Theory Deriv

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theory Deriv
imports Lim
begin

(*  Title       : Deriv.thy
    ID          : $Id: Deriv.thy,v 1.22 2007/06/23 17:33:23 nipkow Exp $
    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 => 'a, 'a, 'a] => 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)"

definition
  differentiable :: "['a::real_normed_field => 'a, 'a] => bool"
    (infixl "differentiable" 60) where
  "f differentiable x = (∃D. DERIV f x :> D)"


consts
  Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)"
primrec
  "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 (λx. k) x :> 0"
by (simp add: deriv_def)

lemma DERIV_ident [simp]: "DERIV (λx. x) x :> 1"
by (simp add: deriv_def cong: LIM_cong)

lemma add_diff_add:
  fixes a b c d :: "'a::ab_group_add"
  shows "(a + c) - (b + d) = (a - b) + (c - d)"
by simp

lemma DERIV_add:
  "[|DERIV f x :> D; DERIV g x :> E|] ==> DERIV (λx. f x + g x) x :> D + E"
by (simp only: deriv_def add_diff_add add_divide_distrib LIM_add)

lemma DERIV_minus:
  "DERIV f x :> D ==> DERIV (λx. - f x) x :> - D"
by (simp only: deriv_def minus_diff_minus divide_minus_left LIM_minus)

lemma DERIV_diff:
  "[|DERIV f x :> D; DERIV g x :> E|] ==> DERIV (λx. f x - g x) x :> D - E"
by (simp only: diff_def DERIV_add DERIV_minus)

lemma DERIV_add_minus:
  "[|DERIV f x :> D; DERIV g x :> E|] ==> DERIV (λx. f x + - g x) x :> D + - E"
by (simp only: DERIV_add DERIV_minus)

lemma DERIV_isCont: "DERIV f x :> D ==> isCont f x"
proof (unfold isCont_iff)
  assume "DERIV f x :> D"
  hence "(λh. (f(x+h) - f(x)) / h) -- 0 --> D"
    by (rule DERIV_D)
  hence "(λh. (f(x+h) - f(x)) / h * h) -- 0 --> D * 0"
    by (intro LIM_mult LIM_ident)
  hence "(λh. (f(x+h) - f(x)) * (h / h)) -- 0 --> 0"
    by simp
  hence "(λh. f(x+h) - f(x)) -- 0 --> 0"
    by (simp cong: LIM_cong)
  thus "(λh. f(x+h)) -- 0 --> f(x)"
    by (simp add: LIM_def)
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 (λx. 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 "(λh. f(x+h)) -- 0 --> f x"
    by (simp only: isCont_iff)
  hence "(λh. 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 LIM_add LIM_mult LIM_const DERIV_D f g)
  thus "(λh. (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_sumr [rule_format (no_asm)]:
     "(∀r. m ≤ r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
      --> DERIV (%x. ∑n=m..<n::nat. f n x :: real) x :> (∑r=m..<n. f' r x)"
apply (induct "n")
apply (auto intro: DERIV_add)
done

text{*Alternative definition for differentiability*}

lemma DERIV_LIM_iff:
     "((%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 inverse_diff_inverse:
  "[|(a::'a::division_ring) ≠ 0; b ≠ 0|]
   ==> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
by (simp add: ring_simps)

lemma DERIV_inverse_lemma:
  "[|a ≠ 0; b ≠ (0::'a::real_normed_field)|]
   ==> (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 ≠ 0"
  shows "DERIV (λx. 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: "!!z. [|z ≠ x; norm (z - x) < s|]
                  ==> norm (f z - f x) < norm (f x)"
    by fast

  show "(λz. (inverse (f z) - inverse (f x)) / (z - x)) -- x --> ?E"
  proof (rule LIM_equal2 [OF s])
    fix z
    assume "z ≠ x" "norm (z - x) < s"
    hence "norm (f z - f x) < norm (f x)" by (rule less_fx)
    hence "f z ≠ 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 "(λz. (f z - f x) / (z - x)) -- x --> D"
      by (unfold DERIV_iff2)
    thus "(λz. - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x)))
          -- x --> ?E"
      by (intro LIM_mult LIM_inverse LIM_minus LIM_const lim_f neq)
  qed
qed

lemma DERIV_divide:
  "[|DERIV f x :> D; DERIV g x :> E; g x ≠ 0|]
   ==> DERIV (λx. 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 => 'a::{real_normed_field,recpower}"
  assumes f: "DERIV f x :> D"
  shows "DERIV (λx. f x ^ Suc n) x :> (1 + of_nat n) * (D * f x ^ n)"
proof (induct n)
case 0
  show ?case by (simp add: power_Suc 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 right_distrib mult_ac add_ac)
    done
qed

lemma DERIV_power:
  fixes f :: "'a => 'a::{real_normed_field,recpower}"
  assumes f: "DERIV f x :> D"
  shows "DERIV (λx. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))"
by (cases "n", simp, simp add: DERIV_power_Suc f)


(* ------------------------------------------------------------------------ *)
(* Caratheodory formulation of derivative at a point: standard proof        *)
(* ------------------------------------------------------------------------ *)

lemma CARAT_DERIV:
     "(DERIV f x :> l) =
      (∃g. (∀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 "∃g. (∀z. f z - f x = g z * (z-x)) ∧ isCont g x ∧ g x = l"
  proof (intro exI conjI)
    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
    show "∀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
    "(∀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 (λx. g (f x)) x :> E * D"
proof (unfold DERIV_iff2)
  obtain d where d: "∀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 "(λz. d (f z)) -- x --> d (f x)"
    by (rule isCont_LIM_compose)
  hence "(λz. d (f z) * ((f z - f x) / (z - x)))
          -- x --> d (f x) * D"
    by (rule LIM_mult [OF _ f [unfolded DERIV_iff2]])
  thus "(λz. (g (f z) - g (f x)) / (z - x)) -- x --> E * D"
    by (simp add: d dfx real_scaleR_def)
qed

(* let's do the standard proof though theorem *)
(* 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)

(* 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 real_scaleR_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)

(*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_scaleR_def real_of_nat_def)
done

text{*Power of -1*}

lemma DERIV_inverse:
  fixes x :: "'a::{real_normed_field,recpower}"
  shows "x ≠ 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
by (drule DERIV_inverse' [OF DERIV_ident]) (simp add: power_Suc)

text{*Derivative of inverse*}
lemma DERIV_inverse_fun:
  fixes x :: "'a::{real_normed_field,recpower}"
  shows "[| DERIV f x :> d; f(x) ≠ 0 |]
      ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
by (drule (1) DERIV_inverse') (simp add: mult_ac power_Suc nonzero_inverse_mult_distrib)

text{*Derivative of quotient*}
lemma DERIV_quotient:
  fixes x :: "'a::{real_normed_field,recpower}"
  shows "[| DERIV f x :> d; DERIV g x :> e; g(x) ≠ 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 power_Suc)


subsection {* Differentiability predicate *}

lemma differentiableD: "f differentiable x ==> ∃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_const: "(λz. a) differentiable x"
  apply (unfold differentiable_def)
  apply (rule_tac x=0 in exI)
  apply simp
  done

lemma differentiable_sum:
  assumes "f differentiable x"
  and "g differentiable x"
  shows "(λx. f x + g x) differentiable x"
proof -
  from prems have "∃D. DERIV f x :> D" by (unfold differentiable_def)
  then obtain df where "DERIV f x :> df" ..
  moreover from prems have "∃D. DERIV g x :> D" by (unfold differentiable_def)
  then obtain dg where "DERIV g x :> dg" ..
  ultimately have "DERIV (λx. f x + g x) x :> df + dg" by (rule DERIV_add)
  hence "∃D. DERIV (λx. f x + g x) x :> D" by auto
  thus ?thesis by (fold differentiable_def)
qed

lemma differentiable_diff:
  assumes "f differentiable x"
  and "g differentiable x"
  shows "(λx. f x - g x) differentiable x"
proof -
  from prems have "f differentiable x" by simp
  moreover
  from prems have "∃D. DERIV g x :> D" by (unfold differentiable_def)
  then obtain dg where "DERIV g x :> dg" ..
  then have "DERIV (λx. - g x) x :> -dg" by (rule DERIV_minus)
  hence "∃D. DERIV (λx. - g x) x :> D" by auto
  hence "(λx. - g x) differentiable x" by (fold differentiable_def)
  ultimately 
  show ?thesis
    by (auto simp: diff_def dest: differentiable_sum)
qed

lemma differentiable_mult:
  assumes "f differentiable x"
  and "g differentiable x"
  shows "(λx. f x * g x) differentiable x"
proof -
  from prems have "∃D. DERIV f x :> D" by (unfold differentiable_def)
  then obtain df where "DERIV f x :> df" ..
  moreover from prems have "∃D. DERIV g x :> D" by (unfold differentiable_def)
  then obtain dg where "DERIV g x :> dg" ..
  ultimately have "DERIV (λx. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult)
  hence "∃D. DERIV (λx. f x * g x) x :> D" by auto
  thus ?thesis by (fold differentiable_def)
qed


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)]: "(∀n. (f::nat=>real) n ≤ f (Suc n)) --> f m ≤ f(m + no)"
apply (induct "no")
apply (auto intro: order_trans)
done

lemma f_inc_g_dec_Beq_f: "[| ∀n. f(n) ≤ f(Suc n);
         ∀n. g(Suc n) ≤ g(n);
         ∀n. f(n) ≤ g(n) |]
      ==> Bseq (f :: nat => real)"
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
apply (induct_tac "n")
apply (auto intro: order_trans)
apply (rule_tac y = "g (Suc na)" in order_trans)
apply (induct_tac [2] "na")
apply (auto intro: order_trans)
done

lemma f_inc_g_dec_Beq_g: "[| ∀n. f(n) ≤ f(Suc n);
         ∀n. g(Suc n) ≤ g(n);
         ∀n. f(n) ≤ g(n) |]
      ==> Bseq (g :: nat => 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 => real"
  shows "[|∀n. f n ≤ f (Suc n); convergent f|] ==> f n ≤ lim f"
apply (rule linorder_not_less [THEN iffD1])
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
apply (drule real_less_sum_gt_zero)
apply (drule_tac x = "f n + - lim f" in spec, safe)
apply (drule_tac P = "%na. no≤na --> ?Q na" and x = "no + n" in spec, auto)
apply (subgoal_tac "lim f ≤ f (no + n) ")
apply (drule_tac no=no and m=n in lemma_f_mono_add)
apply (auto simp add: add_commute)
apply (induct_tac "no")
apply simp
apply (auto intro: order_trans simp add: diff_minus abs_if)
done

lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
apply (rule LIMSEQ_minus [THEN limI])
apply (simp add: convergent_LIMSEQ_iff)
done

lemma g_dec_imp_lim_le:
  fixes g :: "nat => real"
  shows "[|∀n. g (Suc n) ≤ g(n); convergent g|] ==> lim g ≤ g n"
apply (subgoal_tac "- (g n) ≤ - (lim g) ")
apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
done

lemma lemma_nest: "[| ∀n. f(n) ≤ f(Suc n);
         ∀n. g(Suc n) ≤ g(n);
         ∀n. f(n) ≤ g(n) |]
      ==> ∃l m :: real. l ≤ m &  ((∀n. f(n) ≤ l) & f ----> l) &
                            ((∀n. m ≤ 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: "[| ∀n. f(n) ≤ f(Suc n);
         ∀n. g(Suc n) ≤ g(n);
         ∀n. f(n) ≤ g(n);
         (%n. f(n) - g(n)) ----> 0 |]
      ==> ∃l::real. ((∀n. f(n) ≤ l) & f ----> l) &
                ((∀n. l ≤ g(n)) & g ----> l)"
apply (drule lemma_nest, auto)
apply (subgoal_tac "l = m")
apply (drule_tac [2] X = f in LIMSEQ_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 ≤ b ==> ∀n. fst (Bolzano_bisect P a b n) ≤ 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 ≤ b ==>
   ∀n. fst(Bolzano_bisect P a b n) ≤ 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 ≤ b ==>
   ∀n. snd(Bolzano_bisect P a b (Suc n)) ≤ 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 ≤ 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 ≤ b; b ≤ c|] ==> P(a,c)"
      and notP: "~ P(a,b)"
      and le:   "a ≤ 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':
     "[| ∀a b c. P(a,b) & P(b,c) & a ≤ b & b ≤ c --> P(a,c);
         ~ P(a,b);  a ≤ b |] ==>
      ∀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:
     "[| ∀a b c. P(a,b) & P(b,c) & a ≤ b & b ≤ c --> P(a,c);
         ∀x. ∃d::real. 0 < d &
                (∀a b. a ≤ x & x ≤ b & (b-a) < d --> P(a,b));
         a ≤ b |]
      ==> P(a,b)"
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
apply (rule LIMSEQ_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_def)
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: "((∀a b c. (a ≤ b & b ≤ c & P(a,b) & P(b,c)) --> P(a,c)) &
       (∀x. ∃d::real. 0 < d &
                (∀a b. a ≤ x & x ≤ b & (b-a) < d --> P(a,b))))
      --> (∀a b. a ≤ 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) ≤ (y::real); y ≤ f(b);
         a ≤ b;
         (∀x. a ≤ x & x ≤ b --> isCont f x) |]
      ==> ∃x. a ≤ x & x ≤ b & f(x) = y"
apply (rule contrapos_pp, assumption)
apply (cut_tac P = "% (u,v) . a ≤ u & u ≤ v & v ≤ b --> ~ (f (u) ≤ y & y ≤ f (v))" in lemma_BOLZANO2)
apply safe
apply simp_all
apply (simp add: isCont_iff LIM_def)
apply (rule ccontr)
apply (subgoal_tac "a ≤ x & x ≤ 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 --> (∃s>0. ?Q r s) " and x = "¦y - f x¦" 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 ≤ aa", simp_all)
done

lemma IVT2: "[| f(b::real) ≤ (y::real); y ≤ f(a);
         a ≤ b;
         (∀x. a ≤ x & x ≤ b --> isCont f x)
      |] ==> ∃x. a ≤ x & x ≤ b & f(x) = y"
apply (subgoal_tac "- f a ≤ -y & -y ≤ - f b", clarify)
apply (drule IVT [where f = "%x. - f x"], assumption)
apply (auto intro: isCont_minus)
done

(*HOL style here: object-level formulations*)
lemma IVT_objl: "(f(a::real) ≤ (y::real) & y ≤ f(b) & a ≤ b &
      (∀x. a ≤ x & x ≤ b --> isCont f x))
      --> (∃x. a ≤ x & x ≤ b & f(x) = y)"
apply (blast intro: IVT)
done

lemma IVT2_objl: "(f(b::real) ≤ (y::real) & y ≤ f(a) & a ≤ b &
      (∀x. a ≤ x & x ≤ b --> isCont f x))
      --> (∃x. a ≤ x & x ≤ b & f(x) = y)"
apply (blast intro: IVT2)
done

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

lemma isCont_bounded:
     "[| a ≤ b; ∀x. a ≤ x & x ≤ b --> isCont f x |]
      ==> ∃M::real. ∀x::real. a ≤ x & x ≤ b --> f(x) ≤ M"
apply (cut_tac P = "% (u,v) . a ≤ u & u ≤ v & v ≤ b --> (∃M. ∀x. u ≤ x & x ≤ v --> f x ≤ M)" in lemma_BOLZANO2)
apply safe
apply simp_all
apply (rename_tac x xa ya M Ma)
apply (cut_tac x = M and y = Ma in linorder_linear, safe)
apply (rule_tac x = Ma in exI, clarify)
apply (cut_tac x = xb and y = xa in linorder_linear, force)
apply (rule_tac x = M in exI, clarify)
apply (cut_tac x = xb and y = xa in linorder_linear, force)
apply (case_tac "a ≤ x & x ≤ b")
apply (rule_tac [2] x = 1 in exI)
prefer 2 apply force
apply (simp add: LIM_def isCont_iff)
apply (drule_tac x = x in spec, auto)
apply (erule_tac V = "∀M. ∃x. a ≤ x & x ≤ b & ~ f x ≤ M" in thin_rl)
apply (drule_tac x = 1 in spec, auto)
apply (rule_tac x = s in exI, clarify)
apply (rule_tac x = "¦f x¦ + 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: "((∃x. x ∈ S) & (∃y. isUb UNIV S (y::real))) -->
      (∃t. isLub UNIV S t)"
by (blast intro: reals_complete)

lemma isCont_has_Ub: "[| a ≤ b; ∀x. a ≤ x & x ≤ b --> isCont f x |]
         ==> ∃M::real. (∀x::real. a ≤ x & x ≤ b --> f(x) ≤ M) &
                   (∀N. N < M --> (∃x. a ≤ x & x ≤ b & N < f(x)))"
apply (cut_tac S = "Collect (%y. ∃x. a ≤ x & x ≤ 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 ≤ b"
      and con: "∀x::real. a ≤ x & x ≤ b --> isCont f x"
  shows "∃M::real. (∀x. a ≤ x & x ≤ b --> f(x) ≤ M) &
             (∃x. a ≤ x & x ≤ b & f(x) = M)"
proof -
  from isCont_has_Ub [OF le con]
  obtain M where M1: "∀x. a ≤ x ∧ x ≤ b --> f x ≤ M"
             and M2: "!!N. N<M ==> ∃x. a ≤ x ∧ x ≤ b ∧ N < f x"  by blast
  show ?thesis
  proof (intro exI, intro conjI)
    show " ∀x. a ≤ x ∧ x ≤ b --> f x ≤ M" by (rule M1)
    show "∃x. a ≤ x ∧ x ≤ b ∧ f x = M"
    proof (rule ccontr)
      assume "¬ (∃x. a ≤ x ∧ x ≤ b ∧ f x = M)"
      with M1 have M3: "∀x. a ≤ x & x ≤ b --> f x < M"
        by (fastsimp simp add: linorder_not_le [symmetric])
      hence "∀x. a ≤ x & x ≤ b --> isCont (%x. inverse (M - f x)) x"
        by (auto simp add: isCont_inverse isCont_diff con)
      from isCont_bounded [OF le this]
      obtain k where k: "!!x. a ≤ x & x ≤ b --> inverse (M - f x) ≤ k" by auto
      have Minv: "!!x. a ≤ x & x ≤ b --> 0 < inverse (M - f (x))"
        by (simp add: M3 compare_rls)
      have "!!x. a ≤ x & x ≤ b --> inverse (M - f x) < k+1" using k
        by (auto intro: order_le_less_trans [of _ k])
      with Minv
      have "!!x. a ≤ x & x ≤ b --> inverse(k+1) < inverse(inverse(M - f x))"
        by (intro strip less_imp_inverse_less, simp_all)
      hence invlt: "!!x. a ≤ x & x ≤ 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 ≤ x & x ≤ 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 ≤ b; ∀x. a ≤ x & x ≤ b --> isCont f x |]
         ==> ∃M::real. (∀x::real. a ≤ x & x ≤ b --> M ≤ f(x)) &
                   (∃x. a ≤ x & x ≤ b & f(x) = M)"
apply (subgoal_tac "∀x. a ≤ x & x ≤ 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 ≤ b; ∀x. a ≤ x & x ≤ b --> isCont f x |]
      ==> ∃L M::real. (∀x::real. a ≤ x & x ≤ b --> L ≤ f(x) & f(x) ≤ M) &
          (∀y. L ≤ y & y ≤ M --> (∃x. a ≤ x & x ≤ 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


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

lemma DERIV_left_inc:
  fixes f :: "real => real"
  assumes der: "DERIV f x :> l"
      and l:   "0 < l"
  shows "∃d > 0. ∀h > 0. h < d --> f(x) < f(x + h)"
proof -
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
  have "∃s > 0. (∀z. z ≠ 0 ∧ ¦z¦ < s --> ¦(f(x+z) - f x) / z - l¦ < l)"
    by (simp add: diff_minus)
  then obtain s
        where s:   "0 < s"
          and all: "!!z. z ≠ 0 ∧ ¦z¦ < s --> ¦(f(x+z) - f x) / z - l¦ < 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_left_dec:
  fixes f :: "real => real"
  assumes der: "DERIV f x :> l"
      and l:   "l < 0"
  shows "∃d > 0. ∀h > 0. h < d --> f(x) < f(x-h)"
proof -
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
  have "∃s > 0. (∀z. z ≠ 0 ∧ ¦z¦ < s --> ¦(f(x+z) - f x) / z - l¦ < -l)"
    by (simp add: diff_minus)
  then obtain s
        where s:   "0 < s"
          and all: "!!z. z ≠ 0 ∧ ¦z¦ < s --> ¦(f(x+z) - f x) / z - l¦ < -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_local_max:
  fixes f :: "real => real"
  assumes der: "DERIV f x :> l"
      and d:   "0 < d"
      and le:  "∀y. ¦x-y¦ < d --> f(y) ≤ f(x)"
  shows "l = 0"
proof (cases rule: linorder_cases [of l 0])
  case equal thus ?thesis .
next
  case less
  from DERIV_left_dec [OF der less]
  obtain d' where d': "0 < d'"
             and lt: "∀h > 0. h < d' --> f x < f (x-h)" by blast
  from real_lbound_gt_zero [OF d d']
  obtain e where "0 < e ∧ e < d ∧ e < d'" ..
  with lt le [THEN spec [where x="x-e"]]
  show ?thesis by (auto simp add: abs_if)
next
  case greater
  from DERIV_left_inc [OF der greater]
  obtain d' where d': "0 < d'"
             and lt: "∀h > 0. h < d' --> f x < f (x + h)" by blast
  from real_lbound_gt_zero [OF d d']
  obtain e where "0 < e ∧ e < d ∧ 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; ∀y. ¦x-y¦ < d --> f(x) ≤ 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; ∀y. ¦x-y¦ < d --> f(x) = f(y) |] ==> l = 0"
by (auto dest!: DERIV_local_max)

text{*Lemma about introducing open ball in open interval*}
lemma lemma_interval_lt:
     "[| a < x;  x < b |]
      ==> ∃d::real. 0 < d & (∀y. ¦x-y¦ < 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 |] ==>
        ∃d::real. 0 < d &  (∀y. ¦x-y¦ < d --> a ≤ y & y ≤ b)"
apply (drule lemma_interval_lt, auto)
apply (auto intro!: exI)
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 ∈ (a,b)"} such that @{term "f'(x0) = 0"}*}
theorem Rolle:
  assumes lt: "a < b"
      and eq: "f(a) = f(b)"
      and con: "∀x. a ≤ x & x ≤ b --> isCont f x"
      and dif [rule_format]: "∀x. a < x & x < b --> f differentiable x"
  shows "∃z::real. a < z & z < b & DERIV f z :> 0"
proof -
  have le: "a ≤ b" using lt by simp
  from isCont_eq_Ub [OF le con]
  obtain x where x_max: "∀z. a ≤ z ∧ z ≤ b --> f z ≤ f x"
             and alex: "a ≤ x" and xleb: "x ≤ b"
    by blast
  from isCont_eq_Lb [OF le con]
  obtain x' where x'_min: "∀z. a ≤ z ∧ z ≤ b --> f x' ≤ f z"
              and alex': "a ≤ x'" and x'leb: "x' ≤ 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 auto
    from lemma_interval [OF ax xb]
    obtain d where d: "0<d" and bound: "∀y. ¦x-y¦ < d --> a ≤ y ∧ y ≤ b"
      by blast
    hence bound': "∀y. ¦x-y¦ < d --> f y ≤ 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 auto
      from lemma_interval [OF ax' x'b]
      obtain d where d: "0<d" and bound: "∀y. ¦x'-y¦ < d --> a ≤ y ∧ y ≤ b"
  by blast
      hence bound': "∀y. ¦x'-y¦ < d --> f x' ≤ 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: "∀y. ¦r-y¦ < d --> a ≤ y ∧ y ≤ b"
  by blast
      have eq_fb: "∀z. a ≤ z --> z ≤ b --> f z = f b"
      proof (clarify)
        fix z::real
        assume az: "a ≤ z" and zb: "z ≤ b"
        show "f z = f b"
        proof (rule order_antisym)
          show "f z ≤ f b" by (simp add: fb_eq_fx x_max az zb)
          show "f b ≤ f z" by (simp add: fb_eq_fx' x'_min az zb)
        qed
      qed
      have bound': "∀y. ¦r-y¦ < d --> f r = f y"
      proof (intro strip)
        fix y::real
        assume lt: "¦r-y¦ < 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≠b"
  hence ba: "b-a ≠ 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: "∀x. a ≤ x & x ≤ b --> isCont f x"
      and dif [rule_format]: "∀x. a < x & x < b --> f differentiable x"
  shows "∃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: "∀x. a ≤ x ∧ x ≤ b --> isCont ?F x" using con
    by (fast intro: isCont_diff isCont_const isCont_mult isCont_ident)
  have difF: "∀x. a < x ∧ x < b --> ?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 (λx. ?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


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

lemma DERIV_isconst_end:
  fixes f :: "real => real"
  shows "[| a < b;
         ∀x. a ≤ x & x ≤ b --> isCont f x;
         ∀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;
         ∀x. a ≤ x & x ≤ b --> isCont f x;
         ∀x. a < x & x < b --> DERIV f x :> 0 |]
        ==> ∀x. a ≤ x & x ≤ 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;
         ∀x. a ≤ x & x ≤ b --> isCont f x;
         ∀x. a < x & x < b --> DERIV f x :> 0;
         a ≤ x; x ≤ b |]
        ==> f x = f a"
apply (blast dest: DERIV_isconst1)
done

lemma DERIV_isconst_all:
  fixes f :: "real => real"
  shows "∀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 ≠ b; ∀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 ≠ b; ∀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 ≠ (b::real)"
      and der: "∀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


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 => real"
  assumes d: "0 < d"
      and inj [rule_format]: "∀z. ¦z-x¦ ≤ d --> g(f z) = z"
      and cont: "∀z. ¦z-x¦ ≤ d --> isCont f z"
  shows "∃z. ¦z-x¦ ≤ d & f x < f z"
proof (rule ccontr)
  assume  "~ (∃z. ¦z-x¦ ≤ d & f x < f z)"
  hence all [rule_format]: "∀z. ¦z - x¦ ≤ d --> f z ≤ 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) ≤ f x"
     and xlex: "x - d ≤ x"
     and cont': "∀z. x - d ≤ z ∧ z ≤ x --> isCont f z"
       by (auto simp add: abs_if)
    from IVT [OF le flef xlex cont']
    obtain x' where "x-d ≤ x'" "x' ≤ 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) ≤ f x"
     and xlex: "x ≤ x+d"
     and cont': "∀z. x ≤ z ∧ z ≤ x+d --> isCont f z"
       by (auto simp add: abs_if)
    from IVT2 [OF ge flef xlex cont']
    obtain x' where "x ≤ x'" "x' ≤ 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 => real"
  shows "[|0 < d; ∀z. ¦z-x¦ ≤ d --> g(f z) = z;
        ∀z. ¦z-x¦ ≤ d --> isCont f z |]
      ==> ∃z. ¦z-x¦ ≤ 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: isCont_minus 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 => real"
  assumes d: "0 < d"
      and inj: "∀z. ¦z-x¦ ≤ d --> g(f z) = z"
      and cont: "∀z. ¦z-x¦ ≤ d --> isCont f z"
  shows "∃e>0. ∀y. ¦y - f x¦ ≤ e --> (∃z. ¦z-x¦ ≤ d & f z = y)"
proof -
  have "x-d ≤ x+d" "∀z. x-d ≤ z ∧ z ≤ x+d --> 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]: "∀z. x-d ≤ z ∧ z ≤ x+d --> L ≤ f z ∧ f z ≤ M"
    and all2 [rule_format]:
           "∀y. L ≤ y ∧ y ≤ M --> (∃z. x-d ≤ z ∧ z ≤ x+d ∧ f z = y)"
    by auto
  with d have "L ≤ f x & f x ≤ M" by simp
  moreover have "L ≠ f x"
  proof -
    from lemma_isCont_inj2 [OF d inj cont]
    obtain u where "¦u - x¦ ≤ d" "f u < f x"  by auto
    thus ?thesis using all1 [of u] by arith
  qed
  moreover have "f x ≠ M"
  proof -
    from lemma_isCont_inj [OF d inj cont]
    obtain u where "¦u - x¦ ≤ 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 "∀y. ¦y - f x¦ ≤ e --> (∃z. ¦z - x¦ ≤ d ∧ f z = y)"
    proof (intro strip)
      fix y::real
      assume "¦y - f x¦ ≤ e"
      with e have "L ≤ y ∧ y ≤ M" by arith
      from all2 [OF this]
      obtain z where "x - d ≤ z" "z ≤ x + d" "f z = y" by blast
      thus "∃z. ¦z - x¦ ≤ d ∧ 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 => real"
  assumes d: "0 < d"
      and inj: "∀z. ¦z-x¦ ≤ d --> g(f z) = z"
      and cont: "∀z. ¦z-x¦ ≤ d --> isCont f z"
  shows "isCont g (f x)"
proof (simp add: isCont_iff LIM_eq)
  show "∀r. 0 < r -->
         (∃s>0. ∀z. z≠0 ∧ ¦z¦ < s --> ¦g(f x + z) - g(f x)¦ < 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 ∧ e < d" by blast
    with inj cont
    have e_simps: "∀z. ¦z-x¦ ≤ e --> g (f z) = z"
                  "∀z. ¦z-x¦ ≤ e --> isCont f z"   by auto
    from isCont_inj_range [OF e this]
    obtain e' where e': "0 < e'"
        and all: "∀y. ¦y - f x¦ ≤ e' --> (∃z. ¦z - x¦ ≤ e ∧ f z = y)"
          by blast
    show "∃s>0. ∀z. z≠0 ∧ ¦z¦ < s --> ¦g(f x + z) - g(f x)¦ < r"
    proof (intro exI conjI)
      show "0<e'" using e' .
      show "∀z. z ≠ 0 ∧ ¦z¦ < e' --> ¦g (f x + z) - g (f x)¦ < r"
      proof (intro strip)
        fix z::real
        assume z: "z ≠ 0 ∧ ¦z¦ < e'"
        with e e_lt e_simps all [rule_format, of "f x + z"]
        show "¦g (f x + z) - g (f x)¦ < r" by force
      qed
    qed
  qed
qed

text {* Derivative of inverse function *}

lemma DERIV_inverse_function:
  fixes f g :: "real => real"
  assumes der: "DERIV f (g x) :> D"
  assumes neq: "D ≠ 0"
  assumes a: "a < x" and b: "x < b"
  assumes inj: "∀y. a < y ∧ y < b --> 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 simp
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 "(λz. (f z - f (g x)) / (z - g x)) -- g x --> D"
    by (rule der [unfolded DERIV_iff2])
  hence 1: "(λz. (f z - x) / (z - g x)) -- g x --> D"
    using inj a b by simp
  have 2: "∃d>0. ∀y. y ≠ x ∧ norm (y - x) < d --> g y ≠ 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 ≠ x"
    finally show False by simp
  qed
  have "(λy. (f (g y) - x) / (g y - g x)) -- x --> D"
    using cont 1 2 by (rule isCont_LIM_compose2)
  thus "(λy. inverse ((f (g y) - x) / (g y - g x)))
        -- x --> inverse D"
    using neq by (rule LIM_inverse)
qed

theorem GMVT:
  fixes a b :: real
  assumes alb: "a < b"
  and fc: "∀x. a ≤ x ∧ x ≤ b --> isCont f x"
  and fd: "∀x. a < x ∧ x < b --> f differentiable x"
  and gc: "∀x. a ≤ x ∧ x ≤ b --> isCont g x"
  and gd: "∀x. a < x ∧ x < b --> g differentiable x"
  shows "∃g'c f'c c. DERIV g c :> g'c ∧ DERIV f c :> f'c ∧ a < c ∧ c < b ∧ ((f b - f a) * g'c) = ((g b - g a) * f'c)"
proof -
  let ?h = "λx. (f b - f a)*(g x) - (g b - g a)*(f x)"
  from prems have "a < b" by simp
  moreover have "∀x. a ≤ x ∧ x ≤ b --> isCont ?h x"
  proof -
    have "∀x. a <= x ∧ x <= b --> isCont (λx. f b - f a) x" by simp
    with gc have "∀x. a <= x ∧ x <= b --> isCont (λx. (f b - f a) * g x) x"
      by (auto intro: isCont_mult)
    moreover
    have "∀x. a <= x ∧ x <= b --> isCont (λx. g b - g a) x" by simp
    with fc have "∀x. a <= x ∧ x <= b --> isCont (λx. (g b - g a) * f x) x"
      by (auto intro: isCont_mult)
    ultimately show ?thesis
      by (fastsimp intro: isCont_diff)
  qed
  moreover
  have "∀x. a < x ∧ x < b --> ?h differentiable x"
  proof -
    have "∀x. a < x ∧ x < b --> (λx. f b - f a) differentiable x" by (simp add: differentiable_const)
    with gd have "∀x. a < x ∧ x < b --> (λx. (f b - f a) * g x) differentiable x" by (simp add: differentiable_mult)
    moreover
    have "∀x. a < x ∧ x < b --> (λx. g b - g a) differentiable x" by (simp add: differentiable_const)
    with fd have "∀x. a < x ∧ x < b --> (λx. (g b - g a) * f x) differentiable x" by (simp add: differentiable_mult)
    ultimately show ?thesis by (simp add: differentiable_diff)
  qed
  ultimately have "∃l z. a < z ∧ z < b ∧ DERIV ?h z :> l ∧ ?h b - ?h a = (b - a) * l" by (rule MVT)
  then obtain l where ldef: "∃z. a < z ∧ z < b ∧ DERIV ?h z :> l ∧ ?h b - ?h a = (b - a) * l" ..
  then obtain c where cdef: "a < c ∧ c < b ∧ DERIV ?h c :> l ∧ ?h b - ?h a = (b - a) * l" ..

  from cdef have cint: "a < c ∧ c < b" by auto
  with gd have "g differentiable c" by simp
  hence "∃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 ∧ c < b" by auto
  with fd have "f differentiable c" by simp
  hence "∃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 (λx. (f b - f a) * g x) c :> g'c * (f b - f a)"
      apply (insert DERIV_const [where k="f b - f a"])
      apply (drule meta_spec [of _ c])
      apply (drule DERIV_mult [OF _ g'cdef])
      by simp
    moreover have "DERIV (λx. (g b - g a) * f x) c :> f'c * (g b - g a)"
      apply (insert DERIV_const [where k="g b - g a"])
      apply (drule meta_spec [of _ c])
      apply (drule DERIV_mult [OF _ f'cdef])
      by simp
    ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
      by (simp add: DERIV_diff)
  }
  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 "… = (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: mult_ac add_ac right_diff_distrib)
    hence "?h b - ?h a = 0" by auto
  }
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)

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

lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
by auto

end

Derivatives

lemma DERIV_iff:

  DERIV f x :> D = (λh. (f (x + h) - f x) / h) -- 0::'a --> D

lemma DERIV_D:

  DERIV f x :> D ==> (λh. (f (x + h) - f x) / h) -- 0::'a --> D

lemma DERIV_const:

  DERIVx. k) x :> (0::'a)

lemma DERIV_ident:

  DERIVx. x) x :> (1::'a)

lemma add_diff_add:

  a + c - (b + d) = a - b + (c - d)

lemma DERIV_add:

  [| DERIV f x :> D; DERIV g x :> E |] ==> DERIVx. f x + g x) x :> D + E

lemma DERIV_minus:

  DERIV f x :> D ==> DERIVx. - f x) x :> - D

lemma DERIV_diff:

  [| DERIV f x :> D; DERIV g x :> E |] ==> DERIVx. f x - g x) x :> D - E

lemma DERIV_add_minus:

  [| DERIV f x :> D; DERIV g x :> E |] ==> DERIVx. f x + - g x) x :> D + - E

lemma DERIV_isCont:

  DERIV f x :> D ==> isCont f x

lemma DERIV_mult_lemma:

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

lemma DERIV_mult':

  [| DERIV f x :> D; DERIV g x :> E |]
  ==> DERIVx. f x * g x) x :> f x * E + D * g x

lemma DERIV_mult:

  [| DERIV f x :> Da; DERIV g x :> Db |]
  ==> DERIVx. f x * g x) x :> Da * g x + Db * f x

lemma DERIV_unique:

  [| DERIV f x :> D; DERIV f x :> E |] ==> D = E

lemma DERIV_sumr:

  r. m  rr < m + n --> DERIV (f r) x :> f' r x
  ==> DERIVx. ∑n = m..<n. f n x) x :> (∑r = m..<n. f' r x)

lemma DERIV_LIM_iff:

  h. (f (a + h) - f a) / h) -- 0::'a --> D =
  (λx. (f x - f a) / (x - a)) -- a --> D

lemma DERIV_iff2:

  DERIV f x :> D = (λz. (f z - f x) / (z - x)) -- x --> D

lemma inverse_diff_inverse:

  [| a  (0::'a); b  (0::'a) |]
  ==> inverse a - inverse b = - (inverse a * (a - b) * inverse b)

lemma DERIV_inverse_lemma:

  [| a  (0::'a); b  (0::'a) |]
  ==> (inverse a - inverse b) / h = - (inverse a * ((a - b) / h) * inverse b)

lemma DERIV_inverse':

  [| DERIV f x :> D; f x  (0::'a) |]
  ==> DERIVx. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))

lemma DERIV_divide:

  [| DERIV f x :> D; DERIV g x :> E; g x  (0::'a) |]
  ==> DERIVx. f x / g x) x :> (D * g x - f x * E) / (g x * g x)

lemma DERIV_power_Suc:

  DERIV f x :> D
  ==> DERIVx. f x ^ Suc n) x :> ((1::'a) + of_nat n) * (D * f x ^ n)

lemma DERIV_power:

  DERIV f x :> D ==> DERIVx. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))

lemma CARAT_DERIV:

  DERIV f x :> l = (∃g. (∀z. f z - f x = g z * (z - x)) ∧ isCont g xg x = l)

lemma DERIV_chain':

  [| DERIV f x :> D; DERIV g (f x) :> E |] ==> DERIVx. g (f x)) x :> E * D

lemma DERIV_cmult:

  DERIV f x :> D ==> DERIVx. c * f x) x :> c * D

lemma DERIV_chain:

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

lemma DERIV_chain2:

  [| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIVx. f (g x)) x :> Da * Db

lemma DERIV_cmult_Id:

  DERIV (op * c) x :> c

lemma DERIV_pow:

  DERIVx. x ^ n) x :> real n * x ^ (n - Suc 0)

lemma DERIV_inverse:

  x  (0::'a) ==> DERIV inverse x :> - (inverse x ^ Suc (Suc 0))

lemma DERIV_inverse_fun:

  [| DERIV f x :> d; f x  (0::'a) |]
  ==> DERIVx. inverse (f x)) x :> - (d * inverse (f x ^ Suc (Suc 0)))

lemma DERIV_quotient:

  [| DERIV f x :> d; DERIV g x :> e; g x  (0::'a) |]
  ==> DERIVy. f y / g y) x :> (d * g x - e * f x) / g x ^ Suc (Suc 0)

Differentiability predicate

lemma differentiableD:

  f differentiable x ==> ∃D. DERIV f x :> D

lemma differentiableI:

  DERIV f x :> D ==> f differentiable x

lemma differentiable_const:

  z. a) differentiable x

lemma differentiable_sum:

  [| f differentiable x; g differentiable x |]
  ==> (λx. f x + g x) differentiable x

lemma differentiable_diff:

  [| f differentiable x; g differentiable x |]
  ==> (λx. f x - g x) differentiable x

lemma differentiable_mult:

  [| f differentiable x; g differentiable x |]
  ==> (λx. f x * g x) differentiable x

Nested Intervals and Bisection

lemma lemma_f_mono_add:

  n. f n  f (Suc n) ==> f m  f (m + no)

lemma f_inc_g_dec_Beq_f:

  [| ∀n. f n  f (Suc n); ∀n. g (Suc n)  g n; ∀n. f n  g n |] ==> Bseq f

lemma f_inc_g_dec_Beq_g:

  [| ∀n. f n  f (Suc n); ∀n. g (Suc n)  g n; ∀n. f n  g n |] ==> Bseq g

lemma f_inc_imp_le_lim:

  [| ∀n. f n  f (Suc n); convergent f |] ==> f n  lim f

lemma lim_uminus:

  convergent g ==> limx. - g x) = - lim g

lemma g_dec_imp_lim_le:

  [| ∀n. g (Suc n)  g n; convergent g |] ==> lim g  g n

lemma lemma_nest:

  [| ∀n. f n  f (Suc n); ∀n. g (Suc n)  g n; ∀n. f n  g n |]
  ==> ∃l m. l  m ∧ ((∀n. f n  l) ∧ f ----> l) ∧ (∀n. m  g n) ∧ g ----> m

lemma lemma_nest_unique:

  [| ∀n. f n  f (Suc n); ∀n. g (Suc n)  g n; ∀n. f n  g n;
     (λn. f n - g n) ----> 0 |]
  ==> ∃l. ((∀n. f n  l) ∧ f ----> l) ∧ (∀n. l  g n) ∧ g ----> l

lemma Bolzano_bisect_le:

  a  b ==> ∀n. fst (Bolzano_bisect P a b n)  snd (Bolzano_bisect P a b n)

lemma Bolzano_bisect_fst_le_Suc:

  a  b ==> ∀n. fst (Bolzano_bisect P a b n)  fst (Bolzano_bisect P a b (Suc n))

lemma Bolzano_bisect_Suc_le_snd:

  a  b ==> ∀n. snd (Bolzano_bisect P a b (Suc n))  snd (Bolzano_bisect P a b n)

lemma eq_divide_2_times_iff:

  (x = y / (2 * z)) = (2 * x = y / z)

lemma Bolzano_bisect_diff:

  a  b
  ==> snd (Bolzano_bisect P a b n) - fst (Bolzano_bisect P a b n) =
      (b - a) / 2 ^ n

lemma Bolzano_nest_unique:

  [| a1  b1; a1  b1; a1  b1;
     (λn. fst (Bolzano_bisect P1 a1 b1 n) - snd (Bolzano_bisect P1 a1 b1 n))
     ----> 0 |]
  ==> ∃l. ((∀n. fst (Bolzano_bisect P1 a1 b1 n)  l) ∧
           (λn. fst (Bolzano_bisect P1 a1 b1 n)) ----> l) ∧
          (∀n. l  snd (Bolzano_bisect P1 a1 b1 n)) ∧
          (λn. snd (Bolzano_bisect P1 a1 b1 n)) ----> l

lemma not_P_Bolzano_bisect:

  [| !!a b c. [| P (a, b); P (b, c); a  b; b  c |] ==> P (a, c); ¬ P (a, b);
     a  b |]
  ==> ¬ P (fst (Bolzano_bisect P a b n), snd (Bolzano_bisect P a b n))

lemma not_P_Bolzano_bisect':

  [| ∀a b c. P (a, b) ∧ P (b, c) ∧ a  bb  c --> P (a, c); ¬ P (a, b);
     a  b |]
  ==> ∀n. ¬ P (fst (Bolzano_bisect P a b n), snd (Bolzano_bisect P a b n))

lemma lemma_BOLZANO:

  [| ∀a b c. P (a, b) ∧ P (b, c) ∧ a  bb  c --> P (a, c);
     ∀x. ∃d>0. ∀a b. a  xx  bb - a < d --> P (a, b); a  b |]
  ==> P (a, b)

lemma lemma_BOLZANO2:

  (∀a b c. a  bb  cP (a, b) ∧ P (b, c) --> P (a, c)) ∧
  (∀x. ∃d>0. ∀a b. a  xx  bb - a < d --> P (a, b)) -->
  (∀a b. a  b --> P (a, b))

Intermediate Value Theorem

lemma IVT:

  [| f a  y; y  f b; a  b; ∀x. a  xx  b --> isCont f x |]
  ==> ∃xa. x  bf x = y

lemma IVT2:

  [| f b  y; y  f a; a  b; ∀x. a  xx  b --> isCont f x |]
  ==> ∃xa. x  bf x = y

lemma IVT_objl:

  f a  yy  f ba  b ∧ (∀x. a  xx  b --> isCont f x) -->
  (∃xa. x  bf x = y)

lemma IVT2_objl:

  f b  yy  f aa  b ∧ (∀x. a  xx  b --> isCont f x) -->
  (∃xa. x  bf x = y)

lemma isCont_bounded:

  [| a  b; ∀x. a  xx  b --> isCont f x |]
  ==> ∃M. ∀x. a  xx  b --> f x  M

lemma lemma_reals_complete:

  (∃x. xS) ∧ (∃y. isUb UNIV S y) --> (∃t. isLub UNIV S t)

lemma isCont_has_Ub:

  [| a  b; ∀x. a  xx  b --> isCont f x |]
  ==> ∃M. (∀x. a  xx  b --> f x  M) ∧ (∀N<M. ∃xa. x  bN < f x)

lemma isCont_eq_Ub:

  [| a  b; ∀x. a  xx  b --> isCont f x |]
  ==> ∃M. (∀x. a  xx  b --> f x  M) ∧ (∃xa. x  bf x = M)

lemma isCont_eq_Lb:

  [| a  b; ∀x. a  xx  b --> isCont f x |]
  ==> ∃M. (∀x. a  xx  b --> M  f x) ∧ (∃xa. x  bf x = M)

lemma isCont_Lb_Ub:

  [| a  b; ∀x. a  xx  b --> isCont f x |]
  ==> ∃L M. (∀x. a  xx  b --> L  f xf x  M) ∧
            (∀y. L  yy  M --> (∃xa. x  bf x = y))

lemma DERIV_left_inc:

  [| DERIV f x :> l; 0 < l |] ==> ∃d>0. ∀h>0. h < d --> f x < f (x + h)

lemma DERIV_left_dec:

  [| DERIV f x :> l; l < 0 |] ==> ∃d>0. ∀h>0. h < d --> f x < f (x - h)

lemma DERIV_local_max:

  [| DERIV f x :> l; 0 < d; ∀y. ¦x - y¦ < d --> f y  f x |] ==> l = 0

lemma DERIV_local_min:

  [| DERIV f x :> l; 0 < d; ∀y. ¦x - y¦ < d --> f x  f y |] ==> l = 0

lemma DERIV_local_const:

  [| DERIV f x :> l; 0 < d; ∀y. ¦x - y¦ < d --> f x = f y |] ==> l = 0

lemma lemma_interval_lt:

  [| a < x; x < b |] ==> ∃d>0. ∀y. ¦x - y¦ < d --> a < yy < b

lemma lemma_interval:

  [| a < x; x < b |] ==> ∃d>0. ∀y. ¦x - y¦ < d --> a  yy  b

theorem Rolle:

  [| a < b; f a = f b; ∀x. a  xx  b --> isCont f x;
     ∀x. a < xx < b --> f differentiable x |]
  ==> ∃z>a. z < bDERIV f z :> 0

Mean Value Theorem

lemma lemma_MVT:

  f a - (f b - f a) / (b - a) * a = f b - (f b - f a) / (b - a) * b

theorem MVT:

  [| a < b; ∀x. a  xx  b --> isCont f x;
     ∀x. a < xx < b --> f differentiable x |]
  ==> ∃l z. a < zz < bDERIV f z :> lf b - f a = (b - a) * l

lemma DERIV_isconst_end:

  [| a < b; ∀x. a  xx  b --> isCont f x;
     ∀x. a < xx < b --> DERIV f x :> 0 |]
  ==> f b = f a

lemma DERIV_isconst1:

  [| a < b; ∀x. a  xx  b --> isCont f x;
     ∀x. a < xx < b --> DERIV f x :> 0 |]
  ==> ∀x. a  xx  b --> f x = f a

lemma DERIV_isconst2:

  [| a < b; ∀x. a  xx  b --> isCont f x;
     ∀x. a < xx < b --> DERIV f x :> 0; a  x; x  b |]
  ==> f x = f a

lemma DERIV_isconst_all:

  x. DERIV f x :> 0 ==> f x = f y

lemma DERIV_const_ratio_const:

  [| a  b; ∀x. DERIV f x :> k |] ==> f b - f a = (b - a) * k

lemma DERIV_const_ratio_const2:

  [| a  b; ∀x. DERIV f x :> k |] ==> (f b - f a) / (b - a) = k

lemma real_average_minus_first:

  (a + b) / 2 - a = (b - a) / 2

lemma real_average_minus_second:

  (b + a) / 2 - a = (b - a) / 2

lemma DERIV_const_average:

  [| a  b; ∀x. DERIV v x :> k |] ==> v ((a + b) / 2) = (v a + v b) / 2

lemma lemma_isCont_inj:

  [| 0 < d; ∀z. ¦z - x¦  d --> g (f z) = z; ∀z. ¦z - x¦  d --> isCont f z |]
  ==> ∃z. ¦z - x¦  df x < f z

lemma lemma_isCont_inj2:

  [| 0 < d; ∀z. ¦z - x¦  d --> g (f z) = z; ∀z. ¦z - x¦  d --> isCont f z |]
  ==> ∃z. ¦z - x¦  df z < f x

lemma isCont_inj_range:

  [| 0 < d; ∀z. ¦z - x¦  d --> g (f z) = z; ∀z. ¦z - x¦  d --> isCont f z |]
  ==> ∃e>0. ∀y. ¦y - f x¦  e --> (∃z. ¦z - x¦  df z = y)

lemma isCont_inverse_function:

  [| 0 < d; ∀z. ¦z - x¦  d --> g (f z) = z; ∀z. ¦z - x¦  d --> isCont f z |]
  ==> isCont g (f x)

lemma DERIV_inverse_function:

  [| DERIV f (g x) :> D; D  0; a < x; x < b; ∀y. a < yy < b --> f (g y) = y;
     isCont g x |]
  ==> DERIV g x :> inverse D

theorem GMVT:

  [| a < b; ∀x. a  xx  b --> isCont f x;
     ∀x. a < xx < b --> f differentiable x; ∀x. a  xx  b --> isCont g x;
     ∀x. a < xx < b --> g differentiable x |]
  ==> ∃g'c f'c c.
         DERIV g c :> g'cDERIV f c :> f'ca < cc < b ∧ (f b - f a) * g'c = (g b - g a) * f'c

lemma lemma_DERIV_subst:

  [| DERIV f x :> D; D = E |] ==> DERIV f x :> E