29975 ` 674` ```subsection {* Boundedness of continuous functions *} ``` huffman@29975 ` 675` huffman@21164 ` 676` ```text{*By bisection, function continuous on closed interval is bounded above*} ``` huffman@21164 ` 677` huffman@21164 ` 678` ```lemma isCont_bounded: ``` huffman@21164 ` 679` ``` "[| a \ b; \x. a \ x & x \ b --> isCont f x |] ``` huffman@21164 ` 680` ``` ==> \M::real. \x::real. a \ x & x \ b --> f(x) \ M" ``` huffman@21164 ` 681` ```apply (cut_tac P = "% (u,v) . a \ u & u \ v & v \ b --> (\M. \x. u \ x & x \ v --> f x \ M)" in lemma_BOLZANO2) ``` huffman@21164 ` 682` ```apply safe ``` huffman@21164 ` 683` ```apply simp_all ``` huffman@21164 ` 684` ```apply (rename_tac x xa ya M Ma) ``` huffman@36777 ` 685` ```apply (metis linorder_not_less order_le_less order_trans) ``` huffman@21164 ` 686` ```apply (case_tac "a \ x & x \ b") ``` paulson@33654 ` 687` ``` prefer 2 ``` paulson@33654 ` 688` ``` apply (rule_tac x = 1 in exI, force) ``` huffman@31338 ` 689` ```apply (simp add: LIM_eq isCont_iff) ``` huffman@21164 ` 690` ```apply (drule_tac x = x in spec, auto) ``` huffman@21164 ` 691` ```apply (erule_tac V = "\M. \x. a \ x & x \ b & ~ f x \ M" in thin_rl) ``` huffman@21164 ` 692` ```apply (drule_tac x = 1 in spec, auto) ``` huffman@21164 ` 693` ```apply (rule_tac x = s in exI, clarify) ``` huffman@21164 ` 694` ```apply (rule_tac x = "\f x\ + 1" in exI, clarify) ``` huffman@21164 ` 695` ```apply (drule_tac x = "xa-x" in spec) ``` huffman@21164 ` 696` ```apply (auto simp add: abs_ge_self) ``` huffman@21164 ` 697` ```done ``` huffman@21164 ` 698` huffman@21164 ` 699` ```text{*Refine the above to existence of least upper bound*} ``` huffman@21164 ` 700` huffman@21164 ` 701` ```lemma lemma_reals_complete: "((\x. x \ S) & (\y. isUb UNIV S (y::real))) --> ``` huffman@21164 ` 702` ``` (\t. isLub UNIV S t)" ``` huffman@21164 ` 703` ```by (blast intro: reals_complete) ``` huffman@21164 ` 704` huffman@21164 ` 705` ```lemma isCont_has_Ub: "[| a \ b; \x. a \ x & x \ b --> isCont f x |] ``` huffman@21164 ` 706` ``` ==> \M::real. (\x::real. a \ x & x \ b --> f(x) \ M) & ``` huffman@21164 ` 707` ``` (\N. N < M --> (\x. a \ x & x \ b & N < f(x)))" ``` huffman@21164 ` 708` ```apply (cut_tac S = "Collect (%y. \x. a \ x & x \ b & y = f x)" ``` huffman@21164 ` 709` ``` in lemma_reals_complete) ``` huffman@21164 ` 710` ```apply auto ``` huffman@21164 ` 711` ```apply (drule isCont_bounded, assumption) ``` huffman@21164 ` 712` ```apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def) ``` huffman@21164 ` 713` ```apply (rule exI, auto) ``` huffman@21164 ` 714` ```apply (auto dest!: spec simp add: linorder_not_less) ``` huffman@21164 ` 715` ```done ``` huffman@21164 ` 716` huffman@21164 ` 717` ```text{*Now show that it attains its upper bound*} ``` huffman@21164 ` 718` huffman@21164 ` 719` ```lemma isCont_eq_Ub: ``` huffman@21164 ` 720` ``` assumes le: "a \ b" ``` huffman@21164 ` 721` ``` and con: "\x::real. a \ x & x \ b --> isCont f x" ``` huffman@21164 ` 722` ``` shows "\M::real. (\x. a \ x & x \ b --> f(x) \ M) & ``` huffman@21164 ` 723` ``` (\x. a \ x & x \ b & f(x) = M)" ``` huffman@21164 ` 724` ```proof - ``` huffman@21164 ` 725` ``` from isCont_has_Ub [OF le con] ``` huffman@21164 ` 726` ``` obtain M where M1: "\x. a \ x \ x \ b \ f x \ M" ``` huffman@21164 ` 727` ``` and M2: "!!N. N \x. a \ x \ x \ b \ N < f x" by blast ``` huffman@21164 ` 728` ``` show ?thesis ``` huffman@21164 ` 729` ``` proof (intro exI, intro conjI) ``` huffman@21164 ` 730` ``` show " \x. a \ x \ x \ b \ f x \ M" by (rule M1) ``` huffman@21164 ` 731` ``` show "\x. a \ x \ x \ b \ f x = M" ``` huffman@21164 ` 732` ``` proof (rule ccontr) ``` huffman@21164 ` 733` ``` assume "\ (\x. a \ x \ x \ b \ f x = M)" ``` huffman@21164 ` 734` ``` with M1 have M3: "\x. a \ x & x \ b --> f x < M" ``` nipkow@44890 ` 735` ``` by (fastforce simp add: linorder_not_le [symmetric]) ``` huffman@21164 ` 736` ``` hence "\x. a \ x & x \ b --> isCont (%x. inverse (M - f x)) x" ``` huffman@44233 ` 737` ``` by (auto simp add: con) ``` huffman@21164 ` 738` ``` from isCont_bounded [OF le this] ``` huffman@21164 ` 739` ``` obtain k where k: "!!x. a \ x & x \ b --> inverse (M - f x) \ k" by auto ``` huffman@21164 ` 740` ``` have Minv: "!!x. a \ x & x \ b --> 0 < inverse (M - f (x))" ``` nipkow@29667 ` 741` ``` by (simp add: M3 algebra_simps) ``` huffman@21164 ` 742` ``` have "!!x. a \ x & x \ b --> inverse (M - f x) < k+1" using k ``` huffman@21164 ` 743` ``` by (auto intro: order_le_less_trans [of _ k]) ``` huffman@21164 ` 744` ``` with Minv ``` huffman@21164 ` 745` ``` have "!!x. a \ x & x \ b --> inverse(k+1) < inverse(inverse(M - f x))" ``` huffman@21164 ` 746` ``` by (intro strip less_imp_inverse_less, simp_all) ``` huffman@21164 ` 747` ``` hence invlt: "!!x. a \ x & x \ b --> inverse(k+1) < M - f x" ``` huffman@21164 ` 748` ``` by simp ``` huffman@21164 ` 749` ``` have "M - inverse (k+1) < M" using k [of a] Minv [of a] le ``` huffman@21164 ` 750` ``` by (simp, arith) ``` huffman@21164 ` 751` ``` from M2 [OF this] ``` huffman@21164 ` 752` ``` obtain x where ax: "a \ x & x \ b & M - inverse(k+1) < f x" .. ``` huffman@21164 ` 753` ``` thus False using invlt [of x] by force ``` huffman@21164 ` 754` ``` qed ``` huffman@21164 ` 755` ``` qed ``` huffman@21164 ` 756` ```qed ``` huffman@21164 ` 757` huffman@21164 ` 758` huffman@21164 ` 759` ```text{*Same theorem for lower bound*} ``` huffman@21164 ` 760` huffman@21164 ` 761` ```lemma isCont_eq_Lb: "[| a \ b; \x. a \ x & x \ b --> isCont f x |] ``` huffman@21164 ` 762` ``` ==> \M::real. (\x::real. a \ x & x \ b --> M \ f(x)) & ``` huffman@21164 ` 763` ``` (\x. a \ x & x \ b & f(x) = M)" ``` huffman@21164 ` 764` ```apply (subgoal_tac "\x. a \ x & x \ b --> isCont (%x. - (f x)) x") ``` huffman@21164 ` 765` ```prefer 2 apply (blast intro: isCont_minus) ``` huffman@21164 ` 766` ```apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub) ``` huffman@21164 ` 767` ```apply safe ``` huffman@21164 ` 768` ```apply auto ``` huffman@21164 ` 769` ```done ``` huffman@21164 ` 770` huffman@21164 ` 771` huffman@21164 ` 772` ```text{*Another version.*} ``` huffman@21164 ` 773` huffman@21164 ` 774` ```lemma isCont_Lb_Ub: "[|a \ b; \x. a \ x & x \ b --> isCont f x |] ``` huffman@21164 ` 775` ``` ==> \L M::real. (\x::real. a \ x & x \ b --> L \ f(x) & f(x) \ M) & ``` huffman@21164 ` 776` ``` (\y. L \ y & y \ M --> (\x. a \ x & x \ b & (f(x) = y)))" ``` huffman@21164 ` 777` ```apply (frule isCont_eq_Lb) ``` huffman@21164 ` 778` ```apply (frule_tac [2] isCont_eq_Ub) ``` huffman@21164 ` 779` ```apply (assumption+, safe) ``` huffman@21164 ` 780` ```apply (rule_tac x = "f x" in exI) ``` huffman@21164 ` 781` ```apply (rule_tac x = "f xa" in exI, simp, safe) ``` huffman@21164 ` 782` ```apply (cut_tac x = x and y = xa in linorder_linear, safe) ``` huffman@21164 ` 783` ```apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl) ``` huffman@21164 ` 784` ```apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe) ``` huffman@21164 ` 785` ```apply (rule_tac [2] x = xb in exI) ``` huffman@21164 ` 786` ```apply (rule_tac [4] x = xb in exI, simp_all) ``` huffman@21164 ` 787` ```done ``` huffman@21164 ` 788` huffman@21164 ` 789` huffman@29975 ` 790` ```subsection {* Local extrema *} ``` huffman@29975 ` 791` huffman@21164 ` 792` ```text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*} ``` huffman@21164 ` 793` paulson@33654 ` 794` ```lemma DERIV_pos_inc_right: ``` huffman@21164 ` 795` ``` fixes f :: "real => real" ``` huffman@21164 ` 796` ``` assumes der: "DERIV f x :> l" ``` huffman@21164 ` 797` ``` and l: "0 < l" ``` huffman@21164 ` 798` ``` shows "\d > 0. \h > 0. h < d --> f(x) < f(x + h)" ``` huffman@21164 ` 799` ```proof - ``` huffman@21164 ` 800` ``` from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]] ``` huffman@21164 ` 801` ``` have "\s > 0. (\z. z \ 0 \ \z\ < s \ \(f(x+z) - f x) / z - l\ < l)" ``` huffman@21164 ` 802` ``` by (simp add: diff_minus) ``` huffman@21164 ` 803` ``` then obtain s ``` huffman@21164 ` 804` ``` where s: "0 < s" ``` huffman@21164 ` 805` ``` and all: "!!z. z \ 0 \ \z\ < s \ \(f(x+z) - f x) / z - l\ < l" ``` huffman@21164 ` 806` ``` by auto ``` huffman@21164 ` 807` ``` thus ?thesis ``` huffman@21164 ` 808` ``` proof (intro exI conjI strip) ``` huffman@23441 ` 809` ``` show "0 real" ``` huffman@21164 ` 826` ``` assumes der: "DERIV f x :> l" ``` huffman@21164 ` 827` ``` and l: "l < 0" ``` huffman@21164 ` 828` ``` shows "\d > 0. \h > 0. h < d --> f(x) < f(x-h)" ``` huffman@21164 ` 829` ```proof - ``` huffman@21164 ` 830` ``` from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]] ``` huffman@21164 ` 831` ``` have "\s > 0. (\z. z \ 0 \ \z\ < s \ \(f(x+z) - f x) / z - l\ < -l)" ``` huffman@21164 ` 832` ``` by (simp add: diff_minus) ``` huffman@21164 ` 833` ``` then obtain s ``` huffman@21164 ` 834` ``` where s: "0 < s" ``` huffman@21164 ` 835` ``` and all: "!!z. z \ 0 \ \z\ < s \ \(f(x+z) - f x) / z - l\ < -l" ``` huffman@21164 ` 836` ``` by auto ``` huffman@21164 ` 837` ``` thus ?thesis ``` huffman@21164 ` 838` ``` proof (intro exI conjI strip) ``` huffman@23441 ` 839` ``` show "0 real" ``` paulson@33654 ` 857` ``` shows "DERIV f x :> l \ 0 < l \ \d > 0. \h > 0. h < d --> f(x - h) < f(x)" ``` paulson@33654 ` 858` ``` apply (rule DERIV_neg_dec_left [of "%x. - f x" x "-l", simplified]) ``` hoelzl@41368 ` 859` ``` apply (auto simp add: DERIV_minus) ``` paulson@33654 ` 860` ``` done ``` paulson@33654 ` 861` paulson@33654 ` 862` ```lemma DERIV_neg_dec_right: ``` paulson@33654 ` 863` ``` fixes f :: "real => real" ``` paulson@33654 ` 864` ``` shows "DERIV f x :> l \ l < 0 \ \d > 0. \h > 0. h < d --> f(x) > f(x + h)" ``` paulson@33654 ` 865` ``` apply (rule DERIV_pos_inc_right [of "%x. - f x" x "-l", simplified]) ``` hoelzl@41368 ` 866` ``` apply (auto simp add: DERIV_minus) ``` paulson@33654 ` 867` ``` done ``` paulson@33654 ` 868` huffman@21164 ` 869` ```lemma DERIV_local_max: ``` huffman@21164 ` 870` ``` fixes f :: "real => real" ``` huffman@21164 ` 871` ``` assumes der: "DERIV f x :> l" ``` huffman@21164 ` 872` ``` and d: "0 < d" ``` huffman@21164 ` 873` ``` and le: "\y. \x-y\ < d --> f(y) \ f(x)" ``` huffman@21164 ` 874` ``` shows "l = 0" ``` huffman@21164 ` 875` ```proof (cases rule: linorder_cases [of l 0]) ``` huffman@23441 ` 876` ``` case equal thus ?thesis . ``` huffman@21164 ` 877` ```next ``` huffman@21164 ` 878` ``` case less ``` paulson@33654 ` 879` ``` from DERIV_neg_dec_left [OF der less] ``` huffman@21164 ` 880` ``` obtain d' where d': "0 < d'" ``` huffman@21164 ` 881` ``` and lt: "\h > 0. h < d' \ f x < f (x-h)" by blast ``` huffman@21164 ` 882` ``` from real_lbound_gt_zero [OF d d'] ``` huffman@21164 ` 883` ``` obtain e where "0 < e \ e < d \ e < d'" .. ``` huffman@21164 ` 884` ``` with lt le [THEN spec [where x="x-e"]] ``` huffman@21164 ` 885` ``` show ?thesis by (auto simp add: abs_if) ``` huffman@21164 ` 886` ```next ``` huffman@21164 ` 887` ``` case greater ``` paulson@33654 ` 888` ``` from DERIV_pos_inc_right [OF der greater] ``` huffman@21164 ` 889` ``` obtain d' where d': "0 < d'" ``` huffman@21164 ` 890` ``` and lt: "\h > 0. h < d' \ f x < f (x + h)" by blast ``` huffman@21164 ` 891` ``` from real_lbound_gt_zero [OF d d'] ``` huffman@21164 ` 892` ``` obtain e where "0 < e \ e < d \ e < d'" .. ``` huffman@21164 ` 893` ``` with lt le [THEN spec [where x="x+e"]] ``` huffman@21164 ` 894` ``` show ?thesis by (auto simp add: abs_if) ``` huffman@21164 ` 895` ```qed ``` huffman@21164 ` 896` huffman@21164 ` 897` huffman@21164 ` 898` ```text{*Similar theorem for a local minimum*} ``` huffman@21164 ` 899` ```lemma DERIV_local_min: ``` huffman@21164 ` 900` ``` fixes f :: "real => real" ``` huffman@21164 ` 901` ``` shows "[| DERIV f x :> l; 0 < d; \y. \x-y\ < d --> f(x) \ f(y) |] ==> l = 0" ``` huffman@21164 ` 902` ```by (drule DERIV_minus [THEN DERIV_local_max], auto) ``` huffman@21164 ` 903` huffman@21164 ` 904` huffman@21164 ` 905` ```text{*In particular, if a function is locally flat*} ``` huffman@21164 ` 906` ```lemma DERIV_local_const: ``` huffman@21164 ` 907` ``` fixes f :: "real => real" ``` huffman@21164 ` 908` ``` shows "[| DERIV f x :> l; 0 < d; \y. \x-y\ < d --> f(x) = f(y) |] ==> l = 0" ``` huffman@21164 ` 909` ```by (auto dest!: DERIV_local_max) ``` huffman@21164 ` 910` huffman@29975 ` 911` huffman@29975 ` 912` ```subsection {* Rolle's Theorem *} ``` huffman@29975 ` 913` huffman@21164 ` 914` ```text{*Lemma about introducing open ball in open interval*} ``` huffman@21164 ` 915` ```lemma lemma_interval_lt: ``` huffman@21164 ` 916` ``` "[| a < x; x < b |] ``` huffman@21164 ` 917` ``` ==> \d::real. 0 < d & (\y. \x-y\ < d --> a < y & y < b)" ``` chaieb@27668 ` 918` huffman@22998 ` 919` ```apply (simp add: abs_less_iff) ``` huffman@21164 ` 920` ```apply (insert linorder_linear [of "x-a" "b-x"], safe) ``` huffman@21164 ` 921` ```apply (rule_tac x = "x-a" in exI) ``` huffman@21164 ` 922` ```apply (rule_tac [2] x = "b-x" in exI, auto) ``` huffman@21164 ` 923` ```done ``` huffman@21164 ` 924` huffman@21164 ` 925` ```lemma lemma_interval: "[| a < x; x < b |] ==> ``` huffman@21164 ` 926` ``` \d::real. 0 < d & (\y. \x-y\ < d --> a \ y & y \ b)" ``` huffman@21164 ` 927` ```apply (drule lemma_interval_lt, auto) ``` huffman@44921 ` 928` ```apply force ``` huffman@21164 ` 929` ```done ``` huffman@21164 ` 930` huffman@21164 ` 931` ```text{*Rolle's Theorem. ``` huffman@21164 ` 932` ``` If @{term f} is defined and continuous on the closed interval ``` huffman@21164 ` 933` ``` @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"}, ``` huffman@21164 ` 934` ``` and @{term "f(a) = f(b)"}, ``` huffman@21164 ` 935` ``` then there exists @{text "x0 \ (a,b)"} such that @{term "f'(x0) = 0"}*} ``` huffman@21164 ` 936` ```theorem Rolle: ``` huffman@21164 ` 937` ``` assumes lt: "a < b" ``` huffman@21164 ` 938` ``` and eq: "f(a) = f(b)" ``` huffman@21164 ` 939` ``` and con: "\x. a \ x & x \ b --> isCont f x" ``` huffman@21164 ` 940` ``` and dif [rule_format]: "\x. a < x & x < b --> f differentiable x" ``` huffman@21784 ` 941` ``` shows "\z::real. a < z & z < b & DERIV f z :> 0" ``` huffman@21164 ` 942` ```proof - ``` huffman@21164 ` 943` ``` have le: "a \ b" using lt by simp ``` huffman@21164 ` 944` ``` from isCont_eq_Ub [OF le con] ``` huffman@21164 ` 945` ``` obtain x where x_max: "\z. a \ z \ z \ b \ f z \ f x" ``` huffman@21164 ` 946` ``` and alex: "a \ x" and xleb: "x \ b" ``` huffman@21164 ` 947` ``` by blast ``` huffman@21164 ` 948` ``` from isCont_eq_Lb [OF le con] ``` huffman@21164 ` 949` ``` obtain x' where x'_min: "\z. a \ z \ z \ b \ f x' \ f z" ``` huffman@21164 ` 950` ``` and alex': "a \ x'" and x'leb: "x' \ b" ``` huffman@21164 ` 951` ``` by blast ``` huffman@21164 ` 952` ``` show ?thesis ``` huffman@21164 ` 953` ``` proof cases ``` huffman@21164 ` 954` ``` assume axb: "a < x & x < b" ``` huffman@21164 ` 955` ``` --{*@{term f} attains its maximum within the interval*} ``` chaieb@27668 ` 956` ``` hence ax: "ay. \x-y\ < d \ a \ y \ y \ b" ``` huffman@21164 ` 959` ``` by blast ``` huffman@21164 ` 960` ``` hence bound': "\y. \x-y\ < d \ f y \ f x" using x_max ``` huffman@21164 ` 961` ``` by blast ``` huffman@21164 ` 962` ``` from differentiableD [OF dif [OF axb]] ``` huffman@21164 ` 963` ``` obtain l where der: "DERIV f x :> l" .. ``` huffman@21164 ` 964` ``` have "l=0" by (rule DERIV_local_max [OF der d bound']) ``` huffman@21164 ` 965` ``` --{*the derivative at a local maximum is zero*} ``` huffman@21164 ` 966` ``` thus ?thesis using ax xb der by auto ``` huffman@21164 ` 967` ``` next ``` huffman@21164 ` 968` ``` assume notaxb: "~ (a < x & x < b)" ``` huffman@21164 ` 969` ``` hence xeqab: "x=a | x=b" using alex xleb by arith ``` huffman@21164 ` 970` ``` hence fb_eq_fx: "f b = f x" by (auto simp add: eq) ``` huffman@21164 ` 971` ``` show ?thesis ``` huffman@21164 ` 972` ``` proof cases ``` huffman@21164 ` 973` ``` assume ax'b: "a < x' & x' < b" ``` huffman@21164 ` 974` ``` --{*@{term f} attains its minimum within the interval*} ``` chaieb@27668 ` 975` ``` hence ax': "ay. \x'-y\ < d \ a \ y \ y \ b" ``` huffman@21164 ` 978` ``` by blast ``` huffman@21164 ` 979` ``` hence bound': "\y. \x'-y\ < d \ f x' \ f y" using x'_min ``` huffman@21164 ` 980` ``` by blast ``` huffman@21164 ` 981` ``` from differentiableD [OF dif [OF ax'b]] ``` huffman@21164 ` 982` ``` obtain l where der: "DERIV f x' :> l" .. ``` huffman@21164 ` 983` ``` have "l=0" by (rule DERIV_local_min [OF der d bound']) ``` huffman@21164 ` 984` ``` --{*the derivative at a local minimum is zero*} ``` huffman@21164 ` 985` ``` thus ?thesis using ax' x'b der by auto ``` huffman@21164 ` 986` ``` next ``` huffman@21164 ` 987` ``` assume notax'b: "~ (a < x' & x' < b)" ``` huffman@21164 ` 988` ``` --{*@{term f} is constant througout the interval*} ``` huffman@21164 ` 989` ``` hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith ``` huffman@21164 ` 990` ``` hence fb_eq_fx': "f b = f x'" by (auto simp add: eq) ``` huffman@21164 ` 991` ``` from dense [OF lt] ``` huffman@21164 ` 992` ``` obtain r where ar: "a < r" and rb: "r < b" by blast ``` huffman@21164 ` 993` ``` from lemma_interval [OF ar rb] ``` huffman@21164 ` 994` ``` obtain d where d: "0y. \r-y\ < d \ a \ y \ y \ b" ``` huffman@21164 ` 995` ``` by blast ``` huffman@21164 ` 996` ``` have eq_fb: "\z. a \ z --> z \ b --> f z = f b" ``` huffman@21164 ` 997` ``` proof (clarify) ``` huffman@21164 ` 998` ``` fix z::real ``` huffman@21164 ` 999` ``` assume az: "a \ z" and zb: "z \ b" ``` huffman@21164 ` 1000` ``` show "f z = f b" ``` huffman@21164 ` 1001` ``` proof (rule order_antisym) ``` huffman@21164 ` 1002` ``` show "f z \ f b" by (simp add: fb_eq_fx x_max az zb) ``` huffman@21164 ` 1003` ``` show "f b \ f z" by (simp add: fb_eq_fx' x'_min az zb) ``` huffman@21164 ` 1004` ``` qed ``` huffman@21164 ` 1005` ``` qed ``` huffman@21164 ` 1006` ``` have bound': "\y. \r-y\ < d \ f r = f y" ``` huffman@21164 ` 1007` ``` proof (intro strip) ``` huffman@21164 ` 1008` ``` fix y::real ``` huffman@21164 ` 1009` ``` assume lt: "\r-y\ < d" ``` huffman@21164 ` 1010` ``` hence "f y = f b" by (simp add: eq_fb bound) ``` huffman@21164 ` 1011` ``` thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le) ``` huffman@21164 ` 1012` ``` qed ``` huffman@21164 ` 1013` ``` from differentiableD [OF dif [OF conjI [OF ar rb]]] ``` huffman@21164 ` 1014` ``` obtain l where der: "DERIV f r :> l" .. ``` huffman@21164 ` 1015` ``` have "l=0" by (rule DERIV_local_const [OF der d bound']) ``` huffman@21164 ` 1016` ``` --{*the derivative of a constant function is zero*} ``` huffman@21164 ` 1017` ``` thus ?thesis using ar rb der by auto ``` huffman@21164 ` 1018` ``` qed ``` huffman@21164 ` 1019` ``` qed ``` huffman@21164 ` 1020` ```qed ``` huffman@21164 ` 1021` huffman@21164 ` 1022` huffman@21164 ` 1023` ```subsection{*Mean Value Theorem*} ``` huffman@21164 ` 1024` huffman@21164 ` 1025` ```lemma lemma_MVT: ``` huffman@21164 ` 1026` ``` "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)" ``` huffman@21164 ` 1027` ```proof cases ``` huffman@21164 ` 1028` ``` assume "a=b" thus ?thesis by simp ``` huffman@21164 ` 1029` ```next ``` huffman@21164 ` 1030` ``` assume "a\b" ``` huffman@21164 ` 1031` ``` hence ba: "b-a \ 0" by arith ``` huffman@21164 ` 1032` ``` show ?thesis ``` huffman@21164 ` 1033` ``` by (rule real_mult_left_cancel [OF ba, THEN iffD1], ``` huffman@21164 ` 1034` ``` simp add: right_diff_distrib, ``` huffman@21164 ` 1035` ``` simp add: left_diff_distrib) ``` huffman@21164 ` 1036` ```qed ``` huffman@21164 ` 1037` huffman@21164 ` 1038` ```theorem MVT: ``` huffman@21164 ` 1039` ``` assumes lt: "a < b" ``` huffman@21164 ` 1040` ``` and con: "\x. a \ x & x \ b --> isCont f x" ``` huffman@21164 ` 1041` ``` and dif [rule_format]: "\x. a < x & x < b --> f differentiable x" ``` huffman@21784 ` 1042` ``` shows "\l z::real. a < z & z < b & DERIV f z :> l & ``` huffman@21164 ` 1043` ``` (f(b) - f(a) = (b-a) * l)" ``` huffman@21164 ` 1044` ```proof - ``` huffman@21164 ` 1045` ``` let ?F = "%x. f x - ((f b - f a) / (b-a)) * x" ``` huffman@44233 ` 1046` ``` have contF: "\x. a \ x \ x \ b \ isCont ?F x" ``` huffman@44233 ` 1047` ``` using con by (fast intro: isCont_intros) ``` huffman@21164 ` 1048` ``` have difF: "\x. a < x \ x < b \ ?F differentiable x" ``` huffman@21164 ` 1049` ``` proof (clarify) ``` huffman@21164 ` 1050` ``` fix x::real ``` huffman@21164 ` 1051` ``` assume ax: "a < x" and xb: "x < b" ``` huffman@21164 ` 1052` ``` from differentiableD [OF dif [OF conjI [OF ax xb]]] ``` huffman@21164 ` 1053` ``` obtain l where der: "DERIV f x :> l" .. ``` huffman@21164 ` 1054` ``` show "?F differentiable x" ``` huffman@21164 ` 1055` ``` by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"], ``` huffman@21164 ` 1056` ``` blast intro: DERIV_diff DERIV_cmult_Id der) ``` huffman@21164 ` 1057` ``` qed ``` huffman@21164 ` 1058` ``` from Rolle [where f = ?F, OF lt lemma_MVT contF difF] ``` huffman@21164 ` 1059` ``` obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0" ``` huffman@21164 ` 1060` ``` by blast ``` huffman@21164 ` 1061` ``` have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)" ``` huffman@21164 ` 1062` ``` by (rule DERIV_cmult_Id) ``` huffman@21164 ` 1063` ``` hence derF: "DERIV (\x. ?F x + (f b - f a) / (b - a) * x) z ``` huffman@21164 ` 1064` ``` :> 0 + (f b - f a) / (b - a)" ``` huffman@21164 ` 1065` ``` by (rule DERIV_add [OF der]) ``` huffman@21164 ` 1066` ``` show ?thesis ``` huffman@21164 ` 1067` ``` proof (intro exI conjI) ``` huffman@23441 ` 1068` ``` show "a < z" using az . ``` huffman@23441 ` 1069` ``` show "z < b" using zb . ``` huffman@21164 ` 1070` ``` show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp) ``` huffman@21164 ` 1071` ``` show "DERIV f z :> ((f b - f a)/(b-a))" using derF by simp ``` huffman@21164 ` 1072` ``` qed ``` huffman@21164 ` 1073` ```qed ``` huffman@21164 ` 1074` hoelzl@29803 ` 1075` ```lemma MVT2: ``` hoelzl@29803 ` 1076` ``` "[| a < b; \x. a \ x & x \ b --> DERIV f x :> f'(x) |] ``` hoelzl@29803 ` 1077` ``` ==> \z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))" ``` hoelzl@29803 ` 1078` ```apply (drule MVT) ``` hoelzl@29803 ` 1079` ```apply (blast intro: DERIV_isCont) ``` hoelzl@29803 ` 1080` ```apply (force dest: order_less_imp_le simp add: differentiable_def) ``` hoelzl@29803 ` 1081` ```apply (blast dest: DERIV_unique order_less_imp_le) ``` hoelzl@29803 ` 1082` ```done ``` hoelzl@29803 ` 1083` huffman@21164 ` 1084` huffman@21164 ` 1085` ```text{*A function is constant if its derivative is 0 over an interval.*} ``` huffman@21164 ` 1086` huffman@21164 ` 1087` ```lemma DERIV_isconst_end: ``` huffman@21164 ` 1088` ``` fixes f :: "real => real" ``` huffman@21164 ` 1089` ``` shows "[| a < b; ``` huffman@21164 ` 1090` ``` \x. a \ x & x \ b --> isCont f x; ``` huffman@21164 ` 1091` ``` \x. a < x & x < b --> DERIV f x :> 0 |] ``` huffman@21164 ` 1092` ``` ==> f b = f a" ``` huffman@21164 ` 1093` ```apply (drule MVT, assumption) ``` huffman@21164 ` 1094` ```apply (blast intro: differentiableI) ``` huffman@21164 ` 1095` ```apply (auto dest!: DERIV_unique simp add: diff_eq_eq) ``` huffman@21164 ` 1096` ```done ``` huffman@21164 ` 1097` huffman@21164 ` 1098` ```lemma DERIV_isconst1: ``` huffman@21164 ` 1099` ``` fixes f :: "real => real" ``` huffman@21164 ` 1100` ``` shows "[| a < b; ``` huffman@21164 ` 1101` ``` \x. a \ x & x \ b --> isCont f x; ``` huffman@21164 ` 1102` ``` \x. a < x & x < b --> DERIV f x :> 0 |] ``` huffman@21164 ` 1103` ``` ==> \x. a \ x & x \ b --> f x = f a" ``` huffman@21164 ` 1104` ```apply safe ``` huffman@21164 ` 1105` ```apply (drule_tac x = a in order_le_imp_less_or_eq, safe) ``` huffman@21164 ` 1106` ```apply (drule_tac b = x in DERIV_isconst_end, auto) ``` huffman@21164 ` 1107` ```done ``` huffman@21164 ` 1108` huffman@21164 ` 1109` ```lemma DERIV_isconst2: ``` huffman@21164 ` 1110` ``` fixes f :: "real => real" ``` huffman@21164 ` 1111` ``` shows "[| a < b; ``` huffman@21164 ` 1112` ``` \x. a \ x & x \ b --> isCont f x; ``` huffman@21164 ` 1113` ``` \x. a < x & x < b --> DERIV f x :> 0; ``` huffman@21164 ` 1114` ``` a \ x; x \ b |] ``` huffman@21164 ` 1115` ``` ==> f x = f a" ``` huffman@21164 ` 1116` ```apply (blast dest: DERIV_isconst1) ``` huffman@21164 ` 1117` ```done ``` huffman@21164 ` 1118` hoelzl@29803 ` 1119` ```lemma DERIV_isconst3: fixes a b x y :: real ``` hoelzl@29803 ` 1120` ``` assumes "a < b" and "x \ {a <..< b}" and "y \ {a <..< b}" ``` hoelzl@29803 ` 1121` ``` assumes derivable: "\x. x \ {a <..< b} \ DERIV f x :> 0" ``` hoelzl@29803 ` 1122` ``` shows "f x = f y" ``` hoelzl@29803 ` 1123` ```proof (cases "x = y") ``` hoelzl@29803 ` 1124` ``` case False ``` hoelzl@29803 ` 1125` ``` let ?a = "min x y" ``` hoelzl@29803 ` 1126` ``` let ?b = "max x y" ``` hoelzl@29803 ` 1127` ``` ``` hoelzl@29803 ` 1128` ``` have "\z. ?a \ z \ z \ ?b \ DERIV f z :> 0" ``` hoelzl@29803 ` 1129` ``` proof (rule allI, rule impI) ``` hoelzl@29803 ` 1130` ``` fix z :: real assume "?a \ z \ z \ ?b" ``` hoelzl@29803 ` 1131` ``` hence "a < z" and "z < b" using `x \ {a <..< b}` and `y \ {a <..< b}` by auto ``` hoelzl@29803 ` 1132` ``` hence "z \ {a<.. 0" by (rule derivable) ``` hoelzl@29803 ` 1134` ``` qed ``` hoelzl@29803 ` 1135` ``` hence isCont: "\z. ?a \ z \ z \ ?b \ isCont f z" ``` hoelzl@29803 ` 1136` ``` and DERIV: "\z. ?a < z \ z < ?b \ DERIV f z :> 0" using DERIV_isCont by auto ``` hoelzl@29803 ` 1137` hoelzl@29803 ` 1138` ``` have "?a < ?b" using `x \ y` by auto ``` hoelzl@29803 ` 1139` ``` from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y] ``` hoelzl@29803 ` 1140` ``` show ?thesis by auto ``` hoelzl@29803 ` 1141` ```qed auto ``` hoelzl@29803 ` 1142` huffman@21164 ` 1143` ```lemma DERIV_isconst_all: ``` huffman@21164 ` 1144` ``` fixes f :: "real => real" ``` huffman@21164 ` 1145` ``` shows "\x. DERIV f x :> 0 ==> f(x) = f(y)" ``` huffman@21164 ` 1146` ```apply (rule linorder_cases [of x y]) ``` huffman@21164 ` 1147` ```apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+ ``` huffman@21164 ` 1148` ```done ``` huffman@21164 ` 1149` huffman@21164 ` 1150` ```lemma DERIV_const_ratio_const: ``` huffman@21784 ` 1151` ``` fixes f :: "real => real" ``` huffman@21784 ` 1152` ``` shows "[|a \ b; \x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k" ``` huffman@21164 ` 1153` ```apply (rule linorder_cases [of a b], auto) ``` huffman@21164 ` 1154` ```apply (drule_tac [!] f = f in MVT) ``` huffman@21164 ` 1155` ```apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def) ``` nipkow@23477 ` 1156` ```apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus) ``` huffman@21164 ` 1157` ```done ``` huffman@21164 ` 1158` huffman@21164 ` 1159` ```lemma DERIV_const_ratio_const2: ``` huffman@21784 ` 1160` ``` fixes f :: "real => real" ``` huffman@21784 ` 1161` ``` shows "[|a \ b; \x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k" ``` huffman@21164 ` 1162` ```apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1]) ``` huffman@21164 ` 1163` ```apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc) ``` huffman@21164 ` 1164` ```done ``` huffman@21164 ` 1165` huffman@21164 ` 1166` ```lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)" ``` huffman@21164 ` 1167` ```by (simp) ``` huffman@21164 ` 1168` huffman@21164 ` 1169` ```lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)" ``` huffman@21164 ` 1170` ```by (simp) ``` huffman@21164 ` 1171` huffman@21164 ` 1172` ```text{*Gallileo's "trick": average velocity = av. of end velocities*} ``` huffman@21164 ` 1173` huffman@21164 ` 1174` ```lemma DERIV_const_average: ``` huffman@21164 ` 1175` ``` fixes v :: "real => real" ``` huffman@21164 ` 1176` ``` assumes neq: "a \ (b::real)" ``` huffman@21164 ` 1177` ``` and der: "\x. DERIV v x :> k" ``` huffman@21164 ` 1178` ``` shows "v ((a + b)/2) = (v a + v b)/2" ``` huffman@21164 ` 1179` ```proof (cases rule: linorder_cases [of a b]) ``` huffman@21164 ` 1180` ``` case equal with neq show ?thesis by simp ``` huffman@21164 ` 1181` ```next ``` huffman@21164 ` 1182` ``` case less ``` huffman@21164 ` 1183` ``` have "(v b - v a) / (b - a) = k" ``` huffman@21164 ` 1184` ``` by (rule DERIV_const_ratio_const2 [OF neq der]) ``` huffman@21164 ` 1185` ``` hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp ``` huffman@21164 ` 1186` ``` moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k" ``` huffman@21164 ` 1187` ``` by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq) ``` huffman@21164 ` 1188` ``` ultimately show ?thesis using neq by force ``` huffman@21164 ` 1189` ```next ``` huffman@21164 ` 1190` ``` case greater ``` huffman@21164 ` 1191` ``` have "(v b - v a) / (b - a) = k" ``` huffman@21164 ` 1192` ``` by (rule DERIV_const_ratio_const2 [OF neq der]) ``` huffman@21164 ` 1193` ``` hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp ``` huffman@21164 ` 1194` ``` moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k" ``` huffman@21164 ` 1195` ``` by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq) ``` huffman@21164 ` 1196` ``` ultimately show ?thesis using neq by (force simp add: add_commute) ``` huffman@21164 ` 1197` ```qed ``` huffman@21164 ` 1198` paulson@33654 ` 1199` ```(* A function with positive derivative is increasing. ``` paulson@33654 ` 1200` ``` A simple proof using the MVT, by Jeremy Avigad. And variants. ``` paulson@33654 ` 1201` ```*) ``` paulson@33654 ` 1202` ```lemma DERIV_pos_imp_increasing: ``` paulson@33654 ` 1203` ``` fixes a::real and b::real and f::"real => real" ``` paulson@33654 ` 1204` ``` assumes "a < b" and "\x. a \ x & x \ b --> (EX y. DERIV f x :> y & y > 0)" ``` paulson@33654 ` 1205` ``` shows "f a < f b" ``` paulson@33654 ` 1206` ```proof (rule ccontr) ``` wenzelm@41550 ` 1207` ``` assume f: "~ f a < f b" ``` wenzelm@33690 ` 1208` ``` have "EX l z. a < z & z < b & DERIV f z :> l ``` paulson@33654 ` 1209` ``` & f b - f a = (b - a) * l" ``` wenzelm@33690 ` 1210` ``` apply (rule MVT) ``` wenzelm@33690 ` 1211` ``` using assms ``` wenzelm@33690 ` 1212` ``` apply auto ``` wenzelm@33690 ` 1213` ``` apply (metis DERIV_isCont) ``` huffman@36777 ` 1214` ``` apply (metis differentiableI less_le) ``` wenzelm@33690 ` 1215` ``` done ``` wenzelm@41550 ` 1216` ``` then obtain l z where z: "a < z" "z < b" "DERIV f z :> l" ``` paulson@33654 ` 1217` ``` and "f b - f a = (b - a) * l" ``` paulson@33654 ` 1218` ``` by auto ``` wenzelm@41550 ` 1219` ``` with assms f have "~(l > 0)" ``` huffman@36777 ` 1220` ``` by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le) ``` wenzelm@41550 ` 1221` ``` with assms z show False ``` huffman@36777 ` 1222` ``` by (metis DERIV_unique less_le) ``` paulson@33654 ` 1223` ```qed ``` paulson@33654 ` 1224` noschinl@45791 ` 1225` ```lemma DERIV_nonneg_imp_nondecreasing: ``` paulson@33654 ` 1226` ``` fixes a::real and b::real and f::"real => real" ``` paulson@33654 ` 1227` ``` assumes "a \ b" and ``` paulson@33654 ` 1228` ``` "\x. a \ x & x \ b --> (\y. DERIV f x :> y & y \ 0)" ``` paulson@33654 ` 1229` ``` shows "f a \ f b" ``` paulson@33654 ` 1230` ```proof (rule ccontr, cases "a = b") ``` wenzelm@41550 ` 1231` ``` assume "~ f a \ f b" and "a = b" ``` wenzelm@41550 ` 1232` ``` then show False by auto ``` haftmann@37891 ` 1233` ```next ``` haftmann@37891 ` 1234` ``` assume A: "~ f a \ f b" ``` haftmann@37891 ` 1235` ``` assume B: "a ~= b" ``` paulson@33654 ` 1236` ``` with assms have "EX l z. a < z & z < b & DERIV f z :> l ``` paulson@33654 ` 1237` ``` & f b - f a = (b - a) * l" ``` wenzelm@33690 ` 1238` ``` apply - ``` wenzelm@33690 ` 1239` ``` apply (rule MVT) ``` wenzelm@33690 ` 1240` ``` apply auto ``` wenzelm@33690 ` 1241` ``` apply (metis DERIV_isCont) ``` huffman@36777 ` 1242` ``` apply (metis differentiableI less_le) ``` paulson@33654 ` 1243` ``` done ``` wenzelm@41550 ` 1244` ``` then obtain l z where z: "a < z" "z < b" "DERIV f z :> l" ``` haftmann@37891 ` 1245` ``` and C: "f b - f a = (b - a) * l" ``` paulson@33654 ` 1246` ``` by auto ``` haftmann@37891 ` 1247` ``` with A have "a < b" "f b < f a" by auto ``` haftmann@37891 ` 1248` ``` with C have "\ l \ 0" by (auto simp add: not_le algebra_simps) ``` huffman@45051 ` 1249` ``` (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl) ``` wenzelm@41550 ` 1250` ``` with assms z show False ``` paulson@33654 ` 1251` ``` by (metis DERIV_unique order_less_imp_le) ``` paulson@33654 ` 1252` ```qed ``` paulson@33654 ` 1253` paulson@33654 ` 1254` ```lemma DERIV_neg_imp_decreasing: ``` paulson@33654 ` 1255` ``` fixes a::real and b::real and f::"real => real" ``` paulson@33654 ` 1256` ``` assumes "a < b" and ``` paulson@33654 ` 1257` ``` "\x. a \ x & x \ b --> (\y. DERIV f x :> y & y < 0)" ``` paulson@33654 ` 1258` ``` shows "f a > f b" ``` paulson@33654 ` 1259` ```proof - ``` paulson@33654 ` 1260` ``` have "(%x. -f x) a < (%x. -f x) b" ``` paulson@33654 ` 1261` ``` apply (rule DERIV_pos_imp_increasing [of a b "%x. -f x"]) ``` wenzelm@33690 ` 1262` ``` using assms ``` wenzelm@33690 ` 1263` ``` apply auto ``` paulson@33654 ` 1264` ``` apply (metis DERIV_minus neg_0_less_iff_less) ``` paulson@33654 ` 1265` ``` done ``` paulson@33654 ` 1266` ``` thus ?thesis ``` paulson@33654 ` 1267` ``` by simp ``` paulson@33654 ` 1268` ```qed ``` paulson@33654 ` 1269` paulson@33654 ` 1270` ```lemma DERIV_nonpos_imp_nonincreasing: ``` paulson@33654 ` 1271` ``` fixes a::real and b::real and f::"real => real" ``` paulson@33654 ` 1272` ``` assumes "a \ b" and ``` paulson@33654 ` 1273` ``` "\x. a \ x & x \ b --> (\y. DERIV f x :> y & y \ 0)" ``` paulson@33654 ` 1274` ``` shows "f a \ f b" ``` paulson@33654 ` 1275` ```proof - ``` paulson@33654 ` 1276` ``` have "(%x. -f x) a \ (%x. -f x) b" ``` noschinl@45791 ` 1277` ``` apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"]) ``` wenzelm@33690 ` 1278` ``` using assms ``` wenzelm@33690 ` 1279` ``` apply auto ``` paulson@33654 ` 1280` ``` apply (metis DERIV_minus neg_0_le_iff_le) ``` paulson@33654 ` 1281` ``` done ``` paulson@33654 ` 1282` ``` thus ?thesis ``` paulson@33654 ` 1283` ``` by simp ``` paulson@33654 ` 1284` ```qed ``` huffman@21164 ` 1285` huffman@29975 ` 1286` ```subsection {* Continuous injective functions *} ``` huffman@29975 ` 1287` huffman@21164 ` 1288` ```text{*Dull lemma: an continuous injection on an interval must have a ``` huffman@21164 ` 1289` ```strict maximum at an end point, not in the middle.*} ``` huffman@21164 ` 1290` huffman@21164 ` 1291` ```lemma lemma_isCont_inj: ``` huffman@21164 ` 1292` ``` fixes f :: "real \ real" ``` huffman@21164 ` 1293` ``` assumes d: "0 < d" ``` huffman@21164 ` 1294` ``` and inj [rule_format]: "\z. \z-x\ \ d --> g(f z) = z" ``` huffman@21164 ` 1295` ``` and cont: "\z. \z-x\ \ d --> isCont f z" ``` huffman@21164 ` 1296` ``` shows "\z. \z-x\ \ d & f x < f z" ``` huffman@21164 ` 1297` ```proof (rule ccontr) ``` huffman@21164 ` 1298` ``` assume "~ (\z. \z-x\ \ d & f x < f z)" ``` huffman@21164 ` 1299` ``` hence all [rule_format]: "\z. \z - x\ \ d --> f z \ f x" by auto ``` huffman@21164 ` 1300` ``` show False ``` huffman@21164 ` 1301` ``` proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"]) ``` huffman@21164 ` 1302` ``` case le ``` huffman@21164 ` 1303` ``` from d cont all [of "x+d"] ``` huffman@21164 ` 1304` ``` have flef: "f(x+d) \ f x" ``` huffman@21164 ` 1305` ``` and xlex: "x - d \ x" ``` huffman@21164 ` 1306` ``` and cont': "\z. x - d \ z \ z \ x \ isCont f z" ``` huffman@21164 ` 1307` ``` by (auto simp add: abs_if) ``` huffman@21164 ` 1308` ``` from IVT [OF le flef xlex cont'] ``` huffman@21164 ` 1309` ``` obtain x' where "x-d \ x'" "x' \ x" "f x' = f(x+d)" by blast ``` huffman@21164 ` 1310` ``` moreover ``` huffman@21164 ` 1311` ``` hence "g(f x') = g (f(x+d))" by simp ``` huffman@21164 ` 1312` ``` ultimately show False using d inj [of x'] inj [of "x+d"] ``` huffman@22998 ` 1313` ``` by (simp add: abs_le_iff) ``` huffman@21164 ` 1314` ``` next ``` huffman@21164 ` 1315` ``` case ge ``` huffman@21164 ` 1316` ``` from d cont all [of "x-d"] ``` huffman@21164 ` 1317` ``` have flef: "f(x-d) \ f x" ``` huffman@21164 ` 1318` ``` and xlex: "x \ x+d" ``` huffman@21164 ` 1319` ``` and cont': "\z. x \ z \ z \ x+d \ isCont f z" ``` huffman@21164 ` 1320` ``` by (auto simp add: abs_if) ``` huffman@21164 ` 1321` ``` from IVT2 [OF ge flef xlex cont'] ``` huffman@21164 ` 1322` ``` obtain x' where "x \ x'" "x' \ x+d" "f x' = f(x-d)" by blast ``` huffman@21164 ` 1323` ``` moreover ``` huffman@21164 ` 1324` ``` hence "g(f x') = g (f(x-d))" by simp ``` huffman@21164 ` 1325` ``` ultimately show False using d inj [of x'] inj [of "x-d"] ``` huffman@22998 ` 1326` ``` by (simp add: abs_le_iff) ``` huffman@21164 ` 1327` ``` qed ``` huffman@21164 ` 1328` ```qed ``` huffman@21164 ` 1329` huffman@21164 ` 1330` huffman@21164 ` 1331` ```text{*Similar version for lower bound.*} ``` huffman@21164 ` 1332` huffman@21164 ` 1333` ```lemma lemma_isCont_inj2: ``` huffman@21164 ` 1334` ``` fixes f g :: "real \ real" ``` huffman@21164 ` 1335` ``` shows "[|0 < d; \z. \z-x\ \ d --> g(f z) = z; ``` huffman@21164 ` 1336` ``` \z. \z-x\ \ d --> isCont f z |] ``` huffman@21164 ` 1337` ``` ==> \z. \z-x\ \ d & f z < f x" ``` huffman@21164 ` 1338` ```apply (insert lemma_isCont_inj ``` huffman@21164 ` 1339` ``` [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d]) ``` huffman@44233 ` 1340` ```apply (simp add: linorder_not_le) ``` huffman@21164 ` 1341` ```done ``` huffman@21164 ` 1342` huffman@21164 ` 1343` ```text{*Show there's an interval surrounding @{term "f(x)"} in ``` huffman@21164 ` 1344` ```@{text "f[[x - d, x + d]]"} .*} ``` huffman@21164 ` 1345` huffman@21164 ` 1346` ```lemma isCont_inj_range: ``` huffman@21164 ` 1347` ``` fixes f :: "real \ real" ``` huffman@21164 ` 1348` ``` assumes d: "0 < d" ``` huffman@21164 ` 1349` ``` and inj: "\z. \z-x\ \ d --> g(f z) = z" ``` huffman@21164 ` 1350` ``` and cont: "\z. \z-x\ \ d --> isCont f z" ``` huffman@21164 ` 1351` ``` shows "\e>0. \y. \y - f x\ \ e --> (\z. \z-x\ \ d & f z = y)" ``` huffman@21164 ` 1352` ```proof - ``` huffman@21164 ` 1353` ``` have "x-d \ x+d" "\z. x-d \ z \ z \ x+d \ isCont f z" using cont d ``` huffman@22998 ` 1354` ``` by (auto simp add: abs_le_iff) ``` huffman@21164 ` 1355` ``` from isCont_Lb_Ub [OF this] ``` huffman@21164 ` 1356` ``` obtain L M ``` huffman@21164 ` 1357` ``` where all1 [rule_format]: "\z. x-d \ z \ z \ x+d \ L \ f z \ f z \ M" ``` huffman@21164 ` 1358` ``` and all2 [rule_format]: ``` huffman@21164 ` 1359` ``` "\y. L \ y \ y \ M \ (\z. x-d \ z \ z \ x+d \ f z = y)" ``` huffman@21164 ` 1360` ``` by auto ``` huffman@21164 ` 1361` ``` with d have "L \ f x & f x \ M" by simp ``` huffman@21164 ` 1362` ``` moreover have "L \ f x" ``` huffman@21164 ` 1363` ``` proof - ``` huffman@21164 ` 1364` ``` from lemma_isCont_inj2 [OF d inj cont] ``` huffman@21164 ` 1365` ``` obtain u where "\u - x\ \ d" "f u < f x" by auto ``` huffman@21164 ` 1366` ``` thus ?thesis using all1 [of u] by arith ``` huffman@21164 ` 1367` ``` qed ``` huffman@21164 ` 1368` ``` moreover have "f x \ M" ``` huffman@21164 ` 1369` ``` proof - ``` huffman@21164 ` 1370` ``` from lemma_isCont_inj [OF d inj cont] ``` huffman@21164 ` 1371` ``` obtain u where "\u - x\ \ d" "f x < f u" by auto ``` huffman@21164 ` 1372` ``` thus ?thesis using all1 [of u] by arith ``` huffman@21164 ` 1373` ``` qed ``` huffman@21164 ` 1374` ``` ultimately have "L < f x & f x < M" by arith ``` huffman@21164 ` 1375` ``` hence "0 < f x - L" "0 < M - f x" by arith+ ``` huffman@21164 ` 1376` ``` from real_lbound_gt_zero [OF this] ``` huffman@21164 ` 1377` ``` obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto ``` huffman@21164 ` 1378` ``` thus ?thesis ``` huffman@21164 ` 1379` ``` proof (intro exI conjI) ``` huffman@23441 ` 1380` ``` show "0y. \y - f x\ \ e \ (\z. \z - x\ \ d \ f z = y)" ``` huffman@21164 ` 1382` ``` proof (intro strip) ``` huffman@21164 ` 1383` ``` fix y::real ``` huffman@21164 ` 1384` ``` assume "\y - f x\ \ e" ``` huffman@21164 ` 1385` ``` with e have "L \ y \ y \ M" by arith ``` huffman@21164 ` 1386` ``` from all2 [OF this] ``` huffman@21164 ` 1387` ``` obtain z where "x - d \ z" "z \ x + d" "f z = y" by blast ``` chaieb@27668 ` 1388` ``` thus "\z. \z - x\ \ d \ f z = y" ``` huffman@22998 ` 1389` ``` by (force simp add: abs_le_iff) ``` huffman@21164 ` 1390` ``` qed ``` huffman@21164 ` 1391` ``` qed ``` huffman@21164 ` 1392` ```qed ``` huffman@21164 ` 1393` huffman@21164 ` 1394` huffman@21164 ` 1395` ```text{*Continuity of inverse function*} ``` huffman@21164 ` 1396` huffman@21164 ` 1397` ```lemma isCont_inverse_function: ``` huffman@21164 ` 1398` ``` fixes f g :: "real \ real" ``` huffman@21164 ` 1399` ``` assumes d: "0 < d" ``` huffman@21164 ` 1400` ``` and inj: "\z. \z-x\ \ d --> g(f z) = z" ``` huffman@21164 ` 1401` ``` and cont: "\z. \z-x\ \ d --> isCont f z" ``` huffman@21164 ` 1402` ``` shows "isCont g (f x)" ``` huffman@21164 ` 1403` ```proof (simp add: isCont_iff LIM_eq) ``` huffman@21164 ` 1404` ``` show "\r. 0 < r \ ``` huffman@21164 ` 1405` ``` (\s>0. \z. z\0 \ \z\ < s \ \g(f x + z) - g(f x)\ < r)" ``` huffman@21164 ` 1406` ``` proof (intro strip) ``` huffman@21164 ` 1407` ``` fix r::real ``` huffman@21164 ` 1408` ``` assume r: "0 e < d" by blast ``` huffman@21164 ` 1411` ``` with inj cont ``` huffman@21164 ` 1412` ``` have e_simps: "\z. \z-x\ \ e --> g (f z) = z" ``` huffman@21164 ` 1413` ``` "\z. \z-x\ \ e --> isCont f z" by auto ``` huffman@21164 ` 1414` ``` from isCont_inj_range [OF e this] ``` huffman@21164 ` 1415` ``` obtain e' where e': "0 < e'" ``` huffman@21164 ` 1416` ``` and all: "\y. \y - f x\ \ e' \ (\z. \z - x\ \ e \ f z = y)" ``` huffman@21164 ` 1417` ``` by blast ``` huffman@21164 ` 1418` ``` show "\s>0. \z. z\0 \ \z\ < s \ \g(f x + z) - g(f x)\ < r" ``` huffman@21164 ` 1419` ``` proof (intro exI conjI) ``` huffman@23441 ` 1420` ``` show "0z. z \ 0 \ \z\ < e' \ \g (f x + z) - g (f x)\ < r" ``` huffman@21164 ` 1422` ``` proof (intro strip) ``` huffman@21164 ` 1423` ``` fix z::real ``` huffman@21164 ` 1424` ``` assume z: "z \ 0 \ \z\ < e'" ``` huffman@21164 ` 1425` ``` with e e_lt e_simps all [rule_format, of "f x + z"] ``` huffman@21164 ` 1426` ``` show "\g (f x + z) - g (f x)\ < r" by force ``` huffman@21164 ` 1427` ``` qed ``` huffman@21164 ` 1428` ``` qed ``` huffman@21164 ` 1429` ``` qed ``` huffman@21164 ` 1430` ```qed ``` huffman@21164 ` 1431` huffman@23041 ` 1432` ```text {* Derivative of inverse function *} ``` huffman@23041 ` 1433` huffman@23041 ` 1434` ```lemma DERIV_inverse_function: ``` huffman@23041 ` 1435` ``` fixes f g :: "real \ real" ``` huffman@23041 ` 1436` ``` assumes der: "DERIV f (g x) :> D" ``` huffman@23041 ` 1437` ``` assumes neq: "D \ 0" ``` huffman@23044 ` 1438` ``` assumes a: "a < x" and b: "x < b" ``` huffman@23044 ` 1439` ``` assumes inj: "\y. a < y \ y < b \ f (g y) = y" ``` huffman@23041 ` 1440` ``` assumes cont: "isCont g x" ``` huffman@23041 ` 1441` ``` shows "DERIV g x :> inverse D" ``` huffman@23041 ` 1442` ```unfolding DERIV_iff2 ``` huffman@23044 ` 1443` ```proof (rule LIM_equal2) ``` huffman@23044 ` 1444` ``` show "0 < min (x - a) (b - x)" ``` chaieb@27668 ` 1445` ``` using a b by arith ``` huffman@23044 ` 1446` ```next ``` huffman@23041 ` 1447` ``` fix y ``` huffman@23044 ` 1448` ``` assume "norm (y - x) < min (x - a) (b - x)" ``` chaieb@27668 ` 1449` ``` hence "a < y" and "y < b" ``` huffman@23044 ` 1450` ``` by (simp_all add: abs_less_iff) ``` huffman@23041 ` 1451` ``` thus "(g y - g x) / (y - x) = ``` huffman@23041 ` 1452` ``` inverse ((f (g y) - x) / (g y - g x))" ``` huffman@23041 ` 1453` ``` by (simp add: inj) ``` huffman@23041 ` 1454` ```next ``` huffman@23041 ` 1455` ``` have "(\z. (f z - f (g x)) / (z - g x)) -- g x --> D" ``` huffman@23041 ` 1456` ``` by (rule der [unfolded DERIV_iff2]) ``` huffman@23041 ` 1457` ``` hence 1: "(\z. (f z - x) / (z - g x)) -- g x --> D" ``` huffman@23044 ` 1458` ``` using inj a b by simp ``` huffman@23041 ` 1459` ``` have 2: "\d>0. \y. y \ x \ norm (y - x) < d \ g y \ g x" ``` huffman@23041 ` 1460` ``` proof (safe intro!: exI) ``` huffman@23044 ` 1461` ``` show "0 < min (x - a) (b - x)" ``` huffman@23044 ` 1462` ``` using a b by simp ``` huffman@23041 ` 1463` ``` next ``` huffman@23041 ` 1464` ``` fix y ``` huffman@23044 ` 1465` ``` assume "norm (y - x) < min (x - a) (b - x)" ``` huffman@23044 ` 1466` ``` hence y: "a < y" "y < b" ``` huffman@23044 ` 1467` ``` by (simp_all add: abs_less_iff) ``` huffman@23041 ` 1468` ``` assume "g y = g x" ``` huffman@23041 ` 1469` ``` hence "f (g y) = f (g x)" by simp ``` huffman@23044 ` 1470` ``` hence "y = x" using inj y a b by simp ``` huffman@23041 ` 1471` ``` also assume "y \ x" ``` huffman@23041 ` 1472` ``` finally show False by simp ``` huffman@23041 ` 1473` ``` qed ``` huffman@23041 ` 1474` ``` have "(\y. (f (g y) - x) / (g y - g x)) -- x --> D" ``` huffman@23041 ` 1475` ``` using cont 1 2 by (rule isCont_LIM_compose2) ``` huffman@23041 ` 1476` ``` thus "(\y. inverse ((f (g y) - x) / (g y - g x))) ``` huffman@23041 ` 1477` ``` -- x --> inverse D" ``` huffman@44568 ` 1478` ``` using neq by (rule tendsto_inverse) ``` huffman@23041 ` 1479` ```qed ``` huffman@23041 ` 1480` huffman@29975 ` 1481` huffman@29975 ` 1482` ```subsection {* Generalized Mean Value Theorem *} ``` huffman@29975 ` 1483` huffman@21164 ` 1484` ```theorem GMVT: ``` huffman@21784 ` 1485` ``` fixes a b :: real ``` huffman@21164 ` 1486` ``` assumes alb: "a < b" ``` wenzelm@41550 ` 1487` ``` and fc: "\x. a \ x \ x \ b \ isCont f x" ``` wenzelm@41550 ` 1488` ``` and fd: "\x. a < x \ x < b \ f differentiable x" ``` wenzelm@41550 ` 1489` ``` and gc: "\x. a \ x \ x \ b \ isCont g x" ``` wenzelm@41550 ` 1490` ``` and gd: "\x. a < x \ x < b \ g differentiable x" ``` huffman@21164 ` 1491` ``` shows "\g'c f'c c. DERIV g c :> g'c \ DERIV f c :> f'c \ a < c \ c < b \ ((f b - f a) * g'c) = ((g b - g a) * f'c)" ``` huffman@21164 ` 1492` ```proof - ``` huffman@21164 ` 1493` ``` let ?h = "\x. (f b - f a)*(g x) - (g b - g a)*(f x)" ``` wenzelm@41550 ` 1494` ``` from assms have "a < b" by simp ``` huffman@21164 ` 1495` ``` moreover have "\x. a \ x \ x \ b \ isCont ?h x" ``` huffman@44233 ` 1496` ``` using fc gc by simp ``` huffman@44233 ` 1497` ``` moreover have "\x. a < x \ x < b \ ?h differentiable x" ``` huffman@44233 ` 1498` ``` using fd gd by simp ``` huffman@21164 ` 1499` ``` ultimately have "\l z. a < z \ z < b \ DERIV ?h z :> l \ ?h b - ?h a = (b - a) * l" by (rule MVT) ``` huffman@21164 ` 1500` ``` then obtain l where ldef: "\z. a < z \ z < b \ DERIV ?h z :> l \ ?h b - ?h a = (b - a) * l" .. ``` huffman@21164 ` 1501` ``` then obtain c where cdef: "a < c \ c < b \ DERIV ?h c :> l \ ?h b - ?h a = (b - a) * l" .. ``` huffman@21164 ` 1502` huffman@21164 ` 1503` ``` from cdef have cint: "a < c \ c < b" by auto ``` huffman@21164 ` 1504` ``` with gd have "g differentiable c" by simp ``` huffman@21164 ` 1505` ``` hence "\D. DERIV g c :> D" by (rule differentiableD) ``` huffman@21164 ` 1506` ``` then obtain g'c where g'cdef: "DERIV g c :> g'c" .. ``` huffman@21164 ` 1507` huffman@21164 ` 1508` ``` from cdef have "a < c \ c < b" by auto ``` huffman@21164 ` 1509` ``` with fd have "f differentiable c" by simp ``` huffman@21164 ` 1510` ``` hence "\D. DERIV f c :> D" by (rule differentiableD) ``` huffman@21164 ` 1511` ``` then obtain f'c where f'cdef: "DERIV f c :> f'c" .. ``` huffman@21164 ` 1512` huffman@21164 ` 1513` ``` from cdef have "DERIV ?h c :> l" by auto ``` hoelzl@41368 ` 1514` ``` moreover have "DERIV ?h c :> g'c * (f b - f a) - f'c * (g b - g a)" ``` hoelzl@41368 ` 1515` ``` using g'cdef f'cdef by (auto intro!: DERIV_intros) ``` huffman@21164 ` 1516` ``` ultimately have leq: "l = g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique) ``` huffman@21164 ` 1517` huffman@21164 ` 1518` ``` { ``` huffman@21164 ` 1519` ``` from cdef have "?h b - ?h a = (b - a) * l" by auto ``` huffman@21164 ` 1520` ``` also with leq have "\ = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp ``` huffman@21164 ` 1521` ``` finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp ``` huffman@21164 ` 1522` ``` } ``` huffman@21164 ` 1523` ``` moreover ``` huffman@21164 ` 1524` ``` { ``` huffman@21164 ` 1525` ``` have "?h b - ?h a = ``` huffman@21164 ` 1526` ``` ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) - ``` huffman@21164 ` 1527` ``` ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))" ``` nipkow@29667 ` 1528` ``` by (simp add: algebra_simps) ``` huffman@21164 ` 1529` ``` hence "?h b - ?h a = 0" by auto ``` huffman@21164 ` 1530` ``` } ``` huffman@21164 ` 1531` ``` ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto ``` huffman@21164 ` 1532` ``` with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp ``` huffman@21164 ` 1533` ``` hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp ``` huffman@21164 ` 1534` ``` hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac) ``` huffman@21164 ` 1535` huffman@21164 ` 1536` ``` with g'cdef f'cdef cint show ?thesis by auto ``` huffman@21164 ` 1537` ```qed ``` huffman@21164 ` 1538` huffman@29470 ` 1539` huffman@29166 ` 1540` ```subsection {* Theorems about Limits *} ``` huffman@29166 ` 1541` huffman@29166 ` 1542` ```(* need to rename second isCont_inverse *) ``` huffman@29166 ` 1543` huffman@29166 ` 1544` ```lemma isCont_inv_fun: ``` huffman@29166 ` 1545` ``` fixes f g :: "real \ real" ``` huffman@29166 ` 1546` ``` shows "[| 0 < d; \z. \z - x\ \ d --> g(f(z)) = z; ``` huffman@29166 ` 1547` ``` \z. \z - x\ \ d --> isCont f z |] ``` huffman@29166 ` 1548` ``` ==> isCont g (f x)" ``` huffman@29166 ` 1549` ```by (rule isCont_inverse_function) ``` huffman@29166 ` 1550` huffman@29166 ` 1551` ```lemma isCont_inv_fun_inv: ``` huffman@29166 ` 1552` ``` fixes f g :: "real \ real" ``` huffman@29166 ` 1553` ``` shows "[| 0 < d; ``` huffman@29166 ` 1554` ``` \z. \z - x\ \ d --> g(f(z)) = z; ``` huffman@29166 ` 1555` ``` \z. \z - x\ \ d --> isCont f z |] ``` huffman@29166 ` 1556` ``` ==> \e. 0 < e & ``` huffman@29166 ` 1557` ``` (\y. 0 < \y - f(x)\ & \y - f(x)\ < e --> f(g(y)) = y)" ``` huffman@29166 ` 1558` ```apply (drule isCont_inj_range) ``` huffman@29166 ` 1559` ```prefer 2 apply (assumption, assumption, auto) ``` huffman@29166 ` 1560` ```apply (rule_tac x = e in exI, auto) ``` huffman@29166 ` 1561` ```apply (rotate_tac 2) ``` huffman@29166 ` 1562` ```apply (drule_tac x = y in spec, auto) ``` huffman@29166 ` 1563` ```done ``` huffman@29166 ` 1564` huffman@29166 ` 1565` huffman@29166 ` 1566` ```text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*} ``` huffman@29166 ` 1567` ```lemma LIM_fun_gt_zero: ``` huffman@29166 ` 1568` ``` "[| f -- c --> (l::real); 0 < l |] ``` huffman@29166 ` 1569` ``` ==> \r. 0 < r & (\x::real. x \ c & \c - x\ < r --> 0 < f x)" ``` huffman@44209 ` 1570` ```apply (drule (1) LIM_D, clarify) ``` huffman@29166 ` 1571` ```apply (rule_tac x = s in exI) ``` huffman@44209 ` 1572` ```apply (simp add: abs_less_iff) ``` huffman@29166 ` 1573` ```done ``` huffman@29166 ` 1574` huffman@29166 ` 1575` ```lemma LIM_fun_less_zero: ``` huffman@29166 ` 1576` ``` "[| f -- c --> (l::real); l < 0 |] ``` huffman@29166 ` 1577` ``` ==> \r. 0 < r & (\x::real. x \ c & \c - x\ < r --> f x < 0)" ``` huffman@44209 ` 1578` ```apply (drule LIM_D [where r="-l"], simp, clarify) ``` huffman@29166 ` 1579` ```apply (rule_tac x = s in exI) ``` huffman@44209 ` 1580` ```apply (simp add: abs_less_iff) ``` huffman@29166 ` 1581` ```done ``` huffman@29166 ` 1582` huffman@29166 ` 1583` ```lemma LIM_fun_not_zero: ``` huffman@29166 ` 1584` ``` "[| f -- c --> (l::real); l \ 0 |] ``` huffman@29166 ` 1585` ``` ==> \r. 0 < r & (\x::real. x \ c & \c - x\ < r --> f x \ 0)" ``` huffman@44209 ` 1586` ```apply (rule linorder_cases [of l 0]) ``` huffman@44209 ` 1587` ```apply (drule (1) LIM_fun_less_zero, force) ``` huffman@44209 ` 1588` ```apply simp ``` huffman@44209 ` 1589` ```apply (drule (1) LIM_fun_gt_zero, force) ``` huffman@29166 ` 1590` ```done ``` huffman@29166 ` 1591` hoelzl@50327 ` 1592` ```lemma GMVT': ``` hoelzl@50327 ` 1593` ``` fixes f g :: "real \ real" ``` hoelzl@50327 ` 1594` ``` assumes "a < b" ``` hoelzl@50327 ` 1595` ``` assumes isCont_f: "\z. a \ z \ z \ b \ isCont f z" ``` hoelzl@50327 ` 1596` ``` assumes isCont_g: "\z. a \ z \ z \ b \ isCont g z" ``` hoelzl@50327 ` 1597` ``` assumes DERIV_g: "\z. a < z \ z < b \ DERIV g z :> (g' z)" ``` hoelzl@50327 ` 1598` ``` assumes DERIV_f: "\z. a < z \ z < b \ DERIV f z :> (f' z)" ``` hoelzl@50327 ` 1599` ``` shows "\c. a < c \ c < b \ (f b - f a) * g' c = (g b - g a) * f' c" ``` hoelzl@50327 ` 1600` ```proof - ``` hoelzl@50327 ` 1601` ``` have "\g'c f'c c. DERIV g c :> g'c \ DERIV f c :> f'c \ ``` hoelzl@50327 ` 1602` ``` a < c \ c < b \ (f b - f a) * g'c = (g b - g a) * f'c" ``` hoelzl@50327 ` 1603` ``` using assms by (intro GMVT) (force simp: differentiable_def)+ ``` hoelzl@50327 ` 1604` ``` then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c" ``` hoelzl@50327 ` 1605` ``` using DERIV_f DERIV_g by (force dest: DERIV_unique) ``` hoelzl@50327 ` 1606` ``` then show ?thesis ``` hoelzl@50327 ` 1607` ``` by auto ``` hoelzl@50327 ` 1608` ```qed ``` hoelzl@50327 ` 1609` hoelzl@50327 ` 1610` ```lemma lhopital_right_0: ``` hoelzl@50327 ` 1611` ``` fixes f g :: "real \ real" ``` hoelzl@50327 ` 1612` ``` assumes f_0: "isCont f 0" "f 0 = 0" ``` hoelzl@50327 ` 1613` ``` assumes g_0: "isCont g 0" "g 0 = 0" ``` hoelzl@50327 ` 1614` ``` assumes ev: ``` hoelzl@50327 ` 1615` ``` "eventually (\x. g x \ 0) (at_right 0)" ``` hoelzl@50327 ` 1616` ``` "eventually (\x. g' x \ 0) (at_right 0)" ``` hoelzl@50327 ` 1617` ``` "eventually (\x. DERIV f x :> f' x) (at_right 0)" ``` hoelzl@50327 ` 1618` ``` "eventually (\x. DERIV g x :> g' x) (at_right 0)" ``` hoelzl@50327 ` 1619` ``` assumes lim: "((\ x. (f' x / g' x)) ---> x) (at_right 0)" ``` hoelzl@50327 ` 1620` ``` shows "((\ x. f x / g x) ---> x) (at_right 0)" ``` hoelzl@50327 ` 1621` ```proof - ``` hoelzl@50327 ` 1622` ``` from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) ev(4)]] ``` hoelzl@50327 ` 1623` ``` obtain a where [arith]: "0 < a" ``` hoelzl@50327 ` 1624` ``` and g_neq_0: "\x. 0 < x \ x < a \ g x \ 0" ``` hoelzl@50327 ` 1625` ``` and g'_neq_0: "\x. 0 < x \ x < a \ g' x \ 0" ``` hoelzl@50327 ` 1626` ``` and f: "\x. 0 < x \ x < a \ DERIV f x :> (f' x)" ``` hoelzl@50327 ` 1627` ``` and g: "\x. 0 < x \ x < a \ DERIV g x :> (g' x)" ``` hoelzl@50327 ` 1628` ``` unfolding eventually_within eventually_at by (auto simp: dist_real_def) ``` hoelzl@50327 ` 1629` hoelzl@50327 ` 1630` ``` { fix x assume x: "0 \ x" "x < a" ``` hoelzl@50327 ` 1631` ``` from `0 \ x` have "isCont f x" ``` hoelzl@50327 ` 1632` ``` unfolding le_less ``` hoelzl@50327 ` 1633` ``` proof ``` hoelzl@50327 ` 1634` ``` assume "0 = x" with `isCont f 0` show "isCont f x" by simp ``` hoelzl@50327 ` 1635` ``` next ``` hoelzl@50327 ` 1636` ``` assume "0 < x" with f x show ?thesis by (auto intro!: DERIV_isCont) ``` hoelzl@50327 ` 1637` ``` qed } ``` hoelzl@50327 ` 1638` ``` note isCont_f = this ``` hoelzl@50327 ` 1639` hoelzl@50327 ` 1640` ``` { fix x assume x: "0 \ x" "x < a" ``` hoelzl@50327 ` 1641` ``` from `0 \ x` have "isCont g x" ``` hoelzl@50327 ` 1642` ``` unfolding le_less ``` hoelzl@50327 ` 1643` ``` proof ``` hoelzl@50327 ` 1644` ``` assume "0 = x" with `isCont g 0` show "isCont g x" by simp ``` hoelzl@50327 ` 1645` ``` next ``` hoelzl@50327 ` 1646` ``` assume "0 < x" with g x show ?thesis by (auto intro!: DERIV_isCont) ``` hoelzl@50327 ` 1647` ``` qed } ``` hoelzl@50327 ` 1648` ``` note isCont_g = this ``` hoelzl@50327 ` 1649` hoelzl@50327 ` 1650` ``` have "\\. \x\{0 <..< a}. 0 < \ x \ \ x < x \ f x / g x = f' (\ x) / g' (\ x)" ``` hoelzl@50327 ` 1651` ``` proof (rule bchoice, rule) ``` hoelzl@50327 ` 1652` ``` fix x assume "x \ {0 <..< a}" ``` hoelzl@50327 ` 1653` ``` then have x[arith]: "0 < x" "x < a" by auto ``` hoelzl@50327 ` 1654` ``` with g'_neq_0 g_neq_0 `g 0 = 0` have g': "\x. 0 < x \ x < a \ 0 \ g' x" "g 0 \ g x" ``` hoelzl@50327 ` 1655` ``` by auto ``` hoelzl@50327 ` 1656` hoelzl@50327 ` 1657` ``` have "\c. 0 < c \ c < x \ (f x - f 0) * g' c = (g x - g 0) * f' c" ``` hoelzl@50327 ` 1658` ``` using isCont_f isCont_g f g `x < a` by (intro GMVT') auto ``` hoelzl@50327 ` 1659` ``` then guess c .. ``` hoelzl@50327 ` 1660` ``` moreover ``` hoelzl@50327 ` 1661` ``` with g'(1)[of c] g'(2) have "(f x - f 0) / (g x - g 0) = f' c / g' c" ``` hoelzl@50327 ` 1662` ``` by (simp add: field_simps) ``` hoelzl@50327 ` 1663` ``` ultimately show "\y. 0 < y \ y < x \ f x / g x = f' y / g' y" ``` hoelzl@50327 ` 1664` ``` using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c]) ``` hoelzl@50327 ` 1665` ``` qed ``` hoelzl@50327 ` 1666` ``` then guess \ .. ``` hoelzl@50327 ` 1667` ``` then have \: "eventually (\x. 0 < \ x \ \ x < x \ f x / g x = f' (\ x) / g' (\ x)) (at_right 0)" ``` hoelzl@50327 ` 1668` ``` unfolding eventually_within eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def) ``` hoelzl@50327 ` 1669` ``` moreover ``` hoelzl@50327 ` 1670` ``` from \ have "eventually (\x. norm (\ x) \ x) (at_right 0)" ``` hoelzl@50327 ` 1671` ``` by eventually_elim auto ``` hoelzl@50327 ` 1672` ``` then have "((\x. norm (\ x)) ---> 0) (at_right 0)" ``` hoelzl@50327 ` 1673` ``` by (rule_tac real_tendsto_sandwich[where f="\x. 0" and h="\x. x"]) ``` hoelzl@50327 ` 1674` ``` (auto intro: tendsto_const tendsto_ident_at_within) ``` hoelzl@50327 ` 1675` ``` then have "(\ ---> 0) (at_right 0)" ``` hoelzl@50327 ` 1676` ``` by (rule tendsto_norm_zero_cancel) ``` hoelzl@50327 ` 1677` ``` with \ have "filterlim \ (at_right 0) (at_right 0)" ``` hoelzl@50327 ` 1678` ``` by (auto elim!: eventually_elim1 simp: filterlim_within filterlim_at) ``` hoelzl@50327 ` 1679` ``` from this lim have "((\t. f' (\ t) / g' (\ t)) ---> x) (at_right 0)" ``` hoelzl@50327 ` 1680` ``` by (rule_tac filterlim_compose[of _ _ _ \]) ``` hoelzl@50327 ` 1681` ``` ultimately show "((\t. f t / g t) ---> x) (at_right 0)" ``` hoelzl@50327 ` 1682` ``` apply (rule_tac filterlim_cong[THEN iffD1, OF refl refl]) ``` hoelzl@50327 ` 1683` ``` apply (erule_tac eventually_elim1) ``` hoelzl@50327 ` 1684` ``` apply simp_all ``` hoelzl@50327 ` 1685` ``` done ``` hoelzl@50327 ` 1686` ```qed ``` hoelzl@50327 ` 1687` hoelzl@50327 ` 1688` ```lemma lhopital_right_0_at_top: ``` hoelzl@50327 ` 1689` ``` fixes f g :: "real \ real" ``` hoelzl@50327 ` 1690` ``` assumes g_0: "LIM x at_right 0. g x :> at_top" ``` hoelzl@50327 ` 1691` ``` assumes ev: ``` hoelzl@50327 ` 1692` ``` "eventually (\x. g' x \ 0) (at_right 0)" ``` hoelzl@50327 ` 1693` ``` "eventually (\x. DERIV f x :> f' x) (at_right 0)" ``` hoelzl@50327 ` 1694` ``` "eventually (\x. DERIV g x :> g' x) (at_right 0)" ``` hoelzl@50327 ` 1695` ``` assumes lim: "((\ x. (f' x / g' x)) ---> x) (at_right 0)" ``` hoelzl@50327 ` 1696` ``` shows "((\ x. f x / g x) ---> x) (at_right 0)" ``` hoelzl@50327 ` 1697` ``` unfolding tendsto_iff ``` hoelzl@50327 ` 1698` ```proof safe ``` hoelzl@50327 ` 1699` ``` fix e :: real assume "0 < e" ``` hoelzl@50327 ` 1700` hoelzl@50327 ` 1701` ``` with lim[unfolded tendsto_iff, rule_format, of "e / 4"] ``` hoelzl@50327 ` 1702` ``` have "eventually (\t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp ``` hoelzl@50327 ` 1703` ``` from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]] ``` hoelzl@50327 ` 1704` ``` obtain a where [arith]: "0 < a" ``` hoelzl@50327 ` 1705` ``` and g'_neq_0: "\x. 0 < x \ x < a \ g' x \ 0" ``` hoelzl@50327 ` 1706` ``` and f0: "\x. 0 < x \ x \ a \ DERIV f x :> (f' x)" ``` hoelzl@50327 ` 1707` ``` and g0: "\x. 0 < x \ x \ a \ DERIV g x :> (g' x)" ``` hoelzl@50327 ` 1708` ``` and Df: "\t. 0 < t \ t < a \ dist (f' t / g' t) x < e / 4" ``` hoelzl@50327 ` 1709` ``` unfolding eventually_within_le by (auto simp: dist_real_def) ``` hoelzl@50327 ` 1710` hoelzl@50327 ` 1711` ``` from Df have ``` hoelzl@50327 ` 1712` ``` "eventually (\t. t < a) (at_right 0)" ``` hoelzl@50327 ` 1713` ``` "eventually (\t::real. 0 < t) (at_right 0)" ``` hoelzl@50327 ` 1714` ``` unfolding eventually_within eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def) ``` hoelzl@50327 ` 1715` ``` moreover ``` hoelzl@50327 ` 1716` hoelzl@50327 ` 1717` ``` have "eventually (\t. 0 < g t) (at_right 0)" ``` hoelzl@50327 ` 1718` ``` using g_0[unfolded filterlim_at_top, rule_format, of "1"] by eventually_elim auto ``` hoelzl@50327 ` 1719` hoelzl@50327 ` 1720` ``` moreover ``` hoelzl@50327 ` 1721` hoelzl@50327 ` 1722` ``` have "eventually (\t. g a < g t) (at_right 0)" ``` hoelzl@50327 ` 1723` ``` using g_0[unfolded filterlim_at_top, rule_format, of "g a + 1"] by eventually_elim auto ``` hoelzl@50327 ` 1724` ``` moreover ``` hoelzl@50327 ` 1725` ``` have inv_g: "((\x. inverse (g x)) ---> 0) (at_right 0)" ``` hoelzl@50327 ` 1726` ``` using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl] ``` hoelzl@50327 ` 1727` ``` by (rule filterlim_compose) ``` hoelzl@50327 ` 1728` ``` then have "((\x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)" ``` hoelzl@50327 ` 1729` ``` by (intro tendsto_intros) ``` hoelzl@50327 ` 1730` ``` then have "((\x. norm (1 - g a / g x)) ---> 1) (at_right 0)" ``` hoelzl@50327 ` 1731` ``` by (simp add: inverse_eq_divide) ``` hoelzl@50327 ` 1732` ``` from this[unfolded tendsto_iff, rule_format, of 1] ``` hoelzl@50327 ` 1733` ``` have "eventually (\x. norm (1 - g a / g x) < 2) (at_right 0)" ``` hoelzl@50327 ` 1734` ``` by (auto elim!: eventually_elim1 simp: dist_real_def) ``` hoelzl@50327 ` 1735` hoelzl@50327 ` 1736` ``` moreover ``` hoelzl@50327 ` 1737` ``` from inv_g have "((\t. norm ((f a - x * g a) * inverse (g t))) ---> norm ((f a - x * g a) * 0)) (at_right 0)" ``` hoelzl@50327 ` 1738` ``` by (intro tendsto_intros) ``` hoelzl@50327 ` 1739` ``` then have "((\t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)" ``` hoelzl@50327 ` 1740` ``` by (simp add: inverse_eq_divide) ``` hoelzl@50327 ` 1741` ``` from this[unfolded tendsto_iff, rule_format, of "e / 2"] `0 < e` ``` hoelzl@50327 ` 1742` ``` have "eventually (\t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)" ``` hoelzl@50327 ` 1743` ``` by (auto simp: dist_real_def) ``` hoelzl@50327 ` 1744` hoelzl@50327 ` 1745` ``` ultimately show "eventually (\t. dist (f t / g t) x < e) (at_right 0)" ``` hoelzl@50327 ` 1746` ``` proof eventually_elim ``` hoelzl@50327 ` 1747` ``` fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t" ``` hoelzl@50327 ` 1748` ``` assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2" ``` hoelzl@50327 ` 1749` hoelzl@50327 ` 1750` ``` have "\y. t < y \ y < a \ (g a - g t) * f' y = (f a - f t) * g' y" ``` hoelzl@50327 ` 1751` ``` using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+ ``` hoelzl@50327 ` 1752` ``` then guess y .. ``` hoelzl@50327 ` 1753` ``` from this ``` hoelzl@50327 ` 1754` ``` have [arith]: "t < y" "y < a" and D_eq: "(f t - f a) / (g t - g a) = f' y / g' y" ``` hoelzl@50327 ` 1755` ``` using `g a < g t` g'_neq_0[of y] by (auto simp add: field_simps) ``` hoelzl@50327 ` 1756` hoelzl@50327 ` 1757` ``` have *: "f t / g t - x = ((f t - f a) / (g t - g a) - x) * (1 - g a / g t) + (f a - x * g a) / g t" ``` hoelzl@50327 ` 1758` ``` by (simp add: field_simps) ``` hoelzl@50327 ` 1759` ``` have "norm (f t / g t - x) \ ``` hoelzl@50327 ` 1760` ``` norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)" ``` hoelzl@50327 ` 1761` ``` unfolding * by (rule norm_triangle_ineq) ``` hoelzl@50327 ` 1762` ``` also have "\ = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)" ``` hoelzl@50327 ` 1763` ``` by (simp add: abs_mult D_eq dist_real_def) ``` hoelzl@50327 ` 1764` ``` also have "\ < (e / 4) * 2 + e / 2" ``` hoelzl@50327 ` 1765` ``` using ineq Df[of y] `0 < e` by (intro add_le_less_mono mult_mono) auto ``` hoelzl@50327 ` 1766` ``` finally show "dist (f t / g t) x < e" ``` hoelzl@50327 ` 1767` ``` by (simp add: dist_real_def) ``` hoelzl@50327 ` 1768` ``` qed ``` hoelzl@50327 ` 1769` ```qed ``` hoelzl@50327 ` 1770` huffman@21164 ` 1771` ```end ```