(* Title: HOL/Library/Convex_Euclidean_Space.thy
Author: John Harrison
Translation from HOL light: Robert Himmelmann, TU Muenchen *)
header {* Multivariate calculus in Euclidean space. *}
theory Derivative
imports Brouwer_Fixpoint RealVector
begin
(* Because I do not want to type this all the time *)
lemmas linear_linear = linear_conv_bounded_linear[THEN sym]
subsection {* Derivatives *}
text {* The definition is slightly tricky since we make it work over
nets of a particular form. This lets us prove theorems generally and use
"at a" or "at a within s" for restriction to a set (1-sided on R etc.) *}
definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a net \<Rightarrow> bool)"
(infixl "(has'_derivative)" 12) where
"(f has_derivative f') net \<equiv> bounded_linear f' \<and> ((\<lambda>y. (1 / (norm (y - netlimit net))) *\<^sub>R
(f y - (f (netlimit net) + f'(y - netlimit net)))) ---> 0) net"
lemma derivative_linear[dest]:"(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
unfolding has_derivative_def by auto
lemma FDERIV_conv_has_derivative:"FDERIV f (x::'a::{real_normed_vector,perfect_space}) :> f' = (f has_derivative f') (at x)" (is "?l = ?r") proof
assume ?l note as = this[unfolded fderiv_def]
show ?r unfolding has_derivative_def Lim_at apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
fix e::real assume "e>0"
guess d using as[THEN conjunct2,unfolded LIM_def,rule_format,OF`e>0`] ..
thus "\<exists>d>0. \<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
dist ((1 / norm (xa - netlimit (at x))) *\<^sub>R (f xa - (f (netlimit (at x)) + f' (xa - netlimit (at x))))) (0) < e"
apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa - x" in allE)
unfolding vector_dist_norm netlimit_at[of x] by(auto simp add:group_simps) qed next
assume ?r note as = this[unfolded has_derivative_def]
show ?l unfolding fderiv_def LIM_def apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
fix e::real assume "e>0"
guess d using as[THEN conjunct2,unfolded Lim_at,rule_format,OF`e>0`] ..
thus "\<exists>s>0. \<forall>xa. xa \<noteq> 0 \<and> dist xa 0 < s \<longrightarrow> dist (norm (f (x + xa) - f x - f' xa) / norm xa) 0 < e" apply-
apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa + x" in allE)
unfolding vector_dist_norm netlimit_at[of x] by(auto simp add:group_simps) qed qed
subsection {* These are the only cases we'll care about, probably. *}
lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
unfolding has_derivative_def and Lim by(auto simp add:netlimit_within)
lemma has_derivative_at: "(f has_derivative f') (at x) \<longleftrightarrow>
bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within unfolding within_UNIV by auto
subsection {* More explicit epsilon-delta forms. *}
lemma has_derivative_within':
"(f has_derivative f')(at x within s) \<longleftrightarrow> bounded_linear f' \<and>
(\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm(x' - x) \<and> norm(x' - x) < d
\<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
unfolding has_derivative_within Lim_within vector_dist_norm
unfolding diff_0_right norm_mul by(simp add: group_simps)
lemma has_derivative_at':
"(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
(\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm(x' - x) \<and> norm(x' - x) < d
\<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within' by auto
lemma has_derivative_at_within: "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
unfolding has_derivative_within' has_derivative_at' by meson
lemma has_derivative_within_open:
"a \<in> s \<Longrightarrow> open s \<Longrightarrow> ((f has_derivative f') (at a within s) \<longleftrightarrow> (f has_derivative f') (at a))"
unfolding has_derivative_within has_derivative_at using Lim_within_open by auto
subsection {* Derivatives on real = Derivatives on real^1 *}
lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" unfolding vector_dist_norm by(auto simp add:vec1_dest_vec1_simps)
lemma bounded_linear_vec1_dest_vec1: fixes f::"real \<Rightarrow> real"
shows "linear (vec1 \<circ> f \<circ> dest_vec1) = bounded_linear f" (is "?l = ?r") proof-
{ assume ?l guess K using linear_bounded[OF `?l`] ..
hence "\<exists>K. \<forall>x. \<bar>f x\<bar> \<le> \<bar>x\<bar> * K" apply(rule_tac x=K in exI)
unfolding vec1_dest_vec1_simps by (auto simp add:field_simps) }
thus ?thesis unfolding linear_def bounded_linear_def additive_def bounded_linear_axioms_def o_def
unfolding vec1_dest_vec1_simps by auto qed
lemma has_derivative_within_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows
"((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s)
= (f has_derivative f') (at x within s)"
unfolding has_derivative_within unfolding bounded_linear_vec1_dest_vec1[unfolded linear_conv_bounded_linear]
unfolding o_def Lim_within Ball_def unfolding forall_vec1
unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto
lemma has_derivative_at_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows
"((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
using has_derivative_within_vec1_dest_vec1[where s=UNIV, unfolded range_vec1 within_UNIV] by auto
lemma bounded_linear_vec1: fixes f::"'a::real_normed_vector\<Rightarrow>real"
shows "bounded_linear f = bounded_linear (vec1 \<circ> f)"
unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
unfolding vec1_dest_vec1_simps by auto
lemma bounded_linear_dest_vec1: fixes f::"real\<Rightarrow>'a::real_normed_vector"
shows "bounded_linear f = bounded_linear (f \<circ> dest_vec1)"
unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
unfolding vec1_dest_vec1_simps by auto
lemma has_derivative_at_vec1: fixes f::"'a::real_normed_vector\<Rightarrow>real" shows
"(f has_derivative f') (at x) = ((vec1 \<circ> f) has_derivative (vec1 \<circ> f')) (at x)"
unfolding has_derivative_at unfolding bounded_linear_vec1[unfolded linear_conv_bounded_linear]
unfolding o_def Lim_at unfolding vec1_dest_vec1_simps dist_vec1_0 by auto
lemma has_derivative_within_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows
"((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s) = (f has_derivative f') (at x within s)"
unfolding has_derivative_within bounded_linear_dest_vec1 unfolding o_def Lim_within Ball_def
unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto
lemma has_derivative_at_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows
"((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
using has_derivative_within_dest_vec1[where s=UNIV] by(auto simp add:within_UNIV)
lemma derivative_is_linear: fixes f::"real^'a \<Rightarrow> real^'b" shows
"(f has_derivative f') net \<Longrightarrow> linear f'"
unfolding has_derivative_def and linear_conv_bounded_linear by auto
subsection {* Combining theorems. *}
lemma (in bounded_linear) has_derivative: "(f has_derivative f) net"
unfolding has_derivative_def apply(rule,rule bounded_linear_axioms)
unfolding diff by(simp add: Lim_const)
lemma has_derivative_id: "((\<lambda>x. x) has_derivative (\<lambda>h. h)) net"
apply(rule bounded_linear.has_derivative) using bounded_linear_ident[unfolded id_def] by simp
lemma has_derivative_const: "((\<lambda>x. c) has_derivative (\<lambda>h. 0)) net"
unfolding has_derivative_def apply(rule,rule bounded_linear_zero) by(simp add: Lim_const)
lemma (in bounded_linear) cmul: shows "bounded_linear (\<lambda>x. (c::real) *\<^sub>R f x)" proof
guess K using pos_bounded ..
thus "\<exists>K. \<forall>x. norm ((c::real) *\<^sub>R f x) \<le> norm x * K" apply(rule_tac x="abs c * K" in exI) proof
fix x case goal1
hence "abs c * norm (f x) \<le> abs c * (norm x * K)" apply-apply(erule conjE,erule_tac x=x in allE)
apply(rule mult_left_mono) by auto
thus ?case by(auto simp add:field_simps)
qed qed(auto simp add: scaleR.add_right add scaleR)
lemma has_derivative_cmul: assumes "(f has_derivative f') net" shows "((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net"
unfolding has_derivative_def apply(rule,rule bounded_linear.cmul)
using assms[unfolded has_derivative_def] using Lim_cmul[OF assms[unfolded has_derivative_def,THEN conjunct2]]
unfolding scaleR_right_diff_distrib scaleR_right_distrib by auto
lemma has_derivative_cmul_eq: assumes "c \<noteq> 0"
shows "(((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net \<longleftrightarrow> (f has_derivative f') net)"
apply(rule) defer apply(rule has_derivative_cmul,assumption)
apply(drule has_derivative_cmul[where c="1/c"]) using assms by auto
lemma has_derivative_neg:
"(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net"
apply(drule has_derivative_cmul[where c="-1"]) by auto
lemma has_derivative_neg_eq: "((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net \<longleftrightarrow> (f has_derivative f') net"
apply(rule, drule_tac[!] has_derivative_neg) by auto
lemma has_derivative_add: assumes "(f has_derivative f') net" "(g has_derivative g') net"
shows "((\<lambda>x. f(x) + g(x)) has_derivative (\<lambda>h. f'(h) + g'(h))) net" proof-
note as = assms[unfolded has_derivative_def]
show ?thesis unfolding has_derivative_def apply(rule,rule bounded_linear_add)
using Lim_add[OF as(1)[THEN conjunct2] as(2)[THEN conjunct2]] and as
by(auto simp add:group_simps scaleR_right_diff_distrib scaleR_right_distrib) qed
lemma has_derivative_add_const:"(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
apply(drule has_derivative_add) apply(rule has_derivative_const) by auto
lemma has_derivative_sub:
"(f has_derivative f') net \<Longrightarrow> (g has_derivative g') net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_derivative (\<lambda>h. f'(h) - g'(h))) net"
apply(drule has_derivative_add) apply(drule has_derivative_neg,assumption) by(simp add:group_simps)
lemma has_derivative_setsum: assumes "finite s" "\<forall>a\<in>s. ((f a) has_derivative (f' a)) net"
shows "((\<lambda>x. setsum (\<lambda>a. f a x) s) has_derivative (\<lambda>h. setsum (\<lambda>a. f' a h) s)) net"
apply(induct_tac s rule:finite_subset_induct[where A=s]) apply(rule assms(1))
proof- fix x F assume as:"finite F" "x \<notin> F" "x\<in>s" "((\<lambda>x. \<Sum>a\<in>F. f a x) has_derivative (\<lambda>h. \<Sum>a\<in>F. f' a h)) net"
thus "((\<lambda>xa. \<Sum>a\<in>insert x F. f a xa) has_derivative (\<lambda>h. \<Sum>a\<in>insert x F. f' a h)) net"
unfolding setsum_insert[OF as(1,2)] apply-apply(rule has_derivative_add) apply(rule assms(2)[rule_format]) by auto
qed(auto intro!: has_derivative_const)
lemma has_derivative_setsum_numseg:
"\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> ((f i) has_derivative (f' i)) net \<Longrightarrow>
((\<lambda>x. setsum (\<lambda>i. f i x) {m..n::nat}) has_derivative (\<lambda>h. setsum (\<lambda>i. f' i h) {m..n})) net"
apply(rule has_derivative_setsum) by auto
subsection {* somewhat different results for derivative of scalar multiplier. *}
lemma has_derivative_vmul_component: fixes c::"real^'a \<Rightarrow> real^'b" and v::"real^'c"
assumes "(c has_derivative c') net"
shows "((\<lambda>x. c(x)$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$k *\<^sub>R v)) net" proof-
have *:"\<And>y. (c y $ k *\<^sub>R v - (c (netlimit net) $ k *\<^sub>R v + c' (y - netlimit net) $ k *\<^sub>R v)) =
(c y $ k - (c (netlimit net) $ k + c' (y - netlimit net) $ k)) *\<^sub>R v"
unfolding scaleR_left_diff_distrib scaleR_left_distrib by auto
show ?thesis unfolding has_derivative_def and * and linear_conv_bounded_linear[symmetric]
apply(rule,rule linear_vmul_component[of c' k v, unfolded smult_conv_scaleR]) defer
apply(subst vector_smult_lzero[THEN sym, of v]) unfolding scaleR_scaleR smult_conv_scaleR apply(rule Lim_vmul)
using assms[unfolded has_derivative_def] unfolding Lim o_def apply- apply(cases "trivial_limit net")
apply(rule,assumption,rule disjI2,rule,rule) proof-
have *:"\<And>x. x - vec 0 = (x::real^'n)" by auto
have **:"\<And>d x. d * (c x $ k - (c (netlimit net) $ k + c' (x - netlimit net) $ k)) = (d *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net) ))) $k" by(auto simp add:field_simps)
fix e assume "\<not> trivial_limit net" "0 < (e::real)"
then have "eventually (\<lambda>x. dist ((1 / norm (x - netlimit net)) *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net)))) 0 < e) net"
using assms[unfolded has_derivative_def Lim] by auto
thus "eventually (\<lambda>x. dist (1 / norm (x - netlimit net) * (c x $ k - (c (netlimit net) $ k + c' (x - netlimit net) $ k))) 0 < e) net"
proof (rule eventually_elim1)
case goal1 thus ?case apply - unfolding vector_dist_norm apply(rule le_less_trans) prefer 2 apply assumption unfolding * ** and norm_vec1
using component_le_norm[of "(1 / norm (x - netlimit net)) *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net))) - 0" k] by auto
qed
qed(insert assms[unfolded has_derivative_def], auto simp add:linear_conv_bounded_linear) qed
lemma has_derivative_vmul_within: fixes c::"real \<Rightarrow> real" and v::"real^'a"
assumes "(c has_derivative c') (at x within s)"
shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x within s)" proof-
have *:"\<And>c. (\<lambda>x. (vec1 \<circ> c \<circ> dest_vec1) x $ 1 *\<^sub>R v) = (\<lambda>x. (c x) *\<^sub>R v) \<circ> dest_vec1" unfolding o_def by auto
show ?thesis using has_derivative_vmul_component[of "vec1 \<circ> c \<circ> dest_vec1" "vec1 \<circ> c' \<circ> dest_vec1" "at (vec1 x) within vec1 ` s" 1 v]
unfolding * and has_derivative_within_vec1_dest_vec1 unfolding has_derivative_within_dest_vec1 using assms by auto qed
lemma has_derivative_vmul_at: fixes c::"real \<Rightarrow> real" and v::"real^'a"
assumes "(c has_derivative c') (at x)"
shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x)"
using has_derivative_vmul_within[where s=UNIV] and assms by(auto simp add: within_UNIV)
lemma has_derivative_lift_dot:
assumes "(f has_derivative f') net"
shows "((\<lambda>x. inner v (f x)) has_derivative (\<lambda>t. inner v (f' t))) net" proof-
show ?thesis using assms unfolding has_derivative_def apply- apply(erule conjE,rule)
apply(rule bounded_linear_compose[of _ f']) apply(rule inner.bounded_linear_right,assumption)
apply(drule Lim_inner[where a=v]) unfolding o_def
by(auto simp add:inner.scaleR_right inner.add_right inner.diff_right) qed
lemmas has_derivative_intros = has_derivative_sub has_derivative_add has_derivative_cmul has_derivative_id has_derivative_const
has_derivative_neg has_derivative_vmul_component has_derivative_vmul_at has_derivative_vmul_within has_derivative_cmul
bounded_linear.has_derivative has_derivative_lift_dot
subsection {* limit transformation for derivatives. *}
lemma has_derivative_transform_within:
assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x within s)"
shows "(g has_derivative f') (at x within s)"
using assms(4) unfolding has_derivative_within apply- apply(erule conjE,rule,assumption)
apply(rule Lim_transform_within[OF assms(1)]) defer apply assumption
apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
lemma has_derivative_transform_at:
assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x)"
shows "(g has_derivative f') (at x)"
apply(subst within_UNIV[THEN sym]) apply(rule has_derivative_transform_within[OF assms(1)])
using assms(2-3) unfolding within_UNIV by auto
lemma has_derivative_transform_within_open:
assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_derivative f') (at x)"
shows "(g has_derivative f') (at x)"
using assms(4) unfolding has_derivative_at apply- apply(erule conjE,rule,assumption)
apply(rule Lim_transform_within_open[OF assms(1,2)]) defer apply assumption
apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
subsection {* differentiability. *}
definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "differentiable" 30) where
"f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"
definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
"f differentiable_on s \<equiv> (\<forall>x\<in>s. f differentiable (at x within s))"
lemma differentiableI: "(f has_derivative f') net \<Longrightarrow> f differentiable net"
unfolding differentiable_def by auto
lemma differentiable_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
unfolding differentiable_def using has_derivative_at_within by blast
lemma differentiable_within_open: assumes "a \<in> s" "open s" shows
"f differentiable (at a within s) \<longleftrightarrow> (f differentiable (at a))"
unfolding differentiable_def has_derivative_within_open[OF assms] by auto
lemma differentiable_at_imp_differentiable_on: "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI)
lemma differentiable_on_eq_differentiable_at: "open s \<Longrightarrow> (f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x))"
unfolding differentiable_on_def by(auto simp add: differentiable_within_open)
lemma differentiable_transform_within:
assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable (at x within s)"
shows "g differentiable (at x within s)"
using assms(4) unfolding differentiable_def by(auto intro!: has_derivative_transform_within[OF assms(1-3)])
lemma differentiable_transform_at:
assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable at x"
shows "g differentiable at x"
using assms(3) unfolding differentiable_def using has_derivative_transform_at[OF assms(1-2)] by auto
subsection {* Frechet derivative and Jacobian matrix. *}
definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
lemma frechet_derivative_works:
"f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
unfolding frechet_derivative_def differentiable_def and some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..
lemma linear_frechet_derivative: fixes f::"real^'a \<Rightarrow> real^'b"
shows "f differentiable net \<Longrightarrow> linear(frechet_derivative f net)"
unfolding frechet_derivative_works has_derivative_def unfolding linear_conv_bounded_linear by auto
definition "jacobian f net = matrix(frechet_derivative f net)"
lemma jacobian_works: "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow> (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
apply rule unfolding jacobian_def apply(simp only: matrix_works[OF linear_frechet_derivative]) defer
apply(rule differentiableI) apply assumption unfolding frechet_derivative_works by assumption
subsection {* Differentiability implies continuity. *}
lemma Lim_mul_norm_within: fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f(x)) ---> 0) (at a within s)"
unfolding Lim_within apply(rule,rule) apply(erule_tac x=e in allE,erule impE,assumption,erule exE,erule conjE)
apply(rule_tac x="min d 1" in exI) apply rule defer apply(rule,erule_tac x=x in ballE) unfolding vector_dist_norm diff_0_right norm_mul
by(auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])
lemma differentiable_imp_continuous_within: assumes "f differentiable (at x within s)"
shows "continuous (at x within s) f" proof-
from assms guess f' unfolding differentiable_def has_derivative_within .. note f'=this
then interpret bounded_linear f' by auto
have *:"\<And>xa. x\<noteq>xa \<Longrightarrow> (f' \<circ> (\<lambda>y. y - x)) xa + norm (xa - x) *\<^sub>R ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) - ((f' \<circ> (\<lambda>y. y - x)) x + 0) = f xa - f x"
using zero by auto
have **:"continuous (at x within s) (f' \<circ> (\<lambda>y. y - x))"
apply(rule continuous_within_compose) apply(rule continuous_intros)+
by(rule linear_continuous_within[OF f'[THEN conjunct1]])
show ?thesis unfolding continuous_within using f'[THEN conjunct2, THEN Lim_mul_norm_within]
apply-apply(drule Lim_add) apply(rule **[unfolded continuous_within]) unfolding Lim_within and vector_dist_norm
apply(rule,rule) apply(erule_tac x=e in allE) apply(erule impE|assumption)+ apply(erule exE,rule_tac x=d in exI)
by(auto simp add:zero * elim!:allE) qed
lemma differentiable_imp_continuous_at: "f differentiable at x \<Longrightarrow> continuous (at x) f"
by(rule differentiable_imp_continuous_within[of _ x UNIV, unfolded within_UNIV])
lemma differentiable_imp_continuous_on: "f differentiable_on s \<Longrightarrow> continuous_on s f"
unfolding differentiable_on_def continuous_on_eq_continuous_within
using differentiable_imp_continuous_within by blast
lemma has_derivative_within_subset:
"(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
unfolding has_derivative_within using Lim_within_subset by blast
lemma differentiable_within_subset:
"f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable (at x within s)"
unfolding differentiable_def using has_derivative_within_subset by blast
lemma differentiable_on_subset: "f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
unfolding differentiable_on_def using differentiable_within_subset by blast
lemma differentiable_on_empty: "f differentiable_on {}"
unfolding differentiable_on_def by auto
subsection {* Several results are easier using a "multiplied-out" variant. *)
(* (I got this idea from Dieudonne's proof of the chain rule). *}
lemma has_derivative_within_alt:
"(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
(\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm(y - x) < d \<longrightarrow> norm(f(y) - f(x) - f'(y - x)) \<le> e * norm(y - x))" (is "?lhs \<longleftrightarrow> ?rhs")
proof assume ?lhs thus ?rhs unfolding has_derivative_within apply-apply(erule conjE,rule,assumption)
unfolding Lim_within apply(rule,erule_tac x=e in allE,rule,erule impE,assumption)
apply(erule exE,rule_tac x=d in exI) apply(erule conjE,rule,assumption,rule,rule) proof-
fix x y e d assume as:"0 < e" "0 < d" "norm (y - x) < d" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e" "y \<in> s" "bounded_linear f'"
then interpret bounded_linear f' by auto
show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)" proof(cases "y=x")
case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero) next
case False hence "norm (f y - (f x + f' (y - x))) < e * norm (y - x)" using as(4)[rule_format, OF `y\<in>s`]
unfolding vector_dist_norm diff_0_right norm_mul using as(3)
using pos_divide_less_eq[OF False[unfolded dist_nz], unfolded vector_dist_norm]
by(auto simp add:linear_0 linear_sub group_simps)
thus ?thesis by(auto simp add:group_simps) qed qed next
assume ?rhs thus ?lhs unfolding has_derivative_within Lim_within apply-apply(erule conjE,rule,assumption)
apply(rule,erule_tac x="e/2" in allE,rule,erule impE) defer apply(erule exE,rule_tac x=d in exI)
apply(erule conjE,rule,assumption,rule,rule) unfolding vector_dist_norm diff_0_right norm_scaleR
apply(erule_tac x=xa in ballE,erule impE) proof-
fix e d y assume "bounded_linear f'" "0 < e" "0 < d" "y \<in> s" "0 < norm (y - x) \<and> norm (y - x) < d"
"norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
thus "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
apply(rule_tac le_less_trans[of _ "e/2"]) by(auto intro!:mult_imp_div_pos_le simp add:group_simps) qed auto qed
lemma has_derivative_at_alt:
"(f has_derivative f') (at (x::real^'n)) \<longleftrightarrow> bounded_linear f' \<and>
(\<forall>e>0. \<exists>d>0. \<forall>y. norm(y - x) < d \<longrightarrow> norm(f y - f x - f'(y - x)) \<le> e * norm(y - x))"
using has_derivative_within_alt[where s=UNIV] unfolding within_UNIV by auto
subsection {* The chain rule. *}
lemma diff_chain_within:
assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at (f x) within (f ` s))"
shows "((g o f) has_derivative (g' o f'))(at x within s)"
unfolding has_derivative_within_alt apply(rule,rule bounded_linear_compose[unfolded o_def[THEN sym]])
apply(rule assms(2)[unfolded has_derivative_def,THEN conjE],assumption)
apply(rule assms(1)[unfolded has_derivative_def,THEN conjE],assumption) proof(rule,rule)
note assms = assms[unfolded has_derivative_within_alt]
fix e::real assume "0<e"
guess B1 using bounded_linear.pos_bounded[OF assms(1)[THEN conjunct1, rule_format]] .. note B1 = this
guess B2 using bounded_linear.pos_bounded[OF assms(2)[THEN conjunct1, rule_format]] .. note B2 = this
have *:"e / 2 / B2 > 0" using `e>0` B2 apply-apply(rule divide_pos_pos) by auto
guess d1 using assms(1)[THEN conjunct2, rule_format, OF *] .. note d1 = this
have *:"e / 2 / (1 + B1) > 0" using `e>0` B1 apply-apply(rule divide_pos_pos) by auto
guess de using assms(2)[THEN conjunct2, rule_format, OF *] .. note de = this
guess d2 using assms(1)[THEN conjunct2, rule_format, OF zero_less_one] .. note d2 = this
def d0 \<equiv> "(min d1 d2)/2" have d0:"d0>0" "d0 < d1" "d0 < d2" unfolding d0_def using d1 d2 by auto
def d \<equiv> "(min d0 (de / (B1 + 1))) / 2" have "de * 2 / (B1 + 1) > de / (B1 + 1)" apply(rule divide_strict_right_mono) using B1 de by auto
hence d:"d>0" "d < d1" "d < d2" "d < (de / (B1 + 1))" unfolding d_def using d0 d1 d2 de B1 by(auto intro!: divide_pos_pos simp add:min_less_iff_disj not_less)
show "\<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)" apply(rule_tac x=d in exI)
proof(rule,rule `d>0`,rule,rule)
fix y assume as:"y \<in> s" "norm (y - x) < d"
hence 1:"norm (f y - f x - f' (y - x)) \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x))" using d1 d2 d by auto
have "norm (f y - f x) \<le> norm (f y - f x - f' (y - x)) + norm (f' (y - x))"
using norm_triangle_sub[of "f y - f x" "f' (y - x)"] by(auto simp add:group_simps)
also have "\<dots> \<le> norm (f y - f x - f' (y - x)) + B1 * norm (y - x)" apply(rule add_left_mono) using B1 by(auto simp add:group_simps)
also have "\<dots> \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x)) + B1 * norm (y - x)" apply(rule add_right_mono) using d1 d2 d as by auto
also have "\<dots> \<le> norm (y - x) + B1 * norm (y - x)" by auto
also have "\<dots> = norm (y - x) * (1 + B1)" by(auto simp add:field_simps)
finally have 3:"norm (f y - f x) \<le> norm (y - x) * (1 + B1)" by auto
hence "norm (f y - f x) \<le> d * (1 + B1)" apply- apply(rule order_trans,assumption,rule mult_right_mono) using as B1 by auto
also have "\<dots> < de" using d B1 by(auto simp add:field_simps)
finally have "norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 / (1 + B1) * norm (f y - f x)"
apply-apply(rule de[THEN conjunct2,rule_format]) using `y\<in>s` using d as by auto
also have "\<dots> = (e / 2) * (1 / (1 + B1) * norm (f y - f x))" by auto
also have "\<dots> \<le> e / 2 * norm (y - x)" apply(rule mult_left_mono) using `e>0` and 3 using B1 and `e>0` by(auto simp add:divide_le_eq)
finally have 4:"norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
interpret g': bounded_linear g' using assms(2) by auto
interpret f': bounded_linear f' using assms(1) by auto
have "norm (- g' (f' (y - x)) + g' (f y - f x)) = norm (g' (f y - f x - f' (y - x)))"
by(auto simp add:group_simps f'.diff g'.diff g'.add)
also have "\<dots> \<le> B2 * norm (f y - f x - f' (y - x))" using B2 by(auto simp add:group_simps)
also have "\<dots> \<le> B2 * (e / 2 / B2 * norm (y - x))" apply(rule mult_left_mono) using as d1 d2 d B2 by auto
also have "\<dots> \<le> e / 2 * norm (y - x)" using B2 by auto
finally have 5:"norm (- g' (f' (y - x)) + g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
have "norm (g (f y) - g (f x) - g' (f y - f x)) + norm (g (f y) - g (f x) - g' (f' (y - x)) - (g (f y) - g (f x) - g' (f y - f x))) \<le> e * norm (y - x)" using 5 4 by auto
thus "norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)" unfolding o_def apply- apply(rule order_trans, rule norm_triangle_sub) by assumption qed qed
lemma diff_chain_at:
"(f has_derivative f') (at x) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow> ((g o f) has_derivative (g' o f')) (at x)"
using diff_chain_within[of f f' x UNIV g g'] using has_derivative_within_subset[of g g' "f x" UNIV "range f"] unfolding within_UNIV by auto
subsection {* Composition rules stated just for differentiability. *}
lemma differentiable_const[intro]: "(\<lambda>z. c) differentiable (net::'a::real_normed_vector net)"
unfolding differentiable_def using has_derivative_const by auto
lemma differentiable_id[intro]: "(\<lambda>z. z) differentiable (net::'a::real_normed_vector net)"
unfolding differentiable_def using has_derivative_id by auto
lemma differentiable_cmul[intro]: "f differentiable net \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector net)"
unfolding differentiable_def apply(erule exE, drule has_derivative_cmul) by auto
lemma differentiable_neg[intro]: "f differentiable net \<Longrightarrow> (\<lambda>z. -(f z)) differentiable (net::'a::real_normed_vector net)"
unfolding differentiable_def apply(erule exE, drule has_derivative_neg) by auto
lemma differentiable_add: "f differentiable net \<Longrightarrow> g differentiable net
\<Longrightarrow> (\<lambda>z. f z + g z) differentiable (net::'a::real_normed_vector net)"
unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z + f'a z" in exI)
apply(rule has_derivative_add) by auto
lemma differentiable_sub: "f differentiable net \<Longrightarrow> g differentiable net
\<Longrightarrow> (\<lambda>z. f z - g z) differentiable (net::'a::real_normed_vector net)"
unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z - f'a z" in exI)
apply(rule has_derivative_sub) by auto
lemma differentiable_setsum: fixes f::"'a \<Rightarrow> (real^'n \<Rightarrow>real^'n)"
assumes "finite s" "\<forall>a\<in>s. (f a) differentiable net"
shows "(\<lambda>x. setsum (\<lambda>a. f a x) s) differentiable net" proof-
guess f' using bchoice[OF assms(2)[unfolded differentiable_def]] ..
thus ?thesis unfolding differentiable_def apply- apply(rule,rule has_derivative_setsum[where f'=f'],rule assms(1)) by auto qed
lemma differentiable_setsum_numseg: fixes f::"_ \<Rightarrow> (real^'n \<Rightarrow>real^'n)"
shows "\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> (f i) differentiable net \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) {m::nat..n}) differentiable net"
apply(rule differentiable_setsum) using finite_atLeastAtMost[of n m] by auto
lemma differentiable_chain_at:
"f differentiable (at x) \<Longrightarrow> g differentiable (at(f x)) \<Longrightarrow> (g o f) differentiable (at x)"
unfolding differentiable_def by(meson diff_chain_at)
lemma differentiable_chain_within:
"f differentiable (at x within s) \<Longrightarrow> g differentiable (at(f x) within (f ` s))
\<Longrightarrow> (g o f) differentiable (at x within s)"
unfolding differentiable_def by(meson diff_chain_within)
subsection {* Uniqueness of derivative. *)
(* *)
(* The general result is a bit messy because we need approachability of the *)
(* limit point from any direction. But OK for nontrivial intervals etc. *}
lemma frechet_derivative_unique_within: fixes f::"real^'a \<Rightarrow> real^'b"
assumes "(f has_derivative f') (at x within s)" "(f has_derivative f'') (at x within s)"
"(\<forall>i::'a::finite. \<forall>e>0. \<exists>d. 0 < abs(d) \<and> abs(d) < e \<and> (x + d *\<^sub>R basis i) \<in> s)" shows "f' = f''" proof-
note as = assms(1,2)[unfolded has_derivative_def]
then interpret f': bounded_linear f' by auto from as interpret f'': bounded_linear f'' by auto
have "x islimpt s" unfolding islimpt_approachable proof(rule,rule)
guess a using UNIV_witness[where 'a='a] ..
fix e::real assume "0<e" guess d using assms(3)[rule_format,OF`e>0`,of a] ..
thus "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x="x + d*\<^sub>R basis a" in bexI)
using basis_nonzero[of a] norm_basis[of a] unfolding vector_dist_norm by auto qed
hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within) unfolding trivial_limit_within by simp
show ?thesis apply(rule linear_eq_stdbasis) unfolding linear_conv_bounded_linear
apply(rule as(1,2)[THEN conjunct1])+ proof(rule,rule ccontr)
fix i::'a def e \<equiv> "norm (f' (basis i) - f'' (basis i))"
assume "f' (basis i) \<noteq> f'' (basis i)" hence "e>0" unfolding e_def by auto
guess d using Lim_sub[OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
guess c using assms(3)[rule_format,OF d[THEN conjunct1],of i] .. note c=this
have *:"norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R basis i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R basis i)) = norm ((1 / abs c) *\<^sub>R (- (f' (c *\<^sub>R basis i)) + f'' (c *\<^sub>R basis i)))"
unfolding scaleR_right_distrib by auto
also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' (basis i)) + f'' (basis i))))"
unfolding f'.scaleR f''.scaleR unfolding scaleR_right_distrib scaleR_minus_right by auto
also have "\<dots> = e" unfolding e_def norm_mul using c[THEN conjunct1] using norm_minus_cancel[of "f' (basis i) - f'' (basis i)"] by(auto simp add:group_simps)
finally show False using c using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R basis i"] using norm_basis[of i] unfolding vector_dist_norm
unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib by auto qed qed
lemma frechet_derivative_unique_at: fixes f::"real^'a \<Rightarrow> real^'b"
shows "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
apply(rule frechet_derivative_unique_within[of f f' x UNIV f'']) unfolding within_UNIV apply(assumption)+
apply(rule,rule,rule) apply(rule_tac x="e/2" in exI) by auto
lemma "isCont f x = continuous (at x) f" unfolding isCont_def LIM_def
unfolding continuous_at Lim_at unfolding dist_nz by auto
lemma frechet_derivative_unique_within_closed_interval: fixes f::"real^'a \<Rightarrow> real^'b"
assumes "\<forall>i. a$i < b$i" "x \<in> {a..b}" (is "x\<in>?I") and
"(f has_derivative f' ) (at x within {a..b})" and
"(f has_derivative f'') (at x within {a..b})"
shows "f' = f''" apply(rule frechet_derivative_unique_within) apply(rule assms(3,4))+ proof(rule,rule,rule)
fix e::real and i::'a assume "e>0"
thus "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R basis i \<in> {a..b}" proof(cases "x$i=a$i")
case True thus ?thesis apply(rule_tac x="(min (b$i - a$i) e) / 2" in exI)
using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
unfolding mem_interval by(auto simp add:field_simps) next
note * = assms(2)[unfolded mem_interval,THEN spec[where x=i]]
case False moreover have "a $ i < x $ i" using False * by auto
moreover { have "a $ i * 2 + min (x $ i - a $ i) e \<le> a$i *2 + x$i - a$i" by auto
also have "\<dots> = a$i + x$i" by auto also have "\<dots> \<le> 2 * x$i" using * by auto
finally have "a $ i * 2 + min (x $ i - a $ i) e \<le> x $ i * 2" by auto }
moreover have "min (x $ i - a $ i) e \<ge> 0" using * and `e>0` by auto
hence "x $ i * 2 \<le> b $ i * 2 + min (x $ i - a $ i) e" using * by auto
ultimately show ?thesis apply(rule_tac x="- (min (x$i - a$i) e) / 2" in exI)
using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
unfolding mem_interval by(auto simp add:field_simps) qed qed
lemma frechet_derivative_unique_within_open_interval: fixes f::"real^'a \<Rightarrow> real^'b"
assumes "x \<in> {a<..<b}" "(f has_derivative f' ) (at x within {a<..<b})"
"(f has_derivative f'') (at x within {a<..<b})"
shows "f' = f''" apply(rule frechet_derivative_unique_within) apply(rule assms(2-3))+ proof(rule,rule,rule)
fix e::real and i::'a assume "e>0"
note * = assms(1)[unfolded mem_interval,THEN spec[where x=i]]
have "a $ i < x $ i" using * by auto
moreover { have "a $ i * 2 + min (x $ i - a $ i) e \<le> a$i *2 + x$i - a$i" by auto
also have "\<dots> = a$i + x$i" by auto also have "\<dots> < 2 * x$i" using * by auto
finally have "a $ i * 2 + min (x $ i - a $ i) e < x $ i * 2" by auto }
moreover have "min (x $ i - a $ i) e \<ge> 0" using * and `e>0` by auto
hence "x $ i * 2 < b $ i * 2 + min (x $ i - a $ i) e" using * by auto
ultimately show "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R basis i \<in> {a<..<b}"
apply(rule_tac x="- (min (x$i - a$i) e) / 2" in exI)
using `e>0` and assms(1) unfolding mem_interval by(auto simp add:field_simps) qed
lemma frechet_derivative_at: fixes f::"real^'a \<Rightarrow> real^'b"
shows "(f has_derivative f') (at x) \<Longrightarrow> (f' = frechet_derivative f (at x))"
apply(rule frechet_derivative_unique_at[of f],assumption)
unfolding frechet_derivative_works[THEN sym] using differentiable_def by auto
lemma frechet_derivative_within_closed_interval: fixes f::"real^'a \<Rightarrow> real^'b"
assumes "\<forall>i. a$i < b$i" "x \<in> {a..b}" "(f has_derivative f') (at x within {a.. b})"
shows "frechet_derivative f (at x within {a.. b}) = f'"
apply(rule frechet_derivative_unique_within_closed_interval[where f=f])
apply(rule assms(1,2))+ unfolding frechet_derivative_works[THEN sym]
unfolding differentiable_def using assms(3) by auto
subsection {* Component of the differential must be zero if it exists at a local *)
(* maximum or minimum for that corresponding component. *}
lemma differential_zero_maxmin_component: fixes f::"real^'a \<Rightarrow> real^'b"
assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
"f differentiable (at x)" shows "jacobian f (at x) $ k = 0" proof(rule ccontr)
def D \<equiv> "jacobian f (at x)" assume "jacobian f (at x) $ k \<noteq> 0"
then obtain j where j:"D$k$j \<noteq> 0" unfolding Cart_eq D_def by auto
hence *:"abs (jacobian f (at x) $ k $ j) / 2 > 0" unfolding D_def by auto
note as = assms(3)[unfolded jacobian_works has_derivative_at_alt]
guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this
guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this
{ fix c assume "abs c \<le> d"
hence *:"norm (x + c *\<^sub>R basis j - x) < e'" using norm_basis[of j] d by auto
have "\<bar>(f (x + c *\<^sub>R basis j) - f x - D *v (c *\<^sub>R basis j)) $ k\<bar> \<le> norm (f (x + c *\<^sub>R basis j) - f x - D *v (c *\<^sub>R basis j))" by(rule component_le_norm)
also have "\<dots> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j] unfolding D_def[symmetric] by auto
finally have "\<bar>(f (x + c *\<^sub>R basis j) - f x - D *v (c *\<^sub>R basis j)) $ k\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" by simp
hence "\<bar>f (x + c *\<^sub>R basis j) $ k - f x $ k - c * D $ k $ j\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"
unfolding vector_component_simps matrix_vector_mul_component unfolding smult_conv_scaleR[symmetric]
unfolding dot_rmult dot_basis unfolding smult_conv_scaleR by simp } note * = this
have "x + d *\<^sub>R basis j \<in> ball x e" "x - d *\<^sub>R basis j \<in> ball x e"
unfolding mem_ball vector_dist_norm using norm_basis[of j] d by auto
hence **:"((f (x - d *\<^sub>R basis j))$k \<le> (f x)$k \<and> (f (x + d *\<^sub>R basis j))$k \<le> (f x)$k) \<or>
((f (x - d *\<^sub>R basis j))$k \<ge> (f x)$k \<and> (f (x + d *\<^sub>R basis j))$k \<ge> (f x)$k)" using assms(2) by auto
have ***:"\<And>y y1 y2 d dx::real. (y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow> d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
show False apply(rule ***[OF **, where dx="d * D $ k $ j" and d="\<bar>D $ k $ j\<bar> / 2 * \<bar>d\<bar>"])
using *[of "-d"] and *[of d] and d[THEN conjunct1] and j unfolding mult_minus_left
unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding group_simps by (auto intro: mult_pos_pos)
qed
subsection {* In particular if we have a mapping into R^1. *}
lemma differential_zero_maxmin: fixes f::"real^'a \<Rightarrow> real"
assumes "x \<in> s" "open s" "(f has_derivative f') (at x)"
"(\<forall>y\<in>s. f y \<le> f x) \<or> (\<forall>y\<in>s. f x \<le> f y)"
shows "f' = (\<lambda>v. 0)" proof-
note deriv = assms(3)[unfolded has_derivative_at_vec1]
obtain e where e:"e>0" "ball x e \<subseteq> s" using assms(2)[unfolded open_contains_ball] and assms(1) by auto
hence **:"(jacobian (vec1 \<circ> f) (at x)) $ 1 = 0" using differential_zero_maxmin_component[of e x "\<lambda>x. vec1 (f x)" 1]
using assms(4) and assms(3)[unfolded has_derivative_at_vec1 o_def]
unfolding differentiable_def o_def by auto
have *:"jacobian (vec1 \<circ> f) (at x) = matrix (vec1 \<circ> f')" unfolding jacobian_def and frechet_derivative_at[OF deriv] ..
have "vec1 \<circ> f' = (\<lambda>x. 0)" apply(rule) unfolding matrix_works[OF derivative_is_linear[OF deriv],THEN sym]
unfolding Cart_eq matrix_vector_mul_component using **[unfolded *] by auto
thus ?thesis apply-apply(rule,subst vec1_eq[THEN sym]) unfolding o_def apply(drule fun_cong) by auto qed
subsection {* The traditional Rolle theorem in one dimension. *}
lemma vec1_le[simp]:fixes a::real shows "vec1 a \<le> vec1 b \<longleftrightarrow> a \<le> b"
unfolding vector_le_def by auto
lemma vec1_less[simp]:fixes a::real shows "vec1 a < vec1 b \<longleftrightarrow> a < b"
unfolding vector_less_def by auto
lemma rolle: fixes f::"real\<Rightarrow>real"
assumes "a < b" "f a = f b" "continuous_on {a..b} f"
"\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
shows "\<exists>x\<in>{a<..<b}. f' x = (\<lambda>v. 0)" proof-
have "\<exists>x\<in>{a<..<b}. ((\<forall>y\<in>{a<..<b}. f x \<le> f y) \<or> (\<forall>y\<in>{a<..<b}. f y \<le> f x))" proof-
have "(a + b) / 2 \<in> {a .. b}" using assms(1) by auto hence *:"{a .. b}\<noteq>{}" by auto
guess d using continuous_attains_sup[OF compact_real_interval * assms(3)] .. note d=this
guess c using continuous_attains_inf[OF compact_real_interval * assms(3)] .. note c=this
show ?thesis proof(cases "d\<in>{a<..<b} \<or> c\<in>{a<..<b}")
case True thus ?thesis apply(erule_tac disjE) apply(rule_tac x=d in bexI)
apply(rule_tac[3] x=c in bexI) using d c by auto next def e \<equiv> "(a + b) /2"
case False hence "f d = f c" using d c assms(2) by auto
hence "\<And>x. x\<in>{a..b} \<Longrightarrow> f x = f d" using c d apply- apply(erule_tac x=x in ballE)+ by auto
thus ?thesis apply(rule_tac x=e in bexI) unfolding e_def using assms(1) by auto qed qed
then guess x .. note x=this
hence "f' x \<circ> dest_vec1 = (\<lambda>v. 0)" apply(rule_tac differential_zero_maxmin[of "vec1 x" "vec1 ` {a<..<b}" "f \<circ> dest_vec1" "(f' x) \<circ> dest_vec1"])
unfolding vec1_interval defer apply(rule open_interval)
apply(rule assms(4)[unfolded has_derivative_at_dest_vec1[THEN sym],THEN bspec[where x=x]],assumption)
unfolding o_def apply(erule disjE,rule disjI2) by(auto simp add: vector_less_def)
thus ?thesis apply(rule_tac x=x in bexI) unfolding o_def apply rule
apply(drule_tac x="vec1 v" in fun_cong) unfolding vec1_dest_vec1 using x(1) by auto qed
subsection {* One-dimensional mean value theorem. *}
lemma mvt: fixes f::"real \<Rightarrow> real"
assumes "a < b" "continuous_on {a .. b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative (f' x)) (at x)"
shows "\<exists>x\<in>{a<..<b}. (f b - f a = (f' x) (b - a))" proof-
have "\<exists>x\<in>{a<..<b}. (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)"
apply(rule rolle[OF assms(1), of "\<lambda>x. f x - (f b - f a) / (b - a) * x"]) defer
apply(rule continuous_on_intros assms(2) continuous_on_cmul[where 'b=real, unfolded real_scaleR_def])+ proof
fix x assume x:"x \<in> {a<..<b}"
show "((\<lambda>x. f x - (f b - f a) / (b - a) * x) has_derivative (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa)) (at x)"
by(rule has_derivative_intros assms(3)[rule_format,OF x]
has_derivative_cmul[where 'b=real, unfolded real_scaleR_def])+
qed(insert assms(1), auto simp add:field_simps)
then guess x .. thus ?thesis apply(rule_tac x=x in bexI) apply(drule fun_cong[of _ _ "b - a"]) by auto qed
lemma mvt_simple: fixes f::"real \<Rightarrow> real"
assumes "a<b" "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
apply(rule mvt) apply(rule assms(1), rule differentiable_imp_continuous_on)
unfolding differentiable_on_def differentiable_def defer proof
fix x assume x:"x \<in> {a<..<b}" show "(f has_derivative f' x) (at x)" unfolding has_derivative_within_open[OF x open_interval_real,THEN sym]
apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using x by auto qed(insert assms(2), auto)
lemma mvt_very_simple: fixes f::"real \<Rightarrow> real"
assumes "a \<le> b" "\<forall>x\<in>{a..b}. (f has_derivative f'(x)) (at x within {a..b})"
shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)" proof(cases "a = b")
interpret bounded_linear "f' b" using assms(2) assms(1) by auto
case True thus ?thesis apply(rule_tac x=a in bexI)
using assms(2)[THEN bspec[where x=a]] unfolding has_derivative_def
unfolding True using zero by auto next
case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto qed
subsection {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *}
lemma inner_eq_dot: fixes a::"real^'n"
shows "a \<bullet> b = inner a b" unfolding inner_vector_def dot_def by auto
lemma mvt_general: fixes f::"real\<Rightarrow>real^'n"
assumes "a<b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
shows "\<exists>x\<in>{a<..<b}. norm(f b - f a) \<le> norm(f'(x) (b - a))" proof-
have "\<exists>x\<in>{a<..<b}. (op \<bullet> (f b - f a) \<circ> f) b - (op \<bullet> (f b - f a) \<circ> f) a = (f b - f a) \<bullet> f' x (b - a)"
apply(rule mvt) apply(rule assms(1))unfolding inner_eq_dot apply(rule continuous_on_inner continuous_on_intros assms(2))+
unfolding o_def apply(rule,rule has_derivative_lift_dot) using assms(3) by auto
then guess x .. note x=this
show ?thesis proof(cases "f a = f b")
case False have "norm (f b - f a) * norm (f b - f a) = norm (f b - f a)^2" by(simp add:class_semiring.semiring_rules)
also have "\<dots> = (f b - f a) \<bullet> (f b - f a)" unfolding norm_pow_2 ..
also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)" using x by auto
also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))" by(rule norm_cauchy_schwarz)
finally show ?thesis using False x(1) by(auto simp add: real_mult_left_cancel) next
case True thus ?thesis using assms(1) apply(rule_tac x="(a + b) /2" in bexI) by auto qed qed
subsection {* Still more general bound theorem. *}
lemma differentiable_bound: fixes f::"real^'a \<Rightarrow> real^'b"
assumes "convex s" "\<forall>x\<in>s. (f has_derivative f'(x)) (at x within s)" "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
shows "norm(f x - f y) \<le> B * norm(x - y)" proof-
let ?p = "\<lambda>u. x + u *\<^sub>R (y - x)"
have *:"\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
using assms(1)[unfolded convex_alt,rule_format,OF x y] unfolding scaleR_left_diff_distrib scaleR_right_diff_distrib by(auto simp add:group_simps)
hence 1:"continuous_on {0..1} (f \<circ> ?p)" apply- apply(rule continuous_on_intros continuous_on_vmul)+
unfolding continuous_on_eq_continuous_within apply(rule,rule differentiable_imp_continuous_within)
unfolding differentiable_def apply(rule_tac x="f' xa" in exI)
apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) by auto
have 2:"\<forall>u\<in>{0<..<1}. ((f \<circ> ?p) has_derivative f' (x + u *\<^sub>R (y - x)) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u)" proof rule case goal1
let ?u = "x + u *\<^sub>R (y - x)"
have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})"
apply(rule diff_chain_within) apply(rule has_derivative_intros)+
apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using goal1 * by auto
thus ?case unfolding has_derivative_within_open[OF goal1 open_interval_real] by auto qed
guess u using mvt_general[OF zero_less_one 1 2] .. note u = this
have **:"\<And>x y. x\<in>s \<Longrightarrow> norm (f' x y) \<le> B * norm y" proof- case goal1
have "norm (f' x y) \<le> onorm (f' x) * norm y"
using onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]] by assumption
also have "\<dots> \<le> B * norm y" apply(rule mult_right_mono)
using assms(3)[rule_format,OF goal1] by(auto simp add:field_simps)
finally show ?case by simp qed
have "norm (f x - f y) = norm ((f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 1 - (f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 0)"
by(auto simp add:norm_minus_commute)
also have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))" using u by auto
also have "\<dots> \<le> B * norm(y - x)" apply(rule **) using * and u by auto
finally show ?thesis by(auto simp add:norm_minus_commute) qed
(** move this **)
declare norm_vec1[simp]
lemma onorm_vec1: fixes f::"real \<Rightarrow> real"
shows "onorm (\<lambda>x. vec1 (f (dest_vec1 x))) = onorm f" proof-
have "\<forall>x::real^1. norm x = 1 \<longleftrightarrow> x\<in>{vec1 -1, vec1 (1::real)}" unfolding forall_vec1 by(auto simp add:Cart_eq)
hence 1:"{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by(auto simp add:norm_vec1)
have 2:"{norm (vec1 (f (dest_vec1 x))) |x. norm x = 1} = (\<lambda>x. norm (vec1 (f (dest_vec1 x)))) ` {x. norm x=1}" by auto
have "\<forall>x::real. norm x = 1 \<longleftrightarrow> x\<in>{-1, 1}" by auto hence 3:"{x. norm x = 1} = {-1, (1::real)}" by auto
have 4:"{norm (f x) |x. norm x = 1} = (\<lambda>x. norm (f x)) ` {x. norm x=1}" by auto
show ?thesis unfolding onorm_def 1 2 3 4 by(simp add:Sup_finite_Max norm_vec1) qed
lemma differentiable_bound_real: fixes f::"real \<Rightarrow> real"
assumes "convex s" "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)" "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
shows "norm(f x - f y) \<le> B * norm(x - y)"
using differentiable_bound[of "vec1 ` s" "vec1 \<circ> f \<circ> dest_vec1" "\<lambda>x. vec1 \<circ> (f' (dest_vec1 x)) \<circ> dest_vec1" B "vec1 x" "vec1 y"]
unfolding Ball_def forall_vec1 unfolding has_derivative_within_vec1_dest_vec1 image_iff
unfolding convex_vec1 unfolding o_def vec1_dest_vec1_simps onorm_vec1 using assms by auto
subsection {* In particular. *}
lemma has_derivative_zero_constant: fixes f::"real\<Rightarrow>real"
assumes "convex s" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
shows "\<exists>c. \<forall>x\<in>s. f x = c" proof(cases "s={}")
case False then obtain x where "x\<in>s" by auto
have "\<And>y. y\<in>s \<Longrightarrow> f x = f y" proof- case goal1
thus ?case using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\<in>s`
unfolding onorm_vec1[of "\<lambda>x. 0", THEN sym] onorm_const norm_vec1 by auto qed
thus ?thesis apply(rule_tac x="f x" in exI) by auto qed auto
lemma has_derivative_zero_unique: fixes f::"real\<Rightarrow>real"
assumes "convex s" "a \<in> s" "f a = c" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)" "x\<in>s"
shows "f x = c" using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto
subsection {* Differentiability of inverse function (most basic form). *}
lemma has_derivative_inverse_basic: fixes f::"real^'b \<Rightarrow> real^'c"
assumes "(f has_derivative f') (at (g y))" "bounded_linear g'" "g' \<circ> f' = id" "continuous (at y) g"
"open t" "y \<in> t" "\<forall>z\<in>t. f(g z) = z"
shows "(g has_derivative g') (at y)" proof-
interpret f': bounded_linear f' using assms unfolding has_derivative_def by auto
interpret g': bounded_linear g' using assms by auto
guess C using bounded_linear.pos_bounded[OF assms(2)] .. note C = this
(* have fgid:"\<And>x. g' (f' x) = x" using assms(3) unfolding o_def id_def apply()*)
have lem1:"\<forall>e>0. \<exists>d>0. \<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y - g'(z - y)) \<le> e * norm(g z - g y)" proof(rule,rule) case goal1
have *:"e / C > 0" apply(rule divide_pos_pos) using `e>0` C by auto
guess d0 using assms(1)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note d0=this
guess d1 using assms(4)[unfolded continuous_at Lim_at,rule_format,OF d0[THEN conjunct1]] .. note d1=this
guess d2 using assms(5)[unfolded open_dist,rule_format,OF assms(6)] .. note d2=this
guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d=this
thus ?case apply(rule_tac x=d in exI) apply rule defer proof(rule,rule)
fix z assume as:"norm (z - y) < d" hence "z\<in>t" using d2 d unfolding vector_dist_norm by auto
have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
unfolding g'.diff f'.diff unfolding assms(3)[unfolded o_def id_def, THEN fun_cong]
unfolding assms(7)[rule_format,OF `z\<in>t`] apply(subst norm_minus_cancel[THEN sym]) by auto
also have "\<dots> \<le> norm(f (g z) - y - f' (g z - g y)) * C" by(rule C[THEN conjunct2,rule_format])
also have "\<dots> \<le> (e / C) * norm (g z - g y) * C" apply(rule mult_right_mono)
apply(rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]]) apply(cases "z=y") defer
apply(rule d1[THEN conjunct2, unfolded vector_dist_norm,rule_format]) using as d C d0 by auto
also have "\<dots> \<le> e * norm (g z - g y)" using C by(auto simp add:field_simps)
finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)" by simp qed auto qed
have *:"(0::real) < 1 / 2" by auto guess d using lem1[rule_format,OF *] .. note d=this def B\<equiv>"C*2"
have "B>0" unfolding B_def using C by auto
have lem2:"\<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y) \<le> B * norm(z - y)" proof(rule,rule) case goal1
have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))" by(rule norm_triangle_sub)
also have "\<dots> \<le> norm(g' (z - y)) + 1 / 2 * norm (g z - g y)" apply(rule add_left_mono) using d and goal1 by auto
also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)" apply(rule add_right_mono) using C by auto
finally show ?case unfolding B_def by(auto simp add:field_simps) qed
show ?thesis unfolding has_derivative_at_alt proof(rule,rule assms,rule,rule) case goal1
hence *:"e/B >0" apply-apply(rule divide_pos_pos) using `B>0` by auto
guess d' using lem1[rule_format,OF *] .. note d'=this
guess k using real_lbound_gt_zero[OF d[THEN conjunct1] d'[THEN conjunct1]] .. note k=this
show ?case apply(rule_tac x=k in exI,rule) defer proof(rule,rule) fix z assume as:"norm(z - y) < k"
hence "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)" using d' k by auto
also have "\<dots> \<le> e * norm(z - y)" unfolding mult_frac_num pos_divide_le_eq[OF `B>0`]
using lem2[THEN spec[where x=z]] using k as using `e>0` by(auto simp add:field_simps)
finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)" by simp qed(insert k, auto) qed qed
subsection {* Simply rewrite that based on the domain point x. *}
lemma has_derivative_inverse_basic_x: fixes f::"real^'b \<Rightarrow> real^'c"
assumes "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
"continuous (at (f x)) g" "g(f x) = x" "open t" "f x \<in> t" "\<forall>y\<in>t. f(g y) = y"
shows "(g has_derivative g') (at (f(x)))"
apply(rule has_derivative_inverse_basic) using assms by auto
subsection {* This is the version in Dieudonne', assuming continuity of f and g. *}
lemma has_derivative_inverse_dieudonne: fixes f::"real^'a \<Rightarrow> real^'b"
assumes "open s" "open (f ` s)" "continuous_on s f" "continuous_on (f ` s) g" "\<forall>x\<in>s. g(f x) = x"
(**) "x\<in>s" "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
shows "(g has_derivative g') (at (f x))"
apply(rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
using assms(3-6) unfolding continuous_on_eq_continuous_at[OF assms(1)] continuous_on_eq_continuous_at[OF assms(2)] by auto
subsection {* Here's the simplest way of not assuming much about g. *}
lemma has_derivative_inverse: fixes f::"real^'a \<Rightarrow> real^'b"
assumes "compact s" "x \<in> s" "f x \<in> interior(f ` s)" "continuous_on s f"
"\<forall>y\<in>s. g(f y) = y" "(f has_derivative f') (at x)" "bounded_linear g'" "g' \<circ> f' = id"
shows "(g has_derivative g') (at (f x))" proof-
{ fix y assume "y\<in>interior (f ` s)"
then obtain x where "x\<in>s" and *:"y = f x" unfolding image_iff using interior_subset by auto
have "f (g y) = y" unfolding * and assms(5)[rule_format,OF `x\<in>s`] .. } note * = this
show ?thesis apply(rule has_derivative_inverse_basic_x[OF assms(6-8)])
apply(rule continuous_on_interior[OF _ assms(3)]) apply(rule continuous_on_inverse[OF assms(4,1)])
apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+ by(rule, rule *, assumption) qed
subsection {* Proving surjectivity via Brouwer fixpoint theorem. *}
lemma brouwer_surjective: fixes f::"real^'n \<Rightarrow> real^'n"
assumes "compact t" "convex t" "t \<noteq> {}" "continuous_on t f"
"\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t" "x\<in>s"
shows "\<exists>y\<in>t. f y = x" proof-
have *:"\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y" by(auto simp add:group_simps)
show ?thesis unfolding * apply(rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
apply(rule continuous_on_intros assms)+ using assms(4-6) by auto qed
lemma brouwer_surjective_cball: fixes f::"real^'n \<Rightarrow> real^'n"
assumes "0 < e" "continuous_on (cball a e) f"
"\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e" "x\<in>s"
shows "\<exists>y\<in>cball a e. f y = x" apply(rule brouwer_surjective) apply(rule compact_cball convex_cball)+
unfolding cball_eq_empty using assms by auto
text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *}
lemma sussmann_open_mapping: fixes f::"real^'a \<Rightarrow> real^'b"
assumes "open s" "continuous_on s f" "x \<in> s"
"(f has_derivative f') (at x)" "bounded_linear g'" "f' \<circ> g' = id"
(**) "t \<subseteq> s" "x \<in> interior t"
shows "f x \<in> interior (f ` t)" proof-
interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
interpret g':bounded_linear g' using assms by auto
guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this hence *:"1/(2*B)>0" by(auto intro!: divide_pos_pos)
guess e0 using assms(4)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note e0=this
guess e1 using assms(8)[unfolded mem_interior_cball] .. note e1=this
have *:"0<e0/B" "0<e1/B" apply(rule_tac[!] divide_pos_pos) using e0 e1 B by auto
guess e using real_lbound_gt_zero[OF *] .. note e=this
have "\<forall>z\<in>cball (f x) (e/2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
apply(rule,rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
prefer 3 apply(rule,rule) proof-
show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))" unfolding g'.diff
apply(rule continuous_on_compose[of _ _ f, unfolded o_def])
apply(rule continuous_on_intros linear_continuous_on[OF assms(5)])+
apply(rule continuous_on_subset[OF assms(2)]) apply(rule,unfold image_iff,erule bexE) proof-
fix y z assume as:"y \<in>cball (f x) e" "z = x + (g' y - g' (f x))"
have "dist x z = norm (g' (f x) - g' y)" unfolding as(2) and vector_dist_norm by auto
also have "\<dots> \<le> norm (f x - y) * B" unfolding g'.diff[THEN sym] using B by auto
also have "\<dots> \<le> e * B" using as(1)[unfolded mem_cball vector_dist_norm] using B by auto
also have "\<dots> \<le> e1" using e unfolding less_divide_eq using B by auto
finally have "z\<in>cball x e1" unfolding mem_cball by force
thus "z \<in> s" using e1 assms(7) by auto qed next
fix y z assume as:"y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
have "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by auto
also have "\<dots> \<le> e * B" apply(rule mult_right_mono) using as(2)[unfolded mem_cball vector_dist_norm] and B unfolding norm_minus_commute by auto
also have "\<dots> < e0" using e and B unfolding less_divide_eq by auto
finally have *:"norm (x + g' (z - f x) - x) < e0" by auto
have **:"f x + f' (x + g' (z - f x) - x) = z" using assms(6)[unfolded o_def id_def,THEN cong] by auto
have "norm (f x - (y + (z - f (x + g' (z - f x))))) \<le> norm (f (x + g' (z - f x)) - z) + norm (f x - y)"
using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"] by(auto simp add:group_simps)
also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)" using e0[THEN conjunct2,rule_format,OF *] unfolding group_simps ** by auto
also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2" using as(1)[unfolded mem_cball vector_dist_norm] by auto
also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2" using * and B by(auto simp add:field_simps)
also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2" by auto
also have "\<dots> \<le> e/2 + e/2" apply(rule add_right_mono) using as(2)[unfolded mem_cball vector_dist_norm] unfolding norm_minus_commute by auto
finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e" unfolding mem_cball vector_dist_norm by auto
qed(insert e, auto) note lem = this
show ?thesis unfolding mem_interior apply(rule_tac x="e/2" in exI)
apply(rule,rule divide_pos_pos) prefer 3 proof
fix y assume "y \<in> ball (f x) (e/2)" hence *:"y\<in>cball (f x) (e/2)" by auto
guess z using lem[rule_format,OF *] .. note z=this
hence "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by(auto simp add:field_simps)
also have "\<dots> \<le> e * B" apply(rule mult_right_mono) using z(1) unfolding mem_cball vector_dist_norm norm_minus_commute using B by auto
also have "\<dots> \<le> e1" using e B unfolding less_divide_eq by auto
finally have "x + g'(z - f x) \<in> t" apply- apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format])
unfolding mem_cball vector_dist_norm by auto
thus "y \<in> f ` t" using z by auto qed(insert e, auto) qed
text {* Hence the following eccentric variant of the inverse function theorem. *)
(* This has no continuity assumptions, but we do need the inverse function. *)
(* We could put f' o g = I but this happens to fit with the minimal linear *)
(* algebra theory I've set up so far. *}
lemma has_derivative_inverse_strong: fixes f::"real^'n \<Rightarrow> real^'n"
assumes "open s" "x \<in> s" "continuous_on s f"
"\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at x)" "f' o g' = id"
shows "(g has_derivative g') (at (f x))" proof-
have linf:"bounded_linear f'" using assms(5) unfolding has_derivative_def by auto
hence ling:"bounded_linear g'" unfolding linear_conv_bounded_linear[THEN sym]
apply- apply(rule right_inverse_linear) using assms(6) by auto
moreover have "g' \<circ> f' = id" using assms(6) linf ling unfolding linear_conv_bounded_linear[THEN sym]
using linear_inverse_left by auto
moreover have *:"\<forall>t\<subseteq>s. x\<in>interior t \<longrightarrow> f x \<in> interior (f ` t)" apply(rule,rule,rule,rule sussmann_open_mapping )
apply(rule assms ling)+ by auto
have "continuous (at (f x)) g" unfolding continuous_at Lim_at proof(rule,rule)
fix e::real assume "e>0"
hence "f x \<in> interior (f ` (ball x e \<inter> s))" using *[rule_format,of "ball x e \<inter> s"] `x\<in>s`
by(auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
then guess d unfolding mem_interior .. note d=this
show "\<exists>d>0. \<forall>y. 0 < dist y (f x) \<and> dist y (f x) < d \<longrightarrow> dist (g y) (g (f x)) < e"
apply(rule_tac x=d in exI) apply(rule,rule d[THEN conjunct1]) proof(rule,rule) case goal1
hence "g y \<in> g ` f ` (ball x e \<inter> s)" using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]]
by(auto simp add:dist_commute)
hence "g y \<in> ball x e \<inter> s" using assms(4) by auto
thus "dist (g y) (g (f x)) < e" using assms(4)[rule_format,OF `x\<in>s`] by(auto simp add:dist_commute) qed qed
moreover have "f x \<in> interior (f ` s)" apply(rule sussmann_open_mapping)
apply(rule assms ling)+ using interior_open[OF assms(1)] and `x\<in>s` by auto
moreover have "\<And>y. y \<in> interior (f ` s) \<Longrightarrow> f (g y) = y" proof- case goal1
hence "y\<in>f ` s" using interior_subset by auto then guess z unfolding image_iff ..
thus ?case using assms(4) by auto qed
ultimately show ?thesis apply- apply(rule has_derivative_inverse_basic_x[OF assms(5)]) using assms by auto qed
subsection {* A rewrite based on the other domain. *}
lemma has_derivative_inverse_strong_x: fixes f::"real^'n \<Rightarrow> real^'n"
assumes "open s" "g y \<in> s" "continuous_on s f"
"\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at (g y))" "f' o g' = id" "f(g y) = y"
shows "(g has_derivative g') (at y)"
using has_derivative_inverse_strong[OF assms(1-6)] unfolding assms(7) by simp
subsection {* On a region. *}
lemma has_derivative_inverse_on: fixes f::"real^'n \<Rightarrow> real^'n"
assumes "open s" "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)" "\<forall>x\<in>s. g(f x) = x" "f'(x) o g'(x) = id" "x\<in>s"
shows "(g has_derivative g'(x)) (at (f x))"
apply(rule has_derivative_inverse_strong[where g'="g' x" and f=f]) apply(rule assms)+
unfolding continuous_on_eq_continuous_at[OF assms(1)]
apply(rule,rule differentiable_imp_continuous_at) unfolding differentiable_def using assms by auto
subsection {* Invertible derivative continous at a point implies local injectivity. *)
(* It's only for this we need continuity of the derivative, except of course *)
(* if we want the fact that the inverse derivative is also continuous. So if *)
(* we know for some other reason that the inverse function exists, it's OK. *}
lemma bounded_linear_sub: "bounded_linear f \<Longrightarrow> bounded_linear g ==> bounded_linear (\<lambda>x. f x - g x)"
using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g] by(auto simp add:group_simps)
lemma has_derivative_locally_injective: fixes f::"real^'n \<Rightarrow> real^'m"
assumes "a \<in> s" "open s" "bounded_linear g'" "g' o f'(a) = id"
"\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
"\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm(\<lambda>v. f' x v - f' a v) < e"
obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. (f x' = f x) \<longrightarrow> (x' = x)" proof-
interpret bounded_linear g' using assms by auto
note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
have "g' (f' a 1) = 1" using f'g' by auto
hence *:"0 < onorm g'" unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastsimp
def k \<equiv> "1 / onorm g' / 2" have *:"k>0" unfolding k_def using * by auto
guess d1 using assms(6)[rule_format,OF *] .. note d1=this
from `open s` obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using `a\<in>s` ..
obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using assms(2,1) ..
guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\<in>s`] .. note d2=this
guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d = this
show ?thesis proof show "a\<in>ball a d" using d by auto
show "\<forall>x\<in>ball a d. \<forall>x'\<in>ball a d. f x' = f x \<longrightarrow> x' = x" proof(intro strip)
fix x y assume as:"x\<in>ball a d" "y\<in>ball a d" "f x = f y"
def ph \<equiv> "\<lambda>w. w - g'(f w - f x)" have ph':"ph = g' \<circ> (\<lambda>w. f' a w - (f w - f x))"
unfolding ph_def o_def unfolding diff using f'g' by(auto simp add:group_simps)
have "norm (ph x - ph y) \<le> (1/2) * norm (x - y)"
apply(rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
apply(rule_tac[!] ballI) proof- fix u assume u:"u \<in> ball a d" hence "u\<in>s" using d d2 by auto
have *:"(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)" unfolding o_def and diff using f'g' by auto
show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
unfolding ph' * apply(rule diff_chain_within) defer apply(rule bounded_linear.has_derivative[OF assms(3)])
apply(rule has_derivative_intros) defer apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
apply(rule has_derivative_at_within) using assms(5) and `u\<in>s` `a\<in>s`
by(auto intro!: has_derivative_intros derivative_linear)
have **:"bounded_linear (\<lambda>x. f' u x - f' a x)" "bounded_linear (\<lambda>x. f' a x - f' u x)" apply(rule_tac[!] bounded_linear_sub)
apply(rule_tac[!] derivative_linear) using assms(5) `u\<in>s` `a\<in>s` by auto
have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)" unfolding * apply(rule onorm_compose)
unfolding linear_conv_bounded_linear by(rule assms(3) **)+
also have "\<dots> \<le> onorm g' * k" apply(rule mult_left_mono)
using d1[THEN conjunct2,rule_format,of u] using onorm_neg[OF **(1)[unfolded linear_linear]]
using d and u and onorm_pos_le[OF assms(3)[unfolded linear_linear]] by(auto simp add:group_simps)
also have "\<dots> \<le> 1/2" unfolding k_def by auto
finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" by assumption qed
moreover have "norm (ph y - ph x) = norm (y - x)" apply(rule arg_cong[where f=norm])
unfolding ph_def using diff unfolding as by auto
ultimately show "x = y" unfolding norm_minus_commute by auto qed qed auto qed
subsection {* Uniformly convergent sequence of derivatives. *}
lemma has_derivative_sequence_lipschitz_lemma: fixes f::"nat \<Rightarrow> real^'m \<Rightarrow> real^'n"
assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
"\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
shows "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)" proof(default)+
fix m n x y assume as:"N\<le>m" "N\<le>n" "x\<in>s" "y\<in>s"
show "norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
apply(rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF assms(1) _ _ as(3-4)]) apply(rule_tac[!] ballI) proof-
fix x assume "x\<in>s" show "((\<lambda>a. f m a - f n a) has_derivative (\<lambda>h. f' m x h - f' n x h)) (at x within s)"
by(rule has_derivative_intros assms(2)[rule_format] `x\<in>s`)+
{ fix h have "norm (f' m x h - f' n x h) \<le> norm (f' m x h - g' x h) + norm (f' n x h - g' x h)"
using norm_triangle_ineq[of "f' m x h - g' x h" "- f' n x h + g' x h"] unfolding norm_minus_commute by(auto simp add:group_simps)
also have "\<dots> \<le> e * norm h+ e * norm h" using assms(3)[rule_format,OF `N\<le>m` `x\<in>s`, of h] assms(3)[rule_format,OF `N\<le>n` `x\<in>s`, of h]
by(auto simp add:field_simps)
finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h" by auto }
thus "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e" apply-apply(rule onorm(2)) apply(rule linear_compose_sub)
unfolding linear_conv_bounded_linear using assms(2)[rule_format,OF `x\<in>s`, THEN derivative_linear] by auto qed qed
lemma has_derivative_sequence_lipschitz: fixes f::"nat \<Rightarrow> real^'m \<Rightarrow> real^'n"
assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)" "0 < e"
shows "\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> e * norm(x - y)" proof(rule,rule)
case goal1 have *:"2 * (1/2* e) = e" "1/2 * e >0" using `e>0` by auto
guess N using assms(3)[rule_format,OF *(2)] ..
thus ?case apply(rule_tac x=N in exI) apply(rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *]) using assms by auto qed
lemma has_derivative_sequence: fixes f::"nat\<Rightarrow>real^'m\<Rightarrow>real^'n"
assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
"x0 \<in> s" "((\<lambda>n. f n x0) ---> l) sequentially"
shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> (g has_derivative g'(x)) (at x within s)" proof-
have lem1:"\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> e * norm(x - y)"
apply(rule has_derivative_sequence_lipschitz[where e="42::nat"]) apply(rule assms)+ by auto
have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially" apply(rule bchoice) unfolding convergent_eq_cauchy proof
fix x assume "x\<in>s" show "Cauchy (\<lambda>n. f n x)" proof(cases "x=x0")
case True thus ?thesis using convergent_imp_cauchy[OF assms(5)] by auto next
case False show ?thesis unfolding Cauchy_def proof(rule,rule)
fix e::real assume "e>0" hence *:"e/2>0" "e/2/norm(x-x0)>0" using False by(auto intro!:divide_pos_pos)
guess M using convergent_imp_cauchy[OF assms(5), unfolded Cauchy_def, rule_format,OF *(1)] .. note M=this
guess N using lem1[rule_format,OF *(2)] .. note N = this
show " \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e" apply(rule_tac x="max M N" in exI) proof(default+)
fix m n assume as:"max M N \<le>m" "max M N\<le>n"
have "dist (f m x) (f n x) \<le> norm (f m x0 - f n x0) + norm (f m x - f n x - (f m x0 - f n x0))"
unfolding vector_dist_norm by(rule norm_triangle_sub)
also have "\<dots> \<le> norm (f m x0 - f n x0) + e / 2" using N[rule_format,OF _ _ `x\<in>s` `x0\<in>s`, of m n] and as and False by auto
also have "\<dots> < e / 2 + e / 2" apply(rule add_strict_right_mono) using as and M[rule_format] unfolding vector_dist_norm by auto
finally show "dist (f m x) (f n x) < e" by auto qed qed qed qed
then guess g .. note g = this
have lem2:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f n x - f n y) - (g x - g y)) \<le> e * norm(x - y)" proof(rule,rule)
fix e::real assume *:"e>0" guess N using lem1[rule_format,OF *] .. note N=this
show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)" apply(rule_tac x=N in exI) proof(default+)
fix n x y assume as:"N \<le> n" "x \<in> s" "y \<in> s"
have "eventually (\<lambda>xa. norm (f n x - f n y - (f xa x - f xa y)) \<le> e * norm (x - y)) sequentially"
unfolding eventually_sequentially apply(rule_tac x=N in exI) proof(rule,rule)
fix m assume "N\<le>m" thus "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
using N[rule_format, of n m x y] and as by(auto simp add:group_simps) qed
thus "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)" apply-
apply(rule Lim_norm_ubound[OF trivial_limit_sequentially, where f="\<lambda>m. (f n x - f n y) - (f m x - f m y)"])
apply(rule Lim_sub Lim_const g[rule_format] as)+ by assumption qed qed
show ?thesis unfolding has_derivative_within_alt apply(rule_tac x=g in exI)
apply(rule,rule,rule g[rule_format],assumption) proof fix x assume "x\<in>s"
have lem3:"\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially" unfolding Lim_sequentially proof(rule,rule,rule)
fix u and e::real assume "e>0" show "\<exists>N. \<forall>n\<ge>N. dist (f' n x u) (g' x u) < e" proof(cases "u=0")
case True guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
show ?thesis apply(rule_tac x=N in exI) unfolding True
using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0` by auto next
case False hence *:"e / 2 / norm u > 0" using `e>0` by(auto intro!: divide_pos_pos)
guess N using assms(3)[rule_format,OF *] .. note N=this
show ?thesis apply(rule_tac x=N in exI) proof(rule,rule) case goal1
show ?case unfolding vector_dist_norm using N[rule_format,OF goal1 `x\<in>s`, of u] False `e>0`
by (auto simp add:field_simps) qed qed qed
show "bounded_linear (g' x)" unfolding linear_linear linear_def apply(rule,rule,rule) defer proof(rule,rule)
fix x' y z::"real^'m" and c::real
note lin = assms(2)[rule_format,OF `x\<in>s`,THEN derivative_linear]
show "g' x (c *s x') = c *s g' x x'" apply(rule Lim_unique[OF trivial_limit_sequentially])
apply(rule lem3[rule_format]) unfolding smult_conv_scaleR
unfolding lin[unfolded bounded_linear_def bounded_linear_axioms_def,THEN conjunct2,THEN conjunct1,rule_format]
apply(rule Lim_cmul) by(rule lem3[rule_format])
show "g' x (y + z) = g' x y + g' x z" apply(rule Lim_unique[OF trivial_limit_sequentially])
apply(rule lem3[rule_format]) unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format]
apply(rule Lim_add) by(rule lem3[rule_format])+ qed
show "\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)" proof(rule,rule) case goal1
have *:"e/3>0" using goal1 by auto guess N1 using assms(3)[rule_format,OF *] .. note N1=this
guess N2 using lem2[rule_format,OF *] .. note N2=this
guess d1 using assms(2)[unfolded has_derivative_within_alt, rule_format,OF `x\<in>s`, of "max N1 N2",THEN conjunct2,rule_format,OF *] .. note d1=this
show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1]) proof(rule,rule)
fix y assume as:"y \<in> s" "norm (y - x) < d1" let ?N ="max N1 N2"
have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e /3 * norm (y - x)" apply(subst norm_minus_cancel[THEN sym])
using N2[rule_format, OF _ `y\<in>s` `x\<in>s`, of ?N] by auto moreover
have "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)" using d1 and as by auto ultimately
have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)"
using norm_triangle_le[of "g y - g x - (f ?N y - f ?N x)" "f ?N y - f ?N x - f' ?N x (y - x)" "2 * e / 3 * norm (y - x)"]
by (auto simp add:group_simps) moreover
have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)" using N1 `x\<in>s` by auto
ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
using norm_triangle_le[of "g y - g x - f' (max N1 N2) x (y - x)" "f' (max N1 N2) x (y - x) - g' x (y - x)"] by(auto simp add:group_simps)
qed qed qed qed
subsection {* Can choose to line up antiderivatives if we want. *}
lemma has_antiderivative_sequence: fixes f::"nat\<Rightarrow> real^'m \<Rightarrow> real^'n"
assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm h"
shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)" proof(cases "s={}")
case False then obtain a where "a\<in>s" by auto have *:"\<And>P Q. \<exists>g. \<forall>x\<in>s. P g x \<and> Q g x \<Longrightarrow> \<exists>g. \<forall>x\<in>s. Q g x" by auto
show ?thesis apply(rule *) apply(rule has_derivative_sequence[OF assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
apply(rule,rule) apply(rule has_derivative_add_const, rule assms(2)[rule_format], assumption)
apply(rule `a\<in>s`) by(auto intro!: Lim_const) qed auto
lemma has_antiderivative_limit: fixes g'::"real^'m \<Rightarrow> real^'m \<Rightarrow> real^'n"
assumes "convex s" "\<forall>e>0. \<exists>f f'. \<forall>x\<in>s. (f has_derivative (f' x)) (at x within s) \<and> (\<forall>h. norm(f' x h - g' x h) \<le> e * norm(h))"
shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)" proof-
have *:"\<forall>n. \<exists>f f'. \<forall>x\<in>s. (f has_derivative (f' x)) (at x within s) \<and> (\<forall>h. norm(f' x h - g' x h) \<le> inverse (real (Suc n)) * norm(h))"
apply(rule) using assms(2) apply(erule_tac x="inverse (real (Suc n))" in allE) by auto
guess f using *[THEN choice] .. note * = this guess f' using *[THEN choice] .. note f=this
show ?thesis apply(rule has_antiderivative_sequence[OF assms(1), of f f']) defer proof(rule,rule)
fix e::real assume "0<e" guess N using reals_Archimedean[OF `e>0`] .. note N=this
show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h" apply(rule_tac x=N in exI) proof(default+) case goal1
have *:"inverse (real (Suc n)) \<le> e" apply(rule order_trans[OF _ N[THEN less_imp_le]])
using goal1(1) by(auto simp add:field_simps)
show ?case using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]]
apply(rule order_trans) using N * apply(cases "h=0") by auto qed qed(insert f,auto) qed
subsection {* Differentiation of a series. *}
definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> (nat set) \<Rightarrow> bool"
(infixl "sums'_seq" 12) where "(f sums_seq l) s \<equiv> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"
lemma has_derivative_series: fixes f::"nat \<Rightarrow> real^'m \<Rightarrow> real^'n"
assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(setsum (\<lambda>i. f' i x h) (k \<inter> {0..n}) - g' x h) \<le> e * norm(h)"
"x\<in>s" "((\<lambda>n. f n x) sums_seq l) k"
shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) sums_seq (g x)) k \<and> (g has_derivative g'(x)) (at x within s)"
unfolding sums_seq_def apply(rule has_derivative_sequence[OF assms(1) _ assms(3)]) apply(rule,rule)
apply(rule has_derivative_setsum) defer apply(rule,rule assms(2)[rule_format],assumption)
using assms(4-5) unfolding sums_seq_def by auto
subsection {* Derivative with composed bilinear function. *}
lemma has_derivative_bilinear_within: fixes h::"real^'m \<Rightarrow> real^'n \<Rightarrow> real^'p" and f::"real^'q \<Rightarrow> real^'m"
assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at x within s)" "bounded_bilinear h"
shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x within s)" proof-
have "(g ---> g x) (at x within s)" apply(rule differentiable_imp_continuous_within[unfolded continuous_within])
using assms(2) unfolding differentiable_def by auto moreover
interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
interpret g':bounded_linear g' using assms unfolding has_derivative_def by auto
interpret h:bounded_bilinear h using assms by auto
have "((\<lambda>y. f' (y - x)) ---> 0) (at x within s)" unfolding f'.zero[THEN sym]
apply(rule Lim_linear[of "\<lambda>y. y - x" 0 "at x within s" f']) using Lim_sub[OF Lim_within_id Lim_const, of x x s]
unfolding id_def using assms(1) unfolding has_derivative_def by auto
hence "((\<lambda>y. f x + f' (y - x)) ---> f x) (at x within s)"
using Lim_add[OF Lim_const, of "\<lambda>y. f' (y - x)" 0 "at x within s" "f x"] by auto ultimately
have *:"((\<lambda>x'. h (f x + f' (x' - x)) ((1/(norm (x' - x))) *\<^sub>R (g x' - (g x + g' (x' - x))))
+ h ((1/ (norm (x' - x))) *\<^sub>R (f x' - (f x + f' (x' - x)))) (g x')) ---> h (f x) 0 + h 0 (g x)) (at x within s)"
apply-apply(rule Lim_add) apply(rule_tac[!] Lim_bilinear[OF _ _ assms(3)]) using assms(1-2) unfolding has_derivative_within by auto
guess B using bounded_bilinear.pos_bounded[OF assms(3)] .. note B=this
guess C using f'.pos_bounded .. note C=this
guess D using g'.pos_bounded .. note D=this
have bcd:"B * C * D > 0" using B C D by (auto intro!: mult_pos_pos)
have **:"((\<lambda>y. (1/(norm(y - x))) *\<^sub>R (h (f'(y - x)) (g'(y - x)))) ---> 0) (at x within s)" unfolding Lim_within proof(rule,rule) case goal1
hence "e/(B*C*D)>0" using B C D by(auto intro!:divide_pos_pos mult_pos_pos)
thus ?case apply(rule_tac x="e/(B*C*D)" in exI,rule) proof(rule,rule,erule conjE)
fix y assume as:"y \<in> s" "0 < dist y x" "dist y x < e / (B * C * D)"
have "norm (h (f' (y - x)) (g' (y - x))) \<le> norm (f' (y - x)) * norm (g' (y - x)) * B" using B by auto
also have "\<dots> \<le> (norm (y - x) * C) * (D * norm (y - x)) * B" apply(rule mult_right_mono)
apply(rule pordered_semiring_class.mult_mono) using B C D by (auto simp add: field_simps intro!:mult_nonneg_nonneg)
also have "\<dots> = (B * C * D * norm (y - x)) * norm (y - x)" by(auto simp add:field_simps)
also have "\<dots> < e * norm (y - x)" apply(rule mult_strict_right_mono)
using as(3)[unfolded vector_dist_norm] and as(2) unfolding pos_less_divide_eq[OF bcd] by (auto simp add:field_simps)
finally show "dist ((1 / norm (y - x)) *\<^sub>R h (f' (y - x)) (g' (y - x))) 0 < e"
unfolding vector_dist_norm apply-apply(cases "y = x") by(auto simp add:field_simps) qed qed
have "bounded_linear (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))" unfolding linear_linear linear_def
unfolding smult_conv_scaleR unfolding g'.add f'.scaleR f'.add g'.scaleR
unfolding h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right by auto
thus ?thesis using Lim_add[OF * **] unfolding has_derivative_within
unfolding smult_conv_scaleR unfolding g'.add f'.scaleR f'.add g'.scaleR f'.diff g'.diff
h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right h.diff_right h.diff_left
scaleR_right_diff_distrib h.zero_right h.zero_left by(auto simp add:field_simps) qed
lemma has_derivative_bilinear_at: fixes h::"real^'m \<Rightarrow> real^'n \<Rightarrow> real^'p" and f::"real^'p \<Rightarrow> real^'m"
assumes "(f has_derivative f') (at x)" "(g has_derivative g') (at x)" "bounded_bilinear h"
shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x)"
using has_derivative_bilinear_within[of f f' x UNIV g g' h] unfolding within_UNIV using assms by auto
subsection {* Considering derivative R(^1)->R^n as a vector. *}
definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('b) \<Rightarrow> (real net \<Rightarrow> bool)"
(infixl "has'_vector'_derivative" 12) where
"(f has_vector_derivative f') net \<equiv> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
definition "vector_derivative f net \<equiv> (SOME f'. (f has_vector_derivative f') net)"
lemma vector_derivative_works: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net" (is "?l = ?r")
proof assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this
then interpret bounded_linear f' by auto
thus ?r unfolding vector_derivative_def has_vector_derivative_def
apply-apply(rule someI_ex,rule_tac x="f' 1" in exI)
using f' unfolding scaleR[THEN sym] by auto
next assume ?r thus ?l unfolding vector_derivative_def has_vector_derivative_def differentiable_def by auto qed
lemma vector_derivative_unique_at: fixes f::"real\<Rightarrow>real^'n"
assumes "(f has_vector_derivative f') (at x)" "(f has_vector_derivative f'') (at x)" shows "f' = f''" proof-
have *:"(\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1 = (\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1" apply(rule frechet_derivative_unique_at)
using assms[unfolded has_vector_derivative_def] unfolding has_derivative_at_dest_vec1[THEN sym] by auto
show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
hence "((\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1) (vec1 1) = ((\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1) (vec1 1)" using * by auto
ultimately show False unfolding o_def vec1_dest_vec1 by auto qed qed
lemma vector_derivative_unique_within_closed_interval: fixes f::"real \<Rightarrow> real^'n"
assumes "a < b" "x \<in> {a..b}"
"(f has_vector_derivative f') (at x within {a..b})"
"(f has_vector_derivative f'') (at x within {a..b})" shows "f' = f''" proof-
have *:"(\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1 = (\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1"
apply(rule frechet_derivative_unique_within_closed_interval[of "vec1 a" "vec1 b"])
using assms(3-)[unfolded has_vector_derivative_def]
unfolding has_derivative_within_dest_vec1[THEN sym] vec1_interval using assms(1-2) by auto
show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
hence "((\<lambda>x. x *\<^sub>R f') \<circ> dest_vec1) (vec1 1) = ((\<lambda>x. x *\<^sub>R f'') \<circ> dest_vec1) (vec1 1)" using * by auto
ultimately show False unfolding o_def vec1_dest_vec1 by auto qed qed
lemma vector_derivative_at: fixes f::"real \<Rightarrow> real^'a" shows
"(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
apply(rule vector_derivative_unique_at) defer apply assumption
unfolding vector_derivative_works[THEN sym] differentiable_def
unfolding has_vector_derivative_def by auto
lemma vector_derivative_within_closed_interval: fixes f::"real \<Rightarrow> real^'a"
assumes "a < b" "x \<in> {a..b}" "(f has_vector_derivative f') (at x within {a..b})"
shows "vector_derivative f (at x within {a..b}) = f'"
apply(rule vector_derivative_unique_within_closed_interval)
using vector_derivative_works[unfolded differentiable_def]
using assms by(auto simp add:has_vector_derivative_def)
lemma has_vector_derivative_within_subset:
"(f has_vector_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_vector_derivative f') (at x within t)"
unfolding has_vector_derivative_def apply(rule has_derivative_within_subset) by auto
lemma has_vector_derivative_const:
"((\<lambda>x. c) has_vector_derivative 0) net"
unfolding has_vector_derivative_def using has_derivative_const by auto
lemma has_vector_derivative_id: "((\<lambda>x::real. x) has_vector_derivative 1) net"
unfolding has_vector_derivative_def using has_derivative_id by auto
lemma has_vector_derivative_cmul: "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
unfolding has_vector_derivative_def apply(drule has_derivative_cmul) by(auto simp add:group_simps)
lemma has_vector_derivative_cmul_eq: assumes "c \<noteq> 0"
shows "(((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net \<longleftrightarrow> (f has_vector_derivative f') net)"
apply rule apply(drule has_vector_derivative_cmul[where c="1/c"]) defer
apply(rule has_vector_derivative_cmul) using assms by auto
lemma has_vector_derivative_neg:
"(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_vector_derivative (- f')) net"
unfolding has_vector_derivative_def apply(drule has_derivative_neg) by auto
lemma has_vector_derivative_add:
assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
shows "((\<lambda>x. f(x) + g(x)) has_vector_derivative (f' + g')) net"
using has_derivative_add[OF assms[unfolded has_vector_derivative_def]]
unfolding has_vector_derivative_def unfolding scaleR_right_distrib by auto
lemma has_vector_derivative_sub:
assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
shows "((\<lambda>x. f(x) - g(x)) has_vector_derivative (f' - g')) net"
using has_derivative_sub[OF assms[unfolded has_vector_derivative_def]]
unfolding has_vector_derivative_def scaleR_right_diff_distrib by auto
lemma has_vector_derivative_bilinear_within: fixes h::"real^'m \<Rightarrow> real^'n \<Rightarrow> real^'p"
assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at x within s)" "bounded_bilinear h"
shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x within s)" proof-
interpret bounded_bilinear h using assms by auto
show ?thesis using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def has_derivative_within_dest_vec1[THEN sym]], where h=h]
unfolding o_def vec1_dest_vec1 has_vector_derivative_def
unfolding has_derivative_within_dest_vec1[unfolded o_def, where f="\<lambda>x. h (f x) (g x)" and f'="\<lambda>d. h (f x) (d *\<^sub>R g') + h (d *\<^sub>R f') (g x)"]
using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib by auto qed
lemma has_vector_derivative_bilinear_at: fixes h::"real^'m \<Rightarrow> real^'n \<Rightarrow> real^'p"
assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at x)" "bounded_bilinear h"
shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x)"
apply(rule has_vector_derivative_bilinear_within[where s=UNIV, unfolded within_UNIV]) using assms by auto
lemma has_vector_derivative_at_within: "(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f') (at x within s)"
unfolding has_vector_derivative_def apply(rule has_derivative_at_within) by auto
lemma has_vector_derivative_transform_within:
assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_vector_derivative f') (at x within s)"
shows "(g has_vector_derivative f') (at x within s)"
using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within)
lemma has_vector_derivative_transform_at:
assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_vector_derivative f') (at x)"
shows "(g has_vector_derivative f') (at x)"
using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_at)
lemma has_vector_derivative_transform_within_open:
assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_vector_derivative f') (at x)"
shows "(g has_vector_derivative f') (at x)"
using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within_open)
lemma vector_diff_chain_at:
assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at (f x))"
shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
unfolding o_def scaleR.scaleR_left by auto
lemma vector_diff_chain_within:
assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at (f x) within f ` s)"
shows "((g o f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
unfolding o_def scaleR.scaleR_left by auto
end