(*  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)

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 P=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]