(*  Title       : Deriv.thy

     Author      : Jacques D. Fleuriot

     Copyright   : 1998  University of Cambridge

     Conversion to Isar and new proofs by Lawrence C Paulson, 2004

     GMVT by Benjamin Porter, 2005

 *)

 header{* Differentiation *}

 theory Deriv

 imports Limits

 begin

 text{*Standard Definitions*}

 definition

   deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"

     --{*Differentiation: D is derivative of function f at x*}

           ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where

   "DERIV f x :> D = ((%h. (f(x + h) - f x) / h) -- 0 --> D)"

 subsection {* Derivatives *}

 lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)"

 by (simp add: deriv_def)

 lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --> D"

 by (simp add: deriv_def)

 lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"

   by (simp add: deriv_def tendsto_const)

 lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x :> 1"

   by (simp add: deriv_def tendsto_const cong: LIM_cong)

 lemma DERIV_add:

   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"

   by (simp only: deriv_def add_diff_add add_divide_distrib tendsto_add)

 lemma DERIV_minus:

   "DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x :> - D"

   by (simp only: deriv_def minus_diff_minus divide_minus_left tendsto_minus)

 lemma DERIV_diff:

   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x :> D - E"

 by (simp only: diff_minus DERIV_add DERIV_minus)

 lemma DERIV_add_minus:

   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"

 by (simp only: DERIV_add DERIV_minus)

 lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"

 proof (unfold isCont_iff)

   assume "DERIV f x :> D"

   hence "(\<lambda>h. (f(x+h) - f(x)) / h) -- 0 --> D"

     by (rule DERIV_D)

   hence "(\<lambda>h. (f(x+h) - f(x)) / h * h) -- 0 --> D * 0"

     by (intro tendsto_mult tendsto_ident_at)

   hence "(\<lambda>h. (f(x+h) - f(x)) * (h / h)) -- 0 --> 0"

     by simp

   hence "(\<lambda>h. f(x+h) - f(x)) -- 0 --> 0"

     by (simp cong: LIM_cong)

   thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"

     by (simp add: LIM_def dist_norm)

 qed

 lemma DERIV_mult_lemma:

   fixes a b c d :: "'a::real_field"

   shows "(a * b - c * d) / h = a * ((b - d) / h) + ((a - c) / h) * d"

   by (simp add: field_simps diff_divide_distrib)

 lemma DERIV_mult':

   assumes f: "DERIV f x :> D"

   assumes g: "DERIV g x :> E"

   shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x"

 proof (unfold deriv_def)

   from f have "isCont f x"

     by (rule DERIV_isCont)

   hence "(\<lambda>h. f(x+h)) -- 0 --> f x"

     by (simp only: isCont_iff)

   hence "(\<lambda>h. f(x+h) * ((g(x+h) - g x) / h) +

               ((f(x+h) - f x) / h) * g x)

           -- 0 --> f x * E + D * g x"

     by (intro tendsto_intros DERIV_D f g)

   thus "(\<lambda>h. (f(x+h) * g(x+h) - f x * g x) / h)

          -- 0 --> f x * E + D * g x"

     by (simp only: DERIV_mult_lemma)

 qed

 lemma DERIV_mult:

     "DERIV f x :> Da \<Longrightarrow> DERIV g x :> Db \<Longrightarrow> DERIV (\<lambda>x. f x * g x) x :> Da * g x + Db * f x"

   by (drule (1) DERIV_mult', simp only: mult_commute add_commute)

 lemma DERIV_unique:

     "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"

   unfolding deriv_def by (rule LIM_unique) 

 text{*Differentiation of finite sum*}

 lemma DERIV_setsum:

   assumes "finite S"

   and "\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x :> (f' x n)"

   shows "DERIV (%x. setsum (f x) S) x :> setsum (f' x) S"

   using assms by induct (auto intro!: DERIV_add)

 lemma DERIV_sumr [rule_format (no_asm)]:

      "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))

       --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"

   by (auto intro: DERIV_setsum)

 text{*Alternative definition for differentiability*}

 lemma DERIV_LIM_iff:

   fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows

      "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =

       ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"

 apply (rule iffI)

 apply (drule_tac k="- a" in LIM_offset)

 apply (simp add: diff_minus)

 apply (drule_tac k="a" in LIM_offset)

 apply (simp add: add_commute)

 done

 lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"

 by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)

 lemma DERIV_inverse_lemma:

   "\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_field)\<rbrakk>

    \<Longrightarrow> (inverse a - inverse b) / h

      = - (inverse a * ((a - b) / h) * inverse b)"

 by (simp add: inverse_diff_inverse)

 lemma DERIV_inverse':

   assumes der: "DERIV f x :> D"

   assumes neq: "f x \<noteq> 0"

   shows "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))"

     (is "DERIV _ _ :> ?E")

 proof (unfold DERIV_iff2)

   from der have lim_f: "f -- x --> f x"

     by (rule DERIV_isCont [unfolded isCont_def])

   from neq have "0 < norm (f x)" by simp

   with LIM_D [OF lim_f] obtain s

     where s: "0 < s"

     and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z - x) < s\<rbrakk>

                   \<Longrightarrow> norm (f z - f x) < norm (f x)"

     by fast

   show "(\<lambda>z. (inverse (f z) - inverse (f x)) / (z - x)) -- x --> ?E"

   proof (rule LIM_equal2 [OF s])

     fix z

     assume "z \<noteq> x" "norm (z - x) < s"

     hence "norm (f z - f x) < norm (f x)" by (rule less_fx)

     hence "f z \<noteq> 0" by auto

     thus "(inverse (f z) - inverse (f x)) / (z - x) =

           - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x))"

       using neq by (rule DERIV_inverse_lemma)

   next

     from der have "(\<lambda>z. (f z - f x) / (z - x)) -- x --> D"

       by (unfold DERIV_iff2)

     thus "(\<lambda>z. - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x)))

           -- x --> ?E"

       by (intro tendsto_intros lim_f neq)

   qed

 qed

 lemma DERIV_divide:

   "\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk>

    \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)"

 apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) +

           D * inverse (g x) = (D * g x - f x * E) / (g x * g x)")

 apply (erule subst)

 apply (unfold divide_inverse)

 apply (erule DERIV_mult')

 apply (erule (1) DERIV_inverse')

 apply (simp add: ring_distribs nonzero_inverse_mult_distrib)

 done

 lemma DERIV_power_Suc:

   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field}"

   assumes f: "DERIV f x :> D"

   shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (1 + of_nat n) * (D * f x ^ n)"

 proof (induct n)

 case 0

   show ?case by (simp add: f)

 case (Suc k)

   from DERIV_mult' [OF f Suc] show ?case

     apply (simp only: of_nat_Suc ring_distribs mult_1_left)

     apply (simp only: power_Suc algebra_simps)

     done

 qed

 lemma DERIV_power:

   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field}"

   assumes f: "DERIV f x :> D"

   shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))"

 by (cases "n", simp, simp add: DERIV_power_Suc f del: power_Suc)

 text {* Caratheodory formulation of derivative at a point *}

 lemma CARAT_DERIV:

      "(DERIV f x :> l) =

       (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"

       (is "?lhs = ?rhs")

 proof

   assume der: "DERIV f x :> l"

   show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"

   proof (intro exI conjI)

     let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"

     show "\<forall>z. f z - f x = ?g z * (z-x)" by simp

     show "isCont ?g x" using der

       by (simp add: isCont_iff DERIV_iff diff_minus

                cong: LIM_equal [rule_format])

     show "?g x = l" by simp

   qed

 next

   assume "?rhs"

   then obtain g where

     "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast

   thus "(DERIV f x :> l)"

      by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong)

 qed

 lemma DERIV_chain':

   assumes f: "DERIV f x :> D"

   assumes g: "DERIV g (f x) :> E"

   shows "DERIV (\<lambda>x. g (f x)) x :> E * D"

 proof (unfold DERIV_iff2)

   obtain d where d: "\<forall>y. g y - g (f x) = d y * (y - f x)"

     and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"

     using CARAT_DERIV [THEN iffD1, OF g] by fast

   from f have "f -- x --> f x"

     by (rule DERIV_isCont [unfolded isCont_def])

   with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"

     by (rule isCont_tendsto_compose)

   hence "(\<lambda>z. d (f z) * ((f z - f x) / (z - x)))

           -- x --> d (f x) * D"

     by (rule tendsto_mult [OF _ f [unfolded DERIV_iff2]])

   thus "(\<lambda>z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D"

     by (simp add: d dfx)

 qed

 text {*

  Let's do the standard proof, though theorem

  @{text "LIM_mult2"} follows from a NS proof

 *}

 lemma DERIV_cmult:

       "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"

 by (drule DERIV_mult' [OF DERIV_const], simp)

 lemma DERIV_cdivide: "DERIV f x :> D ==> DERIV (%x. f x / c) x :> D / c"

   apply (subgoal_tac "DERIV (%x. (1 / c) * f x) x :> (1 / c) * D", force)

   apply (erule DERIV_cmult)

   done

 text {* Standard version *}

 lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"

 by (drule (1) DERIV_chain', simp add: o_def mult_commute)

 lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"

 by (auto dest: DERIV_chain simp add: o_def)

 text {* Derivative of linear multiplication *}

 lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"

 by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)

 lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"

 apply (cut_tac DERIV_power [OF DERIV_ident])

 apply (simp add: real_of_nat_def)

 done

 text {* Power of @{text "-1"} *}

 lemma DERIV_inverse:

   fixes x :: "'a::{real_normed_field}"

   shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"

 by (drule DERIV_inverse' [OF DERIV_ident]) simp

 text {* Derivative of inverse *}

 lemma DERIV_inverse_fun:

   fixes x :: "'a::{real_normed_field}"

   shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]

       ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"

 by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)

 text {* Derivative of quotient *}

 lemma DERIV_quotient:

   fixes x :: "'a::{real_normed_field}"

   shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]

        ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"

 by (drule (2) DERIV_divide) (simp add: mult_commute)

 text {* @{text "DERIV_intros"} *}

 ML {*

 structure Deriv_Intros = Named_Thms

 (

   val name = @{binding DERIV_intros}

   val description = "DERIV introduction rules"

 )

 *}

 setup Deriv_Intros.setup

 lemma DERIV_cong: "\<lbrakk> DERIV f x :> X ; X = Y \<rbrakk> \<Longrightarrow> DERIV f x :> Y"

   by simp

 declare

   DERIV_const[THEN DERIV_cong, DERIV_intros]

   DERIV_ident[THEN DERIV_cong, DERIV_intros]

   DERIV_add[THEN DERIV_cong, DERIV_intros]

   DERIV_minus[THEN DERIV_cong, DERIV_intros]

   DERIV_mult[THEN DERIV_cong, DERIV_intros]

   DERIV_diff[THEN DERIV_cong, DERIV_intros]

   DERIV_inverse'[THEN DERIV_cong, DERIV_intros]

   DERIV_divide[THEN DERIV_cong, DERIV_intros]

   DERIV_power[where 'a=real, THEN DERIV_cong,

               unfolded real_of_nat_def[symmetric], DERIV_intros]

   DERIV_setsum[THEN DERIV_cong, DERIV_intros]

 subsection {* Differentiability predicate *}

 definition

   differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"

     (infixl "differentiable" 60) where

   "f differentiable x = (\<exists>D. DERIV f x :> D)"

 lemma differentiableE [elim?]:

   assumes "f differentiable x"

   obtains df where "DERIV f x :> df"

   using assms unfolding differentiable_def ..

 lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"

 by (simp add: differentiable_def)

 lemma differentiableI: "DERIV f x :> D ==> f differentiable x"

 by (force simp add: differentiable_def)

 lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x"

   by (rule DERIV_ident [THEN differentiableI])

 lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x"

   by (rule DERIV_const [THEN differentiableI])

 lemma differentiable_compose:

   assumes f: "f differentiable (g x)"

   assumes g: "g differentiable x"

   shows "(\<lambda>x. f (g x)) differentiable x"

 proof -

   from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" ..

   moreover

   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..

   ultimately

   have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2)

   thus ?thesis by (rule differentiableI)

 qed

 lemma differentiable_sum [simp]:

   assumes "f differentiable x"

   and "g differentiable x"

   shows "(\<lambda>x. f x + g x) differentiable x"

 proof -

   from `f differentiable x` obtain df where "DERIV f x :> df" ..

   moreover

   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..

   ultimately

   have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)

   thus ?thesis by (rule differentiableI)

 qed

 lemma differentiable_minus [simp]:

   assumes "f differentiable x"

   shows "(\<lambda>x. - f x) differentiable x"

 proof -

   from `f differentiable x` obtain df where "DERIV f x :> df" ..

   hence "DERIV (\<lambda>x. - f x) x :> - df" by (rule DERIV_minus)

   thus ?thesis by (rule differentiableI)

 qed

 lemma differentiable_diff [simp]:

   assumes "f differentiable x"

   assumes "g differentiable x"

   shows "(\<lambda>x. f x - g x) differentiable x"

   unfolding diff_minus using assms by simp

 lemma differentiable_mult [simp]:

   assumes "f differentiable x"

   assumes "g differentiable x"

   shows "(\<lambda>x. f x * g x) differentiable x"

 proof -

   from `f differentiable x` obtain df where "DERIV f x :> df" ..

   moreover

   from `g differentiable x` obtain dg where "DERIV g x :> dg" ..

   ultimately

   have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult)

   thus ?thesis by (rule differentiableI)

 qed

 lemma differentiable_inverse [simp]:

   assumes "f differentiable x" and "f x \<noteq> 0"

   shows "(\<lambda>x. inverse (f x)) differentiable x"

 proof -

   from `f differentiable x` obtain df where "DERIV f x :> df" ..

   hence "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * df * inverse (f x))"

     using `f x \<noteq> 0` by (rule DERIV_inverse')

   thus ?thesis by (rule differentiableI)

 qed

 lemma differentiable_divide [simp]:

   assumes "f differentiable x"

   assumes "g differentiable x" and "g x \<noteq> 0"

   shows "(\<lambda>x. f x / g x) differentiable x"

   unfolding divide_inverse using assms by simp

 lemma differentiable_power [simp]:

   fixes f :: "'a::{real_normed_field} \<Rightarrow> 'a"

   assumes "f differentiable x"

   shows "(\<lambda>x. f x ^ n) differentiable x"

   apply (induct n)

   apply simp

   apply (simp add: assms)

   done

 subsection {* Local extrema *}

 text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}

 lemma DERIV_pos_inc_right:

   fixes f :: "real => real"

   assumes der: "DERIV f x :> l"

       and l:   "0 < l"

   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"

 proof -

   from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]

   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)"

     by (simp add: diff_minus)

   then obtain s

         where s:   "0 < s"

           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"

     by auto

   thus ?thesis

   proof (intro exI conjI strip)

     show "0<s" using s .

     fix h::real

     assume "0 < h" "h < s"

     with all [of h] show "f x < f (x+h)"

     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]

     split add: split_if_asm)

       assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"

       with l

       have "0 < (f (x+h) - f x) / h" by arith

       thus "f x < f (x+h)"

   by (simp add: pos_less_divide_eq h)

     qed

   qed

 qed

 lemma DERIV_neg_dec_left:

   fixes f :: "real => real"

   assumes der: "DERIV f x :> l"

       and l:   "l < 0"

   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"

 proof -

   from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]

   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)"

     by (simp add: diff_minus)

   then obtain s

         where s:   "0 < s"

           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"

     by auto

   thus ?thesis

   proof (intro exI conjI strip)

     show "0<s" using s .

     fix h::real

     assume "0 < h" "h < s"

     with all [of "-h"] show "f x < f (x-h)"

     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]

     split add: split_if_asm)

       assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"

       with l

       have "0 < (f (x-h) - f x) / h" by arith

       thus "f x < f (x-h)"

   by (simp add: pos_less_divide_eq h)

     qed

   qed

 qed

 lemma DERIV_pos_inc_left:

   fixes f :: "real => real"

   shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"

   apply (rule DERIV_neg_dec_left [of "%x. - f x" x "-l", simplified])

   apply (auto simp add: DERIV_minus)

   done

 lemma DERIV_neg_dec_right:

   fixes f :: "real => real"

   shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"

   apply (rule DERIV_pos_inc_right [of "%x. - f x" x "-l", simplified])

   apply (auto simp add: DERIV_minus)

   done

 lemma DERIV_local_max:

   fixes f :: "real => real"

   assumes der: "DERIV f x :> l"

       and d:   "0 < d"

       and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"

   shows "l = 0"

 proof (cases rule: linorder_cases [of l 0])

   case equal thus ?thesis .

 next

   case less

   from DERIV_neg_dec_left [OF der less]

   obtain d' where d': "0 < d'"

              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast

   from real_lbound_gt_zero [OF d d']

   obtain e where "0 < e \<and> e < d \<and> e < d'" ..

   with lt le [THEN spec [where x="x-e"]]

   show ?thesis by (auto simp add: abs_if)

 next

   case greater

   from DERIV_pos_inc_right [OF der greater]

   obtain d' where d': "0 < d'"

              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast

   from real_lbound_gt_zero [OF d d']

   obtain e where "0 < e \<and> e < d \<and> e < d'" ..

   with lt le [THEN spec [where x="x+e"]]

   show ?thesis by (auto simp add: abs_if)

 qed

 text{*Similar theorem for a local minimum*}

 lemma DERIV_local_min:

   fixes f :: "real => real"

   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"

 by (drule DERIV_minus [THEN DERIV_local_max], auto)

 text{*In particular, if a function is locally flat*}

 lemma DERIV_local_const:

   fixes f :: "real => real"

   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"

 by (auto dest!: DERIV_local_max)

 subsection {* Rolle's Theorem *}

 text{*Lemma about introducing open ball in open interval*}

 lemma lemma_interval_lt:

      "[| a < x;  x < b |]

       ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"

 apply (simp add: abs_less_iff)

 apply (insert linorder_linear [of "x-a" "b-x"], safe)

 apply (rule_tac x = "x-a" in exI)

 apply (rule_tac [2] x = "b-x" in exI, auto)

 done

 lemma lemma_interval: "[| a < x;  x < b |] ==>

         \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"

 apply (drule lemma_interval_lt, auto)

 apply force

 done

 text{*Rolle's Theorem.

    If @{term f} is defined and continuous on the closed interval

    @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},

    and @{term "f(a) = f(b)"},

    then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}

 theorem Rolle:

   assumes lt: "a < b"

       and eq: "f(a) = f(b)"

       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"

       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"

   shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"

 proof -

   have le: "a \<le> b" using lt by simp

   from isCont_eq_Ub [OF le con]

   obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"

              and alex: "a \<le> x" and xleb: "x \<le> b"

     by blast

   from isCont_eq_Lb [OF le con]

   obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"

               and alex': "a \<le> x'" and x'leb: "x' \<le> b"

     by blast

   show ?thesis

   proof cases

     assume axb: "a < x & x < b"

         --{*@{term f} attains its maximum within the interval*}

     hence ax: "a<x" and xb: "x<b" by arith + 

     from lemma_interval [OF ax xb]

     obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"

       by blast

     hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max

       by blast

     from differentiableD [OF dif [OF axb]]

     obtain l where der: "DERIV f x :> l" ..

     have "l=0" by (rule DERIV_local_max [OF der d bound'])

         --{*the derivative at a local maximum is zero*}

     thus ?thesis using ax xb der by auto

   next

     assume notaxb: "~ (a < x & x < b)"

     hence xeqab: "x=a | x=b" using alex xleb by arith

     hence fb_eq_fx: "f b = f x" by (auto simp add: eq)

     show ?thesis

     proof cases

       assume ax'b: "a < x' & x' < b"

         --{*@{term f} attains its minimum within the interval*}

       hence ax': "a<x'" and x'b: "x'<b" by arith+ 

       from lemma_interval [OF ax' x'b]

       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"

   by blast

       hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min

   by blast

       from differentiableD [OF dif [OF ax'b]]

       obtain l where der: "DERIV f x' :> l" ..

       have "l=0" by (rule DERIV_local_min [OF der d bound'])

         --{*the derivative at a local minimum is zero*}

       thus ?thesis using ax' x'b der by auto

     next

       assume notax'b: "~ (a < x' & x' < b)"

         --{*@{term f} is constant througout the interval*}

       hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith

       hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)

       from dense [OF lt]

       obtain r where ar: "a < r" and rb: "r < b" by blast

       from lemma_interval [OF ar rb]

       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"

   by blast

       have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"

       proof (clarify)

         fix z::real

         assume az: "a \<le> z" and zb: "z \<le> b"

         show "f z = f b"

         proof (rule order_antisym)

           show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)

           show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)

         qed

       qed

       have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"

       proof (intro strip)

         fix y::real

         assume lt: "\<bar>r-y\<bar> < d"

         hence "f y = f b" by (simp add: eq_fb bound)

         thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)

       qed

       from differentiableD [OF dif [OF conjI [OF ar rb]]]

       obtain l where der: "DERIV f r :> l" ..

       have "l=0" by (rule DERIV_local_const [OF der d bound'])

         --{*the derivative of a constant function is zero*}

       thus ?thesis using ar rb der by auto

     qed

   qed

 qed

 subsection{*Mean Value Theorem*}

 lemma lemma_MVT:

      "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"

   by (cases "a = b") (simp_all add: field_simps)

 theorem MVT:

   assumes lt:  "a < b"

       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"

       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"

   shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &

                    (f(b) - f(a) = (b-a) * l)"

 proof -

   let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"

   have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"

     using con by (fast intro: isCont_intros)

   have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"

   proof (clarify)

     fix x::real

     assume ax: "a < x" and xb: "x < b"

     from differentiableD [OF dif [OF conjI [OF ax xb]]]

     obtain l where der: "DERIV f x :> l" ..

     show "?F differentiable x"

       by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],

           blast intro: DERIV_diff DERIV_cmult_Id der)

   qed

   from Rolle [where f = ?F, OF lt lemma_MVT contF difF]

   obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"

     by blast

   have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"

     by (rule DERIV_cmult_Id)

   hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z

                    :> 0 + (f b - f a) / (b - a)"

     by (rule DERIV_add [OF der])

   show ?thesis

   proof (intro exI conjI)

     show "a < z" using az .

     show "z < b" using zb .

     show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)

     show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp

   qed

 qed

 lemma MVT2:

      "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]

       ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"

 apply (drule MVT)

 apply (blast intro: DERIV_isCont)

 apply (force dest: order_less_imp_le simp add: differentiable_def)

 apply (blast dest: DERIV_unique order_less_imp_le)

 done

 text{*A function is constant if its derivative is 0 over an interval.*}

 lemma DERIV_isconst_end:

   fixes f :: "real => real"

   shows "[| a < b;

          \<forall>x. a \<le> x & x \<le> b --> isCont f x;

          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]

         ==> f b = f a"

 apply (drule MVT, assumption)

 apply (blast intro: differentiableI)

 apply (auto dest!: DERIV_unique simp add: diff_eq_eq)

 done

 lemma DERIV_isconst1:

   fixes f :: "real => real"

   shows "[| a < b;

          \<forall>x. a \<le> x & x \<le> b --> isCont f x;

          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]

         ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"

 apply safe

 apply (drule_tac x = a in order_le_imp_less_or_eq, safe)

 apply (drule_tac b = x in DERIV_isconst_end, auto)

 done

 lemma DERIV_isconst2:

   fixes f :: "real => real"

   shows "[| a < b;

          \<forall>x. a \<le> x & x \<le> b --> isCont f x;

          \<forall>x. a < x & x < b --> DERIV f x :> 0;

          a \<le> x; x \<le> b |]

         ==> f x = f a"

 apply (blast dest: DERIV_isconst1)

 done

 lemma DERIV_isconst3: fixes a b x y :: real

   assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"

   assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"

   shows "f x = f y"

 proof (cases "x = y")

   case False

   let ?a = "min x y"

   let ?b = "max x y"

   have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"

   proof (rule allI, rule impI)

     fix z :: real assume "?a \<le> z \<and> z \<le> ?b"

     hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto

     hence "z \<in> {a<..<b}" by auto

     thus "DERIV f z :> 0" by (rule derivable)

   qed

   hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"

     and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto

   have "?a < ?b" using `x \<noteq> y` by auto

   from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]

   show ?thesis by auto

 qed auto

 lemma DERIV_isconst_all:

   fixes f :: "real => real"

   shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"

 apply (rule linorder_cases [of x y])

 apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+

 done

 lemma DERIV_const_ratio_const:

   fixes f :: "real => real"

   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"

 apply (rule linorder_cases [of a b], auto)

 apply (drule_tac [!] f = f in MVT)

 apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)

 apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)

 done

 lemma DERIV_const_ratio_const2:

   fixes f :: "real => real"

   shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"

 apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])

 apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)

 done

 lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"

 by (simp)

 lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"

 by (simp)

 text{*Gallileo's "trick": average velocity = av. of end velocities*}

 lemma DERIV_const_average:

   fixes v :: "real => real"

   assumes neq: "a \<noteq> (b::real)"

       and der: "\<forall>x. DERIV v x :> k"

   shows "v ((a + b)/2) = (v a + v b)/2"

 proof (cases rule: linorder_cases [of a b])

   case equal with neq show ?thesis by simp

 next

   case less

   have "(v b - v a) / (b - a) = k"

     by (rule DERIV_const_ratio_const2 [OF neq der])

   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp

   moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"

     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)

   ultimately show ?thesis using neq by force

 next

   case greater

   have "(v b - v a) / (b - a) = k"

     by (rule DERIV_const_ratio_const2 [OF neq der])

   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp

   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"

     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)

   ultimately show ?thesis using neq by (force simp add: add_commute)

 qed

 (* A function with positive derivative is increasing. 

    A simple proof using the MVT, by Jeremy Avigad. And variants.

 *)

 lemma DERIV_pos_imp_increasing:

   fixes a::real and b::real and f::"real => real"

   assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"

   shows "f a < f b"

 proof (rule ccontr)

   assume f: "~ f a < f b"

   have "EX l z. a < z & z < b & DERIV f z :> l

       & f b - f a = (b - a) * l"

     apply (rule MVT)

       using assms

       apply auto

       apply (metis DERIV_isCont)

      apply (metis differentiableI less_le)

     done

   then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"

       and "f b - f a = (b - a) * l"

     by auto

   with assms f have "~(l > 0)"

     by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)

   with assms z show False

     by (metis DERIV_unique less_le)

 qed

 lemma DERIV_nonneg_imp_nondecreasing:

   fixes a::real and b::real and f::"real => real"

   assumes "a \<le> b" and

     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"

   shows "f a \<le> f b"

 proof (rule ccontr, cases "a = b")

   assume "~ f a \<le> f b" and "a = b"

   then show False by auto

 next

   assume A: "~ f a \<le> f b"

   assume B: "a ~= b"

   with assms have "EX l z. a < z & z < b & DERIV f z :> l

       & f b - f a = (b - a) * l"

     apply -

     apply (rule MVT)

       apply auto

       apply (metis DERIV_isCont)

      apply (metis differentiableI less_le)

     done

   then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"

       and C: "f b - f a = (b - a) * l"

     by auto

   with A have "a < b" "f b < f a" by auto

   with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)

     (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)

   with assms z show False

     by (metis DERIV_unique order_less_imp_le)

 qed

 lemma DERIV_neg_imp_decreasing:

   fixes a::real and b::real and f::"real => real"

   assumes "a < b" and

     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"

   shows "f a > f b"

 proof -

   have "(%x. -f x) a < (%x. -f x) b"

     apply (rule DERIV_pos_imp_increasing [of a b "%x. -f x"])

     using assms

     apply auto

     apply (metis DERIV_minus neg_0_less_iff_less)

     done

   thus ?thesis

     by simp

 qed

 lemma DERIV_nonpos_imp_nonincreasing:

   fixes a::real and b::real and f::"real => real"

   assumes "a \<le> b" and

     "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"

   shows "f a \<ge> f b"

 proof -

   have "(%x. -f x) a \<le> (%x. -f x) b"

     apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])

     using assms

     apply auto

     apply (metis DERIV_minus neg_0_le_iff_le)

     done

   thus ?thesis

     by simp

 qed

 text {* Derivative of inverse function *}

 lemma DERIV_inverse_function:

   fixes f g :: "real \<Rightarrow> real"

   assumes der: "DERIV f (g x) :> D"

   assumes neq: "D \<noteq> 0"

   assumes a: "a < x" and b: "x < b"

   assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"

   assumes cont: "isCont g x"

   shows "DERIV g x :> inverse D"

 unfolding DERIV_iff2

 proof (rule LIM_equal2)

   show "0 < min (x - a) (b - x)"

     using a b by arith 

 next

   fix y

   assume "norm (y - x) < min (x - a) (b - x)"

   hence "a < y" and "y < b" 

     by (simp_all add: abs_less_iff)

   thus "(g y - g x) / (y - x) =

         inverse ((f (g y) - x) / (g y - g x))"

     by (simp add: inj)

 next

   have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"

     by (rule der [unfolded DERIV_iff2])

   hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"

     using inj a b by simp

   have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"

   proof (safe intro!: exI)

     show "0 < min (x - a) (b - x)"

       using a b by simp

   next

     fix y

     assume "norm (y - x) < min (x - a) (b - x)"

     hence y: "a < y" "y < b"

       by (simp_all add: abs_less_iff)

     assume "g y = g x"

     hence "f (g y) = f (g x)" by simp

     hence "y = x" using inj y a b by simp

     also assume "y \<noteq> x"

     finally show False by simp

   qed

   have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"

     using cont 1 2 by (rule isCont_LIM_compose2)

   thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))

         -- x --> inverse D"

     using neq by (rule tendsto_inverse)

 qed

 subsection {* Generalized Mean Value Theorem *}

 theorem GMVT:

   fixes a b :: real

   assumes alb: "a < b"

     and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"

     and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"

     and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"

     and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"

   shows "\<exists>g'c f'c c. 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)"

 proof -

   let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"

   from assms have "a < b" by simp

   moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"

     using fc gc by simp

   moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"

     using fd gd by simp

   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)

   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" ..

   then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..

   from cdef have cint: "a < c \<and> c < b" by auto

   with gd have "g differentiable c" by simp

   hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)

   then obtain g'c where g'cdef: "DERIV g c :> g'c" ..

   from cdef have "a < c \<and> c < b" by auto

   with fd have "f differentiable c" by simp

   hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)

   then obtain f'c where f'cdef: "DERIV f c :> f'c" ..

   from cdef have "DERIV ?h c :> l" by auto

   moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"

     using g'cdef f'cdef by (auto intro!: DERIV_intros)

   ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)

   {

     from cdef have "?h b - ?h a = (b - a) * l" by auto

     also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp

     finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp

   }

   moreover

   {

     have "?h b - ?h a =

          ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -

           ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"

       by (simp add: algebra_simps)

     hence "?h b - ?h a = 0" by auto

   }

   ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto

   with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp

   hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp

   hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)

   with g'cdef f'cdef cint show ?thesis by auto

 qed

 lemma GMVT':

   fixes f g :: "real \<Rightarrow> real"

   assumes "a < b"

   assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"

   assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"

   assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"

   assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"

   shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"

 proof -

   have "\<exists>g'c f'c c. 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"

     using assms by (intro GMVT) (force simp: differentiable_def)+

   then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"

     using DERIV_f DERIV_g by (force dest: DERIV_unique)

   then show ?thesis

     by auto

 qed

 subsection {* L'Hopitals rule *}

 lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>

     DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"

   unfolding DERIV_iff2

 proof (rule filterlim_cong)

   assume "eventually (\<lambda>x. f x = g x) (nhds x)"

   moreover then have "f x = g x" by (auto simp: eventually_nhds)

   moreover assume "x = y" "u = v"

   ultimately show "eventually (\<lambda>xa. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)) (at x)"

     by (auto simp: eventually_within at_def elim: eventually_elim1)

 qed simp_all

 lemma DERIV_shift:

   "(DERIV f (x + z) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (x + z)) x :> y)"

   by (simp add: DERIV_iff field_simps)

 lemma DERIV_mirror:

   "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"

   by (simp add: deriv_def filterlim_at_split filterlim_at_left_to_right

                 tendsto_minus_cancel_left field_simps conj_commute)

 lemma lhopital_right_0:

   fixes f0 g0 :: "real \<Rightarrow> real"

   assumes f_0: "(f0 ---> 0) (at_right 0)"

   assumes g_0: "(g0 ---> 0) (at_right 0)"

   assumes ev:

     "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"

     "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"

     "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"

     "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"

   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"

   shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"

 proof -

   def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"

   then have "f 0 = 0" by simp

   def g \<equiv> "\<lambda>x. if x \<le> 0 then 0 else g0 x"

   then have "g 0 = 0" by simp

   have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>

       DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"

     using ev by eventually_elim auto

   then obtain a where [arith]: "0 < a"

     and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"

     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"

     and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"

     and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"

     unfolding eventually_within eventually_at by (auto simp: dist_real_def)

   have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"

     using g0_neq_0 by (simp add: g_def)

   { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"

       by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])

          (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }

   note f = this

   { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"

       by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])

          (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }

   note g = this

   have "isCont f 0"

     using tendsto_const[of "0::real" "at 0"] f_0

     unfolding isCont_def f_def

     by (intro filterlim_split_at_real)

        (auto elim: eventually_elim1

              simp add: filterlim_def le_filter_def eventually_within eventually_filtermap)

   have "isCont g 0"

     using tendsto_const[of "0::real" "at 0"] g_0

     unfolding isCont_def g_def

     by (intro filterlim_split_at_real)

        (auto elim: eventually_elim1

              simp add: filterlim_def le_filter_def eventually_within eventually_filtermap)

   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)"

   proof (rule bchoice, rule)

     fix x assume "x \<in> {0 <..< a}"

     then have x[arith]: "0 < x" "x < a" by auto

     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"

       by auto

     have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"

       using `isCont f 0` f by (auto intro: DERIV_isCont simp: le_less)

     moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"

       using `isCont g 0` g by (auto intro: DERIV_isCont simp: le_less)

     ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"

       using f g `x < a` by (intro GMVT') auto

     then guess c ..

     moreover

     with g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"

       by (simp add: field_simps)

     ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"

       using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c])

   qed

   then guess \<zeta> ..

   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)"

     unfolding eventually_within eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)

   moreover

   from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"

     by eventually_elim auto

   then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"

     by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"])

        (auto intro: tendsto_const tendsto_ident_at_within)

   then have "(\<zeta> ---> 0) (at_right 0)"

     by (rule tendsto_norm_zero_cancel)

   with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"

     by (auto elim!: eventually_elim1 simp: filterlim_within filterlim_at)

   from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"

     by (rule_tac filterlim_compose[of _ _ _ \<zeta>])

   ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)

     by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])

        (auto elim: eventually_elim1)

   also have "?P \<longleftrightarrow> ?thesis"

     by (rule filterlim_cong) (auto simp: f_def g_def eventually_within)

   finally show ?thesis .

 qed

 lemma lhopital_right:

   "((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>

     eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>

     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>

     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>

     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>

     ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>

   ((\<lambda> x. f x / g x) ---> y) (at_right x)"

   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift

   by (rule lhopital_right_0)

 lemma lhopital_left:

   "((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>

     eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>

     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>

     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>

     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>

     ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>

   ((\<lambda> x. f x / g x) ---> y) (at_left x)"

   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror

   by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)

 lemma lhopital:

   "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>

     eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>

     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>

     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>

     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>

     ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>

   ((\<lambda> x. f x / g x) ---> y) (at x)"

   unfolding eventually_at_split filterlim_at_split

   by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])

 lemma lhopital_right_0_at_top:

   fixes f g :: "real \<Rightarrow> real"

   assumes g_0: "LIM x at_right 0. g x :> at_top"

   assumes ev:

     "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"

     "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"

     "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"

   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"

   shows "((\<lambda> x. f x / g x) ---> x) (at_right 0)"

   unfolding tendsto_iff

 proof safe

   fix e :: real assume "0 < e"

   with lim[unfolded tendsto_iff, rule_format, of "e / 4"]

   have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp

   from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]

   obtain a where [arith]: "0 < a"

     and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"

     and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"

     and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"

     and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"

     unfolding eventually_within_le by (auto simp: dist_real_def)

   from Df have

     "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"

     unfolding eventually_within eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)

   moreover

   have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"

     using g_0 by (auto elim: eventually_elim1 simp: filterlim_at_top_dense)

   moreover

   have inv_g: "((\<lambda>x. inverse (g x)) ---> 0) (at_right 0)"

     using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]

     by (rule filterlim_compose)

   then have "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"

     by (intro tendsto_intros)

   then have "((\<lambda>x. norm (1 - g a / g x)) ---> 1) (at_right 0)"

     by (simp add: inverse_eq_divide)

   from this[unfolded tendsto_iff, rule_format, of 1]

   have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"

     by (auto elim!: eventually_elim1 simp: dist_real_def)

   moreover

   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)"

     by (intro tendsto_intros)

   then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)"

     by (simp add: inverse_eq_divide)

   from this[unfolded tendsto_iff, rule_format, of "e / 2"] `0 < e`

   have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"

     by (auto simp: dist_real_def)

   ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"

   proof eventually_elim

     fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"

     assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"

     have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"

       using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+

     then guess y ..

     from this

     have [arith]: "t < y" "y < a" and D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"

       using `g a < g t` g'_neq_0[of y] by (auto simp add: field_simps)

     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"

       by (simp add: field_simps)

     have "norm (f t / g t - x) \<le>

         norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)"

       unfolding * by (rule norm_triangle_ineq)

     also have "\<dots> = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)"

       by (simp add: abs_mult D_eq dist_real_def)

     also have "\<dots> < (e / 4) * 2 + e / 2"

       using ineq Df[of y] `0 < e` by (intro add_le_less_mono mult_mono) auto

     finally show "dist (f t / g t) x < e"

       by (simp add: dist_real_def)

   qed

 qed

 lemma lhopital_right_at_top:

   "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>

     eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>

     eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>

     eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>

     ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>

     ((\<lambda> x. f x / g x) ---> y) (at_right x)"

   unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift

   by (rule lhopital_right_0_at_top)

 lemma lhopital_left_at_top:

   "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>

     eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>

     eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>

     eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>

     ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>

     ((\<lambda> x. f x / g x) ---> y) (at_left x)"

   unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror

   by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)

 lemma lhopital_at_top:

   "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>

     eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>

     eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>

     eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>

     ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>

     ((\<lambda> x. f x / g x) ---> y) (at x)"

   unfolding eventually_at_split filterlim_at_split

   by (auto intro!: lhopital_right_at_top[of g x g' f f'] lhopital_left_at_top[of g x g' f f'])

 lemma lhospital_at_top_at_top:

   fixes f g :: "real \<Rightarrow> real"

   assumes g_0: "LIM x at_top. g x :> at_top"

   assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"

   assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"

   assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"

   assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) at_top"

   shows "((\<lambda> x. f x / g x) ---> x) at_top"

   unfolding filterlim_at_top_to_right

 proof (rule lhopital_right_0_at_top)

   let ?F = "\<lambda>x. f (inverse x)"

   let ?G = "\<lambda>x. g (inverse x)"

   let ?R = "at_right (0::real)"

   let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"

   show "LIM x ?R. ?G x :> at_top"

     using g_0 unfolding filterlim_at_top_to_right .

   show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"

     unfolding eventually_at_right_to_top

     using Dg eventually_ge_at_top[where c="1::real"]

     apply eventually_elim

     apply (rule DERIV_cong)

     apply (rule DERIV_chain'[where f=inverse])

     apply (auto intro!:  DERIV_inverse)

     done

   show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"

     unfolding eventually_at_right_to_top

     using Df eventually_ge_at_top[where c="1::real"]

     apply eventually_elim

     apply (rule DERIV_cong)

     apply (rule DERIV_chain'[where f=inverse])

     apply (auto intro!:  DERIV_inverse)

     done

   show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"

     unfolding eventually_at_right_to_top

     using g' eventually_ge_at_top[where c="1::real"]

     by eventually_elim auto

   show "((\<lambda>x. ?D f' x / ?D g' x) ---> x) ?R"

     unfolding filterlim_at_right_to_top

     apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])

     using eventually_ge_at_top[where c="1::real"]

     by eventually_elim simp

 qed

 end