(*  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 Lim

 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)"

 primrec

   Bolzano_bisect :: "(real \<times> real \<Rightarrow> bool) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real \<times> real" where

   "Bolzano_bisect P a b 0 = (a, b)"

   | "Bolzano_bisect P a b (Suc n) =

       (let (x, y) = Bolzano_bisect P a b n

        in if P (x, (x+y) / 2) then ((x+y)/2, y)

                               else (x, (x+y)/2))"

 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: diff_minus add_divide_distrib [symmetric] ring_distribs)

 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; DERIV g x :> Db |]

       ==> DERIV (%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; DERIV f x :> E |] ==> D = E"

 apply (simp add: deriv_def)

 apply (blast intro: LIM_unique)

 done

 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)

 apply (simp add: mult_ac)

 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 {* Nested Intervals and Bisection *}

 text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).

      All considerably tidied by lcp.*}

 lemma lemma_f_mono_add [rule_format (no_asm)]: "(\<forall>n. (f::nat=>real) n \<le> f (Suc n)) --> f m \<le> f(m + no)"

 apply (induct "no")

 apply (auto intro: order_trans)

 done

 lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);

          \<forall>n. g(Suc n) \<le> g(n);

          \<forall>n. f(n) \<le> g(n) |]

       ==> Bseq (f :: nat \<Rightarrow> real)"

 apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)

 apply (rule conjI)

 apply (induct_tac "n")

 apply (auto intro: order_trans)

 apply (rule_tac y = "g n" in order_trans)

 apply (induct_tac [2] "n")

 apply (auto intro: order_trans)

 done

 lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);

          \<forall>n. g(Suc n) \<le> g(n);

          \<forall>n. f(n) \<le> g(n) |]

       ==> Bseq (g :: nat \<Rightarrow> real)"

 apply (subst Bseq_minus_iff [symmetric])

 apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)

 apply auto

 done

 lemma f_inc_imp_le_lim:

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

   shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"

   by (rule incseq_le, simp add: incseq_SucI, simp add: convergent_LIMSEQ_iff)

 lemma lim_uminus:

   fixes g :: "nat \<Rightarrow> 'a::real_normed_vector"

   shows "convergent g ==> lim (%x. - g x) = - (lim g)"

 apply (rule tendsto_minus [THEN limI])

 apply (simp add: convergent_LIMSEQ_iff)

 done

 lemma g_dec_imp_lim_le:

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

   shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"

   by (rule decseq_le, simp add: decseq_SucI, simp add: convergent_LIMSEQ_iff)

 lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);

          \<forall>n. g(Suc n) \<le> g(n);

          \<forall>n. f(n) \<le> g(n) |]

       ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &

                             ((\<forall>n. m \<le> g(n)) & g ----> m)"

 apply (subgoal_tac "monoseq f & monoseq g")

 prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)

 apply (subgoal_tac "Bseq f & Bseq g")

 prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)

 apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)

 apply (rule_tac x = "lim f" in exI)

 apply (rule_tac x = "lim g" in exI)

 apply (auto intro: LIMSEQ_le)

 apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)

 done

 lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);

          \<forall>n. g(Suc n) \<le> g(n);

          \<forall>n. f(n) \<le> g(n);

          (%n. f(n) - g(n)) ----> 0 |]

       ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &

                 ((\<forall>n. l \<le> g(n)) & g ----> l)"

 apply (drule lemma_nest, auto)

 apply (subgoal_tac "l = m")

 apply (drule_tac [2] f = f in tendsto_diff)

 apply (auto intro: LIMSEQ_unique)

 done

 text{*The universal quantifiers below are required for the declaration

   of @{text Bolzano_nest_unique} below.*}

 lemma Bolzano_bisect_le:

  "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"

 apply (rule allI)

 apply (induct_tac "n")

 apply (auto simp add: Let_def split_def)

 done

 lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>

    \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"

 apply (rule allI)

 apply (induct_tac "n")

 apply (auto simp add: Bolzano_bisect_le Let_def split_def)

 done

 lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>

    \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"

 apply (rule allI)

 apply (induct_tac "n")

 apply (auto simp add: Bolzano_bisect_le Let_def split_def)

 done

 lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"

 apply (auto)

 apply (drule_tac f = "%u. (1/2) *u" in arg_cong)

 apply (simp)

 done

 lemma Bolzano_bisect_diff:

      "a \<le> b ==>

       snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =

       (b-a) / (2 ^ n)"

 apply (induct "n")

 apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)

 done

 lemmas Bolzano_nest_unique =

     lemma_nest_unique

     [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]

 lemma not_P_Bolzano_bisect:

   assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"

       and notP: "~ P(a,b)"

       and le:   "a \<le> b"

   shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"

 proof (induct n)

   case 0 show ?case using notP by simp

  next

   case (Suc n)

   thus ?case

  by (auto simp del: surjective_pairing [symmetric]

              simp add: Let_def split_def Bolzano_bisect_le [OF le]

      P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])

 qed

 (*Now we re-package P_prem as a formula*)

 lemma not_P_Bolzano_bisect':

      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);

          ~ P(a,b);  a \<le> b |] ==>

       \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"

 by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])

 lemma lemma_BOLZANO:

      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);

          \<forall>x. \<exists>d::real. 0 < d &

                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));

          a \<le> b |]

       ==> P(a,b)"

 apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)

 apply (rule tendsto_minus_cancel)

 apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)

 apply (rule ccontr)

 apply (drule not_P_Bolzano_bisect', assumption+)

 apply (rename_tac "l")

 apply (drule_tac x = l in spec, clarify)

 apply (simp add: LIMSEQ_iff)

 apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)

 apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)

 apply (drule real_less_half_sum, auto)

 apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)

 apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)

 apply safe

 apply (simp_all (no_asm_simp))

 apply (rule_tac y = "abs (fst (Bolzano_bisect P a b (no + noa)) - l) + abs (snd (Bolzano_bisect P a b (no + noa)) - l)" in order_le_less_trans)

 apply (simp (no_asm_simp) add: abs_if)

 apply (rule real_sum_of_halves [THEN subst])

 apply (rule add_strict_mono)

 apply (simp_all add: diff_minus [symmetric])

 done

 lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &

        (\<forall>x. \<exists>d::real. 0 < d &

                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))

       --> (\<forall>a b. a \<le> b --> P(a,b))"

 apply clarify

 apply (blast intro: lemma_BOLZANO)

 done

 subsection {* Intermediate Value Theorem *}

 text {*Prove Contrapositive by Bisection*}

 lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);

          a \<le> b;

          (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]

       ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"

 apply (rule contrapos_pp, assumption)

 apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> ~ (f (u) \<le> y & y \<le> f (v))" in lemma_BOLZANO2)

 apply safe

 apply simp_all

 apply (simp add: isCont_iff LIM_eq)

 apply (rule ccontr)

 apply (subgoal_tac "a \<le> x & x \<le> b")

  prefer 2

  apply simp

  apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)

 apply (drule_tac x = x in spec)+

 apply simp

 apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)

 apply safe

 apply simp

 apply (drule_tac x = s in spec, clarify)

 apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)

 apply (drule_tac x = "ba-x" in spec)

 apply (simp_all add: abs_if)

 apply (drule_tac x = "aa-x" in spec)

 apply (case_tac "x \<le> aa", simp_all)

 done

 lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);

          a \<le> b;

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

       |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"

 apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)

 apply (drule IVT [where f = "%x. - f x"], assumption)

 apply simp_all

 done

 (*HOL style here: object-level formulations*)

 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &

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

       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"

 apply (blast intro: IVT)

 done

 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &

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

       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"

 apply (blast intro: IVT2)

 done

 subsection {* Boundedness of continuous functions *}

 text{*By bisection, function continuous on closed interval is bounded above*}

 lemma isCont_bounded:

      "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]

       ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"

 apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> (\<exists>M. \<forall>x. u \<le> x & x \<le> v --> f x \<le> M)" in lemma_BOLZANO2)

 apply safe

 apply simp_all

 apply (rename_tac x xa ya M Ma)

 apply (metis linorder_not_less order_le_less order_trans)

 apply (case_tac "a \<le> x & x \<le> b")

  prefer 2

  apply (rule_tac x = 1 in exI, force)

 apply (simp add: LIM_eq isCont_iff)

 apply (drule_tac x = x in spec, auto)

 apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)

 apply (drule_tac x = 1 in spec, auto)

 apply (rule_tac x = s in exI, clarify)

 apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)

 apply (drule_tac x = "xa-x" in spec)

 apply (auto simp add: abs_ge_self)

 done

 text{*Refine the above to existence of least upper bound*}

 lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->

       (\<exists>t. isLub UNIV S t)"

 by (blast intro: reals_complete)

 lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]

          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &

                    (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"

 apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"

         in lemma_reals_complete)

 apply auto

 apply (drule isCont_bounded, assumption)

 apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)

 apply (rule exI, auto)

 apply (auto dest!: spec simp add: linorder_not_less)

 done

 text{*Now show that it attains its upper bound*}

 lemma isCont_eq_Ub:

   assumes le: "a \<le> b"

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

   shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &

              (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"

 proof -

   from isCont_has_Ub [OF le con]

   obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"

              and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast

   show ?thesis

   proof (intro exI, intro conjI)

     show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)

     show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"

     proof (rule ccontr)

       assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"

       with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"

         by (fastforce simp add: linorder_not_le [symmetric])

       hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"

         by (auto simp add: con)

       from isCont_bounded [OF le this]

       obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto

       have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"

         by (simp add: M3 algebra_simps)

       have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k

         by (auto intro: order_le_less_trans [of _ k])

       with Minv

       have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"

         by (intro strip less_imp_inverse_less, simp_all)

       hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"

         by simp

       have "M - inverse (k+1) < M" using k [of a] Minv [of a] le

         by (simp, arith)

       from M2 [OF this]

       obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..

       thus False using invlt [of x] by force

     qed

   qed

 qed

 text{*Same theorem for lower bound*}

 lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]

          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &

                    (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"

 apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")

 prefer 2 apply (blast intro: isCont_minus)

 apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)

 apply safe

 apply auto

 done

 text{*Another version.*}

 lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]

       ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &

           (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"

 apply (frule isCont_eq_Lb)

 apply (frule_tac [2] isCont_eq_Ub)

 apply (assumption+, safe)

 apply (rule_tac x = "f x" in exI)

 apply (rule_tac x = "f xa" in exI, simp, safe)

 apply (cut_tac x = x and y = xa in linorder_linear, safe)

 apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)

 apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)

 apply (rule_tac [2] x = xb in exI)

 apply (rule_tac [4] x = xb in exI, simp_all)

 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)"

 proof cases

   assume "a=b" thus ?thesis by simp

 next

   assume "a\<noteq>b"

   hence ba: "b-a \<noteq> 0" by arith

   show ?thesis

     by (rule real_mult_left_cancel [OF ba, THEN iffD1],

         simp add: right_diff_distrib,

         simp add: left_diff_distrib)

 qed

 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_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 \<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_nonincreasing [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

 subsection {* Continuous injective functions *}

 text{*Dull lemma: an continuous injection on an interval must have a

 strict maximum at an end point, not in the middle.*}

 lemma lemma_isCont_inj:

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

   assumes d: "0 < d"

       and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"

       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"

   shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"

 proof (rule ccontr)

   assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"

   hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto

   show False

   proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])

     case le

     from d cont all [of "x+d"]

     have flef: "f(x+d) \<le> f x"

      and xlex: "x - d \<le> x"

      and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"

        by (auto simp add: abs_if)

     from IVT [OF le flef xlex cont']

     obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast

     moreover

     hence "g(f x') = g (f(x+d))" by simp

     ultimately show False using d inj [of x'] inj [of "x+d"]

       by (simp add: abs_le_iff)

   next

     case ge

     from d cont all [of "x-d"]

     have flef: "f(x-d) \<le> f x"

      and xlex: "x \<le> x+d"

      and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"

        by (auto simp add: abs_if)

     from IVT2 [OF ge flef xlex cont']

     obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast

     moreover

     hence "g(f x') = g (f(x-d))" by simp

     ultimately show False using d inj [of x'] inj [of "x-d"]

       by (simp add: abs_le_iff)

   qed

 qed

 text{*Similar version for lower bound.*}

 lemma lemma_isCont_inj2:

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

   shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;

         \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]

       ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"

 apply (insert lemma_isCont_inj

           [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])

 apply (simp add: linorder_not_le)

 done

 text{*Show there's an interval surrounding @{term "f(x)"} in

 @{text "f[[x - d, x + d]]"} .*}

 lemma isCont_inj_range:

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

   assumes d: "0 < d"

       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"

       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"

   shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"

 proof -

   have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d

     by (auto simp add: abs_le_iff)

   from isCont_Lb_Ub [OF this]

   obtain L M

   where all1 [rule_format]: "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M"

     and all2 [rule_format]:

            "\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. x-d \<le> z \<and> z \<le> x+d \<and> f z = y)"

     by auto

   with d have "L \<le> f x & f x \<le> M" by simp

   moreover have "L \<noteq> f x"

   proof -

     from lemma_isCont_inj2 [OF d inj cont]

     obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto

     thus ?thesis using all1 [of u] by arith

   qed

   moreover have "f x \<noteq> M"

   proof -

     from lemma_isCont_inj [OF d inj cont]

     obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto

     thus ?thesis using all1 [of u] by arith

   qed

   ultimately have "L < f x & f x < M" by arith

   hence "0 < f x - L" "0 < M - f x" by arith+

   from real_lbound_gt_zero [OF this]

   obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto

   thus ?thesis

   proof (intro exI conjI)

     show "0<e" using e(1) .

     show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"

     proof (intro strip)

       fix y::real

       assume "\<bar>y - f x\<bar> \<le> e"

       with e have "L \<le> y \<and> y \<le> M" by arith

       from all2 [OF this]

       obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast

       thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y" 

         by (force simp add: abs_le_iff)

     qed

   qed

 qed

 text{*Continuity of inverse function*}

 lemma isCont_inverse_function:

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

   assumes d: "0 < d"

       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"

       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"

   shows "isCont g (f x)"

 proof (simp add: isCont_iff LIM_eq)

   show "\<forall>r. 0 < r \<longrightarrow>

          (\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"

   proof (intro strip)

     fix r::real

     assume r: "0<r"

     from real_lbound_gt_zero [OF r d]

     obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast

     with inj cont

     have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"

                   "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto

     from isCont_inj_range [OF e this]

     obtain e' where e': "0 < e'"

         and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"

           by blast

     show "\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r"

     proof (intro exI conjI)

       show "0<e'" using e' .

       show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"

       proof (intro strip)

         fix z::real

         assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"

         with e e_lt e_simps all [rule_format, of "f x + z"]

         show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force

       qed

     qed

   qed

 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

 subsection {* Theorems about Limits *}

 (* need to rename second isCont_inverse *)

 lemma isCont_inv_fun:

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

   shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  

          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  

       ==> isCont g (f x)"

 by (rule isCont_inverse_function)

 lemma isCont_inv_fun_inv:

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

   shows "[| 0 < d;  

          \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  

          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  

        ==> \<exists>e. 0 < e &  

              (\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"

 apply (drule isCont_inj_range)

 prefer 2 apply (assumption, assumption, auto)

 apply (rule_tac x = e in exI, auto)

 apply (rotate_tac 2)

 apply (drule_tac x = y in spec, auto)

 done

 text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}

 lemma LIM_fun_gt_zero:

      "[| f -- c --> (l::real); 0 < l |]  

          ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"

 apply (drule (1) LIM_D, clarify)

 apply (rule_tac x = s in exI)

 apply (simp add: abs_less_iff)

 done

 lemma LIM_fun_less_zero:

      "[| f -- c --> (l::real); l < 0 |]  

       ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"

 apply (drule LIM_D [where r="-l"], simp, clarify)

 apply (rule_tac x = s in exI)

 apply (simp add: abs_less_iff)

 done

 lemma LIM_fun_not_zero:

      "[| f -- c --> (l::real); l \<noteq> 0 |] 

       ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"

 apply (rule linorder_cases [of l 0])

 apply (drule (1) LIM_fun_less_zero, force)

 apply simp

 apply (drule (1) LIM_fun_gt_zero, force)

 done

 end