src/HOL/Deriv.thy
author paulson <lp15@cam.ac.uk>
Thu, 03 Apr 2014 23:51:52 +0100
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permissions -rw-r--r--
removing simprule status for divide_minus_left and divide_minus_right
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(*  Title       : Deriv.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Author      : Brian Huffman
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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    GMVT by Benjamin Porter, 2005
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*)
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header{* Differentiation *}
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theory Deriv
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imports Limits
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begin
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subsection {* Frechet derivative *}
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definition
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  has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow>  bool"
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  (infix "(has'_derivative)" 50)
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where
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  "(f has_derivative f') F \<longleftrightarrow>
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    (bounded_linear f' \<and>
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     ((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) ---> 0) F)"
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text {*
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  Usually the filter @{term F} is @{term "at x within s"}.  @{term "(f has_derivative D)
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  (at x within s)"} means: @{term D} is the derivative of function @{term f} at point @{term x}
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  within the set @{term s}. Where @{term s} is used to express left or right sided derivatives. In
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  most cases @{term s} is either a variable or @{term UNIV}.
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*}
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lemma has_derivative_eq_rhs: "(f has_derivative f') F \<Longrightarrow> f' = g' \<Longrightarrow> (f has_derivative g') F"
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  by simp
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definition 
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  has_field_derivative :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a filter \<Rightarrow> bool"
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  (infix "(has'_field'_derivative)" 50)
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where
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  "(f has_field_derivative D) F \<longleftrightarrow> (f has_derivative op * D) F"
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lemma DERIV_cong: "(f has_field_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_field_derivative Y) F"
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  by simp
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definition
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  has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> real filter \<Rightarrow> bool"
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  (infix "has'_vector'_derivative" 50)
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where
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  "(f has_vector_derivative f') net \<longleftrightarrow> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
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lemma has_vector_derivative_eq_rhs: "(f has_vector_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_vector_derivative Y) F"
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  by simp
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ML {*
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structure Derivative_Intros = Named_Thms
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(
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  val name = @{binding derivative_intros}
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  val description = "structural introduction rules for derivatives"
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)
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*}
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setup {*
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  let
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    val eq_thms = [@{thm has_derivative_eq_rhs}, @{thm DERIV_cong}, @{thm has_vector_derivative_eq_rhs}]
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    fun eq_rule thm = get_first (try (fn eq_thm => eq_thm OF [thm])) eq_thms
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  in
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    Derivative_Intros.setup #>
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    Global_Theory.add_thms_dynamic
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      (@{binding derivative_eq_intros}, map_filter eq_rule o Derivative_Intros.get o Context.proof_of)
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  end;
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*}
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text {*
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  The following syntax is only used as a legacy syntax.
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*}
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abbreviation (input)
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  FDERIV :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow>  ('a \<Rightarrow> 'b) \<Rightarrow> bool"
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  ("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
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where
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  "FDERIV f x :> f' \<equiv> (f has_derivative f') (at x)"
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lemma has_derivative_bounded_linear: "(f has_derivative f') F \<Longrightarrow> bounded_linear f'"
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  by (simp add: has_derivative_def)
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lemma has_derivative_linear: "(f has_derivative f') F \<Longrightarrow> linear f'"
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  using bounded_linear.linear[OF has_derivative_bounded_linear] .
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lemma has_derivative_ident[derivative_intros, simp]: "((\<lambda>x. x) has_derivative (\<lambda>x. x)) F"
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  by (simp add: has_derivative_def tendsto_const)
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lemma has_derivative_const[derivative_intros, simp]: "((\<lambda>x. c) has_derivative (\<lambda>x. 0)) F"
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  by (simp add: has_derivative_def tendsto_const)
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400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
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lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..
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lemma (in bounded_linear) has_derivative:
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  "(g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
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  using assms unfolding has_derivative_def
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  apply safe
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  apply (erule bounded_linear_compose [OF bounded_linear])
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  apply (drule tendsto)
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  apply (simp add: scaleR diff add zero)
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  done
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lemmas has_derivative_scaleR_right [derivative_intros] =
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  bounded_linear.has_derivative [OF bounded_linear_scaleR_right]
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lemmas has_derivative_scaleR_left [derivative_intros] =
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  bounded_linear.has_derivative [OF bounded_linear_scaleR_left]
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lemmas has_derivative_mult_right [derivative_intros] =
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  bounded_linear.has_derivative [OF bounded_linear_mult_right]
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lemmas has_derivative_mult_left [derivative_intros] =
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  bounded_linear.has_derivative [OF bounded_linear_mult_left]
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lemma has_derivative_add[simp, derivative_intros]:
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  assumes f: "(f has_derivative f') F" and g: "(g has_derivative g') F"
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   120
  shows "((\<lambda>x. f x + g x) has_derivative (\<lambda>x. f' x + g' x)) F"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   121
  unfolding has_derivative_def
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   122
proof safe
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   123
  let ?x = "Lim F (\<lambda>x. x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   124
  let ?D = "\<lambda>f f' y. ((f y - f ?x) - f' (y - ?x)) /\<^sub>R norm (y - ?x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   125
  have "((\<lambda>x. ?D f f' x + ?D g g' x) ---> (0 + 0)) F"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   126
    using f g by (intro tendsto_add) (auto simp: has_derivative_def)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   127
  then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) ---> 0) F"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   128
    by (simp add: field_simps scaleR_add_right scaleR_diff_right)
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   129
qed (blast intro: bounded_linear_add f g has_derivative_bounded_linear)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   130
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   131
lemma has_derivative_setsum[simp, derivative_intros]:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   132
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   133
  shows "((\<lambda>x. \<Sum>i\<in>I. f i x) has_derivative (\<lambda>x. \<Sum>i\<in>I. f' i x)) F"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   134
proof cases
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   135
  assume "finite I" from this f show ?thesis
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   136
    by induct (simp_all add: f)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   137
qed simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   138
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   139
lemma has_derivative_minus[simp, derivative_intros]: "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   140
  using has_derivative_scaleR_right[of f f' F "-1"] by simp
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   141
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   142
lemma has_derivative_diff[simp, derivative_intros]:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   143
  "(f has_derivative f') F \<Longrightarrow> (g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f x - g x) has_derivative (\<lambda>x. f' x - g' x)) F"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   144
  by (simp only: diff_conv_add_uminus has_derivative_add has_derivative_minus)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   145
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   146
lemma has_derivative_at_within:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   147
  "(f has_derivative f') (at x within s) \<longleftrightarrow>
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   148
    (bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   149
  by (cases "at x within s = bot") (simp_all add: has_derivative_def Lim_ident_at)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   150
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   151
lemma has_derivative_iff_norm:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   152
  "(f has_derivative f') (at x within s) \<longleftrightarrow>
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   153
    (bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) ---> 0) (at x within s))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   154
  using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric]
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   155
  by (simp add: has_derivative_at_within divide_inverse ac_simps)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   156
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   157
lemma has_derivative_at:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   158
  "(f has_derivative D) (at x) \<longleftrightarrow> (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) -- 0 --> 0)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   159
  unfolding has_derivative_iff_norm LIM_offset_zero_iff[of _ _ x] by simp
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   160
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   161
lemma field_has_derivative_at:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   162
  fixes x :: "'a::real_normed_field"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   163
  shows "(f has_derivative op * D) (at x) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   164
  apply (unfold has_derivative_at)
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   165
  apply (simp add: bounded_linear_mult_right)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   166
  apply (simp cong: LIM_cong add: nonzero_norm_divide [symmetric])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   167
  apply (subst diff_divide_distrib)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   168
  apply (subst times_divide_eq_left [symmetric])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   169
  apply (simp cong: LIM_cong)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   170
  apply (simp add: tendsto_norm_zero_iff LIM_zero_iff)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   171
  done
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   172
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   173
lemma has_derivativeI:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   174
  "bounded_linear f' \<Longrightarrow> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s) \<Longrightarrow>
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   175
  (f has_derivative f') (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   176
  by (simp add: has_derivative_at_within)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   177
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   178
lemma has_derivativeI_sandwich:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   179
  assumes e: "0 < e" and bounded: "bounded_linear f'"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   180
    and sandwich: "(\<And>y. y \<in> s \<Longrightarrow> y \<noteq> x \<Longrightarrow> dist y x < e \<Longrightarrow> norm ((f y - f x) - f' (y - x)) / norm (y - x) \<le> H y)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   181
    and "(H ---> 0) (at x within s)"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   182
  shows "(f has_derivative f') (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   183
  unfolding has_derivative_iff_norm
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   184
proof safe
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   185
  show "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) ---> 0) (at x within s)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   186
  proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   187
    show "(H ---> 0) (at x within s)" by fact
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   188
    show "eventually (\<lambda>n. norm (f n - f x - f' (n - x)) / norm (n - x) \<le> H n) (at x within s)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   189
      unfolding eventually_at using e sandwich by auto
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   190
  qed (auto simp: le_divide_eq tendsto_const)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   191
qed fact
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   192
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   193
lemma has_derivative_subset: "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   194
  by (auto simp add: has_derivative_iff_norm intro: tendsto_within_subset)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   195
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   196
lemmas has_derivative_within_subset = has_derivative_subset 
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   197
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   198
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   199
subsection {* Continuity *}
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   200
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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diff changeset
   201
lemma has_derivative_continuous:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   202
  assumes f: "(f has_derivative f') (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   203
  shows "continuous (at x within s) f"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   204
proof -
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   205
  from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   206
  note F.tendsto[tendsto_intros]
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   207
  let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   208
  have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x))"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   209
    using f unfolding has_derivative_iff_norm by blast
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   210
  then have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x) * norm (y - x))" (is ?m)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   211
    by (rule tendsto_mult_zero) (auto intro!: tendsto_eq_intros)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   212
  also have "?m \<longleftrightarrow> ?L (\<lambda>y. norm ((f y - f x) - f' (y - x)))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   213
    by (intro filterlim_cong) (simp_all add: eventually_at_filter)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   214
  finally have "?L (\<lambda>y. (f y - f x) - f' (y - x))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   215
    by (rule tendsto_norm_zero_cancel)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   216
  then have "?L (\<lambda>y. ((f y - f x) - f' (y - x)) + f' (y - x))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   217
    by (rule tendsto_eq_intros) (auto intro!: tendsto_eq_intros simp: F.zero)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   218
  then have "?L (\<lambda>y. f y - f x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   219
    by simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   220
  from tendsto_add[OF this tendsto_const, of "f x"] show ?thesis
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   221
    by (simp add: continuous_within)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   222
qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   223
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   224
subsection {* Composition *}
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   225
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   226
lemma tendsto_at_iff_tendsto_nhds_within: "f x = y \<Longrightarrow> (f ---> y) (at x within s) \<longleftrightarrow> (f ---> y) (inf (nhds x) (principal s))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   227
  unfolding tendsto_def eventually_inf_principal eventually_at_filter
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   228
  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   229
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   230
lemma has_derivative_in_compose:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   231
  assumes f: "(f has_derivative f') (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   232
  assumes g: "(g has_derivative g') (at (f x) within (f`s))"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   233
  shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   234
proof -
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   235
  from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear)
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   236
  from g interpret G: bounded_linear g' by (rule has_derivative_bounded_linear)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   237
  from F.bounded obtain kF where kF: "\<And>x. norm (f' x) \<le> norm x * kF" by fast
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   238
  from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG" by fast
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   239
  note G.tendsto[tendsto_intros]
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   240
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   241
  let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   242
  let ?D = "\<lambda>f f' x y. (f y - f x) - f' (y - x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   243
  let ?N = "\<lambda>f f' x y. norm (?D f f' x y) / norm (y - x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   244
  let ?gf = "\<lambda>x. g (f x)" and ?gf' = "\<lambda>x. g' (f' x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   245
  def Nf \<equiv> "?N f f' x"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   246
  def Ng \<equiv> "\<lambda>y. ?N g g' (f x) (f y)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   247
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   248
  show ?thesis
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2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   249
  proof (rule has_derivativeI_sandwich[of 1])
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   250
    show "bounded_linear (\<lambda>x. g' (f' x))"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   251
      using f g by (blast intro: bounded_linear_compose has_derivative_bounded_linear)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   252
  next
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   253
    fix y::'a assume neq: "y \<noteq> x"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   254
    have "?N ?gf ?gf' x y = norm (g' (?D f f' x y) + ?D g g' (f x) (f y)) / norm (y - x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   255
      by (simp add: G.diff G.add field_simps)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   256
    also have "\<dots> \<le> norm (g' (?D f f' x y)) / norm (y - x) + Ng y * (norm (f y - f x) / norm (y - x))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   257
      by (simp add: add_divide_distrib[symmetric] divide_right_mono norm_triangle_ineq G.zero Ng_def)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   258
    also have "\<dots> \<le> Nf y * kG + Ng y * (Nf y + kF)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   259
    proof (intro add_mono mult_left_mono)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   260
      have "norm (f y - f x) = norm (?D f f' x y + f' (y - x))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   261
        by simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   262
      also have "\<dots> \<le> norm (?D f f' x y) + norm (f' (y - x))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   263
        by (rule norm_triangle_ineq)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   264
      also have "\<dots> \<le> norm (?D f f' x y) + norm (y - x) * kF"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   265
        using kF by (intro add_mono) simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   266
      finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   267
        by (simp add: neq Nf_def field_simps)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   268
    qed (insert kG, simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   269
    finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   270
  next
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   271
    have [tendsto_intros]: "?L Nf"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   272
      using f unfolding has_derivative_iff_norm Nf_def ..
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   273
    from f have "(f ---> f x) (at x within s)"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   274
      by (blast intro: has_derivative_continuous continuous_within[THEN iffD1])
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   275
    then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   276
      unfolding filterlim_def
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   277
      by (simp add: eventually_filtermap eventually_at_filter le_principal)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   278
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   279
    have "((?N g  g' (f x)) ---> 0) (at (f x) within f`s)"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   280
      using g unfolding has_derivative_iff_norm ..
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   281
    then have g': "((?N g  g' (f x)) ---> 0) (inf (nhds (f x)) (principal (f`s)))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   282
      by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   283
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   284
    have [tendsto_intros]: "?L Ng"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   285
      unfolding Ng_def by (rule filterlim_compose[OF g' f'])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   286
    show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) ---> 0) (at x within s)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   287
      by (intro tendsto_eq_intros) auto
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   288
  qed simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   289
qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   290
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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diff changeset
   291
lemma has_derivative_compose:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   292
  "(f has_derivative f') (at x within s) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   293
  ((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   294
  by (blast intro: has_derivative_in_compose has_derivative_subset)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   295
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   296
lemma (in bounded_bilinear) FDERIV:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
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parents: 55970
diff changeset
   297
  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   298
  shows "((\<lambda>x. f x ** g x) has_derivative (\<lambda>h. f x ** g' h + f' h ** g x)) (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   299
proof -
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   300
  from bounded_linear.bounded [OF has_derivative_bounded_linear [OF f]]
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   301
  obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   302
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   303
  from pos_bounded obtain K where K: "0 < K" and norm_prod:
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   304
    "\<And>a b. norm (a ** b) \<le> norm a * norm b * K" by fast
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   305
  let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   306
  let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   307
  def Ng =="?N g g'" and Nf =="?N f f'"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   308
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   309
  let ?fun1 = "\<lambda>y. norm (f y ** g y - f x ** g x - (f x ** g' (y - x) + f' (y - x) ** g x)) / norm (y - x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   310
  let ?fun2 = "\<lambda>y. norm (f x) * Ng y * K + Nf y * norm (g y) * K + KF * norm (g y - g x) * K"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   311
  let ?F = "at x within s"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   312
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   313
  show ?thesis
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   314
  proof (rule has_derivativeI_sandwich[of 1])
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   315
    show "bounded_linear (\<lambda>h. f x ** g' h + f' h ** g x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   316
      by (intro bounded_linear_add
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   317
        bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left]
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   318
        has_derivative_bounded_linear [OF g] has_derivative_bounded_linear [OF f])
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   319
  next
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   320
    from g have "(g ---> g x) ?F"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   321
      by (intro continuous_within[THEN iffD1] has_derivative_continuous)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   322
    moreover from f g have "(Nf ---> 0) ?F" "(Ng ---> 0) ?F"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   323
      by (simp_all add: has_derivative_iff_norm Ng_def Nf_def)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   324
    ultimately have "(?fun2 ---> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   325
      by (intro tendsto_intros) (simp_all add: LIM_zero_iff)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   326
    then show "(?fun2 ---> 0) ?F"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   327
      by simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   328
  next
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   329
    fix y::'d assume "y \<noteq> x"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   330
    have "?fun1 y = norm (f x ** ?D g g' y + ?D f f' y ** g y + f' (y - x) ** (g y - g x)) / norm (y - x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   331
      by (simp add: diff_left diff_right add_left add_right field_simps)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   332
    also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   333
        norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   334
      by (intro divide_right_mono mult_mono'
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   335
                order_trans [OF norm_triangle_ineq add_mono]
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   336
                order_trans [OF norm_prod mult_right_mono]
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   337
                mult_nonneg_nonneg order_refl norm_ge_zero norm_F
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   338
                K [THEN order_less_imp_le])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   339
    also have "\<dots> = ?fun2 y"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   340
      by (simp add: add_divide_distrib Ng_def Nf_def)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   341
    finally show "?fun1 y \<le> ?fun2 y" .
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   342
  qed simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   343
qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   344
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   345
lemmas has_derivative_mult[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult]
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   346
lemmas has_derivative_scaleR[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   347
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   348
lemma has_derivative_setprod[simp, derivative_intros]:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   349
  fixes f :: "'i \<Rightarrow> 'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   350
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   351
  shows "((\<lambda>x. \<Prod>i\<in>I. f i x) has_derivative (\<lambda>y. \<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x))) (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   352
proof cases
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   353
  assume "finite I" from this f show ?thesis
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   354
  proof induct
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   355
    case (insert i I)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   356
    let ?P = "\<lambda>y. f i x * (\<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x)) + (f' i y) * (\<Prod>i\<in>I. f i x)"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   357
    have "((\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) has_derivative ?P) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   358
      using insert by (intro has_derivative_mult) auto
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   359
    also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   360
      using insert(1,2) by (auto simp add: setsum_right_distrib insert_Diff_if intro!: ext setsum_cong)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   361
    finally show ?case
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   362
      using insert by simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   363
  qed simp  
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   364
qed simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   365
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   366
lemma has_derivative_power[simp, derivative_intros]:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   367
  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   368
  assumes f: "(f has_derivative f') (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   369
  shows "((\<lambda>x. f x^n) has_derivative (\<lambda>y. of_nat n * f' y * f x^(n - 1))) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   370
  using has_derivative_setprod[OF f, of "{..< n}"] by (simp add: setprod_constant ac_simps)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   371
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   372
lemma has_derivative_inverse':
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   373
  fixes x :: "'a::real_normed_div_algebra"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   374
  assumes x: "x \<noteq> 0"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   375
  shows "(inverse has_derivative (\<lambda>h. - (inverse x * h * inverse x))) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   376
        (is "(?inv has_derivative ?f) _")
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   377
proof (rule has_derivativeI_sandwich)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   378
  show "bounded_linear (\<lambda>h. - (?inv x * h * ?inv x))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   379
    apply (rule bounded_linear_minus)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   380
    apply (rule bounded_linear_mult_const)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   381
    apply (rule bounded_linear_const_mult)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   382
    apply (rule bounded_linear_ident)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   383
    done
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   384
next
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   385
  show "0 < norm x" using x by simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   386
next
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   387
  show "((\<lambda>y. norm (?inv y - ?inv x) * norm (?inv x)) ---> 0) (at x within s)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   388
    apply (rule tendsto_mult_left_zero)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   389
    apply (rule tendsto_norm_zero)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   390
    apply (rule LIM_zero)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   391
    apply (rule tendsto_inverse)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   392
    apply (rule tendsto_ident_at)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   393
    apply (rule x)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   394
    done
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   395
next
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   396
  fix y::'a assume h: "y \<noteq> x" "dist y x < norm x"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   397
  then have "y \<noteq> 0"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   398
    by (auto simp: norm_conv_dist dist_commute)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   399
  have "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) = norm ((?inv y - ?inv x) * (y - x) * ?inv x) / norm (y - x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   400
    apply (subst inverse_diff_inverse [OF `y \<noteq> 0` x])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   401
    apply (subst minus_diff_minus)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   402
    apply (subst norm_minus_cancel)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   403
    apply (simp add: left_diff_distrib)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   404
    done
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   405
  also have "\<dots> \<le> norm (?inv y - ?inv x) * norm (y - x) * norm (?inv x) / norm (y - x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   406
    apply (rule divide_right_mono [OF _ norm_ge_zero])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   407
    apply (rule order_trans [OF norm_mult_ineq])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   408
    apply (rule mult_right_mono [OF _ norm_ge_zero])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   409
    apply (rule norm_mult_ineq)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   410
    done
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   411
  also have "\<dots> = norm (?inv y - ?inv x) * norm (?inv x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   412
    by simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   413
  finally show "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) \<le>
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   414
      norm (?inv y - ?inv x) * norm (?inv x)" .
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   415
qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   416
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   417
lemma has_derivative_inverse[simp, derivative_intros]:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   418
  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   419
  assumes x:  "f x \<noteq> 0" and f: "(f has_derivative f') (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   420
  shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   421
  using has_derivative_compose[OF f has_derivative_inverse', OF x] .
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   422
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   423
lemma has_derivative_divide[simp, derivative_intros]:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   424
  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   425
  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" 
55967
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   426
  assumes x: "g x \<noteq> 0"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   427
  shows "((\<lambda>x. f x / g x) has_derivative
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   428
                (\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   429
  using has_derivative_mult[OF f has_derivative_inverse[OF x g]]
55967
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   430
  by (simp add: divide_inverse field_simps)
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   431
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   432
text{*Conventional form requires mult-AC laws. Types real and complex only.*}
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   433
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   434
lemma has_derivative_divide'[derivative_intros]: 
55967
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   435
  fixes f :: "_ \<Rightarrow> 'a::real_normed_field"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   436
  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" and x: "g x \<noteq> 0"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   437
  shows "((\<lambda>x. f x / g x) has_derivative (\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within s)"
55967
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   438
proof -
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   439
  { fix h
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   440
    have "f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   441
          (f' h * g x - f x * g' h) / (g x * g x)"
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   442
      by (simp add: divide_inverse field_simps nonzero_inverse_mult_distrib x)
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   443
   }
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   444
  then show ?thesis
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   445
    using has_derivative_divide [OF f g] x
55967
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   446
    by simp
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   447
qed
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   448
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   449
subsection {* Uniqueness *}
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   450
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   451
text {*
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   452
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   453
This can not generally shown for @{const has_derivative}, as we need to approach the point from
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   454
all directions. There is a proof in @{text Multivariate_Analysis} for @{text euclidean_space}.
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   455
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   456
*}
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   457
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   458
lemma has_derivative_zero_unique:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   459
  assumes "((\<lambda>x. 0) has_derivative F) (at x)" shows "F = (\<lambda>h. 0)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   460
proof -
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   461
  interpret F: bounded_linear F
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   462
    using assms by (rule has_derivative_bounded_linear)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   463
  let ?r = "\<lambda>h. norm (F h) / norm h"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   464
  have *: "?r -- 0 --> 0"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   465
    using assms unfolding has_derivative_at by simp
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   466
  show "F = (\<lambda>h. 0)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   467
  proof
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   468
    fix h show "F h = 0"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   469
    proof (rule ccontr)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 51642
diff changeset
   470
      assume **: "F h \<noteq> 0"
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 51642
diff changeset
   471
      then have h: "h \<noteq> 0"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   472
        by (clarsimp simp add: F.zero)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 51642
diff changeset
   473
      with ** have "0 < ?r h"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   474
        by (simp add: divide_pos_pos)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   475
      from LIM_D [OF * this] obtain s where s: "0 < s"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   476
        and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> ?r x < ?r h" by auto
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   477
      from dense [OF s] obtain t where t: "0 < t \<and> t < s" ..
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   478
      let ?x = "scaleR (t / norm h) h"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   479
      have "?x \<noteq> 0" and "norm ?x < s" using t h by simp_all
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   480
      hence "?r ?x < ?r h" by (rule r)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   481
      thus "False" using t h by (simp add: F.scaleR)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   482
    qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   483
  qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   484
qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   485
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   486
lemma has_derivative_unique:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   487
  assumes "(f has_derivative F) (at x)" and "(f has_derivative F') (at x)" shows "F = F'"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   488
proof -
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   489
  have "((\<lambda>x. 0) has_derivative (\<lambda>h. F h - F' h)) (at x)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   490
    using has_derivative_diff [OF assms] by simp
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   491
  hence "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   492
    by (rule has_derivative_zero_unique)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   493
  thus "F = F'"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   494
    unfolding fun_eq_iff right_minus_eq .
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   495
qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   496
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   497
subsection {* Differentiability predicate *}
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   498
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   499
definition
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   500
  differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
56182
528fae0816ea update syntax of has_*derivative to infix 50; fixed proofs
hoelzl
parents: 56181
diff changeset
   501
  (infix "differentiable" 50)
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   502
where
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   503
  "f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   504
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   505
lemma differentiable_subset: "f differentiable (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable (at x within t)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   506
  unfolding differentiable_def by (blast intro: has_derivative_subset)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   507
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   508
lemmas differentiable_within_subset = differentiable_subset
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   509
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   510
lemma differentiable_ident [simp, derivative_intros]: "(\<lambda>x. x) differentiable F"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   511
  unfolding differentiable_def by (blast intro: has_derivative_ident)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   512
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   513
lemma differentiable_const [simp, derivative_intros]: "(\<lambda>z. a) differentiable F"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   514
  unfolding differentiable_def by (blast intro: has_derivative_const)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   515
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   516
lemma differentiable_in_compose:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   517
  "f differentiable (at (g x) within (g`s)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f (g x)) differentiable (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   518
  unfolding differentiable_def by (blast intro: has_derivative_in_compose)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   519
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   520
lemma differentiable_compose:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   521
  "f differentiable (at (g x)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f (g x)) differentiable (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   522
  by (blast intro: differentiable_in_compose differentiable_subset)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   523
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   524
lemma differentiable_sum [simp, derivative_intros]:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   525
  "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x + g x) differentiable F"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   526
  unfolding differentiable_def by (blast intro: has_derivative_add)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   527
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   528
lemma differentiable_minus [simp, derivative_intros]:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   529
  "f differentiable F \<Longrightarrow> (\<lambda>x. - f x) differentiable F"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   530
  unfolding differentiable_def by (blast intro: has_derivative_minus)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   531
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   532
lemma differentiable_diff [simp, derivative_intros]:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   533
  "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x - g x) differentiable F"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   534
  unfolding differentiable_def by (blast intro: has_derivative_diff)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   535
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   536
lemma differentiable_mult [simp, derivative_intros]:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   537
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_algebra"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   538
  shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x * g x) differentiable (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   539
  unfolding differentiable_def by (blast intro: has_derivative_mult)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   540
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   541
lemma differentiable_inverse [simp, derivative_intros]:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   542
  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   543
  shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> (\<lambda>x. inverse (f x)) differentiable (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   544
  unfolding differentiable_def by (blast intro: has_derivative_inverse)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   545
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   546
lemma differentiable_divide [simp, derivative_intros]:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   547
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   548
  shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   549
  unfolding divide_inverse using assms by simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   550
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   551
lemma differentiable_power [simp, derivative_intros]:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   552
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   553
  shows "f differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   554
  unfolding differentiable_def by (blast intro: has_derivative_power)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   555
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   556
lemma differentiable_scaleR [simp, derivative_intros]:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   557
  "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) differentiable (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   558
  unfolding differentiable_def by (blast intro: has_derivative_scaleR)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   559
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   560
lemma has_derivative_imp_has_field_derivative:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   561
  "(f has_derivative D) F \<Longrightarrow> (\<And>x. x * D' = D x) \<Longrightarrow> (f has_field_derivative D') F"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   562
  unfolding has_field_derivative_def 
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   563
  by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult_commute)
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   564
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   565
lemma has_field_derivative_imp_has_derivative: "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative op * D) F"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   566
  by (simp add: has_field_derivative_def)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   567
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   568
lemma DERIV_subset: 
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   569
  "(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s 
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   570
   \<Longrightarrow> (f has_field_derivative f') (at x within t)"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   571
  by (simp add: has_field_derivative_def has_derivative_within_subset)
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   572
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   573
abbreviation (input)
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   574
  DERIV :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   575
  ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   576
where
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   577
  "DERIV f x :> D \<equiv> (f has_field_derivative D) (at x)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   578
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   579
abbreviation 
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   580
  has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
56182
528fae0816ea update syntax of has_*derivative to infix 50; fixed proofs
hoelzl
parents: 56181
diff changeset
   581
  (infix "(has'_real'_derivative)" 50)
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   582
where
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   583
  "(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   584
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   585
lemma real_differentiable_def:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   586
  "f differentiable at x within s \<longleftrightarrow> (\<exists>D. (f has_real_derivative D) (at x within s))"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   587
proof safe
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   588
  assume "f differentiable at x within s"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   589
  then obtain f' where *: "(f has_derivative f') (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   590
    unfolding differentiable_def by auto
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   591
  then obtain c where "f' = (op * c)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   592
    by (metis real_bounded_linear has_derivative_bounded_linear mult_commute fun_eq_iff)
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   593
  with * show "\<exists>D. (f has_real_derivative D) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   594
    unfolding has_field_derivative_def by auto
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   595
qed (auto simp: differentiable_def has_field_derivative_def)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   596
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   597
lemma real_differentiableE [elim?]:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   598
  assumes f: "f differentiable (at x within s)" obtains df where "(f has_real_derivative df) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   599
  using assms by (auto simp: real_differentiable_def)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   600
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   601
lemma differentiableD: "f differentiable (at x within s) \<Longrightarrow> \<exists>D. (f has_real_derivative D) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   602
  by (auto elim: real_differentiableE)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   603
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   604
lemma differentiableI: "(f has_real_derivative D) (at x within s) \<Longrightarrow> f differentiable (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   605
  by (force simp add: real_differentiable_def)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   606
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   607
lemma DERIV_def: "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   608
  apply (simp add: has_field_derivative_def has_derivative_at bounded_linear_mult_right LIM_zero_iff[symmetric, of _ D])
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   609
  apply (subst (2) tendsto_norm_zero_iff[symmetric])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   610
  apply (rule filterlim_cong)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   611
  apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   612
  done
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   613
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   614
lemma mult_commute_abs: "(\<lambda>x. x * c) = op * (c::'a::ab_semigroup_mult)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   615
  by (simp add: fun_eq_iff mult_commute)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   616
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   617
subsection {* Derivatives *}
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   618
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   619
lemma DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   620
  by (simp add: DERIV_def)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   621
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   622
lemma DERIV_const [simp, derivative_intros]: "((\<lambda>x. k) has_field_derivative 0) F"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   623
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_const]) auto
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   624
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   625
lemma DERIV_ident [simp, derivative_intros]: "((\<lambda>x. x) has_field_derivative 1) F"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   626
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_ident]) auto
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   627
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   628
lemma field_differentiable_add[derivative_intros]:
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   629
  "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow> 
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   630
    ((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   631
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   632
     (auto simp: has_field_derivative_def field_simps mult_commute_abs)
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   633
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   634
corollary DERIV_add:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   635
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   636
  ((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   637
  by (rule field_differentiable_add)
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   638
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   639
lemma field_differentiable_minus[derivative_intros]:
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   640
  "(f has_field_derivative f') F \<Longrightarrow> ((\<lambda>z. - (f z)) has_field_derivative -f') F"
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   641
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   642
     (auto simp: has_field_derivative_def field_simps mult_commute_abs)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   643
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   644
corollary DERIV_minus: "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. - f x) has_field_derivative -D) (at x within s)"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   645
  by (rule field_differentiable_minus)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   646
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   647
lemma field_differentiable_diff[derivative_intros]:
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   648
  "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow> ((\<lambda>z. f z - g z) has_field_derivative f' - g') F"
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   649
  by (simp only: assms diff_conv_add_uminus field_differentiable_add field_differentiable_minus)
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   650
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   651
corollary DERIV_diff:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   652
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   653
  ((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   654
  by (rule field_differentiable_diff)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   655
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   656
lemma DERIV_continuous: "(f has_field_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   657
  by (drule has_derivative_continuous[OF has_field_derivative_imp_has_derivative]) simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   658
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   659
corollary DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   660
  by (rule DERIV_continuous)
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   661
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   662
lemma DERIV_continuous_on:
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   663
  "(\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative D) (at x)) \<Longrightarrow> continuous_on s f"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
   664
  by (metis DERIV_continuous continuous_at_imp_continuous_on)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   665
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   666
lemma DERIV_mult':
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   667
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   668
  ((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   669
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   670
     (auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   671
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   672
lemma DERIV_mult[derivative_intros]:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   673
  "(f has_field_derivative Da) (at x within s) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   674
  ((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   675
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   676
     (auto simp: field_simps dest: has_field_derivative_imp_has_derivative)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   677
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   678
text {* Derivative of linear multiplication *}
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   679
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   680
lemma DERIV_cmult:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   681
  "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   682
  by (drule DERIV_mult' [OF DERIV_const], simp)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   683
55967
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   684
lemma DERIV_cmult_right:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   685
  "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   686
  using DERIV_cmult by (force simp add: mult_ac)
55967
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   687
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   688
lemma DERIV_cmult_Id [simp]: "(op * c has_field_derivative c) (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   689
  by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   690
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   691
lemma DERIV_cdivide:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   692
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. f x / c) has_field_derivative D / c) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   693
  using DERIV_cmult_right[of f D x s "1 / c"] by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   694
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   695
lemma DERIV_unique:
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   696
  "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   697
  unfolding DERIV_def by (rule LIM_unique) 
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   698
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   699
lemma DERIV_setsum[derivative_intros]:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   700
  "(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow> 
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   701
    ((\<lambda>x. setsum (f x) S) has_field_derivative setsum (f' x) S) F"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   702
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_setsum])
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   703
     (auto simp: setsum_right_distrib mult_commute_abs dest: has_field_derivative_imp_has_derivative)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   704
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   705
lemma DERIV_inverse'[derivative_intros]:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   706
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   707
  ((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   708
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_inverse])
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   709
     (auto dest: has_field_derivative_imp_has_derivative)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   710
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   711
text {* Power of @{text "-1"} *}
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   712
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   713
lemma DERIV_inverse:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   714
  "x \<noteq> 0 \<Longrightarrow> ((\<lambda>x. inverse(x)) has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   715
  by (drule DERIV_inverse' [OF DERIV_ident]) simp
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   716
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   717
text {* Derivative of inverse *}
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   718
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   719
lemma DERIV_inverse_fun:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   720
  "(f has_field_derivative d) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   721
  ((\<lambda>x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   722
  by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   723
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   724
text {* Derivative of quotient *}
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   725
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   726
lemma DERIV_divide[derivative_intros]:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   727
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   728
  (g has_field_derivative E) (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   729
  ((\<lambda>x. f x / g x) has_field_derivative (D * g x - f x * E) / (g x * g x)) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   730
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_divide])
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   731
     (auto dest: has_field_derivative_imp_has_derivative simp: field_simps nonzero_inverse_mult_distrib divide_inverse)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   732
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   733
lemma DERIV_quotient:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   734
  "(f has_field_derivative d) (at x within s) \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   735
  (g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> 
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   736
  ((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   737
  by (drule (2) DERIV_divide) (simp add: mult_commute)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   738
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   739
lemma DERIV_power_Suc:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   740
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   741
  ((\<lambda>x. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   742
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   743
     (auto simp: has_field_derivative_def)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   744
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   745
lemma DERIV_power[derivative_intros]:
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   746
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   747
  ((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   748
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   749
     (auto simp: has_field_derivative_def)
31880
6fb86c61747c Added DERIV_intros
hoelzl
parents: 31404
diff changeset
   750
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   751
lemma DERIV_pow: "((\<lambda>x. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   752
  apply (cut_tac DERIV_power [OF DERIV_ident])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   753
  apply (simp add: real_of_nat_def)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   754
  done
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   755
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   756
lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow> 
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   757
  ((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   758
  using has_derivative_compose[of f "op * D" x s g "op * E"]
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   759
  unfolding has_field_derivative_def mult_commute_abs ac_simps .
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   760
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   761
corollary DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   762
  ((\<lambda>x. f (g x)) has_field_derivative Da * Db) (at x within s)"
55967
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   763
  by (rule DERIV_chain')
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   764
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   765
text {* Standard version *}
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   766
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   767
lemma DERIV_chain:
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   768
  "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow> 
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   769
  (f o g has_field_derivative Da * Db) (at x within s)"
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   770
  by (drule (1) DERIV_chain', simp add: o_def mult_commute)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   771
55967
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   772
lemma DERIV_image_chain: 
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   773
  "(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   774
  (f o g has_field_derivative Da * Db) (at x within s)"
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   775
  using has_derivative_in_compose [of g "op * Db" x s f "op * Da "]
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   776
  by (simp add: has_field_derivative_def o_def mult_commute_abs ac_simps)
55967
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   777
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   778
(*These two are from HOL Light: HAS_COMPLEX_DERIVATIVE_CHAIN*)
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   779
lemma DERIV_chain_s:
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   780
  assumes "(\<And>x. x \<in> s \<Longrightarrow> DERIV g x :> g'(x))"
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   781
      and "DERIV f x :> f'" 
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   782
      and "f x \<in> s"
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   783
    shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   784
  by (metis (full_types) DERIV_chain' mult_commute assms)
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   785
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   786
lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*)
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   787
  assumes "(\<And>x. DERIV g x :> g'(x))"
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   788
      and "DERIV f x :> f'" 
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   789
    shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   790
  by (metis UNIV_I DERIV_chain_s [of UNIV] assms)
5dadc93ff3df a few new lemmas
paulson <lp15@cam.ac.uk>
parents: 54230
diff changeset
   791
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   792
declare
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   793
  DERIV_power[where 'a=real, unfolded real_of_nat_def[symmetric], derivative_intros]
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   794
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   795
text{*Alternative definition for differentiability*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   796
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   797
lemma DERIV_LIM_iff:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   798
  fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   799
     "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   800
      ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   801
apply (rule iffI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   802
apply (drule_tac k="- a" in LIM_offset)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53381
diff changeset
   803
apply simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   804
apply (drule_tac k="a" in LIM_offset)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   805
apply (simp add: add_commute)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   806
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   807
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   808
lemma DERIV_iff2: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>z. (f z - f x) / (z - x)) --x --> D"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   809
  by (simp add: DERIV_def DERIV_LIM_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   810
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   811
lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   812
    DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   813
  unfolding DERIV_iff2
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   814
proof (rule filterlim_cong)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 51642
diff changeset
   815
  assume *: "eventually (\<lambda>x. f x = g x) (nhds x)"
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 51642
diff changeset
   816
  moreover from * have "f x = g x" by (auto simp: eventually_nhds)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   817
  moreover assume "x = y" "u = v"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   818
  ultimately show "eventually (\<lambda>xa. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)) (at x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   819
    by (auto simp: eventually_at_filter elim: eventually_elim1)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   820
qed simp_all
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   821
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   822
lemma DERIV_shift:
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   823
  "(DERIV f (x + z) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (x + z)) x :> y)"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   824
  by (simp add: DERIV_def field_simps)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   825
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   826
lemma DERIV_mirror:
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   827
  "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
56409
36489d77c484 removing simprule status for divide_minus_left and divide_minus_right
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   828
  by (simp add: DERIV_def filterlim_at_split filterlim_at_left_to_right divide_minus_right
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   829
                tendsto_minus_cancel_left field_simps conj_commute)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   830
29975
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   831
text {* Caratheodory formulation of derivative at a point *}
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   832
55970
6d123f0ae358 Some new proofs. Tidying up, esp to remove "apply rule".
paulson <lp15@cam.ac.uk>
parents: 55967
diff changeset
   833
lemma CARAT_DERIV: (*FIXME: SUPERSEDED BY THE ONE IN Deriv.thy. But still used by NSA/HDeriv.thy*)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
   834
  "(DERIV f x :> l) \<longleftrightarrow> (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   835
      (is "?lhs = ?rhs")
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   836
proof
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   837
  assume der: "DERIV f x :> l"
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   838
  show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   839
  proof (intro exI conjI)
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   840
    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23412
diff changeset
   841
    show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   842
    show "isCont ?g x" using der
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   843
      by (simp add: isCont_iff DERIV_def cong: LIM_equal [rule_format])
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   844
    show "?g x = l" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   845
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   846
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   847
  assume "?rhs"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   848
  then obtain g where
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
   849
    "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   850
  thus "(DERIV f x :> l)"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
   851
     by (auto simp add: isCont_iff DERIV_def cong: LIM_cong)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   852
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   853
31899
1a7ade46061b fixed document (DERIV_intros);
wenzelm
parents: 31880
diff changeset
   854
text {*
1a7ade46061b fixed document (DERIV_intros);
wenzelm
parents: 31880
diff changeset
   855
 Let's do the standard proof, though theorem
1a7ade46061b fixed document (DERIV_intros);
wenzelm
parents: 31880
diff changeset
   856
 @{text "LIM_mult2"} follows from a NS proof
1a7ade46061b fixed document (DERIV_intros);
wenzelm
parents: 31880
diff changeset
   857
*}
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   858
29975
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   859
subsection {* Local extrema *}
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   860
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   861
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   862
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   863
lemma DERIV_pos_inc_right:
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   864
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   865
  assumes der: "DERIV f x :> l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   866
      and l:   "0 < l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   867
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   868
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   869
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   870
  have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l)"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53381
diff changeset
   871
    by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   872
  then obtain s
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   873
        where s:   "0 < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   874
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   875
    by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   876
  thus ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   877
  proof (intro exI conjI strip)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
   878
    show "0<s" using s .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   879
    fix h::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   880
    assume "0 < h" "h < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   881
    with all [of h] show "f x < f (x+h)"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53381
diff changeset
   882
    proof (simp add: abs_if pos_less_divide_eq split add: split_if_asm)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   883
      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   884
      with l
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   885
      have "0 < (f (x+h) - f x) / h" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   886
      thus "f x < f (x+h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   887
  by (simp add: pos_less_divide_eq h)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   888
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   889
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   890
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   891
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   892
lemma DERIV_neg_dec_left:
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   893
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   894
  assumes der: "DERIV f x :> l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   895
      and l:   "l < 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   896
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   897
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   898
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   899
  have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53381
diff changeset
   900
    by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   901
  then obtain s
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   902
        where s:   "0 < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   903
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   904
    by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   905
  thus ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   906
  proof (intro exI conjI strip)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
   907
    show "0<s" using s .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   908
    fix h::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   909
    assume "0 < h" "h < s"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   910
    with all [of "-h"] show "f x < f (x-h)"
56409
36489d77c484 removing simprule status for divide_minus_left and divide_minus_right
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   911
    proof (simp add: abs_if pos_less_divide_eq divide_minus_right split add: split_if_asm)
36489d77c484 removing simprule status for divide_minus_left and divide_minus_right
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
   912
      assume "- ((f (x-h) - f x) / h) < l" and h: "0 < h"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   913
      with l
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   914
      have "0 < (f (x-h) - f x) / h" by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   915
      thus "f x < f (x-h)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   916
  by (simp add: pos_less_divide_eq h)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   917
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   918
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   919
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   920
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   921
lemma DERIV_pos_inc_left:
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   922
  fixes f :: "real => real"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   923
  shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   924
  apply (rule DERIV_neg_dec_left [of "%x. - f x" "-l" x, simplified])
41368
8afa26855137 use DERIV_intros
hoelzl
parents: 37891
diff changeset
   925
  apply (auto simp add: DERIV_minus)
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   926
  done
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   927
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   928
lemma DERIV_neg_dec_right:
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   929
  fixes f :: "real => real"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   930
  shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
   931
  apply (rule DERIV_pos_inc_right [of "%x. - f x" "-l" x, simplified])
41368
8afa26855137 use DERIV_intros
hoelzl
parents: 37891
diff changeset
   932
  apply (auto simp add: DERIV_minus)
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   933
  done
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   934
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   935
lemma DERIV_local_max:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   936
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   937
  assumes der: "DERIV f x :> l"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   938
      and d:   "0 < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   939
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   940
  shows "l = 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   941
proof (cases rule: linorder_cases [of l 0])
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
   942
  case equal thus ?thesis .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   943
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   944
  case less
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   945
  from DERIV_neg_dec_left [OF der less]
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   946
  obtain d' where d': "0 < d'"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   947
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   948
  from real_lbound_gt_zero [OF d d']
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   949
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   950
  with lt le [THEN spec [where x="x-e"]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   951
  show ?thesis by (auto simp add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   952
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   953
  case greater
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
   954
  from DERIV_pos_inc_right [OF der greater]
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   955
  obtain d' where d': "0 < d'"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   956
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   957
  from real_lbound_gt_zero [OF d d']
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   958
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   959
  with lt le [THEN spec [where x="x+e"]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   960
  show ?thesis by (auto simp add: abs_if)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   961
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   962
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   963
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   964
text{*Similar theorem for a local minimum*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   965
lemma DERIV_local_min:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   966
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   967
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   968
by (drule DERIV_minus [THEN DERIV_local_max], auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   969
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   970
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   971
text{*In particular, if a function is locally flat*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   972
lemma DERIV_local_const:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   973
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   974
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   975
by (auto dest!: DERIV_local_max)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   976
29975
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   977
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   978
subsection {* Rolle's Theorem *}
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
   979
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   980
text{*Lemma about introducing open ball in open interval*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   981
lemma lemma_interval_lt:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   982
     "[| a < x;  x < b |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   983
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
   984
22998
97e1f9c2cc46 avoid using redundant lemmas from RealDef.thy
huffman
parents: 22984
diff changeset
   985
apply (simp add: abs_less_iff)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   986
apply (insert linorder_linear [of "x-a" "b-x"], safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   987
apply (rule_tac x = "x-a" in exI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   988
apply (rule_tac [2] x = "b-x" in exI, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   989
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   990
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   991
lemma lemma_interval: "[| a < x;  x < b |] ==>
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   992
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   993
apply (drule lemma_interval_lt, auto)
44921
58eef4843641 tuned proofs
huffman
parents: 44890
diff changeset
   994
apply force
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   995
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   996
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   997
text{*Rolle's Theorem.
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   998
   If @{term f} is defined and continuous on the closed interval
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
   999
   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1000
   and @{term "f(a) = f(b)"},
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1001
   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1002
theorem Rolle:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1003
  assumes lt: "a < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1004
      and eq: "f(a) = f(b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1005
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
  1006
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1007
  shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1008
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1009
  have le: "a \<le> b" using lt by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1010
  from isCont_eq_Ub [OF le con]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1011
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1012
             and alex: "a \<le> x" and xleb: "x \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1013
    by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1014
  from isCont_eq_Lb [OF le con]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1015
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1016
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1017
    by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1018
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1019
  proof cases
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1020
    assume axb: "a < x & x < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1021
        --{*@{term f} attains its maximum within the interval*}
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
  1022
    hence ax: "a<x" and xb: "x<b" by arith + 
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1023
    from lemma_interval [OF ax xb]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1024
    obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1025
      by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1026
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1027
      by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1028
    from differentiableD [OF dif [OF axb]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1029
    obtain l where der: "DERIV f x :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1030
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1031
        --{*the derivative at a local maximum is zero*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1032
    thus ?thesis using ax xb der by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1033
  next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1034
    assume notaxb: "~ (a < x & x < b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1035
    hence xeqab: "x=a | x=b" using alex xleb by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1036
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1037
    show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1038
    proof cases
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1039
      assume ax'b: "a < x' & x' < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1040
        --{*@{term f} attains its minimum within the interval*}
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
  1041
      hence ax': "a<x'" and x'b: "x'<b" by arith+ 
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1042
      from lemma_interval [OF ax' x'b]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1043
      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1044
  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1045
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1046
  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1047
      from differentiableD [OF dif [OF ax'b]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1048
      obtain l where der: "DERIV f x' :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1049
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1050
        --{*the derivative at a local minimum is zero*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1051
      thus ?thesis using ax' x'b der by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1052
    next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1053
      assume notax'b: "~ (a < x' & x' < b)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1054
        --{*@{term f} is constant througout the interval*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1055
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1056
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1057
      from dense [OF lt]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1058
      obtain r where ar: "a < r" and rb: "r < b" by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1059
      from lemma_interval [OF ar rb]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1060
      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1061
  by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1062
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1063
      proof (clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1064
        fix z::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1065
        assume az: "a \<le> z" and zb: "z \<le> b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1066
        show "f z = f b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1067
        proof (rule order_antisym)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1068
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1069
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1070
        qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1071
      qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1072
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1073
      proof (intro strip)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1074
        fix y::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1075
        assume lt: "\<bar>r-y\<bar> < d"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1076
        hence "f y = f b" by (simp add: eq_fb bound)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1077
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1078
      qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1079
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1080
      obtain l where der: "DERIV f r :> l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1081
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1082
        --{*the derivative of a constant function is zero*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1083
      thus ?thesis using ar rb der by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1084
    qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1085
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1086
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1087
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1088
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1089
subsection{*Mean Value Theorem*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1090
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1091
lemma lemma_MVT:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1092
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
51481
ef949192e5d6 move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents: 51480
diff changeset
  1093
  by (cases "a = b") (simp_all add: field_simps)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1094
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1095
theorem MVT:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1096
  assumes lt:  "a < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1097
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
  1098
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1099
  shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1100
                   (f(b) - f(a) = (b-a) * l)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1101
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1102
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44209
diff changeset
  1103
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56369
diff changeset
  1104
    using con by (fast intro: continuous_intros)
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
  1105
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable (at x)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1106
  proof (clarify)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1107
    fix x::real
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1108
    assume ax: "a < x" and xb: "x < b"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1109
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1110
    obtain l where der: "DERIV f x :> l" ..
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
  1111
    show "?F differentiable (at x)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1112
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1113
          blast intro: DERIV_diff DERIV_cmult_Id der)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1114
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1115
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1116
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1117
    by blast
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1118
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1119
    by (rule DERIV_cmult_Id)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1120
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1121
                   :> 0 + (f b - f a) / (b - a)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1122
    by (rule DERIV_add [OF der])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1123
  show ?thesis
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1124
  proof (intro exI conjI)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1125
    show "a < z" using az .
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23431
diff changeset
  1126
    show "z < b" using zb .
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1127
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1128
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1129
  qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1130
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1131
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1132
lemma MVT2:
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1133
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1134
      ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1135
apply (drule MVT)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1136
apply (blast intro: DERIV_isCont)
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
  1137
apply (force dest: order_less_imp_le simp add: real_differentiable_def)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1138
apply (blast dest: DERIV_unique order_less_imp_le)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1139
done
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1140
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1141
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1142
text{*A function is constant if its derivative is 0 over an interval.*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1143
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1144
lemma DERIV_isconst_end:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1145
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1146
  shows "[| a < b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1147
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1148
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1149
        ==> f b = f a"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1150
apply (drule MVT, assumption)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1151
apply (blast intro: differentiableI)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1152
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1153
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1154
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1155
lemma DERIV_isconst1:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1156
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1157
  shows "[| a < b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1158
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1159
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1160
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1161
apply safe
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1162
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1163
apply (drule_tac b = x in DERIV_isconst_end, auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1164
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1165
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1166
lemma DERIV_isconst2:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1167
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1168
  shows "[| a < b;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1169
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1170
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1171
         a \<le> x; x \<le> b |]
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1172
        ==> f x = f a"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1173
apply (blast dest: DERIV_isconst1)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1174
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1175
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1176
lemma DERIV_isconst3: fixes a b x y :: real
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1177
  assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1178
  assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1179
  shows "f x = f y"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1180
proof (cases "x = y")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1181
  case False
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1182
  let ?a = "min x y"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1183
  let ?b = "max x y"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1184
  
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1185
  have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1186
  proof (rule allI, rule impI)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1187
    fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1188
    hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1189
    hence "z \<in> {a<..<b}" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1190
    thus "DERIV f z :> 0" by (rule derivable)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1191
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1192
  hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1193
    and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1194
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1195
  have "?a < ?b" using `x \<noteq> y` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1196
  from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1197
  show ?thesis by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1198
qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
  1199
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1200
lemma DERIV_isconst_all:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1201
  fixes f :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1202
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1203
apply (rule linorder_cases [of x y])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1204
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1205
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1206
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1207
lemma DERIV_const_ratio_const:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1208
  fixes f :: "real => real"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1209
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1210
apply (rule linorder_cases [of a b], auto)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1211
apply (drule_tac [!] f = f in MVT)
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
  1212
apply (auto dest: DERIV_isCont DERIV_unique simp add: real_differentiable_def)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53381
diff changeset
  1213
apply (auto dest: DERIV_unique simp add: ring_distribs)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1214
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1215
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1216
lemma DERIV_const_ratio_const2:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1217
  fixes f :: "real => real"
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1218
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
56217
dc429a5b13c4 Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents: 56182
diff changeset
  1219
apply (rule_tac c1 = "b-a" in mult_right_cancel [THEN iffD1])
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1220
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1221
done
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1222
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1223
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1224
by (simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1225
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1226
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1227
by (simp)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1228
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1229
text{*Gallileo's "trick": average velocity = av. of end velocities*}
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1230
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1231
lemma DERIV_const_average:
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1232
  fixes v :: "real => real"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1233
  assumes neq: "a \<noteq> (b::real)"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1234
      and der: "\<forall>x. DERIV v x :> k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1235
  shows "v ((a + b)/2) = (v a + v b)/2"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1236
proof (cases rule: linorder_cases [of a b])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1237
  case equal with neq show ?thesis by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1238
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1239
  case less
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1240
  have "(v b - v a) / (b - a) = k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1241
    by (rule DERIV_const_ratio_const2 [OF neq der])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1242
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1243
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1244
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1245
  ultimately show ?thesis using neq by force
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1246
next
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1247
  case greater
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1248
  have "(v b - v a) / (b - a) = k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1249
    by (rule DERIV_const_ratio_const2 [OF neq der])
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1250
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1251
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1252
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1253
  ultimately show ?thesis using neq by (force simp add: add_commute)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1254
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1255
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1256
(* A function with positive derivative is increasing. 
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1257
   A simple proof using the MVT, by Jeremy Avigad. And variants.
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1258
*)
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1259
lemma DERIV_pos_imp_increasing_open:
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1260
  fixes a::real and b::real and f::"real => real"
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1261
  assumes "a < b" and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1262
      and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1263
  shows "f a < f b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1264
proof (rule ccontr)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 41368
diff changeset
  1265
  assume f: "~ f a < f b"
33690
889d06128608 simplified bulky metis proofs;
wenzelm
parents: 33659
diff changeset
  1266
  have "EX l z. a < z & z < b & DERIV f z :> l
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1267
      & f b - f a = (b - a) * l"
33690
889d06128608 simplified bulky metis proofs;
wenzelm
parents: 33659
diff changeset
  1268
    apply (rule MVT)
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1269
      using assms Deriv.differentiableI
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1270
      apply force+
33690
889d06128608 simplified bulky metis proofs;
wenzelm
parents: 33659
diff changeset
  1271
    done
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 41368
diff changeset
  1272
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1273
      and "f b - f a = (b - a) * l"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1274
    by auto
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 41368
diff changeset
  1275
  with assms f have "~(l > 0)"
36777
be5461582d0f avoid using real-specific versions of generic lemmas
huffman
parents: 35216
diff changeset
  1276
    by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 41368
diff changeset
  1277
  with assms z show False
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1278
    by (metis DERIV_unique)
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1279
qed
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1280
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1281
lemma DERIV_pos_imp_increasing:
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1282
  fixes a::real and b::real and f::"real => real"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1283
  assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1284
  shows "f a < f b"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1285
by (metis DERIV_pos_imp_increasing_open [of a b f] assms DERIV_continuous less_imp_le)
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1286
45791
d985ec974815 more systematic lemma name
noschinl
parents: 45600
diff changeset
  1287
lemma DERIV_nonneg_imp_nondecreasing:
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1288
  fixes a::real and b::real and f::"real => real"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1289
  assumes "a \<le> b" and
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1290
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1291
  shows "f a \<le> f b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1292
proof (rule ccontr, cases "a = b")
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 41368
diff changeset
  1293
  assume "~ f a \<le> f b" and "a = b"
efa734d9b221 eliminated global prems;
wenzelm
parents: 41368
diff changeset
  1294
  then show False by auto
37891
c26f9d06e82c robustified metis proof
haftmann
parents: 37888
diff changeset
  1295
next
c26f9d06e82c robustified metis proof
haftmann
parents: 37888
diff changeset
  1296
  assume A: "~ f a \<le> f b"
c26f9d06e82c robustified metis proof
haftmann
parents: 37888
diff changeset
  1297
  assume B: "a ~= b"
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1298
  with assms have "EX l z. a < z & z < b & DERIV f z :> l
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1299
      & f b - f a = (b - a) * l"
33690
889d06128608 simplified bulky metis proofs;
wenzelm
parents: 33659
diff changeset
  1300
    apply -
889d06128608 simplified bulky metis proofs;
wenzelm
parents: 33659
diff changeset
  1301
    apply (rule MVT)
889d06128608 simplified bulky metis proofs;
wenzelm
parents: 33659
diff changeset
  1302
      apply auto
889d06128608 simplified bulky metis proofs;
wenzelm
parents: 33659
diff changeset
  1303
      apply (metis DERIV_isCont)
36777
be5461582d0f avoid using real-specific versions of generic lemmas
huffman
parents: 35216
diff changeset
  1304
     apply (metis differentiableI less_le)
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1305
    done
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 41368
diff changeset
  1306
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
37891
c26f9d06e82c robustified metis proof
haftmann
parents: 37888
diff changeset
  1307
      and C: "f b - f a = (b - a) * l"
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1308
    by auto
37891
c26f9d06e82c robustified metis proof
haftmann
parents: 37888
diff changeset
  1309
  with A have "a < b" "f b < f a" by auto
c26f9d06e82c robustified metis proof
haftmann
parents: 37888
diff changeset
  1310
  with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
45051
c478d1876371 discontinued legacy theorem names from RealDef.thy
huffman
parents: 44921
diff changeset
  1311
    (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 41368
diff changeset
  1312
  with assms z show False
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1313
    by (metis DERIV_unique order_less_imp_le)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1314
qed
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1315
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1316
lemma DERIV_neg_imp_decreasing_open:
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1317
  fixes a::real and b::real and f::"real => real"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1318
  assumes "a < b" and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (EX y. DERIV f x :> y & y < 0)"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1319
      and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1320
  shows "f a > f b"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1321
proof -
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1322
  have "(%x. -f x) a < (%x. -f x) b"
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1323
    apply (rule DERIV_pos_imp_increasing_open [of a b "%x. -f x"])
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1324
    using assms
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1325
    apply auto
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1326
    apply (metis field_differentiable_minus neg_0_less_iff_less)
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1327
    done
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1328
  thus ?thesis
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1329
    by simp
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1330
qed
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1331
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1332
lemma DERIV_neg_imp_decreasing:
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1333
  fixes a::real and b::real and f::"real => real"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1334
  assumes "a < b" and
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1335
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1336
  shows "f a > f b"
56261
918432e3fcfa rearranging some deriv theorems
paulson <lp15@cam.ac.uk>
parents: 56219
diff changeset
  1337
by (metis DERIV_neg_imp_decreasing_open [of a b f] assms DERIV_continuous less_imp_le)
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1338
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1339
lemma DERIV_nonpos_imp_nonincreasing:
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1340
  fixes a::real and b::real and f::"real => real"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1341
  assumes "a \<le> b" and
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1342
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1343
  shows "f a \<ge> f b"
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1344
proof -
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1345
  have "(%x. -f x) a \<le> (%x. -f x) b"
45791
d985ec974815 more systematic lemma name
noschinl
parents: 45600
diff changeset
  1346
    apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
33690
889d06128608 simplified bulky metis proofs;
wenzelm
parents: 33659
diff changeset
  1347
    using assms
889d06128608 simplified bulky metis proofs;
wenzelm
parents: 33659
diff changeset
  1348
    apply auto
33654
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1349
    apply (metis DERIV_minus neg_0_le_iff_le)
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1350
    done
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1351
  thus ?thesis
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1352
    by simp
abf780db30ea A number of theorems contributed by Jeremy Avigad
paulson
parents: 31902
diff changeset
  1353
qed
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1354
56289
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1355
lemma DERIV_pos_imp_increasing_at_bot:
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1356
  fixes f :: "real => real"
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1357
  assumes "\<And>x. x \<le> b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1358
      and lim: "(f ---> flim) at_bot"
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1359
  shows "flim < f b"
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1360
proof -
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1361
  have "flim \<le> f (b - 1)"
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1362
    apply (rule tendsto_ge_const [OF _ lim])
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1363
    apply (auto simp: trivial_limit_at_bot_linorder eventually_at_bot_linorder)
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1364
    apply (rule_tac x="b - 2" in exI)
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1365
    apply (force intro: order.strict_implies_order DERIV_pos_imp_increasing [where f=f] assms)
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1366
    done
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1367
  also have "... < f b"
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1368
    by (force intro: DERIV_pos_imp_increasing [where f=f] assms)
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1369
  finally show ?thesis .
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1370
qed
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1371
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1372
lemma DERIV_neg_imp_decreasing_at_top:
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1373
  fixes f :: "real => real"
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1374
  assumes der: "\<And>x. x \<ge> b \<Longrightarrow> (EX y. DERIV f x :> y & y < 0)"
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1375
      and lim: "(f ---> flim) at_top"
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1376
  shows "flim < f b"
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1377
  apply (rule DERIV_pos_imp_increasing_at_bot [where f = "\<lambda>i. f (-i)" and b = "-b", simplified])
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1378
  apply (metis DERIV_mirror der le_minus_iff neg_0_less_iff_less)
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1379
  apply (metis filterlim_at_top_mirror lim)
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1380
  done
d8d2a2b97168 Some useful lemmas
paulson <lp15@cam.ac.uk>
parents: 56261
diff changeset
  1381
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1382
text {* Derivative of inverse function *}
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1383
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1384
lemma DERIV_inverse_function:
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1385
  fixes f g :: "real \<Rightarrow> real"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1386
  assumes der: "DERIV f (g x) :> D"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1387
  assumes neq: "D \<noteq> 0"
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1388
  assumes a: "a < x" and b: "x < b"
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1389
  assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1390
  assumes cont: "isCont g x"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1391
  shows "DERIV g x :> inverse D"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1392
unfolding DERIV_iff2
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1393
proof (rule LIM_equal2)
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1394
  show "0 < min (x - a) (b - x)"
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
  1395
    using a b by arith 
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1396
next
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1397
  fix y
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1398
  assume "norm (y - x) < min (x - a) (b - x)"
27668
6eb20b2cecf8 Tuned and simplified proofs
chaieb
parents: 26120
diff changeset
  1399
  hence "a < y" and "y < b" 
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1400
    by (simp_all add: abs_less_iff)
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1401
  thus "(g y - g x) / (y - x) =
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1402
        inverse ((f (g y) - x) / (g y - g x))"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1403
    by (simp add: inj)
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1404
next
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1405
  have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1406
    by (rule der [unfolded DERIV_iff2])
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1407
  hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1408
    using inj a b by simp
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1409
  have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
56219
bf80d125406b tuned proofs;
wenzelm
parents: 56217
diff changeset
  1410
  proof (rule exI, safe)
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1411
    show "0 < min (x - a) (b - x)"
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1412
      using a b by simp
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1413
  next
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1414
    fix y
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1415
    assume "norm (y - x) < min (x - a) (b - x)"
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1416
    hence y: "a < y" "y < b"
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1417
      by (simp_all add: abs_less_iff)
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1418
    assume "g y = g x"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1419
    hence "f (g y) = f (g x)" by simp
23044
2ad82c359175 change premises of DERIV_inverse_function lemma
huffman
parents: 23041
diff changeset
  1420
    hence "y = x" using inj y a b by simp
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1421
    also assume "y \<noteq> x"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1422
    finally show False by simp
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1423
  qed
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1424
  have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1425
    using cont 1 2 by (rule isCont_LIM_compose2)
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1426
  thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1427
        -- x --> inverse D"
44568
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44317
diff changeset
  1428
    using neq by (rule tendsto_inverse)
23041
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1429
qed
a0f26d47369b add lemma DERIV_inverse_function
huffman
parents: 22998
diff changeset
  1430
29975
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
  1431
subsection {* Generalized Mean Value Theorem *}
28c5322f0df3 more subsection headings
huffman
parents: 29803
diff changeset
  1432
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1433
theorem GMVT:
21784
e76faa6e65fd changed (ns)deriv to take functions of type 'a::real_normed_field => 'a
huffman
parents: 21404
diff changeset
  1434
  fixes a b :: real
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1435
  assumes alb: "a < b"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 41368
diff changeset
  1436
    and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
  1437
    and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 41368
diff changeset
  1438
    and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
  1439
    and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable (at x)"
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
  1440
  shows "\<exists>g'c f'c c.
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
  1441
    DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1442
proof -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1443
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 41368
diff changeset
  1444
  from assms have "a < b" by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1445
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44209
diff changeset
  1446
    using fc gc by simp
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
  1447
  moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable (at x)"
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44209
diff changeset
  1448
    using fd gd by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1449
  ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" by (rule MVT)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1450
  then obtain l where ldef: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1451
  then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1452
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1453
  from cdef have cint: "a < c \<and> c < b" by auto
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
  1454
  with gd have "g differentiable (at c)" by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1455
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1456
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1457
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1458
  from cdef have "a < c \<and> c < b" by auto
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
  1459
  with fd have "f differentiable (at c)" by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1460
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1461
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1462
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1463
  from cdef have "DERIV ?h c :> l" by auto
41368
8afa26855137 use DERIV_intros
hoelzl
parents: 37891
diff changeset
  1464
  moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
56381
0556204bc230 merged DERIV_intros, has_derivative_intros into derivative_intros
hoelzl
parents: 56371
diff changeset
  1465
    using g'cdef f'cdef by (auto intro!: derivative_eq_intros)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1466
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1467
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1468
  {
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1469
    from cdef have "?h b - ?h a = (b - a) * l" by auto
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 51642
diff changeset
  1470
    also from leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1471
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1472
  }
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1473
  moreover
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1474
  {
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1475
    have "?h b - ?h a =
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1476
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1477
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29472
diff changeset
  1478
      by (simp add: algebra_simps)
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1479
    hence "?h b - ?h a = 0" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1480
  }
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1481
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1482
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1483
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1484
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1485
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1486
  with g'cdef f'cdef cint show ?thesis by auto
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1487
qed
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1488
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1489
lemma GMVT':
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1490
  fixes f g :: "real \<Rightarrow> real"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1491
  assumes "a < b"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1492
  assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1493
  assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1494
  assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1495
  assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1496
  shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1497
proof -
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1498
  have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1499
    a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
56181
2aa0b19e74f3 unify syntax for has_derivative and differentiable
hoelzl
parents: 55970
diff changeset
  1500
    using assms by (intro GMVT) (force simp: real_differentiable_def)+
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1501
  then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1502
    using DERIV_f DERIV_g by (force dest: DERIV_unique)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1503
  then show ?thesis
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1504
    by auto
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1505
qed
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1506
51529
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  1507
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  1508
subsection {* L'Hopitals rule *}
2d2f59e6055a move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
hoelzl
parents: 51526
diff changeset
  1509
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1510
lemma isCont_If_ge:
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1511
  fixes a :: "'a :: linorder_topology"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1512
  shows "continuous (at_left a) g \<Longrightarrow> (f ---> g a) (at_right a) \<Longrightarrow> isCont (\<lambda>x. if x \<le> a then g x else f x) a"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1513
  unfolding isCont_def continuous_within
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1514
  apply (intro filterlim_split_at)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1515
  apply (subst filterlim_cong[OF refl refl, where g=g])
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1516
  apply (simp_all add: eventually_at_filter less_le)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1517
  apply (subst filterlim_cong[OF refl refl, where g=f])
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1518
  apply (simp_all add: eventually_at_filter less_le)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1519
  done
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1520
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1521
lemma lhopital_right_0:
50329
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1522
  fixes f0 g0 :: "real \<Rightarrow> real"
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1523
  assumes f_0: "(f0 ---> 0) (at_right 0)"
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1524
  assumes g_0: "(g0 ---> 0) (at_right 0)"
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1525
  assumes ev:
50329
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1526
    "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1527
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
50329
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1528
    "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1529
    "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1530
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
50329
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1531
  shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1532
proof -
50329
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1533
  def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1534
  then have "f 0 = 0" by simp
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1535
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1536
  def g \<equiv> "\<lambda>x. if x \<le> 0 then 0 else g0 x"
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1537
  then have "g 0 = 0" by simp
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1538
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1539
  have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1540
      DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1541
    using ev by eventually_elim auto
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1542
  then obtain a where [arith]: "0 < a"
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1543
    and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1544
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
50329
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1545
    and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1546
    and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
56219
bf80d125406b tuned proofs;
wenzelm
parents: 56217
diff changeset
  1547
    unfolding eventually_at by (auto simp: dist_real_def)
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1548
50329
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1549
  have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1550
    using g0_neq_0 by (simp add: g_def)
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1551
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1552
  { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1553
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1554
         (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1555
  note f = this
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1556
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1557
  { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1558
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1559
         (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1560
  note g = this
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1561
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1562
  have "isCont f 0"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1563
    unfolding f_def by (intro isCont_If_ge f_0 continuous_const)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1564
50329
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1565
  have "isCont g 0"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1566
    unfolding g_def by (intro isCont_If_ge g_0 continuous_const)
50329
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1567
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1568
  have "\<exists>\<zeta>. \<forall>x\<in>{0 <..< a}. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1569
  proof (rule bchoice, rule)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1570
    fix x assume "x \<in> {0 <..< a}"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1571
    then have x[arith]: "0 < x" "x < a" by auto
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1572
    with g'_neq_0 g_neq_0 `g 0 = 0` have g': "\<And>x. 0 < x \<Longrightarrow> x < a  \<Longrightarrow> 0 \<noteq> g' x" "g 0 \<noteq> g x"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1573
      by auto
50328
25b1e8686ce0 tuned proof
hoelzl
parents: 50327
diff changeset
  1574
    have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
25b1e8686ce0 tuned proof
hoelzl
parents: 50327
diff changeset
  1575
      using `isCont f 0` f by (auto intro: DERIV_isCont simp: le_less)
25b1e8686ce0 tuned proof
hoelzl
parents: 50327
diff changeset
  1576
    moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
25b1e8686ce0 tuned proof
hoelzl
parents: 50327
diff changeset
  1577
      using `isCont g 0` g by (auto intro: DERIV_isCont simp: le_less)
25b1e8686ce0 tuned proof
hoelzl
parents: 50327
diff changeset
  1578
    ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
25b1e8686ce0 tuned proof
hoelzl
parents: 50327
diff changeset
  1579
      using f g `x < a` by (intro GMVT') auto
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 51642
diff changeset
  1580
    then obtain c where *: "0 < c" "c < x" "(f x - f 0) * g' c = (g x - g 0) * f' c"
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 51642
diff changeset
  1581
      by blast
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1582
    moreover
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 51642
diff changeset
  1583
    from * g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1584
      by (simp add: field_simps)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1585
    ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1586
      using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c])
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1587
  qed
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
  1588
  then obtain \<zeta> where "\<forall>x\<in>{0 <..< a}. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)" ..
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1589
  then have \<zeta>: "eventually (\<lambda>x. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)) (at_right 0)"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1590
    unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1591
  moreover
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1592
  from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1593
    by eventually_elim auto
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1594
  then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1595
    by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"])
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1596
       (auto intro: tendsto_const tendsto_ident_at)
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1597
  then have "(\<zeta> ---> 0) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1598
    by (rule tendsto_norm_zero_cancel)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1599
  with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1600
    by (auto elim!: eventually_elim1 simp: filterlim_at)
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1601
  from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1602
    by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
50329
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1603
  ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)
50328
25b1e8686ce0 tuned proof
hoelzl
parents: 50327
diff changeset
  1604
    by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
25b1e8686ce0 tuned proof
hoelzl
parents: 50327
diff changeset
  1605
       (auto elim: eventually_elim1)
50329
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1606
  also have "?P \<longleftrightarrow> ?thesis"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1607
    by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter)
50329
9bd6b6b8a554 weakened assumptions for lhopital_right_0
hoelzl
parents: 50328
diff changeset
  1608
  finally show ?thesis .
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1609
qed
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1610
50330
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1611
lemma lhopital_right:
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1612
  "((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1613
    eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1614
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1615
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1616
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1617
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1618
  ((\<lambda> x. f x / g x) ---> y) (at_right x)"
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1619
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1620
  by (rule lhopital_right_0)
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1621
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1622
lemma lhopital_left:
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1623
  "((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1624
    eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1625
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1626
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1627
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1628
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1629
  ((\<lambda> x. f x / g x) ---> y) (at_left x)"
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1630
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
56409
36489d77c484 removing simprule status for divide_minus_left and divide_minus_right
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
  1631
  by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) 
36489d77c484 removing simprule status for divide_minus_left and divide_minus_right
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
  1632
     (auto simp: DERIV_mirror divide_minus_left divide_minus_right)
50330
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1633
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1634
lemma lhopital:
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1635
  "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1636
    eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1637
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1638
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1639
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1640
    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1641
  ((\<lambda> x. f x / g x) ---> y) (at x)"
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1642
  unfolding eventually_at_split filterlim_at_split
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1643
  by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1644
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1645
lemma lhopital_right_0_at_top:
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1646
  fixes f g :: "real \<Rightarrow> real"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1647
  assumes g_0: "LIM x at_right 0. g x :> at_top"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1648
  assumes ev:
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1649
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1650
    "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1651
    "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1652
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1653
  shows "((\<lambda> x. f x / g x) ---> x) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1654
  unfolding tendsto_iff
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1655
proof safe
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1656
  fix e :: real assume "0 < e"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1657
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1658
  with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1659
  have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1660
  from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1661
  obtain a where [arith]: "0 < a"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1662
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1663
    and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1664
    and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1665
    and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1666
    unfolding eventually_at_le by (auto simp: dist_real_def)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1667
    
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1668
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1669
  from Df have
50328
25b1e8686ce0 tuned proof
hoelzl
parents: 50327
diff changeset
  1670
    "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51529
diff changeset
  1671
    unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1672
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1673
  moreover
50328
25b1e8686ce0 tuned proof
hoelzl
parents: 50327
diff changeset
  1674
  have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
50346
a75c6429c3c3 add filterlim rules for eventually monotone bijective functions; mirror rules for at_top, at_bot; apply them to prove convergence of arctan at infinity and tan at pi/2
hoelzl
parents: 50331
diff changeset
  1675
    using g_0 by (auto elim: eventually_elim1 simp: filterlim_at_top_dense)
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1676
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1677
  moreover
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1678
  have inv_g: "((\<lambda>x. inverse (g x)) ---> 0) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1679
    using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1680
    by (rule filterlim_compose)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1681
  then have "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1682
    by (intro tendsto_intros)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1683
  then have "((\<lambda>x. norm (1 - g a / g x)) ---> 1) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1684
    by (simp add: inverse_eq_divide)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1685
  from this[unfolded tendsto_iff, rule_format, of 1]
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1686
  have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1687
    by (auto elim!: eventually_elim1 simp: dist_real_def)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1688
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1689
  moreover
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1690
  from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) ---> norm ((f a - x * g a) * 0)) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1691
    by (intro tendsto_intros)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1692
  then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1693
    by (simp add: inverse_eq_divide)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1694
  from this[unfolded tendsto_iff, rule_format, of "e / 2"] `0 < e`
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1695
  have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1696
    by (auto simp: dist_real_def)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1697
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1698
  ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1699
  proof eventually_elim
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1700
    fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1701
    assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1702
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1703
    have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1704
      using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
  1705
    then obtain y where [arith]: "t < y" "y < a"
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
  1706
      and D_eq0: "(g a - g t) * f' y = (f a - f t) * g' y"
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
  1707
      by blast
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
  1708
    from D_eq0 have D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
50327
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1709
      using `g a < g t` g'_neq_0[of y] by (auto simp add: field_simps)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1710
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1711
    have *: "f t / g t - x = ((f t - f a) / (g t - g a) - x) * (1 - g a / g t) + (f a - x * g a) / g t"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1712
      by (simp add: field_simps)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1713
    have "norm (f t / g t - x) \<le>
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1714
        norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1715
      unfolding * by (rule norm_triangle_ineq)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1716
    also have "\<dots> = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1717
      by (simp add: abs_mult D_eq dist_real_def)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1718
    also have "\<dots> < (e / 4) * 2 + e / 2"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1719
      using ineq Df[of y] `0 < e` by (intro add_le_less_mono mult_mono) auto
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1720
    finally show "dist (f t / g t) x < e"
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1721
      by (simp add: dist_real_def)
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1722
  qed
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1723
qed
bbea2e82871c add L'Hôpital's rule
hoelzl
parents: 47108
diff changeset
  1724
50330
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1725
lemma lhopital_right_at_top:
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1726
  "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1727
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1728
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1729
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1730
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1731
    ((\<lambda> x. f x / g x) ---> y) (at_right x)"
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1732
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1733
  by (rule lhopital_right_0_at_top)
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1734
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1735
lemma lhopital_left_at_top:
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1736
  "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1737
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1738
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1739
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1740
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1741
    ((\<lambda> x. f x / g x) ---> y) (at_left x)"
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1742
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
56409
36489d77c484 removing simprule status for divide_minus_left and divide_minus_right
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
  1743
  by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"])
36489d77c484 removing simprule status for divide_minus_left and divide_minus_right
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
  1744
     (auto simp: divide_minus_left divide_minus_right DERIV_mirror)
50330
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1745
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1746
lemma lhopital_at_top:
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1747
  "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1748
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1749
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1750
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1751
    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1752
    ((\<lambda> x. f x / g x) ---> y) (at x)"
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1753
  unfolding eventually_at_split filterlim_at_split
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1754
  by (auto intro!: lhopital_right_at_top[of g x g' f f'] lhopital_left_at_top[of g x g' f f'])
d0b12171118e conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
hoelzl
parents: 50329
diff changeset
  1755
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1756
lemma lhospital_at_top_at_top:
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1757
  fixes f g :: "real \<Rightarrow> real"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1758
  assumes g_0: "LIM x at_top. g x :> at_top"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1759
  assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1760
  assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1761
  assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1762
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) at_top"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1763
  shows "((\<lambda> x. f x / g x) ---> x) at_top"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1764
  unfolding filterlim_at_top_to_right
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1765
proof (rule lhopital_right_0_at_top)
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1766
  let ?F = "\<lambda>x. f (inverse x)"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1767
  let ?G = "\<lambda>x. g (inverse x)"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1768
  let ?R = "at_right (0::real)"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1769
  let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1770
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1771
  show "LIM x ?R. ?G x :> at_top"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1772
    using g_0 unfolding filterlim_at_top_to_right .
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1773
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1774
  show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1775
    unfolding eventually_at_right_to_top
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1776
    using Dg eventually_ge_at_top[where c="1::real"]
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1777
    apply eventually_elim
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1778
    apply (rule DERIV_cong)
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1779
    apply (rule DERIV_chain'[where f=inverse])
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1780
    apply (auto intro!:  DERIV_inverse)
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1781
    done
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1782
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1783
  show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1784
    unfolding eventually_at_right_to_top
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1785
    using Df eventually_ge_at_top[where c="1::real"]
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1786
    apply eventually_elim
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1787
    apply (rule DERIV_cong)
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1788
    apply (rule DERIV_chain'[where f=inverse])
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1789
    apply (auto intro!:  DERIV_inverse)
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1790
    done
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1791
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1792
  show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1793
    unfolding eventually_at_right_to_top
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1794
    using g' eventually_ge_at_top[where c="1::real"]
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1795
    by eventually_elim auto
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1796
    
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1797
  show "((\<lambda>x. ?D f' x / ?D g' x) ---> x) ?R"
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1798
    unfolding filterlim_at_right_to_top
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1799
    apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1800
    using eventually_ge_at_top[where c="1::real"]
56409
36489d77c484 removing simprule status for divide_minus_left and divide_minus_right
paulson <lp15@cam.ac.uk>
parents: 56381
diff changeset
  1801
    by eventually_elim (simp add: divide_minus_left divide_minus_right)
50347
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1802
qed
77e3effa50b6 prove tendsto_power_div_exp_0
hoelzl
parents: 50346
diff changeset
  1803
21164
0742fc979c67 new Deriv.thy contains stuff from Lim.thy
huffman
parents:
diff changeset
  1804
end