src/HOL/Multivariate_Analysis/Derivative.thy
author huffman
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generalize some lemmas about derivatives
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(*  Title:                       HOL/Multivariate_Analysis/Derivative.thy
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    Author:                      John Harrison
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    Translation from HOL Light:  Robert Himmelmann, TU Muenchen
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*)
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header {* Multivariate calculus in Euclidean space. *}
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theory Derivative
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imports Brouwer_Fixpoint Operator_Norm
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begin
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(* Because I do not want to type this all the time *)
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lemmas linear_linear = linear_conv_bounded_linear[THEN sym]
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subsection {* Derivatives *}
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text {* The definition is slightly tricky since we make it work over
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  nets of a particular form. This lets us prove theorems generally and use 
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  "at a" or "at a within s" for restriction to a set (1-sided on R etc.) *}
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definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a net \<Rightarrow> bool)"
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(infixl "(has'_derivative)" 12) where
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 "(f has_derivative f') net \<equiv> bounded_linear f' \<and> ((\<lambda>y. (1 / (norm (y - netlimit net))) *\<^sub>R
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   (f y - (f (netlimit net) + f'(y - netlimit net)))) ---> 0) net"
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lemma derivative_linear[dest]:"(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
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  unfolding has_derivative_def by auto
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lemma DERIV_conv_has_derivative:"(DERIV f x :> f') = (f has_derivative op * f') (at (x::real))" (is "?l = ?r") proof 
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  assume ?l note as = this[unfolded deriv_def LIM_def,rule_format]
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  show ?r unfolding has_derivative_def Lim_at apply- apply(rule,rule mult.bounded_linear_right)
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    apply safe apply(drule as,safe) apply(rule_tac x=s in exI) apply safe
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    apply(erule_tac x="xa - x" in allE) unfolding dist_norm netlimit_at[of x] unfolding diff_0_right norm_scaleR
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    by(auto simp add:field_simps) 
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next assume ?r note this[unfolded has_derivative_def Lim_at] note as=conjunct2[OF this,rule_format]
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  have *:"\<And>x xa f'. xa \<noteq> 0 \<Longrightarrow> \<bar>(f (xa + x) - f x) / xa - f'\<bar> = \<bar>(f (xa + x) - f x) - xa * f'\<bar> / \<bar>xa\<bar>" by(auto simp add:field_simps) 
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  show ?l unfolding deriv_def LIM_def apply safe apply(drule as,safe)
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    apply(rule_tac x=d in exI,safe) apply(erule_tac x="xa + x" in allE)
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    unfolding dist_norm diff_0_right norm_scaleR
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    unfolding dist_norm netlimit_at[of x] by(auto simp add:algebra_simps *) qed
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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 
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  assume ?l note as = this[unfolded fderiv_def]
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  show ?r unfolding has_derivative_def Lim_at apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
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    fix e::real assume "e>0"
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    guess d using as[THEN conjunct2,unfolded LIM_def,rule_format,OF`e>0`] ..
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    thus "\<exists>d>0. \<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
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      dist ((1 / norm (xa - netlimit (at x))) *\<^sub>R (f xa - (f (netlimit (at x)) + f' (xa - netlimit (at x))))) (0) < e"
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      apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa - x" in allE)
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      unfolding dist_norm netlimit_at[of x] by (auto simp add: diff_diff_eq) qed next
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  assume ?r note as = this[unfolded has_derivative_def]
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  show ?l unfolding fderiv_def LIM_def apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
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    fix e::real assume "e>0"
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    guess d using as[THEN conjunct2,unfolded Lim_at,rule_format,OF`e>0`] ..
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    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-
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      apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa + x" in allE)
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      unfolding dist_norm netlimit_at[of x] by (auto simp add: diff_diff_eq add.commute) qed qed
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subsection {* These are the only cases we'll care about, probably. *}
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lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
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         bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
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  unfolding has_derivative_def and Lim by(auto simp add:netlimit_within)
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lemma has_derivative_at: "(f has_derivative f') (at x) \<longleftrightarrow>
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         bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
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  apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within unfolding within_UNIV by auto
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subsection {* More explicit epsilon-delta forms. *}
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lemma has_derivative_within':
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  "(f has_derivative f')(at x within s) \<longleftrightarrow> bounded_linear f' \<and>
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        (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm(x' - x) \<and> norm(x' - x) < d
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        \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
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  unfolding has_derivative_within Lim_within dist_norm
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  unfolding diff_0_right by (simp add: diff_diff_eq)
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lemma has_derivative_at':
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 "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
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   (\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm(x' - x) \<and> norm(x' - x) < d
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        \<longrightarrow> norm(f x' - f x - f'(x' - x)) / norm(x' - x) < e)"
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  apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within' by auto
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lemma has_derivative_at_within: "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
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  unfolding has_derivative_within' has_derivative_at' by meson
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lemma has_derivative_within_open:
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  "a \<in> s \<Longrightarrow> open s \<Longrightarrow> ((f has_derivative f') (at a within s) \<longleftrightarrow> (f has_derivative f') (at a))"
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  unfolding has_derivative_within has_derivative_at using Lim_within_open by auto
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lemma bounded_linear_imp_linear: "bounded_linear f \<Longrightarrow> linear f" (* TODO: move elsewhere *)
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proof -
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  assume "bounded_linear f"
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  then interpret f: bounded_linear f .
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  show "linear f"
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    by (simp add: f.add f.scaleR linear_def)
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qed
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lemma derivative_is_linear:
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  "(f has_derivative f') net \<Longrightarrow> linear f'"
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  by (rule derivative_linear [THEN bounded_linear_imp_linear])
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subsection {* Combining theorems. *}
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lemma (in bounded_linear) has_derivative: "(f has_derivative f) net"
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  unfolding has_derivative_def apply(rule,rule bounded_linear_axioms)
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  unfolding diff by(simp add: Lim_const)
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lemma has_derivative_id: "((\<lambda>x. x) has_derivative (\<lambda>h. h)) net"
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hoelzl
parents:
diff changeset
   110
  apply(rule bounded_linear.has_derivative) using bounded_linear_ident[unfolded id_def] by simp
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   111
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   112
lemma has_derivative_const: "((\<lambda>x. c) has_derivative (\<lambda>h. 0)) net"
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parents:
diff changeset
   113
  unfolding has_derivative_def apply(rule,rule bounded_linear_zero) by(simp add: Lim_const)
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parents:
diff changeset
   114
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   115
lemma (in bounded_linear) cmul: shows "bounded_linear (\<lambda>x. (c::real) *\<^sub>R f x)"
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diff changeset
   116
proof -
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   117
  have "bounded_linear (\<lambda>x. c *\<^sub>R x)"
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   118
    by (rule scaleR.bounded_linear_right)
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huffman
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   119
  moreover have "bounded_linear f" ..
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   120
  ultimately show ?thesis
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diff changeset
   121
    by (rule bounded_linear_compose)
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huffman
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diff changeset
   122
qed
33741
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parents:
diff changeset
   123
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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   124
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"
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parents:
diff changeset
   125
  unfolding has_derivative_def apply(rule,rule bounded_linear.cmul)
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parents:
diff changeset
   126
  using assms[unfolded has_derivative_def] using Lim_cmul[OF assms[unfolded has_derivative_def,THEN conjunct2]]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   127
  unfolding scaleR_right_diff_distrib scaleR_right_distrib by auto 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   128
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   129
lemma has_derivative_cmul_eq: assumes "c \<noteq> 0" 
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parents:
diff changeset
   130
  shows "(((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net \<longleftrightarrow> (f has_derivative f') net)"
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hoelzl
parents:
diff changeset
   131
  apply(rule) defer apply(rule has_derivative_cmul,assumption) 
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hoelzl
parents:
diff changeset
   132
  apply(drule has_derivative_cmul[where c="1/c"]) using assms by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   133
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   134
lemma has_derivative_neg:
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parents:
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   135
 "(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   136
  apply(drule has_derivative_cmul[where c="-1"]) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   137
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   138
lemma has_derivative_neg_eq: "((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net \<longleftrightarrow> (f has_derivative f') net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   139
  apply(rule, drule_tac[!] has_derivative_neg) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   140
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   141
lemma has_derivative_add: assumes "(f has_derivative f') net" "(g has_derivative g') net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   142
  shows "((\<lambda>x. f(x) + g(x)) has_derivative (\<lambda>h. f'(h) + g'(h))) net" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   143
  note as = assms[unfolded has_derivative_def]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   144
  show ?thesis unfolding has_derivative_def apply(rule,rule bounded_linear_add)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   145
    using Lim_add[OF as(1)[THEN conjunct2] as(2)[THEN conjunct2]] and as
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   146
    by (auto simp add:algebra_simps scaleR_right_diff_distrib scaleR_right_distrib) qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   147
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
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   148
lemma has_derivative_add_const:"(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
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parents:
diff changeset
   149
  apply(drule has_derivative_add) apply(rule has_derivative_const) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   150
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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   151
lemma has_derivative_sub:
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   152
 "(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"
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
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diff changeset
   153
  apply(drule has_derivative_add) apply(drule has_derivative_neg,assumption) by(simp add:algebra_simps)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   154
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   155
lemma has_derivative_setsum: assumes "finite s" "\<forall>a\<in>s. ((f a) has_derivative (f' a)) net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   156
  shows "((\<lambda>x. setsum (\<lambda>a. f a x) s) has_derivative (\<lambda>h. setsum (\<lambda>a. f' a h) s)) net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   157
  apply(induct_tac s rule:finite_subset_induct[where A=s]) apply(rule assms(1)) 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   158
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" 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   159
  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"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   160
    unfolding setsum_insert[OF as(1,2)] apply-apply(rule has_derivative_add) apply(rule assms(2)[rule_format]) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   161
qed(auto intro!: has_derivative_const)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   162
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   163
lemma has_derivative_setsum_numseg:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   164
  "\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> ((f i) has_derivative (f' i)) net \<Longrightarrow>
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   165
  ((\<lambda>x. setsum (\<lambda>i. f i x) {m..n::nat}) has_derivative (\<lambda>h. setsum (\<lambda>i. f' i h) {m..n})) net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   166
  apply(rule has_derivative_setsum) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   167
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   168
subsection {* somewhat different results for derivative of scalar multiplier. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   169
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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   170
(** move **)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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diff changeset
   171
lemma linear_vmul_component:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   172
  assumes lf: "linear f"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   173
  shows "linear (\<lambda>x. f x $$ k *\<^sub>R v)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   174
  using lf
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   175
  by (auto simp add: linear_def algebra_simps)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   176
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   177
lemma bounded_linear_euclidean_component: "bounded_linear (\<lambda>x. x $$ k)"
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   178
  unfolding euclidean_component_def
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   179
  by (rule inner.bounded_linear_right)
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   180
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   181
lemma has_derivative_vmul_component: fixes c::"'a::real_normed_vector \<Rightarrow> 'b::euclidean_space" and v::"'c::real_normed_vector"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   182
  assumes "(c has_derivative c') net"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   183
  shows "((\<lambda>x. c(x)$$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$$k *\<^sub>R v)) net" proof-
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   184
  have *:"\<And>y. (c y $$ k *\<^sub>R v - (c (netlimit net) $$ k *\<^sub>R v + c' (y - netlimit net) $$ k *\<^sub>R v)) = 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   185
        (c y $$ k - (c (netlimit net) $$ k + c' (y - netlimit net) $$ k)) *\<^sub>R v" 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   186
    unfolding scaleR_left_diff_distrib scaleR_left_distrib by auto
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   187
  show ?thesis unfolding has_derivative_def and *
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   188
    apply (rule conjI)
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   189
    apply (rule bounded_linear_compose [OF scaleR.bounded_linear_left])
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   190
    apply (rule bounded_linear_compose [OF bounded_linear_euclidean_component])
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   191
    apply (rule derivative_linear [OF assms])
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   192
    apply(subst scaleR_zero_left[THEN sym, of v]) unfolding scaleR_scaleR apply(rule Lim_vmul)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   193
    using assms[unfolded has_derivative_def] unfolding Lim o_def apply- apply(cases "trivial_limit net")
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   194
    apply(rule,assumption,rule disjI2,rule,rule) proof-
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   195
    have *:"\<And>x. x - 0 = (x::'a)" by auto 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   196
    have **:"\<And>d x. d * (c x $$ k - (c (netlimit net) $$ k + c' (x - netlimit net) $$ k)) =
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   197
      (d *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net) ))) $$k" by(auto simp add:field_simps)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   198
    fix e assume "\<not> trivial_limit net" "0 < (e::real)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   199
    then have "eventually (\<lambda>x. dist ((1 / norm (x - netlimit net)) *\<^sub>R
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   200
      (c x - (c (netlimit net) + c' (x - netlimit net)))) 0 < e) net"
34103
9095ba4d2cd4 make proof use only abstract properties of eventually
huffman
parents: 33759
diff changeset
   201
      using assms[unfolded has_derivative_def Lim] by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   202
    thus "eventually (\<lambda>x. dist (1 / norm (x - netlimit net) *
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   203
      (c x $$ k - (c (netlimit net) $$ k + c' (x - netlimit net) $$ k))) 0 < e) net"
34103
9095ba4d2cd4 make proof use only abstract properties of eventually
huffman
parents: 33759
diff changeset
   204
      proof (rule eventually_elim1)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   205
      case goal1 thus ?case apply - unfolding dist_norm  apply(rule le_less_trans)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   206
        prefer 2 apply assumption unfolding * **
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   207
        using component_le_norm[of "(1 / norm (x - netlimit net)) *\<^sub>R
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   208
          (c x - (c (netlimit net) + c' (x - netlimit net))) - 0" k] by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   209
    qed
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   210
  qed
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   211
qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   212
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   213
lemma has_derivative_vmul_within: fixes c::"real \<Rightarrow> real"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   214
  assumes "(c has_derivative c') (at x within s)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   215
  shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x within s)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   216
  using has_derivative_vmul_component[OF assms, of 0 v] by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   217
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   218
lemma has_derivative_vmul_at: fixes c::"real \<Rightarrow> real"
33741
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   219
  assumes "(c has_derivative c') (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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diff changeset
   220
  shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   221
  using has_derivative_vmul_within[where s=UNIV] and assms by(auto simp add: within_UNIV)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   222
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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diff changeset
   223
lemma has_derivative_lift_dot:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   224
  assumes "(f has_derivative f') net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   225
  shows "((\<lambda>x. inner v (f x)) has_derivative (\<lambda>t. inner v (f' t))) net" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   226
  show ?thesis using assms unfolding has_derivative_def apply- apply(erule conjE,rule)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   227
    apply(rule bounded_linear_compose[of _ f']) apply(rule inner.bounded_linear_right,assumption)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   228
    apply(drule Lim_inner[where a=v]) unfolding o_def
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   229
    by(auto simp add:inner.scaleR_right inner.add_right inner.diff_right) qed
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diff changeset
   230
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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diff changeset
   231
lemmas has_derivative_intros = has_derivative_sub has_derivative_add has_derivative_cmul has_derivative_id has_derivative_const
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   232
   has_derivative_neg has_derivative_vmul_component has_derivative_vmul_at has_derivative_vmul_within has_derivative_cmul 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   233
   bounded_linear.has_derivative has_derivative_lift_dot
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   234
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   235
subsection {* limit transformation for derivatives. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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diff changeset
   236
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   237
lemma has_derivative_transform_within:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   238
  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)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   239
  shows "(g has_derivative f') (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   240
  using assms(4) unfolding has_derivative_within apply- apply(erule conjE,rule,assumption)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   241
  apply(rule Lim_transform_within[OF assms(1)]) defer apply assumption
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   242
  apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   243
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   244
lemma has_derivative_transform_at:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   245
  assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   246
  shows "(g has_derivative f') (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   247
  apply(subst within_UNIV[THEN sym]) apply(rule has_derivative_transform_within[OF assms(1)])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   248
  using assms(2-3) unfolding within_UNIV by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   249
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   250
lemma has_derivative_transform_within_open:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   251
  assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_derivative f') (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   252
  shows "(g has_derivative f') (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   253
  using assms(4) unfolding has_derivative_at apply- apply(erule conjE,rule,assumption)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   254
  apply(rule Lim_transform_within_open[OF assms(1,2)]) defer apply assumption
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   255
  apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   256
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   257
subsection {* differentiability. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   258
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36334
diff changeset
   259
no_notation Deriv.differentiable (infixl "differentiable" 60)
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36334
diff changeset
   260
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   261
definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "differentiable" 30) where
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   262
  "f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   263
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   264
definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   265
  "f differentiable_on s \<equiv> (\<forall>x\<in>s. f differentiable (at x within s))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   266
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   267
lemma differentiableI: "(f has_derivative f') net \<Longrightarrow> f differentiable net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   268
  unfolding differentiable_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   269
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   270
lemma differentiable_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   271
  unfolding differentiable_def using has_derivative_at_within by blast
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   272
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   273
lemma differentiable_within_open: assumes "a \<in> s" "open s" shows 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   274
  "f differentiable (at a within s) \<longleftrightarrow> (f differentiable (at a))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   275
  unfolding differentiable_def has_derivative_within_open[OF assms] by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   276
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   277
lemma differentiable_on_eq_differentiable_at: "open s \<Longrightarrow> (f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   278
  unfolding differentiable_on_def by(auto simp add: differentiable_within_open)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   279
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   280
lemma differentiable_transform_within:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   281
  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)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   282
  shows "g differentiable (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   283
  using assms(4) unfolding differentiable_def by(auto intro!: has_derivative_transform_within[OF assms(1-3)])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   284
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   285
lemma differentiable_transform_at:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   286
  assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable at x"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   287
  shows "g differentiable at x"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   288
  using assms(3) unfolding differentiable_def using has_derivative_transform_at[OF assms(1-2)] by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   289
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   290
subsection {* Frechet derivative and Jacobian matrix. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   291
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   292
definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   293
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   294
lemma frechet_derivative_works:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   295
 "f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   296
  unfolding frechet_derivative_def differentiable_def and some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
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parents:
diff changeset
   297
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   298
lemma linear_frechet_derivative:
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   299
  shows "f differentiable net \<Longrightarrow> linear(frechet_derivative f net)"
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   300
  unfolding frechet_derivative_works has_derivative_def by (auto intro: bounded_linear_imp_linear)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   301
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   302
subsection {* Differentiability implies continuity. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   303
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   304
lemma Lim_mul_norm_within: fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   305
  shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f(x)) ---> 0) (at a within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   306
  unfolding Lim_within apply(rule,rule) apply(erule_tac x=e in allE,erule impE,assumption,erule exE,erule conjE)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   307
  apply(rule_tac x="min d 1" in exI) apply rule defer apply(rule,erule_tac x=x in ballE) unfolding dist_norm diff_0_right
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   308
  by(auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   309
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   310
lemma differentiable_imp_continuous_within: assumes "f differentiable (at x within s)" 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   311
  shows "continuous (at x within s) f" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   312
  from assms guess f' unfolding differentiable_def has_derivative_within .. note f'=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   313
  then interpret bounded_linear f' by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   314
  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"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   315
    using zero by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   316
  have **:"continuous (at x within s) (f' \<circ> (\<lambda>y. y - x))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   317
    apply(rule continuous_within_compose) apply(rule continuous_intros)+
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   318
    by(rule linear_continuous_within[OF f'[THEN conjunct1]])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   319
  show ?thesis unfolding continuous_within using f'[THEN conjunct2, THEN Lim_mul_norm_within]
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   320
    apply-apply(drule Lim_add) apply(rule **[unfolded continuous_within]) unfolding Lim_within and dist_norm
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   321
    apply(rule,rule) apply(erule_tac x=e in allE) apply(erule impE|assumption)+ apply(erule exE,rule_tac x=d in exI)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   322
    by(auto simp add:zero * elim!:allE) qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   323
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   324
lemma differentiable_imp_continuous_at: "f differentiable at x \<Longrightarrow> continuous (at x) f"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   325
 by(rule differentiable_imp_continuous_within[of _ x UNIV, unfolded within_UNIV])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   326
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   327
lemma differentiable_imp_continuous_on: "f differentiable_on s \<Longrightarrow> continuous_on s f"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   328
  unfolding differentiable_on_def continuous_on_eq_continuous_within
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   329
  using differentiable_imp_continuous_within by blast
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   330
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   331
lemma has_derivative_within_subset:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   332
 "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   333
  unfolding has_derivative_within using Lim_within_subset by blast
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   334
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   335
lemma differentiable_within_subset:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   336
  "f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   337
  unfolding differentiable_def using has_derivative_within_subset by blast
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   338
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   339
lemma differentiable_on_subset: "f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   340
  unfolding differentiable_on_def using differentiable_within_subset by blast
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   341
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   342
lemma differentiable_on_empty: "f differentiable_on {}"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   343
  unfolding differentiable_on_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   344
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   345
subsection {* Several results are easier using a "multiplied-out" variant.              *)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   346
(* (I got this idea from Dieudonne's proof of the chain rule). *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   347
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   348
lemma has_derivative_within_alt:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   349
 "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   350
  (\<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")
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   351
proof assume ?lhs thus ?rhs unfolding has_derivative_within apply-apply(erule conjE,rule,assumption)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   352
    unfolding Lim_within apply(rule,erule_tac x=e in allE,rule,erule impE,assumption)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   353
    apply(erule exE,rule_tac x=d in exI) apply(erule conjE,rule,assumption,rule,rule) proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   354
    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>
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   355
      dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e" "y \<in> s" "bounded_linear f'"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   356
    then interpret bounded_linear f' by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   357
    show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)" proof(cases "y=x")
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   358
      case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero) next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   359
      case False hence "norm (f y - (f x + f' (y - x))) < e * norm (y - x)" using as(4)[rule_format, OF `y\<in>s`]
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   360
	unfolding dist_norm diff_0_right using as(3)
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   361
	using pos_divide_less_eq[OF False[unfolded dist_nz], unfolded dist_norm]
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   362
	by (auto simp add: linear_0 linear_sub)
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   363
      thus ?thesis by(auto simp add:algebra_simps) qed qed next
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   364
  assume ?rhs thus ?lhs unfolding has_derivative_within Lim_within apply-apply(erule conjE,rule,assumption)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   365
    apply(rule,erule_tac x="e/2" in allE,rule,erule impE) defer apply(erule exE,rule_tac x=d in exI)
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   366
    apply(erule conjE,rule,assumption,rule,rule) unfolding dist_norm diff_0_right norm_scaleR
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   367
    apply(erule_tac x=xa in ballE,erule impE) proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   368
    fix e d y assume "bounded_linear f'" "0 < e" "0 < d" "y \<in> s" "0 < norm (y - x) \<and> norm (y - x) < d"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   369
        "norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   370
    thus "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   371
      apply(rule_tac le_less_trans[of _ "e/2"]) by(auto intro!:mult_imp_div_pos_le simp add:algebra_simps) qed auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   372
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   373
lemma has_derivative_at_alt:
35172
579dd5570f96 Added integration to Multivariate-Analysis (upto FTC)
himmelma
parents: 35028
diff changeset
   374
  "(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   375
  (\<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))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   376
  using has_derivative_within_alt[where s=UNIV] unfolding within_UNIV by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   377
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   378
subsection {* The chain rule. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   379
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   380
lemma diff_chain_within:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   381
  assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at (f x) within (f ` s))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   382
  shows "((g o f) has_derivative (g' o f'))(at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   383
  unfolding has_derivative_within_alt apply(rule,rule bounded_linear_compose[unfolded o_def[THEN sym]])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   384
  apply(rule assms(2)[unfolded has_derivative_def,THEN conjE],assumption)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   385
  apply(rule assms(1)[unfolded has_derivative_def,THEN conjE],assumption) proof(rule,rule)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   386
  note assms = assms[unfolded has_derivative_within_alt]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   387
  fix e::real assume "0<e"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   388
  guess B1 using bounded_linear.pos_bounded[OF assms(1)[THEN conjunct1, rule_format]] .. note B1 = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   389
  guess B2 using bounded_linear.pos_bounded[OF assms(2)[THEN conjunct1, rule_format]] .. note B2 = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   390
  have *:"e / 2 / B2 > 0" using `e>0` B2 apply-apply(rule divide_pos_pos) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   391
  guess d1 using assms(1)[THEN conjunct2, rule_format, OF *] .. note d1 = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   392
  have *:"e / 2 / (1 + B1) > 0" using `e>0` B1 apply-apply(rule divide_pos_pos) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   393
  guess de using assms(2)[THEN conjunct2, rule_format, OF *] .. note de = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   394
  guess d2 using assms(1)[THEN conjunct2, rule_format, OF zero_less_one] .. note d2 = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   395
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   396
  def d0 \<equiv> "(min d1 d2)/2" have d0:"d0>0" "d0 < d1" "d0 < d2" unfolding d0_def using d1 d2 by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   397
  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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   398
  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)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   399
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   400
  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)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   401
    proof(rule,rule `d>0`,rule,rule) 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   402
    fix y assume as:"y \<in> s" "norm (y - x) < d"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   403
    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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   404
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   405
    have "norm (f y - f x) \<le> norm (f y - f x - f' (y - x)) + norm (f' (y - x))"
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   406
      using norm_triangle_sub[of "f y - f x" "f' (y - x)"] by(auto simp add:algebra_simps)
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   407
    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:algebra_simps)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   408
    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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   409
    also have "\<dots> \<le> norm (y - x) + B1 * norm (y - x)" by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   410
    also have "\<dots> = norm (y - x) * (1 + B1)" by(auto simp add:field_simps)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   411
    finally have 3:"norm (f y - f x) \<le> norm (y - x) * (1 + B1)" by auto 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   412
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   413
    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 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   414
    also have "\<dots> < de" using d B1 by(auto simp add:field_simps) 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   415
    finally have "norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 / (1 + B1) * norm (f y - f x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   416
      apply-apply(rule de[THEN conjunct2,rule_format]) using `y\<in>s` using d as by auto 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   417
    also have "\<dots> = (e / 2) * (1 / (1 + B1) * norm (f y - f x))" by auto 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   418
    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)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   419
    finally have 4:"norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   420
    
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   421
    interpret g': bounded_linear g' using assms(2) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   422
    interpret f': bounded_linear f' using assms(1) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   423
    have "norm (- g' (f' (y - x)) + g' (f y - f x)) = norm (g' (f y - f x - f' (y - x)))"
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   424
      by(auto simp add:algebra_simps f'.diff g'.diff g'.add)
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   425
    also have "\<dots> \<le> B2 * norm (f y - f x - f' (y - x))" using B2 by(auto simp add:algebra_simps)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   426
    also have "\<dots> \<le> B2 * (e / 2 / B2 * norm (y - x))" apply(rule mult_left_mono) using as d1 d2 d B2 by auto 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   427
    also have "\<dots> \<le> e / 2 * norm (y - x)" using B2 by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   428
    finally have 5:"norm (- g' (f' (y - x)) + g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   429
    
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   430
    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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   431
    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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   432
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   433
lemma diff_chain_at:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   434
  "(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)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   435
  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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   436
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   437
subsection {* Composition rules stated just for differentiability. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   438
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   439
lemma differentiable_const[intro]: "(\<lambda>z. c) differentiable (net::'a::real_normed_vector net)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   440
  unfolding differentiable_def using has_derivative_const by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   441
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   442
lemma differentiable_id[intro]: "(\<lambda>z. z) differentiable (net::'a::real_normed_vector net)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   443
    unfolding differentiable_def using has_derivative_id by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   444
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   445
lemma differentiable_cmul[intro]: "f differentiable net \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector net)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   446
  unfolding differentiable_def apply(erule exE, drule has_derivative_cmul) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   447
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   448
lemma differentiable_neg[intro]: "f differentiable net \<Longrightarrow> (\<lambda>z. -(f z)) differentiable (net::'a::real_normed_vector net)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   449
  unfolding differentiable_def apply(erule exE, drule has_derivative_neg) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   450
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   451
lemma differentiable_add: "f differentiable net \<Longrightarrow> g differentiable net
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   452
   \<Longrightarrow> (\<lambda>z. f z + g z) differentiable (net::'a::real_normed_vector net)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   453
    unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z + f'a z" in exI)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   454
    apply(rule has_derivative_add) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   455
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   456
lemma differentiable_sub: "f differentiable net \<Longrightarrow> g differentiable net
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   457
  \<Longrightarrow> (\<lambda>z. f z - g z) differentiable (net::'a::real_normed_vector net)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   458
  unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z - f'a z" in exI)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   459
    apply(rule has_derivative_sub) by auto 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   460
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   461
lemma differentiable_setsum:
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   462
  assumes "finite s" "\<forall>a\<in>s. (f a) differentiable net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   463
  shows "(\<lambda>x. setsum (\<lambda>a. f a x) s) differentiable net" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   464
  guess f' using bchoice[OF assms(2)[unfolded differentiable_def]] ..
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   465
  thus ?thesis unfolding differentiable_def apply- apply(rule,rule has_derivative_setsum[where f'=f'],rule assms(1)) by auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   466
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   467
lemma differentiable_setsum_numseg:
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   468
  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"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   469
  apply(rule differentiable_setsum) using finite_atLeastAtMost[of n m] by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   470
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   471
lemma differentiable_chain_at:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   472
  "f differentiable (at x) \<Longrightarrow> g differentiable (at(f x)) \<Longrightarrow> (g o f) differentiable (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   473
  unfolding differentiable_def by(meson diff_chain_at)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   474
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   475
lemma differentiable_chain_within:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   476
  "f differentiable (at x within s) \<Longrightarrow> g differentiable (at(f x) within (f ` s))
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   477
   \<Longrightarrow> (g o f) differentiable (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   478
  unfolding differentiable_def by(meson diff_chain_within)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   479
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   480
subsection {* Uniqueness of derivative.                                                 *)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   481
(*                                                                           *)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   482
(* The general result is a bit messy because we need approachability of the  *)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   483
(* limit point from any direction. But OK for nontrivial intervals etc. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   484
    
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   485
lemma frechet_derivative_unique_within: fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   486
  assumes "(f has_derivative f') (at x within s)" "(f has_derivative f'') (at x within s)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   487
  "(\<forall>i<DIM('a). \<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-
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   488
  note as = assms(1,2)[unfolded has_derivative_def]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   489
  then interpret f': bounded_linear f' by auto from as interpret f'': bounded_linear f'' by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   490
  have "x islimpt s" unfolding islimpt_approachable proof(rule,rule)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   491
    fix e::real assume "0<e" guess d using assms(3)[rule_format,OF DIM_positive `e>0`] ..
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   492
    thus "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x="x + d *\<^sub>R basis 0" in bexI)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   493
      unfolding dist_norm by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   494
  hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within) unfolding trivial_limit_within by simp
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   495
  show ?thesis  apply(rule linear_eq_stdbasis) unfolding linear_conv_bounded_linear
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   496
    apply(rule as(1,2)[THEN conjunct1])+ proof(rule,rule,rule ccontr)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   497
    fix i assume i:"i<DIM('a)" def e \<equiv> "norm (f' (basis i) - f'' (basis i))"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   498
    assume "f' (basis i) \<noteq> f'' (basis i)" hence "e>0" unfolding e_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   499
    guess d using Lim_sub[OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   500
    guess c using assms(3)[rule_format,OF i d[THEN conjunct1]] .. note c=this
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   501
    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)))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   502
      unfolding scaleR_right_distrib by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   503
    also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' (basis i)) + f'' (basis i))))"  
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   504
      unfolding f'.scaleR f''.scaleR unfolding scaleR_right_distrib scaleR_minus_right by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   505
    also have "\<dots> = e" unfolding e_def using c[THEN conjunct1] using norm_minus_cancel[of "f' (basis i) - f'' (basis i)"] by (auto simp add: add.commute ab_diff_minus)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   506
    finally show False using c using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R basis i"] 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   507
      unfolding dist_norm unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   508
        scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib using i by auto qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   509
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   510
lemma frechet_derivative_unique_at: fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   511
  shows "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   512
  apply(rule frechet_derivative_unique_within[of f f' x UNIV f'']) unfolding within_UNIV apply(assumption)+
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   513
  apply(rule,rule,rule,rule) apply(rule_tac x="e/2" in exI) by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   514
 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   515
lemma "isCont f x = continuous (at x) f" unfolding isCont_def LIM_def
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   516
  unfolding continuous_at Lim_at unfolding dist_nz by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   517
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   518
lemma frechet_derivative_unique_within_closed_interval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   519
  assumes "\<forall>i<DIM('a). a$$i < b$$i" "x \<in> {a..b}" (is "x\<in>?I") and
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   520
  "(f has_derivative f' ) (at x within {a..b})" and
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   521
  "(f has_derivative f'') (at x within {a..b})"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   522
  shows "f' = f''" apply(rule frechet_derivative_unique_within) apply(rule assms(3,4))+ proof(rule,rule,rule,rule)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   523
  fix e::real and i assume "e>0" and i:"i<DIM('a)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   524
  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")
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   525
    case True thus ?thesis apply(rule_tac x="(min (b$$i - a$$i)  e) / 2" in exI)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   526
      using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   527
      unfolding mem_interval euclidean_simps basis_component using i by(auto simp add:field_simps)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   528
  next note * = assms(2)[unfolded mem_interval,THEN spec[where x=i]]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   529
    case False moreover have "a $$ i < x $$ i" using False * by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   530
    moreover { have "a $$ i * 2 + min (x $$ i - a $$ i) e \<le> a$$i *2 + x$$i - a$$i" by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   531
    also have "\<dots> = a$$i + x$$i" by auto also have "\<dots> \<le> 2 * x$$i" using * by auto 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   532
    finally have "a $$ i * 2 + min (x $$ i - a $$ i) e \<le> x $$ i * 2" by auto }
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   533
    moreover have "min (x $$ i - a $$ i) e \<ge> 0" using * and `e>0` by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   534
    hence "x $$ i * 2 \<le> b $$ i * 2 + min (x $$ i - a $$ i) e" using * by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   535
    ultimately show ?thesis apply(rule_tac x="- (min (x$$i - a$$i) e) / 2" in exI)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   536
      using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   537
      unfolding mem_interval euclidean_simps basis_component using i by(auto simp add:field_simps) qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   538
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   539
lemma frechet_derivative_unique_within_open_interval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   540
  assumes "x \<in> {a<..<b}" "(f has_derivative f' ) (at x within {a<..<b})"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   541
                         "(f has_derivative f'') (at x within {a<..<b})"
37650
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
   542
  shows "f' = f''"
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
   543
proof -
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
   544
  from assms(1) have *: "at x within {a<..<b} = at x"
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
   545
    by (simp add: at_within_interior interior_open open_interval)
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
   546
  from assms(2,3) [unfolded *] show "f' = f''"
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
   547
    by (rule frechet_derivative_unique_at)
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
   548
qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   549
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   550
lemma frechet_derivative_at: fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   551
  shows "(f has_derivative f') (at x) \<Longrightarrow> (f' = frechet_derivative f (at x))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   552
  apply(rule frechet_derivative_unique_at[of f],assumption)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   553
  unfolding frechet_derivative_works[THEN sym] using differentiable_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   554
37648
41b7dfdc4941 generalize more euclidean_space lemmas
huffman
parents: 37606
diff changeset
   555
lemma frechet_derivative_within_closed_interval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   556
  assumes "\<forall>i<DIM('a). a$$i < b$$i" "x \<in> {a..b}" "(f has_derivative f') (at x within {a.. b})"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   557
  shows "frechet_derivative f (at x within {a.. b}) = f'"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   558
  apply(rule frechet_derivative_unique_within_closed_interval[where f=f]) 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   559
  apply(rule assms(1,2))+ unfolding frechet_derivative_works[THEN sym]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   560
  unfolding differentiable_def using assms(3) by auto 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   561
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   562
subsection {* The traditional Rolle theorem in one dimension. *}
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   563
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   564
lemma linear_componentwise:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   565
  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   566
  assumes lf: "linear f"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   567
  shows "(f x) $$ j = (\<Sum>i<DIM('a). (x$$i) * (f (basis i)$$j))" (is "?lhs = ?rhs")
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   568
proof -
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   569
  have fA: "finite {..<DIM('a)}" by simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   570
  have "?rhs = (\<Sum>i<DIM('a). x$$i *\<^sub>R f (basis i))$$j"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   571
    by (simp add: euclidean_simps)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   572
  then show ?thesis
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   573
    unfolding linear_setsum_mul[OF lf fA, symmetric]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   574
    unfolding euclidean_representation[symmetric] ..
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   575
qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   576
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   577
text {* We do not introduce @{text jacobian}, which is defined on matrices, instead we use
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   578
  the unfolding of it. *}
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   579
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   580
lemma jacobian_works:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   581
  "(f::('a::euclidean_space) \<Rightarrow> ('b::euclidean_space)) differentiable net \<longleftrightarrow>
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   582
   (f has_derivative (\<lambda>h. \<chi>\<chi> i.
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   583
      \<Sum>j<DIM('a). frechet_derivative f net (basis j) $$ i * h $$ j)) net"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   584
  (is "?differentiable \<longleftrightarrow> (f has_derivative (\<lambda>h. \<chi>\<chi> i. ?SUM h i)) net")
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   585
proof
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   586
  assume *: ?differentiable
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   587
  { fix h i
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   588
    have "?SUM h i = frechet_derivative f net h $$ i" using *
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   589
      by (auto intro!: setsum_cong
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   590
               simp: linear_componentwise[of _ h i] linear_frechet_derivative) }
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   591
  thus "(f has_derivative (\<lambda>h. \<chi>\<chi> i. ?SUM h i)) net"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   592
    using * by (simp add: frechet_derivative_works)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   593
qed (auto intro!: differentiableI)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   594
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   595
lemma differential_zero_maxmin_component:
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   596
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   597
  assumes k: "k < DIM('b)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   598
    and ball: "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)$$k))"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   599
    and diff: "f differentiable (at x)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   600
  shows "(\<chi>\<chi> j. frechet_derivative f (at x) (basis j) $$ k) = (0::'a)" (is "?D k = 0")
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   601
proof (rule ccontr)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   602
  assume "?D k \<noteq> 0"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   603
  then obtain j where j: "?D k $$ j \<noteq> 0" "j < DIM('a)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   604
    unfolding euclidean_lambda_beta euclidean_eq[of _ "0::'a"] by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   605
  hence *: "\<bar>?D k $$ j\<bar> / 2 > 0" by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   606
  note as = diff[unfolded jacobian_works has_derivative_at_alt]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   607
  guess e' using as[THEN conjunct2, rule_format, OF *] .. note e' = this
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   608
  guess d using real_lbound_gt_zero[OF ball(1) e'[THEN conjunct1]] .. note d = this
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   609
  { fix c assume "abs c \<le> d"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   610
    hence *:"norm (x + c *\<^sub>R basis j - x) < e'" using norm_basis[of j] d by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   611
    let ?v = "(\<chi>\<chi> i. \<Sum>l<DIM('a). ?D i $$ l * (c *\<^sub>R basis j :: 'a) $$ l)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   612
    have if_dist: "\<And> P a b c. a * (if P then b else c) = (if P then a * b else a * c)" by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   613
    have "\<bar>(f (x + c *\<^sub>R basis j) - f x - ?v) $$ k\<bar> \<le>
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   614
        norm (f (x + c *\<^sub>R basis j) - f x - ?v)" by (rule component_le_norm)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   615
    also have "\<dots> \<le> \<bar>?D k $$ j\<bar> / 2 * \<bar>c\<bar>"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   616
      using e'[THEN conjunct2, rule_format, OF *] and norm_basis[of j] by fastsimp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   617
    finally have "\<bar>(f (x + c *\<^sub>R basis j) - f x - ?v) $$ k\<bar> \<le> \<bar>?D k $$ j\<bar> / 2 * \<bar>c\<bar>" by simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   618
    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>"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   619
      unfolding euclidean_simps euclidean_lambda_beta using j k
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   620
      by (simp add: if_dist setsum_cases field_simps) } note * = this
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   621
  have "x + d *\<^sub>R basis j \<in> ball x e" "x - d *\<^sub>R basis j \<in> ball x e"
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   622
    unfolding mem_ball dist_norm using norm_basis[of j] d by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   623
  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>
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   624
         ((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 ball by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   625
  have ***: "\<And>y y1 y2 d dx::real.
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   626
    (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
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   627
  show False apply(rule ***[OF **, where dx="d * ?D k $$ j" and d="\<bar>?D k $$ j\<bar> / 2 * \<bar>d\<bar>"])
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   628
    using *[of "-d"] and *[of d] and d[THEN conjunct1] and j unfolding mult_minus_left
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   629
    unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding algebra_simps by (auto intro: mult_pos_pos)
34906
bb9dad7de515 spurious proof failure
haftmann
parents: 34291
diff changeset
   630
qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   631
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   632
subsection {* In particular if we have a mapping into @{typ "real"}. *}
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   633
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   634
lemma differential_zero_maxmin:
37650
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
   635
  fixes f::"'a\<Colon>euclidean_space \<Rightarrow> real"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   636
  assumes "x \<in> s" "open s"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   637
  and deriv: "(f has_derivative f') (at x)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   638
  and mono: "(\<forall>y\<in>s. f y \<le> f x) \<or> (\<forall>y\<in>s. f x \<le> f y)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   639
  shows "f' = (\<lambda>v. 0)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   640
proof -
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   641
  obtain e where e:"e>0" "ball x e \<subseteq> s" using `open s`[unfolded open_contains_ball] and `x \<in> s` by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   642
  with differential_zero_maxmin_component[where 'b=real, of 0 e x f, simplified]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   643
  have "(\<chi>\<chi> j. frechet_derivative f (at x) (basis j)) = (0::'a)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   644
    unfolding differentiable_def using mono deriv by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   645
  with frechet_derivative_at[OF deriv, symmetric]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   646
  have "\<forall>i<DIM('a). f' (basis i) = 0"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   647
    by (simp add: euclidean_eq[of _ "0::'a"])
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   648
  with derivative_is_linear[OF deriv, THEN linear_componentwise, of _ 0]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   649
  show ?thesis by (simp add: expand_fun_eq)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   650
qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   651
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   652
lemma rolle: fixes f::"real\<Rightarrow>real"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   653
  assumes "a < b" "f a = f b" "continuous_on {a..b} f"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   654
  "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   655
  shows "\<exists>x\<in>{a<..<b}. f' x = (\<lambda>v. 0)" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   656
  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-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   657
    have "(a + b) / 2 \<in> {a .. b}" using assms(1) by auto hence *:"{a .. b}\<noteq>{}" by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   658
    guess d using continuous_attains_sup[OF compact_interval * assms(3)] .. note d=this
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   659
    guess c using continuous_attains_inf[OF compact_interval * assms(3)] .. note c=this
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   660
    show ?thesis proof(cases "d\<in>{a<..<b} \<or> c\<in>{a<..<b}")
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   661
      case True thus ?thesis apply(erule_tac disjE) apply(rule_tac x=d in bexI)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   662
	apply(rule_tac[3] x=c in bexI) using d c by auto next def e \<equiv> "(a + b) /2"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   663
      case False hence "f d = f c" using d c assms(2) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   664
      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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   665
      thus ?thesis apply(rule_tac x=e in bexI) unfolding e_def using assms(1) by auto qed qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   666
  then guess x .. note x=this
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   667
  hence "f' x = (\<lambda>v. 0)" apply(rule_tac differential_zero_maxmin[of x "{a<..<b}" f "f' x"])
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   668
    defer apply(rule open_interval)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   669
    apply(rule assms(4)[unfolded has_derivative_at[THEN sym],THEN bspec[where x=x]],assumption)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   670
    unfolding o_def apply(erule disjE,rule disjI2) by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   671
  thus ?thesis apply(rule_tac x=x in bexI) unfolding o_def apply rule
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   672
    apply(drule_tac x=v in fun_cong) using x(1) by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   673
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   674
subsection {* One-dimensional mean value theorem. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   675
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   676
lemma mvt: fixes f::"real \<Rightarrow> real"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   677
  assumes "a < b" "continuous_on {a .. b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative (f' x)) (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   678
  shows "\<exists>x\<in>{a<..<b}. (f b - f a = (f' x) (b - a))" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   679
  have "\<exists>x\<in>{a<..<b}. (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   680
    apply(rule rolle[OF assms(1), of "\<lambda>x. f x - (f b - f a) / (b - a) * x"]) defer
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   681
    apply(rule continuous_on_intros assms(2) continuous_on_cmul[where 'b=real, unfolded real_scaleR_def])+ proof
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   682
    fix x assume x:"x \<in> {a<..<b}"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   683
    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)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   684
      by(rule has_derivative_intros assms(3)[rule_format,OF x]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   685
        has_derivative_cmul[where 'b=real, unfolded real_scaleR_def])+ 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   686
  qed(insert assms(1), auto simp add:field_simps)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   687
  then guess x .. thus ?thesis apply(rule_tac x=x in bexI) apply(drule fun_cong[of _ _ "b - a"]) by auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   688
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   689
lemma mvt_simple: fixes f::"real \<Rightarrow> real"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   690
  assumes "a<b"  "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   691
  shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   692
  apply(rule mvt) apply(rule assms(1), rule differentiable_imp_continuous_on)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   693
  unfolding differentiable_on_def differentiable_def defer proof 
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   694
  fix x assume x:"x \<in> {a<..<b}" show "(f has_derivative f' x) (at x)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   695
    unfolding has_derivative_within_open[OF x open_interval,THEN sym] 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   696
    apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using x by auto qed(insert assms(2), auto)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   697
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   698
lemma mvt_very_simple: fixes f::"real \<Rightarrow> real"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   699
  assumes "a \<le> b" "\<forall>x\<in>{a..b}. (f has_derivative f'(x)) (at x within {a..b})"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   700
  shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)" proof(cases "a = b")
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   701
  interpret bounded_linear "f' b" using assms(2) assms(1) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   702
  case True thus ?thesis apply(rule_tac x=a in bexI)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   703
    using assms(2)[THEN bspec[where x=a]] unfolding has_derivative_def
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   704
    unfolding True using zero by auto next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   705
  case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   706
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   707
subsection {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   708
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   709
lemma mvt_general: fixes f::"real\<Rightarrow>'a::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   710
  assumes "a<b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   711
  shows "\<exists>x\<in>{a<..<b}. norm(f b - f a) \<le> norm(f'(x) (b - a))" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   712
  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)"
35542
8f97d8caabfd replaced \<bullet> with inner
himmelma
parents: 35290
diff changeset
   713
    apply(rule mvt) apply(rule assms(1)) apply(rule continuous_on_inner continuous_on_intros assms(2))+ 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   714
    unfolding o_def apply(rule,rule has_derivative_lift_dot) using assms(3) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   715
  then guess x .. note x=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   716
  show ?thesis proof(cases "f a = f b")
36844
5f9385ecc1a7 Removed usage of normalizating locales.
hoelzl
parents: 36725
diff changeset
   717
    case False
5f9385ecc1a7 Removed usage of normalizating locales.
hoelzl
parents: 36725
diff changeset
   718
    have "norm (f b - f a) * norm (f b - f a) = norm (f b - f a)^2" by(simp add: power2_eq_square)
35542
8f97d8caabfd replaced \<bullet> with inner
himmelma
parents: 35290
diff changeset
   719
    also have "\<dots> = (f b - f a) \<bullet> (f b - f a)" unfolding power2_norm_eq_inner ..
37606
b47dd044a1f1 inner_simps is not enough, need also local facts
haftmann
parents: 37489
diff changeset
   720
    also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)" using x unfolding inner_simps by (auto simp add: inner_diff_left)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   721
    also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))" by(rule norm_cauchy_schwarz)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   722
    finally show ?thesis using False x(1) by(auto simp add: real_mult_left_cancel) next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   723
    case True thus ?thesis using assms(1) apply(rule_tac x="(a + b) /2" in bexI) by auto qed qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   724
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   725
subsection {* Still more general bound theorem. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   726
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   727
lemma differentiable_bound: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   728
  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"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   729
  shows "norm(f x - f y) \<le> B * norm(x - y)" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   730
  let ?p = "\<lambda>u. x + u *\<^sub>R (y - x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   731
  have *:"\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   732
    using assms(1)[unfolded convex_alt,rule_format,OF x y] unfolding scaleR_left_diff_distrib scaleR_right_diff_distrib by(auto simp add:algebra_simps)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   733
  hence 1:"continuous_on {0..1} (f \<circ> ?p)" apply- apply(rule continuous_on_intros continuous_on_vmul)+
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   734
    unfolding continuous_on_eq_continuous_within apply(rule,rule differentiable_imp_continuous_within)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   735
    unfolding differentiable_def apply(rule_tac x="f' xa" in exI)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   736
    apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   737
  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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   738
    let ?u = "x + u *\<^sub>R (y - x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   739
    have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})" 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   740
      apply(rule diff_chain_within) apply(rule has_derivative_intros)+ 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   741
      apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using goal1 * by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   742
    thus ?case unfolding has_derivative_within_open[OF goal1 open_interval] by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   743
  guess u using mvt_general[OF zero_less_one 1 2] .. note u = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   744
  have **:"\<And>x y. x\<in>s \<Longrightarrow> norm (f' x y) \<le> B * norm y" proof- case goal1
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   745
    have "norm (f' x y) \<le> onorm (f' x) * norm y"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   746
      using onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]] by assumption
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   747
    also have "\<dots> \<le> B * norm y" apply(rule mult_right_mono)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   748
      using assms(3)[rule_format,OF goal1] by(auto simp add:field_simps)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   749
    finally show ?case by simp qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   750
  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)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   751
    by(auto simp add:norm_minus_commute) 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   752
  also have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))" using u by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   753
  also have "\<dots> \<le> B * norm(y - x)" apply(rule **) using * and u by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   754
  finally show ?thesis by(auto simp add:norm_minus_commute) qed 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   755
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   756
lemma differentiable_bound_real: fixes f::"real \<Rightarrow> real"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   757
  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"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   758
  shows "norm(f x - f y) \<le> B * norm(x - y)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   759
  using differentiable_bound[of s f f' B x y]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   760
  unfolding Ball_def image_iff o_def using assms by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   761
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   762
subsection {* In particular. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   763
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   764
lemma has_derivative_zero_constant: fixes f::"real\<Rightarrow>real"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   765
  assumes "convex s" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   766
  shows "\<exists>c. \<forall>x\<in>s. f x = c" proof(cases "s={}")
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   767
  case False then obtain x where "x\<in>s" by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   768
  have "\<And>y. y\<in>s \<Longrightarrow> f x = f y" proof- case goal1
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   769
    thus ?case using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\<in>s`
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   770
    unfolding onorm_const by auto qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   771
  thus ?thesis apply(rule_tac x="f x" in exI) by auto qed auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   772
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   773
lemma has_derivative_zero_unique: fixes f::"real\<Rightarrow>real"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   774
  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"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   775
  shows "f x = c" using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   776
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   777
subsection {* Differentiability of inverse function (most basic form). *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   778
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   779
lemma has_derivative_inverse_basic: fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   780
  assumes "(f has_derivative f') (at (g y))" "bounded_linear g'" "g' \<circ> f' = id" "continuous (at y) g"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   781
  "open t" "y \<in> t" "\<forall>z\<in>t. f(g z) = z"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   782
  shows "(g has_derivative g') (at y)" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   783
  interpret f': bounded_linear f' using assms unfolding has_derivative_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   784
  interpret g': bounded_linear g' using assms by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   785
  guess C using bounded_linear.pos_bounded[OF assms(2)] .. note C = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   786
(*  have fgid:"\<And>x. g' (f' x) = x" using assms(3) unfolding o_def id_def apply()*)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   787
  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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   788
    have *:"e / C > 0" apply(rule divide_pos_pos) using `e>0` C by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   789
    guess d0 using assms(1)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note d0=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   790
    guess d1 using assms(4)[unfolded continuous_at Lim_at,rule_format,OF d0[THEN conjunct1]] .. note d1=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   791
    guess d2 using assms(5)[unfolded open_dist,rule_format,OF assms(6)] .. note d2=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   792
    guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   793
    thus ?case apply(rule_tac x=d in exI) apply rule defer proof(rule,rule)
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   794
      fix z assume as:"norm (z - y) < d" hence "z\<in>t" using d2 d unfolding dist_norm by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   795
      have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   796
        unfolding g'.diff f'.diff unfolding assms(3)[unfolded o_def id_def, THEN fun_cong] 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   797
	unfolding assms(7)[rule_format,OF `z\<in>t`] apply(subst norm_minus_cancel[THEN sym]) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   798
      also have "\<dots> \<le> norm(f (g z) - y - f' (g z - g y)) * C" by(rule C[THEN conjunct2,rule_format]) 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   799
      also have "\<dots> \<le> (e / C) * norm (g z - g y) * C" apply(rule mult_right_mono)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   800
	apply(rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]]) apply(cases "z=y") defer
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   801
	apply(rule d1[THEN conjunct2, unfolded dist_norm,rule_format]) using as d C d0 by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   802
      also have "\<dots> \<le> e * norm (g z - g y)" using C by(auto simp add:field_simps)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   803
      finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)" by simp qed auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   804
  have *:"(0::real) < 1 / 2" by auto guess d using lem1[rule_format,OF *] .. note d=this def B\<equiv>"C*2"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   805
  have "B>0" unfolding B_def using C by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   806
  have lem2:"\<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y) \<le> B * norm(z - y)" proof(rule,rule) case goal1
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   807
    have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))" by(rule norm_triangle_sub)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   808
    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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   809
    also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)" apply(rule add_right_mono) using C by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   810
    finally show ?case unfolding B_def by(auto simp add:field_simps) qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   811
  show ?thesis unfolding has_derivative_at_alt proof(rule,rule assms,rule,rule) case goal1
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   812
    hence *:"e/B >0" apply-apply(rule divide_pos_pos) using `B>0` by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   813
    guess d' using lem1[rule_format,OF *] .. note d'=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   814
    guess k using real_lbound_gt_zero[OF d[THEN conjunct1] d'[THEN conjunct1]] .. note k=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   815
    show ?case apply(rule_tac x=k in exI,rule) defer proof(rule,rule) fix z assume as:"norm(z - y) < k"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   816
      hence "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)" using d' k by auto
36721
bfd7f5c3bf69 former free-floating field_comp_conv now in structure Normalizer
haftmann
parents: 36593
diff changeset
   817
      also have "\<dots> \<le> e * norm(z - y)" unfolding times_divide_eq_left pos_divide_le_eq[OF `B>0`]
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   818
	using lem2[THEN spec[where x=z]] using k as using `e>0` by(auto simp add:field_simps)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   819
      finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)" by simp qed(insert k, auto) qed qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   820
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   821
subsection {* Simply rewrite that based on the domain point x. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   822
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   823
lemma has_derivative_inverse_basic_x: fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   824
  assumes "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   825
  "continuous (at (f x)) g" "g(f x) = x" "open t" "f x \<in> t" "\<forall>y\<in>t. f(g y) = y"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   826
  shows "(g has_derivative g') (at (f(x)))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   827
  apply(rule has_derivative_inverse_basic) using assms by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   828
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   829
subsection {* This is the version in Dieudonne', assuming continuity of f and g. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   830
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   831
lemma has_derivative_inverse_dieudonne: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   832
  assumes "open s" "open (f ` s)" "continuous_on s f" "continuous_on (f ` s) g" "\<forall>x\<in>s. g(f x) = x"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   833
  (**) "x\<in>s" "(f has_derivative f') (at x)"  "bounded_linear g'" "g' o f' = id"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   834
  shows "(g has_derivative g') (at (f x))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   835
  apply(rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   836
  using assms(3-6) unfolding continuous_on_eq_continuous_at[OF assms(1)]  continuous_on_eq_continuous_at[OF assms(2)] by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   837
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   838
subsection {* Here's the simplest way of not assuming much about g. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   839
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   840
lemma has_derivative_inverse: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   841
  assumes "compact s" "x \<in> s" "f x \<in> interior(f ` s)" "continuous_on s f"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   842
  "\<forall>y\<in>s. g(f y) = y" "(f has_derivative f') (at x)" "bounded_linear g'" "g' \<circ> f' = id"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   843
  shows "(g has_derivative g') (at (f x))" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   844
  { fix y assume "y\<in>interior (f ` s)" 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   845
    then obtain x where "x\<in>s" and *:"y = f x" unfolding image_iff using interior_subset by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   846
    have "f (g y) = y" unfolding * and assms(5)[rule_format,OF `x\<in>s`] .. } note * = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   847
  show ?thesis apply(rule has_derivative_inverse_basic_x[OF assms(6-8)])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   848
    apply(rule continuous_on_interior[OF _ assms(3)]) apply(rule continuous_on_inverse[OF assms(4,1)])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   849
    apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+ by(rule, rule *, assumption)  qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   850
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   851
subsection {* Proving surjectivity via Brouwer fixpoint theorem. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   852
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   853
lemma brouwer_surjective: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   854
  assumes "compact t" "convex t"  "t \<noteq> {}" "continuous_on t f"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   855
  "\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t" "x\<in>s"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   856
  shows "\<exists>y\<in>t. f y = x" proof-
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   857
  have *:"\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y" by(auto simp add:algebra_simps)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   858
  show ?thesis  unfolding * apply(rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   859
    apply(rule continuous_on_intros assms)+ using assms(4-6) by auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   860
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   861
lemma brouwer_surjective_cball: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   862
  assumes "0 < e" "continuous_on (cball a e) f"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   863
  "\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e" "x\<in>s"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   864
  shows "\<exists>y\<in>cball a e. f y = x" apply(rule brouwer_surjective) apply(rule compact_cball convex_cball)+
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   865
  unfolding cball_eq_empty using assms by auto 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   866
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   867
text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   868
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   869
lemma sussmann_open_mapping: fixes f::"'a::euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   870
  assumes "open s" "continuous_on s f" "x \<in> s" 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   871
  "(f has_derivative f') (at x)" "bounded_linear g'" "f' \<circ> g' = id"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   872
  "t \<subseteq> s" "x \<in> interior t"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   873
  shows "f x \<in> interior (f ` t)" proof- 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   874
  interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   875
  interpret g':bounded_linear g' using assms by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   876
  guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this hence *:"1/(2*B)>0" by(auto intro!: divide_pos_pos)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   877
  guess e0 using assms(4)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note e0=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   878
  guess e1 using assms(8)[unfolded mem_interior_cball] .. note e1=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   879
  have *:"0<e0/B" "0<e1/B" apply(rule_tac[!] divide_pos_pos) using e0 e1 B by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   880
  guess e using real_lbound_gt_zero[OF *] .. note e=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   881
  have "\<forall>z\<in>cball (f x) (e/2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   882
    apply(rule,rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   883
    prefer 3 apply(rule,rule) proof- 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   884
    show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))" unfolding g'.diff
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   885
      apply(rule continuous_on_compose[of _ _ f, unfolded o_def])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   886
      apply(rule continuous_on_intros linear_continuous_on[OF assms(5)])+
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   887
      apply(rule continuous_on_subset[OF assms(2)]) apply(rule,unfold image_iff,erule bexE) proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   888
      fix y z assume as:"y \<in>cball (f x) e"  "z = x + (g' y - g' (f x))"
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   889
      have "dist x z = norm (g' (f x) - g' y)" unfolding as(2) and dist_norm by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   890
      also have "\<dots> \<le> norm (f x - y) * B" unfolding g'.diff[THEN sym] using B by auto
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   891
      also have "\<dots> \<le> e * B" using as(1)[unfolded mem_cball dist_norm] using B by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   892
      also have "\<dots> \<le> e1" using e unfolding less_divide_eq using B by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   893
      finally have "z\<in>cball x e1" unfolding mem_cball by force
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   894
      thus "z \<in> s" using e1 assms(7) by auto qed next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   895
    fix y z assume as:"y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   896
    have "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by auto
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   897
    also have "\<dots> \<le> e * B" apply(rule mult_right_mono) using as(2)[unfolded mem_cball dist_norm] and B unfolding norm_minus_commute by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   898
    also have "\<dots> < e0" using e and B unfolding less_divide_eq by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   899
    finally have *:"norm (x + g' (z - f x) - x) < e0" by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   900
    have **:"f x + f' (x + g' (z - f x) - x) = z" using assms(6)[unfolded o_def id_def,THEN cong] by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   901
    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)"
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   902
      using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"] by(auto simp add:algebra_simps)
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   903
    also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)" using e0[THEN conjunct2,rule_format,OF *] unfolding algebra_simps ** by auto 
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   904
    also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2" using as(1)[unfolded mem_cball dist_norm] by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   905
    also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2" using * and B by(auto simp add:field_simps)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   906
    also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2" by auto
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   907
    also have "\<dots> \<le> e/2 + e/2" apply(rule add_right_mono) using as(2)[unfolded mem_cball dist_norm] unfolding norm_minus_commute by auto
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   908
    finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e" unfolding mem_cball dist_norm by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   909
  qed(insert e, auto) note lem = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   910
  show ?thesis unfolding mem_interior apply(rule_tac x="e/2" in exI)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   911
    apply(rule,rule divide_pos_pos) prefer 3 proof 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   912
    fix y assume "y \<in> ball (f x) (e/2)" hence *:"y\<in>cball (f x) (e/2)" by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   913
    guess z using lem[rule_format,OF *] .. note z=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   914
    hence "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by(auto simp add:field_simps)
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   915
    also have "\<dots> \<le> e * B" apply(rule mult_right_mono) using z(1) unfolding mem_cball dist_norm norm_minus_commute using B by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   916
    also have "\<dots> \<le> e1"  using e B unfolding less_divide_eq by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   917
    finally have "x + g'(z - f x) \<in> t" apply- apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format]) 
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
   918
      unfolding mem_cball dist_norm by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   919
    thus "y \<in> f ` t" using z by auto qed(insert e, auto) qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   920
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   921
text {* Hence the following eccentric variant of the inverse function theorem.    *)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   922
(* This has no continuity assumptions, but we do need the inverse function.  *)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   923
(* We could put f' o g = I but this happens to fit with the minimal linear   *)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   924
(* algebra theory I've set up so far. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   925
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   926
(* move  before left_inverse_linear in Euclidean_Space*)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   927
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   928
 lemma right_inverse_linear: fixes f::"'a::euclidean_space => 'a"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   929
   assumes lf: "linear f" and gf: "f o g = id"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   930
   shows "linear g"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   931
 proof-
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   932
   from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def expand_fun_eq)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   933
     by metis 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   934
   from linear_surjective_isomorphism[OF lf fi]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   935
   obtain h:: "'a => 'a" where
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   936
     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   937
   have "h = g" apply (rule ext) using gf h(2,3)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   938
     apply (simp add: o_def id_def expand_fun_eq)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   939
     by metis
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   940
   with h(1) show ?thesis by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   941
 qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   942
 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   943
lemma has_derivative_inverse_strong: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   944
  assumes "open s" "x \<in> s" "continuous_on s f"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   945
  "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at x)" "f' o g' = id"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   946
  shows "(g has_derivative g') (at (f x))" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   947
  have linf:"bounded_linear f'" using assms(5) unfolding has_derivative_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   948
  hence ling:"bounded_linear g'" unfolding linear_conv_bounded_linear[THEN sym]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   949
    apply- apply(rule right_inverse_linear) using assms(6) by auto 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   950
  moreover have "g' \<circ> f' = id" using assms(6) linf ling unfolding linear_conv_bounded_linear[THEN sym]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   951
    using linear_inverse_left by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   952
  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 )
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   953
    apply(rule assms ling)+ by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   954
  have "continuous (at (f x)) g" unfolding continuous_at Lim_at proof(rule,rule)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   955
    fix e::real assume "e>0"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   956
    hence "f x \<in> interior (f ` (ball x e \<inter> s))" using *[rule_format,of "ball x e \<inter> s"] `x\<in>s`
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   957
      by(auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   958
    then guess d unfolding mem_interior .. note d=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   959
    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"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   960
      apply(rule_tac x=d in exI) apply(rule,rule d[THEN conjunct1]) proof(rule,rule) case goal1
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   961
      hence "g y \<in> g ` f ` (ball x e \<inter> s)" using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   962
	by(auto simp add:dist_commute)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   963
      hence "g y \<in> ball x e \<inter> s" using assms(4) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   964
      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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   965
  moreover have "f x \<in> interior (f ` s)" apply(rule sussmann_open_mapping)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   966
    apply(rule assms ling)+ using interior_open[OF assms(1)] and `x\<in>s` by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   967
  moreover have "\<And>y. y \<in> interior (f ` s) \<Longrightarrow> f (g y) = y" proof- case goal1
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   968
    hence "y\<in>f ` s" using interior_subset by auto then guess z unfolding image_iff ..
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   969
    thus ?case using assms(4) by auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   970
  ultimately show ?thesis apply- apply(rule has_derivative_inverse_basic_x[OF assms(5)]) using assms by auto qed 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   971
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   972
subsection {* A rewrite based on the other domain. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   973
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   974
lemma has_derivative_inverse_strong_x: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   975
  assumes "open s" "g y \<in> s" "continuous_on s f"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   976
  "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at (g y))" "f' o g' = id" "f(g y) = y"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   977
  shows "(g has_derivative g') (at y)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   978
  using has_derivative_inverse_strong[OF assms(1-6)] unfolding assms(7) by simp
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   979
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   980
subsection {* On a region. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   981
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   982
lemma has_derivative_inverse_on: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   983
  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"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   984
  shows "(g has_derivative g'(x)) (at (f x))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   985
  apply(rule has_derivative_inverse_strong[where g'="g' x" and f=f]) apply(rule assms)+
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   986
  unfolding continuous_on_eq_continuous_at[OF assms(1)]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   987
  apply(rule,rule differentiable_imp_continuous_at) unfolding differentiable_def using assms by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   988
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   989
subsection {* Invertible derivative continous at a point implies local injectivity.     *)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   990
(* It's only for this we need continuity of the derivative, except of course *)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   991
(* if we want the fact that the inverse derivative is also continuous. So if *)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   992
(* we know for some other reason that the inverse function exists, it's OK. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   993
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   994
lemma bounded_linear_sub: "bounded_linear f \<Longrightarrow> bounded_linear g ==> bounded_linear (\<lambda>x. f x - g x)"
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
   995
  using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g] by(auto simp add:algebra_simps)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   996
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   997
lemma has_derivative_locally_injective: fixes f::"'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   998
  assumes "a \<in> s" "open s" "bounded_linear g'" "g' o f'(a) = id"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
   999
  "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1000
  "\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm(\<lambda>v. f' x v - f' a v) < e"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1001
  obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. (f x' = f x) \<longrightarrow> (x' = x)" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1002
  interpret bounded_linear g' using assms by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1003
  note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1004
  have "g' (f' a (\<chi>\<chi> i.1)) = (\<chi>\<chi> i.1)" "(\<chi>\<chi> i.1) \<noteq> (0::'n)" defer 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1005
    apply(subst euclidean_eq) using f'g' by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1006
  hence *:"0 < onorm g'" unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastsimp
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1007
  def k \<equiv> "1 / onorm g' / 2" have *:"k>0" unfolding k_def using * by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1008
  guess d1 using assms(6)[rule_format,OF *] .. note d1=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1009
  from `open s` obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using `a\<in>s` ..
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1010
  obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using assms(2,1) ..
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1011
  guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\<in>s`] .. note d2=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1012
  guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1013
  show ?thesis proof show "a\<in>ball a d" using d by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1014
    show "\<forall>x\<in>ball a d. \<forall>x'\<in>ball a d. f x' = f x \<longrightarrow> x' = x" proof(intro strip)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1015
      fix x y assume as:"x\<in>ball a d" "y\<in>ball a d" "f x = f y"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1016
      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))"
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
  1017
	unfolding ph_def o_def unfolding diff using f'g' by(auto simp add:algebra_simps)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1018
      have "norm (ph x - ph y) \<le> (1/2) * norm (x - y)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1019
	apply(rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1020
	apply(rule_tac[!] ballI) proof- fix u assume u:"u \<in> ball a d" hence "u\<in>s" using d d2 by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1021
	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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1022
	show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1023
	  unfolding ph' * apply(rule diff_chain_within) defer apply(rule bounded_linear.has_derivative[OF assms(3)])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1024
	  apply(rule has_derivative_intros) defer apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1025
	  apply(rule has_derivative_at_within) using assms(5) and `u\<in>s` `a\<in>s`
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36334
diff changeset
  1026
          by(auto intro!: has_derivative_intros derivative_linear)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1027
	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)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1028
	  apply(rule_tac[!] derivative_linear) using assms(5) `u\<in>s` `a\<in>s` by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1029
	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)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1030
	  unfolding linear_conv_bounded_linear by(rule assms(3) **)+ 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1031
	also have "\<dots> \<le> onorm g' * k" apply(rule mult_left_mono) 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1032
	  using d1[THEN conjunct2,rule_format,of u] using onorm_neg[OF **(1)[unfolded linear_linear]]
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
  1033
	  using d and u and onorm_pos_le[OF assms(3)[unfolded linear_linear]] by(auto simp add:algebra_simps) 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1034
	also have "\<dots> \<le> 1/2" unfolding k_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1035
	finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" by assumption qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1036
      moreover have "norm (ph y - ph x) = norm (y - x)" apply(rule arg_cong[where f=norm])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1037
	unfolding ph_def using diff unfolding as by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1038
      ultimately show "x = y" unfolding norm_minus_commute by auto qed qed auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1039
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1040
subsection {* Uniformly convergent sequence of derivatives. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1041
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1042
lemma has_derivative_sequence_lipschitz_lemma: fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1043
  assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1044
  "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1045
  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)+ 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1046
  fix m n x y assume as:"N\<le>m" "N\<le>n" "x\<in>s" "y\<in>s"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1047
  show "norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1048
    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-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1049
    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)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1050
      by(rule has_derivative_intros assms(2)[rule_format] `x\<in>s`)+
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1051
    { 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)"
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
  1052
	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:algebra_simps) 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1053
      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]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1054
	by(auto simp add:field_simps)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1055
      finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h" by auto }
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1056
    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)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1057
      unfolding linear_conv_bounded_linear using assms(2)[rule_format,OF `x\<in>s`, THEN derivative_linear] by auto qed qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1058
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1059
lemma has_derivative_sequence_lipschitz: fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1060
  assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1061
  "\<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"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1062
  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)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1063
  case goal1 have *:"2 * (1/2* e) = e" "1/2 * e >0" using `e>0` by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1064
  guess N using assms(3)[rule_format,OF *(2)] ..
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1065
  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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1066
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1067
lemma has_derivative_sequence: fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1068
  assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1069
  "\<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)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1070
  "x0 \<in> s"  "((\<lambda>n. f n x0) ---> l) sequentially"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1071
  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-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1072
  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)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1073
    apply(rule has_derivative_sequence_lipschitz[where e="42::nat"]) apply(rule assms)+ by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1074
  have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially" apply(rule bchoice) unfolding convergent_eq_cauchy proof
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1075
    fix x assume "x\<in>s" show "Cauchy (\<lambda>n. f n x)" proof(cases "x=x0")
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1076
      case True thus ?thesis using convergent_imp_cauchy[OF assms(5)] by auto next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1077
      case False show ?thesis unfolding Cauchy_def proof(rule,rule)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1078
	fix e::real assume "e>0" hence *:"e/2>0" "e/2/norm(x-x0)>0" using False by(auto intro!:divide_pos_pos)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1079
	guess M using convergent_imp_cauchy[OF assms(5), unfolded Cauchy_def, rule_format,OF *(1)] .. note M=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1080
	guess N using lem1[rule_format,OF *(2)] .. note N = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1081
	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+)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1082
	  fix m n assume as:"max M N \<le>m" "max M N\<le>n"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1083
	  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))"
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
  1084
	    unfolding dist_norm by(rule norm_triangle_sub)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1085
	  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
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
  1086
	  also have "\<dots> < e / 2 + e / 2" apply(rule add_strict_right_mono) using as and M[rule_format] unfolding dist_norm by auto 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1087
	  finally show "dist (f m x) (f n x) < e" by auto qed qed qed qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1088
  then guess g .. note g = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1089
  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)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1090
    fix e::real assume *:"e>0" guess N using lem1[rule_format,OF *] .. note N=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1091
    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+)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1092
      fix n x y assume as:"N \<le> n" "x \<in> s" "y \<in> s"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1093
      have "eventually (\<lambda>xa. norm (f n x - f n y - (f xa x - f xa y)) \<le> e * norm (x - y)) sequentially" 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1094
	unfolding eventually_sequentially apply(rule_tac x=N in exI) proof(rule,rule)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1095
	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)"
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
  1096
	  using N[rule_format, of n m x y] and as by(auto simp add:algebra_simps) qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1097
      thus "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)" apply-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1098
	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)"])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1099
	apply(rule Lim_sub Lim_const g[rule_format] as)+ by assumption qed qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1100
  show ?thesis unfolding has_derivative_within_alt apply(rule_tac x=g in exI)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1101
    apply(rule,rule,rule g[rule_format],assumption) proof fix x assume "x\<in>s"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1102
    have lem3:"\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially" unfolding Lim_sequentially proof(rule,rule,rule)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1103
      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")
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1104
	case True guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1105
	show ?thesis apply(rule_tac x=N in exI) unfolding True 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1106
	  using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0` by auto next
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1107
	case False hence *:"e / 2 / norm u > 0" using `e>0` by(auto intro!: divide_pos_pos)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1108
	guess N using assms(3)[rule_format,OF *] .. note N=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1109
	show ?thesis apply(rule_tac x=N in exI) proof(rule,rule) case goal1
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
  1110
	  show ?case unfolding dist_norm using N[rule_format,OF goal1 `x\<in>s`, of u] False `e>0`
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1111
	    by (auto simp add:field_simps) qed qed qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1112
    show "bounded_linear (g' x)" unfolding linear_linear linear_def apply(rule,rule,rule) defer proof(rule,rule)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1113
      fix x' y z::"'m" and c::real
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1114
      note lin = assms(2)[rule_format,OF `x\<in>s`,THEN derivative_linear]
36593
fb69c8cd27bd define linear algebra concepts using scaleR instead of (op *s); generalized many lemmas, though a few theorems that used to work on type int^'n are a bit less general
huffman
parents: 36587
diff changeset
  1115
      show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'" apply(rule Lim_unique[OF trivial_limit_sequentially])
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1116
	apply(rule lem3[rule_format])
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1117
        unfolding lin[unfolded bounded_linear_def bounded_linear_axioms_def,THEN conjunct2,THEN conjunct1,rule_format]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1118
	apply(rule Lim_cmul) by(rule lem3[rule_format])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1119
      show "g' x (y + z) = g' x y + g' x z" apply(rule Lim_unique[OF trivial_limit_sequentially])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1120
	apply(rule lem3[rule_format]) unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1121
        apply(rule Lim_add) by(rule lem3[rule_format])+ qed 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1122
    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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1123
      have *:"e/3>0" using goal1 by auto guess N1 using assms(3)[rule_format,OF *] .. note N1=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1124
      guess N2 using lem2[rule_format,OF *] .. note N2=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1125
      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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1126
      show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1]) proof(rule,rule)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1127
	fix y assume as:"y \<in> s" "norm (y - x) < d1" let ?N ="max N1 N2"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1128
	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])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1129
	  using N2[rule_format, OF _ `y\<in>s` `x\<in>s`, of ?N] by auto moreover
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1130
	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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1131
	have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)" 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1132
	  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)"] 
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
  1133
	  by (auto simp add:algebra_simps) moreover
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1134
	have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)" using N1 `x\<in>s` by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1135
	ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
  1136
	  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:algebra_simps)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1137
	qed qed qed qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1138
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1139
subsection {* Can choose to line up antiderivatives if we want. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1140
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1141
lemma has_antiderivative_sequence: fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1142
  assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1143
  "\<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"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1144
  shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)" proof(cases "s={}")
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1145
  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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1146
  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)"])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1147
    apply(rule,rule) apply(rule has_derivative_add_const, rule assms(2)[rule_format], assumption)  
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1148
    apply(rule `a\<in>s`) by(auto intro!: Lim_const) qed auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1149
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1150
lemma has_antiderivative_limit: fixes g'::"'m::euclidean_space \<Rightarrow> 'm \<Rightarrow> 'n::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1151
  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))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1152
  shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1153
  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))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1154
    apply(rule) using assms(2) apply(erule_tac x="inverse (real (Suc n))" in allE) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1155
  guess f using *[THEN choice] .. note * = this guess f' using *[THEN choice] .. note f=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1156
  show ?thesis apply(rule has_antiderivative_sequence[OF assms(1), of f f']) defer proof(rule,rule)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1157
    fix e::real assume "0<e" guess  N using reals_Archimedean[OF `e>0`] .. note N=this 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1158
    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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1159
      have *:"inverse (real (Suc n)) \<le> e" apply(rule order_trans[OF _ N[THEN less_imp_le]])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1160
	using goal1(1) by(auto simp add:field_simps) 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1161
      show ?case using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]] 
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1162
	apply(rule order_trans) using N * apply(cases "h=0") by auto qed qed(insert f,auto) qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1163
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1164
subsection {* Differentiation of a series. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1165
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1166
definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> (nat set) \<Rightarrow> bool"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1167
(infixl "sums'_seq" 12) where "(f sums_seq l) s \<equiv> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1168
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1169
lemma has_derivative_series: fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1170
  assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1171
  "\<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)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1172
  "x\<in>s" "((\<lambda>n. f n x) sums_seq l) k"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1173
  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)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1174
  unfolding sums_seq_def apply(rule has_derivative_sequence[OF assms(1) _ assms(3)]) apply(rule,rule)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1175
  apply(rule has_derivative_setsum) defer apply(rule,rule assms(2)[rule_format],assumption)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1176
  using assms(4-5) unfolding sums_seq_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1177
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1178
subsection {* Derivative with composed bilinear function. *}
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1179
37650
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
  1180
lemma has_derivative_bilinear_within:
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1181
  assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at x within s)" "bounded_bilinear h"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1182
  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-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1183
  have "(g ---> g x) (at x within s)" apply(rule differentiable_imp_continuous_within[unfolded continuous_within])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1184
    using assms(2) unfolding differentiable_def by auto moreover
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1185
  interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1186
  interpret g':bounded_linear g' using assms unfolding has_derivative_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1187
  interpret h:bounded_bilinear h using assms by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1188
  have "((\<lambda>y. f' (y - x)) ---> 0) (at x within s)" unfolding f'.zero[THEN sym]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1189
    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]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1190
    unfolding id_def using assms(1) unfolding has_derivative_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1191
  hence "((\<lambda>y. f x + f' (y - x)) ---> f x) (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1192
    using Lim_add[OF Lim_const, of "\<lambda>y. f' (y - x)" 0 "at x within s" "f x"] by auto ultimately
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1193
  have *:"((\<lambda>x'. h (f x + f' (x' - x)) ((1/(norm (x' - x))) *\<^sub>R (g x' - (g x + g' (x' - x))))
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1194
             + 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)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1195
    apply-apply(rule Lim_add) apply(rule_tac[!] Lim_bilinear[OF _ _ assms(3)]) using assms(1-2)  unfolding has_derivative_within by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1196
  guess B using bounded_bilinear.pos_bounded[OF assms(3)] .. note B=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1197
  guess C using f'.pos_bounded .. note C=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1198
  guess D using g'.pos_bounded .. note D=this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1199
  have bcd:"B * C * D > 0" using B C D by (auto intro!: mult_pos_pos)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1200
  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
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1201
    hence "e/(B*C*D)>0" using B C D by(auto intro!:divide_pos_pos mult_pos_pos)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1202
    thus ?case apply(rule_tac x="e/(B*C*D)" in exI,rule) proof(rule,rule,erule conjE)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1203
      fix y assume as:"y \<in> s" "0 < dist y x" "dist y x < e / (B * C * D)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1204
      have "norm (h (f' (y - x)) (g' (y - x))) \<le> norm (f' (y - x)) * norm (g' (y - x)) * B" using B by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1205
      also have "\<dots> \<le> (norm (y - x) * C) * (D * norm (y - x)) * B" apply(rule mult_right_mono)
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34981
diff changeset
  1206
	apply(rule mult_mono) using B C D by (auto simp add: field_simps intro!:mult_nonneg_nonneg)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1207
      also have "\<dots> = (B * C * D * norm (y - x)) * norm (y - x)" by(auto simp add:field_simps)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1208
      also have "\<dots> < e * norm (y - x)" apply(rule mult_strict_right_mono)
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
  1209
	using as(3)[unfolded dist_norm] and as(2) unfolding pos_less_divide_eq[OF bcd] by (auto simp add:field_simps)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1210
      finally show "dist ((1 / norm (y - x)) *\<^sub>R h (f' (y - x)) (g' (y - x))) 0 < e"
36587
534418d8d494 remove redundant lemma vector_dist_norm
huffman
parents: 36581
diff changeset
  1211
	unfolding dist_norm apply-apply(cases "y = x") by(auto simp add:field_simps) qed qed
37650
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
  1212
  have "bounded_linear (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))"
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
  1213
    apply (rule bounded_linear_add)
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
  1214
    apply (rule bounded_linear_compose [OF h.bounded_linear_right `bounded_linear g'`])
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
  1215
    apply (rule bounded_linear_compose [OF h.bounded_linear_left `bounded_linear f'`])
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
  1216
    done
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1217
  thus ?thesis using Lim_add[OF * **] unfolding has_derivative_within 
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1218
    unfolding g'.add f'.scaleR f'.add g'.scaleR f'.diff g'.diff
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1219
     h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right h.diff_right h.diff_left
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1220
    scaleR_right_diff_distrib h.zero_right h.zero_left by(auto simp add:field_simps) qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1221
37650
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
  1222
lemma has_derivative_bilinear_at:
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1223
  assumes "(f has_derivative f') (at x)" "(g has_derivative g') (at x)" "bounded_bilinear h"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1224
  shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1225
  using has_derivative_bilinear_within[of f f' x UNIV g g' h] unfolding within_UNIV using assms by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1226
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1227
subsection {* Considering derivative @{typ "real \<Rightarrow> 'b\<Colon>real_normed_vector"} as a vector. *}
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1228
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1229
definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> (real net \<Rightarrow> bool)"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1230
(infixl "has'_vector'_derivative" 12) where
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1231
 "(f has_vector_derivative f') net \<equiv> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1232
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1233
definition "vector_derivative f net \<equiv> (SOME f'. (f has_vector_derivative f') net)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1234
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1235
lemma vector_derivative_works: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1236
  shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net" (is "?l = ?r")
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1237
proof assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1238
  then interpret bounded_linear f' by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1239
  thus ?r unfolding vector_derivative_def has_vector_derivative_def
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1240
    apply-apply(rule someI_ex,rule_tac x="f' 1" in exI)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1241
    using f' unfolding scaleR[THEN sym] by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1242
next assume ?r thus ?l  unfolding vector_derivative_def has_vector_derivative_def differentiable_def by auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1243
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1244
lemma vector_derivative_unique_at: fixes f::"real\<Rightarrow> 'n::euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1245
  assumes "(f has_vector_derivative f') (at x)" "(f has_vector_derivative f'') (at x)" shows "f' = f''" proof-
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1246
  have *:"(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')" apply(rule frechet_derivative_unique_at)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1247
    using assms[unfolded has_vector_derivative_def] by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1248
  show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1249
    hence "(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')" using * by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1250
    ultimately show False unfolding expand_fun_eq by auto qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1251
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1252
lemma vector_derivative_unique_within_closed_interval: fixes f::"real \<Rightarrow> 'n::ordered_euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1253
  assumes "a < b" "x \<in> {a..b}"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1254
  "(f has_vector_derivative f') (at x within {a..b})"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1255
  "(f has_vector_derivative f'') (at x within {a..b})" shows "f' = f''" proof-
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1256
  have *:"(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1257
    apply(rule frechet_derivative_unique_within_closed_interval[of "a" "b"])
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1258
    using assms(3-)[unfolded has_vector_derivative_def] using assms(1-2) by auto
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1259
  show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1260
    hence "(\<lambda>x. x *\<^sub>R f') 1 = (\<lambda>x. x *\<^sub>R f'') 1" using * by (auto simp: expand_fun_eq)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1261
    ultimately show False unfolding o_def by auto qed qed
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1262
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1263
lemma vector_derivative_at: fixes f::"real \<Rightarrow> 'a::euclidean_space" shows
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1264
 "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1265
  apply(rule vector_derivative_unique_at) defer apply assumption
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1266
  unfolding vector_derivative_works[THEN sym] differentiable_def
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1267
  unfolding has_vector_derivative_def by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1268
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1269
lemma vector_derivative_within_closed_interval: fixes f::"real \<Rightarrow> 'a::ordered_euclidean_space"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1270
  assumes "a < b" "x \<in> {a..b}" "(f has_vector_derivative f') (at x within {a..b})"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1271
  shows "vector_derivative f (at x within {a..b}) = f'"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1272
  apply(rule vector_derivative_unique_within_closed_interval)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1273
  using vector_derivative_works[unfolded differentiable_def]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1274
  using assms by(auto simp add:has_vector_derivative_def)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1275
34981
475aef44d5fb Removed explicit type annotations
himmelma
parents: 34964
diff changeset
  1276
lemma has_vector_derivative_within_subset: 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1277
 "(f has_vector_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_vector_derivative f') (at x within t)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1278
  unfolding has_vector_derivative_def apply(rule has_derivative_within_subset) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1279
34981
475aef44d5fb Removed explicit type annotations
himmelma
parents: 34964
diff changeset
  1280
lemma has_vector_derivative_const: 
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1281
 "((\<lambda>x. c) has_vector_derivative 0) net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1282
  unfolding has_vector_derivative_def using has_derivative_const by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1283
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1284
lemma has_vector_derivative_id: "((\<lambda>x::real. x) has_vector_derivative 1) net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1285
  unfolding has_vector_derivative_def using has_derivative_id by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1286
34981
475aef44d5fb Removed explicit type annotations
himmelma
parents: 34964
diff changeset
  1287
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"
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36334
diff changeset
  1288
  unfolding has_vector_derivative_def apply(drule has_derivative_cmul) by(auto simp add:algebra_simps)
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1289
34981
475aef44d5fb Removed explicit type annotations
himmelma
parents: 34964
diff changeset
  1290
lemma has_vector_derivative_cmul_eq: assumes "c \<noteq> 0"
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1291
  shows "(((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net \<longleftrightarrow> (f has_vector_derivative f') net)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1292
  apply rule apply(drule has_vector_derivative_cmul[where c="1/c"]) defer
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1293
  apply(rule has_vector_derivative_cmul) using assms by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1294
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1295
lemma has_vector_derivative_neg:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1296
 "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_vector_derivative (- f')) net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1297
  unfolding has_vector_derivative_def apply(drule has_derivative_neg) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1298
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1299
lemma has_vector_derivative_add:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1300
  assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1301
  shows "((\<lambda>x. f(x) + g(x)) has_vector_derivative (f' + g')) net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1302
  using has_derivative_add[OF assms[unfolded has_vector_derivative_def]]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1303
  unfolding has_vector_derivative_def unfolding scaleR_right_distrib by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1304
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1305
lemma has_vector_derivative_sub:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1306
  assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1307
  shows "((\<lambda>x. f(x) - g(x)) has_vector_derivative (f' - g')) net"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1308
  using has_derivative_sub[OF assms[unfolded has_vector_derivative_def]]
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1309
  unfolding has_vector_derivative_def scaleR_right_diff_distrib by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1310
37650
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
  1311
lemma has_vector_derivative_bilinear_within:
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1312
  assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at x within s)" "bounded_bilinear h"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1313
  shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x within s)" proof-
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1314
  interpret bounded_bilinear h using assms by auto 
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1315
  show ?thesis using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def], of h]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1316
    unfolding o_def has_vector_derivative_def
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1317
    using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib by auto qed
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1318
37650
181a70d7b525 generalize some lemmas about derivatives
huffman
parents: 37648
diff changeset
  1319
lemma has_vector_derivative_bilinear_at:
33741
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1320
  assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at x)" "bounded_bilinear h"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1321
  shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1322
  apply(rule has_vector_derivative_bilinear_within[where s=UNIV, unfolded within_UNIV]) using assms by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1323
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1324
lemma has_vector_derivative_at_within: "(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f') (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1325
  unfolding has_vector_derivative_def apply(rule has_derivative_at_within) by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1326
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1327
lemma has_vector_derivative_transform_within:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1328
  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)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1329
  shows "(g has_vector_derivative f') (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1330
  using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1331
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1332
lemma has_vector_derivative_transform_at:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1333
  assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_vector_derivative f') (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1334
  shows "(g has_vector_derivative f') (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1335
  using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_at)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1336
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1337
lemma has_vector_derivative_transform_within_open:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1338
  assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_vector_derivative f') (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1339
  shows "(g has_vector_derivative f') (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1340
  using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within_open)
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1341
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1342
lemma vector_diff_chain_at:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1343
  assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at (f x))"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1344
  shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1345
  using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1346
  unfolding o_def scaleR.scaleR_left by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1347
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1348
lemma vector_diff_chain_within:
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1349
  assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at (f x) within f ` s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1350
  shows "((g o f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1351
  using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1352
  unfolding o_def scaleR.scaleR_left by auto
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1353
4c414d0835ab Added derivation and Brouwer's fixpoint theorem in Multivariate Analysis (translated by Robert Himmelmann from HOL-light)
hoelzl
parents:
diff changeset
  1354
end